Cellular Automata and Groups (Springer Monographs in Mathematics)

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Cellular Automata and Groups (Springer Monographs in Mathematics)

Springer Monographs in Mathematics For further volumes: www.springer.com/series/3733 Tullio Ceccherini-Silberstein r

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Springer Monographs in Mathematics

For further volumes: www.springer.com/series/3733

Tullio Ceccherini-Silberstein r Michel Coornaert

Cellular Automata and Groups

Tullio Ceccherini-Silberstein Dipartimento di Ingegneria Università del Sannio C.so Garibaldi 107 82100 Benevento Italy [email protected]

Michel Coornaert Institut de Recherche Mathématique Avancée Université de Strasbourg 7 rue René-Descartes 67084 Strasbourg Cedex France [email protected]

ISSN 1439-7382 ISBN 978-3-642-14033-4 e-ISBN 978-3-642-14034-1 DOI 10.1007/978-3-642-14034-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010934641 Mathematics Subject Classification (2010): 37B15, 68Q80, 20F65, 43A07, 16S34, 20C07 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Katiuscia, Giacomo, and Tommaso To Martine and Nathalie

Preface

Two seemingly unrelated mathematical notions, namely that of an amenable group and that of a cellular automaton, were both introduced by John von Neumann in the first half of the last century. Amenability, which originated from the study of the Banach-Tarski paradox, is a property of groups generalizing both commutativity and finiteness. Nowadays, it plays an important role in many areas of mathematics such as representation theory, harmonic analysis, ergodic theory, geometric group theory, probability theory, and dynamical systems. Von Neumann used cellular automata to serve as theoretical models for self-reproducing machines. About twenty years later, the famous cellular automaton associated with the Game of Life was invented by John Horton Conway and popularized by Martin Gardner. The theory of cellular automata flourished as one of the main branches of computer science. Deep connections with complexity theory and logic emerged from the discovery that some cellular automata are universal Turing machines. A group G is said to be amenable (as a discrete group) if the set of all subsets of G admits a right-invariant finitely additive probability measure. All finite groups, all solvable groups (and therefore all abelian groups), and all finitely generated groups of subexponential growth are amenable. Von Neumann observed that the class of amenable groups is closed under the operation of taking subgroups and that the free group of rank two F2 is nonamenable. It follows that a group which contains a subgroup isomorphic to F2 is non-amenable. However, there are examples of groups which are nonamenable and contain no subgroups isomorphic to F2 (the first examples of such groups were discovered by Alexander Y. Ol’shanskii and by Sergei I. Adyan). Loosely speaking, a general cellular automaton can be described as follows. A configuration is a map from a set called the universe into another set called the alphabet. The elements of the universe are called cells and the elements of the alphabet are called states. A cellular automaton is then a map from the set of all configurations into itself satisfying the following local property: the state of the image configuration at a given cell only depends on

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the states of the initial configuration on a finite neighborhood of the given cell. In the classical setting, for instance in the cellular automata constructed by von Neumann and the one associated with Conway’s Game of Life, the alphabet is finite, the universe is the two dimensional infinite square lattice, and the neighborhood of a cell consists of the cell itself and its eight adjacent cells. By iterating a cellular automaton one gets a discrete dynamical system. Such dynamical systems have proved very useful to model complex systems arising from natural sciences, in particular physics, biology, chemistry, and population dynamics. ∗





In this book, the universe will always be a group G (in the classical setting the corresponding group was G = Z2 ) and the alphabet may be finite or infinite. The left multiplication in G induces a natural action of G on the set of configurations which is called the G-shift and all cellular automata will be required to commute with the shift. It was soon realized that the question whether a given cellular automaton is surjective or not needs a special attention. From the dynamical viewpoint, surjectivity means that each configuration may be reached at any time. The first important result in this direction is the celebrated theorem of Moore and Myhill which gives a necessary and sufficient condition for the surjectivity of a cellular automaton with finite alphabet over the group G = Z2 . Edward F. Moore and John R. Myhill proved that such a cellular automaton is surjective if and only if it is pre-injective. As the term suggests it, pre-injectivity is a weaker notion than injectivity. More precisely, a cellular automaton is said to be pre-injective if two configurations are equal whenever they have the same image and coincide outside a finite subset of the group. Moore proved the “surjective ⇒ pre-injective” part and Myhill proved the converse implication shortly after. One often refers to this result as to the Garden of Eden theorem. This biblical terminology is motivated by the fact that, regarding a cellular automaton as a dynamical system with discrete time, a configuration which is not in the image of the cellular automaton may only appear as an initial configuration, that is, at time t = 0. The surprising connection between amenability and cellular automata was established in 1997 when Antonio Mach`ı, Fabio Scarabotti and the first author proved the Garden of Eden theorem for cellular automata with finite alphabets over amenable groups. At the same time, and completely independently, Misha Gromov, using a notion of spacial entropy, presented a more general form of the Garden of Eden theorem where the universe is an amenable graph with a dense holonomy and cellular automata are called maps of bounded propagation. Mach`ı, Scarabotti and the first author also showed that both implications in the Garden of Eden theorem become false if the underlying group contains a subgroup isomorphic to F2 . The question whether the Garden of Eden theorem could be extended beyond the class of

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amenable groups remained open until 2008 when Laurent Bartholdi proved that the Moore implication fails to hold for non-amenable groups. As a consequence, the whole Garden of Eden theorem only holds for amenable groups. This gives a new characterization of amenable groups in terms of cellular automata. Let us mention that, up to now, the validity of the Myhill implication for non-amenable groups is still an open problem. Following Walter H. Gottschalk, a group G is said to be surjunctive if every injective cellular automaton with finite alphabet over G is surjective. Wayne Lawton proved that all residually finite groups are surjunctive and that every subgroup of a surjunctive group is surjunctive. Since injectivity implies pre-injectivity, an immediate consequence of the Garden of Eden theorem for amenable groups is that every amenable group is surjunctive. Gromov and Benjamin Weiss introduced a class of groups, called sofic groups, which includes all residually finite groups and all amenable groups, and proved that every sofic group is surjunctive. Sofic groups can be defined in three equivalent ways: in terms of local approximation by finite symmetric groups equipped with their Hamming distance, in terms of local approximation of their Cayley graphs by finite labelled graphs, and, finally, as being the groups that can be embedded into ultraproducts of finite symmetric groups (this last characterization is due to G´ abor Elek and Endre Szab´ o). The class of sofic groups is the largest known class of surjunctive groups. It is not known, up to now, whether all groups are surjunctive (resp. sofic) or not. Stimulated by Gromov ideas, we considered cellular automata whose alphabets are vector spaces. In this framework, the space of configurations has a natural structure of a vector space and cellular automata are required to be linear. An analogue of the Garden of Eden theorem was proved for linear cellular automata with finite dimensional alphabets over amenable groups. In the proof, the role of entropy, used in the finite alphabet case, is now played by the mean dimension, a notion introduced by Gromov. Also, examples of linear cellular automata with finite dimensional alphabets over groups containing F2 showing that the linear version of the Garden of Eden theorem may fail to hold in this case, were provided. It is not known, up to now, if the Garden of Eden theorem for linear cellular automata with finite dimensional alphabet only holds for amenable groups or not. We also introduced the notion of linear surjunctivity: a group G is said to be L-surjunctive if every injective linear cellular automaton with finite dimensional alphabet over G is surjective. We proved that every sofic group is L-surjunctive. Linear cellular automata over a group G with alphabet of finite dimension d over a field K may be represented by d × d matrices with entries in the group ring K[G]. This leads to the following characterization of L-surjunctivity: a group is L-surjunctive if and only if it satisfies Kaplansky’s conjecture on the stable finiteness of group rings (a ring is said to be stably finite if one-sided invertible finite dimensional square matrices with coefficients in that ring are in fact two-sided invertible). As a corollary, one has that group rings of sofic groups are stably finite, a result previously established by Elek and Szab´ o

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using different methods. Moreover, given a group G and a field K, the preinjectivity of all nonzero linear cellular automata with alphabet K over G is equivalent to the absence of zero-divisors in K[G]. As a consequence, another important problem on the structure of group rings also formulated by Irving Kaplansky may be expressed in terms of cellular automata. Is every nonzero linear cellular automaton with one-dimensional alphabet over a torsion-free group always pre-injective? ∗





The material presented in this book is entirely self-contained. In fact, its reading only requires some acquaintance with undergraduate general topology and abstract algebra. Each chapter begins with a brief overview of its contents and ends with some historical notes and a list of exercises at various difficulty levels. Some additional topics, such as subshifts and cellular automata over subshifts, are treated in these exercises. Hints are provided each time help may be needed. In order to improve accessibility, a few appendices are included to quickly introduce the reader to facts he might be not too familiar with. In the first chapter, we give the definition of a cellular automaton. We present some basic examples and discuss general methods for constructing cellular automata. We equip the set of configurations with its prodiscrete uniform structure and prove the generalized Curtis-Hedlund theorem: a necessary and sufficient condition for a self-mapping of the configuration space to be a cellular automaton is that it is uniformly continuous and commutes with the shift. Chapter 2 is devoted to residually finite groups. We give several equivalent characterizations of residual finiteness and prove that the class of residually finite groups is closed under taking subgroups and projective limits. We establish in particular the theorems, respectively due to Anatoly I. Mal’cev and Gilbert Baumslag, which assert that finitely generated residually finite groups are Hopfian and that their automorphism group is residually finite. Surjunctive groups are introduced in Chap. 3. We show that every subgroup of a surjunctive group is surjunctive and that locally residually finite groups are surjunctive. We also prove a theorem of Gromov which says that limits of surjunctive marked groups are surjunctive. The theory of amenable groups is developed in Chap. 4. The class of amenable groups is closed under taking subgroups, quotients, extensions, and inductive limits. We prove the theorems due to Erling Følner and Alfred Tarski which state the equivalence between amenability, the existence of a Følner net, and the non-existence of a paradoxical decomposition. The Garden of Eden theorem is established in Chap. 5. It is proved by showing that both surjectivity and pre-injectivity of the cellular automaton are equivalent to the fact that the image of the configuration space has maxi-

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mal entropy. We give an example of a cellular automaton with finite alphabet over F2 which is pre-injective but not surjective. Following Bartholdi’s construction, we also prove the existence of a surjective but not pre-injective cellular automaton with finite alphabet over any non-amenable group. In Chap. 6 we present the basic elementary notions and results on growth of finitely generated groups. We prove that finitely generated nilpotent groups have polynomial growth. We then introduce the Grigorchuk group and show that it is an infinite finitely generated periodic group of intermediate growth. We show that every finitely generated group of subexponential growth is amenable. We also establish the Kesten-Day characterization of amenability which asserts that a group with a finite (not necessarily symmetric) generating subset is amenable if and only if 0 is in the 2 -spectrum of the associated Laplacian. Finally, we consider the notion of quasi-isometry for not necessarily countable groups and we show that amenability is a quasi-isometry invariant. In Chap. 7 we consider the notion of local embeddability of groups into a class of groups. For the class of finite groups, this gives the class of LEF groups introduced by Anatoly M. Vershik and Edward I. Gordon. We discuss several stability properties of local embeddability and show that locally embeddable groups are closed in marked groups spaces. The remaining of the chapter is devoted to the class of sofic groups. We show that the three definitions, namely analytic, geometric, and algebraic, we alluded to before, are equivalent. We then prove the Gromov-Weiss theorem which states that every sofic group is surjunctive. The last chapter is devoted to linear cellular automata. We prove the linear version of the Garden of Eden theorem and show that every sofic group is L-surjunctive. We end the chapter with a discussion on the stable finiteness and the zero-divisors conjectures of Kaplansky and their reformulation in terms of linear cellular automata. Appendix A gives a quick overview of a few fundamental notions and results of topology (nets, compactness, product topology, and the Tychonoff product theorem). Appendix B is devoted to Andr´e Weil’s theory of uniform spaces. It includes also a detailed exposition of the Hausdorff-Bourbaki uniform structure on subsets of a uniform space. In Appendix C, we establish some basic properties of symmetric groups and prove the simplicity of the alternating groups. The definition and the construction of free groups are given in Appendix D. The proof of Klein’s ping-pong lemma is also included there. In Appendix E we shortly describe the constructions of inductive and projective limits of groups. Appendix F treats topological vector spaces, the weak-∗ topology, and the Banach-Alaoglu theorem. The proof of the MarkovKakutani fixed point theorem is presented in Appendix G. In the subsequent appendix, of a pure graph-theoretical and combinatorial flavour, we consider bipartite graphs and their matchings. We prove Hall’s marriage theorem and its harem version which plays a key role in the proof of Tarski’s theorem on amenability. The Baire theorem, the open mapping theorem, as well as other

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complements of functional analysis including uniform convexity are treated in Appendix I. The last appendix deals with the notions of filters and ultrafilters. We would like to express our deep gratitude to Dr. Catriona Byrne, Dr. Marina Reizakis and Annika Eling from Springer Verlag and to Donatas Akmanaviˇcius for their constant and kindest help at all stages of the editorial process. Rome and Strasbourg

Tullio Ceccherini-Silberstein Michel Coornaert

Contents

1

Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Configuration Set and the Shift Action . . . . . . . . . . . . . . . . 1.2 The Prodiscrete Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Periodic Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Minimal Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Cellular Automata over Quotient Groups . . . . . . . . . . . . . . . . . . 1.7 Induction and Restriction of Cellular Automata . . . . . . . . . . . . 1.8 Cellular Automata with Finite Alphabets . . . . . . . . . . . . . . . . . . 1.9 The Prodiscrete Uniform Structure . . . . . . . . . . . . . . . . . . . . . . . 1.10 Invertible Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 3 6 14 15 16 20 22 24 27 29

2

Residually Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition and First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stability Properties of Residually Finite Groups . . . . . . . . . . . . 2.3 Residual Finiteness of Free Groups . . . . . . . . . . . . . . . . . . . . . . . 2.4 Hopfian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Automorphism Groups of Residually Finite Groups . . . . . . . . . 2.6 Examples of Finitely Generated Groups Which Are Not Residually Finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Dynamical Characterization of Residual Finiteness . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 40 42 44 45 47 50 51 52

Surjunctive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stability Properties of Surjunctive Groups . . . . . . . . . . . . . . . . . 3.3 Surjunctivity of Locally Residually Finite Groups . . . . . . . . . . .

57 57 58 59

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3.4 Marked Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Expansive Actions on Uniform Spaces . . . . . . . . . . . . . . . . . . . . . 3.6 Gromov’s Injectivity Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Closedness of Marked Surjunctive Groups . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4

Amenable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Measures and Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Properties of the Set of Means . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Measures and Means on Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Definition of Amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Stability Properties of Amenable Groups . . . . . . . . . . . . . . . . . . 4.6 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 The Følner Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Paradoxical Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 The Theorems of Tarski and Følner . . . . . . . . . . . . . . . . . . . . . . . 4.10 The Fixed Point Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 82 83 85 88 92 94 98 99 103 105 106

5

The Garden of Eden Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Garden of Eden Configurations and Garden of Eden Patterns 5.2 Pre-injective Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Statement of the Garden of Eden Theorem . . . . . . . . . . . . . . . . 5.4 Interiors, Closures, and Boundaries . . . . . . . . . . . . . . . . . . . . . . . 5.5 Mutually Erasable Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Proof of the Garden of Eden Theorem . . . . . . . . . . . . . . . . . . . . 5.9 Surjunctivity of Locally Residually Amenable Groups . . . . . . . 5.10 A Surjective but Not Pre-injective Cellular Automaton over F2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 A Pre-injective but Not Surjective Cellular Automaton over F2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 A Characterization of Amenability in Terms of Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Garden of Eden Patterns for Life . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 112 114 115 121 122 125 128 131

Finitely Generated Amenable Groups . . . . . . . . . . . . . . . . . . . . 6.1 The Word Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Labeled Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Cayley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Growth Functions and Growth Types . . . . . . . . . . . . . . . . . . . . .

151 151 153 156 160

6

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6.5 The Growth Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Growth of Subgroups and Quotients . . . . . . . . . . . . . . . . . . . . . . 6.7 A Finitely Generated Metabelian Group with Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Growth of Finitely Generated Nilpotent Groups . . . . . . . . . . . . 6.9 The Grigorchuk Group and Its Growth . . . . . . . . . . . . . . . . . . . . 6.10 The Følner Condition for Finitely Generated Groups . . . . . . . . 6.11 Amenability of Groups of Subexponential Growth . . . . . . . . . . 6.12 The Theorems of Kesten and Day . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Quasi-Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

8

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168 170 173 175 178 191 192 193 204 214 217

Local Embeddability and Sofic Groups . . . . . . . . . . . . . . . . . . . . 7.1 Local Embeddability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Local Embeddability and Ultraproducts . . . . . . . . . . . . . . . . . . . 7.3 LEF-Groups and LEA-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The Hamming Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Sofic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Sofic Groups and Metric Ultraproducts of Finite Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 A Characterization of Finitely Generated Sofic Groups . . . . . . 7.8 Surjunctivity of Sofic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

233 234 243 246 251 254

Linear Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Algebra of Linear Cellular Automata . . . . . . . . . . . . . . . . . 8.2 Configurations with Finite Support . . . . . . . . . . . . . . . . . . . . . . . 8.3 Restriction and Induction of Linear Cellular Automata . . . . . . 8.4 Group Rings and Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Group Ring Representation of Linear Cellular Automata . . . . 8.6 Modules over a Group Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Matrix Representation of Linear Cellular Automata . . . . . . . . . 8.8 The Closed Image Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 The Garden of Eden Theorem for Linear Cellular Automata . 8.10 Pre-injective but not Surjective Linear Cellular Automata . . . 8.11 Surjective but not Pre-injective Linear Cellular Automata . . . 8.12 Invertible Linear Cellular Automata . . . . . . . . . . . . . . . . . . . . . . 8.13 Pre-injectivity and Surjectivity of the Discrete Laplacian . . . . 8.14 Linear Surjunctivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.15 Stable Finiteness of Group Algebras . . . . . . . . . . . . . . . . . . . . . . 8.16 Zero-Divisors in Group Algebras and Pre-injectivity of One-Dimensional Linear Cellular Automata . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283 284 288 289 291 294 299 301 305 308 314 315 317 321 324 327

260 265 272 275 278

330 335 338

xvi

Contents

A

Nets and the Tychonoff Product Theorem . . . . . . . . . . . . . . . . A.1 Directed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Nets in Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Initial Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Product Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 The Tychonoff Product Theorem . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

343 343 343 346 346 347 349

B

Uniform Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Uniformly Continuous Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Product of Uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 The Hausdorff-Bourbaki Uniform Structure on Subsets . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351 351 353 355 356 358

C

Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 The Symmetric Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Permutations with Finite Support . . . . . . . . . . . . . . . . . . . . . . . . C.3 Conjugacy Classes in Sym0 (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4 The Alternating Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

359 359 360 362 363

D

Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1 Concatenation of Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Definition and Construction of Free Groups . . . . . . . . . . . . . . . . D.3 Reduced Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4 Presentations of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.5 The Klein Ping-Pong Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .

367 367 367 373 375 376

E

Inductive Limits and Projective Limits of Groups . . . . . . . . 379 E.1 Inductive Limits of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 E.2 Projective Limits of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

F

The Banach-Alaoglu Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2 The Weak-∗ Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.3 The Banach-Alaoglu Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .

383 383 384 384

G

The Markov-Kakutani Fixed Point Theorem . . . . . . . . . . . . . . G.1 Statement of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.2 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

387 387 387 389

H

The Hall Harem Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 H.1 Bipartite Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 H.2 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Contents

xvii

H.3 The Hall Marriage Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 H.4 The Hall Harem Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 I

Complements of Functional Analysis . . . . . . . . . . . . . . . . . . . . . . I.1 The Baire Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2 The Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.3 Spectra of Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.4 Uniform Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

403 403 404 406 407

J

Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.1 Filters and Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.2 Limits Along Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

409 409 412 415

Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

Notation

Throughout this book, the following conventions are used: • N is the set of nonnegative integers so that 0 ∈ N; • the notation A ⊂ B means that each element in the set A is also in the set B so that A and B may coincide; • a countable set is a set which admits a bijection onto a subset of N so that finite sets are countable; • all group actions are left actions; • all rings are assumed to be associative (but not necessarily commutative) with a unity element; • a field is a nonzero commutative ring in which each nonzero element is invertible.

xix

Chapter 1

Cellular Automata

In this chapter we introduce the notion of a cellular automaton. We fix a group and an arbitrary set which will be called the alphabet. A configuration is defined as being a map from the group into the alphabet. Thus, a configuration is a way of attaching an element of the alphabet to each element of the group. There is a natural action of the group on the set of configurations which is called the shift action (see Sect. 1.1). A cellular automaton is a self-mapping of the set of configurations defined from a system of local rules commuting with the shift (see Definition 1.4.1). We equip the configuration set with the prodiscrete topology, that is, the topology of pointwise convergence associated with the discrete topology on the alphabet (see Sect. 1.2). It turns out that every cellular automaton is continuous with respect to the prodiscrete topology (Proposition 1.4.8) and commutes with the shift (Proposition 1.4.4). Conversely, when the alphabet is finite, every continuous self-mapping of the configuration space which commutes with the shift is a cellular automaton (Theorem 1.8.1). Another important fact in the finite alphabet case is that every bijective cellular automaton is invertible, in the sense that its inverse map is also a cellular automaton (Theorem 1.10.2). We give examples showing that, when the alphabet is infinite, a continuous self-mapping of the configuration space which commutes with the shift may fail to be a cellular automaton and a bijective cellular automaton may fail to be invertible. In Sect. 1.9, we introduce the prodiscrete uniform structure on the configuration space. We show that a self-mapping of the configuration space is a cellular automaton if and only if it is uniformly continuous and commutes with the shift (Theorem 1.9.1).

1.1 The Configuration Set and the Shift Action Let G be a group. For g ∈ G, denote by Lg the left multiplication by g in G, that is, the map Lg : G → G given by T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 1, © Springer-Verlag Berlin Heidelberg 2010

1

2

1 Cellular Automata

Lg (g  ) = gg 

for all g  ∈ G.

Observe that for all g1 , g2 , g  ∈ G one has (Lg1 ◦ Lg2 ) (g  ) = Lg1 (Lg2 (g  )) = Lg1 (g2 g  ) = g1 g2 g  = Lg1 g2 (g  ) which shows that Lg1 ◦ Lg2 = Lg1 g2 .

(1.1)

Let A be a set. Consider the set AG consisting of all maps from G to A:  AG = A = {x : G → A}. g∈G

The set A is called the alphabet. The elements of A are called the letters, or the states, or the symbols, or the colors. The group G is called the universe. The set AG is called the set of configurations. Given an element g ∈ G and a configuration x ∈ AG , we define the configuration gx ∈ AG by (1.2) gx = x ◦ Lg−1 . Thus one has

gx(g  ) = x(g −1 g  )

for all g  ∈ G.

The map G × AG → AG (g, x) → gx

is a left action of G on AG . Indeed, for all g1 , g2 ∈ G and x ∈ AG , one has g1 (g2 x) = g1 (x ◦ Lg−1 ) = x ◦ Lg−1 ◦ Lg−1 = x ◦ Lg−1 g−1 = x ◦ L(g1 g2 )−1 2

2

1

2

1

= (g1 g2 )x, where the third equality follows from (1.1). Also, denoting by 1G the identity element of G and by IdG : G → G the identity map, one has 1G x = x ◦ L1G = x ◦ IdG = x. This left action of G on AG is called the G-shift on AG . A pattern over the group G and the alphabet A is a map p : Ω → G defined on some finite subset Ω of G. The set Ω is then called the support of p.

1.3 Periodic Configurations

3

1.2 The Prodiscrete Topology Let G be a group and let A be a set. We equip each factor A of AG with the discrete topology (all subsets of A are open) and AG with the associated product topology (see Sect. A.4). This topology is called the prodiscrete topology on AG . This is the smallest topology on AG for which the projection map πg : AG → A, given by πg (x) = x(g), is continuous for every g ∈ G (cf. Sect. A.4). The elementary cylinders C(g, a) = πg−1 ({a}) = {x ∈ AG : x(g) = a}

(g ∈ G, a ∈ A)

are both open and closed in AG . A subset U ⊂ AG is open if and only if U can be expressed as a (finite or infinite) union of finite intersections of elementary cylinders. For a subset Ω ⊂ G and a configuration x ∈ AG let x|Ω ∈ AΩ denote the restriction of x to Ω, that is, the map x|Ω : Ω → A defined by x|Ω (g) = x(g) for all g ∈ Ω. If x ∈ AG , a neighborhood base of x is given by the sets  V (x, Ω) = {y ∈ AG : x|Ω = y|Ω } = C(g, x(g)), (1.3) g∈Ω

where Ω runs over all finite subsets of G. Proposition 1.2.1. The space AG is Hausdorff and totally disconnected. Proof. The discrete topology on A is Hausdorff and totally disconnected, and, by Proposition A.4.1 and Proposition A.4.2, a product of Hausdorff (resp. totally disconnected) topological spaces is Hausdorff (resp. totally disconnected).   Recall that an action of a group G on a topological space X is said to be continuous if the map ϕg : X → X given by ϕg (x) = gx is continuous on X for each g ∈ G. Proposition 1.2.2. The action of G on AG is continuous. Proof. Let g ∈ G and consider the map ϕg : AG → AG defined by ϕg (x) = gx. The map πh ◦ϕg is equal to πg−1 h and is therefore continuous on AG for every   h ∈ G. Consequently, ϕg is continuous (cf. Sect. A.4).

1.3 Periodic Configurations Let G be a group and let A be a set. Let H be a subgroup of G. A configuration x ∈ AG is called H-periodic if x is fixed by H, that is, if one has

4

1 Cellular Automata

hx = x

for all h ∈ H.

Let Fix(H) denote the subset of AG consisting of all H-periodic configurations. Examples 1.3.1. (a) One has Fix({1G }) = AG . (b) The set Fix(G) consists of all constant configurations and may be therefore identified with A. (c) For G = Z and H = nZ, n ≥ 1, the set Fix(H) is the set of sequences x : Z → A which admit n as a (not necessarily minimal) period, that is, such that x(i + n) = x(i) for all i ∈ Z. Proposition 1.3.2. Let H be a subgroup of G. Then the set Fix(H) is closed in AG for the prodiscrete topology. Proof. We have Fix(H) =



{x ∈ AG : hx = x}.

(1.4)

h∈H

The space AG is Hausdorff by Proposition 1.2.1 and the action of G on AG is continuous by Proposition 1.2.2. Thus the set of fixed points of the map x → gx is closed in AG for each g ∈ G. Therefore Fix(H) is closed in AG by (1.4).   Consider the set H\G = {Hg : g ∈ G} consisting of all right cosets of H in G and the canonical surjective map ρ : G → H\G g → Hg. Given an element y ∈ AH\G , i.e., a map y : H\G → A, we can form the composite map y ◦ ρ : G → A which is an element of AG . In fact, we have y ◦ ρ ∈ Fix(H) since (h(y ◦ ρ))(g) = y ◦ ρ(h−1 g) = y(ρ(h−1 g)) = y(ρ(g)) = y ◦ ρ(g) for all g ∈ G and h ∈ H. Proposition 1.3.3. Let H be a subgroup of G and let denote by ρ : G → H\G the canonical surjection. Then the map ρ∗ : AH\G → Fix(H) defined by ρ∗ (y) = y ◦ ρ for all y ∈ AH\G is bijective. Proof. If y1 , y2 ∈ AH\G satisfy y1 ◦ρ = y2 ◦ρ, then y1 = y2 since ρ is surjective. Thus ρ∗ is injective. If x ∈ Fix(H), then hx = x for all h ∈ H, that is, x(h−1 g) = x(g)

for all h ∈ H, g ∈ G.

Thus, the configuration x is constant on each right coset of G modulo H,   that is, x is in the image of ρ∗ . This shows that ρ∗ is surjective.

1.3 Periodic Configurations

5

Corollary 1.3.4. If the set A is finite and H is a subgroup of finite index of G, then the set Fix(H) is finite and one has | Fix(H)| = |A|[G:H] , where [G : H] denotes the index of H in G.   Example 1.3.5. Let G = Z and H = nZ, where n ≥ 1. If A is finite of cardinality k, then | Fix(H)| = k n . Suppose now that N is a normal subgroup of G, that is, gN = N g for all g ∈ G. Then, there is a natural group structure on G/N = N \G for which the canonical surjection ρ : G → G/N is a homomorphism. Proposition 1.3.6. Let N be a normal subgroup of G. Then Fix(N ) is a G-invariant subset of AG . Proof. Let x ∈ Fix(N ) and g ∈ G. Given h ∈ N , then there exists h ∈ N such that hg = gh , since N is normal in G. Thus, we have h(gx) = g(h x) = gx which shows that gx ∈ Fix(N ).

 

Since every element of Fix(N ) is fixed by N , the action of G on Fix(N ) induces an action of G/N on Fix(N ) which satisfies ρ(g)x = gx for all g ∈ G and x ∈ Fix(N ). Suppose that a group Γ acts on two sets X and Y . A map ϕ : X → Y is called Γ -equivariant if one has ϕ(γx) = γϕ(x) for all γ ∈ Γ and x ∈ X. Proposition 1.3.7. Let N be a normal subgroup of G and let ρ : G → G/N denote the canonical epimorphism. Then the map ρ∗ : AG/N → Fix(N ) defined by ρ∗ (y) = y ◦ ρ for all y ∈ AG/N is a G/N -equivariant bijection. Proof. We already know that ρ∗ is bijective (Proposition 1.3.3). Let g ∈ G and y ∈ AG/N . For all g  ∈ G, we have ρ(g)ρ∗ (y)(g  ) = gρ∗ (y)(g  ) = ρ∗ (y)(g −1 g  ) = (y ◦ ρ)(g −1 g  ) = y(ρ(g −1 g  )) = y((ρ(g))−1 ρ(g  )) = ρ(g)y(ρ(g  )) = ρ∗ (ρ(g)y)(g  ). Thus ρ(g)ρ∗ (y) = ρ∗ (ρ(g)y). This shows that ρ∗ is G/N -equivariant.

 

6

1 Cellular Automata

1.4 Cellular Automata Let G be a group and let A be a set. Definition 1.4.1. A cellular automaton over the group G and the alphabet A is a map τ : AG → AG satisfying the following property: there exist a finite subset S ⊂ G and a map μ : AS → A such that τ (x)(g) = μ((g −1 x)|S )

(1.5)

for all x ∈ AG and g ∈ G, where (g −1 x)|S denotes the restriction of the configuration g −1 x to S. Such a set S is called a memory set and μ is called a local defining map for τ . Observe that formula (1.5) says that the value of the configuration τ (x) at an element g ∈ G is the value taken by the local defining map μ at the pattern obtained by restricting to the memory set S the shifted configuration g −1 x. Remark 1.4.2. (a) Equality (1.5) may also be written τ (x)(g) = μ((x ◦ Lg )|S )

(1.6)

by (1.2). (b) For g = 1G , formula (1.5) gives us τ (x)(1G ) = μ(x|S ).

(1.7)

As the restriction map AG → AS , x → x|S , is surjective, this shows that if S is a memory set for the cellular automaton τ , then there is a unique map μ : AS → A which satisfies (1.5). Thus one says that this unique μ is the local defining map for τ associated with the memory set S. Examples 1.4.3. (a) The cellular automaton associated with the Game of Life. Consider an infinite two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, live or dead. Every cell c interacts with its eight neighboring cells, namely the North, North-East, East, South-East, South, South-West, West and North-West cells (see Fig. 1.1). At each step in time, the following rules for the evolution of the states of the cells are applied (in Figs. 1.2–1.5 we label with a “•” a live cell and with a “◦” a dead cell): • (birth): a cell that is dead at time t becomes alive at time t + 1 if and only if three of its neighbors are alive at time t (cf. Fig. 1.2); • (survival): a cell that is alive at time t will remain alive at time t + 1 if and only if it has exactly two or three live neighbors at time t (cf. Fig. 1.3);

1.4 Cellular Automata

7

Fig. 1.1 The cell c and its eight neighboring cells

Fig. 1.2 A cell that is dead at time t becomes alive at time t + 1 if and only if three of its neighbors are alive at time t

Fig. 1.3 A cell that is alive at time t will remain alive at time t + 1 if and only if it has exactly two or three live neighbors at time t

8

1 Cellular Automata

Fig. 1.4 A live cell that has at most one live neighbor at time t will be dead at time t + 1

• (death by loneliness): a live cell that has at most one live neighbor at time t will be dead at time t + 1 (cf. Fig. 1.4); • (death by overcrowding): a cell that is alive at time t and has four or more live neighbors at time t, will be dead at time t + 1 (cf. Fig. 1.5).

Fig. 1.5 A cell that is alive at time t and has four or more live neighbors at time t, will be dead at time t + 1

Let us show that the map which transforms a configuration of cells at time t into the configuration at time t + 1 according to the above rules is indeed a cellular automaton. Consider the group G = Z2 and the finite set S = {−1, 0, 1}2 ⊂ G. Then there is a one-to-one correspondence between the cells in the grid and the elements in G in such a way that the following holds. If c is a given cell, then c+(0, 1) is the neighboring North cell, c+(1, 1) is the neighboring North-East cell, and so on; in other words, c and its eight neighboring cells correspond to the group elements c + s with s ∈ S (see Fig. 1.6). Consider the alphabet A = {0, 1}. The state 0 (resp. 1) corresponds to absence (resp. presence) of life. With each configuration of the states of the cells in the grid we associate a map x ∈ AG defined as follows. Given a cell c we set x(c) = 1 (resp. 0) if the cell c is alive (resp. dead).

1.4 Cellular Automata

9

Fig. 1.6 The cell c and its eight neighboring cells c + s, s ∈ S = {−1, 0, 1}2

Consider the map μ : AS → A given by ⎧ ⎧ ⎪ ⎪ ⎨ s∈S y(s) = 3 ⎪ ⎪ ⎨1 if or ⎪ μ(y) = ⎩ ⎪ ⎪ s∈S y(s) = 4 and y((0, 0)) = 1, ⎪ ⎩ 0 otherwise

(1.8)

for all y ∈ AS . A moment of thought tells us that μ just expresses the rules for the Game of Life. The cellular automaton τ : AG → AG with memory set S and local defining map μ is called the cellular automaton associated with the Game of Life. (b) The Discrete Laplacian. Let G = Z and A = R. Consider the map Δ : RZ → RZ defined by Δ(x)(n) = 2x(n) − x(n − 1) − x(n + 1). Then Δ is the cellular automaton over Z with memory set S = {−1, 0, 1} and local defining map μ : RS → R given by μ(y) = 2y(0) − y(−1) − y(1)

for all y ∈ RS .

This may be generalized in the following way. Let G be an arbitrary group and let S be a nonempty finite subset of G. Let K be a field. Consider the G G map ΔS = ΔG S : K → K defined by x(gs). ΔS (x)(g) = |S|x(g) − s∈S

10

1 Cellular Automata

Then ΔS is a cellular automaton over G with memory set S ∪ {1G } and local defining map μ : KS∪{1G } → K given by μ(y) = |S|y(1G ) − y(s) for all y ∈ KS∪{1G } . s∈S

This cellular automaton is called the discrete Laplacian over K associated with G and S. (c) The Majority action cellular automaton. Let G be a group and let S be a finite subset of G. Take A = {0, 1} and consider the map τ : AG → AG defined by ⎧  ⎪ x(gs) > |S| if ⎨1 2 s∈S |S| τ (x)(g) = 0 if x(gs) < 2 ⎪ s∈S ⎩ |S| x(g) if s∈S x(gs) = 2 for all x ∈ AG . Then τ is a cellular automaton over G with memory set S ∪ {1G } and local defining map μ : AS∪{1G } → A given by ⎧  ⎪ y(s) > |S| if ⎨1 2 s∈S |S| μ(y) = 0 if y(s) < 2 ⎪ s∈S ⎩ |S| y(1G ) if s∈S y(s) = 2 for all y ∈ AS∪{1G } . The cellular automaton τ is called the majority action cellular automaton associated with G and S (see Figs. 1.7–1.8). The terminology comes from the fact that given x ∈ AG and g ∈ G, the value τ (x)(g) is equal to a ∈ {0, 1} if there is a strict majority of elements of gS at which the configuration x takes the value a, or to x(g) if no such majority exists. (d) Let G be a group, A a set, and f : A → A a map from A into itself. Then the map τ : AG → AG defined by τ (x) = f ◦ x is a cellular automaton with memory set S = {1G } and local defining map μ : AS → A given by μ(y) = f (y(1G )). Note that, if f is the identity map IdA on A, then τ equals the identity map IdAG on AG . (e) Let G be a group, A a set, and s0 an element of G. Let Rs0 : G → G denote the right multiplication by s0 in G, that is, the map Rs0 : G → G defined by Rs0 (g) = gs0 . Then the map τ : AG → AG defined by τ (x) = x ◦ Rs0 is a cellular automaton with memory set S = {s0 } and local defining map μ : AS → A given by μ(y) = y(s0 ). Proposition 1.4.4. Let G be a group and let A be a set. Then every cellular automaton τ : AG → AG is G-equivariant.

1.4 Cellular Automata

11

Fig. 1.7 The local defining map μ for the majority action on Z associated with S = {+1, −1}

Fig. 1.8 The majority action τ on Z associated with S = {+1, −1}

Proof. Let S be a memory set for τ and let μ : AS → A be the associated local defining map. For all g, h ∈ G and x ∈ AG , we have τ (gx)(h) = μ((h−1 gx)|S ) = μ(((g −1 h)−1 x)|S ) = τ (x)(g −1 h) = gτ (x)(h). Thus τ (gx) = gτ (x).

 

Corollary 1.4.5. Let τ : AG → AG be a cellular automaton and let H be a subgroup of G. Then one has τ (Fix(H)) ⊂ Fix(H). Proof. Let x ∈ Fix(H). By the previous Proposition, we have, for every h ∈ H, hτ (x) = τ (hx) = τ (x). Thus τ (x) ∈ Fix(H).

 

The following characterization of cellular automata will be useful in the sequel.

12

1 Cellular Automata

Proposition 1.4.6. Let G be a group and let A be a set. Consider a map τ : AG → AG . Let S be a finite subset of G and let μ : AS → A. Then the following conditions are equivalent: (a) τ is a cellular automaton admitting S as a memory set and μ as the associated local defining map; (b) τ is G-equivariant and one has τ (x)(1G ) = μ(x|S ) for every x ∈ AG . Proof. The fact that (a) implies (b) follows from Proposition 1.4.4 and formula (1.7) Conversely, suppose (b). Then, by using the G-equivariance of τ , we get τ (x)(g) = τ (g −1 x)(1G ) = μ((g −1 x)|S ) for all x ∈ AG and g ∈ G. Consequently, τ satisfies (a).

 

An important feature of cellular automata is their continuity (with respect to the prodiscrete topology). In the proof of this property, we shall use the following. Lemma 1.4.7. Let G be a group and let A be a set. Let τ : AG → AG be a cellular automaton with memory set S and let g ∈ G. Then τ (x)(g) depends only on the restriction of x to gS. Proof. This is an immediate consequence of (1.5) since (g −1 x)(s) = x(gs) for all s ∈ S.   Proposition 1.4.8. Let G be a group and let A be a set. Then every cellular automaton τ : AG → AG is continuous. Proof. Let S be a memory set for τ . Let x ∈ AG and let W be a neighborhood of τ (x) in AG . Then we can find a finite subset Ω ⊂ G such that (cf. equation (1.3)) V (τ (x), Ω) ⊂ W. Consider the finite set ΩS = {gs : g ∈ Ω, s ∈ S}. If y ∈ AG coincide with x on ΩS, then τ (y) and τ (x) coincide on Ω by Lemma 1.4.7. Thus, we have τ (V (x, ΩS)) ⊂ V (τ (x), Ω) ⊂ W. This shows that τ is continuous.

 

Proposition 1.4.9. Let G be a group and let A be a set. Let σ : AG → AG and τ : AG → AG be cellular automata. Then the composite map σ ◦ τ : AG → AG is a cellular automaton. Moreover, if S (resp. T ) is a memory set for σ (resp. τ ), then ST = {st : s ∈ S, t ∈ T } is a memory set for σ ◦ τ . Proof. It is clear that the map σ ◦ τ is G-equivariant since σ and τ are G-equivariant (by Proposition 1.4.4). Let S (resp. T ) be a memory set for

1.4 Cellular Automata

13

σ (resp. τ ). For every x ∈ AG , we have σ ◦ τ (x)(1G ) = σ(τ (x))(1G ). By Lemma 1.4.7, σ(τ (x))(1G ) depends only on the restriction of τ (x) to S. By using Lemma 1.4.7 again, we deduce that, for every s ∈ S, the element τ (x)(s) depends only on the restriction of x to sT . Therefore, σ ◦ τ (x)(1G ) depends only on the restriction of x to ST . By applying Proposition 1.4.6, we conclude that σ ◦ τ is a cellular automaton admitting ST as a memory set.   Remark 1.4.10. With the hypotheses and notation of the previous proposition, denote by μ : AS → A and ν : AT → A the local defining maps for σ and τ , respectively. Then, the local defining map κ : AST → A for σ ◦ τ may be described in the following way. For y ∈ AST and s ∈ S define ys ∈ AT by setting ys (t) = y(st) for all t ∈ T . Also, denote by y ∈ AS the map defined by y(s) = ν(ys ) for all s ∈ S. We finally define the map κ : AST → A by setting (1.9)

κ(y) = μ(y) for all y ∈ AST . Let x ∈ AG , g ∈ G, s ∈ S, and t ∈ T . We then have (s−1 g −1 x)|T (t) = s−1 g −1 x(t) = g −1 x(st) = (g −1 x)|ST (st)

= (g −1 x)|ST s (t). This shows that



(s−1 g −1 x)|T = (g −1 x)|ST s

and therefore



τ (g −1 x)(s) = ν (s−1 g −1 x)|T = ν (g −1 x)|ST s = (g −1 x)|ST (s). As a consequence,

τ (g −1 x)|S = (g −1 x)|ST .

(1.10)

Finally, one has (σ ◦ τ )(x)(g) = σ(τ (x))(g)

= μ (g −1 τ (x))|S = μ(τ (g −1 x)|S )

(1.11)

(by (1.10)) = μ((g −1 x)|ST )

(by (1.9)) = κ (g −1 x)|ST . Recall that a monoid is a set equipped with an associative binary operation admitting an identity element. Denote by CA(G; A) the set consisting of all

14

1 Cellular Automata

cellular automata τ : AG → AG . In Example 1.4.3(d) we have seen that the identity map IdAG : AG → AG is a cellular automaton. Thus we have: Corollary 1.4.11. The set CA(G; A) is a monoid for the composition of maps.  

1.5 Minimal Memory Let G be a group and let A be a set. Let τ : AG → AG be a cellular automaton. Let S be a memory set for τ and let μ : AS → A be the associated defining map. If S  is a finite subset of G such that S ⊂ S  , then S  is also a memory set  for τ and the local defining map associated with S  is the map μ : AS → A  given by μ = μ◦p, where p : AS → AS is the canonical projection (restriction map). This shows that the memory set of a cellular automaton is not unique in general. However, we shall see that every cellular automaton admits a unique memory set of minimal cardinality. Let us first establish the following result. Lemma 1.5.1. Let τ : AG → AG be a cellular automaton. Let S1 and S2 be memory sets for τ . Then S1 ∩ S2 is also a memory set for τ . Proof. Let x ∈ AG . Let us show that τ (x)(1G ) depends only on the restriction of x to S1 ∩ S2 . To see this, consider an element y ∈ AG such that x|S1 ∩S2 = y|S1 ∩S2 . Let us choose an element z ∈ AG such that z|S1 = x|S1 and z|S2 = y|S2 (we may take for instance the configuration z ∈ AG which coincides with x on S1 and with y on G \ S1 ). We have τ (x)(1G ) = τ (z)(1G ) since x and z coincide on S1 , which is a memory set for τ . On the other hand, we have τ (y)(1G ) = τ (z)(1G ) since y and z coincide on S2 , which is also a memory set for τ . It follows that τ (x)(1G ) = τ (y)(1G ). Thus there exists a map μ : AS1 ∩S2 → A such that τ (x)(1G ) = μ(x|S1 ∩S2 )

for all x ∈ AG .

As τ is G-equivariant (Proposition 1.4.4), we deduce that S1 ∩S2 is a memory set for τ by using Proposition 1.4.6.   Proposition 1.5.2. Let τ : AG → AG be a cellular automaton. Then there exists a unique memory set S0 ⊂ G for τ of minimal cardinality. Moreover, if S is a finite subset of G, then S is a memory set for τ if and only if S0 ⊂ S. Proof. Let S0 be a memory set for τ of minimal cardinality. As we have seen at the beginning of this section, every finite subset of G containing S0 is also a memory set for τ . Conversely, let S be a memory set for τ . As S ∩ S0 is a memory set for τ by Lemma 1.5.1, we have |S ∩ S0 | ≥ |S0 |. This implies

1.6 Cellular Automata over Quotient Groups

15

S ∩ S0 = S0 , that is, S0 ⊂ S. In particular, S0 is the unique memory set of minimal cardinality.   The memory set of minimal cardinality of a cellular automaton is called its minimal memory set. Remark 1.5.3. A map F : AG → AG is constant if there exists a configuration x0 ∈ AG such that F (x) = x0 for all x ∈ AG . By G-equivariance, a cellular automaton τ : AG → AG is constant if and only if there exists a ∈ A such that τ (x)(g) = a for all x ∈ AG and g ∈ G. Observe that a cellular automaton τ : AG → AG is constant if and only if its minimal memory set is the empty set.

1.6 Cellular Automata over Quotient Groups Let G be a group and let A be a set. Let τ : AG → AG be a cellular automaton. Suppose that N is a normal subgroup of G and let ρ : G → G/N denote the canonical epimorphism. It follows from Proposition 1.3.7 that the map ρ∗ : AG/N → Fix(N ), defined by ρ∗ (y) = y ◦ ρ for all y ∈ AG/N , is a bijection from the set AG/N of configurations over the group G/N onto the set Fix(N ) ⊂ AG of N -periodic configurations over G. On the other hand, the set Fix(N ) satisfies τ (Fix(N )) ⊂ Fix(N ) by Corollary 1.4.5. Thus, we can define a map τ : AG/N → AG/N by setting τ = (ρ∗ )−1 ◦ τ |Fix(N ) ◦ ρ∗ .

(1.12)

In other words, the map τ is obtained by conjugating by ρ∗ the restriction of τ to Fix(N ), so that the diagram ρ∗

AG/N −−−−→ Fix(N ) ⊂ AG ⏐ ⏐ ⏐ ⏐τ | τ

Fix(N ) AG/N −−−∗−→ ρ

Fix(N )

is commutative. Suppose that S ⊂ G is a memory set for τ and that μ : AS → A is the associated local defining map. Consider the finite subset S = ρ(S) ⊂ G/N and the map μ : AS → A defined by μ = μ ◦ π, where π : AS → AS is the injective map induced by ρ. Proposition 1.6.1. The map τ : AG/N → AG/N is a cellular automaton over the group G/N admitting S as a memory set and μ : AS → A as the associated local defining map.

16

1 Cellular Automata

Proof. Let y ∈ AG/N , g ∈ G, and g = ρ(g). We have τ (y)(g) = τ (y ◦ ρ)(g) = μ((g −1 (y ◦ ρ))|S ) = μ((g −1 y)|S ). Thus τ is a cellular automaton with memory set S and local defining map μ : AS → A.   Consider now the map Φ : CA(G; A) → CA(G/N ; A) given by Φ(τ ) = τ , where τ is defined by (1.12). We have the following: Proposition 1.6.2. The map Φ : CA(G; A) → CA(G/N ; A) is a monoid epimorphism. Proof. Let σ : AG/N → AG/N be a cellular automaton over G/N with memory set T ⊂ G/N and local defining map ν : AT → A. Let S ⊂ G be a finite set such that ρ induces a bijection φ : S → T . Consider the map μ : AS → A defined by μ(y) = ν(y ◦ φ−1 ) for all y ∈ AS . Let τ : AG → AG be the cellular automaton over G with memory set S and local defining map μ. We have μ(z) = (μ ◦ π)(z) = ν(π(z) ◦ φ−1 ) = ν(z) for all z ∈ AS . It follows that μ = ν and τ = σ. This shows that Φ is surjective. The fact that Φ is a monoid morphism immediately follows from (1.12).   Examples 1.6.3. Let G be a group, S ⊂ G a finite subset, and N a normal subgroup of G. Denote by ρ : G → G/N the canonical epimorphism and suppose that ρ induces a bijection between S and S = ρ(S) ⊂ G/N . (a) Consider the discrete laplacian ΔS : RG → RG associated with G and S (cf. Example 1.4.3(b)). Then Φ(ΔS ) : RG/N → RG/N is the discrete laplacian associated with G/N and S. (b) Consider the majority action cellular automaton τ : {0, 1}G → {0, 1}G associated with G and S (cf. Example 1.4.3(c)). Then Φ(τ ) : {0, 1}G/N → {0, 1}G/N is the majority action cellular automaton associated with G/N and S.

1.7 Induction and Restriction of Cellular Automata Let G be a group and let A be a set. Let H be a subgroup of G. Let CA(G, H; A) denote the set consisting of all cellular automata τ : AG → G A admitting a memory set S such that S ⊂ H. Thus, CA(G, H; A) is the

1.7 Induction and Restriction of Cellular Automata

17

subset of CA(G; A) consisting of the cellular automata whose minimal memory set is contained in H. Recall that a subset N of a monoid M is called a submonoid if the identity element 1M is in N and N is stable under the monoid operation (that is, xy ∈ N for all x, y ∈ N ). If N is a submonoid of a monoid M , then the monoid operation induces by restriction a monoid structure on N . Proposition 1.7.1. The set CA(G, H; A) is a submonoid of CA(G; A). Proof. The identity element of CA(G; A) is the identity map IdAG . We have IdAG ∈ CA(G, H; A) since {1G } is a memory set for IdAG and {1G } ⊂ H. Let σ, τ ∈ CA(G, H; A). Let S (resp. T ) be a memory set for σ (resp. τ ) such that S ⊂ H (resp. T ⊂ H). It follows from Proposition 1.4.9 that ST is a memory set for σ ◦ τ . Since ST ⊂ H, this implies that σ ◦ τ ∈ CA(G, H; A). This shows that CA(G, H; A) is a submonoid of CA(G; A).   Let τ ∈ CA(G, H; A). Let S be a memory set for τ such that S ⊂ H and let μ : AS → A denote the associated local defining map. Then, the map τH : AH → AH defined by τH (x)(h) = μ((h−1 x)|S )

for all x ∈ AH , h ∈ H,

is a cellular automaton over the group H with memory set S and local defining map μ. Observe that if x  ∈ AG is such that x |H = x, then x)(h) τH (x)(h) = τ (

for all h ∈ H.

(1.13)

This shows in particular that τH does not depend on the choice of the memory set S ⊂ H. One says that τH is the restriction of the cellular automaton τ to H. Conversely, let σ : AH → AH be a cellular automaton with memory set S and local defining map μ : AS → A. Then the map σ G : AG → AG defined by x)(g) = μ((g −1 x )|S ) σ G (

for all x  ∈ AG , g ∈ G,

is a cellular automaton over G with memory set S and local defining map μ. If S0 is the minimal memory set of σ and μ0 : AS0 → A is the associated local defining map then μ = μ0 ◦ π, where π : AS → AS0 is the restriction map (see Sect. 1.5). Thus, one has x)(g) = μ((g −1 x )|S ) = μ0 ◦ π((g −1 x )|S ) = μ0 ((g −1 x )|S0 ) σ G ( for all x  ∈ AG and g ∈ G. This shows in particular that σ G does not depend on the choice of the memory set S ⊂ H. One says that σ G ∈ CA(G, H; A) is the cellular automaton induced by σ ∈ CA(H; A). Proposition 1.7.2. The map τ → τH is a monoid isomorphism from CA(G, H; A) onto CA(H; A) whose inverse is the map σ → σ G .

18

1 Cellular Automata

Proof. To simplify notation, denote by α : CA(G, H; A) → CA(H; A) and β : CA(H; A) → CA(G, H; A) the maps defined by α(τ ) = τH and β(σ) = σ G respectively. It is clear from the definitions given above that β ◦ α and α ◦ β are the identity maps. Therefore, α is bijective with inverse β. It remains to show that α is a monoid homomorphism.  ∈ AG extending x. By applying (1.13), we get Let x ∈ AH and let x x)(h) = x (h) = x(h) α(IdAG )(x)(h) = IdAG ( for all h ∈ H. This shows that α(IdAG )(x) = x for all x ∈ AH , that is, α(IdAG ) = IdAH . Let σ, τ ∈ CA(G, H; A). Let x ∈ AH and let x  ∈ AG extending x. By applying (1.13) again, we have α(σ ◦ τ )(x)(h) = (σ ◦ τ )( x)(h) = σ(τ ( x))(h)

(1.14)

for all h ∈ H. On the other hand, since τ ( x) extends α(τ )(x), we have α(σ)(α(τ )(x))(h) = σ(τ ( x))(h) that is, (α(σ) ◦ α(τ ))(x)(h) = σ(τ ( x))(h)

(1.15)

for all h ∈ H. From (1.14) and (1.15), we deduce that α(σ ◦ τ )(x) = (α(σ) ◦   α(τ ))(x) for all x ∈ AH , that is, α(σ ◦ τ ) = α(σ) ◦ α(τ ). Let τ ∈ CA(G, H; A). In order to analyze the way τ transforms a configuration x  ∈ AG , we now introduce the set G/H = {gH : g ∈ G} consisting of all left cosets of H in G. Since the cosets c ∈ G/H form a partition of G, we  have a natural identification AG = c∈G/H Ac . With this identification, we have x  = ( x|c )c∈G/H for each x  ∈ AG , where x |c ∈ Ac denotes the restriction of x  to c. Observe now that if c ∈ G/H and g ∈ c, then τ ( x)(g) depends only on x |c (this directly follows from Lemma 1.4.7 since if S is a memory set for τ with S ⊂ H, then gS ⊂ c). This implies that τ may be written as a product  τc , (1.16) τ= c∈G/H

x|c ) = (τ ( x))|c for where τc : Ac → Ac is the unique map which satisfies τc ( all x  ∈ AG . Note that the notation is coherent when c = H, since, in this case, τc = τH : AH → AH is the cellular automaton obtained by restriction of τ to H. Given a coset c ∈ G/H and an element g ∈ c, denote by φg : H → c the bijective map defined by φg (h) = gh for all h ∈ H. Then φg induces a bijective map φ∗g : Ac → AH given by

1.7 Induction and Restriction of Cellular Automata

φ∗g (x) = x ◦ φg

19

(1.17)

for all x ∈ Ac . It turns out that the maps τc and τH are conjugate by φ∗g : Proposition 1.7.3. With the above notation, we have, τc = (φ∗g )−1 ◦ τH ◦ φ∗g .

(1.18)

In other words, the following diagram τ

Ac −−−c−→ ⏐ ⏐ φ∗ g

Ac ⏐ ⏐φ∗

g

AH −−−−→ AH τH

is commutative. Proof. Let x ∈ Ac and let x  ∈ AG extending x. For all h ∈ H, we have (φ∗g ◦ τc )(x)(h) = φ∗g (τc (x))(h) = (τc (x) ◦ φg )(h) = τc (x)(gh) = τ ( x)(gh) x)(h) = g −1 τ ( = τ (g −1 x )(h),

where the last equality follows from the G-equivariance of τ (Proposition  ∈ AG extends x ◦ φg ∈ AH . 1.4.4). Now observe that the configuration g −1 x Thus, we have (φ∗g ◦ τc )(x)(h) = τH (x ◦ φg )(h) = τH (φ∗g (x))(h) = (τH ◦ φ∗g )(x)(h).  This shows that φ∗g ◦ τc = τH ◦ φ∗g , which gives (1.18) since φ∗g is bijective.  The following statement will be used in the proof of Proposition 3.2.1: Proposition 1.7.4. Let G be a group and let A be a set. Let H be a subgroup of G and let τ ∈ CA(G, H; A). Let τH : AH → AH denote the cellular automaton obtained by restriction of τ to H. Then the following hold: (i) τ is injective if and only if τH is injective; (ii) τ is surjective if and only if τH is surjective; (iii) τ is bijective if and only if τH is bijective.

20

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Proof. It immediately follows from (1.16) that τ is injective (resp. surjective, resp. bijective) if and only if τc is injective (resp. surjective, resp. bijective) for all c ∈ G/H. Now, (1.18) says that, given c ∈ G/H and g ∈ G, the map τc and τH are conjugate by the bijection φg . We deduce that τc is injective (resp. surjective, resp. bijective) if and only if τH is injective (resp. surjective, resp. bijective). Thus, τ is injective (resp. surjective, resp. bijective) if and only if τH is injective (resp. surjective, resp. bijective).  

1.8 Cellular Automata with Finite Alphabets Let G be a group and let A be a finite alphabet. As a product of finite spaces is compact by Tychonoff theorem (see Corollary A.5.3), it follows that AG is compact. This topological property is very useful in the study of cellular automata over finite alphabets. In particular, it may be used to prove the following: Theorem 1.8.1 (Curtis-Hedlund theorem). Let G be a group and let A be a finite set. Let τ : AG → AG be a map and equip AG with its prodiscrete topology. Then the following conditions are equivalent: (a) the map τ is a cellular automaton; (b) the map τ is continuous and G-equivariant. Proof. The fact that (a) implies (b) directly follows from Proposition 1.4.4 and Proposition 1.4.8 (this implication does not require the finiteness assumption on the alphabet A). Conversely, suppose (b). Let us show that τ is a cellular automaton. As the map ϕ : AG → A defined by ϕ(x) = τ (x)(1G ) is continuous, we can find, for each x ∈ AG , a finite subset Ωx ⊂ G such that if y ∈ AG coincide with x on Ωx , that is, if y ∈ V (x, Ωx ), then τ (y)(1G ) = τ (x)(1G ). The sets V (x, Ωx ) form an open cover of AG . As AG is compact, there is a finite subset F ⊂ AG such that the sets V (x, Ωx ), x ∈ F , cover AG . Let us set S = ∪x∈F Ωx and suppose that two configurations y, z ∈ AG coincide on S. Let x0 ∈ F be such that y ∈ V (x0 , Ωx0 ), that is, y|Ωx0 = x0 |Ωx0 . As S ⊃ Ωx0 we have y|Ωx0 = z|Ωx0 and therefore τ (y)(1G ) = τ (x0 )(1G ) = τ (z)(1G ). Thus there is a map μ : AS → A such that τ (x)(1G ) = μ(x|S ) for all x ∈ AG . As τ is G-equivariant, it follows from Proposition 1.4.6 that τ is a cellular automaton with memory set S and local defining map μ.   When the alphabet A is infinite, a continuous and G-equivariant map τ : AG → AG may fail to be a cellular automaton. In other words, the implication (b) ⇒ (a) in Theorem 1.8.1 becomes false if we suppress the finiteness hypothesis on A. This is shown by the following example.

1.8 Cellular Automata with Finite Alphabets

21

Example 1.8.2. Let G be an arbitrary infinite group and take A = G as the alphabet set. To avoid confusion, we denote by g · h the product of two elements g and h in G. Consider the map τ : AG → AG defined by τ (x)(g) = x(g · x(g)) for all x ∈ AG and g ∈ G. Given x ∈ AG and g, h ∈ G we have g(τ (x))(h) = τ (x)(g −1 · h) = x(g −1 · h · x(g −1 · h)) = x(g −1 · h · [gx](h)) = gx(h · [gx](h)) = τ (gx)(h). This shows that g(τ (x)) = τ (gx) for all x ∈ AG and g ∈ G. Therefore, τ is G-equivariant. Moreover, τ is continuous. Indeed, given x ∈ AG and a finite set K ⊂ G, let us show that there exists a finite set F ⊂ G such that, if y ∈ AG and y ∈ V (x, F ), then τ (y) ∈ V (τ (x), K). Set F = K ∪ {k · x(k) : k ∈ K}. Then, if y ∈ V (x, F ), then, for all k ∈ K one has τ (x)(k) = x(k · x(k)) = y(k · x(k)) = y(k · y(k)) = τ (y)(k). This shows that τ (y) ∈ V (τ (x), K). Thus, τ is continuous. However, τ is not a cellular automaton. Indeed, fix g0 ∈ G \ {1G } and, for all g ∈ G, consider the configurations xg and yg in AG defined by ⎧ ⎪ if h = 1G ⎨g xg (h) = g0 if h = g ⎪ ⎩ 1G otherwise 

and yg (h) =

g 1G

if h = 1G otherwise

for all h ∈ G. Note that xg |G\{g} = yg |G\{g} . Let F ⊂ G be a finite set and choose g ∈ G \ F (this is possible because G is infinite). Then one has xg |F = yg |F while τ (xg )(1G ) = xg (xg (1G )) = xg (g) = g0 and τ (yg )(1G ) = yg (yg (1G )) = yg (g) = 1G ,

22

1 Cellular Automata

so that τ (xg )(1G ) = τ (yg )(1G ). It follows that there is no finite set F ⊂ G such that, for all x ∈ AG , the value of τ (x) at 1G only depends on the values of x|F . This shows that τ is not a cellular automaton (cf. Remark 1.4.2(b)).

1.9 The Prodiscrete Uniform Structure Let G be a group and let A be a set. The prodiscrete uniform structure on AG is the product uniform structure  obtained by taking the discrete uniform structure on each factor A of AG = g∈G A (see Appendix B for definition and basic facts about uniform structures). A base of entourages for the prodiscrete uniform structure on AG is given by the sets WΩ ⊂ AG × AG , where WΩ = {(x, y) ∈ AG × AG : x|Ω = y|Ω }

(1.19)

and Ω runs over all finite subsets of G. Observe that, using the notation introduced in (1.3), we have V (x, Ω) = {y ∈ AG : (x, y) ∈ WΩ } for all x ∈ AG . The following statement gives a global characterization of cellular automata in terms of the prodiscrete uniform structure and the G-shift on AG . Theorem 1.9.1. Let A be a set and let G be a group. Let τ : AG → AG be a map and equip AG with its prodiscrete uniform structure. Then the following conditions are equivalent: (a) τ is a cellular automaton; (b) τ is uniformly continuous and G-equivariant. Proof. Suppose that τ : AG → AG is a cellular automaton. We already know that τ is G-equivariant by Proposition 1.4.4. Let us show that τ is uniformly continuous. Let S be a memory set for τ . It follows from Lemma 1.4.7 that if two configurations x, y ∈ AG coincide on gS for some g ∈ G, then τ (x)(g) = τ (y)(g). Consequently, if the configurations x and y coincide on ΩS = {gs : g ∈ Ω, s ∈ S} for some subset Ω ⊂ G, then τ (x) and τ (y) coincide on Ω. Observe that ΩS is finite whenever Ω is finite. Using the notation introduced in (1.19), we deduce that (τ × τ )(WΩS ) ⊂ WΩ for every finite subset Ω of G. As the sets WΩ , where Ω runs over all finite subsets of G, form a base of entourages for the prodiscrete uniform structure

1.9 The Prodiscrete Uniform Structure

23

on AG , it follows that τ is uniformly continuous. This shows that (a) implies (b). Conversely, suppose that τ is uniformly continuous and G-equivariant. Let us show that τ is a cellular automaton. Consider the subset Ω = {1G } ⊂ G. Since τ is uniformly continuous, there exists a finite subset S ⊂ G such that (τ ×τ )(WS ) ⊂ WΩ . This means that τ (x)(1G ) only depends on the restriction of x to S. Thus, there is a map μ : AS → A such that τ (x)(1G ) = μ(x|S ) for all x ∈ AG . Using the G-equivariance of τ , we get τ (x)(g) = [g −1 τ (x)](1G ) = τ (g −1 x)(1G ) = μ((g −1 x)|S ) for all x ∈ AG and g ∈ G. This shows that τ is a cellular automaton with memory set S and local defining map μ. Consequently, (b) implies (a).   Every uniformly continuous map between uniform spaces is continuous with respect to the associated topologies, and the converse is true when the source space is compact (Theorem B.2.3). The topology defined by the prodiscrete uniform structure on AG is the prodiscrete topology (see Example (1) in Sect. B.3). In the case when A is finite, the prodiscrete topology on AG is compact by Tychonoff theorem (Theorem A.5.2). Thus Theorem 1.9.1 reduces to the Curtis-Hedlund theorem (Theorem 1.8.1) in this case. Remark 1.9.2. Suppose that G is countable and A is an arbitrary set. Then the prodiscrete uniform structure (and hence the prodiscrete topology) on AG is metrizable. To see this, choose an increasing sequence ∅ = E 0 ⊂ E 1 ⊂ · · · ⊂ En ⊂ · · ·  of finite subsets of G such that n≥0 En = G. Then the sets WEn , n ≥ 0, form a base of entourages for the prodiscrete uniform structure on AG . Consider now the metric d on AG defined by  0 if x = y, d(x, y) = − max{n≥0: x|En =y|En } 2 if x = y. for all x, y ∈ AG . Then we have WEn = {(x, y) ∈ AG × AG : d(x, y) < 2−n+1 } for every n ≥ 0. Consequently, d defines the prodiscrete uniform structure on AG . Let G be a group and let A be a set. Let H be a subgroup of G. Let us equip AH\G with its prodiscrete uniform structure and Fix(H) ⊂ AG with the

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uniform structure induced by the prodiscrete uniform structure on AG . Recall from Proposition 1.3.7 that there is a natural bijection ρ∗ : AH\G → AG defined by ρ∗ (y) = y ◦ ρ, where ρ : G → H\G is the canonical surjection. Proposition 1.9.3. The map ρ∗ : AH\G → Fix(H) is a uniform isomorphism. Proof. For g ∈ G, let πg : AG → A and πg : AH\G → A denote the projection maps given by x → x(g) and y → y(ρ(g)) respectively. Observe that πg ◦ ρ∗ = πg is uniformly continuous for all g ∈ G. This shows that ρ∗ is uniformly continuous. Similarly, the uniform continuity of (ρ∗ )−1 follows from the fact that πg ◦ (ρ∗ )−1 = πg |Fix(H) is uniformly continuous for each g ∈ G.   Consequently, ρ∗ is a uniform isomorphism.

1.10 Invertible Cellular Automata Let G be a group and let A be a set. One says that a cellular automaton τ : AG → AG is invertible (or reversible) if τ is bijective and the inverse map τ −1 : AG → AG is also a cellular automaton. This is equivalent to the existence of a cellular automaton σ : AG → AG such that τ ◦ σ = σ ◦ τ = IdAG . Thus, the set of invertible cellular automata over the group G and the alphabet A is exactly the group ICA(G; A) consisting of all invertible elements of the monoid CA(G; A). Theorem 1.10.1. Let A be a set and let G be a group. Let τ : AG → AG be a map and equip AG with its prodiscrete uniform structure. Then the following conditions are equivalent: (a) τ is an invertible cellular automaton; (b) τ is a G-equivariant uniform automorphism of AG . Proof. It is clear that the inverse map of a bijective G-equivariant map from AG onto itself is also G-equivariant. Therefore, the equivalence of conditions (a) and (b) follows from the characterization of cellular automata given in Theorem 1.9.1.   Bijective cellular automata over finite alphabets are always invertible: Theorem 1.10.2. Let G be a group and let A be a finite set. Then every bijective cellular automaton τ : AG → AG is invertible. Proof. Let τ : AG → AG be a bijective cellular automaton. The map τ −1 is G-equivariant since τ is G-equivariant. On the other hand, τ −1 is continuous with respect to the prodiscrete topology by compactness of AG . Consequently,   τ −1 is a cellular automaton by Theorem 1.8.1.

1.10 Invertible Cellular Automata

25

The following example shows that Theorem 1.10.2 becomes false if we omit the finiteness hypothesis on the alphabet set A. Example 1.10.3. Let K be a field. Let us take as the alphabet set the ring A = K[[t]] of all formal power series in one indeterminate t with coefficients in K. Thus, an element of A is just a sequence a = (ki )i∈N of elements of K written in the form ki ti , a = k0 + k1 t + k2 t2 + k3 t3 + · · · = i∈N

 i and the addition and multiplication of two elements a = i∈N ki t and b =    i  i  i respectively given by a+b = (k +k )t and ab = i i i∈N ki t are i∈N i∈N ki t  with ki = i1 +i2 =i ki1 ki2 for all i ∈ N. We take G = Z. Thus, a configuration x ∈ AG is a map x : Z → K[[t]]. Consider the map τ : AG → AG defined by τ (x)(n) = x(n) − tx(n + 1) for all x ∈ AG and n ∈ Z. Clearly τ is a cellular automaton admitting S = {0, 1} as a memory set (the local defining map associated with S is the map μ : AS → A defined by μ(x0 , x1 ) = x0 − tx1 for all x0 , x1 ∈ A). Let us show that τ is bijective. Consider the map σ : AG → AG given by σ(x)(n) = x(n) + tx(n + 1) + t2 x(n + 2) + t3 x(n + 3) + · · · for all x ∈ AG and n ∈ Z. Observe that σ(x)(n) ∈ K[[t]] is well defined by the preceding formula. In fact, if we develop x(n) ∈ K[[t]] in the form x(n) = xn,i ti (n ∈ Z, xn,i ∈ K), i∈N

then σ(x)(n) =

i∈N

⎛ ⎞ i ⎝ xn+j,i−j ⎠ ti . j=0

One immediately checks that σ ◦ τ = τ ◦ σ = IdAG . Therefore, τ is bijective with inverse map τ −1 = σ. Let us show that the map σ : AG → AG is not a cellular automaton. Let F be a finite subset of Z and choose an integer M ≥ 0 such that F ⊂ (−∞, M ]. Consider the configuration y defined by y(n) = 0 if n ≤ M and y(n) = 1 if n ≥ M + 1, and the configuration z defined by z(n) = 0 for all n ∈ Z. Then y and z coincide on F . However, the value at 0 of σ(y) is σ(y)(0) = tM +1 + tM +2 + tM +3 + · · · while the value of σ(z) at 0 is σ(z)(0) = 0. It follows that there is no finite subset F ⊂ Z such that σ(x)(0) only depends on the restriction of x ∈ AG

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to F . This shows that σ is not a cellular automaton. Consequently, τ is a bijective cellular automaton which is not invertible. In the next proposition we show that invertibility is preserved under the operations of induction and restriction. Proposition 1.10.4. Let G be a group and let A be a set. Let H be a subgroup of G and let τ ∈ CA(G, H; A). Let τH ∈ CA(H; A) denote the cellular automaton obtained by restriction of τ to H. Then the following conditions are equivalent: (a) τ is invertible; (b) τH is invertible. Moreover, if τ is invertible, then τ −1 ∈ CA(G, H; A) and one has (τ −1 )H = (τH )−1 .

(1.20)

Proof. First recall from (1.16) the factorizations   AG = Ac and τ = τc , c∈G/H

(1.21)

c∈G/H

where τc : Ac → Ac satisfies τc ( x|c ) = (τ ( x))|c for all x  ∈ AG . Suppose that τ is invertible. Denote by σ ∈ CA(G; A) the inverse cellular automaton τ −1 . It follows from (1.21) that the map τc : Ac → Ac is bijective for each c ∈ G/H and that  σ= (τc )−1 , (1.22) c∈G/H

where (τc )−1 : Ac → Ac is the inverse map of τc . Let us show that σ ∈ CA(G, H; A). Let S ⊂ G be a memory set for σ. Let x  ∈ AG . It follows from −1 x|H ). Thus, we have (1.22) that (σ( x))|H = (τH ) ( x))|H (1G ) = (τH )−1 ( x|H )(1G ). σ( x)(1G ) = (σ( |H . Arguing as in the proof of This shows that σ( x)(1G ) only depends on x Lemma 1.5.1, we deduce that S ∩ H is a memory set for σ. Indeed, suppose that two configurations x , y ∈ AG coincide on S ∩ H. Consider the G  on S and with y on G \ S. We configuration z ∈ A which coincide with x z )(1G ) since x  and z coincide on S. On the other hand, have σ( x)(1G ) = σ( we have σ( y )(1G ) = σ( z )(1G ) since y and z coincide on H. This implies y )(1G ). Thus, there is a map μ : AS∩H → A such that σ( x)(1G ) = σ( x|S∩H ) σ( x)(1G ) = μ( for all x  ∈ AG . By applying Proposition 1.4.6, it follows that S ∩ H is a memory set for σ. Since S ∩ H ⊂ H, this shows that τ −1 = σ ∈ CA(G, H; A).

Notes

27

Moreover, it follows from (1.22) that (τ −1 )H = σH = (τH )−1 which gives us (1.20). The equivalence (a) ⇔ (b) is then an immediate consequence of Proposition 1.7.2 which tells us that the restriction map CA(G, H; A) → CA(H; A) is a monoid isomorphism.  

Notes Cellular automata were introduced by J. von Neumann (see [vNeu2]) who used them to describe theoretical models of self-reproducing machines. He first attempted to get such models by means of partial differential equations in R3 . Later he changed the perspective and tried to use ideas and methods coming from robotics and electrical engineering. Eventually, in 1952, following a suggestion of S. Ulam, his former colleague at the Los Alamos Laboratories, he constructed a cellular automaton over the group Z2 with an alphabet consisting of 29 states. He then outlined the construction of a pattern, containing approximatively 200,000 cells, which would reproduce itself. The details were later filled in by A.W. Burks in the 1960s [Bur]. The branch of mathematics which is concerned with the study of the dynamical properties of the shift action is known as symbolic dynamics. Many authors trace the birth of symbolic dynamics back to a paper published in 1898 by J. Hadamard [Had] in which words on two letters were used to code geodesics on certain surfaces with negative curvature. However, as it was pointed out by E.M. Coven and Z.W. Nitecki [CovN], Hadamard’s symbolic description of geodesics is purely static and involves only finite words. According to the authors of [CovN], the beginning of symbolic dynamics should be placed in a paper by G.A. Hedlund [Hed-1] published in 1944. Symbolic dynamics has important applications in dynamical systems, especially in the study of hyperbolic dynamical systems for which symbolic codings may be obtained from Markov partitions. One of the first examples of such an application was the use of the properties of the Thue-Morse sequence (see Exercise 3.41) by M. Morse [Mors] in 1921 to prove the existence of non-periodic recurrent geodesics on surfaces with negative curvature. A detailed exposition of symbolic dynamics over Z may be found for example in the books by B. Kitchens [Kit], by P. K˚ urka [Kur], and by D. Lind and B. Marcus [LiM]. In the mid-1950s, Hedlund studied the so-called shift-commuting block maps which turn out to be exactly cellular automata over the group Z. The Curtis-Hedlund theorem (Theorem 1.8.1), also called Curtis-HedlundLyndon’s theorem or Hedlund’s theorem, is named after Hedlund [Hed-3]

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who proved it in 1969. Its generalization to infinite alphabets, as stated as in Theorem 1.9.1, was proved by the authors in [CeC7]. Cellular automata were intensely studied from the 1960s, both by pure and applied mathematicians, under different names such as tessellation automata, parallel maps, cellular spaces, iterative automata, homogeneous structures, universal spaces, and sliding block codes (cf. [LiM, Section 1.1]). In most cases, these researches focused on cellular automata with finite alphabet over the groups Z or Z2 (see the surveys [BanMS], [BKM], [Kar3], [Wolfr3]). The Game of Life was invented by the British mathematician J.H. Conway. This cellular automaton was described for the first time by M. Gardner [Gar-1] in the October 1970 issue of the Scientific American. From a theoretical computer science point of view, it is important because it has the power of a universal Turing machine, that is, anything that can be computed algorithmically can be computed by using the Game of Life. In the 1980s, S. Wolfram [Wolfr1], [Wolfr2] started a systematic study and empirical classification of elementary cellular automata, that is, of cellular automata over Z with alphabet A = {0, 1} and memory set S = {−1, 0, 1}. 3 There are 2(2 ) = 256 such elementary cellular automata. Wolfram introduced a naming scheme for them which is nowadays widely used. Each elementary cellular automaton τ : AZ → AZ is uniquely determined by the eight bit sequence μ(111)μ(110)μ(101)μ(100)μ(011)μ(010)μ(001)μ(000) ∈ A8 , where μ : AS → A is the associated local defining map. This bit sequence is the binary expansion of an integer in the interval [0, 255], called the Wolfram number of τ . For example, the majority action τ : AZ → AZ associated with S = {−1, 0, 1} (cf. Example 1.4.3(c)) is an elementary cellular automaton. Its local defining map μ gives μ(111)μ(110)μ(101)μ(100)μ(011)μ(010)μ(001)μ(000) = 11101000 (cf. Fig. 1.7). It follows that the Wolfram number of τ is 232. Let us also mention that the elementary cellular automaton with Wolfram number 110 was recently proved computationally universal by M. Cook. Wolfram introduced an empirical classification of elementary cellular automata into four classes according to the behavior of random initial configurations under iterations. These are known as Wolfram classes and are defined as follows: (W1) Almost all initial configurations lead to the same uniform fixed-point configuration, (W2) Almost all initial configurations lead to a periodic configuration, (W3) Almost all initial configurations lead to chaos, (W4) Localized structures with complex interactions emerge. The survey paper [BKM] contains other dynamical classifications of cellular automata over Z with finite alphabet due to R. Gilman and to M. Hurley and P. K˚ urka.

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Invertible cellular automata are used to model time-reversible processes occurring in physics and biology. A group G is called periodic if every element g ∈ G has finite order. In [CeC11] it is shown that if G is a nonperiodic group, then for every infinite set A there exists a bijective cellular automaton τ : AG → AG which is not invertible (cf. Theorem 1.10.2 and Example 1.10.3). It was shown by Amoroso and Patt in 1972 [Amo] that it is decidable whether a given cellular automaton with finite alphabet over Z is invertible. This means that there exists an algorithm which establishes, after a finite number of steps, whether the cellular automaton corresponding to a given local defining map is invertible or not. On the other hand, J. Kari [Kar1], [Kar2], [Kar3] proved that the similar problem for cellular automata with finite alphabet over Zd , d ≥ 2, is undecidable. Its proof is based on R. Berger’s undecidability result for the tiling problem of Wang tiles.

Exercises 1.1. An action of a group Γ on a topological space X is said to be topologically mixing if for each pair of nonempty subsets U and V of X there exists a finite subset F ⊂ Γ such that U ∩ γV = ∅ for all γ ∈ Γ \ F . Show that if G is a group and A is a set then the G-shift on AG is topologically mixing for the prodiscrete topology on AG . 1.2. Let G be a group and let A be a set. Let x ∈ AG and let Ω1 and Ω2 be two subsets of G. Show that V (x, Ω1 ∪ Ω2 ) = V (x, Ω1 ) ∩ V (x, Ω2 ) and WΩ1 ∪Ω2 = WΩ1 ∩ WΩ2 (see (1.3) and (1.19) for the definition of V (x, Ω) and WΩ ). 1.3. Let G be a countable group and let A be a set. Show that the metric d on AG introduced in Remark 1.9.2 is complete. 1.4. Let G be an uncountable group and let A be a set having at least two elements. Prove that the prodiscrete topology on AG is not metrizable. Hint: Prove that this topology does not satisfy the first axiom of countability. 1.5. Let G be a group. Let A and B be two sets. Let τA : AG → AG and τB : B G → B G be cellular automata. For x ∈ (A × B)G , let xA ∈ AG and xB ∈ B G be the configurations defined by x(g) = (xA (g), xB (g)) for all g ∈ G. Show that the map τ : (A × B)G → (A × B)G given by τ (x)(g) = (τA (xA )(g), τB (xB )(g)) for all g ∈ G is a cellular automaton. 1.6. Let G be a group and let S be a finite subset of G of cardinality k. Let k A be a finite set of cardinality n. Show that there are exactly nn cellular automata τ : AG → AG admitting S as a memory set.

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1.7. Let G = Z2 and A = {0, 1}. Let τ : AG → AG denote the cellular automaton associated with the Game of Life. Let y ∈ AG be the constant configuration defined by y(g) = 1 for all g ∈ G (all cells are alive). Find a configuration x ∈ AG such that y = τ (x). 1.8. Let A be a set and suppose that G is a trivial group. Show that the monoid CA(G; A) is canonically isomorphic to the monoid consisting of all maps from A to A (with composition of maps as the monoid operation). Also show that the group ICA(G; A) is canonically isomorphic to the symmetric group of A. 1.9. Let G be a group and let A be a set. Let τ : AG → AG be a cellular automaton. Show that τ admits a memory set which is reduced to a single element if and only if there exist an element s ∈ G and a map f : A → A such that one has τ (x)(g) = f (x(gs)) for all x ∈ AG and g ∈ G. 1.10. Prove that there are exactly 218 cellular automata τ : {0, 1}Z → {0, 1}Z whose minimal memory set is {−1, 0, 1}. 1.11. Let τ ∈ CA(Z2 ; {0, 1}) denote the cellular automaton associated with the Game of Life. Show that the minimal memory set of τ is the set {−1, 0, 1}2 . 1.12. Let G be a group and let A be a set. Let σ, τ ∈ CA(G; A). Let S0 (resp. T0 , resp. C0 ) denote the minimal memory set of σ (resp. τ , resp. σ ◦ τ ). Prove that C0 ⊂ S0 T0 . Give an example showing that this inclusion may be strict. 1.13. Let G be a group and let A be a set. Let H be a subgroup of G and let τ ∈ CA(G, H; A). Show that τ and τH have the same minimal memory set. 1.14. Prove Proposition 1.4.9 by applying Theorem 1.9.1. Hint: Observe that the composite of two uniformly continuous maps is a uniformly continuous map. 1.15. Let G be a group and let A be a set. Let F be a nonempty finite subset of G and set B = AF . The sets AG and B G are equipped with their prodiscrete uniform structures and with the G-shift action. Show that the map ΦF : AG → B G defined by ΦF (x)(g) = (g −1 x)|F for all x ∈ AG and g ∈ G is a G-equivariant uniform embedding. 1.16. Let G be a group, H ⊂ G a subgroup of G, and let A be a set. Let H\G = {Hg : g ∈ G} be the set of all right cosets of H in G and set B = AH\G . The set AG (resp. B H ) is equipped with its prodiscrete uniform structure and with the G-shift (resp. H-shift) action. Let T ⊂ G be a complete set of representatives for the right cosets of H in G so that G = t∈T Ht. Show that the map Ψ = Ψ (H, T ) : AG → B H defined by Ψ (x)(h)(Ht) = x(ht) for all x ∈ AG , h ∈ H and t ∈ T is an H-equivariant uniform isomorphism.

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1.17. Let G be a group and let A be a set. For each s ∈ G, let τs : AG → AG be the cellular automaton defined by τs (x)(g) = x(gs) for all x ∈ AG , g ∈ G (cf. Example 1.4.3(e)). (a) Show that τs ∈ ICA(G; A) for every s ∈ G. (b) Prove that the map Φ : G → ICA(G; A) defined by φ(s) = τs for all s ∈ G is a group homomorphism. (c) Prove that if A has at least two elements, then Φ is injective but not surjective. 1.18. Let G be a group and let A be a set. Prove that the set consisting of all invertible cellular automata τ : AG → AG admitting a memory set which is reduced to a single element is a subgroup of ICA(G; A) isomorphic to the direct product G × Sym(A). 1.19. (cf. [Amo]) Let G = Z and A = {0, 1}. Fix an integer n ≥ 3 and let S = {−1, 0, 1, . . . , n}. Consider the element α ∈ AS (resp. β ∈ AS ) defined by α(−1) = α(n) = 0 and α(k) = 1 for 0 ≤ k ≤ n − 1 (resp. β(−1) = β(0) = β(n) = 0 and β(k) = 1 for 1 ≤ k ≤ n − 1) and the map μ : AS → A defined by μ(α) = 0, μ(β) = 1 and μ(y) = y(0) for y ∈ AS \{α, β}. Let τ : AG → AG be the cellular automaton with memory set S and local defining map μ. (a) Show that S is the minimal memory set of τ . (b) Show that τ is an invertible cellular automaton and that τ −1 = τ . 1.20. Show that the inverse map of the bijective cellular automaton τ : AZ → AZ studied in Example 1.10.3 is discontinuous, with respect to the prodiscrete topology on AZ , at every configuration x ∈ AZ . 1.21. Let G be a group and let A be a set. A subshift of the configuration space AG is a subset X ⊂ AG which is G-invariant (i.e., such that gx ∈ X for all x ∈ X and g ∈ G) and closed in AG with respect to the prodiscrete topology. (a) Show that ∅ and AG are subshifts of AG . (b) Show that if x ∈ AG then its orbit closure Gx ⊂ AG is a subshift.  (c) Show that if (Xi )i∈I is a family of subshifts of AG then i∈I Xi is a subshift of AG .  (d) Show that if (Xi )i∈I is a finite family of subshifts of AG then i∈I Xi is a subshift of AG . (e) Suppose that A is finite. Show that if X ⊂ AG is a subshift and τ : AG → AG is a cellular automaton then τ (X) is a subshift of AG . Note: This last statement becomes false when A is infinite (see Example 3.3.3). 1.22. Let G be a group and let A and B be two sets. Let f : A → B be a map and consider the map f∗ : AG → B G defined by f∗ (x) = f ◦ x for all x ∈ AG . (a) Show that if A is finite and X is a subshift of AG then f∗ (X) is a subshift of B G . Hint: Use the compactness of the configuration space AG .

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(b) Let G = Z, A = Z, B = {0, 1} and let f : A → B be defined by f (n) = 0 if n < 0 and f (n) = 1 otherwise. Let X = {xn : n ∈ Z} ⊂ AZ , where xn (m) = n + m for all n, m ∈ Z. In other words, X is the Z-orbit Zx0 of the configurations x0 . Show that X is a subshift of AZ but f ∗ (X) is not a subshift of B Z .  1.23. Let G be a group and let A be a set. Given a set of patterns P ⊂ AΩ , where the union runs over all finite subsets Ω of G, we set / P for all g ∈ G and all finite subsets Ω ⊂ G}, XP = {x ∈ AG : (gx)|Ω ∈ the restriction of the configuration gx to Ω. where (gx)|Ω denotes  (a) Let P ⊂ AΩ be a set of patterns. Show that XP is a subshift of AG . G (b) Conversely,  Ω show that if X ⊂ A is a subshift, then there exists a subset P ⊂ A such that X = XP . Such a set P is called a defining set of forbidden patterns for X. 1.24. Let G be a group and let A be a set. Given a finite subset Ω ⊂ G and a finite subset A ⊂ AΩ we set X(Ω, A) = {x ∈ AG : (gx)|Ω ∈ A for all g ∈ G}. (a) Let Ω ⊂ G and A ⊂ AΩ be finite subsets. Show that X(Ω, A) is a subshift of AG . A subshift X ⊂ AG is said to be of finite type if there exists a finite subset Ω ⊂ G and a finite subset A ⊂ AΩ such that X = X(Ω, A). Such a set A is then called a defining set of admissible patterns for X and the subset Ω is called a memory set for X. (b) Suppose that A is finite. Show that a subshift X ⊂ AG is of finite type if and only if it admits a finite defining set of forbidden patterns. 1.25. Let G be a countable group and let A be a finite set. Show that there are at most countably many distinct subshifts X ⊂ AG of finite type. 1.26. Let G be a group and let A be a finite set. (a) Let τ1 , τ2 : AG → AG be two cellular automata. Show that the set {x ∈ AG : τ1 (x) = τ2 (x)} ⊂ AG is a subshift of finite type. (b) Deduce from (a) that if τ : AG → AG is a cellular automaton then the set Fix(τ ) = {x ∈ AG : τ (x) = x} ⊂ AG is a subshift of finite type. (c) Conversely, show that if X ⊂ AG is a subshift of finite type then there exists a cellular automaton τ : AG → AG such that X = Fix(τ ). 1.27. Suppose that a group Γ acts continuously on a topological space Z. One says that the action of Γ on Z is topologically transitive if for any pair of nonempty open subsets U and V of Z there exists an element γ ∈ Γ such that U ∩ γV = ∅. Let G be a group and let A be a set. A subshift X ⊂ AG is said to be irreducible if for any finite subset Ω of G and any two elements x1 , x2 ∈ X,

Exercises

33

there exist a configuration x ∈ X and an element g ∈ G such that x|Ω = x1 |Ω and (gx)|Ω = x2 |Ω . Suppose that X ⊂ AG is a subshift. Show that the action of G on X induced by the G-shift is topologically transitive if and only if X is irreducible. 1.28. Let G be a group and let A be a set. Let B ⊂ A and consider the subsets X, Y ⊂ AG defined by X = {b : b ∈ B} ⊂ AG , where b denotes the constant configuration given by b(g) = b for all g ∈ G, and Y = {y ∈ AG : y(g) ∈ B for all g ∈ G}. (a) Show that X and Y are subshifts of AG . (b) Show that if G is infinite then Y is irreducible. (c) Suppose that B has at least two distinct elements. Show that X is not irreducible. 1.29. Let G be a group acting continuously on a nonempty complete metric space X whose topology satisfies the second axiom of countability (i.e., admitting a countable base of open subsets). Show that the following conditions are equivalent: (i) the action of G on X is topologically transitive; (ii) there is a point x ∈ X whose G-orbit is dense in X; (iii) there is a dense subset D ⊂ X such that the G-orbit of each point x ∈ D is dense in X. Hint: The implications (iii) ⇒ (ii) and (ii) ⇒ (i) are straightforward. To prove (i) ⇒ (iii), consider a sequence (Un )n∈N of nonempty open subsets of X which form a base of the topology and denote by Ωn the set of points x ∈ X whose G-orbit meets Un . Then observe that each Ωn is and open dense subset of X if (i) is satisfied and apply Baire’s theorem (Theorem I.1.1, also cf. Remark I.1.2(ii)). 1.30. Let G be a countable group and let A be a countable (e.g. finite) set. Let X ⊂ AG be a nonempty subshift. Show that the following conditions are equivalent: (i) the subshift X is irreducible; (ii) there is a configuration x ∈ X whose G-orbit is dense in X; (iii) there is a dense subset D ⊂ X such that the G-orbit of each configuration x ∈ D is dense in X. Hint: Use the results of Exercises 1.3 and 1.29. 1.31. Let G be a group and let A be a set. One says that a subshift X ⊂ AG is topologically mixing if the action of G on X induced by the G-shift is topologically mixing (cf. Exercise 1.1). (a) Let X ⊂ AG be a subshift. Show that X is topologically mixing if and only if for any finite subset Ω of G and any two configurations x1 , x2 ∈ X, there exists a finite subset F ⊂ G such that, for all g ∈ G \ F , there exists a configuration x ∈ X satisfying x|Ω = x1 |Ω and (gx)|Ω = x2 |Ω . (b) Show that if G is infinite then every topologically mixing subshift X ⊂ AG is irreducible.

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1 Cellular Automata

1.32. Let G be a group and let A be a set. Let Δ ⊂ G be a finite subset. A subshift X ⊂ AG is said to be Δ-irreducible if it satisfies the following condition: if Ω1 and Ω2 are two finite subsets of G such that Ω1 and Ω2 δ are disjoint for all δ ∈ Δ, then, given any two configurations x1 , x2 ∈ X, there exists a configuration x ∈ X which satisfies x|Ω1 = x1 |Ω1 and x|Ω2 = x2 |Ω2 . A subshift X ⊂ AG is said to be strongly irreducible if there exists a finite subset Δ ⊂ G such that X is Δ-irreducible. (a) Show that the subshift Y ⊂ AG described in Exercise 1.28 is {1G }irreducible and therefore strongly irreducible. (b) Show that every strongly irreducible subshift is topologically mixing. 1.33. Let G be a group and let A be a set. Let H be a subgroup of G. H the configuration defined by For x ∈ AG and g ∈ G, denote by xH g ∈ A H xg = (gx)|H . H G (a) Check that hxH g = xhg for all x ∈ A , h ∈ H and g ∈ G. (b) Let X ⊂ AH be a subshift. Show that the set X (G) defined by X (G) = G {x ∈ AG : xH g ∈ X for all g ∈ G} is a subshift of A . (G) (c) Show that if H = G then the subshift X ⊂ AG is irreducible for H any subshift X ⊂ A . (d) Let X ⊂ AH be a subshift. Show that X (G) ⊂ AG is of finite type (resp. topologically mixing, resp. strongly irreducible) if and only if X is of finite type (resp. topologically mixing, resp. strongly irreducible). (e) Suppose that σ : AH → AH is a cellular automaton and let σ G : AG → G A be the induced cellular automaton (cf. Sect. 1.7). Check that one has H G σ G (x)H g = σ(xg ) for all x ∈ A and g ∈ G. H (f) Show that if X ⊂ A is a subshift such that σ(X) ⊂ X then one has σ G (X (G) ) ⊂ X (G) . 1.34. Let G be a group and let A be a set. Let F be a nonempty finite subset of G and consider the map ΦF : AG → B G defined in Exercise 1.15, where B = AF . Let X ⊂ AG be a subshift and set X [F ] = ΦF (X). (a) Show that X [F ] is a subshift of B G . (b) Show that X is irreducible (resp. topologically mixing, resp. strongly irreducible) if and only if X [F ] is irreducible (resp. topologically mixing, resp. strongly irreducible). (c) Show that if X is of finite type then X [F ] is of finite type. 1.35. Let G be a group, H ⊂ G a subgroup of G, and let A be a set. Let also T ⊂ G be a complete set of representatives for the right cosets of H in G and consider the map Ψ : AG → B H defined in Exercise 1.16, where B = AH\G . Let X ⊂ AG be a subshift and set X (H,T ) = Ψ (X). (a) Show that X (H,T ) is a subshift of B H . (b) Show that if X is of finite type then X (H,T ) is of finite type. 1.36. Let A be a set. Let A∗ denote the monoid consisting of all words in the alphabet A (cf. Sect. D.1). Recall that any word w ∈ A∗ can be uniquely

Exercises

35

written in the form w = a1 a2 · · · an , where n ≥ 0 and ai ∈ A for 1 ≤ i ≤ n. The integer n is called the length of the word w and it is denoted by (w). In the sequel, we shall identify the word w = a1 a2 · · · an with the pattern p : {1, 2, . . . , n} → A defined by p(i) = ai for 1 ≤ i ≤ n. Given a subshift X ⊂ AZ and an integer n ≥ 0, we denote by Ln (X) ⊂ A∗ the set consisting of all words w ∈ A∗ for which there exists an element x ∈ X such that w = x(1)x(2) · · · x(n). The set L(X) = ∪n∈N Ln (X) is called the language of X. The elements w ∈ L(X) are called the admissible words of X (or, simply, the X-admissible words). The elements w ∈ A∗ \ L(X) are called the forbidden words of X. (a) Let X and Y be two subshifts of AZ . Show that one has X ⊂ Y (resp. X = Y ) if and only if L(X) ⊂ L(Y ) (resp. L(X) = L(Y )). (b) One says that a word u ∈ A∗ is a subword of a word w ∈ A∗ if there exist v1 , v2 ∈ A∗ such that w = v1 uv2 . Let X ⊂ AZ be a subshift and let L = L(X). Show that L satisfies the following conditions: (i) if w ∈ L, then u ∈ L for every subword u of w; (ii) if w ∈ L, then there exist a, a ∈ A such that awa ∈ L. (c) Conversely, show that if a subset L ⊂ A∗ satisfies conditions (i) and (ii) in (b), then there exists a unique subshift X ⊂ AZ such that L = L(X). 1.37. Let A be a set and let X ⊂ AZ be a subshift. (a) Show that X is of finite type if and only if the following holds: there exists an integer n0 ≥ 0 such that if the words u, v, w ∈ A∗ satisfy (v) ≥ n0 and uv, vw ∈ L(X), then one has uvw ∈ L(X). (b) Show that X is irreducible if and only if for every pair of words u and v in L(X), there exists a word w ∈ A∗ such that uwv ∈ L(X). (c) Show that X is topologically mixing if and only if the following holds: for every pair of words u and v in L(X), there exists an integer n0 ≥ 0 such that for every integer n ≥ n0 there exists a word w ∈ A∗ of length (w) = n satisfying uwv ∈ L(X). (d) Show that X is strongly irreducible if and only if the following holds: there exists an integer n0 ≥ 0 such that, for every pair of words u and v in L(X), and for every integer n ≥ n0 , there exists a word w ∈ A∗ of length (w) = n such that uwv ∈ L(X). 1.38. Let A = {0, 1} and let X ⊂ AZ be the set of all x ∈ AZ such that the following holds: if x(n) = 1, x(n + 1) = x(n + 2) = · · · = x(n + k) = 0, x(n + k + 1) = 1, for some n ∈ Z and k ∈ N, then k is even. (a) Show that X is a subshift (it is called the even subshift). (b) Show that X is not of finite type. (c) Show that X is strongly irreducible (and therefore topologically mixing and irreducible). 1.39. Let A = {0, 1} and consider the subshift of finite type X ⊂ AZ defined by X = X{11} = {x ∈ AZ : (x(n), x(n + 1)) = (1, 1) for all n ∈ Z}. Show that X is strongly irreducible (and therefore topologically mixing and irreducible). The subshift X is called the golden mean subshift.

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1 Cellular Automata

1.40. Let A = {0, 1} and let X ⊂ AZ be the set consisting of the two configurations x, y ∈ AZ defined by   0 if n is even 1 if n is even and y(n) = x(n) = 1 otherwise 0 otherwise, for all n ∈ Z. Show that X is an irreducible subshift of finite type which is not topologically mixing (and therefore not strongly irreducible either). 1.41. Let A be a set. Let X ⊂ AZ be a subshift of finite type. Show that X is topologically mixing if and only if X is strongly irreducible. 1.42. Let A = {0, 1} and let X ⊂ AZ be the subshift with defining set of forbidden words {01k 0h 1 : 1 ≤ h ≤ k, k = 1, 2, . . .}. Show that X is topologically mixing (and therefore irreducible) but not strongly irreducible. 1.43. Cellular automata between subshifts. Let G be a group and let A be a set. The set AG is equipped with its prodiscrete uniform structure and with the G-shift action. Let X, Y ⊂ AG be two subshifts and τ : X → Y a map. Then the following are equivalent: (i) there exists a cellular automaton τ : AG → AG such that τ (x) = τ (x) for all x ∈ X; (ii) τ is G-equivariant and uniformly continuous. One says that τ : X → Y is a cellular automaton if the two equivalent conditions above are satisfied. 1.44. Let G be a group and let A be a finite set. Let τ : X → Y be a map between subshifts X, Y ⊂ AG . Show that the following conditions are equivalent: (i) τ is a cellular automaton, (ii) τ is G-equivariant and continuous (with respect to the topologies induced on X and Y by the prodiscrete topology on AG ). 1.45. Let G be a group and let A be a finite set. Let τ : AG → AG be a cellular automaton and let X ⊂ AG be an irreducible (resp. topologically mixing, resp. strongly irreducible) subshift. Show that τ (X) is an irreducible (resp. topologically mixing, resp. strongly irreducible) subshift of AG .

Chapter 2

Residually Finite Groups

This chapter is devoted to the study of residually finite groups, which form a class of groups of special importance in several branches of mathematics. As their name suggests it, residually finite groups generalize finite groups. They are defined as being the groups whose elements can be distinguished after taking finite quotients (see Sect. 2.1). There are many other equivalent definitions. For example, a group is residually finite if and only if it can be embedded into the direct product of a family of finite groups (Corollary 2.2.6). The class of residually finite groups is closed under taking subgroups and taking projective limits. It contains in particular all finite groups, all finitely generated abelian groups, and all free groups (Theorem 2.3.1). Every finitely generated residually finite group is Hopfian (Theorem 2.4.3) and the automorphism group of a finitely generated residually finite group is itself residually finite (Theorem 2.5.1). Examples of finitely generated groups which are not residually finite are presented in Sect. 2.6. The following dynamical characterization of residually finite groups is given in Sect. 2.7: a group is residually finite if and only if there is a Hausdorff topological space on which the group acts continuously and faithfully with a dense subset of points with finite orbit.

2.1 Definition and First Examples Definition 2.1.1. A group G is called residually finite if for each element g ∈ G with g = 1G , there exist a finite group F and a homomorphism φ : G → F such that φ(g) = 1F . Proposition 2.1.2. Let G be a group. Then the following conditions are equivalent: (a) G is residually finite; T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 2, © Springer-Verlag Berlin Heidelberg 2010

37

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(b) for all g, h ∈ G with g =  h, there exist a finite group F and a homomorphism φ : G → F such that φ(g) = φ(h). Proof. The fact that (b) implies (a) is obvious, since (b) gives (a) by taking h = 1G . Conversely, suppose that G is residually finite. Let g, h ∈ G with g = h. As gh−1 = 1G , there exist a finite group F and a homomorphism φ : G → F such that φ(gh−1 ) = 1F . Since φ(gh−1 ) = φ(g)(φ(h))−1 , it follows that φ(g) = φ(h). Therefore, (a) implies (b).   Proposition 2.1.3. Every finite group is residually finite. Proof. Let G be a finite group and consider g ∈ G such that g = 1G . If   φ = IdG : G → G is the identity map, we have φ(g) = g = 1G . Proposition 2.1.4. The additive group Z is residually finite. Proof. Consider k ∈ Z such that k = 0. Choose an integer m such that |k| < m. Then the canonical homomorphism φ : Z → Z/mZ (reduction modulo m) satisfies φ(k) = 0.   A similar argument gives us the following: Proposition 2.1.5. The group GLn (Z) is residually finite for every n ≥ 1. Proof. Let A = (aij ) ∈ GLn (Z) with A = In = 1GLn (Z) . Choose an integer m such that |aij | < m for all i, j. Then the homomorphism φ : GLn (Z) →  GLn (Z/mZ) given by reduction modulo m satisfies φ(A) = 1GLn (Z/mZ) .  We shall now give examples of groups which are not residually finite. A group G is called divisible if for each g ∈ G and each integer n ≥ 1, there is an element h ∈ G such that hn = g. Example 2.1.6. The additive groups Q, R, and C are divisible. More generally, every Q-vector space, with its additive underlying group structure, is divisible. In particular, every field of characteristic 0, with its underlying additive group structure, is divisible. Lemma 2.1.7. Let G be a divisible group and let F be a finite group. Then every homomorphism φ : G → F is trivial. Proof. Set n = |F |. Let g ∈ G. As G is divisible, we can find h ∈ G such that g = hn . If φ : G → F is a homomorphism, then we have φ(g) = φ(hn ) =   φ(h)n = 1F . The preceding lemma immediately yields the following result. Proposition 2.1.8. A nontrivial divisible group cannot be residually finite.  

2.1 Definition and First Examples

39

Example 2.1.9. The additive group underlying a field of characteristic 0 is not residually finite. In particular, the additive group Q is not residually finite. In order to give another characterization of residually finiteness, we shall use the following: Lemma 2.1.10. Let G be a group and let H be a subgroup of G. Let K =  −1 . Then K is a normal subgroup of G contained in H. Moreover, g∈G gHg if H is of finite index in G, then Kis of finite index in G. Proof. Since gHg −1 = H for g = 1G , we have K ⊂ H. Let Sym(G/H) denote the group of permutations of G/H. Consider the action of G on G/H given by left multiplication and let ρ : G → Sym(G/H) be the associated homomorphism. For each g ∈ G, the stabilizer of gH is gHg −1 . Consequently, we have K = Ker(ρ), which shows that K is a normal subgroup of G. The group G/K is isomorphic to Im(ρ) ⊂ Sym(G/H). Suppose that H is of finite index in G. This means that the set G/H is finite. This implies that the group Sym(G/H) is finite. We deduce that the group G/K is finite, that is, K is of finite index in G.   Given a group G, the intersection of all subgroups of finite index of G is called the residual subgroup (or profinite kernel ) of G. Proposition 2.1.11. Let G be a group and let N denote the residual subgroup of G. Then: (i) N is equal to the intersection of all normal subgroups of finite index in G; (ii) N is a normal subgroup of G; (iii) G is residually finite if and only if N = {1G }. Proof. Denote by N  the intersection of all normal subgroups of finite index of G. The inclusion N ⊂ N  is trivial. If H is a subgroup of finite index of G, then K = ∩g∈G gHg −1 is a normal subgroup of finite index of G contained in H, by Lemma 2.1.10. This implies N  ⊂ K ⊂ H. It follows that N  ⊂ N . This shows (i). Assertion (ii) follows from (i) since the intersection of a family of normal subgroups of G is a normal subgroup of G. Suppose that G is residually finite. Let g ∈ G such that g = 1G . Then there exist a finite group F and a homomorphism φ : G → F such that φ(g) = 1F . Thus g ∈ / Ker(φ). As the group G/Ker(φ) is isomorphic to F , the subgroup Ker(φ) is of finite index in G. This shows that N = {1G }. Conversely, suppose that N = {1G }. Let g ∈ G such that g = 1G . By (i), we can find a normal subgroup of finite index K ⊂ G such that g ∈ / K. If φ : G → G/K is the canonical homomorphism, we have φ(g) = 1G/K . This shows that G is residually finite.  

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2.2 Stability Properties of Residually Finite Groups Proposition 2.2.1. Every subgroup of a residually finite group is residually finite. Proof. Let G be a residually finite group and let H be a subgroup of G. Let h ∈ H such that h = 1G . Since G is residually finite, there exist a finite group F and a homomorphism φ : G → F such that φ(h) = 1F . If φ : H → F is the restriction of φ to H, we have φ (h) = φ(h) = 1F . Consequently, H is residually finite.   Proposition 2.2.2. Let  (Gi )i∈I be a family of residually finite groups. Then their direct product G = i∈I Gi is residually finite. Proof. Let g = (gi )i∈I ∈ G such that g = 1G . Then there exists i0 ∈ I such that gi0 = 1Gi0 . Since Gi0 is residually finite, we can find a finite group F and a homomorphism φ : Gi0 → F such that φ(gi0 ) = 1F . Consider the homomorphism φ : G → F defined by φ = π ◦ φ, where π : G → Gi0 is the projection onto Gi0 . We have φ (g) = φ(gi0 ) = 1F . Consequently, G is residually finite.   Corollary 2.2.3. Let (Gi )i∈I be a family of residually finite groups. Then their direct sum G = i∈I Gi is residually finite. Proof. This follows immediately from Proposition 2.2.1  and Proposition 2.2.2, since G is the subgroup of the direct product P = i∈I Gi consisting of all   g = (gi ) ∈ P for which gi = 1Gi for all but finitely many i ∈ I. Corollary 2.2.4. Every finitely generated abelian group is residually finite. Proof. If G is a finitely generated abelian group, then there exist an integer r ≥ 0 and a finite abelian group T such that G is isomorphic to Zr × T . By using Proposition 2.1.3 and Proposition 2.1.4, we deduce that G is residually finite.   Remark 2.2.5. An arbitrary abelian group need not be residually finite. For example, the additive group Q is not residually finite (Example 2.1.9). Corollary 2.2.6. Let G be a group. Then the following conditions are equivalent: (a) the group G is residually finite; (b) there exist a family (Fi )i∈I of finite groups such that  the group G is isomorphic to a subgroup of the direct product group i∈I Fi . Proof. The fact that (b) implies (a) follows from Proposition 2.1.3, Proposition 2.2.2 and Proposition 2.2.1. Conversely, suppose that G is residually finite. Then, for each g ∈ G \ {1G }, we can find a finite group Fg and a homomorphism φg : G → Fg such that φg (g) = 1Fg . Consider the group

2.2 Stability Properties of Residually Finite Groups

H=



41

Fg .

g∈G\{1G }

The homomorphism ψ : G → H defined by  ψ= φg g∈G\{1G }

is injective. Therefore, G is isomorphic to a subgroup of H. This shows that (a) implies (b).   The class of residually finite groups is closed under taking projective limits (see Sect. E.2 for the definition of the limit of a projective system of groups): Proposition 2.2.7. If a group G is the limit of a projective system of residually finite groups, then G is residually finite. Proof. Let (Gi )i∈I be a projective system of residually finite groups such that of a projective limit (see Appendix E), G is a G = lim Gi . By construction  ←− subgroup of the group i∈I Gi . We deduce that G is residually finite by using Proposition 2.2.2 and Proposition 2.2.1.   A group G is called profinite if G is the limit of some projective system of finite groups. An immediate consequence of Proposition 2.2.7 is the following: Corollary 2.2.8. Every profinite group is residually finite.

 

Example 2.2.9. Let p be a prime number. Given integers n ≥ m ≥ 0, let φn,m : Z/pm Z → Z/pn Z denote reduction modulo pn . Then (Z/pn Z, φn,m ) is a projective system of groups over N. The limit of this projective system is called the group of p-adic integers and is denoted by Zp . Since Zp is the projective limit of finite groups, it is profinite and hence residually finite by Corollary 2.2.8. Remark 2.2.10. A group which is the limit of an inductive system of residually finite groups need not be residually finite. For instance, we have seen in Example 2.1.9 that the additive group underlying a field of characteristic 0 is not residually finite. However, such a group is the limit of the inductive system formed by its finitely generated subgroups, which are all residually finite by Corollary 2.2.4. If P is a property of groups, one says that a group G is virtually P if G contains a subgroup of finite index which satisfies P. Lemma 2.2.11. Let G be a group. Let H be a subgroup of finite index of G and let K be a subgroup of finite index of H. Then K is a subgroup of finite index of G.

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Proof. Let h1 , . . . , hn be a complete set of representatives of the left cosets of G modulo H and let k1 , . . . , kp be a complete set of representatives of the left cosets of H modulo K. Observe that the elements hi kj , 1 ≤ i ≤ n, 1 ≤ j ≤ p, form a complete set of representatives of the left cosets of G modulo K. Therefore [G : K] = np = [G : H][H : K] < ∞.   Proposition 2.2.12. Every virtually residually finite group is residually finite. Proof. Let G be a group and let H be a subgroup of finite index of G. By Lemma 2.2.11, the intersection of the subgroups of finite index of G is contained in the intersection of the subgroups of finite index of H. Since a group is residually finite if and only if the intersection of its subgroups of finite index is reduced to the identity element (Proposition 2.1.11), we deduce that G is residually finite if H is residually finite.  

2.3 Residual Finiteness of Free Groups The goal of this section is to establish the following result: Theorem 2.3.1. Every free group is residually finite. To prove this theorem, we shall use the following: Lemma 2.3.2. The subgroup of SL2 (Z) generated by the matrices     12 10 a= and b = 01 21 is a free group of rank 2. Proof. The group GL2 (R) naturally acts on the set of lines of R2 passing through the origin, that is, on the projective line P1 (R). In nonhomogeneous coordinates this action is given by gt =

g11 t + g12 g21 t + g22

for g = (gij )1≤i,j≤2 ∈ GL2 (R) and t ∈ P1 (R) = R ∪ {∞} representing the line of R2 with slope 1/t passing through (0, 0) (see Fig. 2.1). Note that we have ak t = t + 2k

and

bk t =

t 2kt + 1

for all k ∈ Z. Consider the subsets Y and Z of P1 (R) defined by Y = ] − 1, 1[

and

Z = P1 (R) \ [−1, 1].

2.3 Residual Finiteness of Free Groups

43

Fig. 2.1 The action of GL2 (R) on P1 (R). Here, t = 1 and g = a

One immediately checks that, for all k ∈ Z \ {0}, one has ak Y = ]2k − 1, 2k + 1[ ⊂ Z

and

bk Z = ]1/(2k + 1), 1/(2k − 1)[ ⊂ Y.

By applying the Klein Ping-Pong theorem (Theorem D.5.1), we deduce that a and b generate a free group of rank 2.   Proof of Theorem 2.3.1. Since GL2 (Z) is residually finite by Proposition 2.1.5, it follows from Lemma 2.3.2, Corollary D.5.3, and Proposition 2.2.1 that every free group of finite rank is residually finite. Consider now an arbitrary set X and let F (X) denote the free group based on X. Let g ∈ F (X) such that g = 1F (X) . Let Y ⊂ X denote the set of elements x ∈ X such that xk appear in the reduced form of g for some k ∈ Z \ {0}. The subgroup F (Y ) ⊂ F (X) generated by Y is a free group with base Y (see Proposition D.2.4). We have g ∈ F (Y ). As the group F (Y ) is free of finite rank, it is residually finite by the first part of the proof. Thus we can find a finite group H and a homomorphism φ : F (Y ) → H such that φ(g) = 1H . Consider the unique homomorphism π : F (X) → F (Y ) such that π(y) = y for every y ∈ Y and π(x) = 1F (Y ) for every x ∈ X \ Y . Then the homomorphism φ ◦ π : F (X) → H satisfies φ ◦ π(g) = φ(π(g)) = φ(g) = 1H . This shows that F (X) is residually finite.  

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2.4 Hopfian Groups Definition 2.4.1. A group G is called Hopfian if every surjective endomorphism of G is injective. Examples 2.4.2. (a) Every finite group is Hopfian. (b) The additive group Q is Hopfian. Indeed, every endomorphism of the group Q is of the form x → ax for some a ∈ Q, and it is clear that such an endomorphism is surjective (resp. injective) if and only if a = 0. (c) Every simple group is Hopfian. (we recall that a simple group is a nontrivial group G such that the only normal subgroups of G are {1G } and G). (d) The additive group Q/Z is not Hopfian. Indeed, the endomorphism ψ : Q/Z → Q/Z defined by ψ(x) = 2x is surjective but not injective. Theorem 2.4.3. Every finitely generated residually finite group is Hopfian. Given groups G1 and G2 , we shall denote by Hom(G1 , G2 ) the set of all homomorphisms from G1 to G2 . We begin by establishing the following result. Lemma 2.4.4. Let G be a finitely generated group and let F be a finite group. Then the set Hom(G, F ) is finite. Proof. Let A be a finite generating subset of G. Let us set n = |A| and p = |F |. As A generates G, any homomorphism u : G → F is completely determined by the elements u(a), a ∈ A. Thus the set Hom(G, F ) contains   at most pn elements. Proof of Theorem 2.4.3. Let G be a finitely generated residually finite group. Suppose that ψ : G → G is a surjective endomorphism of G. Let K be a normal subgroup of finite index of G and let ρ : G → G/K denote the canonical homomorphism. Consider the map Φ : Hom(G, G/K) → Hom(G, G/K) defined by Φ(u) = u ◦ ψ for all u ∈ Hom(G, G/K). The map Φ is injective since ψ is surjective by our hypothesis. As the set Hom(G, G/K) is finite by Lemma 2.4.4, we deduce that Φ is also surjective. In particular, there exists a homomorphism u0 ∈ Hom(G, G/K) such that ρ = u0 ◦ ψ. This implies Ker(ψ) ⊂ Ker(ρ) = K. It follows that Ker(ψ) is contained in the intersection of all normal subgroups of finite index of G. As G is residually finite, we deduce that Ker(ψ) = {1G } by Proposition 2.1.11(iii). Thus ψ is injective. This shows that G is Hopfian.   Since every free group is residually finite by Theorem 2.3.1, an immediate consequence of Theorem 2.4.3 is the following: Corollary 2.4.5. Every free group of finite rank is Hopfian.

2.5 Automorphism Groups of Residually Finite Groups

45

Remarks 2.4.6. (a) Let X be an infinite set and let F (X) denote the free group based on X. Every surjective but non injective map f : X → X extends to a surjective endomorphism ψ : F (X) → F (X) which is not injective. Consequently, the group F (X) is not Hopfian. Since F (X) is residually finite by Theorem 2.3.1, this shows that we cannot suppress the hypothesis that G is finitely generated in Theorem 2.4.3. (b) An example of a finitely generated Hopfian group which is not residually finite will be given in Sect. 2.6 (see Proposition 2.6.1).

2.5 Automorphism Groups of Residually Finite Groups Let G be a group. Recall that an automorphism of G is a bijective homomorphism α : G → G. Clearly, the set Aut(G) consisting of all automorphisms of G is a subgroup of the symmetric group of G. The group Aut(G) is called the automorphism group of G. Theorem 2.5.1. Let G be a finitely generated residually finite group. Then the group Aut(G) is residually finite. Let us first establish the following result. Lemma 2.5.2. Let G be a group. Let H1 and H2 be subgroups of finite index of G. Then the subgroup H = H1 ∩ H2 is of finite index in G. Proof. Two elements in G are left congruent modulo H if and only if they are both left congruent modulo H1 and left congruent modulo H2 . Therefore, there is an injective map from G/H into G/H1 × G/H2 given by gH → (gH1 , gH2 ). As the sets G/H1 and G/H2 are finite by hypothesis, we deduce that G/H is finite, that is, H is of finite index in G.   Proof of Theorem 2.5.1. Let α0 ∈ Aut(G) such that α0 = IdG . Then we can find an element g0 ∈ G such that α0 (g0 ) = g0 . As G is residually finite, there exist a finite group F and a homomorphism φ : G → F satisfying φ(α0 (g0 )) = φ(g0 ). Consider the set H defined by  H= Ker(ψ), ψ∈Hom(G,F )

where Hom(G, F ) denotes, as above, the set of all homomorphisms from G to F . Observe that H is a normal subgroup of G since it is the intersection of a family of normal subgroups of G. On the other hand, for every α ∈ Aut(G), one has

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α(H) = α

Ker(ψ)

ψ∈Hom(G,F )

=



α(Ker(ψ))

ψ∈Hom(G,F )

=



Ker(ψ ◦ α−1 ).

ψ∈Hom(G,F )

As the map from Hom(G, F ) to itself defined by ψ → ψ ◦ α−1 is bijective (with ψ → ψ ◦ α as inverse map), we get  α(H) = Ker(ψ) = H. ψ∈Hom(G,F )

Therefore α induces an automorphism α of G/H, given by α(gH) = α(g)H for all g ∈ G. The map α → α is clearly a homomorphism from Aut(G) to Aut(G/H). Let us show that the group Aut(G/H) is finite and that α0 = 1Aut(G/H) = IdG/H . Observe first that the set Hom(G, F ) is finite by Lemma 2.4.4. As Ker(ψ) is of finite index in G for every ψ ∈ Hom(G, F ), we deduce that H is of finite index in G by applying Lemma 2.5.2. This implies that the group Aut(G/H) is finite. On the other hand, we have α0 (g0 H) = α0 (g0 )H = g0 H since H is a subgroup of Ker(φ) and φ(g0 ) = g0 . Therefore α0 = IdG/H . This shows that the group Aut(G) is residually finite.   Every free group is residually finite by Theorem 2.3.1. Therefore we deduce from Theorem 2.5.1 the following result. Corollary 2.5.3. The group Aut(Fn ) is residually finite for every n ≥ 1.

 

Every finitely generated abelian group is residually finite by Corollary 2.2.4. Thus we have: Corollary 2.5.4. The automorphism group of a finitely generated abelian group is residually finite.   Remark 2.5.5. The automorphism group of a free abelian group of finite rank n is isomorphic to GLn (Z). Thus, the residual finiteness of GLn (Z) (Proposition 2.1.5) may also be deduced from Corollary 2.5.4. Corollary 2.5.6. Let R be a ring and let M be a left (or right) module over R. Suppose that M is finitely generated as a Z-module. Then the automorphism group AutR (M ) of the R-module M is residually finite. Proof. The group AutR (M ) is a subgroup of AutZ (M ). Since AutZ (M ) is residually finite by Corollary 2.5.4, we deduce that AutR (M ) is residually finite by applying Proposition 2.2.1.  

2.6 Examples of Finitely Generated Groups Which Are Not Residually Finite

47

Corollary 2.5.7. Let R be a ring. Suppose that R is finitely generated as a Z-module. Then the group GLn (R) is residually finite for every n ≥ 1. Proof. This is an immediate consequence of the preceding corollary since the group GLn (R) is isomorphic to AutR (Rn ), where Rn is viewed as a left module over R.   Example 2.5.8. The ring of Gaussian integers Z[i] = {a + bi : a, b ∈ Z} is a free abelian Z-module of rank 2. Thus, the group GLn (Z[i]) is residually finite for every n ≥ 1 by Corollary 2.5.7. More generally, consider a number field K, that is, a field extension of Q such that d = dimQ K < ∞. Let A denote the ring of algebraic integers of K (we recall that an element x ∈ K is called an algebraic integer of K if x is a root of a monic polynomial with integral coefficients). It is a standard fact in algebraic number theory that A is a free Z-module of rank d. Thus, the group GLn (A) is residually finite for every n ≥ 1 by Corollary 2.5.7.

2.6 Examples of Finitely Generated Groups Which Are Not Residually Finite We have seen in Example 2.1.9 that the additive group Q is not residually finite. Observe that the group Q is not finitely generated. In fact, any finitely generated abelian group is residually finite by Corollary 2.2.4. The purpose of this section is to give two examples of finitely generated groups G1 and G2 which are not residually finite. Our first example is a subgroup of the symmetric group Sym(Z) generated by two elements: Proposition 2.6.1. Let G1 denote the subgroup of Sym(Z) generated by the translation T : n → n + 1 and the transposition S = (0 1). Then G1 is a finitely generated Hopfian group which is not residually finite. Let us first establish the following lemmas: Lemma 2.6.2. Let G be an infinite simple group. Then G is not residually finite. Proof. The only normal subgroup of finite index of G is G itself. Therefore G is not residually finite by Proposition 2.1.11(iii).   Given a set X, we recall that Sym0 (X) denotes the subgroup of Sym(X) consisting of all permutations of X whose support is finite (see Appendix C). Lemma 2.6.3. Let X be an infinite set. Then the group Sym0 (X) is not residually finite.

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Proof. The subgroup Sym+ 0 (X) ⊂ Sym(X) consisting of all permutations of X with finite support and signature 1 is an infinite simple group by Theorem C.4.3. Therefore Sym+ 0 (X) is not residually finite by Lemma 2.6.2. We deduce that Sym0 (X) is not residually finite by using Proposition 2.2.1.   Lemma 2.6.4. The group G1 contains Sym0 (Z) as a normal subgroup. Moreover, G1 is the semidirect product of Sym0 (Z) with the infinite cyclic subgroup of G1 generated by T . Proof. For all i ∈ Z, we have T i ST −i = (i i + 1),

(2.1)

which shows that (i i + 1) ∈ G1 . If i < j, we also have (i j + 1) = (j j + 1)(i j)(j j + 1) which, by induction on j, shows that (i j) ∈ G1 for all i, j ∈ Z with i < j. Since the transpositions (i j), with i, j ∈ Z and i < j, generate Sym0 (Z) by Corollary C.2.4, it follows that G1 contains Sym0 (Z). The fact that Sym0 (Z) is normal in Sym(Z) (Proposition C.2.2) implies that Sym0 (Z) is normal in G1 . Let H denote the subgroup of G1 generated by T . It is clear that H ∩ Sym0 (Z) is reduced to the identity map IdZ . On the other hand, by using (2.1), we see that every element g ∈ G1 may be written in the form g = hσ, where h ∈ H and σ ∈ Sym0 (Z). Therefore, G1 is the semidirect product of Sym0 (Z) and H.   Proof of Proposition 2.6.1. The group G1 contains Sym0 (Z) as a subgroup by Lemma 2.6.4. As the group Sym0 (Z) is not residually finite by Lemma 2.6.3, we conclude that G1 is not residually finite by applying Proposition 2.2.1. It remains to show that G1 is Hopfian. We start by observing that there are exactly two surjective homomorphisms from G1 onto Z. Indeed, as G1 is the semidirect product of Sym0 (Z) with the subgroup generated by T (Lemma 2.6.4), every element g ∈ G1 can be uniquely written in the form g = T k σ, where k ∈ Z and σ ∈ Sym0 (Z), and the map u : G1 → Z defined by u(g) = k is a surjective homomorphism. As all elements of Sym0 (Z) have finite order, it immediately follows that the only homomorphisms from G1 onto Z are u and −u. Now, let φ : G1 → G1 be a surjective homomorphism. Then u◦φ : G1 → Z is a surjective homomorphism so that, by our preceding observation, we have u◦φ = u or −u. As Ker(u) = Sym0 (Z) and φ : G1 → G1 is onto, it follows that Ker(φ) ⊂ Sym0 (Z) and φ(Sym0 (Z)) = Sym0 (Z). The simplicity of Sym+ 0 (Z) + (Z) is either equal to Sym (Theorem C.4.3) implies that Ker(φ) ∩ Sym+ 0 0 (Z) + + or reduced to the identity. We cannot have Ker(φ) ∩ Sym0 (Z) = Sym0 (Z), that is, Sym+ 0 (Z) ⊂ Ker(φ), since this would imply that φ(Sym0 (Z)) is either reduced to the identity or cyclic of order 2, which contradicts φ(Sym0 (Z)) = Sym0 (Z). Consequently, we have Ker(φ) ∩ Sym+ 0 (Z) = {IdZ }. If Ker(φ) is not

2.6 Examples of Finitely Generated Groups Which Are Not Residually Finite

49

reduced to the identity, it follows that Ker(φ) is a normal subgroup of order 2 of Sym0 (Z). But this is impossible since Proposition C.2.3 combined with Proposition C.3.2 implies that, if X is an infinite set, then every non-trivial element in Sym0 (X) has an infinite number of conjugates in Sym0 (X). Thus Ker(φ) must be reduced to the identity, that is, φ is injective. This shows   that G1 is Hopfian. Our second example of a finitely generated but not residually finite group will also show that the class of residually finite groups is not closed under extensions. In other words, a group G admitting a normal subgroup N such that both N and G/N are residually finite may fail to be residually finite. In order to construct it we consider first the group H= Hi , i∈Z

where each Hi is a copy of the alternating group Sym+ 5 . Let ψ : Z → Aut(H) be the homomorphism defined by ψ(n) = αn , where α ∈ Aut(H) is the one-step shift given by α(h) = (hi−1 )i∈Z for all h = (hi )i∈Z ∈ H. Proposition 2.6.5. The semidirect product G2 = H ψ Z is a finitely generated but not residually finite group. Proof. By definition of a semidirect product, G2 is the group with underlying set H × Z and group operation given by (h, n)(h , n ) = (hαn (h ), n + n ) for all (h, n), (h , n ) ∈ H × Z. Let t be the element of G2 defined by t = (1H , 1) and identify each element h ∈ H with the element (h, 0) ∈ G. Then H is a normal subgroup of G2 and the quotient group G2 /H is an infinite cyclic group generated by the class of t. One has (h, n) = htn (2.2) and

tn ht−n = αn (h)

(2.3)

for all h ∈ H and n ∈ Z. It follows from (2.2) that G2 is generated by t and the elements of H. In fact, since H is the direct sum of the groups Hi and Hi = ti H0 t−i

(2.4)

by (2.3), we deduce that G2 is generated by t and the elements of H0 . As the group H0 is finite, this shows that G is finitely generated.

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Let us prove now that G2 is not residually finite. Suppose on the contrary that G2 is residually finite. Let g be an element in H0 such that g = 1H0 . The residual finiteness of G2 implies the existence of a finite group F and a homomorphism φ : G2 → F such that φ(g) = 1F . As H0 is a simple group, the restriction of φ to H0 is injective. Therefore, the group φ(H0 ) is isomorphic m m to H0 and hence to Sym+ 5 . Let m = |F |. We have φ(t ) = φ(t) = 1F . Thus, it follows from (2.4) that φ(Hm ) = φ(H0 ). Since xy = yx for all x ∈ H0 and y ∈ Hm , this implies that φ(H0 ) is abelian. This contradicts the fact that   φ(H0 ) is isomorphic to Sym+ 5 , which is not abelian. Remark 2.6.6. Observe that the group H is residually finite by Corollary 2.2.3. The quotient group G2 /H is also residually finite since it is isomorphic to Z. This shows that an extension of a residually finite group by a residually finite group need not to be residually finite.

2.7 Dynamical Characterization of Residual Finiteness Let G be a group and let A be a set. Recall that the set AG = {x : G → A} is equipped with the prodiscrete topology and that G acts on AG by the left shift defined by (1.2). Theorem 2.7.1. Let G be a group. Then the following conditions are equivalent: (a) the group G is residually finite; (b) for every set A, the set of points of AG which have a finite G-orbit is dense in AG ; (c) there exists a set A having at least two elements such that the set of points of AG which have a finite G-orbit is dense in AG ; (d) there exists a Hausdorff topological space X equipped with a continuous and faithful action of G such that the set of points of X which have a finite G-orbit is dense in X. We recall that an action of a group G on a set X is called faithful if 1G is the only element of G fixing all points of X. Lemma 2.7.2. Let G be a group and let A be a set having at least two elements. Then the action of G on AG is faithful. Proof. Let a and b be two distinct elements in A. Consider an element g0 in G such that g0 = 1G . Let x ∈ AG be the configuration defined by x(g) = a if g = 1G and x(g) = b otherwise. We have g0 x = x since g0 x(1G ) = x(g0−1 ) = b   and x(1G ) = a. Consequently, the action of G on AG is faithful. Lemma 2.7.3. Let G be a residually finite group and let Ω be a finite subset of G. Then there exists a normal subgroup of finite index K of G such that the restriction of the canonical homomorphism ρ : G → G/K to Ω is injective.

Notes

51

Proof. Consider the finite subset S = {g −1 h : g, h ∈ Ω and g = h} ⊂ G. Since G is residually finite, we can find, for every s ∈ S, a normal subgroup / Ns . The set K = ∩s∈S Ns is a normal of finite index Ns ⊂ G such that s ∈ subgroup of finite index in G by Lemma 2.5.2. Let ρ : G → G/K be the canonical homomorphism. If g and h are distinct elements in Ω, then g −1 h ∈ / K and hence ρ(g) = ρ(h).   Proof of Theorem 2.7.1. Suppose that G is residually finite. Let A be a set and let W be a neighborhood of a point x in AG . Let us show that W contains a configuration with finite G-orbit. Consider a finite subset Ω ⊂ G such that V (x, Ω) = {y ∈ AG : y|Ω = x|Ω } ⊂ W. By Lemma 2.7.3, we can find a normal subgroup of finite index K ⊂ G such that the restriction to Ω of the canonical homomorphism ρ : G → G/K(= K\G) is injective. This implies that the map Φ : AG/K → AΩ defined by Φ(z) = (z ◦ ρ)|Ω is surjective. Thus we can find an element z0 ∈ AG/K such that the configurations z0 ◦ ρ and x coincide on Ω, that is, such that z0 ◦ ρ ∈ V (x, Ω). On the other hand, the configuration z0 ◦ ρ is K-periodic by Proposition 1.3.3 (observe that K\G = G/K as K is normal in G). As K is of finite index in G, we deduce that the G-orbit of z0 ◦ ρ is finite. Thus W contains a configuration whose G-orbit is finite. This shows that (a) implies (b). Implication (b) ⇒ (c) is trivial. The fact that (c) implies (d) follows from Proposition 1.2.1, Proposition 1.2.2, and Lemma 2.7.2. Let us show that (d) implies (a). Suppose that the group G acts continuously and faithfully on a Hausdorff topological space X and let E denote the set of points of X whose G-orbit is finite. For x ∈ E, let Stab(x) = {g ∈ G : gx = x} denote the stabilizer of x in G. Observe that x ∈ E if and only if Stab(x) is of finite index in G. If E is dense in X, then ∩x∈E Stab(x) = {1G } since the action of G on X is continuous and faithful. Thus, by Proposition 2.1.11 we have that G is residually finite.  

Notes A survey article on residually finite groups has been written by W. Magnus [Mag]. A group G is called linear if there exist an integer n ≥ 1 and a field K such that G is isomorphic to a subgroup of GLn (K). A theorem of A.I. Mal’cev [Mal1] asserts that every finitely generated linear group is residually finite.

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2 Residually Finite Groups

An example of a finitely presented group which is residually finite but not linear was recently given by C. Drut¸u and M. Sapir [DrS]. The residual finiteness of free groups was established by F. Levi [Lev]. The proof presented in Sect. 2.3 is based on the fact that the matrices 10 21

and 12 01 generate a free subgroup of GL2 (Z). This last result is due to I.N. Sanov [San]. The reader may find other proofs of the residual finiteness of free groups in [RobD, p. 158] and [MaKS, p. 116]. Hopfian groups are named after H. Hopf who used topological methods to prove that fundamental groups of closed orientable surfaces are Hopfian (see [MaKS, p. 415]) and raised the question of the existence of finitely generated non-Hopfian groups. The fact that every finitely generated residually finite group is Hopfian (Theorem 2.4.3) was discovered by Mal’cev [Mal1]. The first example of a finitely generated non-Hopfian group was given by B.H. Neumann [Neu2]. Shortly after, a finitely presented non-Hopfian group was found by G. Higman [Hig]. The simplest example of a finitely presented non-Hopfian group is certainly provided by the Baumslag-Solitar group BS(2, 3) = x, y : yx2 y −1 = x3 , that is, the quotient of the free group F2 based on two generators x and y by the smallest normal subgroup of F2 containing the element x−3 yx2 y −1 (see [BaS], [MaKS], [LS]). On the other hand, the Baumslag-Solitar group BS(2, 4) = x, y : yx2 y −1 = x4  is an example of a finitely presented Hopfian group which is not residually finite (the incorrect statement in [BaS] about the residual finiteness of BS(2, 4) was corrected by S. Meskin in [Mes]). The residual finiteness of the automorphism group of a finitely generated residually finite group (Theorem 2.5.1) was proved by G. Baumslag in [Bau]. The group G1 of Sect. 2.6 was considered by Mal’cev in [Mal1]. Given groups A and G, the wreath product A  G is the semidirect product H ψ G, where H = ⊕g∈G A and ψ : G → Aut(A) is the group homomorphism associated with the action of G on H ⊂ AG induced by the G-shift (see [RobD], [Rot]). Thus, the group G2 of Sect. 2.6 is the wreath product G2 = Sym+ 5 Z. The proof of Proposition 2.6.5 shows that, if A is a finitely generated nonabelian simple group and G is a finitely generated infinite group, then the wreath product A  G is a finitely generated group which is not residually finite. Examples of finitely generated nonabelian simple groups are provided by the (finite) groups Sym+ n , where n ≥ 5, or PSLn (K), where n ≥ 2 and K a finite field having at least 4 elements, or one of the famous infinite finitely presented simple Thompson groups T or V (see [CFP]).

Exercises 2.1. Show that every quotient of a divisible group is a divisible group. 2.2. Show that every torsionfree divisible abelian group G is isomorphic to a direct sum of copies of Q. Hint: Prove that there is a natural Q-vector space structure on G.

Exercises

53

2.3. Let G be a group. (a) Show that it is possible to define a topology on G by taking as open sets the subsets Ω ⊂ G which satisfy the following property: for each g ∈ Ω there is a subgroup of finite index H ⊂ G such that gH ⊂ Ω. This topology is called the profinite topology on G. (b) Show that G is residually finite if and only if the profinite topology on G is Hausdorff. 2.4. Show that the class of residually finite groups is not closed under taking quotients. Hint: Any group is isomorphic to a quotient of a free group (see Corollary D.4.2). 2.5. Let X be an infinite set. Show that the symmetric group Sym(X) is not residually finite. Hint: Use Cayley’s theorem (Theorem C.1.2) to prove that Sym(X) contains a subgroup isomorphic to Q or use Lemma 2.6.3. 2.6. Let G be a residually finite group and let A be a finite set. (a) Show that there exists a canonical injective  homomorphism of the group ICA(G; A) (cf. Sect. 1.10) into the group H Sym(Fix(H)), where H runs over all finite index subgroups of G. Hint. Use the fact that the configurations with finite G-orbit are dense in AG (cf. Theorem 2.7.1). (b) Deduce from (a) that ICA(G; A) is residually finite. 2.7. Let m be an integer such that |m| ≥ 2. Let G denote the quotient of the free group F2 on two generators x and y by the normal closure of the single element xyx−1 y −m . Thus, G is the group given by the presentation G = a, b : aba−1 = bm , where a and b denote the images in G of x and y by the quotient homomorphism. (a) Show that every element g ∈ G may be (not uniquely) written in the form g = ai bj ak for some i, j, k ∈ Z. (b) Show that there is a unique group homomorphism φ : G → GL2 (Q) which satisfies     m0 11 φ(a) = and φ(b) = . 0 1 01 (c) Use (a) to show that φ is injective and that the image of φ is the subgroup φ(G) ⊂ GL2 (Q) given by   n  m r : n ∈ Z, r ∈ Z[1/m] , φ(G) = 0 1 where Z[1/m] denotes the set of all rationals r ∈ Q which can be written in the form r = umv for some u, v ∈ Z. (d) Use (c) to prove that G is torsionfree. (e) Prove that G is residually finite. Hint: You have to show that, for any element g ∈ G \ {1G }, there exist a finite group F and a homomorphism ψ : G → F such that ψ(g) = 1F . Write g = ai bj ak , where i, j, k ∈ Z, and

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2 Residually Finite Groups

treat first the case i + k = 0 by using determinants. If i + k = 0, choose a prime number p which is not a divisor of m nor a divisor of j, and consider the homomorphism ψ : G → GL2 (Z/pZ) defined by     m0 11 , and ψ(b) = ψ(a) = 0 1 01 where n ∈ Z/pZ denotes the class of n ∈ Z modulo p. 2.8. Let m ≥ 2 be an integer. Denote by Z[1/m] the subgroup of the additive group Q consisting of all rationals r which can be written in the form r = kmn for some k, n ∈ Z. Show that the group Z[1/m] is residually finite. Hint: Observe that Z[1/m] is isomorphic to a subgroup of the group G studied in Exercise 2.7. 2.9. Let K be a field of characteristic p > 0. Show that the additive group K is residually finite. Hint: Use a base of K seen as a Z/pZ vector space and apply Corollary 2.2.3. 2.10. Show that all elements of the additive group Q/Z have finite order but that Q/Z is not residually finite. 2.11. Show that the multiplicative group Q∗ of nonzero rational numbers is residually finite. Hint: Use the unique factorization of elements of Q∗ into powers of prime numbers and apply Corollary 2.2.3. 2.12. Show that the automorphism group Aut(Q) of the additive group Q is isomorphic to the multiplicative group Q∗ . 2.13. Show that the additive group R is neither residually finite nor Hopfian. 2.14. Let p be a prime. Show that the additive group Zp of p-adic integers is not Hopfian. 2.15. Show that the multiplicative group R∗ of nonzero real numbers is neither residually finite nor Hopfian. 2.16. Let F be a free group of finite rank n. Suppose that S is a finite generating subset of F of cardinality |S| ≤ n. Use the fact that F is Hopfian (Corollary 2.4.5) to prove that |S| = n and that S is a base for F . 2.17. (M. Hall [Hal-M-2]) Let G be a finitely generated group and let n ≥ 1 be an integer. Show that G contains only a finite number of subgroups of index n. Hint: Use Lemma 2.1.10 and Lemma 2.4.4. 2.18. Show that the group GLn (Z) is finitely generated. Hint: For 1 ≤ i, j ≤ n, let Eij denote the n × n matrix all of whose entries are 0 except the entry located on the ith row and the jth column which is equal to 1. Use Euclidean division in Z to show that GLn (Z) is generated by the n2 matrices In − 2Eii , In + Eij , where 1 ≤ i, j ≤ n and i = j.

Exercises

55

2.19. Show that GLn (Z) is Hopfian. 2.20. Give a direct proof of the fact that the automorphism group of a finitely generated abelian group G is residually finite (Corollary 2.5.4). Hint: Show that there is a finite group F and an integer n ≥ 0 such that Aut(G) is isomorphic to F × GLn (Z). 2.21. Show that the group G1 considered in Sect. 2.6 contains no normal subgroup H such that both H and G/H are residually finite. 2.22. Show that the group G2 considered in Sect. 2.6 is Hopfian. 2.23. A group G is called almost perfect if all nontrivial finite quotients of G are nonabelian. Show that, if A is a finitely generated almost perfect nontrivial group and G is a finitely generated infinite group, then the wreath product group A  G is finitely generated but not residually finite. Hint: Follow the proof of Proposition 2.6.5. 2.24. Let G be a group. Denote by Nf q the set of all normal subgroups of finite index of G, partially ordered by reverse inclusion. (a) Show that Nf q is a directed set. (b) For H, K ∈ Nf q with H ⊂ K, let ϕK,H : G/H → G/K denote the canonical homomorphism. Show that the directed set Nf q together with the homomorphisms ϕK,H form a projective system of groups. The limit of this projective system is called the profinite completion of the group G and is  denoted by G.  and that the (c) Show that there is a canonical homomorphism η : G → G kernel of η is the residual subgroup of G. (d) Prove that G is residually finite if and only if the canonical homomor is injective. phism η : G → G

Chapter 3

Surjunctive Groups

Surjunctive groups are defined in Sect. 3.1 as being the groups on which all injective cellular automata with finite alphabet are surjective. In Sect. 3.2 it is shown that every subgroup of a surjunctive group is a surjunctive group and that every locally surjunctive group is surjunctive. Every locally residually finite group is surjunctive (Corollary 3.3.6). The class of locally residually finite groups is quite large and includes in particular all finite groups, all abelian groups, and all free groups (a still wider class of surjunctive groups, namely the class of sofic groups, will be described in Chap. 7). In Sect. 3.4, given an arbitrary group Γ , we introduce a natural topology on the set of its quotient groups. In Sect. 3.7, it is shown that the set of surjunctive quotients is closed in the space of all quotients of Γ .

3.1 Definition A set X is finite if and only if every injective map f : X → X is surjective. The definition given below is related to this characterization of finite sets. Definition 3.1.1. A group G is said to be surjunctive if it satisfies the following condition: if A is a finite set, then every injective cellular automaton τ : AG → AG is surjective (and hence bijective). Remark 3.1.2. Given a group G and a finite set A, it follows from Theorem 1.8.1 that a map f : AG → AG is a cellular automaton if and only if f is G-equivariant (with respect to the G-shift) and continuous (with respect to the prodiscrete topology on AG ). Thus, the definition of a surjunctive group may be reformulated as follows: a group G is surjunctive if and only if, for any finite set A, every injective G-equivariant continuous map f : AG → AG is surjective. Proposition 3.1.3. Every finite group is surjunctive. T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 3, © Springer-Verlag Berlin Heidelberg 2010

57

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3 Surjunctive Groups

Proof. If G is a finite group and A is a finite set, then the set AG is finite.  Therefore, every injective cellular automaton τ : AG → AG is surjective. 

3.2 Stability Properties of Surjunctive Groups Proposition 3.2.1. Every subgroup of a surjunctive group is surjunctive. Proof. Suppose that H is a subgroup of a surjunctive group G. Let A be a finite set and let τ : AH → AH be an injective cellular automaton over H. Consider the cellular automaton τ G : AG → AG over G obtained from τ by induction (see Sect. 1.7). The fact that τ is injective implies that τ G is injective by Proposition 1.7.4(i). Since G is surjunctive, it follows that τ G is surjective. By applying Proposition 1.7.4(ii), we deduce that τ is surjective.   Proposition 3.2.2. Let G be a group. Then the following conditions are equivalent: (a) G is surjunctive; (b) every finitely generated subgroup of G is surjunctive. Proof. The fact that (a) implies (b) follows from Proposition 3.2.1. Conversely, let G be a group all of whose finitely generated subgroups are surjunctive. Let A be a finite set and let τ : AG → AG be an injective cellular automaton with memory set S. Let H denote the subgroup of G generated by S and consider the cellular automaton τH : AH → AH obtained by restriction of τ (see Sect. 1.7). The fact that τ is injective implies that τH is injective by Proposition 1.7.4(i). As H is finitely generated, it is surjunctive by our hypothesis on G. It follows that τH is surjective. By applying Proposition 1.7.4(ii), we deduce that τ is also surjective. This shows that (b) implies (a).   If P is a property of groups (e.g. being finite, being nilpotent, being solvable, being free, etc.), a group G is called locally P if all finitely generated subgroups of G satisfy P. With this terminology, Proposition 3.2.2 may be rephrased by saying that the class of surjunctive groups and the class of locally surjunctive groups coincide. Corollary 3.2.3. Every locally finite group is surjunctive. Proof. This immediately follows from Proposition 3.2.2 since every finite group is surjunctive by Proposition 3.1.3.   Example 3.2.4. Let X be a set and consider the group Sym0 (X) consisting of all permutations of X which have finite support (see Sect. C.2). The group Sym0 (X) is locally finite. Indeed, if Σ is a finite subset of Sym0 (X), then

3.3 Surjunctivity of Locally Residually Finite Groups

59

the subgroup H ⊂ Sym0 (X) generated by Σ is finite since it is isomorphic to a subgroup of Sym(A), where A denotes the union of the supports of the elements of Σ. Consequently, the group Sym0 (X) is surjunctive. Observe that Sym0 (X) is infinite when X is infinite. This yields our first examples of infinite surjunctive groups. Finally, note that it follows from Lemma 2.6.3 that Sym0 (X) is not residually finite whenever X is infinite.

3.3 Surjunctivity of Locally Residually Finite Groups Theorem 3.3.1. Every residually finite group is surjunctive. Let us first establish an important property of cellular automata with finite alphabet, namely the fact that they always have a closed image with respect to the prodiscrete topology on the set of configurations: Lemma 3.3.2. Let G be a group and let A be a finite set. Let τ : AG → AG be a cellular automaton. Then the set τ (AG ) is closed in AG for the prodiscrete topology. Proof. Since A is finite, the space AG is compact by Tychonoff theorem (see Corollary A.5.3). As τ is continuous by Proposition 1.4.8, we deduce that τ (AG ) is a compact subset of AG . This implies that τ (AG ) is closed in AG since AG is Hausdorff by Proposition 1.2.1.   The following example shows that Lemma 3.3.2 becomes false if we suppress the hypothesis that the alphabet is finite. Example 3.3.3. Consider the map τ : NZ → NZ given by τ (x)(n) = max(0, x(n) − x(n + 1)) for all x ∈ NZ and n ∈ Z. Clearly, τ is a cellular automaton over the group Z and the alphabet N with memory set {0, 1}. Consider the configuration y : Z → N defined by y(n) = 1 if n ≥ 0 and y(n) = 0 otherwise. Let F be a finite subset of Z and choose an integer M ≥ 1 such that F ⊂ [−M +1, M −1]. Consider the configuration xM : Z → N defined by xM (n) = max(0, M − n) if n ≥ 0 and xM (n) = M otherwise. Observe that yM = τ (xM ) is such that yM (n) = 1 if 0 ≤ n ≤ M − 1 and yM (n) = 0 otherwise. Therefore, the configurations yM and y coincide on [−M + 1, M − 1] and hence on F . Thus y is in the closure of τ (NZ ) in NZ . On the other hand, it is clear that y is not in the image of τ . This shows that τ (NZ ) is not closed in NZ for the prodiscrete topology. In the proof of the surjunctivity of residually finite groups, we shall also use the following result:

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Lemma 3.3.4. Let G be a group. Suppose that G satisfies the following property: for each finite subset Ω ⊂ G, there exist a surjunctive group Γ and a homomorphism φ : G → Γ such that the restriction of φ to Ω is injective. Then G is surjunctive. Proof. Let A be a finite set and equip AG with its prodiscrete topology. Suppose that τ : AG → AG is an injective cellular automaton. Let us show that τ is surjective, that is, τ (AG ) = AG . Since τ (AG ) is closed in AG by Lemma 3.3.2, it suffices to prove that τ (AG ) is dense in AG . Consider a configuration x ∈ AG and a neighborhood W of x in AG . Then there is a finite subset Ω ⊂ G such that V (x, Ω) = {y ∈ AG : y|Ω = x|Ω } ⊂ W. By our hypothesis, we may find a surjunctive group Γ and a homomorphism φ : G → Γ such that the restriction of φ to Ω is injective. Let K denote the image of φ and let N denote its kernel. By Proposition 1.3.3, there is a bijective map ψ : AK → Fix(N ) given by ψ(z) = z ◦ φ for all z ∈ AK . Moreover, if we identify AK with Fix(N ) via ψ, the restriction of τ to Fix(N ) yields a cellular automaton τ : AK → AK over the group K (see Proposition 1.6.1). Since the group Γ is surjunctive, the group K is surjunctive by Proposition 3.2.1. We deduce that τ is surjective and hence τ (Fix(N )) = Fix(N ). On the other hand, as the restriction of φ to Ω is injective, we may find z0 ∈ AK such that ψ(z0 ) = x|Ω . We then have ψ(z0 ) ∈ Fix(N ) ∩ V (x, Ω) = τ (Fix(N )) ∩ V (x, Ω). This shows that W meets the image of τ . Thus τ (AG ) is dense in AG .

 

Proof of Theorem 3.3.1. Let G be a residually finite group. If Ω is a finite subset of G, then there exist, By Lemma 2.7.3, a finite group Γ and a homomorphism φ : G → Γ such that the restriction of φ to Ω is injective. As finite groups are surjunctive by Proposition 3.1.3, it follows that G satisfies the hypothesis of Lemma 3.3.4. Consequently, G is surjunctive.   Corollary 3.3.5. Every free group is surjunctive. Proof. Free groups are residually finite by Theorem 2.3.1.

 

Corollary 3.3.6. Every locally residually finite group is surjunctive. Proof. This immediately follows from Theorem 3.3.1 by using Proposition 3.2.2.   Corollary 3.3.7. Every abelian group is surjunctive. Proof. This is an immediate consequence of Corollary 3.3.6 since every abelian group is locally residually finite by Corollary 2.2.4.  

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61

When X is a finite set, every surjective map f : X → X is injective. Therefore, a surjective cellular automaton with finite alphabet over a finite group is necessarily injective. However, a surjective cellular automaton with finite alphabet τ : AG → AG may fail to be injective when the group G is infinite as shown by the following example. Example 3.3.8. Take A = Z/2Z = {0, 1} and G = Z. Let τ : AZ → AZ be the map defined by τ (x)(n) = x(n) + x(n + 1) for all x ∈ AZ and n ∈ Z. Clearly, τ is a cellular automaton with memory set S = {0, 1} ⊂ Z and local defining map μ : AS → A given by μ(y) = y(0) + y(1) for all y ∈ AS . The cellular automaton τ is surjective. Indeed, given an element y ∈ AG , the configuration x : Z → Z/2Z defined by ⎧ ⎪ if n = 0, ⎨0 x(n) = y(0) + y(1) + · · · + y(n − 1) if n > 0, ⎪ ⎩ y(n) + y(n + 1) + · · · + y(−1) if n < 0, satisfies τ (x) = y. However τ is not injective since the two constant configurations have the same image under τ .

3.4 Marked Groups Let Γ be a group. A Γ -quotient is a pair (G, ρ), where G is a group and ρ : Γ → G is a surjective homomorphism. We define an equivalence relation on the class of all Γ -quotients by declaring that two Γ -quotients (G1 , ρ1 ) and (G2 , ρ2 ) are equivalent when there exists a group isomorphism φ : G2 → G1 such that the following diagram is commutative: G2 ρ2 φ∼ =

Γ ρ1

G1 that is, such that ρ1 = φ ◦ ρ2 . An equivalence class of Γ -quotients is called a Γ -marked group. Observe that two Γ -quotients (G1 , ρ1 ) and (G2 , ρ2 ) are equivalent if and only if Ker(ρ1 ) = Ker(ρ2 ). Thus, the set of Γ -marked groups may be identified with the set N (Γ ) consisting of all normal subgroups of Γ . Let us identify the set P(Γ ) consisting of all subsets of Γ with the set {0, 1}Γ by means of the bijection from P(Γ ) onto {0, 1}Γ given by A → χA ,

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where χA : Γ → {0, 1} is the characteristic map of A ⊂ Γ . We equip the set P(Γ ) = {0, 1}Γ with its prodiscrete uniform structure and N (Γ ) ⊂ P(Γ ) with the induced uniform structure. Thus, a base of entourages of N (Γ ) is provided by the sets VF = {(N1 , N2 ) ∈ N (Γ ) × N (Γ ) : N1 ∩ F = N2 ∩ F }, where F runs over all finite subsets of Γ . Intuitively, two normal subgroups of Γ are “close” in N (Γ ) when their intersection with a large finite subset of Γ coincide. Proposition 3.4.1. Let Γ be a group. Then the space N (Γ ) of Γ -marked groups is a totally disconnected compact Hausdorff topological space. Proof. The space P(Γ ) = {0, 1}Γ is totally disconnected and Hausdorff by Proposition 1.2.1. Moreover, P(Γ ) is compact by Corollary A.5.3 since it is a product of finite spaces. Thus, it suffices to show that N (Γ ) is closed in P(Γ ). To see this, observe that a subset E ∈ P(Γ ) is a normal subgroup of Γ if and only if it satisfies (1) 1Γ ∈ E; (2) αβ −1 ∈ E for all α, β ∈ E; (3) γαγ −1 ∈ E for all α ∈ E and γ ∈ Γ . Denoting, for each γ ∈ Γ , by πγ : P(Γ ) = {0, 1}Γ → {0, 1} the projection map corresponding to the γ-factor, these conditions are equivalent to (1’) π1Γ (E) = 1; (2’) πα (E)πβ (E)(παβ −1 (E) − 1) = 0 for all α, β ∈ Γ ; (3’) πα (E)(πγαγ −1 (E) − 1) = 0 for all α, γ ∈ Γ . As all projection maps are continuous on P(Γ ), this shows that N (Γ ) is closed in P(Γ ).   Remark 3.4.2. If Γ is countable then the uniform structure on N (Γ ) is metrizable. Indeed, the uniform structure on P(Γ ) = {0, 1}Γ is metrizable when Γ is countable (cf. Remark 1.9.2). Let P be a property of groups. A group Γ is called residually P if for each element γ ∈ Γ with γ = 1Γ , there exist a group Γ  satisfying P and an epimorphism φ : Γ → Γ  such that φ(γ) = 1Γ  . Observe that every group which satisfies P is residually P. Proposition 3.4.3. Let Γ be a group and let P be a property of groups. Suppose that the class of groups which satisfy P is closed under taking finite products and subgroups. Then the following conditions are equivalent: (a) Γ is residually P; (b) there exists a net (Ni )i∈I in N (Γ ) which converges to {1Γ } such that Γ/Ni satisfies P for all i ∈ I.

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63

Proof. Suppose that Γ is residually P. For all γ ∈ Γ \{1Γ } we can find a group Γγ satisfying P and an epimorphism φγ : Γ → Γγ such that φγ (γ) = 1Γγ . Let I ⊂ P(Γ ) be the directed set consisting of all finite subsets of Γ  not containing 1 partially ordered by inclusion. For i ∈ I we denote by φ = Γ i γ∈i φγ : Γ →     γ∈i Γγ the homomorphism defined by φi (γ ) = (φγ (γ ))γ∈I for all γ ∈ Γ . Set Ni = ker(φi ) and let us show that the net (Ni )i∈I in N (Γ ) converges to {1Γ }. Given a finite set F ⊂ Γ set iF = F \ {1Γ }. Let f ∈ F \ {1Γ } and i ∈ I such that iF ⊂ i. Then φi (f ) = 1Γi since φf (f ) = 1Γf . Thus / {1Γ } ∩ F. f∈ / Ni and f ∈

(3.1)

/ F then Ni ∩ F = ∅ = {1Γ } ∩ F . On the other hand, It follows that if 1Γ ∈ if 1Γ ∈ F , from (3.1) we deduce Ni ∩ F = {1Γ } = {1Γ } ∩ F . In either cases that limi Ni = {1Γ }. Finally, by our we have Ni ∩ F = {1Γ } ∩ F . We deduce  assumptions on P, we have that γ∈i Γγ and its subgroup φi (Γ ) ∼ = Γ/Ni satisfy P for all i ∈ I. This shows (a) ⇒ (b). Suppose now that (b) holds. Let γ ∈ Γ \ {1Γ }. Consider the finite set F = {γ}. Then there exists iF ∈ I such that NiF ∩ F = {1Γ } ∩ F and / NiF . In other words, Γ/NiF satisfy P. As {1Γ } ∩ F = ∅ we have that γ ∈ if φF : Γ → Γ/NiF denotes the canonical epimorphism, we have φF (γ) =   1Γ/NiF . We deduce that Γ is residually P. This shows (b) ⇒ (a). Note that in the above proposition, the assumptions on P are not needed for the implication (b) ⇒ (a) Let A be a set. Consider the set AΓ equipped with its prodiscrete uniform structure and the Γ -shift action. For each N ∈ N (Γ ), let Fix(N ) = {x ∈ AΓ : γx = x for all γ ∈ N } ⊂ AΓ denote the set of configurations which are fixed by N . Recall from Proposition 1.3.6 that Fix(N ) is a closed Γ -invariant subset of AΓ . Let us equip P(AΓ ) with the Hausdorff-Bourbaki uniform structure associated with the prodiscrete uniform structure on AΓ (see Sect. B.4 for the definition of the Hausdorff-Bourbaki uniform structure on the set of subsets of a uniform space). Theorem 3.4.4. Let Γ be a group and let A be a set. Then the map Ψ : N (Γ ) → P(AΓ ) defined by Ψ (N ) = Fix(N ) is uniformly continuous. Moreover, if A contains at least two elements then Ψ is a uniform embedding. Proof. Let N0 ∈ N (Γ ) and let W be an entourage of P(AΓ ). Let us show that there exists an entourage V of N (Γ ) such that Ψ (V [N0 ]) ⊂ W [Ψ (N0 )].

(3.2)

This will prove that Ψ is continuous. By definition of the Hausdorff-Bourbaki uniform structure on P(AΓ ), there is an entourage T of AΓ such that

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T = {(X, Y ) ∈ P(AΓ ) × P(AΓ ) : Y ⊂ T [X] and X ⊂ T [Y ]} ⊂ W.

(3.3)

Since AΓ is endowed with its prodiscrete uniform structure, there is a finite subset F ⊂ Γ such that U = {(x, y) ∈ AΓ × AΓ : πF (x) = πF (y)} ⊂ T,

(3.4)

where πF : AΓ → AF is the projection map. Consider now the finite subset E ⊂ Γ defined by E = F · F −1 = {γη −1 : γ, η ∈ F }, and the entourage V of N (Γ ) given by V = {(N1 , N2 ) ∈ N (Γ ) × N (Γ ) : N1 ∩ E = N2 ∩ E}. We claim that V satisfies (3.2). To prove our claim, suppose that N ∈ V [N0 ]. Let x ∈ Fix(N ). The fact that N ∩ E = N0 ∩ E implies that if γ and η are elements of F with γ = νη for some ν ∈ N , then ν ∈ N0 and therefore x(γ) = x(η). Denoting by ρ0 : Γ → Γ/N0 the canonical epimorphism, we deduce that we may find an element x0 ∈ AΓ/N0 such that x(γ) = x0 ◦ ρ0 (γ) for all γ ∈ F . We have x0 ◦ ρ0 ∈ Fix(N0 ) and (x, x0 ◦ ρ0 ) ∈ U . Since U ⊂ T by (3.4), this shows that Fix(N ) ⊂ T [Fix(N0 )]. Therefore Ψ is continuous. As N (Γ ) is compact by Proposition 3.4.1, we deduce that Ψ is uniformly continuous by applying Theorem B.2.3. Suppose now that A has at least two elements. Let us show that Ψ is injective. Let N1 , N2 ∈ N (Γ ). Fix two elements a, b ∈ A with a = b and consider the map x : Γ → A defined by x(γ) = a if γ ∈ N1 and x(γ) = b otherwise. We clearly have x ∈ Fix(N1 ). Suppose that Ψ (N1 ) = Ψ (N2 ), that is, Fix(N1 ) = Fix(N2 ). Then for all ν ∈ N2 , we have ν −1 x = x since x ∈ Fix(N1 ) = Fix(N2 ), and hence x(ν) = ν −1 x(1Γ ) = x(1Γ ) = a. This implies N2 ⊂ N1 . By symmetry, we also have N1 ⊂ N2 . Therefore N1 = N2 . This shows that Ψ is injective. As N (Γ ) is compact and P(AΓ ) is Hausdorff, we conclude that Ψ is a uniform embedding by applying Proposition B.2.5.  

3.5 Expansive Actions on Uniform Spaces Let X be a uniform space and let Γ be a group acting on X. We consider the diagonal action of Γ on X × X defined by γ(x, y) = (γx, γy) for all γ ∈ Γ and x, y ∈ X. One says that the action of Γ on X is uniformly continuous if the orbit map X → X, x → γx, is uniformly continuous for each γ ∈ Γ . This is

3.6 Gromov’s Injectivity Lemma

65

equivalent to saying that γ −1 V is an entourage of X for all entourage V of X and γ ∈ Γ . The action of Γ on X is said to be expansive if there exists an entourage W0 of X such that  γ −1 W0 = ΔX , (3.5) γ∈Γ

where ΔX = {(x, x) : x ∈ X} denotes the diagonal in X × X. Equality (3.5) means that if x, y ∈ X satisfy (γx, γy) ∈ W0 for all γ ∈ Γ , then x = y. An entourage W0 satisfying (3.5) is then called an expansivity entourage for the action of Γ on X. Remarks 3.5.1. (a) If a uniform space X admits an expansive uniformly continuous action of a group Γ , then the topology on X must be Hausdorff. Indeed, if W0 is an expansivity entourage for such an action then ΔX is equal to the intersection of the entourages γ −1 W0 , γ ∈ Γ , by (3.5). (b) Suppose that B is a base of the uniform structure on X. Then an action of a group Γ on X is expansive if and only if it admits an expansivity entourage B0 ∈ B. (c) Let (X, d) be a metric space. Then an action of a group Γ on X is expansive if and only if there exists a real number ε0 > 0 with the following property: if x, y ∈ X satisfy d(γx, γy) < ε0 for all γ ∈ Γ , then x = y. Such an ε is called an expansivity constant for the action of Γ on X. Our basic example of a uniformly continuous and expansive action is provided by the following: Proposition 3.5.2. Let G be a group and let A be a set. Then the G-shift on AG is uniformly continuous and expansive with respect to the prodiscrete uniform structure on AG . Proof. Uniform continuity follows from the fact that G acts on AG by permuting coordinates. Expansiveness is due to the fact that the action of G on itself by left multiplication is transitive. Indeed, consider the entourage W0 of AG defined by W0 = {(x, y) ∈ AG × AG : x(1G ) = y(1G )}. Given g ∈ G, we have (x, y) ∈ g −1 W0 if and only if x(g −1 ) = y(g −1 ). Thus −1 W0 is equal to the diagonal in AG × AG .   g∈G g

3.6 Gromov’s Injectivity Lemma The following result will be used in the next section to prove that surjunctive groups define a closed subset of the space of Γ -marked groups for any group Γ .

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Theorem 3.6.1 (Gromov’s injectivity lemma). Let X be a uniform space endowed with a uniformly continuous and expansive action of a group Γ . Let f : X → X be a uniformly continuous and Γ -equivariant map. Suppose that Y is a subset of X such that the restriction of f to Y is a uniform embedding. Then there exists an entourage V of X satisfying the following property: if Z is a Γ -invariant subset of X such that Z ⊂ V [Y ], then the restriction of f to Z is injective. (We recall that the notation Z ⊂ V [Y ] means that for each z ∈ Z there exists y ∈ Y such that (z, y) ∈ V .) Proof. By expansivity of the action of Γ , there is an entourage W0 of X such that  γ −1 W0 = ΔX . (3.6) γ∈Γ

It follows from the axioms of a uniform structure that we can find a symmetric entourage S of X such that S ◦ S ◦ S ⊂ W0 .

(3.7)

Since the restriction of f to Y is a uniform embedding, we can find an entourage T of X such that (f (y1 ), f (y2 )) ∈ T ⇒ (y1 , y2 ) ∈ S

(3.8)

for all y1 , y2 ∈ Y . Let U be a symmetric entourage of X such that U ◦ U ⊂ T.

(3.9)

Since f is uniformly continuous, we can find an entourage E of X such that (x1 , x2 ) ∈ E ⇒ (f (x1 ), f (x2 )) ∈ U

(3.10)

for all x1 , x2 ∈ X. Let us show that the entourage V = S ∩ E has the required property. So let Z be a Γ -invariant subset of X such that Z ⊂ V [Y ] and let us show that the restriction of f to Z is injective. Let z  and z  be points in Z such that f (z  ) = f (z  ). Since f is Γ equivariant, we have (3.11) f (γz  ) = f (γz  ) for all γ ∈ Γ . As the points γz  and γz  stay in Z, the fact that Z ⊂ V [Y ] implies that there are points yγ and yγ in Y such that (γz  , yγ ) ∈ V and (γz  , yγ ) ∈ V . Since V ⊂ E, it follows from (3.10) that (f (γz  ), f (yγ )) and (f (γz  ), f (yγ )) are in U . As U is symmetric, we also have (f (yγ ), f (γz  )) ∈ U . We deduce that (f (yγ ), f (yγ )) ∈ U ◦ U ⊂ T by using (3.9) and (3.11). This implies (yγ , yγ ) ∈ S by (3.8). On the other hand, we also have (γz  , yγ ) ∈ S and (yγ , γz  ) ∈ S since V ⊂ S and S is symmetric. It follows that

3.7 Closedness of Marked Surjunctive Groups

67

(γz  , γz  ) ∈ S ◦ S ◦ S ⊂ W0 by (3.7). This gives us (z  , z  ) ∈



γ −1 W0 ,

γ∈Γ 



and hence z = z by (3.6). Thus the restriction of f to Z is injective.

 

3.7 Closedness of Marked Surjunctive Groups Theorem 3.7.1. Let Γ be a group. Then the set of normal subgroups N ⊂ Γ such that the quotient group Γ/N is surjunctive is closed in N (Γ ). Proof. Let N ∈ N (Γ ) and let (Ni )i∈I be a net in N (Γ ) converging to N . Suppose that the groups Γ/Ni are surjunctive for all i ∈ I. Let us show that the group Γ/N is also surjunctive. Let A be a finite set and let τ : AΓ/N → AΓ/N be an injective cellular automaton over the group Γ/N . Let S ⊂ Γ/N be a memory set for τ with associated local defining map μ : AS → A. Choose a subset S ⊂ Γ such that the canonical epimorphism ρ : Γ → Γ/N gives a bijection ψ : S → S, and let e π : AS → AS denote the bijective map induced by ψ. Consider the cellular automaton τ : AΓ → AΓ over Γ with memory set S and local defining map e μ

= μ ◦ π −1 : AS → A. To simplify notation, let us set X = AΓ , f = τ , Y = Fix(N ) and Zi = Fix(Ni ). We claim that the hypotheses of Theorem 3.6.1 are satisfied by X, f and Y . Indeed, we first observe that the action of Γ on X is uniformly continuous and expansive by Proposition 3.5.2. On the other hand, the cellular automaton f : X → X is uniformly continuous and Γ -equivariant by Theorem 1.9.1. Finally, the restriction of f to Y is injective since this restriction is conjugate to τ by Proposition 1.6.1. As Y is a closed subset of X and hence compact, it follows from Proposition B.2.5 that the restriction of f to Y is a uniform embedding. By applying Theorem 3.6.1, it follows that there exists an entourage V of X such that if Z is a Γ -invariant subset of X with Z ⊂ V [Y ], then the restriction of f to Z is injective. Since the net (Zi )i∈I converges to Y for the Hausdorff-Bourbaki topology on P(X) by Theorem 3.4.4, there is an element i0 ∈ I such that Zi ⊂ V [Y ] for all i ≥ i0 . As the sets Zi are Γ -invariant by Proposition 1.3.6, it follows that the restriction of f to Zi is injective for all i ≥ i0 . On the other hand, f (Zi ) ⊂ Zi and the restriction of f to Zi is conjugate to a cellular automaton τi : AΓ/Ni → AΓ/Ni over the group Γ/Ni for all i ∈ I by Proposition 1.6.1. As the groups Γ/Ni are surjunctive by our hypotheses, we deduce that f (Zi ) = Zi for all i ≥ i0 . Now, it follows from Proposition B.4.6 that the net (f (Zi ))i∈I converges to f (Y ) in P(X). Thus, the net (Zi )i∈I converges to both Y and f (Y ). As Y and f (Y ) are closed in X (by compactness of Y ), we deduce that

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Y = f (Y ) by applying Proposition B.4.3. This shows that τ is surjective since τ is conjugate to the restriction of f to Y . Consequently, the group Γ/N is surjunctive.  

Notes Surjunctive groups were introduced by W. Gottschalk. The first results on these groups are due to W. Lawton who proved in particular Proposition 3.2.1, Corollary 3.2.3, Theorem 3.3.1, and Corollary 3.3.7 (see [Got], [Law]). The characterization of infinite sets by the existence of injective but not surjective self-maps is known as Dedekind’s definition of infinite sets. Proposition 3.3.6 together with Mal’cev theorem [Mal1] which says that every finitely generated linear group is residually finite implies that every linear group is surjunctive. The problem of the existence of a non surjunctive group was raised by Gottschalk in [Got] and remains open up to now. Note that to prove that all groups are surjunctive it would suffice to prove that the symmetric group Sym(N) is surjunctive (this immediately follows from Proposition 3.2.1, Proposition 3.2.2, and the fact that every finitely generated group is countable and hence isomorphic to a subgroup of Sym(N) by Cayley’s theorem). In [CeC11] it is shown that if G is a non-periodic group, then for every infinite set A there exists a cellular automaton τ : AG → AG whose image τ (AG ) is not closed in AG with respect to the prodiscrete topology (cf. Lemma 3.3.2 and Example 3.3.3). Theorem 3.6.1 is a uniform version of Lemma 4.H” in [Gro5]. The proof presented in this chapter of the surjunctivity of limits of surjunctive groups (Theorem 3.7.1) closely follows Sect. 4 of [Gro5] (see also [CeC10]). There is another proof based on techniques from model theory (see [Gro5] and [GlG]).

Exercises 3.1. Let K be an algebraically closed field. Show that every injective polynomial map f : K → K is surjective. 3.2. Show that every injective polynomial map f : R → R is surjective. 3.3. Show that the polynomial map f : Q → Q defined by f (x) = x3 is injective but not surjective. 3.4. Show that every injective holomorphic map f : C → C is surjective. 3.5. Give an example of a real analytic map f : R → R which is injective but not surjective.

Exercises

69

3.6. Let K be a field and let V be a vector space over K. Show that V is finite-dimensional if and only if every injective endomorphism f : V → V is surjective. 3.7. Let R be a ring and let M be a left (or right) R-module. One says that M is Artinian if every descending chain N0 ⊃ N1 ⊃ N2 ⊃ . . . eventually stabilizes (i.e., there is an integer n0 ≥ 0 such that NN = Nn+1 for all n ≥ n0 ). Show that if M is Artinian then every injective endomorphism f : M → M is surjective. Hint: Consider the sequence of submodules defined by Nn = Im(f n ) for n ≥ 0. 3.8. Let U = {z ∈ C : |z| = 1}. Show that every injective continuous map f : U → U is surjective. 3.9. Let n ≥ 0 be an integer. An n-dimensional topological manifold is a nonempty Hausdorff topological space X such that each point in X admits a neighborhood homeomorphic to Rn . Show that if X is a compact n-dimensional topological manifold, then every injective continuous map f : X → X is surjective. Hint: Use Brouwer’s invariance of domain to prove that f (X) is open in X. 3.10. Let P be a property of groups. Show that every subgroup of a locally P group is itself locally P. 3.11. Let P be a property of groups. Let G be a group. Show that G is locally P if and only if all its finitely generated subgroups are locally P. 3.12. Show that the additive group Q is locally cyclic. 3.13. Show that in a locally finite group every element has finite order. 3.14. Let G be an abelian group. Show that G is locally finite if and only if every element of G has finite order. 3.15. Show that every subgroup and every quotient of a locally finite group is a locally finite group. 3.16. Let G be a group. Suppose that G contains a normal subgroup H such that both H and G/H are locally finite. Show that G is locally finite. 3.17. Let G be a group which is the limit of an inductive system of locally finite groups. Show that G is locally finite. 3.18. Show that the direct sum of any family of locally finite groups is a locally finite group. 3.19. Let G be a group. Let S denote the set consisting of all normal locally finite subgroups of G. (a) Show that if H ∈ S and K ∈ S then HK ∈ S. (b) Show that M = H∈S H is a normal locally finite subgroup of G and that every normal locally finite subgroup of G is contained in M .

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3.20. Let G be a locally finite group and let A be a set. Show that every bijective cellular automaton τ : AG → AG is invertible. 3.21. Let G be a locally finite group and let A be a finite set. Show that every surjective cellular automaton τ : AG → AG is injective. 3.22. Let G be a locally finite group and let A be a set. Let τ : AG → AG be a cellular automaton. Show that τ (AG ) is closed in AG with respect to the prodiscrete topology. Hint: First treat the case when G is finite, then use Sect. 1.7 and Proposition A.4.3. 3.23. Let G be a group and let A be a nonzero abelian group. Suppose that s0 ∈ G is an element of infinite order. Show that the map τ : AG → AG defined by τ (x)(g) = x(gs0 ) − x(g), for all x ∈ AG and g ∈ G, is a cellular automaton which is surjective but not injective. 3.24. Let X be an infinite set. Show that the symmetric group Sym(X) is not locally residually finite. 3.25. Let G be a group. Suppose that there exists a family (Ni )i∈I of normal subgroups of G satisfying the following properties: (1) for all i and j in I, there exists k in I such that Nk ⊂ Ni ∩ Nj ; (2) i∈I Ni = {1G }; (3) the group G/Ni is surjunctive for each i ∈ I. Show that G is surjunctive. Hint: Apply Lemma 3.3.4. 3.26. It follows from Theorem 3.3.1 that every residually finite group is surjunctive. The goal of this exercise is to present an alternative proof of this result. Let G be a residually finite group and let A be a finite set. Let τ : AG → AG be an injective cellular automaton. Fix a family (Γi )i∈I of subgroups of finite index of G such that i∈I Γi = {1G } (the existence of such a family follows from the residual finiteness of G). (a) Show that, for each i ∈ I, the set Fix(Γi ) = {x ∈ AG : gx = x for all g ∈ Γi } is finite. (b) Show that τ (Fix(Γi )) = Fix(Γi ) for all i ∈ I. (c) Prove that i∈I Fix(Γi ) is dense in AG and conclude. 3.27. Let X be a set. Let (Ai )i∈I be a net of subsets of X. (a) Show that   Aj ⊂ Aj . i j≥i

i j≥i

(b) Let B be a subset of X. Show that the net (Ai )i∈I converges to B with respect to the prodiscrete topology on P(X) = {0, 1}X if and only if   B= Aj = Aj . i j≥i

i j≥i

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71

3.28. Let G be a group and let A be a set. Let H be a subgroup of G. In Exercise 1.33, we associated with each subshift X ⊂ AH the subshift G H X (G) = {x ∈ AG : xH g ∈ X for all g ∈ G} ⊂ A , where xg = (gx)|H for G H H all x ∈ A , h ∈ H and g ∈ G. Let σ : A → A be a cellular automaton and denote by σ G : AG → AG the induced cellular automaton. Show that if X, Y ⊂ AH are two subshifts such that σ(X) ⊂ Y , then the cellular automaton σ G |X (G) : X (G) → Y (G) is injective (resp. surjective) if and only if the cellular automaton σ|X : X → Y is injective (resp. surjective). 3.29. Let G be a group and let A be a finite set. Given a subshift Z ⊂ AG we denote by Zf the set of all configurations in Z whose G-orbit is finite (cf. Example 1.3.1(c)). We say that Z is surjunctive if every injective cellular automaton σ : Z → Z is surjective. Let X ⊂ AG be a subshift such that Xf is dense in X. (a) Let τ : AG → AG be a cellular automaton. Consider the subshift Y = τ (X). Show that Yf is dense in Y . (b) Deduce from (a) that X is surjunctive. 3.30. Let A be a finite set and let X ⊂ AZ be an irreducible subshift of finite type. (a) Show that Xf is dense in X. (b) Deduce from (a) and Exercise 3.29(b) that X is surjunctive (compare with its stronger version in Exercise 6.36). (c) Let A = {0, 1} and consider the subshift of finite type Y ⊂ AZ defined by Y = {x ∈ AZ : (x(n), x(n + 1)) = (0, 1) for all n ∈ Z}. Show that Yf consists of the two constant configurations x0 and x1 and therefore it is closed but not dense in Y . (d) Show that the subshift Y is surjunctive. (e) Let A = {0, 1} and let Y as in (c). Consider the associated subshift 2 2 Z = Y (Z ) ⊂ AZ defined in Exercise 1.33. Show that Z is an irreducible subshift of finite type but that Zf is not dense in Z. (f) Show that the subshift Z is surjunctive. 3.31. Show that the golden mean subshift is surjunctive. Hint: Use Exercise 1.39 and Exercise 3.30(b). 3.32. Show that the even subshift is surjunctive. Hint: Use Exercise 3.30. 3.33. A subshift of finite type which is not surjunctive (cf. [Weiss, Sect. 4]). Let A = {0, 1, 2} and consider the subshift of finite type X = X(Ω, A) ⊂ AZ with memory set Ω = {0, 1} ⊂ Z and defining set of admissible words A = {00, 01, 11, 12, 22} ⊂ AΩ . (a) Show that X is not irreducible. (b) Consider the cellular automaton σ : AZ → AZ with memory set S = {0, 1} and local defining map μ : AS → A defined by

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μ(y) =

1 y(0)

if y(0)y(1) = 01 otherwise.

Show that σ(X) ⊂ X. (c) Show that the cellular automaton τ = σ|X : X → X is injective but not surjective. Deduce that X is not surjunctive. 3.34. Let G be a group and let A be a finite set. Suppose that G contains an element of infinite order and that A contains at least two distinct elements. (a) Show that there exists a subshift X ⊂ AG which is not surjunctive. (b) Show that if, in addition, G is not infinite cyclic then there exists an irreducible subshift X ⊂ AG which is not surjunctive. Hint: Use Exercise 3.33 and Exercise 1.33(c). 3.35. Given a group Γ acting continuously on a topological space Z, a subset M ⊂ Z is called a minimal set if M is a nonempty Γ -invariant subset of Z and the Γ -orbit of every point m ∈ M is dense in M . Let G be a group and let A be a set. A subshift X ⊂ AG is called minimal if X is a minimal set with respect to the G-shift action on AG . (a) Show that a subshift X ⊂ AG is minimal if and only if X = ∅ and there is no subshift Y in AG such that ∅ = Y  X. (b) Show that every minimal subshift X ⊂ AG is irreducible. (c) Show that the finite minimal subshifts in AG are precisely the finite G-orbits. (d) Show that if X ⊂ AG is an infinite minimal subshift then the G-orbit of every configuration x ∈ X is infinite. (e) Show that if X and Y are minimal subshifts in AG then either X = Y or X ∩ Y = ∅. 3.36. Let G be a group and let A be a finite set. Let X, Y ⊂ AG be two subshifts. Show that if X is nonempty and Y is minimal then every cellular automaton τ : X → Y is surjective. 3.37. Let G be a group and let A be a finite set. Show that every minimal subshift X ⊂ AG is surjunctive. 3.38. Let G be a group and let A be a finite set. Show that every nonempty subshift X ⊂ AG contains a minimal subshift. Hint: Apply Zorn’s lemma to the set of nonempty subshifts contained in X. 3.39. Let G be a group and let A be a set. A subset R ⊂ G is called syndetic if there exists a finite subset K ⊂ G such that the set Kg meets R for every g ∈ G. A configuration x ∈ AG is called almost periodic if for every finite subset Ω ⊂ G the set R(x, Ω) = {g ∈ G : (gx)|Ω = x|Ω } is syndetic in G. (a) Show that if x ∈ AG is an almost periodic configuration then its orbit closure X = Gx ⊂ AG is a minimal subshift. Hint: Suppose that the orbit closure X = Gx of a configuration x ∈ AG is not minimal. Then there exists

Exercises

73

a nonempty subshift Y ⊂ X with x ∈ / Y . This implies that there is a finite subset Ω ⊂ G such that x|Ω = y|Ω for all y ∈ Y . Let K be a finite subset of G and consider the set Ω  = K −1 Ω. Choose an arbitrary configuration y0 ∈ Y . Since y0 ∈ X, there is an element g0 ∈ G such that (g0 x)|Ω  = y0 |Ω  . This implies that (kg0 x)|Ω = (ky0 )|Ω = x|Ω for all k ∈ K. Thus Kg0 does not meet R(x, Ω) so that x is not almost-periodic. (b) Suppose that the set A is finite. Show that if X ⊂ AG is a minimal subshift, then every configuration x ∈ X is almost-periodic. Hint: Observe that if Ω ⊂ G is a finite subset and x ∈ X, then the open sets gV , where V = {y ∈ AG : y|Ω = x|Ω } and g runs over G, cover X by minimality and use the compactness of X. (c) Suppose that the set A is finite. Show that a nonempty subshift X ⊂ AG is minimal if and only if X is irreducible and every configuration x ∈ X is almost-periodic. 3.40. Let A be a set. The language of a configuration x ∈ AZ is the subset L(x) ⊂ A∗ consisting of all words w ∈ A∗ which can be written in the form w = x(n + 1)x(n + 2) . . . x(n + m) for some integers n ∈ Z and m ≥ 0. (a) Let X ⊂ AZ be a subshift and let x ∈ X. Show that L(x) ⊂ L(X) and that one has L(x) = L(X) if and only if the Z-orbit of x is dense in X. (b) Show that a nonempty subshift X ⊂ AZ is minimal if and only if one has L(x) = L(X) for all x ∈ X. (c) Show that a configuration x ∈ AZ is almost periodic if and only if it satisfies the following condition: for every word u ∈ L(x), there exists an integer n = n(u) ≥ 0 such that u is a subword of any word v ∈ L(x) of length n. 3.41. The Thue-Morse sequence. Let A = {0, 1}. The Thue-Morse sequence is the map x : N → A defined by

1 if the number of ones in the binary expansion of n is odd x(n) = 0 otherwise. (a) Check that x(0)x(1) . . . x(16) = 0110100110010110. (b) Show that one has x(k + 3m2n ) = x(k) for all k, m, n ∈ N such that 0 ≤ k ≤ 2n . (c) Show that x satisfies the recurrence relations ⎧ ⎪ ⎨x(0) = 0 x(2n) = x(n) ⎪ ⎩ x(2n + 1) = 1 − x(n) and that these relations uniquely determine x.

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(d) Consider the unique monoid homomorphism ϕ : A∗ → A∗ which satisfies ϕ(0) = 01 and ϕ(1) = 10. Show that ϕ(x(0)x(1) . . . x(2n − 1)) = x(0)x(1) . . . x(2n+1 − 1) for all n ∈ N. (e) Consider the unique monoid isomorphism ι : A∗ → A∗ which satisfies ι(0) = 1 and ι(1) = 0. Show that x(0)x(1) . . . x(2n+1 − 1) = x(0)x(1) . . . x(2n − 1)ι(x(0)x(1) . . . x(2n − 1)) for all n ∈ N. Note: The Thue-Morse sequence was first studied by E. Prouhet in a 1851 paper dealing with problems in number theory. It is an example of an automatic sequence (see [AlS, Sect. 5.1]). 3.42. The Morse subshift. Let A = {0, 1} and let L = {(x(n), x(n + 1), . . . , x(n + m)) : n, m ∈ N} ⊂ A∗ , where x : N → A is the Thue-Morse sequence. (a) Show that the set L satisfies the conditions (i) and (ii) in Exercise 1.36(b). (b) Let X ⊂ AZ denote the unique subshift whose language is L(X) = L (cf. Exercise 1.36(c)). Show that X is an infinite minimal subshift (it is called the Morse subshift). Hint: Use Exercises 3.39(a), 3.40 and 3.41(b). 3.43. Toeplitz subshifts. Let A be a finite set. A configuration x ∈ AZ is said to be a Toeplitz configuration if for every n ∈ Z there exists k ≥ 1 such that x(n) = x(n + kr) for all r ∈ Z. In other words, x is a Toeplitz configuration if there exists a partition of Z into arithmetic progressions such that x is constant on each element of the partition. Given a Toeplitz configuration x ∈ AZ and n ∈ Z we set k(x, n) = min{k ≥ 1 : x(n) = x(n + kr) for all r ∈ Z}. One says that a subshift X ⊂ AZ is a Toeplitz subshift if it is the orbit closure of some Toeplitz configuration x ∈ AZ . (a) Let x ∈ AZ be a Toeplitz configuration. Show that the Z-orbit of x is finite if and only if the set {k(x, n) : n ∈ Z} is finite. (b) Show that every Toeplitz configuration x ∈ AZ is almost-periodic. Hint: Use Exercise 3.40. (c) Show that every Toeplitz subshift X ⊂ AZ is minimal.

Exercises

3.44. Let A be a finite set and let X ⊂ AZ be a Toeplitz subshift. (a) Show that X is irreducible. Hint: Use Exercises 3.43(d) and 3.35. (b) Show that X is surjunctive. Hint: Use Exercises 3.43(d) and 3.37.

75

Chapter 4

Amenable Groups

This chapter is devoted to the class of amenable groups. This is a class of groups which plays an important role in many areas of mathematics such as ergodic theory, harmonic analysis, representation theory, dynamical systems, geometric group theory, probability theory and statistics. As residually finite groups, amenable groups generalize finite groups but there are residually finite groups which are not amenable and there are amenable groups which are not residually finite. An amenable group is a group whose subsets admit an invariant finitely additive probability measure (see Sect. 4.4). The class of amenable groups contains in particular all finite groups, all abelian groups and, more generally, all solvable groups (Theorem 4.6.3). It is closed under the operations of taking subgroups, taking quotients, taking extensions, and taking inductive limits (Sect. 4.5). The notion of a Følner net, roughly speaking, a net of almost invariant finite subsets of a group, is introduced in Sect. 4.7. Paradoxical decompositions are defined in Sect. 4.8. The Følner-Tarski theorem (Theorem 4.9.1) asserts that amenability, existence of a Følner net, and non-existence of a paradoxical decomposition are three equivalent conditions for groups. Another characterization of amenable groups is given in Sect. 4.10: a group is amenable if and only if every continuous affine action of the group on a nonempty compact convex subset of a Hausdorff topological vector space admits a fixed point (Corollary 4.10.2).

4.1 Measures and Means Let E be a set. We shall denote by P(E) the set of all subsets of E. Definition 4.1.1. A map μ : P(E) → [0, 1] is called a finitely additive probability measure on E if it satisfies the following properties: (1) μ(E) = 1, (2) μ(A ∪ B) = μ(A) + μ(B) for all A, B ∈ P(E) such that A ∩ B = ∅. T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 4, © Springer-Verlag Berlin Heidelberg 2010

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4 Amenable Groups

Example 4.1.2. Let F be a nonempty finite subset of E. Then the map μF : P(E) → [0, 1] defined by μF (A) =

|A ∩ F | |F |

for all A ⊂ E, is a finitely additive probability measure on E. Proposition 4.1.3. Let μ : P(E) → [0, 1] be a finitely additive probability measure on E. Then one has: (i) μ(∅) = 0; (ii) μ(A ∪ B) = μ(A) + μ(B) − μ(A ∩ B); (iii) μ(A ∪ B) ≤ μ(A) + μ(B); (iv) A ⊂ B ⇒ μ(B \ A) = μ(B) − μ(A); (v) A ⊂ B ⇒ μ(A) ≤ μ(B); for all A, B ∈ P(E). Proof. We have μ(E) = μ(E ∪ ∅) = μ(E) + μ(∅), which implies (i). For all A, B ∈ P(E), we have μ(A ∪ B) = μ(A) + μ(B \ A) and μ(B) = μ(A ∩ B) + μ(B \ A), which gives (ii). Since μ(A ∩ B) ≥ 0, equality (ii) implies (iii). If A ⊂ B, we have μ(B) = μ(B \ A) + μ(A), which implies (iv). Finally, inequality (v) follows from (iv) since μ(B \ A) ≥ 0.

Consider now the real vector space ∞ (E) consisting of all bounded functions x : E → R. Recall that ∞ (E) is a Banach space for the norm · ∞ defined by x ∞ = sup |x(a)|. a∈E



We equip  (E) with the partial ordering ≤ given by x ≤ y ⇐⇒ (x(a) ≤ y(a) for all a ∈ E). For each λ ∈ R, we shall also denote by λ the element of ∞ (E) which is identically equal to λ on E. Definition 4.1.4. A mean on E is a linear map m : ∞ (E) → R such that: (1) m(1) = 1, (2) x ≥ 0 ⇒ m(x) ≥ 0 for all x ∈ ∞ (E). Example 4.1.5. Let S be a countable (finite or infinite) subset of E and let f : S → R such that: (C1) f(s) > 0 for all s ∈ S; (C2) s∈S f (s) = 1.

4.1 Measures and Means

79

Then the map mf : ∞ (E) → R defined by  f (s)x(s) mf (x) = s∈S

is a mean on E. One says that a mean m on E has finite (resp. countable) support if there exists a finite (resp. countable) subset S ⊂ E and a map f : S → R satisfying conditions (C1) and (C2) above such that m = mf . Proposition 4.1.6. Let m : ∞ (E) → R be a mean on E. Then one has: (i) m(λ) = λ, (ii) x ≤ y ⇒ m(x) ≤ m(y), (iii) inf E x ≤ m(x) ≤ supE x, (iv) |m(x)| ≤ x ∞ , for all λ ∈ R and x, y ∈ ∞ (E). Proof. (i) By linearity, we have m(λ) = λm(1) = λ. (ii) If x ≤ y then m(y) − m(x) = m(y − x) ≥ 0 and thus m(x) ≤ m(y). (iii) We have inf E x ≤ x ≤ supE x and therefore inf E x ≤ m(x) ≤ supE x by applying (i) and (ii). (iv) From (iii) we get − x ∞ ≤ m(x) ≤ x ∞ , that is, |m(x)| ≤ x ∞ .

Consider the topological dual of ∞ (E), that is, the vector space (∞ (E))∗ consisting of all continuous linear maps u : ∞ (E) → R. Recall that (∞ (E))∗ is a Banach space for the operator norm · defined by (4.1) u = sup |u(x)| x∞ ≤1

for all u ∈ (∞ (E))∗ . Proposition 4.1.7. Let m : ∞ (E) → R be a mean on E. Then m ∈ (∞ (E))∗ and m = 1. Proof. By definition m is linear. Inequality (iv) in Proposition 4.1.6 shows that m is continuous and satisfies m ≤ 1. Since m(1) = 1, we have m = 1.

Let PM(E) (resp. M(E)) denote the set of all finitely additive probability measures (resp. of all means) on E. We are going to show that there is a natural bijection between the sets PM(E) and M(E). This natural bijection, which is analogous to the Riesz representation in measure theory, may be used to view finitely additive measures on E as points in the dual space (∞ (E))∗ where classical techniques of functional analysis may be applied. For each subset A ⊂ E, we denote by χA the characteristic map of A, that is, the map χA : E → R defined by χA (x) = 1 if x ∈ A and χA (x) = 0 otherwise.

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Let m ∈ M(E). Consider the map m  : P(E) → R given by m(A)  = m(χA ).  ∈ [0, 1] by Proposition 4.1.6(ii). Observe that 0 ≤ χA ≤ 1 and therefore m(A) We have m(E)  = m(χE ) = m(1) = 1. On the other hand, if A and B are disjoint subsets of E, we have χA∪B = χA + χB and thus m(A  ∪ B) = m(A)  + m(B)  by linearity of m. It follows that m  ∈ PM(E). Theorem 4.1.8. The map Φ : M(E) → PM(E) defined by Φ(m) = m  is bijective. The proof will be divided in several steps. Denote by E(E) the set of all maps x : E → R which take only finitely many values. Observe that E(E) is the vector subspace of ∞ (E) spanned by the set of characteristic maps {χA : A ⊂ E}. Lemma 4.1.9. The vector subspace E(E) is dense in ∞ (E). Proof. Let x ∈ ∞ (E) and ε > 0. Set α = inf E x and β = supE x. Choose an integer n ≥ 1 such that (β − α)/n < ε and set λi = α + i(β − α)/n for i = 1, 2, . . . , n. Consider the map y : E → R defined by y(a) = min{λi : x(a) ≤ λi } for all a ∈ E. The map y take its values in the set {λ1 , . . . , λn }. Thus y ∈ E(E). We have x − y ∞ ≤ (β − α)/n < ε by construction. Consequently,

E(E) is dense in ∞ (E). Let μ ∈ PM(E). Let us set, for all x ∈ E(E),  μ(x) = μ(x−1 (λ))λ.

(4.2)

λ∈R

Observe that there is only a finite number of nonzero terms in the right-hand side of (4.2) since x ∈ E(E) and μ(∅) = 0. Lemma 4.1.10. Let x ∈ E(E). Suppose that there is a finite partition (Ai )i∈I of E such that the restriction of x to Ai is constant equal to αi for each i ∈ I. Then one has  μ(x) = μ(Ai )αi . i∈I

Proof. Since μ is finitely additive, we have       μ(Ai )αi = μ(Ai ) λ = μ(x−1 (λ))λ = μ(x). i∈I

λ∈R

αi =λ

λ∈R



4.1 Measures and Means

81

Lemma 4.1.11. The map μ : E(E) → R is linear and continuous. Proof. Let x, y ∈ E(E) and ξ, η ∈ R. Denote by V (resp. W ) the set of values taken by x (resp. y). The subsets x−1 (α) ∩ y −1 (β), (α, β) ∈ V × W , form a finite partition of E. By applying Lemma 4.1.10, we get  μ(ξx + ηy) = μ(x−1 (α) ∩ y −1 (β))(ξα + ηβ) (α,β)∈V ×W







= ξ⎝

μ(x−1 (α) ∩ y −1 (β))α⎠

(α,β)∈V ×W



+ ⎝η



⎞ μ(x−1 (α) ∩ y −1 (β))β ⎠

(α,β)∈V ×W

= ξμ(x) + ημ(y). Consequently, μ is linear. Formula (4.2) implies inf E x ≤ μ(x) ≤ supE x since  μ(x−1 (λ)) = μ(E) = 1. λ∈R

It follows that |μ(x)| ≤ x ∞ for all x ∈ E(E). This shows that the linear

map μ is continuous. Lemma 4.1.12. Let X be a normed vector space and let Y be a dense vector subspace of X. Suppose that ϕ : Y → R is a continuous linear map. Then there exists a continuous linear map ϕ : X → R extending ϕ (that is, such that ϕ| Y = ϕ). Proof. Let x ∈ X. Since Y is dense in X, we can find a sequence (yn )n≥0 of points of Y which converges to x. The sequence (ϕ(yn )) is a Cauchy sequence since |ϕ(yp ) − ϕ(yq )| = |ϕ(yp − yq )| ≤ ϕ yp − yq for all p, q ≥ 0. As R is complete, this sequence converges. Set ϕ(x) = lim ϕ(yn ). If (yn ) is another sequence of points in Y converging to x, then |ϕ(yn ) − ϕ(yn )| ≤ ϕ yn − yn . Thus we have lim ϕ(yn ) = lim ϕ(yn ). This shows that ϕ(x) does not depend of the choice of the sequence (yn ). The linearity of ϕ gives the linearity of ϕ by taking limits. We also get ϕ(x) ≤ ϕ x for all x ∈ X by taking limits. This shows that the linear map ϕ is continuous. If x ∈ Y , we have ϕ(x) = ϕ(x) since we can take as extends ϕ.

(yn ) the constant sequence yn = x in this case. Thus ϕ Proof of Theorem 4.1.8. Let μ ∈ PM(E). By Lemma 4.1.9, Lemma 4.1.11 and Lemma 4.1.12, the map μ : E(E) → R can be extended to a continuous linear map μ : ∞ (E) → R. We have

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μ (1) = μ(1) = μ(E) = 1. If y ∈ E(E) and y ≥ 0, then μ (y) = μ(y) ≥ 0 by (4.2). Consider now an element x ∈ ∞ (E) such that x ≥ 0. Let (yn )n≥0 be a sequence of elements of E(E) converging to x. Let us set zn = |yn |. Since zn ∈ E(E) and zn ≥ 0, we have μ (zn ) ≥ 0. On the other hand, the sequence (zn ) converges to x since, by the triangle inequality, x − zn ∞ ≤ x − yn ∞ . It follows that μ (x) = lim μ(zn ) ≥ 0. n→∞

Thus μ ∈ M(E). For every subset A ⊂ E, we have μ (χA ) = μ(χA ) = μ(A). Therefore Φ( μ) = μ. This proves that Φ is surjective. It remains to show that Φ is injective. To see this, consider m1 , m2 ∈ M(E) such that Φ(m1 ) = Φ(m2 ). This means that m1 (χA ) = m2 (χA ) for every subset A ⊂ E. By linearity, this implies that m1 and m2 coincide on E(E). Since E(E) is dense in ∞ (E) by Lemma 4.1.9, we deduce that m1 = m2 by

continuity of m1 and m2 . This shows that Φ is injective.

4.2 Properties of the Set of Means Let E be a set. The topology on (∞ (E))∗ associated with the operator norm · defined in (4.1) is called the strong topology on (∞ (E))∗ . Another topology that is commonly used is the weak-∗ topology on (∞ (E))∗ . Recall that the weak-∗ topology on (∞ (E))∗ is by definition the smallest topology for which the evaluation map ψx : (∞ (E))∗ → R u → u(x) is continuous for each x ∈ ∞ (E) (see Sect. F.2). By Proposition 4.1.7, the set M(E) is contained in the unit sphere {u : u = 1} ⊂ (∞ (E))∗ . The following result will play an important role in the sequel. Theorem 4.2.1. The set M(E) is a convex compact subset of (∞ (E))∗ with respect to the weak-∗ topology.

4.3 Measures and Means on Groups

83

Proof. Let m1 , m2 ∈ M(E) and t ∈ [0, 1]. Then (tm1 + (1 − t)m2 )(1) = tm1 (1) + (1 − t)m2 (1) = t + (1 − t) = 1 and (tm1 + (1 − t)m2 )(x) = tm1 (x) + (1 − t)m2 (x) ≥ 0 for all x ∈ ∞ (E) such that x ≥ 0. Thus tm1 + (1 − t)m2 ∈ M(E). This shows that M(E) is convex. Equip now (∞ (E))∗ with the weak-∗ topology and suppose that (mi )i∈I is a net in M(E) converging to u ∈ (∞ (E))∗ . Then, for every i ∈ I, we have ψ1 (mi ) = mi (1) = 1 and ψx (mi ) = mi (x) ≥ 0 for all x ∈ ∞ (E) such that x ≥ 0. By taking limits, we get u(1) = ψ1 (u) = 1 and u(x) = ψx (u) ≥ 0 for all x ∈ ∞ (E) such that x ≥ 0. Thus u ∈ M(E). This shows that M(E) is closed in (∞ (E))∗ (Proposition A.2.1). As M(E) is contained in the unit ball of (∞ (E))∗ , which is compact for the weak-∗ topology by the Banach-Alaoglu Theorem (Theorem F.3.1), it follows that M(E) is compact.

4.3 Measures and Means on Groups In this section, we shall see that the set of finitely additive measures and the set of means carry additional structure when the underlying set is a group. Indeed, in this case, the group naturally acts on both sets. Moreover, there are involutions coming from the operation of taking inverses in the group. More precisely, let G be a group. The group G naturally acts on the left and on the right on each of the sets PM(G) and M(G) in the following way. Firstly, for μ ∈ PM(G) and g ∈ G, we define the maps gμ : P(G) → [0, 1] and μg : P(G) → [0, 1] by gμ(A) = μ(g −1 A)

and

μg(A) = μ(Ag −1 )

for all A ∈ P(G). One clearly has gμ ∈ PM(G) and μg ∈ PM(G). Moreover, it is straightforward to check that the map (g, μ) → gμ (resp. (μ, g) → μg) defines a left (resp. right) action of G on PM(G). Note that these two actions commute in the sense that g(μh) = (gμ)h for all g, h ∈ G and μ ∈ PM(G). On the other hand, recall that we introduced in Sect. 1.1 a left action of G on RG (the G-shift) by defining, for all g ∈ G and x ∈ RG , the element gx ∈ RG by gx(g ) = x(g −1 g ) for all g ∈ G. Similarly, we make G act on the right on RG by defining the element xg ∈ RG by xg(g ) = x(g g −1 ) for all g ∈ G. These two actions of G on RG are linear and commute. Moreover, the vector subspace ∞ (G) ⊂ RG is left invariant by both actions. Observe that gx ∞ = xg ∞ = x ∞ for all g ∈ G and x ∈ ∞ (G). Thus the left (resp. right) action of G on ∞ (G) is isometric (and therefore continuous).

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4 Amenable Groups

By duality, we also get a left and a right action of G on (∞ (G))∗ . More precisely, for g ∈ G and u ∈ (∞ (G))∗ , we define the elements gu and ug by gu(x) = u(g −1 x)

and

ug(x) = u(xg −1 ),

for all x ∈ ∞ (G). We have gu ∈ (∞ (G))∗ and ug ∈ (∞ (G))∗ since the left and right actions of G on ∞ (G) are linear and continuous. Note that the set M(G) is invariant under both actions of G on (∞ (G))∗ . Proposition 4.3.1. The left (resp. right) action of G on M(G) is affine and continuous with respect to the weak-∗ topology on M(G). Proof. It is clear that the left (resp. right) action of G on (∞ (G))∗ is linear. On the other hand, these actions are continuous if we equip (∞ (G))∗ with the weak-∗ topology. Indeed, if we fix g ∈ G, the map u → gu is continuous on (∞ (G))∗ since, for each x ∈ ∞ (G), the map u → gu(x) is the evaluation map at g −1 x, which is continuous by definition of the weak-∗ topology. Similarly, the map u → ug is continuous. Since M(G) is a convex subset of (∞ (G))∗ , we deduce that the restrictions of both actions to M(G) are affine and continuous.

Given μ ∈ PM(G), we define the map μ∗ : P(G) → [0, 1] by μ∗ (A) = μ(A−1 ) for all A ∈ P(G). It is clear that μ∗ ∈ PM(G) and that the map μ → μ∗ is an involution of PM(G). For x ∈ ∞ (G), define x∗ ∈ ∞ (G) by x∗ (g) = x(g −1 ) for all g ∈ G. The map x → x∗ is an isometric involution of ∞ (G). By duality, it gives an isometric involution u → u∗ of (∞ (G))∗ defined by u∗ (x) = u(x∗ )

for all x ∈ ∞ (G).

Note that m∗ ∈ M(G) for all m ∈ M(G). Proposition 4.3.2. Let g ∈ G, x ∈ ∞ (G), μ ∈ PM(G) and u ∈ (∞ (G))∗ . Then one has (i) (gx)∗ = x∗ g −1 ; (ii) (xg)∗ = g −1 x∗ ; (iii) (gμ)∗ = μ∗ g −1 ; (iv) (μg)∗ = g −1 μ∗ ; (v) (gu)∗ = u∗ g −1 ; (vi) (ug)∗ = g −1 u∗ .

4.4 Definition of Amenability

85

Proof. For every h ∈ G, we have (gx)∗ (h) = gx(h−1 ) = x(g −1 h−1 ) = x∗ (hg) = x∗ g −1 (h), which gives (gx)∗ = x∗ g −1 . The proofs of the other properties are similar.

Proposition 4.3.3. Let g ∈ G and m ∈ M(G). Then one has: (i) gm = g m;  (ii) mg = mg;  ∗ = m  ∗. (iii) m We need the following result. Lemma 4.3.4. Let g ∈ G and A ⊂ G. Then one has: (i) (χA )∗ = χA−1 ; (ii) gχA = χgA . Proof. (i) For h ∈ G one has (χA )∗ (h) = χA (h−1 ) = 1 ⇔ h−1 ∈ A ⇔ h ∈ A−1 ⇔ χA−1 (h) = 1. This shows that (χA )∗ = χA−1 . (ii) For h ∈ G one has (gχA )(h) = χA (g −1 h) = 1 ⇔ g −1 h ∈ A ⇔ h ∈ gA ⇔ χgA (h) = 1. This shows that gχA = χgA .

Proof of Proposition 4.3.3. For every A ∈ P(G), we have, using Lemma 4.3.4(ii), gm(A) = gm(χA ) = m(g −1 χA ) = m(χg−1 A ) = m(g  −1 A) = g m(A).  Thus gm = g m.  Similarly, we get mg = mg.  On the other hand, for every A ∈ P(G), using Lemma 4.3.4(i) one obtains ∗ (A)  −1 ) = m(χA−1 ) = m((χA )∗ ) = m∗ (χA ) = m m  ∗ (A) = m(A ∗ . and this shows that m ∗ = m



Remark 4.3.5. Properties (i) and (ii) in Proposition 4.3.3 say that the bijective map Φ : M(G) → PM(G) given by m → m  (see Theorem 4.1.8) is bi-equivariant.

4.4 Definition of Amenability Let G be a group. A finitely additive probability measure μ ∈ PM(G) is called left-invariant (resp. right-invariant) if μ is fixed under the left (resp. right) action of G on PM(G), that is, if it satisfies gμ = μ (resp. μg = μ) for all g ∈ G. One says that μ is bi-invariant if μ is both left and right-invariant.

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4 Amenable Groups

Similarly, a mean m ∈ M(G) is called left-invariant (resp. right-invariant) if m is fixed under the left (resp. right) action of G on M(G). One says that m is bi-invariant if m is both left and right invariant. Proposition 4.4.1. Let μ ∈ PM(G). Then μ is left-invariant (resp. rightinvariant) if and only if μ∗ is right-invariant (resp. left-invariant). Proof. This immediately follows from Proposition 4.3.2(ii).



Proposition 4.4.2. Let m ∈ M(G). Then m is left-invariant (resp. rightinvariant) if and only if m∗ is right-invariant (resp. left-invariant). Proof. This immediately follows from Proposition 4.3.2(iii).



Proposition 4.4.3. Let m ∈ M(G). Then m is left-invariant (resp. rightinvariant, resp. bi-invariant) if and only if the associated finitely additive probability measure m  ∈ PM(G) is left-invariant (resp. right-invariant, resp. bi-invariant). Proof. This immediately follows from assertions (i) and (ii) of Proposition 4.3.3.

Proposition 4.4.4. Let G be a group. Then the following conditions are equivalent: (a) there exists a left-invariant finitely additive probability measure μ : P(G) → [0, 1] on G; (b) there exists a right-invariant finitely additive probability measure μ : P(G) → [0, 1] on G; (c) there exists a bi-invariant finitely additive probability measure μ : P(G) → [0, 1] on G; (d) there exists a left-invariant mean m : ∞ (G) → R on G; (e) there exists a right-invariant mean m : ∞ (G) → R on G; (f) there exists a bi-invariant mean m : ∞ (G) → R on G. Proof. Equivalences (a) ⇔ (d), (b) ⇔ (e) and (c) ⇔ (f) immediately follow from Proposition 4.4.3. The equivalence between (a) and (b) follows from Proposition 4.4.1. Implication (f) ⇒ (d) is trivial. Thus it suffices to show (d) ⇒ (f) to conclude. Suppose that there exists a left-invariant mean m : ∞ (G) → R. For each x ∈ ∞ (G), define the map x : G → R by x (g) = m(xg) for all g ∈ G. By Proposition 4.1.6(iv), we have | x(g)| = |m(xg)| ≤ xg ∞ = x ∞ . Therefore x ∈ ∞ (G) for all x ∈ ∞ (G). Consider now the map M : ∞ (G) → R defined by

4.4 Definition of Amenability

87

M (x) = m( x) for all x ∈ ∞ (G). Clearly M is a mean on G. Let h ∈ G and x ∈ ∞ (G). For all g ∈ G, we have

hx(g) = m(hxg) = m(xg) = x (g),

=x since m is left-invariant. We deduce that hx . This implies

= m( M (hx) = m(hx) x) = M (x). Consequently M is left-invariant. On the other hand, for all g ∈ G, we have

xh(g) = m((xh)g) = m(x(hg)) = x (hg) = h−1 x (g).

= h−1 x It follows that xh . Therefore, by using again the fact that m is left-invariant, we get

= m(h−1 x M (xh) = m(xh) ) = m( x) = M (x). Therefore the mean M is also right-invariant. This shows that (d) implies (f).

Definition 4.4.5. A group G is called amenable if it satisfies one of the equivalent conditions of Proposition 4.4.4. Proposition 4.4.6. Every finite group is amenable. Proof. If G is a finite group, then the map μ : P(G) → [0, 1] defined by μ(A) =

|A| |G|

for all A ∈ P(G)

is a bi-invariant finitely additive probability measure on G. Theorem 4.4.7. The free group on two generators F2 is not amenable.

Proof. Suppose that there exists a left-invariant finitely additive probability measure μ : P(F2 ) → [0, 1]. Denote by a and b the canonical generators of F2 . Let A ⊂ F2 be the set of elements of F2 whose reduced form begins by a nonzero (positive or negative) power of a. We have F2 = A ∪ aA and hence μ(F2 ) ≤ μ(A) + μ(aA) = 2μ(A). Since μ(F2 ) = 1, this implies μ(A) ≥

1 . 2

(4.3)

On the other hand, the subsets A, bA and b2 A are pairwise disjoint. Thus one has μ(F2 ) ≥ μ(A) + μ(bA) + μ(b2 A) = 3μ(A). This gives

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4 Amenable Groups

μ(A) ≤

1 3

contradicting (4.3).

4.5 Stability Properties of Amenable Groups Proposition 4.5.1. Every subgroup of an amenable group is amenable. Proof. Let G be an amenable group and let H be a subgroup of G. Let μ : P(G) → [0, 1] be a left-invariant finitely additive probability measure on G. Choose a complete set of representatives of the right cosets of G modulo H, that is, a subset R ⊂ G such that, for each g ∈ G, there exists a unique element (h, r) ∈ H × R satisfying g = hr. Let us check that the map μ : P(H) → [0, 1] defined by    Ar for all A ∈ P(H) μ (A) = μ r∈R

is a left-invariant finitely additive probability measure on H. Firstly, we have    Hr = μ(G) = 1. μ (H) = μ r∈R

On the other hand, if A and B are disjoint subsets of H, then    μ (A ∪ B) = μ (A ∪ B)r r∈R

 =μ













Ar

r∈R

Ar







 +μ

 r∈R

=μ (A) + μ (B),  r∈R

are disjoint.

Ar

and

Br

r∈R

r∈R

since the sets



 r∈R

Br

 Br

4.5 Stability Properties of Amenable Groups

89

Finally, for all h ∈ H and A ∈ P(H), we have          μ (hA) = μ hAr = μ h Ar = μ Ar = μ (A). r∈R

r∈R

r∈R



This shows that H is amenable. Combining Proposition 4.5.1 with Theorem 4.4.7, we get:

Corollary 4.5.2. If G is a group containing a subgroup isomorphic to the free group on two generators F2 , then G is not amenable.

Examples 4.5.3. (a) Let X be a set having at least two elements. Then the free group F (X) based on X is not amenable. Indeed, the subgroup of F (X) generated by two distinct elements a, b ∈ X is isomorphic to F2 . (b) The group SL(n, Z) is not amenable for n ≥ 2. Indeed, as we haveseen 12 in Lemma  1 0  2.3.2, the subgroup of SL(2, Z) generated by the matrices 0 1 and 2 1 is isomorphic to F2 . Proposition 4.5.4. Every quotient of an amenable group is amenable. Proof. Let G be an amenable group and let H be a normal subgroup of G. Let ρ : G → G/H denote the canonical homomorphism. Consider a left-invariant finitely additive probability measure μ : P(G) → [0, 1] on G. Let us show that the map μ  : P(G/H) → [0, 1] defined by μ (A) = μ(ρ−1 (A))

for all A ∈ P(G/H)

is a left-invariant finitely additive probability measure on G/H. Firstly, we have μ (G/H) = μ((ρ−1 (G/H)) = μ(G) = 1. On the other hand, if A and B are disjoint subsets of G/H, then μ (A ∪ B) = μ(ρ−1 (A ∪ B)) = μ(ρ−1 (A) ∪ ρ−1 (B)) = μ(ρ−1 (A)) + μ(ρ−1 (B)) (A ∪ B) = μ (A) + μ (B). since ρ−1 (A) ∩ ρ−1 (B) = ∅. Thus we have μ Finally, if g ∈ G and A ⊂ G/H, we have (A). μ (ρ(g)A) = μ(ρ−1 (ρ(g)A)) = μ(gρ−1 (A)) = μ(ρ−1 (A)) = μ As ρ is surjective, it follows that μ  is left-invariant with respect to G/H. This shows that G/H is amenable.

In the next proposition we show that the class of amenable groups is closed under the operation of taking extensions of amenable groups by amenable groups.

90

4 Amenable Groups

Proposition 4.5.5. Let G be a group and let H be a normal subgroup of G. Suppose that the groups H and G/H are amenable. Then the group G is amenable. Proof. Since H is amenable, we can find a H-left-invariant mean m0 : ∞ (H) → : G/H → R defined by R. Let x ∈ ∞ (G). Consider the map x x (g) = m0 ((g −1 x)|H )

for all g ∈ G,

is where g = gH = Hg ∈ G/H denote the class of g modulo H. The map x well defined. Indeed, if g1 and g2 are elements of G such that g1 = g2 , then g2 = g1 h for some h ∈ H, and hence m0 ((g2−1 x)|H ) = m0 ((h−1 g1−1 x)|H ) = m0 (h−1 (g1−1 x)|H ) = m0 ((g1−1 x)|H ) ∈ ∞ (G/H) since, by Proposince m0 is H-left-invariant. Observe also that x sition 4.1.6(iv), | x(g)| = |m0 ((g −1 x)|H )| ≤ sup |(g −1 x)(h)| ≤ g −1 x ∞ = x ∞ h∈H

for all g ∈ G. As the group G/H is amenable, there exists a G/H-left-invariant mean m1 : ∞ (G/H) → R. Let us set, for each x ∈ ∞ (G), x). m(x) = m1 ( Clearly m is a mean on G. Let us show that it is G-left-invariant. Let g ∈ G and x ∈ ∞ (G). For all g ∈ G, we have g x(g ) = m0 ((g

−1

gx)|H ) = m0 (((g −1 g )−1 x)|H ) = x (g −1 g ) = x (g −1 g )

x(g ). = g x. As m1 is G/H-left-invariant, it follows that Thus g x = g g x) = m1 (g x) = m1 ( x) = m(x), m(gx) = m1 ( which shows that m is G-left-invariant.Therefore G is amenable.



Corollary 4.5.6. Suppose that G1 and G2 are amenable groups. Then the group G = G1 × G2 is amenable. Proof. The set H = {(g1 , 1G2 ) : g1 ∈ G1 } is a normal subgroup of G isomor

phic to G1 with quotient G/H isomorphic to G2 . Remark 4.5.7. It immediately follows from the preceding corollary that every direct product of a finite number of amenable groups is amenable. However, a direct product of infinitely many amenable groups is not necessarily amenable. For example, it follows from Theorem 2.3.1 and Corollary 2.2.6

4.5 Stability Properties of Amenable Groups

91

that there exists a (countable) family  of finite, and hence amenable, groups (Gi )i∈I such that the group G = i∈I Gi contains a subgroup isomorphic to F2 . Such a group G is not amenable by Corollary 4.5.2. Corollary 4.5.8. Every virtually amenable group is amenable. Proof. Let G be a virtually amenable group. Let H be an amenable subgroup of finite index of G. By Lemma 2.1.10, the set K = ∩g∈G gHg −1 is a normal subgroup of finite index of G contained in H. The group K is amenable by Proposition 4.5.1. On the other hand the group G/K is finite and hence amenable. Consequently, G is amenable by Proposition 4.5.5.

Our next goal is to show that an inductive limit of amenable groups is amenable. In the proof we shall use the following: Lemma 4.5.9. Let G be a group. Suppose that there is a net (mi )i∈I in M(G) such that, for each g ∈ G, the net (gmi − mi )i∈I converges to 0 in (∞ (G))∗ for the weak-∗ topology. Then G is amenable. Proof. Since M(G) is compact for the weak-∗ topology by Theorem 4.2.1, we may assume, after taking a subnet if necessary, that the net (mi ) converges to a mean m ∈ M(G). Let g ∈ G. Since the left action of G on M(G) is continuous by Proposition 4.3.1, we deduce that, for every x ∈ ∞ (G) the net (gmi − mi )(x) = gmi (x) − mi (x) converges to 0. By taking limits, we get gm(x) − m(x) = 0. Therefore gm = m. This shows that the mean m is left-invariant. Thus G is amenable.

Proposition 4.5.10. Every group which is the limit of an inductive system of amenable groups is amenable. Proof. Let (Gi )i∈I be an inductive system of amenable groups and set G = lim Gi . Consider the family (Hi )i∈I of subgroups G defined by Hi = hi (Gi ), −→ where hi : Gi → G is the canonical homomorphism. As Hi is amenable by Proposition 4.5.4, we can find, for each i ∈ I, a Hi -left-invariant mean m

i : ∞ (Hi ) → R. Consider the family mi : ∞ (G) → R of means on G defined by

i (x|Hi ) for all x ∈ ∞ (G). mi (x) = m Let g ∈ G. Since G = lim Gi , there exists i0 (g) ∈ I such that g ∈ Hi for all −→ i ≥ i0 (g). For x ∈ ∞ (G) and i ≥ i0 (g), we have

i (x|Hi ) − m

i (x|Hi ) = 0, (gmi − mi )(x) = gmi (x) − mi (x) = g m since m

i is Hi -left-invariant. Thus gmi −mi = 0 for all i ≥ i0 (g). By applying Lemma 4.5.9, we deduce that G is amenable.

Every group is the inductive limit of its finitely generated subgroups. Therefore we have:

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4 Amenable Groups

Corollary 4.5.11. Every locally amenable group is amenable.



Since every finite group is amenable, we obtain in particular the following: Corollary 4.5.12. Every locally finite group is amenable.



Example 4.5.13. Let X be a set. Then the group Sym0 (X) consisting of all permutations of X with finite support is locally finite (see Example 3.2.4). Consequently, the group Sym0 (X) is amenable. The groups Sym0 (X), where X is a infinite set, are our first examples of infinite amenable groups. Recall from Lemma 2.6.3 that Sym0 (X) is not residually finite whenever X is infinite. Corollary 4.5.14.  Let (Gi )i∈I be a family of amenable groups. Then their direct sum G = i∈I Gi is amenable. Proof. Every finitely generated subgroup of G is a subgroup of some finite product of the groups Gi and hence amenable by Corollary 4.5.6 and Proposition 4.5.1. Thus G is locally amenable and therefore amenable by Corollary 4.5.11.

4.6 Solvable Groups Theorem 4.6.1. Every abelian group is amenable. Proof. Let G be an abelian group. Equip (∞ (G))∗ with the weak-∗ topology. By Theorem 4.2.1, the set M(G) is a nonempty convex compact subset of (∞ (G))∗ . On the other hand, it follows from Proposition 4.3.1 that the action of G on M(G) is affine and continuous (note that the left and the right actions coincide since G is Abelian). By applying the Markov-Kakutani fixed-point Theorem (Theorem G.1.1), we deduce that G has at least one fixed point in M(G). Such a fixed point is clearly a bi-invariant mean on G. This shows that G is amenable.

Let G be a group. Recall the following definitions. The commutator of two elements h and k in G is the element [h, k] ∈ G defined by [h, k] = hkh−1 k −1 . If H and K are subgroups of G, we denote by [H, K] the subgroup of G generated by all commutators [h, k], where h ∈ H and k ∈ K. Note that [H, K] ⊂ K if K is normal in G. Note also that [H, K] is normal in G if H and K are both normal in G. The subgroup D(G) = [G, G] is called the derived subgroup, or commutator subgroup, of G. The subgroup D(G) is normal in G and the quotient group G/D(G) is abelian. Observe that G is abelian if and only if [G, G] = {1G }. A group is called metabelian if its derived subgroup is abelian. The derived series of a group G is the sequence (Di (G))i≥0 of subgroups of G inductively defined by D0 (G) = G and Di+1 (G) = D(Di (G)) for all i ≥ 0. One has

4.6 Solvable Groups

93

G = D0 (G) ⊃ D1 (G) ⊃ D2 (G) ⊃ . . . with Di+1 (G) normal in Di (G) and Di (G)/Di+1 (G) abelian for all i ≥ 0. The group G is said to be solvable if there is an integer i ≥ 0 such that Di (G) = {1G }. The smallest integer i ≥ 0 such that Di (G) = {1G } is then called the solvability degree of G. Examples 4.6.2. (a) A group is solvable of degree 0 if and only if it is reduced to the identity element. (b) The solvable groups of degree 1 are the nontrivial abelian groups. (c) The solvable groups of degree 2 are the nonabelian metabelian groups. (d) Let K be a field. The affine group over K is the subgroup Aff(K) of Sym(K) consisting of all permutations of K which are of the form x → ax + b, where a, b ∈ K and a = 0. The group D1 (Aff(K)) is the group of translations x → x + b, b ∈ K. Since D1 (Aff(K)) is abelian, we have Di (Aff(K)) = {1Aff(K) } for all i ≥ 2. Consequently, the group Aff(K) is solvable of degree 2. + (e) The alternating groups Sym+ 2 and Sym3 are abelian and hence solvable + of degree 1. The group Sym4 is solvable of degree 2. For n ≥ 5, the alternating + + i group Sym+ n is simple (see Remark C.4.4) and therefore D (Symn ) = Symn + for all i ≥ 0. Thus Symn is not solvable for n ≥ 5. (f) The symmetric group Symn is solvable of degree 1 for n = 2, solvable of degree 2 for n = 3, solvable of degree 3 for n = 4, and not solvable for n ≥ 5. Theorem 4.6.3. Every solvable group is amenable. Proof. We proceed by induction on the solvability degree i of the group. For i = 0, the group is reduced to the identity element and there is nothing to prove. Suppose now that the statement is true for solvable groups of degree i for some i ≥ 0. Let G be a solvable group of degree i + 1. Then its derived subgroup D(G) is solvable of degree i and hence amenable by our induction hypothesis. As G/D(G) is abelian and therefore amenable by Theorem 4.6.1, we deduce that G is amenable by applying Proposition 4.5.5.

Remark 4.6.4. The alternating group Sym+ 5 is amenable since it is finite. However, as mentioned above, Sym+ 5 is not solvable. An example of an infinite amenable group which is not solvable is provided by the group Z × Sym+ 5. Let G be a group. The lower central series of G is the sequence (C i (G))i≥0 of subgroups of G defined by C 0 (G) = G and C i+1 (G) = [C i (G), G] for all i ≥ 0. An easy induction shows that C i (G) is normal in G and that C i+1 (G) ⊂ C i (G) for all i. The group G is said to be nilpotent if there is an integer i ≥ 0 such that C i (G) = {1G }. The smallest integer i ≥ 0 such that C i (G) = {1G } is then called the nilpotency degree of G.

94

4 Amenable Groups

Example 4.6.5. Let R be a nontrivial commutative ring. The Heisenberg group with coefficients in R is the subgroup HR of GL3 (R) consisting of all matrices of the form ⎛ ⎞ 1yz M (x, y, z) = ⎝0 1 x⎠ (x, y, z ∈ R). 001 One easily checks that the center Z(HR ) of HR consists of all matrices of the form M (0, 0, z), z ∈ R, and that Z(HR ) is isomorphic to the additive group (R, +). Moreover, one has D(HR ) = Z(HR ). It follows that HR is nilpotent of degree 2. Proposition 4.6.6. Every nilpotent group is solvable. Proof. An easy induction yields Di (G) ⊂ C i (G) for all i ≥ 0.



Remark 4.6.7. We have seen in Example 4.6.2(d) that the affine group Aff(K) over a field K is solvable. However, the group G = Aff(K) is not nilpotent. Indeed, the lower central series satisfies C i (G) = D(G) = {1G } for all i ≥ 1. From Theorem 4.6.3 and Proposition 4.6.6, we get: Corollary 4.6.8. Every nilpotent group is amenable.



4.7 The Følner Conditions Proposition 4.7.1. Let G be a group. Then the following conditions are equivalent: (a) for every finite subset K ⊂ G and every real number ε > 0, there exists a nonempty finite subset F ⊂ G such that |F \ kF | 0, there exists a nonempty finite subset F ⊂ G such that |F \ F k| 0. We equip J with the partial ordering ≤ defined by (K, ε) ≤ (K , ε ) ⇔ (K ⊂ K and ε ≥ ε ). Note that (J, ≤) is a lattice, that is, a partially ordered set in which any two elements admit a supremum (also called a join) and an infimum (also called a meet). Indeed, one has sup{(K, ε), (K , ε )} = (K ∪ K , min(ε, ε )) and inf{(K, ε), (K , ε )} = (K ∩ K , max(ε, ε )). In particular, J is a directed set. By (a), for each j = (K, ε) ∈ J, there exists a nonempty finite subset Fj ⊂ G such that |Fj \ kFj | 0. Set j0 = ({g}, ε0 ). If j = (K, ε) satisfies j ≥ j0 , then g ∈ K and ε ≤ ε0 . Thus we have |Fj \ gFj | < ε0 , |Fj |

(4.10)

for all j ≥ j0 by (4.9). This shows that the net (Fj )j∈J satisfies (4.5). Consequently, (a) implies (b). Finally, let us show (b) ⇒ (a). Suppose (b). Let K ⊂ G be a finite subset and let ε > 0. By (4.5), for every k ∈ K there exist j(k) ∈ J such that |Fj \ kFj | 0 such that for every nonempty finite subset F ⊂ G one has |F \ k0 F | ≥ ε0 |F |

(4.16)

for some k0 = k0 (F ) ∈ K0 . Consider the set K1 = K0 ∪ {1G }. Let F be a nonempty finite subset of G. Observe that K1 F ⊃ F and K1 F \F = K0 F \F . Thus, we have |K1 F | − |F | = |K1 F \ F | = |K0 F \ F | ≥ |k0 F \ F | = |F \ k0 F | (since |F | = |k0 F |) ≥ ε0 |F | (by (4.16)), which gives |K1 F | ≥ (1 + ε0 )|F |.

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4 Amenable Groups

Choose n0 ∈ N such that (1 + ε0 )n0 ≥ 2 and set K = K1n0 . Then, we have |KF | = |K1n0 F | ≥ (1 + ε0 )|K1n0 −1 F | ≥ · · · ≥ (1 + ε0 )n0 |F | so that |KF | ≥ 2|F | for every finite subset F ⊂ G. This shows that (b) implies (c). (c) ⇒ (e). Suppose that G satisfies condition (c), that is, there exists a finite subset K ⊂ G such that |KF | ≥ 2|F | for every finite subset F ⊂ G.

(4.17)

Consider the bipartite graph GK (G) = (G, G, E) (see Appendix H), where the set E ⊂ G × G of edges consists of all the pairs (g, h) such that g ∈ G and h ∈ Kg. We claim that GK (G) satisfies the Hall 2-harem conditions. Indeed, if F is a finite subset of G, then, using the terminology for bipartite graphs introduced in Sect. H.1, the right and left neighborhoods of F in GK (G) are the sets NR (F ) = KF and NL (F ) = K −1 F respectively. Therefore, we have |NR (F )| = |KF | ≥ 2|F |, by applying (4.17). On the other hand, if k ∈ K, then NL (F ) ⊃ k−1 F so that 1 |NL (F )| ≥ |k−1 F | = |F | ≥ |F |. 2 This proves our claim. Thus, by virtue of the Hall harem Theorem (Theorem H.4.2), we deduce the existence of a perfect (1, 2)-matching M for GK (G). In other words, there exists a 2-to-one surjective map ϕ : G → G such that (ϕ(g), g) ∈ E, that is, g(ϕ(g))−1 ∈ K for all g ∈ G. This shows that (c) implies (e). (e) ⇒ (g). Suppose (e), that is, there exist a 2-to-one surjective map ϕ : G → G and a finite set K ⊂ G such that g(ϕ(g))−1 ∈ K for all g ∈ G.

(4.18)

By the axiom of choice, we can find maps ψ1 , ψ2 : G → G such that, for every g ∈ G, the elements ψ1 (g) and ψ2 (g) are the two preimages of g for ϕ. Observe that θ1 (g) = ψ1 (g)g −1 and θ2 (g) = ψ2 (g)g −1 belong to K for every g ∈ G by (4.18). For each k ∈ K, define Ak and Bk by Ak = {g ∈ G : θ1 (g) = k} We have G=

 k∈K

and Bk = {g ∈ G : θ2 (g) = k}.

Ak =

 k∈K

Bk .

(4.19)

4.10 The Fixed Point Property

103

ψ1 (G) =  On the other hand, observe that if g ∈Ak then ψ1 (g) = kg. Thus ψ2 (G), k∈K kAk . Similarly, we have ψ2 (G) = k∈K kBk . As G = ψ1 (G) we deduce that        (4.20) kAk kBk . G= k∈K

k∈K

Combining together (4.19) and (4.20), ,we deduce that (K, (Ak )k∈K , (Bk )k∈K ) is a left paradoxical decomposition for G. This shows that (e) implies (g). (g) ⇒ (a). Suppose that G admits a left paradoxical decomposition (K, (Ak )k∈K , (Bk )k∈K ). If μ : P(G) → [0, 1] is a left invariant finitely additive probability measure on G, then (4.11) gives 1 = μ(G) 



=μ =



 kAk

k∈K

μ(kAk ) +

k∈K

=









 kBk

k∈K

μ(kBk )

k∈K

μ(Ak ) +

k∈K

= μ(







μ(Bk )

k∈K

Ak ) + μ(

k∈K



Bk )

k∈K

= μ(G) + μ(G) = 2, which is clearly absurd. Therefore G is not amenable. This shows that (g) implies (a).

4.10 The Fixed Point Property The following fixed point theorem is a generalization of the Markov-Kakutani theorem (Theorem G.1.1). Theorem 4.10.1. Let G be an amenable group acting affinely and continuously on a nonempty convex compact subset C of a Hausdorff topological vector space X. Then G fixes at least one point in C. Proof. Let (Fj )j∈J be a left Følner net for G. Choose an arbitrary point x ∈ C and set xj =

1  hx. |Fj | h∈Fj

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for each j ∈ J. Observe that xj ∈ C since C is convex. By compactness of C, we can assume, after taking a subnet, that the net (xj )j∈J converges to a point c ∈ C. Let g ∈ G. For every j ∈ J, we have ⎛ ⎞  1  1 gxj − xj = g ⎝ hx⎠ − hx |Fj | |Fj | h∈Fj

h∈Fj

1  1  = ghx − hx (since the action of G on C is affine) |Fj | |Fj | h∈Fj

h∈Fj

1  1  = hx − hx, |Fj | |Fj | h∈gFj

h∈Fj

which yields, after simplification, gxj − xj =

1 |Fj |



hx −

h∈gFj \Fj

1 |Fj |



hx.

(4.21)

h∈Fj \gFj

Consider the points yj =

1 |gFj \ Fj |



hx,

and

zj =

h∈gFj \Fj

1 |Fj \ gFj |



hx.

h∈Fj \gFj

Note that yj and zj belong to C by convexity of C. We have |Fj \ gFj | = |gFj \ Fj | since the sets Fj and gFj have the same cardinality (cf. (4.15)). Setting λj =

|gFj \ Fj | |Fj \ gFj | = , |Fj | |Fj |

equality (4.21) gives us gxj − xj = λj yj − λj zj . The net (λj ) converges to 0 since (Fj ) is a left Følner net. As C is compact, it follows that lim λj yj = lim λj zj = 0, j

j

by using Lemma G.2.2. Therefore, we have gc − c = lim(gxj − xj ) = 0. j

This shows that c is fixed by G.



Notes

105

Corollary 4.10.2. Let G be a group. Then the following conditions are equivalent: (a) the group G is amenable; (b) every continuous affine action of G on a nonempty convex compact subset of a Hausdorff topological vector space admits a fixed point; (c) every continuous affine action of G on a nonempty convex compact subset of a Hausdorff locally convex topological vector space admits a fixed point. Proof. Implication (a) ⇒ (b) follows from the preceding theorem. Implication (b) ⇒ (c) is trivial. Finally,(c) ⇒ (a) follows from Theorem 4.2.1, Proposition 4.3.1 and the fact that (∞ (G))∗ is a locally convex Hausdorff topological vector space for the weak-∗ topology (see Sect. F.2).

Notes The theory of amenable groups emerged from the study of the axiomatic properties of the Lebesgue integral and the discovery of the Banach-Tarski paradox at the beginning of the last century (see [Har3], [Pat], [Wag]). The first definition of an amenable group, by the existence of an invariant finitely additive probability measure, is due to J. von Neumann in [vNeu1]. Von Neumann proved in particular that every abelian group is amenable (Theorem 4.6.1) and that an amenable group cannot contain a subgroup isomorphic to F2 (Corollary 4.5.2). The term amenable was introduced in the 1950s by M.M. Day, who played a central role in the development of the modern theory of amenable groups by using means and applying techniques from functional analysis. Moreover, Day extended the notion of amenability to semigroups, for which one has to distinguish between right amenability and left amenability. The question of the existence of a non-amenable group containing no subgroup isomorphic to F2 , which is called by some authors the von Neumann conjecture or Day’s problem, was answered in the affirmative by A.Yu. Ol’shanskii [Ols] who gave an example of a finitely generated non-amenable group all of whose proper subgroups are cyclic. Other examples of nonamenable groups with no subgroup isomorphic to F2 were found by S.I. Adyan [Ady] who showed that the free Burnside group B(m, n) is non-amenable for m ≥ 2 and n odd with n ≥ 665. The free Burnside group B(m, n) is the quotient of the free group Fm by the subgroup of Fm generated by all n-powers, that is, all elements of the form wn for some w ∈ Fm . The order of every element of B(m, n) divide n. In particular, B(m, n) is a periodic group, that is, a group in which every element has finite order. It is clear that a periodic group cannot contain a subgroup isomorphic to F2 . A geometric method for constructing finitely generated non-amenable periodic groups was described by M. Gromov in [Gro3]. Examples of finitely presented non-amenable groups

106

4 Amenable Groups

which contain no subgroup isomorphic to F2 were given by A.Yu Ol’shanskii and M. Sapir [OlS]. There is also a more general notion of amenability for locally compact groups and actions of locally compact groups (see [Gre], [Pat]). A group G is called polycyclic if it admits a finite sequence of subgroups {1G } = H0 ⊂ H1 ⊂ H2 ⊂ · · · ⊂ Hn = G such that Hi is normal in Hi+1 and Hi+1 /Hi is a (finite or infinite) cyclic group for each 0 ≤ i ≤ n − 1. Clearly, every polycyclic group is solvable. It is not hard to prove that every finitely generated nilpotent group is polycyclic. It was shown by L. Auslander [Aus] that every polycyclic group is isomorphic to a subgroup of SLn (Z) for some integer n ≥ 1. It follows in particular that every polycyclic group is residually finite. This last result is due to K.A. Hirsch [Hir] (see also [RobD, p. 154]). P. Hall [Hall2] proved that every finitely generated metabelian group is residually finite and gave an example of a finitely generated solvable group of degree 3 which is not residually finite. The equivalence between Følner conditions and amenability was established by E. Følner in [Føl]. The proof was later simplified by I. Namioka in [Nam]. The equivalence between amenability and the non-existence of a paradoxical decomposition is due to A. Tarski (see [Tar1], [Tar2] and [CGH1]). Let G be a group. Given a left (or right) paradoxical decomposition P = (K, (Ak )k∈K , (Bk )k∈K ) of G, the integer number c(P) = m + n, where m = |{k ∈ K : Ak = ∅}| and n = |{k ∈ K : Bk = ∅}|, is called the complexity of P. Then the quantity T (G) = inf c(P), where the infimum is taken over all left (or right) paradoxical decompositions P of G, is called the Tarski number of G. One uses the convention that T (G) = ∞ if G admits no paradoxical decompositions, that is, if G is amenable (cf. Theorem 4.9.1). It was proved by B. Jonsson, a student of Tarski in the 1940s, that a group G has Tarski number T (G) = 4 if and only if G contains a subgroup isomorphic to F2 , the free group of rank 2. In [CGH1, CGH2] it was shown that for the free Burnside groups B(m, n) with m ≥ 2 and n ≥ 665 odd one has 6 ≤ T (B(m, n)) ≤ 14. The computations involve spectral analysis and Cheeger-Buser type isoperimetric inequalities (see Sect. 6.10 and Sect. 6.12) and Adyan’s cogrowth estimates [Ady] for the free Burnside groups B(m, n) (see (6.117) in the Notes for Chap. 6). The extension of the Markov-Kakutani fixed point theorem to amenable groups (cf. Theorem 4.10.1) is due to Day [Day2].

Exercises 4.1. Let G be an infinite group and let μ : P(G) → [0, 1] be a left (or right) invariant finitely additive probability measure on G. Show that every finite subset A ⊂ G satisfies μ(A) = 0.

Exercises

107

4.2. Let X be an infinite set. Prove that the symmetric group Sym(X) is not amenable. Hint: Show that Sym(X) contains a subgroup isomorphic to the free group F2 . 4.3. Show that the finitely generated and non residually finite groups G1 and G2 described in Sect. 2.6 are amenable. Hint: Each of these groups is the semidirect product of a locally finite group and an infinite cyclic group. 4.4. Let G be a group. (a) Suppose that H and K are normal subgroups of G. Prove that HK is a normal subgroup of G and that the groups HK/K and H/(H ∩ K) are isomorphic. (b) Suppose that H and K are normal amenable subgroups of G. Prove that HK is a normal amenable subgroup of G. (c) Show that the set of all normal amenable subgroups of G has a maximal element for inclusion. This maximal element is called the amenable radical of the group G. Hint: Use (b) and Proposition 4.5.10 to show that the union of all normal amenable subgroups of G is a normal amenable subgroup of G. 4.5. Let G be a group. Show that G is metabelian if and only if it contains a normal subgroup N such that the group G/N is abelian. 4.6. Let G be a finitely generated solvable group. Show that if all elements of G have finite order then G is finite. Hint: Use induction on the solvability degree of G. 4.7. Show that every subgroup of a solvable (resp. nilpotent) group is solvable (resp. nilpotent). 4.8. Show that every quotient of a solvable (resp. nilpotent) group is solvable (resp. nilpotent). 4.9. Let G be a group containing a normal subgroup H such that both H and G/H are solvable. Show that G is solvable. 4.10. Show that the direct product of two nilpotent groups is a nilpotent group. 4.11. Show that a semidirect product of two nilpotent groups may fail to be nilpotent. Hint: Take for example the symmetric group Sym3 , which is the semidirect product of two cyclic groups. 4.12. Let G be a group. Show that G is solvable if and only if there is a finite sequence {1G } = H0 ⊂ H1 ⊂ H2 ⊂ · · · ⊂ Hn = G of subgroups of G such that Hi is normal in Hi+1 and Hi+1 /Hi is abelian for all 0 ≤ i ≤ n − 1.

108

4 Amenable Groups

4.13. Let G be a group and i ≥ 0. Show that the group C i (G)/C i+1 (G) is contained in the center of G/C i+1 (G). 4.14. Show that every nontrivial nilpotent group has a nontrivial center. 4.15. Let G be a nilpotent group of nilpotency degree d ≥ 1. Show that the quotient group G/C d−1 (G) is nilpotent of nilpotency degree d − 1. 4.16. Let G be a finite group whose order is a power of a prime number. Show that G is nilpotent. Hint: Prove that the center of G is nontrivial by considering the action of G on itself by conjugation and then proceed by induction. 4.17. Let G be a group. Denote by Nnq (resp. Nsq , resp. Naq ) the set of all normal subgroups N ⊂ G such that the quotient group G/N is nilpotent (resp. solvable, resp. amenable). The sets Nnq , Nsq and Naq are partially ordered by reverse inclusion. (a) Show that if N1 and N2 are two elements of Nnq (resp. Nsq , resp. Naq ), then N1 ∩ N2 is an element of Nnq (resp. Nsq , resp. Naq ). Hint: observe that if ρ1 : G → G/N1 and ρ2 : G → G/N2 are the canonical homomorphisms, then the map ψ : G → G/N1 × G/N2 defined by ψ(g) = (ρ1 (g), ρ2 (g)) is a homomorphism whose kernel is N1 ∩ N2 . (b) Show that Nnq (resp. Nsq , resp. Naq ) gives rise to a projective system of groups in a natural way. The limit of this projective system is called the pronilpotent completion (resp. prosolvable completion, resp. proamenable  s , resp. G  a ).  n (resp. G completion) of the group G and is denoted by G  s , (c) Show that there is a canonical homomorphism G → Gn (resp. G → G  a ) and that this homomorphism is injective if and only if G is resp. G → G residually nilpotent (resp. residually solvable, resp. residually amenable). 4.18. Recall that a group G is called polycyclic if it admits a finite sequence of subgroups {1G } = H0 ⊂ H1 ⊂ H2 ⊂ · · · ⊂ Hn = G such that Hi is normal in Hi+1 and Hi+1 /Hi is a (finite or infinite) cyclic group for each 0 ≤ i ≤ n − 1. (a) Show that every polycyclic group is finitely generated. (b) Show that every subgroup of a polycyclic group is polycyclic. (c) Deduce from (a) and (b) that every subgroup of a polycyclic group is finitely generated. 4.19. The lamplighter group is the wreath product L = (Z/2Z) Z. Thus, L is the semidirect product of the groups H = ⊕n∈Z An and Z, where An = Z/2Z for all n ∈ Z and Z acts on H by the Z-shift. (a) Show that the group L is metabelian (and therefore solvable) and residually finite.

Exercises

109

(b) Prove that L is finitely generated. Hint: Show that L is generated by the two elements s and t corresponding respectively to the nontrivial element of A0 and to the canonical generator of Z. (c) Show that L is not polycyclic. Hint: Observe that H is not finitely generated and use Exercise 4.18(c). 4.20. Show that every finite solvable group is polycyclic. 4.21. Let m be an integer such that |m| ≥ 2. Let G be the group given by the presentation G = a, b : aba−1 = bm . Use the results in Exercise 2.7 to prove that the commutator subgroup [G, G] is isomorphic to the additive group Z[1/m] and that the quotient group G/[G, G] is infinite cyclic. 4.22. Let G be a locally finite group. Let S denote the directed set consisting of all finitely generated subgroups of G partially ordered by inclusion. Prove that the net (H)H∈S is a Følner net for G. 4.23. Show that the sequence (Fn )n∈N , where Fn consists of all rational numbers of the form k/n! with k ∈ N and k ≤ (n + 1)!, is a Følner sequence for the additive group Q. 4.24. Let G = HZ denote the integral Heisenberg group (cf. Example 4.6.5). For each integer n ≥ 0, define the subset Fn ⊂ G by ⎧⎛ ⎫ ⎞ ⎨ 1xz ⎬ Fn = ⎝0 1 y ⎠ ∈ G : 1 ≤ x ≤ n, 1 ≤ y ≤ n, 1 ≤ z ≤ n2 . ⎩ ⎭ 001 Show that the sequence (Fn )n≥0 is a Følner sequence for G. 4.25. Let G be a group. Show that G is amenable if and only if the following condition holds: for every finite subset K ⊂ G and every ε > 0, there exists a finite subset F ⊂ G such that |KF | < (1 + ε)|F |. 4.26. Let G be a countable amenable group. Show that  G admits a left (resp. right) Følner sequence (Fn )n∈N which satisfies G = n∈N Fn and Fn ⊂ Fn+1 for all n ∈ N. 4.27. Let (K, (Ak )k∈K , (Bk )k∈K ) be a left (or right) paradoxical decomposition of a group G. Show that |K| ≥ 3. 4.28. Let G be a group and H ⊂ G a subgroup. Let (K, (Ak )k∈K , (Bk )k∈K ) be a left paradoxical decomposition of H and T ⊂ G a set of representa tives for the right cosets of H in G. For each k ∈ K set A k = t∈T Ak t  and Bk = t∈T Bk t. Show that (K, (A k )k∈K , (Bk )k∈K ) is a left paradoxical decomposition of G.

110

4 Amenable Groups

4.29. Let T (G) ∈ N ∪ {∞} denote the Tarski number of a group G. (a) Show that T (G) ≥ 4 for all groups G. (b) Show that T (G) = 4 if and only if G contains a subgroup isomorphic to F2 . Hint: Use Example 4.8.2 and the Klein Ping-Pong theorem (Theorem D.5.1). (c) Let H be a subgroup of a group G. Show that T (G) ≤ T (H). (d) Let N be a normal subgroup of a group G. Show that T (G) ≤ T (G/N ). (e) Let G be a group. Show that there exists a finitely generated subgroup H ⊂ G such that T (H) = T (G). (f) Let G be a group. Suppose that all elements of G have finite order. Show that T (G) ≥ 6.

Chapter 5

The Garden of Eden Theorem

The Garden of Eden Theorem gives a necessary and sufficient condition for the surjectivity of a cellular automaton with finite alphabet over an amenable group. It states that such an automaton is surjective if and only if it is preinjective. As the name suggests it, pre-injectivity is a weaker notion than injectivity. It means that any two configurations which have the same image under the automaton must be equal if they coincide outside a finite subset of the underlying group (see Sect. 5.2). We shall establish the Garden of Eden theorem in Sect. 5.8 by showing that both the surjectivity and the preinjectivity are equivalent to the maximality of the entropy of the image of the cellular automaton. The entropy of a set of configurations with respect to a Følner net of an amenable group is defined in Sect. 5.7. Another important tool in the proof of the Garden of Eden theorem is a notion of tiling for groups introduced in Sect. 5.6. The Garden of Eden theorem is used in Sect. 5.9 to prove that every residually amenable (and hence every amenable) group is surjunctive. In Sect. 5.10 and Sect. 5.11, we give simple examples showing that both implications in the Garden of Eden theorem become false over a free group of rank two. In Sect. 5.12 it is shown that a group G is amenable if and only if every surjective cellular automaton with finite alphabet over G is pre-injective. This last result gives a characterization of amenability in terms of cellular automata.

5.1 Garden of Eden Configurations and Garden of Eden Patterns Let G be a group and let A be a set. Let τ : AG → AG be a cellular automaton. A configuration y ∈ AG is called a Garden of Eden configuration for τ if y is not in the image of τ . Thus the surjectivity of τ is equivalent to the nonexistence of Garden of Eden configurations. T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 5, © Springer-Verlag Berlin Heidelberg 2010

111

112

5 The Garden of Eden Theorem

The biblical terminology “Garden of Eden” (a peaceful place where we will never return) comes from the fact that one often regards a cellular automaton τ : AG → AG from a dynamical viewpoint. This means that one thinks of a configuration as evolving with time according to τ : if x ∈ AG is the configuration at time t = 0, 1, 2, . . ., then τ (x) is the configuration at time t + 1. A configuration x ∈ AG \ τ (AG ) is called a Garden of Eden configuration because it may only appear at time t = 0. A pattern p : Ω → A is called a Garden of Eden pattern for τ if there is no configuration x ∈ AG such that τ (x)|Ω = p. It follows from this definition that if p : Ω → G is a Garden of Eden pattern for τ , then any configuration y ∈ AG such that y|Ω = p is a Garden of Eden configuration for τ . Thus, the existence of a Garden of Eden pattern implies the existence of Garden of Eden configurations, that is, the non-surjectivity of τ . It turns out that the converse is also true when the alphabet set is finite: Proposition 5.1.1. Let G be a group and let A be a finite set. Let τ : AG → AG be a cellular automaton. Suppose that τ is not surjective. Then τ admits a Garden of Eden pattern. Proof. We know that the set τ (AG ) is closed in AG for the prodiscrete topology by Lemma 3.3.2. It follows that the set AG \ τ (AG ) is open in AG . Therefore, if y ∈ AG is a Garden of Eden configuration for τ , we may find a finite subset Ω ⊂ G such that V (y, Ω) = {x ∈ AG : x|Ω = y|Ω } ⊂ AG \ τ (AG ). In other words, every configuration extending y|Ω is not in τ (AG ), that is,   y|Ω is a Garden of Eden pattern for τ .

5.2 Pre-injective Maps Let G be a group and let A be a set. Two configurations x1 , x2 ∈ AG are called almost equal if the set {g ∈ G : x1 (g) = x2 (g)} is finite. It is clear that being almost equal defines an equivalence relation on the set AG . Given a subset X ⊂ AG and a set Z, a map f : X → Z is called preinjective if it satisfies the following condition: if two configurations x1 , x2 ∈ X are almost equal and such that f (x1 ) = f (x2 ), then x1 = x2 . It immediately follows from this definition that the injectivity of f implies its pre-injectivity. The converse is trivially true when the group G is finite. However, a preinjective map f : AG → Z may fail to be injective when G is infinite. Examples 5.2.1. (a) Let us take G = Z and A = Z/3Z. Consider the cellular automaton τ : AG → AG defined by τ (x)(n) = x(n−1)+x(n)+x(n+1) for all

5.2 Pre-injective Maps

113

x ∈ AG and n ∈ G. Then τ is pre-injective. Indeed, suppose that x1 , x2 ∈ AG are two configurations such that the set Ω = {n ∈ G : x1 (n) = x2 (n)} is a nonempty finite subset of Z. Let n0 denote the largest element in Ω. Then τ (x1 )(n0 + 1) = τ (x2 )(n0 + 1) and hence τ (x1 ) = τ (x2 ). This sows that τ is pre-injective. However, τ is not injective since the constant configurations c0 , c1 ∈ AG given by c0 (n) = 0 and c1 (n) = 1 for all n ∈ Z have the same image c0 by τ . (b) Let G = Z2 and A = {0, 1}. Consider the cellular automaton τ : AG → G A associated with the Game of Life (cf. Example 1.4.3(a)). Let x1 ∈ AG be the configuration with no live cells (x1 (g) = 0 for all g ∈ G) and let x2 ∈ AG be the configuration with only one live cell at the origin (x2 (g) = 1 if g = (0, 0) and x2 (g) = 0 otherwise). Then x1 and x2 are almost equal and one has τ (x1 ) = τ (x2 ) = x1 . Therefore τ is not pre-injective. (c) Let G be a group and let A = {0, 1}. Let S be a finite subset of G having at least 3 elements. Let τ : AG → AG be the majority action cellular automaton associated with G and S (cf. Example 1.4.3(c)). Let x0 ∈ AG be the configuration defined by x0 (g) = 0 for all g ∈ G. Let x1 ∈ AG be the configuration defined by x1 (1G ) = 1 and x1 (g) = 0 if g = 1G . One has τ (x0 ) = τ (x1 ) = x0 and {g ∈ G : x0 (g) = x1 (g)} = {1G }. Consequently, τ is not pre-injective. The operations of induction and restriction of cellular automata with respect to a subgroup of the underlying group have been introduced in Sect. 1.7. It turns out that pre-injectivity, like injectivity and surjectivity (see Proposition 1.7.4), is preserved by these operations. More precisely, we have the following result (which will not be used in the proof of the Garden of Eden theorem given below): Proposition 5.2.2. Let G be a group and let A be a set. Let H be a subgroup of G and let τ ∈ CA(G, H; A). Let τH ∈ CA(H; A) denote the cellular automaton obtained by restriction of τ to H. Then, τ is pre-injective if and only if τH is pre-injective. Proof. First recall from (1.16) the factorizations   AG = Ac and τ = τc , c∈G/H

(5.1)

c∈G/H

x|c ) = (τ ( x))|c for all x  ∈ Ac . where τc : Ac → Ac satisfies τc ( Suppose that τ is pre-injective. We want to show that τH is pre-injective. So let x, y ∈ AH be almost equal configurations over H such that τH (x) = τH (y). Let us fix an arbitrary element a0 ∈ A and extend x and y to configurations x  and y in AG by setting   x(g) if g ∈ H, y(g) if g ∈ H, x (g) = and y(g) = a0 otherwise a0 otherwise

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5 The Garden of Eden Theorem

for all g ∈ G. Note that the configurations x  and y are almost equal since {g ∈ G : x (g) = y(g)} = {h ∈ H : x(h) = y(h)}. By construction, x |c = y|c for all c ∈ G/H \ {H}, while x |H = x and y|H = y. From (5.1), we deduce that τ ( x) = τ ( y ). It follows that x  = y, by pre-injectivity of τ . This implies that x = x |H equals y|H = y. This shows that τH is pre-injective. , y ∈ AG be almost equal Conversely, suppose that τH is pre-injective. Let x configurations over G such that τ ( x) = τ ( y ). For each g ∈ G, consider the g (h) = x (gh) and yg (h) = y(gh) for configurations x g , yg ∈ AH defined by x all h ∈ H. Observe that the configurations x g and yg are almost equal since x  and y are almost equal. On the other hand, we have x g = φ∗g ( x|c ) and yg = ∗ ∗ c H φg ( y |c ), where c = gH ∈ G/H and φg : A → A is the bijective map defined x) = τ ( y ), we have τc ( x|c ) = τc ( y |c ) by φ∗g (u)(h) = u(gh) for all u ∈ Ac . As τ ( xg ) = τH ( yg ) since φ∗g conjugates τc and τH by by (5.1). We deduce that τH ( Proposition 1.18. Consequently, we have x g = yg for all g ∈ G by the pre = y. Therefore, τ is pre-injective.   injectivity of τH . This implies that x

5.3 Statement of the Garden of Eden Theorem The Garden of Eden theorem gives a necessary and sufficient condition for the surjectivity of a cellular automaton with finite alphabet over an amenable group. Its name comes from the fact that the surjectivity of a cellular automaton is equivalent to the absence of Garden of Eden configurations (see Sect. 5.1). Theorem 5.3.1 (The Garden of Eden theorem). Let G be an amenable group and let A be a finite set. Let τ : AG → AG be a cellular automaton. Then one has τ is surjective ⇐⇒ τ is pre-injective. The proof of Theorem 5.3.1 will be given in Sect. 5.8 (see Theorem 5.8.1). Let us first present some applications of this theorem. Examples 5.3.2. (a) Let G = Z and A = Z/3Z. We have seen in Example 5.2.1(a) that the cellular automaton τ : AG → AG defined by τ (x)(n) = x(n − 1) + x(n) + x(n + 1) is pre-injective. Since Z is amenable (cf. Theorem 4.6.1), it follows from the Garden of Eden theorem that τ is surjective. In fact, a direct proof of the surjectivity of τ is not difficult (see Exercise 5.5). (b) Let G = Z2 and A = {0, 1}. Consider the cellular automaton τ : AG → G A associated with the Game of Life. We have seen in Example 5.2.1(b) that τ is not pre-injective. The group Z2 is amenable by Theorem 4.6.1. By applying the Garden of Eden theorem, we deduce that τ is not surjective (see Sect. 5.13 for a direct proof).

5.4 Interiors, Closures, and Boundaries

115

(c) Let G be a group and let A = {0, 1}. Let S be a finite subset of G having at least 3 elements. Let τ : AG → AG be the majority action cellular automaton associated with G and S. We have seen in Example 5.2.1(c) that τ is not pre-injective. Thus it follows from the Garden of Eden theorem that if G is amenable then τ is not surjective. We shall see in Sect. 5.10 that if G is the free group F2 , then τ is surjective.

5.4 Interiors, Closures, and Boundaries Let G be a group. Let E and Ω be subsets of G. The E-interior Ω −E and the E-closure Ω +E of Ω are the subsets of G defined respectively by Ω −E = {g ∈ G : gE ⊂ Ω} Ω

+E

and

= {g ∈ G : gE ∩ Ω = ∅}

(see Figs. 5.1–5.2). Observe that Ω −E =



Ωe−1

(5.2)

Ωe−1 = ΩE −1 .

(5.3)

e∈E

and Ω +E =

 e∈E

Fig. 5.1 The E-interior Ω −E of a set Ω. Here, Ω ⊂ R2 , E = {e, f } with e = (2, 1) and f = (−1, −1)

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5 The Garden of Eden Theorem

Fig. 5.2 The E-closure Ω +E of a set Ω. Here, Ω ⊂ R2 , E = {e, f } with e = (2, 1) and f = (−1, −1)

The E-boundary of Ω is the subset ∂E (Ω) of G defined by ∂E (Ω) = Ω +E \ Ω −E (see Fig. 5.3). Examples 5.4.1. (a) If E = ∅, then Ω −E = G and Ω +E = ∂E (Ω) = ∅. (b) If E = {1G }, then Ω −E = Ω +E = Ω and ∂E (Ω) = ∅. (c) If a ∈ G and E = {1G , a}, then Ω −E = Ω ∩ Ωa−1 , Ω +E = Ω ∪ Ωa−1 , so that ∂E (Ω) = (Ω ∪ Ωa−1 ) \ (Ω ∩ Ωa−1 ) = (Ω \ Ωa−1 ) ∪ (Ωa−1 \ Ω)

(5.4)

is the symmetric difference between the sets Ω and Ωa−1 (see Fig. 5.4). (d) Let us take G = Z2 and E = {−1, 0, 1}2 . Let a, b, c, d ∈ Z and consider the rectangle Ω = [a, b] × [c, d] = {(x, y) ∈ Z2 : a ≤ x ≤ b, c ≤ y ≤ d}. Then one has Ω −E = [a + 1, b − 1] × [c + 1, d − 1]

and

Ω +E = [a − 1, b + 1] × [c − 1, d + 1]

(see Fig. 5.5). Here are some general properties of the sets Ω −E , Ω +E and ∂E (Ω) which we shall frequently use in the sequel. Proposition 5.4.2. Let G be a group. Let E, E1 , E2 and Ω be subsets of G. Then the following hold: (i) (G \ Ω)−E = G \ Ω +E ; (ii) (G \ Ω)+E = G \ Ω −E ;

5.4 Interiors, Closures, and Boundaries

117

Fig. 5.3 The E-boundary ∂E (Ω) of a set Ω. Here, Ω ⊂ R2 , E = {e, f } with e = (2, 1) and f = (−1, −1)

Fig. 5.4 The E-interior Ω −E = Ω ∩ Ωa−1 , the E-closure Ω +E = Ω ∪ Ωa−1 , and the E-boundary ∂E (Ω) = (Ω ∪ Ωa−1 ) \ (Ω ∩ Ωa−1 ) of a set Ω when E = {1G , a}

Fig. 5.5 The E-interior Ω −E , the E-closure Ω +E , and the E-boundary ∂E (Ω) of a rectangle Ω ⊂ Z2 . Here, Ω = [a, b] × [b, c] and E = {−1, 0, −1}2

(iii) if (iv) if (v) if (vi) if

a ∈ E, then Ω −E ⊂ Ωa−1 ⊂ Ω +E ; 1G ∈ E, then Ω −E ⊂ Ω ⊂ Ω +E ; E is nonempty and Ω is finite, then Ω −E is finite; E and Ω are both finite, then Ω +E and ∂E (Ω) are finite;

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5 The Garden of Eden Theorem

(vii) if E1 ⊂ E2 , then Ω −E2 ⊂ Ω −E1 , Ω +E1 ⊂ Ω +E2 and ∂E1 (Ω) ⊂ ∂E2 (Ω); (viii) if h ∈ G, then h(Ω −E ) = (hΩ)−E , h(Ω +E ) = (hΩ)+E and h(∂E (Ω)) = ∂E (hΩ). Proof. (i) By definition, we have g ∈ G \ Ω +E if and only if gE does not meet Ω, that is, if and only if g ∈ (G \ Ω)−E . (ii) By replacing Ω by G \ Ω in (i) we get Ω −E = G \ (G \ Ω)+E which gives (ii) after taking complements. (iii) If a ∈ E, then Ω −E ⊂ Ωa−1 by (5.2) and Ωa−1 ⊂ Ω +E by (5.3). (iv) Assertion (iii) gives (iv) by taking a = 1G . (v) If a ∈ E and Ω is finite, then |Ω −E | ≤ |Ωa−1 | = |Ω| by (iii). (vi) If E and Ω are both finite, then |Ω +E | ≤ |Ω||E| by (5.3) so that Ω +E is finite. The set ∂E (Ω) is then also finite since it is contained in Ω +E . (vii) The first two statements follow immediately from (5.2) and (5.3), respectively, and imply that ∂E1 (Ω) = Ω +E1 \ Ω −E1 ⊂ Ω +E2 \ Ω −E1 ⊂ Ω +E2 \ Ω −E2 = ∂E2 (Ω). (viii) By using (5.2) we have h(Ω −E ) = h ∩e∈E Ωe−1 = ∩e∈E hΩe−1 = (hΩ)−E , which gives the first statement. Similarly, from (5.3) we get h(Ω +E ) = h(ΩE −1 ) = (hΩ)E −1 = (hΩ)+E . Finally, we have h(∂E (Ω)) = h(Ω +E \ Ω −E ) = hΩ −E − hΩ −E = ∂E (hΩ).   Proposition 5.4.3. Let G be a group and let A be a set. Let τ : AG → AG be a cellular automaton with memory set S. Let x and x be elements of AG . Suppose that there is a subset Ω of G such that x and x coincide on Ω (resp. on G \ Ω). Then the configurations τ (x) and τ (x ) coincide on Ω −S (resp. on G \ Ω +S ). Proof. Suppose that x and x coincide on Ω. If g ∈ Ω −S , then gS ⊂ Ω and therefore τ (x)(g) = τ (x )(g) by Lemma 1.4.7. It follows that τ (x) and τ (x ) coincide on Ω −S . Suppose now x and x coincide on G \ Ω. Then τ (x) and τ (x ) coincide on (G \ Ω)−S = G \ Ω +S by the first part of the proof and Proposition 5.4.2(i).   Proposition 5.4.4. Let G be a group and let (Fj )j∈J be a net of nonempty finite subsets of G. Then the following conditions are equivalent:

5.4 Interiors, Closures, and Boundaries

119

(a) the net (Fj )j∈J is a right Følner net for G; (b) one has lim j

|∂E (Fj )| =0 |Fj |

for every finite subset E ⊂ G.

Proof. (b) ⇒ (a). Suppose (b). Let g ∈ G and take E = {1G , g −1 }. By (5.4), we have Fj \ Fj g ⊂ ∂E (Fj ), and hence |Fj \ Fj g| ≤ |∂E (Fj )|. Therefore, property (b) implies lim j

|Fj \ Fj g| = 0. |Fj |

This shows that (Fj ) is a right Følner net for G. (a) ⇒ (b). Let E be a finite subset of G. By (5.2) and (5.3) we have       −1 −1 \ ∂E (Fj ) = Fj a Fj b a∈E

 =



 Fj a

−1

a∈E

 =



 Fj a

−1

b∈E







G\ 

a∈E

=





 Fj b

−1

b∈E



(G \ Fj b

 −1

)

b∈E

(Fj a−1 \ Fj b−1 ).

a,b∈E

This implies |∂E (Fj )| ≤



|Fj a−1 \ Fj b−1 |.

(5.5)

a,b∈E

Now observe that, for all a, b ∈ E, we have |Fj a−1 \ Fj b−1 | = |Fj \ Fj b−1 a|, since right multiplication by a is bijective on G. Therefore, inequality (5.5) gives us |Fj \ Fj g| |∂E (Fj )| ≤ |E|2 max , g∈K |Fj | |Fj | where K is the finite subset of G defined by K = {b−1 a : a, b ∈ E}. This shows that (a) implies (b).  

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5 The Garden of Eden Theorem

Corollary 5.4.5. Let G be a group. Then the following conditions are equivalent: (a) G is amenable; (b) for every finite subset E ⊂ G and every real number ε > 0, there exists a nonempty finite subset F ⊂ G such that |∂E (F )| < ε. |F |

(5.6)

Proof. Suppose first that G is amenable. It follows from the Tarski-Følner theorem (Theorem 4.9.1) and Proposition 4.7.1 that there exists a right |∂ (F )| Følner net (Fj )j∈J in G. By Proposition 5.4.4 we have that limj E|Fj |j = 0 for every finite subset E ⊂ G. Thus, given ε > 0 and a finite subset S ⊂ G, |∂ (F )| there exists j0 ∈ J such that E|Fj |j < ε for all j ≥ j0 . Taking F = Fj0 we deduce (5.6). This shows (a) ⇒ (b). Conversely, suppose (b). Let J denote the set of all pairs (E, ε), where E is a finite subset of G and ε > 0. We equip J with the partial ordering ≤ defined by (E, ε) ≤ (E  , ε ) ⇔ (E ⊂ E  and ε ≤ ε). Then J is a directed set. By (b), for every j = (E, ε) ∈ J, there exists a nonempty finite subset Fj ⊂ G such that |∂E (Fj )| < ε. |Fj |

(5.7)

Let us show that lim j

|∂E (Fj )| = 0 for every finite subset E ⊂ G. |Fj |

(5.8)

Fix a finite subset E0 ⊂ G and ε0 > 0. Let j ∈ J and suppose that j ≥ j0 , where j0 = (E0 , ε0 ) ∈ J. By virtue of Proposition 5.4.2(vii) we have ∂E0 (Fj ) ⊂ ∂E (Fj ) so that, from (5.7), we deduce that |∂E (Fj )| |∂E0 (Fj )| ≤ < ε ≤ ε0 . |Fj | |Fj | This shows (5.8). From Proposition 5.4.4 and Proposition 4.7.1 we deduce that G satisfies the Følner conditions. Thus G is amenable by virtue of the Tarski-Følner theorem (Theorem 4.9.1).  

5.5 Mutually Erasable Patterns

121

5.5 Mutually Erasable Patterns In this section, we give a characterization of pre-injective cellular automata based on the notion of mutually erasable patterns. This leads to an equivalent definition of pre-injectivity which is frequently used in the literature. However, the material contained in this section will not be used in the proof of the Garden of Eden theorem so that the reader who is only interested in this proof may go directly to the next section. Let G be a group and let A be a set. Let Z be a set and let f : AG → Z be a map. Two distinct patterns p1 , p2 : Ω → Z with the same support Ω ⊂ G are called mutually erasable (with respect to f ) if they satisfy the following condition: if x1 , x2 ∈ AG are configurations such that x1 |Ω = p1 , x2 |Ω = p2 and x1 |G\Ω = x2 |G\Ω , then f (x1 ) = f (x2 ). Example 5.5.1. Let G = Z2 and A = {0, 1}. Consider the cellular automaton τ : AG → AG associated with the Game of Life (see Example 1.4.3(a)). Let Ω = {−1, 0, 1}2 be the 3 × 3 square in Z2 centered at the origin and consider the pattern p1 (resp. p2 ) with support Ω defined by p1 (g) = 0 for all g ∈ Ω (resp. p2 (g) = 1 if g = (0, 0) and p2 (g) = 0 otherwise). Then it is clear that p1 and p2 are mutually erasable patterns for τ (see Fig. 5.6).

Fig. 5.6 Two mutually erasable patterns for the Game of Life (recall that ◦ denotes a dead cell, while • denotes a live cell)

Suppose that p1 and p2 are mutually erasable patterns for a map f : AG → Z. Let Ω denote their common support and consider two configurations x1 and x2 which coincide outside Ω and such that x1 |Ω = p1 and x2 |Ω = p2 . Then x1 and x2 are almost equal and f (x1 ) = f (x2 ). On the other hand, x1 = x2 since p1 = p2 and therefore f is not pre-injective. This shows that a pre-injective map f : AG → Z admits no mutually erasable patterns. It turns out that for cellular automata, the converse is true:

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5 The Garden of Eden Theorem

Proposition 5.5.2. Let G be a group and let A be a set. Let τ : AG → AG be a cellular automaton. Then τ is pre-injective if and only if it does not admit mutually erasable patterns. Proof. It remains only to show that if τ is not pre-injective then it admits mutually erasable patterns. So let us assume that τ is not pre-injective. This means that there exist configurations x1 , x2 ∈ AG satisfying τ (x1 ) = τ (x2 ) such that Σ = {g ∈ G : x1 (g) = x2 (g)} is a nonempty finite subset of G. Let S be a memory set for τ such that S = S −1 and 1G ∈ S. Consider the 2 finite sets S 2 = {s1 s2 : s1 , s2 ∈ S} and Ω = Σ +S . Let us show that the patterns p1 = x1 |Ω and p2 = x2 |Ω are mutually erasable. First observe that p1 = p2 since Σ ⊂ Ω. Suppose now that y1 , y2 ∈ AG are two configurations coinciding outside Ω such that y1 |Ω = p1 and y2 |Ω = p2 . Then y1 and y2 coincide outside Σ since p1 and p2 coincide on Ω \ Σ. This implies τ (y1 )(g) = τ (y2 )(g) for all g ∈ G \ Σ +S

(5.9)

by Proposition 5.4.3. On the other hand, for i = 1, 2, the configurations yi and xi coincide on Ω by construction. Thus, it follows from Proposition 5.4.3 that τ (yi ) and τ (xi ) coincide on Ω −S . As τ (x1 ) = τ (x2 ), we deduce that the configurations τ (y1 ) and τ (y2 ) coincide on Ω −S . Now observe that Σ +S ⊂ Ω −S . Indeed, if g ∈ Σ +S , that is, gs0 ∈ Σ for some s0 ∈ S, then gs ∈ Ω for all s ∈ S since gss−1 s0 = gs0 ∈ gsS 2 ∩ Σ. It follows that τ (y1 ) and τ (y2 ) coincide on Σ +S . Combining this with (5.9), we conclude that τ (y1 ) = τ (y2 ).   This sows that p1 and p2 are mutually erasable patterns.

5.6 Tilings Let G be a group. Let E and E  be subsets of G. A subset T ⊂ G is called an (E, E  )-tiling of G if the sets gE, g ∈ T are pairwise disjoint and if the sets gE  , g ∈ T cover G. In other words, T ⊂ G is an (E, E  )-tiling if and only if the following conditions are satisfied: (T1) g1 E ∩ g2 E = ∅ for all g1 , g2 ∈ T such that g1 = g2 ; (T2) G = g∈T gE  . Examples 5.6.1. (a) In the additive group R, the set Z is a ([0, 1[, [0, 1[)-tiling and the set [0, 1] is a (2Z, Z)-tiling. (b) If G is a group and H is a subgroup of G, then every complete set of representatives for left cosets of G modulo H is an (H, H)-tiling. Remark 5.6.2. Let G be a group and let E and E  be subsets of G. If T is an (E, E  )-tiling of G and if E1 and E1 are subsets of G such that E1 ⊂ E and E  ⊂ E1 , then it is clear that T is also an (E1 , E1 )-tiling of G.

5.6 Tilings

123

The Zorn lemma may be used to prove the existence of (E, E  )-tilings for any subset E of G and for E  ⊂ G “large enough”. More precisely, we have the following: Proposition 5.6.3. Let G be a group and let E be a nonempty subset of G. Let E  = {g1 g2−1 : g1 , g2 ∈ E}. Then there is an (E, E  )-tiling T ⊂ G. Proof. Consider the set S consisting of all subsets S ⊂ G such that the sets (gE)g∈S are pairwise disjoint. Observe that S is not empty since {1G } ∈ S. On the other hand, the set S, partially ordered by inclusion, is inductive.

Indeed, if S  is a totally ordered subset of S, then the set M = S∈S  S belongs to S and is an upper bound for S  . By applying Zorn’s lemma, we deduce that S admits a maximal element T . The sets (gE)g∈T are pairwise disjoint since T ∈ S. On the other hand, consider an arbitrary element h ∈ G. By maximality of T , we can find g ∈ T such that the set hE meets gE. This implies h ∈ gE  . This shows that the sets (gE  )g∈T cover G. Consequently,   T is an (E, E  )-tiling of G. Proposition 5.6.4. Let G be an amenable group and let (Fj )j∈J be a right Følner net for G. Let E and E  be finite subsets of G and suppose that T ⊂ G is an (E, E  )-tiling of G. Let us set, for each j ∈ J, Tj = T ∩ Fj−E = {g ∈ T : gE ⊂ Fj }. Then there exist a real number α > 0 and an element j0 ∈ J such that |Tj | ≥ α|Fj |

for all j ≥ j0 .

Proof. After possibly replacing E  by E ∪ E  , we can assume that E ⊂ E  .  Let us set Tj+ = T ∩ Fj+E = {g ∈ T : gE  ∩ Fj = ∅}. As the sets gE  , g ∈ Tj+ , cover Fj , we have |Fj | ≤ |Tj+ | · |E  |, which gives |Tj+ | 1 ≥  |Fj | |E |

(5.10)

for all j ∈ J. Observe now that   Tj+ \ Tj = T ∩ Fj+E \ T ∩ Fj−E 

= T ∩ (Fj+E \ Fj−E ) 



⊂ T ∩ (Fj+E \ Fj−E ) ⊂ T ∩ ∂E  (Fj ) ⊂ ∂E  (Fj ) where the first inclusion follows from E ⊂ E  . Thus we have |∂E  (Fj )| ≥ |Tj+ | − |Tj |.

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5 The Garden of Eden Theorem

Using (5.10), we then deduce |Tj+ | |∂E  (Fj )| 1 |∂E  (Fj )| |Tj | ≥ − ≥  − . |Fj | |Fj | |Fj | |E | |Fj | Hence we have

1 |Tj | ≥α= |Fj | 2|E  |  

for j large enough, by Proposition 5.4.4 (see Fig. 5.7).

Fig. 5.7 The set T7 = T ∩ F7−E ⊂ Z2 , where E = {(0, 0), (1, 0)}, E  = {0, 1} × {0, 1} and

T = (2Z + 1) × (2Z + 1) ⊂ Z2 is an (E, E  )-tiling in Z2 . Note that 1 8

=

1 2|E  |



|T7 | |F7 |

=

12 64

=

3 16



2 16

=

5.7 Entropy

125

5.7 Entropy In this section, G is an amenable group, F = (Fj )j∈J is a right Følner net for G, and A is a finite set. For E ⊂ G, we denote by πE : AG → AE the canonical projection (restriction map). We thus have πE (x) = x|E for all x ∈ AG . Definition 5.7.1. Let X ⊂ AG . The entropy entF (X) of X with respect to the right Følner net F = (Fj )j∈J is defined by entF (X) = lim sup j

log |πFj (X)| . |Fj |

Here are some immediate properties of entropy. Proposition 5.7.2. One has (i) entF (AG ) = log |A|; (ii) entF (X) ≤ entF (Y ) if X ⊂ Y ⊂ AG ; (iii) entF (X) ≤ log |A| for all X ⊂ AG . Proof. (i) If X = AG , then, for every j, we have πFj (X) = AFj and therefore log |πFj (X)| log |A||Fj | |Fj | log |A| = = = log |A|. |Fj | |Fj | |Fj | Thus we have entF (X) = log |A|. (ii) If X ⊂ Y , then πFj (X) ⊂ πFj (Y ) and hence |πFj (X)| ≤ |πFj (Y )| for all j. This implies entF (X) ≤ entF (Y ). (iii) This follows immediately from (i) and (ii).   An important property of cellular automata is the fact that applying a cellular automaton to a set of configurations cannot increase the entropy of the set. More precisely, we have the following: Proposition 5.7.3. Let τ : AG → AG be a cellular automaton and let X ⊂ AG . Then one has entF (τ (X)) ≤ entF (X). Proof. Let Y = τ (X). Let S ⊂ G be a memory set for τ . After replacing S by S ∪ {1G }, we can assume that 1G ∈ S. Let Ω be a finite subset of G. Observe first that τ induces a map τΩ : πΩ (X) → πΩ −S (Y ) defined as follows. If u ∈ πΩ (X), then τΩ (u) = (τ (x))|Ω −S , where x is an element of X such that x|Ω = u. Note that the fact that τΩ (u) does not depend on the choice of such an x follows from Proposition 5.4.3.

126

5 The Garden of Eden Theorem

Clearly τΩ is surjective. Indeed, if v ∈ πΩ −S (Y ), then there exists x ∈ X such that (τ (x))|Ω −S = v. Then, setting u = πΩ (x) we have, by construction, τΩ (u) = v. Therefore, we have |πΩ −S (Y )| ≤ |πΩ (X)|.

(5.11)

Observe now that Ω −S ⊂ Ω, since 1G ∈ S (cf. Proposition 5.4.2(iv)). Thus −S πΩ (Y ) ⊂ πΩ −S (Y ) × AΩ\Ω . This implies log |πΩ (Y )| ≤ log |πΩ −S (Y ) × AΩ\Ω

−S

|

Ω\Ω −S

= log |πΩ −S (Y )| + log |A | = log |πΩ −S (Y )| + |Ω \ Ω −S | log |A| ≤ log |πΩ (X)| + |Ω \ Ω −S | log |A|, by (5.11). As Ω \ Ω −S ⊂ ∂S (Ω), we deduce that log |πΩ (Y )| ≤ log |πΩ (X)| + |∂S (Ω)| log |A|. By taking Ω = Fj , this gives us log |πFj (X)| |∂S (Fj )| log |πFj (Y )| ≤ + log |A|. |Fj | |Fj | |Fj | Since lim j

|∂S (Fj )| =0 |Fj |

by Proposition 5.4.4, we finally get entF (Y ) = lim sup j

log |πFj (Y )| log |πFj (X)| ≤ lim sup = entF (X). |Fj | |Fj | j  

It follows from Proposition 5.7.2 that the maximal value for the entropy of a subset X ⊂ AG is log |A|. The following result gives a sufficient condition on X which implies that its entropy is strictly less than log |A|. Proposition 5.7.4. Let X ⊂ AG . Suppose that there exist finite subsets E and E  of G and an (E, E  )-tiling T ⊂ G such that πgE (X)  AgE for all g ∈ T . Then one has entF (X) < log |A|. Proof. For each j ∈ J, let us define, as in Proposition 5.6.4 (see Fig. 5.7), the subset Tj ⊂ T by Tj = T ∩ Fj−E = {g ∈ T : gE ⊂ Fj } and set  gE Fj∗ = Fj \ g∈Tj

(see Fig. 5.8). By hypothesis, we have |πgE (X)| ≤ |AgE | − 1 = |A||gE| − 1 for all g ∈ T.

(5.12)

5.7 Entropy

127

‘ Fig. 5.8 The set F7∗ = F7 \ g∈T7 gE = F7 \ T7 E ⊂ Z2 , where E and T are as in Fig. 5.7 (note that here F7 is the whole square, while F7∗ ⊂ F7 consists of its •-points)

As



πFj (X) ⊂ AFj ×



πgE (X),

g∈Tj

we get ∗

log |πFj (X)| ≤ log |AFj ×



πgE (X)|

g∈Tj

= |Fj∗ | log |A| +



log |πgE (X)|

g∈Tj

≤ |Fj∗ | log |A| +



log(|A||gE| − 1) (by (5.12))

g∈Tj

=

|Fj∗ | log |A|

+



g∈Tj

|gE| log |A| +

g∈Tj

= |Fj | log |A| + |Tj | log(1 − |A|−|E| ),

log(1 − |A|−|gE| )

128

since

5 The Garden of Eden Theorem

|Fj | = |Fj∗ | +



|gE|

|gE| = |E|.

and

g∈Tj

By setting c = − log(1 − |A|−|E| ) (note that c > 0), this gives us log |πFj (X)| ≤ |Fj | log |A| − c|Tj |

for all j ∈ J.

Now, by Proposition 5.6.4, there exist α > 0 and j0 ∈ J such that |Tj | ≥ α|Fj | for all j ≥ j0 . Thus log |πFj (X)| ≤ log |A| − cα |Fj |

for all j ≥ j0 .

This implies that entF X = lim sup j

log |πFj (X)| ≤ log |A| − cα < log |A|. |Fj |  

Recall from Sect. 1.1 that G acts on the left on A by the shift (g, x) → gx defined by gx(g  ) = x(g −1 g  ) for g, g  ∈ G and x ∈ AG . G

Corollary 5.7.5. Let X be a G-invariant subset of AG . Suppose that there exists a finite subset E ⊂ G such that πE (X)  AE . Then one has entF (X) < log |A|. Proof. Let E  = {g1 g2−1 : g1 , g2 ∈ E}. By Proposition 5.6.3, we may find an (E, E  )-tiling T ⊂ G. Since πE (X)  AE and X is G-invariant, we have πgE (X)  AgE for all g ∈ G. This implies entF (X) < log |A| by Proposition 5.7.4.  

5.8 Proof of the Garden of Eden Theorem The purpose of this section is to establish the following: Theorem 5.8.1. Let G be an amenable group and let A be a finite set. Let F = (Fj )j∈J be a right Følner net for G. Let τ : AG → AG be a cellular automaton. Then the following conditions are equivalent: (a) τ is surjective; (b) entF (τ (AG )) = log |A|; (c) τ is pre-injective. Note that this will prove the Garden of Eden theorem (Theorem 5.3.1) since the Garden of Eden theorem asserts the equivalence of conditions (a) and (c) in Theorem 5.8.1.

5.8 Proof of the Garden of Eden Theorem

129

We divide the proof of Theorem 5.8.1 into several lemmas. In these lemmas, it is assumed that the hypotheses of Theorem 5.8.1 are satisfied: G is an amenable group, F = (Fj )j∈J is a right Følner net for G, A is a finite set, and τ : AG → AG is a cellular automaton. Lemma 5.8.2. Suppose that τ is not surjective. Then one has entF (τ (AG )) < log |A|. Proof. By Proposition 5.1.1, τ admits a Garden of Eden pattern. This means that there is a finite subset E ⊂ G such that πE (τ (AG ))  AE . The set τ (AG ) is G-invariant since τ is G-equivariant by Proposition 1.4.4. We deduce that entF (τ (AG )) < log |A| by applying Corollary 5.7.5.   Lemma 5.8.3. Suppose that ent(τ (AG )) < log |A|.

(5.13)

Then τ is not pre-injective. Proof. Let S be a memory set for τ such that 1G ∈ S. Let Y = τ (AG ). We have Fj−S ⊂ Fj ⊂ Fj+S by Proposition 5.4.2(iv) and therefore Fj+S \ Fj ⊂ +S

∂S (Fj ). As πF +S (Y ) ⊂ πFj (Y ) × AFj

\Fj

, it follows that

j

log |πF +S (Y )| ≤ log |πFj (Y )| + |Fj+S \ Fj | log |A| j

≤ log |πFj (Y )| + |∂S (Fj )| log |A|. This implies log |πF +S (Y )| j

|Fj |



log |πFj (Y )| |∂S (Fj )| + log |A|. |Fj | |Fj |

As ent(Y ) = lim sup j

by hypothesis, and lim j

(5.14)

log |πFj (Y )| < log |A| |Fj |

|∂S (Fj )| =0 |Fj |

by Proposition 5.4.4, we deduce from inequality (5.14) that there exists j0 ∈ J such that log |πF +S (Y )| j0 < log |A|. (5.15) |Fj0 | Let us fix an arbitrary element a0 ∈ A and denote by Z the finite set of configurations z ∈ AG such that z(g) = a0 for all g ∈ G\Fj0 . Inequality (5.15) gives us

130

5 The Garden of Eden Theorem

|πF +S (Y )| < |A||Fj0 | = |Z|. j0

Observe that τ (z1 ) and τ (z2 ) coincide outside Fj +S for all z1 , z2 ∈ Z. Thus 0

|τ (Z)| = |πF

+S j0

(τ (Z))| ≤ |πF

+S j0

(Y )| < |Z|.

This implies that we may find distinct configurations z1 , z2 ∈ Z such that τ (z1 ) and τ (z2 ). Since z1 and z2 coincide outside the finite set Fj0 , this shows that τ is not pre-injective.   Lemma 5.8.4. Suppose that τ is not pre-injective. Then one has entF (τ (AG )) < log |A|.

(5.16)

Proof. Since τ is not pre-injective, we may find two configurations x1 , x2 ∈ AG satisfying τ (x1 ) = τ (x2 ) such that the set Ω = {g ∈ G : x1 (g) = x2 (g)} is a nonempty finite subset of G. Observe that, for each h ∈ G, the configurations hx1 and hx2 satisfy τ (hx1 ) = τ (hx2 ) (since τ is G-equivariant by Proposition 1.4.4) and {g ∈ G : hx1 (g) = hx2 (g)} = hΩ. Let S be a memory set for τ such that 1G ∈ S. Then the set R = {s−1 s : s, s ∈ S} is finite and we have 1G ∈ R. Let E = Ω +R . By Proposition 5.6.3, we may find a finite subset E  ⊂ G and an (E, E  )-tiling T ⊂ G. Consider the subset Z ⊂ AG consisting of all configurations z ∈ AG such that z|hE = (hx1 )|hE

for all h ∈ T.

Observe that, for each h ∈ T , we have πhE (Z)  AhE since (hx1 )|hE ∈ / πhE (Z). We deduce that entF (Z) < log |A| by applying Proposition 5.7.4. As entF (τ (Z)) ≤ entF (Z) by Proposition 5.7.3, this implies entF (τ (Z)) < log |A|.

(5.17)

Thus, to establish inequality (5.16), it suffices to prove that τ (AG ) = τ (Z). To see this, consider an arbitrary configuration x ∈ AG and let us show that there is a configuration z ∈ Z such that τ (x) = τ (z). Let T  = {h ∈ T : x|hE = (hx1 )|hE }.

5.9 Surjunctivity of Locally Residually Amenable Groups

131

Let z ∈ AG be the configuration defined by  hx2 (g) if there is h ∈ T  such that g ∈ hE, z(g) = x(g) otherwise. Notice that the configuration z is obtained from x by modifying the values taken by x only on the subsets of the form hΩ, where h ∈ T  , (since, as we have seen above, hx1 and hx2 coincide outside hΩ). By construction, we have z ∈ Z. Let g ∈ G. Let us shows that τ (x)(g) = τ (z)(g). Suppose first that gS does not meet any of the sets hΩ, h ∈ T  . Then we have z|gS = x|gS . We deduce that τ (z)(g) = τ (x)(g) by applying Lemma 1.4.7. Suppose now that there is an element h ∈ T  such that gS meets hΩ. This means that there exists an element s0 ∈ S such that gs0 ∈ hΩ. For each s ∈ S, we have gss−1 s0 = gs0 ∈ hΩ. As s−1 s0 ∈ R, this implies gs ∈ (hΩ)+R = hΩ +R = hE.

(by Proposition 5.4.2(viii))

We deduce that gS ⊂ hE. Thus we have τ (x)(g) = τ (hx1 )(g) since x|hE = hx1 |hE . Similarly, by applying Lemma 1.4.7, we get τ (z)(g) = τ (hx2 )(g), since z and hx2 coincide on hE. As τ (hx1 ) = τ (hx2 ), we deduce that τ (x)(g) = τ (z)(g). Thus τ (z) = τ (x). This shows that τ (AG ) = τ (Z) and completes the proof of the lemma.   Proof of Theorem 5.8.1. If τ is surjective, then τ (AG ) = AG and hence entF (τ (AG )) = entF (AG ) = log |A|. Thus (a) implies (b). Since the converse implication follows from Lemma 5.8.2, we deduce that conditions (a) and (b) are equivalent. The fact that (c) implies (b) follows from Lemma 5.8.3 and the converse implication follows from Lemma 5.8.4. Thus, conditions (b) and (c) are also equivalent.  

5.9 Surjunctivity of Locally Residually Amenable Groups The notion of a residually finite group was introduced in Chap. 2. More generally, if P is a property of groups, a group G is called residually P if for each element g ∈ G with g = 1G , there exist a group Γ satisfying P and an epimorphism φ : G → Γ such that φ(g) = 1Γ . Observe that every group which satisfies P is residually P.

132

5 The Garden of Eden Theorem

According to the preceding definition, a group G is called residually amenable if for each element g ∈ G with g = 1G , there exist an amenable group Γ and an epimorphism φ : G → Γ such that φ(g) = 1Γ . Note that, as every subgroup of an amenable group is amenable, it is not necessary to require that the homomorphism φ is surjective in this definition. Observe also that every subgroup of a residually amenable group is residually amenable and that the fact that every finite group is amenable (Proposition 4.4.6) implies that every residually finite group is residually amenable. Theorem 5.9.1. Every residually amenable group is surjunctive. Let us first establish the following: Lemma 5.9.2. Let G be a residually amenable group and let Ω be a finite subset of G. Then there exist an amenable group Γ and a homomorphism ρ : G → Γ such that the restriction of ρ to Ω is injective. Proof. Consider the finite subset S ⊂ G defined by S = {g −1 h : g, h ∈ Ω and g = h}. Since G is residually amenable, we can find, for each s ∈ S, an amenable group Λs and a homomorphism φs : G → Λs such that φs (s) = 1Λs . Let us show that the group  Λs Γ = s∈S

and the homomorphism ρ : G → Γ given by  ρ= φs s∈S

have the required properties. The fact that the group Γ is amenable follows from Corollary 4.5.6. On the other hand, suppose that g and h are distinct elements of Ω. Then s = g −1 h ∈ S and φs (g) = φs (h) since (φs (g))−1 φs (h) = φs (g −1 h) = φs (s) = 1Λs . This implies ρ(g) = ρ(h). Therefore, the restriction of ρ to Ω is injective.   Proof of Theorem 5.9.1. Since every injective cellular automaton is preinjective, the Garden of Eden theorem (Theorem 5.3.1) implies that every amenable group is surjunctive. By applying Lemma 3.3.4 and Lemma 5.9.2, it follows that every residually amenable group is surjunctive.   As an immediate consequence of Theorem 5.9.1 and Proposition 3.2.2, we obtain the following: Corollary 5.9.3. Every locally residually amenable group is surjunctive.

 

5.11 A Pre-injective but Not Surjective Cellular Automaton over F2

133

5.10 A Surjective but Not Pre-injective Cellular Automaton over F2 The Garden of Eden theorem (Theorem 5.3.1) implies that every surjective cellular automaton with finite alphabet over an amenable group is necessarily pre-injective. In this section, we give an example of a surjective but not preinjective cellular automaton with finite alphabet over the free group F2 . Let G = F2 be the free group on two generators a and b. Let S = {a, b, a−1 , b−1 } and A = {0, 1}. Consider the cellular automaton τ : AG → AG defined by ⎧  ⎪ if x(gs) > 2, ⎨1 s∈S τ (x)(g) = 0 x(gs) < 2, if ⎪ s∈S ⎩ x(g) if s∈S x(gs) = 2. Thus τ is the majority action automaton associated with G and S (see Example 1.4.3(c)). We have already seen in Example 5.2.1(c) that τ is not pre-injective. Let us show that τ is surjective. Let y ∈ AG be an arbitrary configuration. We may construct a configuration x ∈ AG such that y = τ (x) in the following way. Consider the map ψ : G\{1G } → G which associates to each g ∈ G\{1G } the element of G obtained by suppressing the last factor in the reduced form of g. Thus if g = s1 s2 . . . sn with si ∈ S for 1 ≤ i ≤ n and si si+1 = 1G for 1 ≤ i ≤ n − 1, then ψ(g) = s1 s2 . . . sn−1 . Let x ∈ AG be the configuration defined by x(g) = y(ψ(g)) for all g ∈ G \ {1G } and x(1G ) = 0. Then, for each g ∈ G, the configuration x takes the value y(g) at (at least) three of the four elements gs, s ∈ S. it follows that τ (x) = y. Thus τ is surjective. More generally, let G be a group containing a free subgroup H based on two elements a and b. Let τ : {0, 1}G → {0, 1}G be the majority action cellular automaton over G associated with the set S = {a, b, a−1 , b−1 }. We have seen in Example 5.2.1(c) that τ is not pre-injective. Consider its restriction τH : {0, 1}H → {0, 1}H . Note that τH is the majority action cellular automaton over H associated with S. We have seen above that τH is surjective. It follows from Proposition 1.7.4(ii) that τ is also surjective. Thus, every group containing a free subgroup of rank two admits a cellular automaton with finite alphabet which is surjective but not pre-injective. This will be extended to all non-amenable groups in Sect. 5.12.

5.11 A Pre-injective but Not Surjective Cellular Automaton over F2 It follows from the Garden of Eden theorem (Theorem 5.3.1) that every preinjective cellular automaton with finite alphabet over an amenable group is

134

5 The Garden of Eden Theorem

necessarily surjective. In this section, we give examples of cellular automata with finite alphabet over the free group F2 which are pre-injective but not surjective. Let G = F2 be the free group on two generators a and b. Let H be a nontrivial abelian group. Our alphabet will be the group A = H ×H. We shall use additive notation for the group operations on H and A. Let p1 , p2 : A → A be the group endomorphisms defined respectively by p1 (u) = (h1 , 0) and p2 (u) =(h2 , 0) for all u = (h1 , h2 ) ∈ A. Let us equip the Cartesian product AG = g∈G A with its natural Abelian group structure. Consider the map τ : AG → AG given by τ (x)(g) = p1 (x(ga)) + p2 (x(gb)) + p1 (x(ga−1 )) + p2 (x(gb−1 ))

(5.18)

for all g ∈ G and x ∈ AG . It is clear that τ is a cellular automaton over the group G and the alphabet A with memory set S = {a, b, a−1 , b−1 } and a group endomorphism of AG . Proposition 5.11.1. The cellular automaton τ : AG → AG defined by (5.18) is pre-injective but not surjective. Proof. The image of τ is contained in (H × {0})G  (H × H)G = AG . Thus τ is not surjective. Let us show that τ is pre-injective. Suppose not. Then there exist configurations x1 , x2 ∈ AG satisfying τ (x1 ) = τ (x2 ) such that the set Ω = {g ∈ G : x1 (g) = x2 (g)} is a nonempty finite subset of G. The configuration x0 = x1 − x2 ∈ AG satisfies τ (x0 ) = τ (x1 ) − τ (x2 ) = 0 and one has Ω = {g ∈ G : x0 (g) = 0}. Consider an element g0 ∈ Ω whose reduced form has maximal length, say n0 . Then x0 (g0 ) is a nonzero element (h0 , k0 ) of A. If h0 = 0, take s0 ∈ {a, a−1 } such that the reduced form of g0 s0 has length n0 + 1. Then, for each s ∈ S \ {s−1 0 }, the length of the reduced form of g0 s0 s is n0 + 2 and hence x(g0 s0 s) = 0. By applying 5.18, we deduce that τ (x0 )(g0 s0 ) = p1 (x0 (g0 )) = (h0 , 0) = 0, which contradicts the fact that τ (x0 ) = 0. If h0 = 0, then k0 = 0 and we proceed similarly by taking s0 ∈ {b, b−1 } such that the reduced form of g0 s0 has length n0 + 1. This gives us τ (x0 )(g0 s0 ) = p2 (x0 (g0 )) = (k0 , 0) = 0, which yields again a contradiction. This shows that τ is pre-injective.

 

If we take for H a finite abelian group of cardinality |H| = n ≥ 2 (e.g., the group H = Z/nZ), this gives us a pre-injective but not surjective cellular automaton over F2 whose alphabet is finite of cardinality n2 . From Proposition 1.7.4 we deduce the following:

5.12 A Characterization of Amenability in Terms of Cellular Automata

135

Proposition 5.11.2. Let G be a group containing a free subgroup of rank two. Then there exist a finite set A and a cellular automaton τ : AG → AG which is pre-injective but not surjective.  

5.12 A Characterization of Amenability in Terms of Cellular Automata In Sect. 5.10, we gave an example of a cellular automaton with finite alphabet over the free group F2 which is surjective but not pre-injective. In fact, the existence of such an automaton holds for any non-amenable group: Theorem 5.12.1. Let G be a non-amenable group. Then there exists a finite set A and a cellular automaton τ : AG → AG which is surjective but not pre-injective. Proof. Since G is non-amenable, it follows from Theorem 4.9.2 that there exist a 2-to-one surjective map ϕ : G → G and a finite subset S ⊂ G such that (5.19) (ϕ(g))−1 g ∈ S for all g ∈ G. Our alphabet will be the Cartesian product A = S × S. Let us fix some total order ≤ on S and an arbitrary element s0 ∈ S. Define the map μ : AS → A by ⎧   ⎪ ⎨(s , t ) if there exists a unique element (s, t) ∈ S × S with s < t μ(y) = such that y(s) = (s, s ) and y(t) = (t, t ), where s , t ∈ S, ⎪ ⎩ (s0 , s0 ) otherwise, (5.20) for all y ∈ AS . Let us show that the cellular automaton τ : AG → AG with memory set S and local defining map μ has the required properties. We fist observe that S has at least two elements since otherwise (5.19) would imply that ϕ is bijective. Let s1 ∈ S such that s1 = s0 . Consider the configurations x0 , x1 ∈ AG , where x0 is defined by x0 (g) = (s0 , s0 ) for all g ∈ G, and x1 is defined by x1 (g) = (s0 , s0 ) if g = 1G and x1 (1G ) = (s0 , s1 ). The configurations x0 and x1 are almost equal since they differ only at 1G . On the other hand, it is clear that x0 and x1 have the same image, namely x0 , by τ . Thus τ is not pre-injective. We use the properties of ϕ to prove that τ is surjective. Let x ∈ AG be an arbitrary configuration. Let us show that there is a configuration z ∈ AG such that x = τ (z). We construct z in the following way. Let u : G → S and v : G → S be the maps defined by x(g) = (u(g), v(g)) for all g ∈ G. For each g ∈ G, there are exactly two elements sg , tg ∈ S such that sg < tg and ϕ(gsg ) = ϕ(gtg ) = g. Let us set z(gsg ) = (sg , u(g)) and z(gtg ) = (tg , v(g)).

136

5 The Garden of Eden Theorem

Observe that z : G → A is well defined and that the value of z at g ∈ G is either ((ϕ(g))−1 g, u(ϕ(g)) or ((ϕ(g))−1 g, v(ϕ(g)). It immediately follows from the definition of τ that x = τ (z). This shows that τ is surjective.   Combining Theorem 5.12.1 with Theorem 5.3.1, we obtain the following characterization of amenable groups in terms of cellular automata: Corollary 5.12.2. Let G be a group. Then the following conditions are equivalent: (a) G is amenable; (b) every surjective cellular automaton with finite alphabet over G is preinjective.  

5.13 Garden of Eden Patterns for Life Let τ : {0, 1}Z → {0, 1}Z denote the cellular automaton associated with the Game of Life (see Example 1.4.3(a)). As it was observed in Example 5.3.2(b), it follows from the Garden of Eden theorem that τ is not surjective. Thus, Proposition 5.1.1 implies that τ admits Garden of Eden patterns. The purpose of this section is to present a direct proof of the existence of such patterns. This construction will provide a concrete illustration of the ideas underlying the proof that surjectivity implies pre-injectivity in the Garden of Eden theorem. Given an integer n ≥ 1, one says that a subset Ω ⊂ Z2 is a square of size n × n in Z2 if there exist p, q ∈ Z such that 2

2

Ω = {p, p + 1, . . . , p + n − 1} × {q, q + 1, . . . , q + n − 1}. Proposition 5.13.1. Let τ : {0, 1}Z → {0, 1}Z be the cellular automaton associated with the Game of Life. Then every square Ω ⊂ Z2 of size n × n with n ≥ 3 × 109 is the support of a Garden of Eden pattern for τ . 2

2

Proof. Let n ≥ 1 be an integer. Consider a square Cn ⊂ Z2 of size 5n × 5n. Let Dn ⊂ Cn be the square of size (5n − 2) × (5n − 2) which is the S-interior of Dn for S = {−1, 0, 1}2 ⊂ Z2 . Thus Dn is obtained from Cn by removing the 20n − 4 points of Z2 located on the (usual) boundary of Cn . Let us set Xn = {0, 1}Cn and Yn = {0, 1}Dn . The map τ induces a map τn : Xn → Yn defined as follows. If u ∈ Xn , we set τn (u) = (τ (x))|Dn , where 2 x ∈ {0, 1}Z satisfies x|Cn = u (the fact that (τ (x))|Dn does not depend of the choice of x follows from Proposition 5.4.3 since S is a memory set for τ and Dn is the S-interior of Cn ). We have 2

|Yn | = 2(5n−2) = 225n

2

−20n+4

.

Let us divide the square Cn into n2 squares of size 5 × 5 (see Fig. 5.10).

5.13 Garden of Eden Patterns for Life

137

Fig. 5.9 A square Cn ⊂ Z2 of size 5n × 5n and its S-interior Dn ⊂ Cn , a square of size (5n − 2) × (5n − 2) where S = {−1, 0, 1}2 ⊂ Z2 ; here n = 3

There are 225 maps from each square of size 5 × 5 to the set {0, 1}. Now observe that if the restriction of an element u ∈ Xn to one of these 5 × 5 squares is identically 0, then we may replace the 0 value at the center of this 5 × 5 square by 1 without changing τn (u). We deduce that 2

|τn (Xn )| ≤ (225 − 1)n = (2log2 (2

25

−1) n2

)

= 2(25+log2 (1−2

−25

Therefore we have |τn (Xn )| < |Yn | if (25 + log2 (1 − 2−25 ))n2 < 25n2 − 20n + 4. This inequality is equivalent to −n2 log2 (1 − 2−25 ) − 20n + 4 > 0, which is verified if and only if n>

 2 (5 + 25 + log2 (1 − 2−25 )), −25 − log2 (1 − 2 )

that is, if and only if n ≥ 465 163 744.

))n2

.

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Fig. 5.10 The square Cn ⊂ Z2 of size 5n × 5n is divided into n2 squares of size 5×5; here, as in Fig. 5.9, n = 3

For such values of n, the map τn is not surjective, i.e., there exists a Garden of Eden pattern whose support is Dn . This shows the existence of a Garden of Eden pattern of size 2 325 818 718 × 2 325 818 718.

 

Notes According to M. Gardner (see [Gar-2, p.230]), it was J. Tuckey who introduced the term “Garden of Eden” in the theory of cellular automata. The first contribution to the Garden of Eden theorem goes back to E.F. Moore [Moo] who proved that every surjective cellular automaton with finite alphabet over Z2 is pre-injective. The converse implication was established shortly after by J. Myhill [Myh]. This is the reason why the Garden of Eden theorem for Z2 is often referred to as the Moore-Myhill theorem. The next step in the proof of the Garden of Eden theorem was done by A. Mach`ı and F. Mignosi [MaM] who extended it to finitely generated groups of subexponential growth (see Chap. 6 for the definition of finitely generated groups of subexponential growth). Then, the Garden of Eden theorem was proved for

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all finitely generated amenable groups by A. Mach`ı, F. Scarabotti and the first author [CMS1]. The general case may be reduced to the case of finitely generated groups by considering the restriction of the cellular automaton to the subgroup generated by a memory set and applying Proposition 1.7.4(ii) and Proposition 5.2.2 (see [CeC8]). The proof based on Følner nets which is presented in this chapter is more direct. The term “pre-injective” was introduced by M. Gromov in the appendix of [Gro5]. It follows from a result due to D.S. Ornstein and B. Weiss (see [OrW], [Gro6], [Kri]) that the lim sup appearing in the definition of entropy (Definition 5.7.1) is in fact a true limit and is independent of the particular choice of the right Følner net F in the group. Versions of the Garden of Eden theorem for cellular automata over certain classes of subshifts (closed invariant subsets of the full shift) may be found in the appendix of [Gro5] and in two papers of F. Fiorenzi [Fio1], [Fio2]. See also [CFS]. The first examples of pre-injective (resp. surjective) cellular automata which are not surjective (resp. not pre-injective) where described by D.E. Muller (unpublished class notes). The underlying group was the modular group G = P SL(2, Z) = (Z/2Z) ∗ (Z/3Z) (note that G contains the free group F2 ). Theorem 5.12.1 is due to L. Bartholdi [Bar]. The computation in Sect. 5.13 is taken from [BCG, Page 828]. The size of the corresponding Garden of Eden pattern is far from being optimal. The first explicit example of a Garden of Eden pattern for the cellular automaton associated with Conway’s Game of Life was found by R. Banks in 1971. Banks’ pattern is supported by a rectangle of size 33 × 9 and has 226 alive cells. The smallest known Garden of Eden pattern for Life, found by N. Beluchenko on September 2009, has as support a square 11 × 11 and bears 69 alive cells. It was proved that there exist no Garden of Eden patterns for Life with support contained in a rectangle of size 6 × 5.

Exercises 5.1. Let G be a group and let A be a finite set. Let τ : AG → AG be a cellular automaton admitting a memory set M ⊂ G such that |M | = 1. Show that the following conditions are equivalent: (i) τ is pre-injective; (ii) τ is injective; (iii) τ is surjective. 5.2. Let G be a group and let A be a finite set. Let τ : AG → AG be a nonsurjective cellular automaton. Show that there are uncountably many Garden of Eden configurations in AG .

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5.3. Life on Z. Let A = {0, 1} and consider the cellular automaton τ : AZ → AZ with memory set S = {−1, 0, 1} and local defining map μ : AS → A given by   1 if s∈S y(s) = 2 μ(y) = 0 otherwise for all y ∈ AS . (a) Show that τ is not pre-injective. (b) Deduce from (a) that τ is not surjective either. Hint: Use Theorem 4.6.1 and the Garden of Eden Theorem (cf. Exemple 5.3.2(c)). (c) It follows from Proposition 5.1.1 that τ admits a Garden of Eden pattern. Check that the map p : {0, 1, 2, 3, 4, 5, 6, 7, 8} → A defined by p(0) = p(1) = p(3) = p(5) = p(8) = 1 and p(2) = p(4) = p(6) = p(7) = 0 is a Garden of Eden pattern for τ . 5.4. Let G = Z, A = {0, 1} and let τ : AG → AG be the majority action cellular automaton (cf. Example 1.4.3(c)). We have seen in Example 5.2.1(c) that τ is not pre-injective so that, by amenability of the group G (cf. Theorem 4.6.1) and the Garden of Eden Thoerem, τ is not surjective either (cf. Exemple 5.3.2(c)). It follows from Proposition 5.1.1 that τ admits a Garden of Eden pattern. Check that the map p : {1, 2, 3, 4, 5} → A defined by p(1) = p(3) = p(4) = 0 and p(2) = p(5) = 1 is a Garden of Eden pattern for τ . 5.5. Let (A, +) be an abelian group (not necessarily finite). Let τ : AZ → AZ be the cellular automaton defined by τ (x)(n) = x(n − 1) + x(n) + x(n + 1) for all x ∈ AZ and n ∈ Z. Show that τ is surjective. Hint: Given an arbitrary configuration y ∈ AZ , construct a configuration x ∈ AZ such that x(0) = x(1) = 0A and τ (x) = y. 5.6. Take G = Z and A = Z/2Z = {0, 1}. Let x1 ∈ AG be the configuration defined by x1 (n) = 1 for all n ∈ Z. Let x2 ∈ AG be the configuration defined by x2 (0) = 1 and x2 (n) = 0 for all n ∈ Z \ {0}. Let f : AG → AG be the map defined by f (x2 ) = x1 and f (x) = x for all x ∈ AG \ {x2 }. Let g : AG → AG be the map defined by g(x)(n) = x(n + 1) + x(n) for all n ∈ Z and x ∈ AG . Verify that the maps f and g are both pre-injective but that the map g ◦ f is not pre-injective. 5.7. Let G be group and let A be a set. Suppose that σ : AG → AG and τ : AG → AG are pre-injective cellular automata. Show that the cellular automaton τ ◦ σ is pre-injective. 5.8. Let G be a locally finite group and let A be a set. Show that every pre-injective cellular automaton τ : AG → AG is injective. 5.9. Give a direct proof of the Garden of Eden theorem (Theorem 5.3.1) in the case when the group G is locally finite.

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5.10. Life in a tree. Let G = F4 denote the free group based on four generators a, b, c, d. Let A = {0, 1}. Consider the cellular automaton τ : AG → AG with memory set S = {1G , a, b, c, d, a−1 , b−1 , c−1 , d−1 } and local defining map μ : AS → A given by ⎧ ⎧ ⎪ ⎪ ⎨ s∈S y(s) = 3 ⎪ ⎪ ⎨1 if or ⎪ μ(y) = ⎩ ⎪ ⎪ s∈S y(s) = 4 and y((0, 0)) = 1, ⎪ ⎩ 0 otherwise for all y ∈ AS (cf. Example 1.4.3(a)). Show that τ is surjective but not pre-injective. 5.11. Let G be a group. Let E, F and Ω be subsets of G. (a) Show that (Ω −E )−F = Ω −F E and (Ω +E )+F = Ω +F E . (b) Let G = Z, E = {−2, −1, 0, 1, 2}, F = {−1, 0, 1} and Ω = {−10, −9, . . . , −1, 0, 1, . . . , 9, 10}. Check that ∂F (∂E (Ω)) = {−13, −12, −9, −8, 8, 9, 12, 13} and ∂F E (Ω) = {−13, −12, −11, −10, −9, −8, 8, 9, 10, 11, 12, 13}. Deduce that, in general, one has ∂F (∂E (Ω)) = ∂EF (Ω). 5.12. Let G be a group and let E ⊂ G. Determine the sets ∂E (∅) and ∂E (G). 5.13. Let G be a group. Let E and Ω be subsets of G. Show that ∂E (G\Ω) = ∂E (Ω). 5.14. Let G be a group. Let E, Ω1 and Ω2 be subsets of G. Show that ∂E (Ω1 ∪ Ω2 ) ⊂ ∂E (Ω1 ) ∪ ∂E (Ω2 ). Give an example showing that one may have ∂E (Ω1 ∪ Ω2 ) = ∂E (Ω1 ) ∪ ∂E (Ω2 ). 5.15. Let G be a group. Let E, Ω1 and Ω2 be subsets of G. Show that ∂E (Ω1 \ Ω2 ) ⊂ ∂E (Ω1 ) ∪ ∂E (Ω2 ). Give an example showing that one may have ∂E (Ω1 \ Ω2 ) = ∂E (Ω1 ) ∪ ∂E (Ω2 ). 5.16. Let G be a group. Let E and Ω be subsets of G. Show that Ω −gE = Ω −E g −1 , Ω +gE = Ω +E g −1 , and ∂gE (Ω) = ∂E (Ω)g −1 for all g ∈ G. 5.17. Let G be a group. Show that a subset R ⊂ G is syndetic (cf. Exercise 3.39) if and only if there exists a finite subset S ⊂ G such that the set Ω = R−1 satisfies Ω +S = G. 5.18. Let G be a group. Let E and Ω be subsets of G. Show that Ω ⊂ −1 (Ω +E )−E . 5.19. Let G be a group and let A be a set. Let τ : AG → AG be a cellular automaton with memory set M . Let Ω be a subset of G. Show that if two −1 configurations x1 , x2 ∈ AG coincide on Ω +M then the configurations τ (x1 ) and τ (x2 ) coincide on Ω.

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5.20. Let (A, +) be a finite abelian group and let ϕ : A × A → A be a map. Let τ : AZ → AZ be the map defined by τ (x)(n) = x(n)+ϕ(x(n+1), x(n+2)) for all n ∈ Z and x ∈ AZ . (a) Show that τ is a cellular automaton over the group Z and the alphabet A. (b) Show that τ is pre-injective. (c) Show that τ is surjective. 5.21. Let G = Z and A = {0, 1}. Verify that, among the 16 cellular automata τ : AZ → AZ with memory set S = {0, 1} ⊂ Z, there are exactly 6 of them which are surjective, 4 of them which are injective, 4 of them which are bijective, and 10 of them which are neither surjective nor injective. Hint: The Garden of Eden Theorem may help. 5.22. Topological entropy. Let G be an amenable group and let F = (Fj )j∈J be a right Følner net for G. If X is a compact topological space equipped with a continuous action of G, the topological entropy hF (X, G), 0 ≤ hF (X, G) ≤ ∞, of X is defined as follows. Suppose that U = (Ui )i∈I is an open cover of X. We denote by N (U) the smallest integer

n ≥ 0 such that there is a finite subset I0 ⊂ I of cardinality n such that i∈I0 Ui = X (note that there exists such a finite subset I0 ⊂ I by compactness of X). Given a nonempty subset F ⊂ G, we define the open cover UF = (Wα )α∈I F indexed by the set I F = {α : F → I} by setting  Wα = gUα(g) g∈F

for all α ∈ I F . Finally, we set hF (U) = lim sup j

log N (UFj ) |Fj |

and hF (X, G) = sup hF (U), U

where U ranges over all open covers of X. Observe that it is clear from this definition that if X and Y are two compact topological spaces, each equipped with a continuous action of G, such that there exists a G-equivariant homeomorphism f : X → Y , then one has hF (X, G) = hF (Y, G). (a) Let X be a compact space and let U = (Ui )i∈I and V = (Vk )k∈K be two open covers of X. One says that the open cover V is finer than U if, for each k ∈ K, there exists i ∈ I such that Vk ⊂ Ui . Show that if the open cover V is finer than the open cover U then one has N (U) ≤ N (V). (b) Let X be a compact space and let U = (Ui )i∈I be an open cover of X which forms a partition of X (i.e., such that Ui1 ∩ Ui2 = ∅ for all i1 , i2 ∈ I with i1 = i2 ). Show that N (U) = |{i ∈ I : Ui = ∅}|.

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(c) Let A be a finite set and let X ⊂ AG be a subshift. Show that if X is equipped with the action of G induced by the G-shift on AG , then one has hF (X, G) = entF (X). Hint: Consider the open cover T = (Ta )a∈A of X defined by Ta = {x ∈ X : x(1G ) = a} and deduce from (b) that N (TFj ) = |πFj (X)|, where πFj : AG → AFj denotes the projection map. This gives hF (X, G) ≥ hF (T ) = entF (X). To prove hF (X, G) ≤ entF (X), observe that if U = (Ui )i∈I is an arbitrary open cover of X, then there exists a finite subset Ω ⊂ G such that the open cover TΩ is finer than U. Apply (a) to get N (UFj ) ≤ |πFj Ω (X)| and finally use the fact that F is a right Følner net to conclude. Note: It can be shown by using a result due to Ornstein and Weiss (cf. the notes above) that the topological entropy hF (X, G) of a compact space X equipped with a continuous action of an amenable group G is in fact independent of the choice of the right Følner net F. 5.23. Let G be an amenable group and let F = (Fj )j∈J be a right Følner net for G. Let A and B be finite sets. Suppose that X ⊂ AG and Y ⊂ B G are two subshifts such that there exists a bijective continuous G-equivariant map ϕ : X → Y . Show that entF (X) = entF (Y ). Hint: Use Exercise 5.22. 5.24. Let G be an amenable group, F a right Følner net for G, and A a finite set. Let X be a subset of AG and let X denote the closure of X in AG for the prodiscrete topology. Show that one has entF (X) = entF (X). 5.25. Let G be an amenable group and let (Fj )j∈J be a right Følner net for G. Let A be a finite set. Suppose that X (resp. Y ) is a subset of AG having at least two distinct elements. Show that entF (X ∪ Y ) ≤ entF (X) + entF (Y ). 5.26. Let G be an amenable group, A a finite set, F be a right Følner net for G, and X ⊂ AG a subshift. Let F be a nonempty finite subset of G and consider the subshift X [F ] ⊂ B G , where B = AF (cf. Exercise 1.34). Show that entF (X [F ] ) = entF (X). 5.27. Let G be a group, A be a set and X ⊂ AG a subshift. Let also H ⊂ G be a subgroup of G and T ⊂ G a complete set of representatives for the right cosets of H in G, and consider the subshift X (H,T ) ⊂ B H , where B = AH\G (cf. Exercise 1.35). Suppose that G and H are isomorphic and denote by φ : G → H an isomorphism. Let F = (Fj )j∈J be a right Følner net in G so that F  = (φ(Fj ))j∈J is a right Følner net in G. Show that entF (X (H,T ) ) = entF  (X). 5.28. Let G = Z and A = {0, 1}. Set Fn = {0, 1, . . . , n − 1} and recall from Example 4.7.4(b) that F = (Fn )n∈N is a Følner sequence in Z. Show that if X ⊂ AZ is either the even subshift considered in Exercise 1.38 or the golden mean subshift considered in Exercise 1.39 then entF (X) = log ϕ, √ 1+ 5 where ϕ = 2 is the golden number.

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5 The Garden of Eden Theorem

5.29. Entropy of the Morse subshift. Let A = {0, 1} and let x ∈ AN and X ⊂ AZ denote the Thue-Morse sequence (cf. Exercises 3.41) and the Morse subshift (cf. Exercise 3.42) respectively. (a) Show that |Ln (X)| ≤ 8n for all n ≥ 1. Hint: Let n ≥ 1 and denote by k the unique integer such that 2k−1 ≤ n < 2k . Consider the words u = x(0)x(1) · · · x(2k − 1) and v = ι(u) (cf. Exercise 3.41(e)). Observe that any word w ∈ Ln (X) is necessarily a subword of one of the words uv, vv, vu, or uu. Altogether, this gives at most 4 · 2k = 8 · 2k−1 ≤ 8n distinct possibilities for w. (b) Let F = (Fn )n≥1 denote the Følner sequence of Z where Fn = {0, 1, . . . , n − 1}. Deduce from (a) that entF (X) = 0. 5.30. Let A = {0, 1, 2} and set Y = {y ∈ AZ : y(g) ∈ {1, 2} for all g ∈ Z} and X = Y ∪ {x0 }, where x0 ∈ AZ denotes the constant configuration defined by x0 (g) = 0 for all g ∈ Z. (a) Show that X and Y are subshifts of finite type of AZ and that one has the strict inclusion Y  X. (b) Show that X is not irreducible. (c) Check that X and Y have the same entropy entF (X) = entF (Y ) = log 2 with respect to the Følner sequence F = (Fn )n∈N of Z given by Fn = {0, 1, . . . , n − 1}. 5.31. Let G be an amenable group, A a finite set, X ⊂ AG a strongly irreducible subshift, and Y ⊂ AG a nonempty subshift such that Y  X. Let also F = (Fj )j∈J be a right Følner net in G. For a subset Ω ⊂ G we denote by πΩ : AG → AΩ the projection (restriction) map. (a) Show that there exists a finite subset Ω0 ⊂ G such that πΩ0 (Y )  πΩ0 (X). (b) Let Δ ⊂ G be a finite subset such that 1G ∈ Δ and X is Δ-irreducible. Let E = Ω0+Δ and set ξ = |πE (X)|−1 and E  = EE −1 = {ab−1 : a, b ∈ E}. By virtue of Proposition 5.6.3 we can find an (E, E  )-tiling T ⊂ G. For j ∈ J set Tj = {g ∈ T : gE ⊂ Fj }. Show that ξ|πFj (X)| ≤ |πFj \gE (X)| for all g ∈ Tj . (c) Let p ∈ πΩ0 (X) \ πΩ0 (Y ) (cf. (a)) and let x ∈ X such that x|Ω0 = p. For g ∈ G set pg = (gx)|gΩ0 ∈ AgΩ0 . Show that pg ∈ πgΩ0 (X) \ πgΩ0 (Y ). F (d) For j ∈ J and g ∈ Tj denote by πgΩj 0 : πFj (X) → πgΩ0 (X) the projection (restriction) map and let pg ∈ πgΩ0 (X) \ πgΩ0 (Y ) (cf. (c)). Using (b) and F the Δ-irreducibility of X show that |πFj (X)\(πgΩj 0 )−1 (pg )| ≤ (1−ξ)|πFj (X)| for all j ∈ J and g ∈ Tj . (e) For j ∈ J denote by πFj (X)∗ the set of patterns p ∈ πFj (X) such F that πgΩj 0 (p) = pg for all g ∈ Tj . Observe that πFj (X)∗ = πFj (X) \

Fj −1 (pg ) and using the Δ-irreducibility of X and an inductive arg∈Tj (πgΩ0 ) gument based on (d), show that |πFj (X)∗ | ≤ (1 − ξ)|Tj | |πFj (X)|.

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(f) Observe that πFj (Y ) ⊂ πFj (X)∗ and deduce from (e) that log |πFj (Y )| ≤ |Tj | log(1 − ξ) + log |πFj (X)|. (g) By Proposition 5.6.4 there exists a real number α > 0 and an element j0 ∈ J such that |Tj | ≥ α|Fj | for all j ≥ j0 . Deduce from (f) that entF (Y ) ≤ α log(1 − ξ) + entF (X). (h) Deduce from (g) that entF (Y ) < entF (X). 5.32. Let G be an amenable group and let A be a finite set. Let F = (Fj )j∈J be a right Følner net in G. Let X ⊂ AG be a nonempty strongly irreducible subshift. Show that if X is not minimal then one has entF (X) > 0. Hint: Show that, with the notation introduced in Exercise 5.31, one has entF (X) ≥ α log 2. 5.33. Let G be a group and let A be a finite set. Let Δ be a finite subset of G and let X ⊂ AG be a Δ-irreducible subshift. Suppose that (Ωi )i∈I is a (possibly infinite) family of (possibly infinite) subsets of G such that ⎛ ⎞  Ωi+Δ ∩ ⎝ Ωk ⎠ = ∅ for all i ∈ I. k∈I\{i}

Let also (xi )i∈I be a family of configurations in X. Denote by Pf (G) the set of all finite subsets of G. For each Λ ∈ Pf (G) let X(Λ) ⊂ X denote the set consisting of all configurations in X which coincide with xi on Λ ∩ Ωi for all i ∈ I. (a) Show that X(Λ) is closed in X for each Λ ∈ Pf (G). (b) Show that if we fix Λ ∈ Pf (G), then the subsets Ψi = Λ ∩ Ωi are all contained in Λ and satisfy ⎛ ⎞  Ψk ⎠ = ∅ for all i ∈ I. Ψi+Δ ∩ ⎝ k∈I\{i}

(c) Fix Λ ∈ Pf (G) and set IΛ = {i ∈ I : Λ ∩ Ωi = ∅}. Show that IΛ is finite. Then, applying induction on the cardinality of IΛ , show that, by Δ-irreducibility of X, one has X(Λ) = ∅. (d) Show that X(Λ1 ) ∩ X(Λ2 ) ∩ · · · ∩ X(Λn ) = X(Λ1 ∪ Λ2 ∪ · · · ∪ Λn ) and deduce that X(Λ1 )∩X(Λ2 )∩· · ·∩X(Λn ) = ∅ for all Λ1 , Λ2 , . . . , Λn ∈ Pf (G). (e) Deduce from (d) that the family (X(Λ))Λ∈Pf (G) has a nonempty intersection. (f) Let x ∈ X be such that x ∈ X(Λ) for each finite subset Λ ⊂ G (whose existence is guaranteed by (e)). Show that x coincides with xi on Ωi for all i ∈ I. 5.34. Let G be an amenable group and let A be a finite set. Let F = (Fj )j∈J be a right Følner net in G. Let X ⊂ AG be a (possibly minimal) strongly irreducible subshift containing at least two distinct configurations x0 and x1 .

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5 The Garden of Eden Theorem

(a) Show that there exists a finite subset D ⊂ G such that (gx0 )|gD = (gx1 )|gD for all g ∈ G. (b) Let Δ be a finite subset of G such that X is Δ-irreducible and 1G ∈ Δ. Set E = D+Δ . By Proposition 5.6.3 there exists an (E, E  )-tiling T ⊂ G for some finite subset E  ⊂ G. Consider the subset Z ⊂ X consisting of all the configurations z ∈ X such that, for all g ∈ T , one has either z|gD = (gx0 )|gD or z|gD = (gx1 )|gD . Applying Exercise 5.33 to the family (gD)g∈T show that, given any map ι : T → {0, 1}, there exists a configuration x ∈ X such that x|gD = (gxι(g) )|gD for all g ∈ T . (c) Deduce that |ZF j | ≥ 2|Tj | for all j ∈ J, where, Tj = {g ∈ T : gE ⊂ Fj }. (d) By Proposition 5.6.4 there exists a real number α > 0 and an element j0 ∈ J such that |Tj | ≥ α|Fj | for all j ≥ j0 . Deduce from (c) that entF (Z) ≥ α log 2. (e) Deduce that entF (X) > 0. 5.35. Let G be a group and let A be a finite set. Let X ⊂ AG be a strongly irreducible subshift. Suppose that Δ is a finite subset of G such that X is Δ-irreducible. Show that if Ω1 and Ω2 are (possibly infinite) subsets of G such that Ω1+Δ ∩ Ω2 = ∅, then, given any two configurations x1 and x2 in X, there exists a configuration x ∈ X which coincides with x1 on Ω1 and with x2 on Ω2 . Hint: For each finite subset Ω ⊂ G, consider the subset X(Ω) ⊂ X consisting of all configurations in X which coincide with x1 on Ω ∩ Ω1 and with x2 on Ω ∩ Ω2 . Observe that the family formed by the sets X(Ω), where Ω runs over all finite subsets of G is a family of closed subsets of X having the finite intersection property and use the compactness of X. 5.36. (cf. Lemma 5.8.3) Let G be an amenable group, F = (Fj )j∈J a right Følner net for G, and A a finite set. Let τ : X → Y be a cellular automaton, where X, Y ⊂ AG are subshifts such that X is strongly irreducible and entF (Y ) < entF (X). (a) Let S ⊂ G be a memory set for τ . Up to enlarging the subset S if necessary, we can also suppose that 1G ∈ S and that X is S-irreducible. Deduce from the hypothesis entF (Y ) < entF (X) that there exists j0 ∈ J such that |πF +S2 (Y )| < |πFj0 (X)|. j0

(b) Fix an arbitrary configuration x0 ∈ X and consider the finite subset Z ⊂ X consisting of all configurations z ∈ X which coincide with x0 on G \ . Show that πFj0 (Z) = πFj0 (X). Hint: Use the result of Exercise 5.35(a) Fj+S 0 by taking Ω1 = Fj0 and Ω2 = G \ Fj+S . 0 (c) Use Proposition 5.4.3 to show that τ (z) coincide with τ (x0 ) on G\Fj+S 0 for all z ∈ Z. (d) Deduce from (c) that |τ (Z)| = |πF +S2 (τ (Z))|. j0

2

(e) Using the fact that τ (Z) ⊂ Y , deduce from (c), (a) and (b) that there exist configurations z1 = z2 in Z such that τ (z1 ) = τ (z2 ). (f) Deduce from (e) that τ is not pre-injective.

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5.37. Let G be an amenable group, F = (Fj )j∈J a right Følner net for G, and A a finite set. Let X, Y ⊂ AG be two strongly irreducible subshifts such that entF (X) = entF (Y ). Show that every pre-injective cellular automaton τ : X → Y is surjective. Hint: Use the results of Exercise 5.31 and Exercise 5.36. 5.38. Let G be an amenable group and let A be a finite set. Let X ⊂ AG be a strongly irreducible subshift. Show that every pre-injective cellular automaton τ : X → X is surjective. Hint: Use Exercise 5.37 with Y = X. 5.39. Let G be an amenable group and let A be a finite set. Show that every strongly irreducible subshift X ⊂ AG is surjunctive. Hint: This immediately follows from Exercise 5.38. 5.40. (cf. Lemma 5.8.4) Let G be an amenable group, F = (Fj )j∈J a right Følner net for G, and A a finite set. Let X ⊂ AG be a strongly irreducible subshift of finite type and let τ : X → AG be a cellular automaton which is not pre-injective. Let x1 , x2 ∈ X such that τ (x1 ) = τ (x2 ) and Ω = {g ∈ G : x1 (g) = x2 (g)} is a nonempty finite subset of G. Let S be a memory set for both τ and X. Up to enlarging the subset S if necessary, we can also suppose −1 that 1G ∈ S and that X is S-irreducible. Set E = Ω +(S S) = {ωs−1 s : ω ∈ Ω, s, s ∈ S} and E  = EE −1 = {ab−1 : a, b, ∈ E}. By Proposition 5.6.3 there exists an (E, E  )-tiling T ⊂ G. Let Z ⊂ X be the set consisting of all configurations z in X such that z|tE = (tx1 )|tE for all t ∈ T . (a) Using the S-irreducibility of X, show that entF (Z) < entF (X). Hint: Use the arguments in Exercise 5.31. (b) From (a) and Proposition 5.7.3 deduce that entF (τ (Z)) ≤ entF (X). (c) Using the fact that S is a memory set for X, show that for every x ∈ X the configuration z defined by  tx2 (g) if there is t ∈ T such that g ∈ tE and x|tE = tx1 |tE , z(g) = x(g) otherwise. satisfies z ∈ Z and τ (z) = τ (x) (cf. the end of the proof of Lemma 5.8.4). (d) Deduce from (c) that τ (Z) = τ (X). (e) Deduce from (b) and (d) that entF (τ (X)) < entF (X). 5.41. Let G be an amenable group, F = (Fj )j∈J a right Følner net for G, and A a finite set. Let X, Y ⊂ AG be two subshifts such that X is strongly irreducible of finite type and entF (X) = entF (Y ). Show that every surjective cellular automaton τ : X → Y is pre-injective. Hint: Use the result of Exercise 5.40. 5.42. The Garden of Eden theorem for strongly irreducible subshifts of finite type. (cf. [Fio2, Theorem 4.7]). Let G be an amenable group, A a finite set, X ⊂ AG a strongly irreducible subshift of finite type, and τ : X → X a cellular automaton. Show that τ is pre-injective if and only if it is surjective.

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5 The Garden of Eden Theorem

Hint: Combine together the results from Exercise 5.41 with Y = X and Exercise 5.38. 5.43. Let G be a group. Let F be a nontrivial finite abelian group and set A = F × F . Suppose that G contains an element h0 ∈ G of infinite order (e.g. G = Z). Denote by H ⊂ G the infinite cyclic subgroup generated by h0 . For each x ∈ AG , let x1 , x2 : G → F be the maps defined by x(g) = (x1 (g), x2 (g)) for all g ∈ G. Consider the subset X ⊂ AG consisting of all configurations x ∈ AG which satisfy x2 (gh0 ) = x2 (g) for any g ∈ G. In other words, X is formed by all the configurations x ∈ AG such that x2 is constant on each left coset of H in G. (a) Show that X is a subshift of finite type. (b) Consider the map τ : X → X defined by τ (x) = (x1 + x2 , 0) for all x = (x1 , x2 ) ∈ X (here 0 denotes the zero configuration, i.e., the constant configuration whose value at each element of G is equal to the identity element of F ). Show that τ is a cellular automaton. (c) Show that the cellular automaton τ defined in (b) is pre-injective but not surjective. (d) Show that if H is of infinite index in G (this is the case, for example, when G = Z2 ) then the subshift X is irreducible. 5.44. Let G be a group and let A be a set. Let H be a subgroup of G. Suppose that σ : AH → AH is a cellular automaton and let σ G : AG → AG be the induced cellular automaton (cf. Sect. 1.7). Let X ⊂ AH be a subshift and denote by X (G) ⊂ AG the associated subshift defined in Exercise 1.33. Suppose that σ(X) ⊂ X. Show that the cellular automaton σ G |X (G) : X (G) → X (G) is pre-injective if and only if the cellular automaton σ|X : X → X is pre-injective. 5.45. Let A = {0, 1}. Let x0 , x1 ∈ AZ denote the two constant configurations defined by x0 (n) = 0 and x1 (n) = 1 for all n ∈ Z. (a) Show that X = {x0 , x1 } is a subshift of finite type. (b) Show that the map τ : X → X given by τ (x0 ) = τ (x1 ) = x0 is a cellular automaton which is pre-injective but neither surjective nor injective. 5.46. Let A = {0, 1}, G = Z2 , and H = Z × {0} ⊂ G. Consider the subset X = Fix(H) ⊂ AG consisting of all the configurations x ∈ AG which are fixed by each element of H. (a) Show that X is an irreducible subshift of finite type of AG . (b) Consider the map τ : X → X defined by τ (x)(g) = 0 for all x ∈ X and g ∈ G. Show that τ is a cellular automaton which is pre-injective but neither surjective nor injective. 5.47. ([Fio1, Counterexample 4.27]) Let A = {0, 1, 2} and let X ⊂ AZ be the subshift of finite type with defining set of forbidden words {01, 02}. (a) Show that X is not irreducible.

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(b) Consider the cellular automaton σ : AZ → AZ with memory set S = {0, 1} and local defining map  y(0) if y(1) = 0 μ(y) = 0 otherwise. Show that σ(X) ⊂ X. (c) Show that the cellular automaton σ|X : X → X is surjective but not pre-injective. 5.48. Let A, X ⊂ AZ and σ : AZ → AZ as in Exercise 5.47. Let H = Z×{0} ⊂ 2 Z2 and let X(Z2 ) be the associated irreducible subshift of finite type in AZ 2 2 2 defined as in Exercise 1.33 and let σ Z : AZ → AZ be the induced cellular 2 2 Z2 automaton. Show that the cellular automaton σ |X (Z2 ) : X (Z ) → X (Z ) is surjective but not pre-injective. 5.49. ([Fio1, Section 3]) Let A = {0, 1} and let X ⊂ AZ be the even subshift (cf. Exercise 1.38). Let also σ : AZ → AZ be the cellular automaton with memory set S = {0, 1, 2, 3, 4} and local defining map μ : AS → A given by  1 if y(0)y(1)y(2) ∈ {000, 111} or y(0)y(1)y(2)y(3)y(4) = 00100 μ(y) = 0 otherwise. (a) Show that σ(X) ⊂ X. (b) Set τ = σ|X : X → X. Show that τ is surjective. (c) Show that τ is not pre-injective. Hint: Show that the configurations x1 = · · · 0000(100100)0000 · · · and x2 = · · · 0000(011100)0000 · · · satisfy τ (x1 ) = τ (x2 ) = · · · 1111(100100)1111 · · ·

Chapter 6

Finitely Generated Amenable Groups

This chapter is devoted to the growth and amenability of finitely generated groups. The choice of a finite symmetric generating subset for a finitely generated group defines a word metric on the group and a labelled graph, which is called a Cayley graph. The associated growth function counts the number of group elements in a ball of radius n with respect to the word metric. We define a notion of equivalence for such growth functions and observe that the growth functions associated with different finite symmetric generating subsets are in the same equivalence class (Corollary 6.4.5). This equivalence class is called the growth type of the group. The notions of polynomial, subexponential and exponential growth are introduced in Sect. 6.5. In Sect. 6.7 we give an example of a finitely generated metabelian group with exponential growth. We prove that finitely generated nilpotent groups have polynomial growth (Theorem 6.8.1). In Sect. 6.9 we consider the Grigorchuk group. It is shown that it is an infinite periodic (Theorem 6.9.8), residually finite (Corollary 6.9.5) finitely generated group of intermediate growth (Theorem 6.9.17). In the subsequent section, we show that every finitely generated group of subexponential growth is amenable (Theorem 6.11.2). In Sect. 6.12 we prove the Kesten-Day characterization of amenability (Theorem 6.12.9) which asserts that a group with a finite (not necessarily symmetric) generating subset is amenable if and only if 0 is in the 2 -spectrum of the associated Laplacian. Finally, in Sect. 6.13 we consider the notion of quasi-isometry for not necessarily countable groups and we show that amenability is a quasi-isometry invariant (Theorem 6.13.23).

6.1 The Word Metric Let G be a group. One says that a subset S ⊂ G generates G, or that S is a generating subset of G, if every element g ∈ G can be expressed as a product of elements in T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 6, © Springer-Verlag Berlin Heidelberg 2010

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S ∪ S −1 , that is, for each g ∈ G there exist n ≥ 0, s1 , s2 , . . . , sn ∈ S and ε1 , ε2 , . . . , εn ∈ {1, −1} such that g = sε11 sε22 · · · sεnn .

(6.1)

A subset S ⊂ G is called symmetric if S = S −1 , that is, s−1 ∈ S whenever s ∈ S. Observe that if S ⊂ G is a generating subset of G, then S ∪ S −1 is a symmetric generating subset of G. Recall that G is said to be finitely generated if it admits a finite generating subset. If S ⊂ G is a finite generating subset of G, then S ∪ S −1 is a finite symmetric generating subset of G. Thus any finitely generated group admits a finite symmetric generating subset. Let G be a finitely generated group and let S be a finite symmetric generating subset of G. The S-word-length S (g) = G S (g) of an element g ∈ G is the minimal integer n ≥ 0 such that g can be expressed as a product of n elements in S, that is, S (g) = min{n ≥ 0 : g = s1 s2 · · · sn , si ∈ S, 1 ≤ i ≤ n}.

(6.2)

It immediately follows from the definition that for g ∈ G one has S (g) = 0 if an only if g = 1G .

(6.3)

Proposition 6.1.1. One has S (g −1 ) = S (g)

(6.4)

S (gh) ≤ S (g) + S (h)

(6.5)

and for all g, h ∈ G. Proof. Let g, h ∈ G. Set m = S (g) and n = S (h). Then there exist s1 , s2 , . . . , sm and t1 , t2 , . . . , tn in S such that g = s1 s2 · · · sm and h = −1 −1 so that S (g −1 ) ≤ m = S (g). t1 t2 · · · tn . We have g −1 = s−1 m · · · s2 s1 −1 Exchanging the roles of g and g we get S (g) ≤ S (g −1 ) and therefore −1 S (g ) = S (g). On the other hand, we have gh = s1 s2 · · · sm t1 t2 · · · tn so that S (gh) ≤   m + n = S (g) + S (h). Consider the map dS = dG S : G × G → N defined by dS (g, h) = S (g −1 h) for all g, h ∈ G. Proposition 6.1.2. The map dS is a metric on G.

(6.6)

6.2 Labeled Graphs

153

Proof. Let g, h, k ∈ G. It follows from (6.3) that dS (g, h) = 0 if and only if g = h. Moreover, from (6.4) we deduce that dS (g, h) = dS (h, g). Finally, using (6.5) we get dS (g, k) + dS (k, h) = S (g −1 k) + S (k−1 h) ≥ S ((g −1 k)(k−1 h)) = S (g −1 h) = dS (g, h).   The metric dS is called the word metric on G associated with the finite symmetric generating subset S. Proposition 6.1.3. The metric dS is invariant by left multiplication, that is, dS (gg1 , gg2 ) = dS (g1 , g2 ) for all g, g1 , g2 ∈ G. Proof. We have dS (gg1 , gg2 ) = S ((gg1 )−1 gg2 ) = S (g1−1 g −1 gg2 ) = S (g1−1 g2 ) = dS (g1 , g2 ).   We may rephrase the previous result by saying that the action of G on itself by left multiplication is an isometric action with respect to the word metric. For g ∈ G and n ∈ N, we denote by BSG (g, n) = {h ∈ G : dS (g, h) ≤ n} the ball of radius n in G centered at the element g ∈ G. When g = 1G we have BSG (1G , n) = {h ∈ G : S (h) ≤ n} and we simply write BSG (n) instead of BSG (1G , n). Also, when there is no ambiguity on the group G, we omit the supscript “G” and we simply write BS (g, n) and BS (n) instead of BSG (g, n) and BSG (n).

6.2 Labeled Graphs Let S be a set. An S-labeled graph is a pair Q = (Q, E), where Q is a set, called the set of vertices, and E is a subset of Q×S ×Q, called the set of edges. The projection map λ : E → S, defined by λ(e) = s for all e = (q, s, q  ) ∈ E, is called the labelling map. Also consider the projection maps α, ω : E → Q defined by α(e) = q and ω(e) = q  for all e = (q, s, q  ) ∈ E. Then, α(e) and ω(e) are called the initial and terminal vertices of the edge e ∈ E.

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Let Q1 = (Q1 , E1 ) and Q2 = (Q2 , E2 ) be two S-labeled graphs. An Slabeled graph homomorphism from Q1 to Q2 is a map φ : Q1 → Q2 such that (φ(q), s, φ(q  )) ∈ E2 for all (q, s, q  ) ∈ E1 . An S-labeled graph homomorphism φ : Q1 → Q2 which is bijective and such that the inverse map φ−1 : Q2 → Q1 is also an S-labeled graph homomorphism is called a S-labeled graph isomorphism. If such an S-labeled graph isomorphism exists one says that the S-labeled graphs Q1 and Q2 are isomorphic. An S-labeled graph Q = (Q, E) is said to be finite if the sets Q and E are both finite. Let Q = (Q, E) be an S-labeled graph. An S-labeled subgraph of Q is an S-labeled graph Q = (Q , E  ) such that  Q ⊂ Q and E  ⊂ E. Let Q ⊂ Q and set E  = E ∩ (Q × S × Q ). Then the S-labeled graph  Q = (Q , E  ) is called the S-labeled subgraph of Q induced on Q . Sometimes, by abuse of language, we shall simply denote by Q the S-labeled graph Q = (Q , E  ). If there exist vertices q, q  ∈ Q and labels s1 = s2 in S such that (q, s1 , q  ), (q, s2 , q  ) ∈ E, in other words, if there exist two edges with the same initial and terminal vertices but with different labels, then one says that Q has multiple edges. An edge of the form (q, s, q), q ∈ Q, s ∈ S, is called a loop at q. A path in Q is a finite sequence of edges π = (e1 , e2 , . . . , en ), e1 , e2 , . . . , en ∈ E, such that ω(ei ) = α(ei+1 ) for all i = 1, . . . , n − 1. The integer n is called the length of the path π and it is denoted by (π). The vertices π − = α(e1 ) and π + = ω(en ) are called the initial and terminal vertices of π and one says that π connects π − to π + . The label of a path π = (e1 , e2 , . . . , en ) is defined by λ(π) = (λ(e1 ), λ(e2 ), . . . , λ(en )) ∈ S n . For q ∈ Q, we also admit the empty path starting and ending at q. It has length 0 and its label is the empty word . Let π1 = (e1 , e2 , . . . , en ) and π2 = (e1 , e2 , . . . , em ) be two paths with π1+ = π2− . Then the path π1 π2 = (e1 , e2 , . . . , en , e1 , e2 , . . . , em ) is called the composition of the paths π1 and π2 . Note that (π1 π2 ) = (π1 ) + (π2 ) and that λ(π1 π2 ) = λ(π1 )λ(π2 ). Let q, q  ∈ Q. If there is no path connecting q to q  we set dQ (q, q  ) = ∞, otherwise, we set dQ (q, q  ) = min{(π) : π a path connecting q to q  }.

(6.7)

A path π connecting q to q  with minimal length, that is, such that (π) = dQ (q, q  ), is called a geodesic path from q to q  . Proposition 6.2.1. Let Q = (Q, E) be a labeled graph. Then, for all q, q  , q  ∈ Q one has:

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155

(i) dQ (q, q  ) ∈ N ∪ {∞}; (ii) dQ (q, q  ) = 0 if and only if q = q  ; (iii) dQ (q, q  ) < ∞ if and only if there exists a path which connects q to q  ; (iv) dQ (q, q  ) ≤ dQ (q, q  ) + dQ (q  , q  ) (triangular inequality). Proof. The statements (i), (ii), and (iii) are trivial. Let us prove (iv). If dQ (q, q  ) = ∞ or dQ (q  , q  ) = ∞ there is nothing to prove. Otherwise, let π1 be a geodesic path connecting q to q  and π2 a geodesic path connecting q  to q  . Then π1 π2 connects q to q  and therefore dQ (q, q  ) ≤ (π1 π2 ) = (π1 ) + (π2 ) = dQ (q, q  ) + dQ (q  , q  ).   Note that, in general, for q, q  ∈ Q one has dQ (q, q  ) = dQ (q  , q). For q ∈ Q and r ∈ N we denote by B(q, r) = {q  ∈ Q : dQ (q, q  ) ≤ r} the ball of radius r centered at q. The labeled graph Q is said to be connected if given any two vertices q, q  ∈ Q there exists a path which connects q to q  . Equivalently, Q is connected if and only if dQ (q, q  ) < ∞ for all q, q  ∈ Q. A path π in Q such that π − = π + is said to be closed. The valence (or degree) δ(q) of a vertex q ∈ Q is the cardinality of the set {e ∈ E : α(e) = q}. If δ(q) < ∞ for all q ∈ Q, one says that Q is locally finite. If all vertices have the same finite valence k then one says that Q is regular of degree k. Suppose that S is equipped with an involution s → s. Then the labeled graph Q is said to be edge-symmetric if for all e = (q, s, q  ) ∈ E one has that the inverse edge e−1 = (q  , s, q) also belongs to E. If Q is edge-symmetric and (q, s, q  ) ∈ E one says that the vertices q and q  are neighbors. Suppose that Q is edge-symmetric. If π = (e1 , e2 , . . . , en ) is a path in Q connecting a vertex q to a vertex q  , then π −1 = (en −1 , en−1 −1 , . . . , e2 −1 , e1 −1 ) is a path in Q connecting q  to q. The path π −1 is called the inverse of π. Note that (π −1 ) = (π). Corollary 6.2.2. Suppose that Q is edge-symmetric and connected. Then the map dQ : Q × Q → N defined in (6.7) is a metric on the set Q of vertices of Q. Proof. This follows immediately from Proposition 6.2.1 and the fact that dQ (q, q  ) = dQ (q  , q) for all q, q  ∈ Q. The latter immediately follows from the fact that the map π → π −1 yields a length-preserving bijection from the   set of paths connecting q to q  onto the set of paths connecting q  to q. The metric dQ is called the graph metric on Q. Let π = (e1 , e2 , . . . , en ) be a path in Q. We associate with π the sequence πQ = (q0 , q1 , . . . , qn ) ∈ Qn+1 defined by qi = α(ei+1 ) for all i = 0, 1, . . . , n and qn = ω(en ). The vertices q0 , q1 , . . . , qn are called the vertices visited by π.

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One says that π is simple if qi = qj for all 0 ≤ i = j ≤ n. If π is closed, one says that π is a closed simple path if qi = qj for all 0 ≤ i, j ≤ n such that (i, j) = (0, n). One says that π is proper, or with no back–tracking, if ei+1 = e−1 for i = 1, 2, . . . , n − 1. i An edge-symmetric, connected labeled graph with no loops, no multiple edges and with no closed proper paths is called a tree.

6.3 Cayley Graphs Let G be a finitely generated group and S a finite symmetric generating subset of G. The Cayley graph of G with respect to S is the S-labeled graph CS (G) = (Q, E) whose vertices are the group elements, that is, Q = G and the edge set is E = {(g, s, gs) : g ∈ G and s ∈ S}.

Fig. 6.1 An edge in the Cayley graph CS (G) is a triple (g, s, h), where g ∈ G, s ∈ S, and h = gs

Note that, as S is symmetric, the inverse map s → s−1 is an involution on S. Since we have (h, s−1 , g) ∈ E for all (g, s, h) ∈ E, it follows that the Cayley graph CS (G) is edge-symmetric with respect to the inverse map on S. Moreover, the Cayley graph is connected. For, given g, h ∈ G, as S generates G, we can find a nonnegative integer n and s1 , s2 , . . . , sn ∈ S such that g −1 h = s1 s2 · · · sn . Then the path π = (e1 , e2 , . . . , en ), where ei = (gs1 s2 · · · si−1 , si , gs1 s2 · · · si−1 si ), i = 1, 2, . . . , n, connects g to h = gs1 s2 · · · sn . Also observe that if 1G ∈ S then CS (G) has a loop at each vertex and that, on the contrary, CS (G) has no loops if 1G ∈ / S. On the other hand, CS (G) has no multiple edges, that is, for all g, h ∈ G, there exists at most one s ∈ S such that (g, s, h) ∈ E, namely s = g −1 h if g −1 h ∈ S. Proposition 6.3.1. Let G be a finitely generated group. Let S ⊂ G be a finite symmetric generating subset. Then, the S-distance of two group elements equals the graph distance of the same elements viewed as vertices in the associated Cayley graph CS (G). In other words, dS (g, h) = dCS (G) (g, h)

(6.8)

for all g, h ∈ G. Proof. Let g, h ∈ G and suppose that π = (e1 , e2 , . . . , en ) is a geodesic path connecting g and h. Let λ(π) = (s1 , s2 , . . . , sn ) be the label of π. It then follows that dS (g, h) = S (g −1 h) = S (s1 s2 · · · sn ) ≤ n = (π) = dCS (G) (g, h).

6.3 Cayley Graphs

157

Conversely, suppose that dS (g, h) = m. Then we can find s1 , s2 , . . . , sm ∈ S such that g −1 h = s1 s2 · · · sm . Then the unique path π  = (e1 , e2 , . . . , em ) with (π  )− = g and label λ(π  ) = s1 s2 · · · sm clearly satisfies (π  )+ = h, that is, it connects g to h. We deduce that dCS (G) (g, h) ≤ (π  ) = m = dS (g, h). Then (6.8) follows.   Note that, in particular, two distinct elements g and h in G are neighbors in the graph CS (G) if and only if dS (g, h) = 1. For all g ∈ G, the map s → gs is a bijection from S onto the set gS of all neighbors of g in CS (G). In particular, all vertices g in CS (G) have the same degree δ(g) = |S|. Summarizing, we have that a Cayley graph is a connected, edge-symmetric and regular labeled graph. Examples 6.3.2. We graphically represent Cayley graphs by connecting two neighboring vertices by a single directed labeled arc e (see Fig. 6.1). Thus one should think of the inverse edge e−1 as the oppositely directed arc with label λ(e)−1 . (a) Let G = Z and take S = {1, −1} as a finite symmetric generating subset of G. Then the Cayley graph CS (Z) is represented in Fig. 6.2.

Fig. 6.2 The Cayley graph of G = Z for S = {1, −1}

(b) Let G = Z and take S = {1, 0, −1} as a finite symmetric generating subset of G. Then the Cayley graph CS (Z) is represented in Fig. 6.3. Note that as 0 = 1Z ∈ S, we have a loop at each vertex in CS (Z).

Fig. 6.3 The Cayley graph of G = Z for S = {1, 0, −1}

(c) Let G = Z and S = {2, −2, 3, −3}. Then, the corresponding Cayley graph CS (Z) is represented in Fig. 6.4. (d) Let G = Z×(Z/2Z), where Z/2Z = {0, 1} and take S = {(1, 0), (−1, 0), (0, 1)}. Then the Cayley graph CS (G) is represented by the bi-infinite ladder as in Fig. 6.5. (e) Let G = D∞ be the infinite dihedral group, that is, the group of isometries of the real line R generated by the reflections r : R → R and

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Fig. 6.4 The Cayley graph of G = Z for S = {2, −2, 3, −3}

Fig. 6.5 The Cayley graph of G = Z × (Z/2Z) for S = {(1, 0), (−1, 0), (0, 1)}

s : R → R defined by r(x) = −x s(x) = 1 − x

(symmetry with respect to 0) (symmetry with respect to 1/2)

for all x ∈ R. Note that r2 = s2 = 1G . Taking S = {r, s}, the Cayley graph CS (G) is as in Fig. 6.6.

Fig. 6.6 The Cayley graph of G = D∞ for S = {r, s}

(f) Let G be the infinite dihedral group, as in the previous example. Denote by t : R → R the map defined by t(x) = x + 1 for all x ∈ R (translation). Then we have t = sr and hence s = tr. It follows that the set S = {r, t, t−1 } is also a symmetric generating subset of G and the corresponding Cayley graph CS (G) is as in Fig. 6.7. (g) Let G = Z2 and take S = {(1, 0), (−1, 0), (0, 1), (0, −1)} as a finite and symmetric generating subset of G. Then, the corresponding Cayley graph CS (Z2 ) is given in Fig. 6.8. (h) Let G = Z2 and take

6.3 Cayley Graphs

159

Fig. 6.7 The Cayley graph of G = D∞ for S = {r, t, t−1 }

Fig. 6.8 The Cayley graph of G = Z2 for S = {(1, 0), (−1, 0), (0, 1), (0, −1)}

S = {(1, 0), (−1, 0), (0, 1), (0, −1), (1, 1), (−1, 1), (1, −1), (−1, −1)}. Then, the corresponding Cayley graph CS (Z2 ) is as in Fig. 6.9. (i) Let G = FX be the free group based on a nonempty finite set X and take S = X ∪X −1 . Then, the Cayley graph CS (FX ) is a regular tree of degree / S, the Cayley graph CS (FX ) |S| = 2|X| (see Fig. 6.10). Indeed, since 1FX ∈ does not have loops. Moreover, if π is a nontrivial proper path in CS (FX ) with label λ(π) = (s1 , s2 , . . . , sn ) ∈ S ∗ , then, the word w = s1 s2 · · · sn ∈ S ∗ is nonempty and, by definition of properness, it is reduced. Therefore 1FX and h = s1 s2 · · · sn ∈ FX are distinct and we have π + = π − h = π − , that is, π is not closed. This shows that CS (FX ) is a tree.

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6 Finitely Generated Amenable Groups

Fig. 6.9 The Cayley graph of G = Z2 for S = {(1, 0), (−1, 0), (0, 1), (0, −1), (1, 1), (−1, 1), (1, −1), (−1, −1)}

6.4 Growth Functions and Growth Types Let G be a finitely generated group and S ⊂ G a finite and symmetric generating subset of G. The growth function of G relative to S is the function γSG : N → N defined by γSG (n) = |BSG (n)| = |{g ∈ G : S (g) ≤ n}|

(6.9)

for all n ∈ N. When there is no ambiguity we omit the supscript “G” and simply denote it by γS . Note that γS (0) = |BS (0)| = |{1G }| = 1 and that γS (n) ≤ γS (n + 1) for all n ∈ N. Also, as the map (s1 , s2 , . . . , sn ) → s1 s2 · · · sn is a surjection from (S ∪ {1G })n to BS (n), one has γS (n) ≤ |S ∪ {1G }|n

(6.10)

for all n ∈ N. Proposition 6.4.1. Let G be a finitely generated group and let S and S  be two finite symmetric generating subsets of G. Let c = max{S  (s) : s ∈ S}.

6.4 Growth Functions and Growth Types

161

Fig. 6.10 The Cayley graph of G = F2 for S = {a, a−1 , b, b−1 }

Then the following holds. (i) S  (g) ≤ cS (g) for all g ∈ G; (ii) dS  (g, h) ≤ cdS (g, h) for all g, h ∈ G; (iii) BS (n) ⊂ BS  (cn) for all n ∈ N; (iv) γS (n) ≤ γS  (cn) for all n ∈ N. Proof. (i) Let g ∈ G. Suppose that S (g) = n. Then there exist s1 , s2 , . . . , sn ∈ S such that g = s1 s2 · · · sn . Using (6.5) we get S  (g) = S  (s1 s2 · · · sn ) ≤

n 

S  (si ) ≤ cn.

i=1

(ii) By applying (i) we have, for all g, h ∈ G, dS  (g, h) = S  (g −1 h) ≤ cS (g −1 h) = cdS (g, h).

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6 Finitely Generated Amenable Groups

Finally, from (ii), we have that dS  (g, 1G ) ≤ cn if dS (g, 1G ) ≤ n for all g ∈ G. This gives (iii). Thus γS (n) = |BS (n)| ≤ |BS  (cn)| = γS  (cn) for all n ∈ N.

 

Two metrics d and d on a set X are said to be Lipschitz-equivalent if there exist constants c1 , c2 > 0 such that c1 d(x, y) ≤ d (x, y) ≤ c2 d(x, y) for all x, y ∈ X. Corollary 6.4.2. Let G be a finitely generated group and let S and S  be two finite symmetric generating subsets of G. Then, the word metrics dS and dS  are Lipschitz-equivalent.   A non-decreasing function γ : N → [0, +∞) is called a growth function. Let γ, γ  : N → [0, +∞) be two growth functions. One says that γ  dominates γ, and one writes γ  γ  , if there exists an integer c ≥ 1 such that γ(n) ≤ cγ  (cn) for all n ≥ 1. One says that γ and γ  are equivalent and one writes γ ∼ γ  if γ  γ  and γ   γ. Proposition 6.4.3. We have the following: (i)  is reflexive and transitive; (ii) ∼ is an equivalence relation; (iii) let γ1 , γ2 , γ1 , γ2 : N → [0, +∞) be growth functions. Suppose that γ1 ∼ γ1 , γ2 ∼ γ2 and that γ1  γ2 . Then γ1  γ2 . Proof. It is clear that  is reflexive. Let γ1 , γ2 , γ3 : N → [0, +∞) be growth functions. Suppose that γ1  γ2 and that γ2  γ3 . Let c1 and c2 be positive integers such that γ1 (n) ≤ c1 γ2 (c1 n) and γ2 (n) ≤ c2 γ3 (c2 n) for all n ≥ 1. Then, taking c = c1 c2 one has γ1 (n) ≤ c1 γ2 (c1 n) ≤ c1 c2 γ3 (c2 c1 n) = cγ3 (cn) for all n ≥ 1. Thus  is also transitive. This shows (i). Property (ii) immediately follows from (i) and the definition of ∼. Finally, suppose that γ1 , γ2 , γ1 , γ2 satisfy the hypotheses of (iii). Then we have in particular, γ1  γ1 , γ1  γ2 and γ2  γ2 . From (i) we deduce that   γ1  γ2 . Let γ : N → [0, +∞) be a growth function. We denote by [γ] the ∼equivalence class of γ. By abuse of notation, we shall also write [γ] ∼ γ(n).

6.4 Growth Functions and Growth Types

163

If γ1 and γ2 are two growth functions, we write [γ1 ]  [γ2 ] if γ1  γ2 . This definition makes sense by virtue of Proposition 6.4.3(iii). Note that, this way,  becomes a partial ordering on the set of equivalence classes of growth functions. Examples 6.4.4. (a) Let α and β be nonnegative real numbers. Then nα  nβ if and only if α ≤ β, and nα ∼ nβ if and only if α = β. (b) Let γ : N → [0, +∞) be a growth function. Suppose that γ is a polynomial of degree d for some d ≥ 0. Then one has γ(n) ∼ nd . (c) Let a, b ∈ (1, +∞). Then an ∼ bn .

(6.11)

Indeed, suppose for instance that a ≤ b. On the one hand we have an ≤ bn for all n ≥ 1, so that trivially an  bn . On the other, setting c = [loga b] + 1 > 1 (here [ · ] denotes the integer part), one has bn = (aloga b )n = a(loga b)n ≤ acn ≤ cacn for all n ≥ 1, so that bn  an . We deduce that an ∼ bn . In particular, we have an ∼ exp(n) for all a ∈ (1, +∞). (d) Let d ≥ 0 be an integer. Then nd  exp(n) and nd ∼ exp(n). As a consequence if γ : N → [0, +∞) is a growth function such that γ(n)  nd , nd then γ  exp(n) and γ ∼ exp(n). Indeed, since limn→∞ exp(n) = 0, the d

n sequence ( exp(n) )n≥1 is bounded and we can find an integer c ≥ 1 such that nd exp(n)

≤ c for all n ≥ 1. It follows that nd ≤ c exp(n) ≤ c exp(cn) for all n ≥ 1 and therefore nd  exp(n). On the other hand, suppose by contradiction that exp(n)  nd . Then we can find an integer c > 0 such that exp(n) ≤ c(cn)d for all n ≥ 1. But then exp(n) ≤ cd+1 for all n ≥ 1 contradicting the fact that limn→∞ exp(n) = +∞. nd nd d Thus n ∼ exp(n). The remaining statements concerning the growth function γ immediately follow from transitivity of  (cf. Proposition 6.4.3(i)) and symmetry of ∼ (cf. Proposition 6.4.3(ii)). From Proposition 6.4.1(iii) and (6.10) we deduce the following: Corollary 6.4.5. Let G be a finitely generated group and let S and S  be two finite symmetric generating subsets of G. Then, the growth functions associated with S and S  are equivalent, that is, γS ∼ γS  . Moreover, γS (n)  exp(n).   Let G be a finitely generated group. The equivalence class [γS ] of the growth functions associated with the finite symmetric generating subsets S of G is called the growth type of G and we denote it by γ(G).

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6 Finitely Generated Amenable Groups

Proposition 6.4.6. Let γ : N → [0, +∞) be a growth function with γ(0) > 0. Then γ ∼ 1 if and only if γ is bounded. Proof. Suppose first that γ is bounded. Then we can find an integer c ≥ 1 such that γ(n) ≤ c for all n ≥ 1. It follows that γ(n) ≤ c1(n) ≤ c1(cn) for 1 all n ≥ 1. Thus γ  1. On the other hand, setting c = [ γ(0) ] + 1, we have 1(n) = 1 ≤ cγ(0) ≤ cγ(n) ≤ cγ(cn) for all n ≥ 1 so that 1  γ. This shows that γ ∼ 1. Conversely, suppose that γ ∼ 1. Then γ  1 and we can find an integer c ≥ 1 such that γ(n) ≤ c1(cn) = c for all n ≥ 1. This shows that γ is bounded.   Corollary 6.4.7. Let G be a finitely generated group. Then γ(G) ∼ 1 if and only if G is finite. As a consequence, all finite groups have the same growth type. Proof. Let S be a finite and symmetric generating subset of G. Suppose that γ(G) ∼ γS (n) ∼ 1. Then, by Proposition 6.4.6 we deduce that γS is bounded, say by an integer c ≥ 1. This shows that |G| ≤ c and therefore G is finite. Conversely, if G is finite, we have γS (n) = |BS (n)| ≤ |G| for all n ≥ 1, that is,  γS is bounded. From Proposition 6.4.6 we deduce that γ(G) ∼ γS (n) ∼ 1.  Proposition 6.4.8. Let G be an infinite finitely generated group. Then n  γ(G). Proof. Let S be a finite symmetric generating subset of G. Consider the inclusions {1G } = BS (0) ⊂ BS (1) ⊂ BS (2) ⊂ · · · ⊂ BS (n) ⊂ BS (n + 1) ⊂ · · · (6.12) Let us show that if BS (n) = BS (n+1) for some n ∈ N , then BS (n) = BS (m) for all m ≥ n. We proceed by induction on m. Suppose that BS (n) = BS (m) for some m ≥ n + 1. For all g ∈ BS (m + 1) there exist g  ∈ BS (m) and s ∈ S such that g = g  s. By the inductive hypothesis, g  ∈ BS (m − 1) so that g = g  s ∈ BS (m − 1)S ⊂ BS (m). Since BS (m) ⊂ BS (m + 1), it follows that BS (m + 1) = BS (m) = BS (n). As a consequence, if BS (n) = BS (n + 1) for some n ∈ N, then we have G = BS (n). Since, by our assumptions, G is infinite, we deduce that all the inclusions in (6.12) are strict. It follows that for all n ∈ N we have  n ≤ |BS (n)| = γS (n). This shows that n  γS (n) and therefore n  γ(G).  Definition 6.4.9. Let G be a finitely generated group. One says that G has exponential (resp. subexponential ) growth if γ(G) ∼ exp(n) (resp. γ(G) ∼ exp(n)). One says that G has polynomial growth if there exists an integer d ≥ 0 such that γ(G)  nd .

6.4 Growth Functions and Growth Types

165

Proposition 6.4.10. Every finitely generated group of polynomial growth has subexponential growth. Proof. Let G be a finitely generated group. It follows from Example 6.4.4(d) that if γ(G)  nd for some integer d ≥ 0, then γ(G) ∼ exp(n).   Examples 6.4.11. (a) Let G = Z. With S = {1, −1} one has that the ball of radius r centered at the element g ∈ G is the interval [g − r, g + r] = {n ∈ Z : g − r ≤ n ≤ g + r}, see Fig. 6.11. We have γS (n) = 2n + 1. It follows that γ(Z) ∼ γS (n) ∼ n. In particular, Z has polynomial growth.

Fig. 6.11 The ball BS (n, r) ⊂ Z with S = {1, −1}; here r = 2

(b) Let G = Z2 and S = {(1, 0), (−1, 0), (0, 1), (0, −1)}. Then the ball of radius r centered at the element g = (n, m) ∈ G is the diagonal square with vertices (n + r, m), (n − r, m), (n, m − r), (n, m + r), see Fig. 6.12. In the 2 language of cellular automata, the ball BSZ (g, 1) is  commonly called the von n Neumann neighborhood of g. We have γS (n) = 1 + k=1 4k = 2n2 + 2n + 1. 2 2 2 It follows that γ(Z ) ∼ γS (n) ∼ n . In particular, Z has polynomial growth.

Fig. 6.12 The ball BS (n, r) ⊂ Z2 with S = {(1, 0), (−1, 0), (0, 1), (0, −1)} and r = 2

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6 Finitely Generated Amenable Groups

(c) Let G = Z2 and S = {(1, 0), (−1, 0), (0, 1), (0, −1), (1, 1), (−1, 1), (1, −1), (−1, −1)}. Then the ball of radius r centered at the element g = (n, m) ∈ G is the square [n−r, n+r]×[m−r, m+r], see Fig. 6.13. In the language of cellular automata, 2 the ball BSZ (g, 1) is commonly called the Moore neighborhood of g. We have 2 γSZ (n) = 4n2 + 4n + 1 = (2n + 1)2 = (γSZ (n))2 . Note that this yields again γ(Z2 ) ∼ γS (n) ∼ n2 and the polynomial growth of Z2 (cf. Example 6.4.11(b) above).

Fig. 6.13 The ball BS (n, r) ⊂ Z2 with S = {±(1, 0), ±(0, 1), ±(1, 1), ±(1, −1)} and r = 2

¯ (−1, 0), ¯ (1, ¯1), (−1, ¯1)}. Then the (d) Let G = Z × (Z/2Z) and S = {(1, 0), ball of radius r centered at the element (n, ¯ 0) is represented in Fig. 6.14. We deduce that γS (n) = (2n + 1) + 2(n − 1) + 1 = 4n. Thus γ(Z × (Z/2Z)) ∼ γS (n) ∼ n. In particular, Z × (Z/2Z) has polynomial growth. (e) Let G = D∞ be the infinite dihedral group and S = {r, s}. Then the ball of radius n centered at the element g ∈ G is represented in Fig. 6.15. It follows that γS (n) = 2n + 1. Thus γ(D∞ ) ∼ γS (n) ∼ n. In particular, D∞ has polynomial growth.

6.4 Growth Functions and Growth Types

167

Fig. 6.14 The ball BS ((n, 0), r) ⊂ Z × (Z/2Z) with S = {(1, 0), (−1, 0), (0, 1)}; here r = 2

Fig. 6.15 The ball BS (g, n) ⊂ D∞ with S = {r, s}; here n = 3

(f) Let G = D∞ be the infinite dihedral group and S = {r, t, t−1 }. Then the ball of radius r centered at the element g ∈ G is represented in Fig. 6.16. It follows that γS (n) = (2n + 1) + 2(n − 1) + 1 = 4n. Note that this yields again γ(D∞ ) ∼ γS (n) ∼ n and the polynomial growth of D∞ .

Fig. 6.16 The ball BS (g, n) ⊂ D∞ with S = {r, t, t−1 }; here n = 2

(g) Let G = Fk be the free group of rank k ≥ 2. Let {a1 , a2 , . . . , ak } be a −1 −1 free basis and set S = {a1 , a−1 1 , a2 , a2 , . . . , ak , ak }. Then the ball of radius r centered at the element g ∈ G is the finite tree rooted at g of depth r (see Fig. 6.17). We have γSFk (n) = 1 + 2k

n−1 

(2k − 1)j =

j=0

k(2k − 1)n − 1 . k−1

It follows that γ(Fk ) ∼ γS (n) ∼ (2k − 1)n ∼ exp(n). In particular, Fk has exponential growth.

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6 Finitely Generated Amenable Groups

Fig. 6.17 The ball BS (a, 2) ⊂ F2 with S = {a, a−1 , b, b−1 }

6.5 The Growth Rate Lemma 6.5.1. Let (an )n≥1 be a sequence of positive real numbers such that an+m ≤ an am for all n, m ≥ 1. Then the limit lim

n→∞

exists and equals inf n≥1

√ n

an

√ n a . n

Proof. Fix an integer t ≥ 1 and, for all n ≥ 1 write n = qt+r, with 0 ≤ r < t. √ q/n √ Then an ≤ aqt ar ≤ aqt ar and n an ≤ at n ar . As 0 ≤ r/n < t/n and limn→∞ t/n = 0, we have that limn→∞ r/n = 0 and limn→∞ q/n = 1/t. In √ √ √ q/n 1/n 1/t particular limn→∞ at ar = at = t at . Thus, lim supn→∞ n an ≤ t at . √ √ √ This gives lim supn→∞ n an ≤ inf t≥1 t at ≤ lim inf t→∞ t at completing the proof.  

6.5 The Growth Rate

169

Proposition 6.5.2. Let G be a finitely generated group and let S be a finite symmetric generating subset of G. Then, the limit  (6.13) λS = lim n γS (n) n→∞

exists and λS ∈ [1, +∞). Proof. From (6.5) we deduce that BS (n + m) ⊆ BS (n)BS (m) and therefore γS (n + m) = |BS (n + m)| ≤ |BS (n)BS (m)| ≤ |BS (n)| · |BS (m)| = γS (n)γS (m). Thus, the sequence {γS (n)}n≥1 satisfies the hypotheses of the previous lemma and λS exists and is finite. As γS (n) ≥ 1 for all n ∈ N we also have λS ≥ 1.   Definition 6.5.3. The number λS = λG S in (6.13) is called the growth rate of G with respect to S. Proposition 6.5.4. Let G be a finitely generated group and let S be a finite symmetric generating subset of G. Then λS > 1 if and only if G has exponential growth. Proof. Suppose that γ(G) ∼ exp(n). We then have exp(n)  γS so that there exists an integer c ≥ 1 such that en ≤ cγS (cn) for all n ≥ 1. This implies     √ √ √  1 < c e = lim cn en ≤ lim cn cγS (cn) = lim cn c · lim cn γS (cn) = λS . n→∞

n→∞

n→∞

n→∞

Conversely, suppose that λS > 1. By Lemma 6.5.1 we have that γS (n) ≥ λnS .

 n

γS (n) ≥ λS so

This shows that exp(n) ∼ λnS  γS (n). By Corollary 6.4.5 we have γS (n)  exp(n) and therefore γ(G) ∼ γS ∼ exp(n).   Corollary 6.5.5. Let G be a finitely generated group and let S be a finite symmetric generating subset of G. Then λS = 1 if and only if G has subexponential growth.   As an immediate consequence of Proposition 6.5.4 and Proposition 6.4.3(ii) we deduce the following. Corollary 6.5.6. Let G be a finitely generated group and let S and S  be two finite symmetric generating subsets of G. Then λS = 1 (resp. λS > 1) if and   only if λS  = 1 (resp. λS  > 1).

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6 Finitely Generated Amenable Groups

6.6 Growth of Subgroups and Quotients Proposition 6.6.1. Let G be a group and let N be a normal subgroup of G. Suppose that N and the quotient group G/N are both finitely generated. Then G is finitely generated. Proof. Denote by π : G → G/N the quotient homomorphism. Let U ⊂ N and T ⊂ G/N be two finite symmetric generating subsets of N and G/N respectively. Let S ⊂ G be a finite symmetric set such that π(S) ⊃ T and S ⊃ U . Let us prove that S generates G. Let g ∈ G. Then there exist h ≥ 0 and t1 , t2 , . . . , th ∈ T such that π(g) = t1 t2 · · · th . Let s1 , s2 , . . . , sh ∈ S be such that π(si ) = ti for i = 1, 2, . . . , h. Setting g  = s1 s2 · · · sh we then have π(g  ) = π(g). It follows that n = (g  )−1 g ∈ ker(π) = N . Therefore, there exist k ≥ 0 and sh+1 , sh+2 , . . . , sh+k ∈ S such that n = sh+1 sh+2 · · · sh+k . It follows that g = g  n = s1 s2 · · · sh sh+1 , sh+2 · · · sh+k . This shows that S generates G.   Proposition 6.6.2. Let G be a group and let H be a subgroup of finite index in G. Then, G is finitely generated if and only if H is finitely generated. Proof. Let R ⊂ G be a complete set of representatives of the right cosets of H in G such that 1G ∈ R. Note that |R| = [G : H] < ∞. Suppose first that H is finitely generated. Let S ⊂ H be a finite symmetric generating subset of H. Given g ∈ G, there exist r ∈ R and h ∈ H such that g = hr. Let s1 , s2 , . . . , sn ∈ S be such that h = s1 s2 · · · sn . Then g = s1 s2 · · · sn r. This shows that the set S ∪ R is a finite generating subset of G. Suppose now that G is finitely generated and let S be a finite symmetric generating subset of G. Let us show that the finite set S  = RSR−1 ∩ H

(6.14)

generates H. Let h ∈ H and write h = s1 s2 · · · sn where si ∈ S. Then, there exists h1 ∈ H and r1 ∈ R such that s1 = h1 r1 . Note that h1 = 1G s1 r1−1 ∈ S  . By induction, for all i = 2, 3, . . . , n−1 there exist hi ∈ H and ri ∈ R such that ri−1 si = hi ri . Moreover, hi = ri−1 si ri−1 ∈ S  . Thus, setting hn = rn−1 sn we have h = s1 s2 · · · sn −1 = (1G s1 r1−1 )(r1 s2 r2−1 ) · · · (rn−2 sn−1 rn−1 )(rn−1 sn )

= h1 h2 · · · hn−1 hn . −1 −1 Note that hn = h−1 n−1 · · · h2 h1 h ∈ H and, on the other hand, hn = −1 rn−1 sn = rn−1 sn 1G ∈ RSR . Thus also hn ∈ S  . This shows that S  generates H.  

6.6 Growth of Subgroups and Quotients

171

Proposition 6.6.3. Let G be a finitely generated group and let H be a finitely generated subgroup of G. Then γ(H)  γ(G). Proof. Let SG (resp. SH ) be a finite symmetric generating subset of G (resp. H). Then the set S = SH ∪ SG is a finite symmetric generating subset of G. As SH ⊂ S we have BSHH (n) ⊂ BSG (n) and therefore γSHH (n) ≤ γSG (n) for all n ∈ N. Thus, γ(H)  γ(G).   Corollary 6.6.4. Every finitely generated group which contains a finitely generated subgroup of exponential growth has exponential growth.   From Example 6.4.11(d) and Corollary 6.6.4 we deduce the following. Corollary 6.6.5. Every finitely generated group which contains a subgroup   isomorphic to the free group F2 has exponential growth. In the following proposition we show that finitely generated groups and their finite index subgroups (which are finitely generated as well, by Proposition 6.6.2) have the same growth type. Proposition 6.6.6. Let G be a finitely generated group and let H be a finite index subgroup of G. Then H is finitely generated and γ(H) = γ(G). Proof. Since [G : H] < ∞, we deduce from Proposition 6.6.2 that H is finitely generated. It follows from Proposition 6.6.3 that γ(H)  γ(G). Let now S be a finite symmetric generating subset of G and consider the finite symmetric set S  = RSR−1 ∩ H. It follows from the proof of Proposition 6.6.2 that S  generates H. Let g ∈ BSG (n) and write g = s1 s2 · · · sn , where s1 , s2 , . . . , sn ∈ S. As in the proof of Proposition 6.6.2 we may find r0 = 1G , r1 , . . . , rn ∈ R such that g = s1 s2 · · · sn −1 = (1G s1 r1−1 )(r1 s2 r2−1 ) · · · (rn−2 sn−1 rn−1 )(rn−1 sn rn−1 )rn

= h1 h2 · · · hn−1 hn rn where hi = ri−1 si ri−1 ∈ S  . It follows that BSG (n) ⊂ BSH (n)R so that, taking cardinalities, γSG (n) = |BSG (n)| ≤ |BSH (n)||R| = [G : H]γSH (n) ≤ [G : H]γSH ([G : H]n). This shows that γ(G)  γ(H). It follows that γ(G) = γ(H).

 

Two groups G1 and G2 are called commensurable if there exist finite index subgroups H1 ⊂ G1 and H2 ⊂ G2 such that H1 and H2 are isomorphic. From the previous proposition we immediately deduce: Corollary 6.6.7. If G1 and G2 are commensurable groups and G2 is finitely   generated, then G1 is finitely generated and one has γ(G1 ) = γ(G2 ).

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6 Finitely Generated Amenable Groups

Proposition 6.6.8. Let G be a finitely generated group and let N be a normal subgroup of G. Then the quotient group G/N is finitely generated and one has γ(G/N )  γ(G). If in addition N is finite, then γ(G/N ) = γ(G). Proof. Let S be a finite symmetric generating subset of G and let π : G → G/N denote the canonical quotient homomorphism. Then S  = π(S) is a finite symmetric generating subset of G/N . Thus, for all n ∈ N one has G/N

BS 

(n) = π(BSG (n))

(6.15)

and therefore G/N

γS 

G/N

(n) = |BS 

(n)| ≤ |BSG (n)| = γSG (n).

This shows that γ(G/N )  γ(G). Suppose now that N is finite. From (6.15) we deduce that BSG (n) ⊂ G/N −1 π (BS  (n)) and therefore G/N

γSG (n) = |BSG (n)| ≤ |N ||BS 

G/N

(n)| = |N |γS 

G/N

(n) ≤ |N |γS 

(|N |n).

This shows that γ(G)  γ(G/N ). It follows that γ(G) = γ(G/N ).

 

Lemma 6.6.9. Let γ1 , γ2 , γ1 , γ2 : N → [0, +∞) be growth functions. Suppose that γ1  γ1 , γ2  γ2 . Then the products γ1 γ2 , γ1 γ2 : N → [0, ∞) are also growth functions and one has γ1 γ2  γ1 γ2 . Proof. Since the product of non-decreasing functions is also non-decreasing, it is clear that γ1 γ2 and γ1 γ2 are also growth functions. Let c1 and c2 be positive integers such that γ1 (n) ≤ c1 γ1 (c1 n) and γ2 (n) ≤ c2 γ2 (c2 n) for all n ≥ 1. Taking c = c1 c2 we have (γ1 γ2 )(n) = γ1 (n)γ2 (n) ≤ c1 c2 γ1 (c1 n)γ2 (c2 n) ≤ c1 c2 γ1 (c1 c2 n)γ2 (c1 c2 n) = c(γ1 γ2 )(cn) for all n ≥ 1. This shows γ1 γ2  γ1 γ2 .

 

Given two growth functions γ1 and γ2 we set [γ1 ] · [γ2 ] = [γ1 γ2 ]. This definition makes sense since if γ1 ∼ γ1 and γ2 ∼ γ2 then γ1 γ2 ∼ γ1 γ2 as it immediately follows from Lemma 6.6.9. Proposition 6.6.10. Let G1 and G2 be two finitely generated groups. Then the direct product G1 × G2 is also finitely generated and γ(G1 × G2 ) = γ(G1 )γ(G2 ). Proof. Let S1 and S2 be finite symmetric generating subsets of G1 and G2 . Then the set

6.7 A Finitely Generated Metabelian Group with Exponential Growth

173

S = (S1 × {1G2 }) ∪ ({1G1 } × S2 ) is a finite symmetric generating subset of G1 × G2 . Let (g1 , g2 ) ∈ BSG1 ×G2 (n). Then there exist s1,1 , s2,1 , . . . , sk,1 ∈ S1 and s1,2 , s2,2 , . . . , sh,2 ∈ S2 , where h + k ≤ n, such that (g1 , g2 ) = (s1,1 , 1G2 )(s2,1 , 1G2 ) · · · (sk,1 , 1G2 ) · (1G1 , s1,2 )(1G1 , s2,2 ) · · · (1G1 , sh,2 ) = (s1,1 s2,1 · · · sk,1 , s1,2 s2,2 · · · sh,s ). Thus, BSG1 ×G2 (n) ⊂ BSG11 (n) × BSG22 (n) and γSG1 ×G2 (n) ≤ γSG11 (n)γSG22 (n). This shows that γ(G1 × G2 )  γ(G1 )γ(G2 ). On the other hand, if g1 ∈ BSG11 (n) and g2 ∈ BSG22 (n), then (g1 , g2 ) ∈ BSG1 ×G2 (2n) and one has γSG11 (n)γSG22 (n) ≤ γSG1 ×G2 (2n) ≤ 2γSG1 ×G2 (2n). This shows that γ(G1 )γ(G2 )  γ(G1 × G2 ). It follows that γ(G1 × G2 ) =   γ(G1 )γ(G2 ). From Example 6.4.11(a) and the previous proposition one immediately deduces the following. Corollary 6.6.11. Let d be a positive integer. Then γ(Zd ) ∼ nd .

 

Corollary 6.6.12. Every finitely generated abelian group has polynomial growth. Proof. Let G be a finitely generated abelian group. Then there exist an integer d ≥ 0 and a finite group F such that G is isomorphic to the cartesian product Zd × F . It follows from Proposition 6.6.10, Corollary 6.6.11 and Corollary 6.4.7 that γ(G) = γ(Zd )γ(F ) ∼ nd . This shows that G has polynomial growth.  

6.7 A Finitely Generated Metabelian Group with Exponential Growth We have seen in Corollary 6.6.12 that every finitely generated abelian group has polynomial growth. The purpose of the present section is to give an example of a finitely generated metabelian group with exponential growth. As every metabelian group is solvable and hence amenable (Theorem 4.6.3), this will show in particular that there exist finitely generated amenable groups, and even finitely generated solvable groups, whose growth is exponential. Proposition 6.7.1. Let G denote the subgroup of GL2 (Q) generated by the two matrices     20 11 A= and B = . 01 01 Then G is a finitely generated metabelian group with exponential growth.

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6 Finitely Generated Amenable Groups

Proof. The group G is finitely generated by definition. We have G=

 k  2 r : k ∈ Z, r ∈ Z[1/2] , 0 1

(6.16)

where Z[1/2] is the subring of Q consisting of all dyadic rationals. Indeed, if we denote by H the right-hand side of (6.16), we first observe that G ⊂ H since H is clearly a subgroup of GL2 (Q) containing A and B. On the other hand, we also have the inclusion H ⊂ G since any r ∈ Z[1/2] can be written in the form r = 2m n for some m, n ∈ Z, so that 

2k r 0 1

 = Am B n Ak−m ∈ G.

From (6.16), we deduce that the determinant map yields a surjective homomorphism from G onto an infinite cyclic group whose kernel is isomorphic to the additive group Z[1/2] and is therefore abelian. Since any element in [G, G] has determinant 1, this shows that [G, G] is abelian, that is, that G is metabelian. It remains to show that G has exponential growth. To see this, let us estimate from below the cardinality of the ball BS (3n − 2), where S is the finite symmetric generating subset of G defined by S = {A, B, A−1 , B −1 } and n ≥ 1 is a fixed integer. Consider the subset E ⊂ G consisting of all matrices of the form   1q M (q) = , 01 where q is an integer such that 0 ≤ q ≤ 2n − 1. Developing q in base 2, we get n  q= ui 2i−1 , i=1

where ui ∈ {0, 1} for 1 ≤ i ≤ n. This gives us  M (q) = As we have

 u  u  u  n−1 un 1 1 1 1 2 2 1 22 3 12 ··· . 01 01 0 1 0 1 

1 2i−1 0 1



= Ai−1 BA−(i−1)

for all 1 ≤ i ≤ n, it follows that we can write M (q) in the form M (q) = B u1 AB u2 A−1 A2 B u3 A−2 · · · An−2 B un−1 A−(n−2) An−1 B un A−(n−1) = B u1 AB u2 AB u3 A · · · AB un−1 AB un A−(n−1) .

6.8 Growth of Finitely Generated Nilpotent Groups

175

This shows that S (M (q)) ≤ (2n − 1) + (n − 1) = 3n − 2. We deduce that |BS (3n − 2)| ≥ |E| = 2n . It follows that the growth rate of G with respect to S satisfies  √ √ 3n−2 3 2n = 2 > 1. λS = lim 3n−2 |BS (3n − 2)| ≥ lim n→∞

n→∞

 

Thus G has exponential growth.

6.8 Growth of Finitely Generated Nilpotent Groups In this section we prove the following Theorem 6.8.1. Every finitely generated nilpotent group has polynomial growth. In order to prove this result we need some preliminaries. Lemma 6.8.2. Let G be any group. Let H and K be two normal subgroups of G. Suppose that S ⊆ H and T ⊆ K generate H and K respectively. Then [H, K] is equal to the normal closure in G of the set {[s, t] : s ∈ S, t ∈ T }. Proof. Denote by N the normal closure in G of the set {[s, t] : s ∈ S, t ∈ T } and let us show that N = [H, K]. Since {[s, t] : s ∈ S, t ∈ T } ⊂ [H, K] and [H, K] is a normal subgroup, we have N ⊂ [H, K].

(6.17)

Consider the quotient homomorphism π : G → G/N . For all s ∈ S and t ∈ T we have [π(s), π(t)] = π([s, t]) = 1G/N , that is, π(s) and π(t) commute. It follows that all elements of π(H) commute with all elements of π(K), since S generates H and T generates K. In other words, π([h, k]) = [π(h), π(k)] = 1G/N , for all h ∈ H and k ∈ K. This gives [h, k] ∈ N for all h ∈ H and k ∈ K and therefore [H, K] ⊂ N . From (6.17) we deduce that [H, K] = N .   Let G be a group. Recall that the lower central series of G is the sequence (C i (G))i≥0 of normal subgroups of G defined by C 0 (G) = G and C i+1 (G) = [C i (G), G] for all i ≥ 0. Given elements g1 , g2 , . . . , gi ∈ G, i ≥ 3, we inductively set [g1 , g2 , . . . , gi ] = [[g1 , g2 , . . . , gi−1 ], gi ] ∈ C i−1 (G). (i)

If S ⊂ G and i ≥ 2 we then denote by SG the set consisting of all elements of the form

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6 Finitely Generated Amenable Groups

[s1 , s2 , . . . , si ], s1 , s2 , . . . , si ∈ S.

(6.18)

The elements (6.18) are called simple S-commutators of weight i. Note that (i) SG ⊂ C i−1 (G). Lemma 6.8.3. Let G be any group. Let SG ⊂ G be a generating subset of G. Then for all i ≥ 1, the subgroup C i (G) is the normal closure in G of the set (i+1) SG . Proof. Let us prove the statement by induction on i. We have that C 1 (G) = (2) [G, G] is the normal closure in G of SG = {[s1 , s2 ] : s1 , s2 ∈ SG }, as it follows from Lemma 6.8.2 by taking H = K = G and S = T = SG . Thus, the statement holds for i = 1. Suppose by induction that C i−1 (G) is the normal (i) (i+1) closure in G of SG , denote by N the normal closure in G of SG , and (i+1) let us show that C i (G) = N . Since SG ⊂ C i (G) and C i (G) is a normal subgroup, we deduce that N ⊂ C i (G). (6.19) (i)

Denote by π : G → G/N the quotient map. Let w ∈ SG and s ∈ SG . (i+1) Then, by definition, [w, s] ∈ SG and therefore [w, s] ∈ N . It follows that [π(w), π(s)] = π([w, s]) = 1G/N , that is, π(w) and π(s) commute. As SG generates G, we have that π(SG ) generates G/N and therefore π(w) ∈ Z(G/N ). It follows that π(hwh−1 ) = π(h)π(w)π(h)−1 = π(w) for all h ∈ G, so that π([hwh−1 , s]) = [π(hwh−1 ), π(s)] = [π(w), π(s)] = 1G/N . It follows that [hwh−1 , s] ∈ N.

(6.20)

By the inductive hypothesis, every element in C i−1 (G) can be expressed (i) −1 −1 as a product (h1 w1 h−1 1 )(h2 w2 h2 ) · · · (hm wm hm ) with wk ∈ SG and hk ∈ G, k = 1, 2, . . . , m, for some m ∈ N. By taking H = C i−1 (G), K = G, (i) S = {wh : w ∈ SG , h ∈ G} and T = SG in Lemma 6.8.2, we deduce that i i−1 C (G) = [C (G), G] is the normal closure in G of the set {[hwh−1 , s] : (i) w ∈ SG , h ∈ G, s ∈ SG }. Thus from (6.20) it follows that C i (G) ⊂ N . By using (6.19) this shows that C i (G) = N .   Lemma 6.8.4. Let G be a finitely generated nilpotent group of nilpotency degree d ≥ 1. Then the subgroups C i (G), i = 1, 2, . . . , d − 1 are finitely generated. Proof. Let SG ⊂ G be a finite generating subset of G. We first show that the quotient groups C i (G)/C i+1 (G) are finitely generated, i = 0, 1, . . . , d − 1. From Lemma 6.8.3 we have that C i (G) is the normal closure in G of the (i+1) finite set SG . Therefore, if π : G → G/C i+1 (G) denotes the quotient homomorphism, we have that C i (G)/C i+1 (G) = π(C i (G)) is the normal clo(i+1) (i+1) sure in G/C i+1 (G) of the set π(SG ). But π(SG ) = (π(SG ))(i+1) ⊂ (i+1) Z(C i (G)/C i+1 (G)) and therefore π(SG ) generates C i (G)/C i+1 (G).

6.8 Growth of Finitely Generated Nilpotent Groups

177

Let us now prove the statement by reverse induction starting from i = d−1. It follows from the first part of the proof that the subgroup C d−1 (G) ∼ = C d−1 (G)/{1G } = C d−1 (G)/C d (G) is finitely generated. Suppose by induction that the subgroup C i+1 (G) is finitely generated for some i ≤ d − 2. From the first part of the proof we have that C i (G)/C i+1 (G) is also finitely generated. Therefore from Proposition 6.6.1 we deduce that C i (G) is finitely generated.   We are now in position to prove the main result of this section. Proof of Theorem 6.8.1. Let G be a finitely generated nilpotent group of nilpotency degree d. Let us prove the statement by induction on d. If d = 0 then G = {1G } and therefore G has polynomial growth. Suppose now that d ≥ 1 and that all finitely generated nilpotent groups of nilpotency degree ≤ d − 1 have polynomial growth. We first observe that the subgroup H = C 1 (G) is nilpotent of nilpotency degree ≤ d − 1. Indeed, as one immediately checks by induction, we have C i (H) ⊂ C i+1 (G) for all i = 0, 1, . . . , d − 1, so that C d−1 (H) ⊂ C d (G) = {1G }. Moreover, by Lemma 6.8.4, we have that H is finitely generated. Thus, by the inductive hypothesis we have that H has polynomial growth. Let T ⊂ H be a finite symmetric generating subset of H. Then there exist integers c1 > 0 and q ≥ 0 such that γTH (n) ≤ c1 (c1 n)q

(6.21)

for all n ≥ 1. Let S = {s1 , s2 , . . . , sk } ⊂ G be a finite symmetric generating subset of G. Let g ∈ G and suppose that m = G S (g) ≤ n. Then there exist 1 ≤ i1 , i2 , . . . , im ≤ k such that g = si1 si2 · · · sim .

(6.22)

Since g2 g1 = g1 g2 [g2−1 , g1−1 ] and [g2−1 , g1−1 ] ∈ H for all g1 , g2 ∈ G, we can permute the generators in (6.22) and express g in the form g = sj1 sj2 · · · sjm h

(6.23)

where 1 ≤ j1 ≤ j2 ≤ · · · ≤ jm ≤ k and h ∈ H. Let us estimate H T (h). We set (i) L = max{H T (w) : w ∈ S , 2 ≤ i ≤ d}.

(6.24)

As we observed above, exchanging a pair of consecutive generators produces a simple S-commutator of weight two on their right. It is clear that one needs at most n such exchanges to bring sj1 , where j1 = ip1 = min{ip : p = 1, 2, . . . , m}, to the leftmost place. Analogously, one needs at most n such exchanges to bring sj2 , where j2 = min{ip : p = 1, 2, . . . , m; p = p1 }, to the leftmost but one place (in fact on the right of sj1 ). And so on. Altogether,

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6 Finitely Generated Amenable Groups

there are at most mn ≤ n2 such exchanges. This produces at most n2 simple S-commutators of weight two. But at each step, when moving a generator sjt to the left, we also have to exchange it with all the simple S-commutators on its left that were produced before (namely by sj1 , sj2 , . . . , sjt−1 ). As one easily checks by induction, that this produces, altogether, at most n3 simple S-commutators of weight three, n4 simple S-commutators of weight four, and so on. Continuing this way, since G is nilpotent of nilpotency degree d, all simple S-commutators of weight d+1 are equal to 1G . Thus, the total number of simple S-commutators that are eventually produced in this process is at most n2 + n3 + · · · + nd ≤ dnd . It follows from (6.24) that d H T (h) ≤ Ldn .

(6.25)

On the other hand, we can bound the number of elements in G which are of the form sj1 sj2 · · · sjm , where 1 ≤ j1 ≤ j2 ≤ · · · ≤ jm ≤ k, by c2 nk , where c2 > 0 is a constant independent of n. Indeed, each such group element can be written in the form sn1 1 sn2 2 · · · snk k , where 0 ≤ ni ≤ n for all i = 1, 2, . . . , k. We deduce from (6.21) and (6.25) that γSG (n) ≤ c2 nk c1 (Ldnd )q = Cnδ ≤ C(Cn)δ for all n ≥ 1, where δ = k + qd and C = c1 c2 (Ld)q . Note that C > 0 is a constant independent of n. It follows that G has polynomial growth.   From Theorem 6.8.1 and Proposition 6.6.6 we deduce the following: Corollary 6.8.5. Every finitely generated virtually nilpotent group has polynomial growth.  

6.9 The Grigorchuk Group and Its Growth In this section we present the Grigorchuk group and some of its main properties, namely being infinite, periodic, residually finite and of intermediate growth. Let Σ = {0, 1}. We denote by Σ ∗ = ∪n∈N Σ n the set of all words on the alphabet Σ. Recall that Σ ∗ is a monoid for the word concatenation whose identity element is the empty word (cf. Sect. D.1). Every word w ∈ Σ ∗ may be uniquely written in the form w = σ1 σ2 · · · σn , where σ1 , σ2 , . . . , σn ∈ Σ and n = (w) ∈ N is the length of w. Denote by Sym(Σ ∗ ) the symmetric group on Σ ∗ (cf. Appendix C). We introduce a partial order  in Σ ∗ by setting u  v, u, v ∈ Σ ∗ , if there exists w ∈ Σ ∗ such that uw = v. We then set Sym(Σ ∗ , ) = {g ∈ Sym(Σ ∗ ) : g(u)  g(v) for all u, v ∈ Σ ∗ such that u  v}. (6.26)

6.9 The Grigorchuk Group and Its Growth

179

Note that Sym(Σ ∗ , ) is a subgroup of Sym(Σ ∗ ). Moreover, for w ∈ Σ ∗ the length (w) equals the maximum of n ∈ N such that there exists a sequence (wk )0≤k≤n of distinct words in Σ ∗ such that = w0  w1  · · ·  wn = w. It follows that if g ∈ Sym(Σ ∗ , ) then (g(w)) = (w) for all w ∈ Σ ∗ . Consider the elements a, b, c, d ∈ Sym(Σ ∗ ) defined as follows. We first define a by setting a( ) = (6.27) and a(0w) = 1w,

a(1w) = 0w

(6.28)

for all w ∈ Σ ∗ . Then, for all w ∈ Σ ∗ , we define b(w), c(w), d(w) by induction on (w). We start by setting b( ) = c( ) = d( ) = .

(6.29)

b(0w) = 0a(w), b(1w) = 1c(w), c(0w) = 0a(w), c(1w) = 1d(w), d(0w) = 0w, d(1w) = 1b(w)

(6.30)

Then we set

for all w ∈ Σ ∗ . By an obvious induction we have a, b, c, d ∈ Sym(Σ ∗ , ).

(6.31)

Example 6.9.1. Let w = 10110 ∈ Σ ∗ . Then we have a(w) = a(10110) = 00110, b(w) = b(10110) = 1c(0110) = 10a(110) = 10010, c(w) = c(10110) = 1d(0110) = 10110, d(w) = d(10110) = 1b(0110) = 10a(110) = 10010. Definition 6.9.2. The Grigorchuk group is the subgroup G of Sym(Σ ∗ ) generated by the elements a, b, c, d. Observe that by (6.31) we have G ⊂ Sym(Σ ∗ , ).

(6.32)

Proposition 6.9.3. The following relations hold in G. a2 = b2 = c2 = d2 = 1G

(6.33)

bc = cb = d, dc = cd = b, db = bd = c.

(6.34)

and Proof. To prove (6.33), we have to show that g 2 (w) = w

(6.35)

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6 Finitely Generated Amenable Groups

for all w ∈ Σ ∗ and g ∈ {a, b, c, d}. We have a(ε) = ε by (6.27). Moreover from (6.28) we have a2 (0w) = a(1w) = 0w

and a2 (1w) = a(0w) = 1w,

(6.36)

for all w ∈ Σ ∗ . This shows that a2 = 1G . To prove (6.35) for g = b, c and d, we use induction on (w). By virtue of (6.29) this holds when (w) = 0. Suppose that (6.35) holds when (w) = n. From (6.30) and the fact that a2 = 1G we deduce, for all w ∈ Σ ∗ with (w) = n, b2 (0w) = b(0a(w)) = 0a2 (w) = 0w, c2 (0w) = c(0a(w)) = 0a2 (w) = 0w, d2 (0w) = d(0w) = 0w,

b2 (1w) = b(1c(w)) = 1c2 (w) = 1w, c2 (1w) = c(1d(w)) = 1d2 (w) = 1w, d2 (1w) = d(1b(w)) = 1b2 (w) = 1w.

This shows that (6.35) holds for all w ∈ Σ ∗ with (w) = n + 1. This proves (6.35) for g = b, c, d and (6.33) follows. To prove (6.34), we have to show that ij(w) = k(w)

(6.37)

for all w ∈ Σ ∗ and all distinct i, j, k ∈ {b, c, d}. Again, we can use induction on (w). If (w) = 0 then (6.37) follows from (6.29). Suppose by induction that (6.37) holds when (w) = n. From (6.28),(6.30) and (6.33) we deduce, for all w ∈ Σ ∗ with (w) = n, bc(0w) = b(0a(w)) = 0a2 (w) = 0w = d(0w), bc(1w) = b(1d(w)) = 1cd(w) = 1b(w) = d(1w), cd(0w) = c(0w) = 0a(w) = b(0w), cd(1w) = c(1b(w)) = 1db(w) = 1c(w) = b(1w), db(0w) = d(0a(w)) = 0a(w) = c(0w), db(1w) = d(1c(w)) = 1bc(w) = 1d(w) = c(1w). It follows that (6.37) holds for all w ∈ Σ ∗ with (w) = n + 1 whenever (i, j, k) = (b, c, d), (c, d, b), (d, b, c). Thus, by induction bc = d, cd = b and db = c. Finally, using (6.33) we deduce cb = cd2 b = (cd)(db) = bc, dc = db2 c = (db)(bc) = cd and bd = bc2 d = (bc)(cd) = db. This completes the proof of (6.34).   It follows from (6.33) that the set S = {a, b, c, d} is a symmetric generating subset of G. We denote by S : G → N the corresponding word-length function. Every group element g ∈ G can be expressed in the form g = s1 s2 · · · sn

(6.38)

6.9 The Grigorchuk Group and Its Growth

181

with s1 , s2 , . . . , sn ∈ S. We say that the expression (6.38) is a reduced form of g provided that for all i, j = 1, 2, . . . , n − 1 one has that if si = a then si+1 ∈ {b, c, d}, and if sj ∈ {b, c, d} then sj+1 = a. It immediately follows from (6.33) and (6.34) that every group element g ∈ G can be expressed (not necessarily in a unique way) as in (6.38) in reduced form. For all n ∈ N we set Hn = {g ∈ G : g(w) = w for all w ∈ Σ n }.

(6.39)

Proposition 6.9.4. For all n ∈ N, the set Hn is a normal subgroup of G and [G : Hn ] < ∞. (6.40) Moreover, G = H0 ⊃ H1 ⊃ H2 ⊃ · · · ⊃ Hn ⊃ Hn+1 ⊃ · · · .

(6.41)

Proof. Since g(Σ n ) ⊂ Σ n for all g ∈ G (cf. (6.32)) we may consider the (restriction) map θn : G → Sym(Σ n ) defined by θn (g)(w) = g(w) for all g ∈ G and w ∈ Σ n . Clearly, θn is a homomorphism and ker(θn ) = Hn . This shows that Hn is a normal subgroup and that [G : Hn ] = |G/Hn | = |θn (G)| ≤ | Sym(Σ n )| < ∞. Finally, let n ∈ N, u ∈ Σ n and g ∈ Hn+1 . Let σ ∈ Σ and set w = uσ ∈ n+1 so that u  w. From (6.32) we deduce that g(u)  g(w) = uσ and Σ therefore g(u) = u. This shows that g ∈ Hn . Thus Hn ⊃ Hn+1 and (6.41) follows.   Corollary 6.9.5. The Grigorchuk group G is residually finite. Proof. Let h ∈ ∩n∈N Hn . We have h(w) = w for all w ∈ Σ ∗ and therefore h = 1G . This shows that ∩n∈N Hn = {1G }. As [G : Hn ] < ∞ for all n ∈ N we deduce from Proposition 2.1.11 that G is residually finite.   Proposition 6.9.6. We have: (i) the subgroup H1 consists of all group elements g ∈ G which can be expressed in the form (6.38) (not necessarily reduced) with an even number of occurrences of the generator a; (ii) [G : H1 ] = 2; (iii) the group H1 is generated by the elements b, c, d, aba, aca, ada; (iv) the group H1 equals the normal closure in G of the elements b, c and d. Proof. It follows from (6.28) and (6.30) that a generator s ∈ S satisfies s(σ) = σ for all σ ∈ Σ if and only if s ∈ {b, c, d}. Thus, g = s1 s2 · · · sn , si ∈ S, belongs to H1 if and only if |{i : si = a}| is an even number. This also shows that [G : H1 ] = 2. Finally, let g ∈ H1 . Then it can be expressed in one of the following reduced forms:

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6 Finitely Generated Amenable Groups

g = t0 at1 at2 at3 at4 · · · at2k−1 at2k = t0 (at1 a)t2 (at3 a)t4 · · · (at2k−1 a)t2k , g = at1 at2 at3 at 4 · · · at2k−1 at2k = (at1 a)t2 (at3 a)t4 · · · (at2k−1 a)t2k , g = t0 at1 at2 at3 at4 · · · at2k−1 a = t0 (at1 a)t2 (at3 a)t4 · · · (at2k−1 a), g = at1 at2 at3 at4 · · · at2k−1 a = (at1 a)t2 (at3 a)t4 · · · (at2k−1 a), where t0 , t1 , . . . , t2k+1 ∈ {b, c, d} and k ∈ N (observe that the occurrences of the a’s is 2k). This shows that the set {b, c, d, aba, aca, ada} generates H1 . Now, the normal closure in G of the generators b, c, d is the subgroup H ⊂ G generated by all the conjugates gtg −1 with g ∈ G and t ∈ {b, c, d}. Thus taking g = 1G , a we deduce from (iii) that H1 ⊂ H. On the other hand, since   b, c, d ∈ H1 we also have H ⊂ H1 . Thus H = H1 . Let h ∈ H1 . For all w ∈ Σ ∗ there exist w0 , w1 ∈ Σ ∗ , (w0 ) = (w1 ) = (w) such that h(0w) = 0w0 and h(1w) = 1w1 . Denote by h0 , h1 ∈ Sym(Σ ∗ , ) the maps defined by h0 (w) = w0 and h1 (w) = w1 . We thus have h(0w) = 0h0 (w) and h(1w) = 1h1 (w) for all w ∈ Σ ∗ . We denote by φ0 : H1 → Sym(Σ ∗ , ) (resp. φ1 : H → Sym(Σ ∗ , )) the map defined by φ0 (h) = h0 (resp. φ1 (h) = h1 ) and by φ : H1 → Sym(Σ ∗ , ) × Sym(Σ ∗ , ) the product map φ(h) = (h0 , h1 ). From (6.30) we immediately deduce φ(b) = (a, c), φ(c) = (a, d), φ(d) = (1G , b),

φ(aba) = (c, a), φ(aca) = (d, a), φ(ada) = (b, 1G ).

(6.42)

Proposition 6.9.7. We have: (i) the maps φ0 , φ1 : H1 → G are surjective homomorphisms; (ii) the map φ : H1 → G × G is an injective homomorphism. Proof. Let σ ∈ {0, 1}. For all h, h ∈ H and w ∈ Σ ∗ we have σφσ (hh )(w) = hh (σw) = h(σφσ (h )(w)) = σφσ (h)φσ (h )(w). This shows that φσ (hh ) = φσ (h)φσ (h ). It follows that φ0 and φ1 are homomorphisms. From (6.42) and Proposition 6.9.6(iii) we immediately deduce that φ0 (H1 ) = φ1 (H1 ) = G. This shows (i). To show the injectivity of φ, let h ∈ H1 and suppose that φ(h) = 1G×G = (1G , 1G ). We have h( ) = and h(σw) = σhσ (w) = σw for all σ ∈ {0, 1} and   w ∈ Σ ∗ . It follows that h = 1G = 1H . This shows (ii). As a consequence of Proposition 6.9.7(ii), we can identify each element h ∈ H1 with its image φ(h) ∈ G × G. Thus, we shall write h = (h0 , h1 ) if h ∈ H1 and φ(h) = (h0 , h1 ). Recall that a group is called periodic if it contains no elements of infinite order.

6.9 The Grigorchuk Group and Its Growth

183

Theorem 6.9.8. The Grigorchuk group G is an infinite finitely generated periodic group. Proof. Since H1 is a proper subset of G and φ0 : H1 → G is surjective (cf. Proposition 6.9.7(i)), we deduce that G is infinite. Let now show that G is a 2–group, that is, that the order of every element g ∈ G is a power of 2. The proof is by induction on S (g). By (6.33) the statement is true for S (g) = 1. Let us first show that every element g ∈ G \ {1G } is conjugate either to an element in S or to an element g  which can be expressed in a reduced form g  = at1 at2 · · · atk

(6.43)

with t1 , t2 , . . . , tk ∈ {b, c, d}, k ≥ 1, and such that S (g  ) ≤ S (g). Indeed, if g is not conjugate to an element in S, then any element of minimal S-length in the conjugacy class of g necessarily admits one of the following two reduced forms, besides (6.43): t1 at2 · · · atk a and t0 at1 · · · atk with t0 , t1 , . . . , tk ∈ {b, c, d}, t0 = tk , k ≥ 1. Conjugating by a and t0 respectively and replacing tk t0 with t ∈ {b, c, d} according to (6.34), we transform the above two expressions into one of the form (6.43). It is clear that such a process does not increase the word lengths so that S (g  ) ≤ S (g). Suppose first that g ∈ H1 and (g) > 1. Since the order of any element equals the order of all its conjugates, we may suppose, up to conjugacy, that g admits a reduced expression of the form (6.43) (with k ≥ 2 even, since g ∈ H1 ). Now, the image of each quadruple at2i−1 at2i , i = 1, 2, . . . , k/2, via φ0 or φ1 has word-length ≤ 2. Thus, S (g0 ), S (g1 ) ≤ S (g)/2 < S (g). By induction, g0 e g1 are 2−elements, and thus g = (g0 , g1 ) is a 2−element as well. Suppose now that g ∈ / H1 . As before, we may suppose, up to conjugacy, that g admits a reduced expression of the form (6.43) (with k odd, since g ∈ H1 ). We distinguish three cases. Case 1. The generator d appears in the expression of g, say d = ti for some 1 ≤ i ≤ k. Then, up to conjugating by the element ti−1 ati−2 a · · · at1 a, we can suppose that d = t1 . Now g 2 = (adat2 )(at3 at4 ) · · · (atk ad)(at2 at3 ) · · · (atk−1 atk ) ∈ H. The image of each quadruple via φ0 (resp. φ1 ) has length 2, with the only exception for those of the form atj ad = (atj a)d (resp. adatj = (ada)tj ) since φ0 (d) = 1G (resp. φ1 (ada) = 1G ). But at least one of these quadruples occurs in g 2 , for instance for j = k (resp. j = 1), so that S (φ0 (g 2 )) ≤ 2k − 1 < 2k = S (g) (resp. S (φ1 (g 2 )) < S (g)). By induction, φ0 (g 2 ) and φ1 (g 2 ) are 2-elements so that g 2 = (φ0 (g 2 ), φ1 (g 2 )) and therefore g are 2-elements as well. Case 2. Suppose now that d doesn’t occur in the expression of g but c does. As before, up to conjugacy, we may suppose that t1 = c. We then have φ0 (abat ) = ca, φ0 (acat ) = da and φ1 (abat ) = ac, φ1 (acat ) = ad for all t ∈ {b, c}. It follows that S (φ0 (g 2 )) = S (φ1 (g 2 )) = 2k = (g). Observe that φ0 (g 2 ) = da · · · a (resp. φ1 (g 2 ) = a · · · ad) so that, by conjugating by a (resp.

184

6 Finitely Generated Amenable Groups

ad) we fall into Case 1. This shows that φ0 (g 2 ) and φ1 (g 2 ) are 2-elements. Thus g 2 = (φ0 (g 2 ), φ1 (g 2 )) and therefore g are 2-elements. Case 3. Finally, suppose that neither d nor c appear in (6.43) so that necessarily g = (ab)2h+1 for some h ≥ 0. We have g 2 = (ab)4h+2 = ((aba)b)2h+1 ∈ H, so that φ0 (g 2 ) = (ca)2h+1 and φ1 (g 2 ) = (ac)2h+1 . Since S (φ0 (g 2 )) = S (φ1 (g 2 )) = 4h + 2 = S (g), we are in Case 2 and we deduce that φ0 (g 2 ) and φ1 (g 2 ) are 2-elements. Thus g 2 = (φ0 (g 2 ), φ1 (g 2 )) and therefore g itself are 2-elements.   Theorem 6.9.9. The Grigorchuk group G does not have polynomial growth. In order to prove this theorem, let us prove some preliminary results. Lemma 6.9.10. The subgroup of G generated by a and d is dihedral of order 8. Proof. The dihedral group D8 of order 8 has presentation D8 = x, y : x2 , y 2 , (xy)4 . Since a2 = d2 = 1G by (6.33), we are only left to verify that the order of the element ad is 4. We have (ad)2 = (ada)d = (b, 1)(1, b) =   (b, b) = 1G , while (ad)4 = ((ad)2 )2 = (b, b)2 = (b2 , b2 ) = (1G , 1G ) = 1G . Recall that two groups G1 and G2 are commensurable if there exist two subgroups of finite index K1 ⊂ G1 and K2 ⊂ G2 such that K1 and K2 are isomorphic. Lemma 6.9.11. The Grigorchuk group G is commensurable with its own square G × G. Proof. Let us start by showing that the index of φ(H1 ) inside G × G is finite. Since b, c, d, aba, aca and ada generate H1 (cf. Proposition 6.9.4), we deduce that the elements in (6.42) generate φ(H1 ). Denote by B ⊂ G the normal closure in G of b. The quotient G/B is generated by the images of the generators of G and since cd = b ∈ B, it is generated by the images of a and d. From Lemma 6.9.10 we deduce [G : B] = |G/B| ≤ 8.

(6.44)

Let g ∈ G and consider the element gbg −1 . Since φ0 (resp. φ1 ) is surjective, there exists an element h ∈ H (resp. h ∈ H) such that φ0 (h) = g (resp. φ1 (h ) = g). It follows that (gbg −1 , 1G ) = φ(hadah−1 ) ∈ φ(H1 ) (resp. (1G , gbg −1 ) = φ(h d(h )−1 ) ∈ φ(H1 )). As g varies in G the elements (gbg −1 , 1) (resp. (1, gbg −1 )) generate the subgroup B0 = B × {1G } ⊂ φ(H1 ) (resp. B1 = {1G } × B ⊂ φ(H1 )). Observe that B0  B1  B and that B0 and B1 are normal subgroups of G × G. Moreover, B0 ∩ B1 = {1G×G }, so that (G×G)/(B0 B1 )  G/B ×G/B. From (6.44) and the fact that B0 B1 ⊂ φ(H1 ) we deduce that [G × G : φ(H1 )] ≤ [G × G : B0 B1 ] = [G : B]2 = 64. This shows that φ(H1 ) has finite index in G × G.

(6.45)

6.9 The Grigorchuk Group and Its Growth

185

On the other hand [G : H1 ] = 2 (Proposition 6.9.6(ii)) and since φ is an injective homomorphism (Proposition 6.9.7(ii)), we have that H1 and φ(H1 ) are isomorphic. It follows that G and G × G are commensurable.   Proof of Theorem 6.9.9. Since G is infinite (Theorem 6.9.8) we have n  γ(G) (cf. Proposition 6.4.8). Since G and G×G are commensurable (Lemma 6.9.11) we deduce from Corollary 6.6.7 that γ(G) = γ(G×G). On the other hand, by Proposition 6.6.10 we have γ(G × G) = γ(G)2 . Using Lemma 6.6.9 it follows that n2  γ(G)2 = γ(G). By induction, we have that n2h  γ(G) for all h ∈ N. Thus G cannot have polynomial growth.   The remaining of this section is devoted to showing the following: Theorem 6.9.12. The Grigorchuk group G has subexponential growth. To prove this theorem we need some preliminaries. We start with a useful criterion for detecting that certain finitely generated groups have subexponential growth. Lemma 6.9.13. Let G be a finitely generated group. Let S ⊂ G be a finite symmetric generating subset and suppose that there exist an integer M ≥ 2, two constants 0 < k < 1 and K ≥ 0, and an injective homomorphism ψ : H → GM g → (gi )M i=1 where H ⊂ G is a finite index subgroup of G, such that M 

S (gi ) ≤ kS (g) + K

(6.46)

i=1

for all g ∈ H. Then G has subexponential growth. 1

Proof. Let us show that λS = limn→∞ γSG (n) n equals 1. Fix ε > 0. Then there exists an integer n0 ≥ 1 such that γSG (n) < (λS +ε)n for all n ≥ n0 . Since λS ≥ 1, it follows that γSG (n) ≤ γSG (n0 )(λS + ε)n

(6.47)

for all n ∈ N. Let γSH (n) = |{h ∈ H : S (h) ≤ n}| and fix a system T of left coset representatives of H in G. Set C = maxt∈T S (t). Then, given g ∈ G, there exist unique h ∈ H and t ∈ T such that g = th. Therefore S (h) ≤ S (t) + S (g) ≤ C + (g), so that BSG (n) ⊂ T BSH (n + C). We deduce that γSG (n) ≤ [G : H]γSH (n + C).

(6.48)

186

6 Finitely Generated Amenable Groups

On the other hand, by (6.46) we have  γSH (n) ≤ γSG (n1 )γSG (n2 ) · · · γSG (nM ) where the sum runs over all M −tuples n1 , n2 , . . . , nM such that kn + K. From (6.47) we then deduce:  γSH (n) ≤ γSG (n0 )M (λS + )n1 (λS + )n2 · · · (λS + )nM  = γSG (n0 )M (λS + )n1 +n2 +···+nM M ≤ γSG (n0 )(kn + K) (λS + )kn+K .



ni ≤

Using (6.48) we then obtain M  (λS + ε)kn+K γSG (n) ≤ [G : H]γSH (n + C) ≤ [G : H] γSG (n0 )(kn + K  ) (6.49) where K  = K +kC. Taking the nth roots and passing to the limit for n → ∞ in (6.49), the first term on the left tends to λS , while the last term on the right tends to (λS + ε)k . Thus λS ≤ (λS + ε)k . Since ε was arbitrary we deduce that λS ≤ λkS . Since by hypothesis k < 1, we have that λS = 1. Finally, from Corollary 6.5.5 we deduce that G has subexponential growth.   Lemma 6.9.14. Let n ≥ 1. Then φσ (Hn+1 ) ⊂ Hn for σ = 0, 1. Proof. We proceed by induction on n. We have already noticed that φσ (H1 ) ⊂ G = H0 (Proposition 6.9.7(i)). Suppose that φσ (Hn ) ⊂ Hn−1 and let us show that φσ (Hn+1 ) ⊂ Hn . Let g ∈ Hn+1 and w = σu, with u ∈ Σ n and σ ∈ {0, 1}. Then σu = w = g(w) = σφσ (g)(u), so that φσ (g)(u) = u. This shows that φσ (g) ∈ Hn . It follows that φσ (Hn+1 ) ⊂ Hn .   As a consequence of the previous lemma, for all n ≥ 1 and w = σ1 σ2 · · · σn ∈ Σ n the homomorphisms φw : Hn → G defined by φw = φσ1 ◦ φσ2 ◦ · · · ◦ φσn

are well defined. We then define the homomorphism ψn : Hn → w∈Σ n G by setting ψn (g) = (φw (g))w∈Σ n for all g ∈ Hn . Note that by Proposition 6.9.7(ii) ψn is injective. Thus, identifying Hn with its image ψn (Hn ) ⊂

n w∈Σ n G, we simply write g = (gw )w∈Σ for all g ∈ Hn . In the following, we consider the alphabet Λ = {α, β, γ, δ} and the monoid Λ∗ . The map Λ → S given by α → a, β → b, γ → c and δ → d uniquely extends to a surjective monoid homomorphism π : Λ∗ → G. We say that a

6.9 The Grigorchuk Group and Its Growth

187

word w = λ1 λ2 · · · λn ∈ Λ∗ is reduced if for all i, j = 1, 2, . . . , n − 1 one has λi = α implies λi+1 ∈ {β, γ, δ} and λj ∈ {β, γ, δ} implies λj+1 = α. Consider the following transformation p : Λ2 → Λ∗ defined by setting ⎧ if λ = λ ⎪ ⎪ ⎪ ⎪ if λ = β, λ = γ or λ = γ, λ = β ⎨δ  if λ = β, λ = δ or λ = δ, λ = β (6.50) p(λλ ) = γ ⎪   ⎪ = δ or λ = δ, λ = γ β if λ = γ, λ ⎪ ⎪ ⎩  λλ otherwise. Given a word u = λ1 λ2 · · · λn ∈ Λ∗ , and 1 ≤ i ≤ n, we set u(i) = λ1 λ2 · · · λi−1 p(λi λi+1 )λi+2 · · · λn ∈ Λ∗ . Note that u is reduced if and only if u = u(i) for all i = 1, 2, . . . , n − 1. By induction we define u(i1 ,i2 ,...,ik ) = (u(i1 ,i2 ,...,ik−1 ) )ik for 1 ≤ ik ≤ (u(i1 ,i2 ,...,ik−1 ) ) − 1. Let now w = λ1 λ2 · · · λn ∈ Λ∗ . If w is reduced we set w = w. Otherwise, let i1 be the minimal integer such that w = w(i1 ) . If w(i1 ) is reduced we set w = w(i1 ) . Otherwise, let i2 be the minimal integer such that w(i1 ) = w(i1 ,i2 ) . If w(i1 ,i2 ) is reduced we set w = w(i1 ,i2 ) . And so on. Since (w) > (w(i1 ) ) > (w(i1 ,i2 ) > · · · > (w(i1 ,i2 ,...,ik−1 ) ) > (w(i1 ,i2 ,...,ik ) ) > · · · there exists an integer k such that w(i1 ,i2 ,...,ik ) is reduced. We then set w = w(i1 ,i2 ,...,ik ) . A transformation w(i1 ,i2 ,...,ij−1 ) → w(i1 ,i2 ,...,ij ) is called a leftmost cancellation. It follows that every word in Λ∗ can be transformed into a reduced word by a finite sequence of leftmost cancellations. We denote by Θ1 ⊂ Λ∗ the subset consisting of all reduced words w on Λ which contain an even number α (w) of occurrences of the letter α. Thus, a word w ∈ Θ1 is an alternate product of terms of the form (αλα) and λ , with λ, λ ∈ {β, γ, δ}. Consider the maps Φ0 : Θ1 → Λ∗ and Φ1 : Θ1 → Λ∗ defined by setting Φσ ((αλ1 α)λ2 · · · ) = Φσ (αλ1 α)Φσ (λ2 ) · · · (resp. Φσ (λ1 (αλ2 α) · · · ) = Φσ (λ1 )Φσ (αλ2 α) · · · for all λ1 , λ2 , . . . ∈ {β, γ, δ} where Φ0 (β) = α, Φ0 (αβα) = γ, Φ1 (β) = γ, Φ1 (αβα) = α, Φ0 (γ) = α, Φ0 (αγα) = δ, Φ1 (γ) = δ, Φ1 (αβα) = α, Φ0 (δ) = , Φ0 (αδα) = β, Φ1 (β) = δ, Φ1 (αβα) = . For n ≥ 1 we inductively define Θn+1 ⊂ Λ∗ as the subset consisting of all reduced words w in Θn such that the reduced words Φ0 (w) and Φ1 (w) belong to Θn . Also, given w ∈ Θn we recursively set wσ0 σ1 ···σn−1 σ = Φσ (wσ0 σ1 ···σn−1 ) for all σ, σ1 , σ2 , . . . , σn−1 ∈ {0, 1}. Lemma 6.9.15. For all w ∈ Θ1 one has (w0 ) + (w1 ) ≤ (w) + 1.

(6.51)

188

6 Finitely Generated Amenable Groups

Proof. Let w ∈ Θ1 . We distinguish a few cases. Case 1. Suppose that w starts and ends with α so that w = (αλ1 α)λ2 · · · λk−1 (αλk α) with λi ∈ {β, γ, δ}. Note that (w) = 2k + 1. Then wσ = Φσ (w) and Φσ (w) = Φσ (αλ1 α)Φσ (λ2 ) · · · Φσ (λk−1 )Φσ (αλk α) and therefore (wσ ) = (Φσ (w)) ≤ (Φσ (w)) ≤ (Φσ (αλ1 α)) + (Φσ (λ2 )) + · · · + (Φσ (λk−1 )) + (Φσ (αλk α)) (2k + 1) − 1 2 (w) − 1 . = 2 ≤k=

Case 2. Suppose that w starts and ends with letters in {β, γ, δ}, that is, w = λ1 (αλ2 α)λ3 · · · (αλk−1 α)λk with λi ∈ {β, γ, δ}. Note that (w) = 2k − 1. Then wσ = Φσ (w) and Φσ (w) = Φσ (λ1 )Φσ (αλ2 α) · · · Φσ (αλk−1 α)Φσ (λk ) and therefore (wσ ) = (Φσ (w)) ≤ (Φσ (w)) ≤ (Φσ (λ1 )) + (Φσ (αλ2 α)) + · · · + (Φσ (αλk−1 α)) + (Φσ (λk )) (2k − 1) + 1 2 (w) + 1 . = 2 ≤k=

Case 3. Finally, suppose that w starts with α and ends with λ ∈ {β, γ, δ}, or viceversa. To fix ideas, suppose that w = (αλ1 α)λ2 · · · (αλk−1 α)λk with λi ∈ {β, γ, δ}. Passing to the inverse word w−1 one handles the other possibility. Note that (w) = 2k. Then wσ = Φσ (w) and Φσ (w) = Φσ (αλ1 α)Φσ (λ2 ) · · · Φσ (αλk−1 α)Φσ (λk ) and therefore (wσ ) = (Φσ (w)) ≤ (Φσ (w)) ≤ (Φσ (αλ1 α)) + (Φσ (λ2 )) + · · · + (Φσ (αλk−1 α)) + (Φσ (λk )) 2k 2 (w) . = 2 ≤k=

 

This shows (6.51). Lemma 6.9.16. For all g ∈ H3 one has 1  i,j,k=0

S (gijk ) ≤

5 S (g) + 8. 6

(6.52)

6.9 The Grigorchuk Group and Its Growth

189

Proof. Let g ∈ H3 and let w ∈ Θ3 be such that g = π(w) and S (g) = (w). As a consequence of Lemma 6.9.15 we have 1 

(wij ) ≤

i,j=0

1  ((wi ) + 1) i=0

=

1 

(6.53) (wi ) + 2

i=0

≤ (w) + 3 and therefore 1 

(wijk ) ≤

1 

((wij ) + 1)

i,j=0

i,j,k=0



1 

(wij ) + 4

i,j=0

(by (6.53)) ≤

 1  i=0

(6.54)

 (wi ) + 2

+4=

1 

(wi ) + 6

i=0

(by (6.51)) ≤ (w) + 1 + 6 = (w) + 7. Now we observe that by definition of the maps Φ0 and Φ1 , the inequalities in (6.51), (6.53) and (6.54) can be sharpened as follows. (w0 ) + (w1 ) ≤ (w) + 1 − δ (w),

(6.55)

where δ (w) denotes the number of occurrences of the letter δ in the word w. Indeed, every such δ, may appear in w either isolated or in a triplet (αδα). Then, in the first case we have Φ0 (δ) = , while in the second one, Φ1 (αδα) = . Similarly, since every letter γ in the word w will give rise via Φ0 or Φ1 to a letter δ in one of w0 and w1 , the above argument shows that, even if some cancellation occurs, (w00 ) + (w01 ) + (w10 ) + (w11 ) ≤ (w) + 3 − γ (w),

(6.56)

where γ (w) denotes the number of occurrences of the letter γ in the word w. Finally, since every letter β in the word w will give rise via Φ0 or Φ1 to a letter γ in one of w0 and w1 , and therefore to a letter δ in one of w00 , w01 , w10 and w11 , the above arguments show that indeed, even if some cancellation occurs,

190

6 Finitely Generated Amenable Groups 1 

(wijk ) ≤ (w) + 7 − β (w),

(6.57)

i,j,k=0

where β (w) denotes the number of occurrences of the letter β in the word w. Since (w) = α (w)+β (w)+γ (w)+δ (w) and w is reduced, we necessarily and therefore have β (w) + γ (w) + δ (w) ≥ (w)−1 2 max λ∈{β,γ,δ}

λ (w) >

(w) − 1. 6

Taking into account the inequalities in (6.54) we obtain ⎧ ⎫ 1 1 1 ⎨ ⎬   (wijk ) ≤ min (wi ) + 6, (wij ) + 4 . ⎩ ⎭ i=0

i,j,k=0

(6.58)

(6.59)

i,j=0

Observe that π(wijk ) = gijk so that S (gijk ) ≤ (wijk ) for all i, j, k = 0, 1. Thus, using first (6.59), (6.55), (6.56) and (6.57), and then (6.58) we deduce 1  i,j,k=0

1 

S (gijk ) ≤

(wijk )

i,j,k=0

≤ min{(w) + 7 − β (w), (w) + 7 − γ (w), (w) + 7 − δ (w)} ≤ (w) + 7 − max{β (w), γ (w), δ (w)}   (w) −1 ≤ (w) + 7 − 6 5 ≤ (w) + 8 6 5 = (g) + 8. 6 This shows (6.52).

 

Proof of Theorem 6.9.12. Let H = H3 . By Proposition 6.9.4 we have [G : H] < ∞. Moreover, by Lemma 6.9.16 we can take ψ = ψ3 , M = 8, k = 5/6 and K = 8, and apply Lemma 6.9.13 to obtain that G has subexponential growth.   A finitely generated group G is said to have intermediate growth if G has subexponential growth but does not have polynomial growth. Note that every finitely generated group of intermediate growth contains no subgroups isomorphic to F2 , by Corollary 6.6.5. From Theorem 6.9.9 and Theorem 6.9.12 we then get: Theorem 6.9.17. The Grigorchuk group G has intermediate growth.

 

6.10 The Følner Condition for Finitely Generated Groups

191

6.10 The Følner Condition for Finitely Generated Groups Let G be a group. As right multiplication by elements in G is bijective we have, for all A, B, F ⊂ G, F finite, and g ∈ G, (A \ B)g = Ag \ Bg,

(6.60)

|F \ F g| = |F g \ F | = |F \ F g −1 | = |F g −1 \ F |.

(6.61)

Lemma 6.10.1. Let A, B, C ⊂ G be three finite sets. Then we have: |A \ B| ≤ |A \ C| + |C \ B|.

(6.62)

Proof. Suppose that x ∈ A \ B, that is, (i) x ∈ A and (ii) x ∈ / B. We distinguish two cases. If x ∈ / C, then, by (i), x ∈ A \ C. If x ∈ C, then, by (ii), x ∈ C \ B. In both cases, x ∈ (A \ C) ∪ (C \ B). It follows that A \ B ⊂ (A \ C) ∪ (C \ B) and (6.62) follows.   Let now S ⊂ G be a finite subset. The isoperimetric constant of G with respect to S is the nonnegative number ιS (G) = inf F

|F S \ F | |F |

(6.63)

where F runs over all non empty finite subsets of G. Observe that ιS (G) = ιS∪{1G } (G). Proposition 6.10.2. Let G be a finitely generated group and let S ⊂ G be a finite generating subset. Then the following conditions are equivalent: (a) G is amenable; (b) for all ε > 0 there is a finite subset F = F (ε) ⊂ G such that |F \F s| < ε|F |

for all s ∈ S;

(6.64)

(c) ιS (G) = 0. Proof. Suppose (a) and fix ε > 0. By Theorem 4.9.1, G satisfies the Følner conditions. For our convenience, we express the Følner conditions as follows. Given any finite subset K ⊂ G and ε > 0 there exists a finite subset F = F (K, ε ) ⊂ G such that |F \F k| < ε |F |

for all k ∈ K.

(6.65)

Taking ε = ε, K = S and F = F (S, ε) in 6.65, we then immediately obtain (6.64), and the implication (a) ⇒ (b) follows.

192

6 Finitely Generated Amenable Groups

Suppose (b). Taking g = s in (6.61) we immediately deduce |F \F s| < ε|F |

for all s ∈ S ∪ S −1 .

(6.66)

Fix a finite set K ⊂ G and ε > 0. Since S generates G we can find n ∈ N such that K ⊂ (S ∪ S −1 )n . Set ε = ε /n. Let F = F (ε) ⊂ G be a finite set such that (6.66) holds. Given k ∈ K we can find a1 , a2 , . . . , an ∈ S ∪ S −1 such that k = a1 a2 · · · an . Then, recalling (6.62) and (6.60), we have |F \ F k| = |F \ F a1 a2 · · · an | ≤ |F \ F an | + |F an \ F an−1 an | + |F an−1 an \ F an−2 an−1 an | + · · · + |F a2 a3 · · · an \ F a1 a2 · · · an | = |F \ F an | + |F \ F an−1 | + |F \ F an−2 | + · · · + |F \ F a1 | < nε|F | ≤ ε |F | for all k ∈ K. This shows that G satisfies the Følner conditions (6.65) and therefore G is amenable by Theorem 4.9.1. Thus (b) ⇒ (a). Finally, the equivalence (b) ⇔ (c) immediately follows from (6.61) and the inequalities   |F s \ F | = |F \ F s |. |F \ F s| = |F s \ F | ≤ |F S \ F | ≤ s ∈S

s ∈S

for any s ∈ S and any finite subset F ⊂ G.

 

6.11 Amenability of Groups of Subexponential Growth In this section we show that every finitely generated group of subexponential growth is amenable. We first prove the following: Lemma 6.11.1. Let (an )n≥1 be a sequence of positive real numbers. Then lim inf n→∞

√ an+1 ≤ lim inf n an . n→∞ an

(6.67)

Proof. Set α = lim inf n→∞ aan+1 . If α = 0 there is nothing to prove. Othern ≥β wise, let 0 < β < α. Then, there exists an integer N ≥ 1 such that aan+1 n for all n ≥ N . Thus, for all p = 1, 2 . . . one has aN +p aN +p−1 aN +1 aN +p = · ··· ≥ βp. aN aN +p−1 aN +p−2 aN This gives aN +p ≥ β p aN . It follows that setting n = N + p, one has, for all n≥N

6.12 The Theorems of Kesten and Day

193

an ≥ β n−N aN = β n (β −N aN ).  √ Thus, taking the nth roots we obtain n an ≥ β n β −N aN and therefore, lim inf n→∞

√ n an ≥ β.

As (6.68) holds for all β < α, we have lim inf n→∞

(6.68) √ n a n ≥ α = lim inf n→∞

an+1 an .

 

We are now in position to prove the main result of this section. Theorem 6.11.2. Every finitely generated group of subexponential growth is amenable. Proof. Let G be a finitely generated group of subexponential growth. Let S ⊂ G be a finite symmetric generating subset of G. Then, by virtue of the  n previous lemma we have 1 ≤ lim inf n→∞ γSγ(n+1) ≤ lim γ n→∞ S (n) = 1, so S (n) that lim inf n→∞

γS (n+1) γS (n)

= 1. Fix ε > 0 and let n0 ∈ N be such that γS (n0 + 1) < 1 + ε. γS (n0 )

(6.69)

Set F = BS (n0 ) and let us show that |F \ F s| < ε|F | for all s ∈ S. Let s ∈ S. Then, as BS (n0 )s ⊂ BS (n0 + 1) for all s ∈ S, we have |F \ F s| = |F s \ F | = |BS (n0 )s \ BS (n0 )| ≤ |BS (n0 + 1) \ BS (n0 )| = γS (n0 + 1) − γS (n0 ) < εγS (n0 ) = ε|F |, where the last inequality follows from (6.69). From Proposition 6.10.2 we deduce that G is amenable.   From Theorem 6.9.12 and Theorem 6.11.2 we deduce the following: Corollary 6.11.3. The Grigorchuk group G is amenable.

6.12 The Theorems of Kesten and Day Let G be a group. Let p ∈ [1, +∞). We consider the real Banach space    p G p  (G) = x ∈ R : |x(g)| < ∞ g∈G

 

194

6 Finitely Generated Amenable Groups

consisting of all p-summable real functions on G. For x ∈ p (G), the nonneg 1  p p is called the p -norm of x. ative number xp = g∈G |x(g)| The support of a configuration x ∈ RG is the set {g ∈ G : x(g) = 0}. We denote by R[G] ⊂ RG the vector subspace consisting of all finitely supported configurations in RG . Note that R[G] is a dense subspace in p (G). When p = 2, it is possible to endow 2 (G) with a scalar product ·, · defined by setting  x(g)y(g) x, y = g∈G

for all x,y ∈ 2 (G). Then, the 2 -norm of an element x ∈ 2 (G) is given by x2 = x, x. The space (2 (G), ·, ·) is a real Hilbert space. The p -norm of a linear map T : p (G) → p (G) is defined by sup T xp =

T p→p =

p

x∈ (G)

x p ≤1

T xp . (G) xp

sup p

x∈ x =0

Then, T is continuous if and only if T p→p < ∞. We denote by L(p (G)) the space of all continuous linear maps T : p (G) → p (G) and by I : p (G) → p (G) the identity map. Note that, by density of R[G] in p (G), we have T xp x∈R[G] xp

T p→p = sup T xp = sup x∈R[G]

x p ≤1

(6.70)

x =0

for all T ∈ L(p (G)). Let p ∈ [1, +∞) and s ∈ G. For all x ∈ p (G) and g ∈ G we set (Ts(p) x)(g) = x(gs). We then have Ts(p) xpp =



|Ts(p) x(g)|p

g∈G

=



|x(gs)|p

g∈G

= =



|x(h)|p (by setting h = gs)

h∈G xpp (p)

for all x ∈ p (G). We deduce that Ts x ∈ p (G) and that the linear map (p) Ts : p (G) → p (G) has p -norm

6.12 The Theorems of Kesten and Day

195

Ts(p) p→p = 1.

(6.71)

(p)

In particular, Ts ∈ L(p (G)). (p) Let now S ⊂ G be a non-empty finite set. We denote by MS : p (G) →  (p) (p) 1 p (G) the map defined by MS = |S| s∈S Ts . In other words, (p)

(MS x)(g) =

1  x(gs) |S| s∈S

(p)

for all x ∈ p (G) and g ∈ G. The map MS on p (G) associated with S.

is called the p -Markov operator

Proposition 6.12.1. Let G be a group and S ⊂ G a non-empty finite set. (p) Then the p -Markov operator MS : p (G) → p (G) is linear and continuous. Moreover, (p)

MS p→p ≤ 1. (p)

Proof. Since Ts have

(p)

(6.72) (p)

∈ L(p (G)), we deduce that MS

MS p→p = 

∈ L(p (G)). Finally, we

1  (p) 1  (p) Ts p→p ≤ Ts p→p = 1 |S| |S| s∈S

s∈S

 

where the last equality follows from (6.71).

Proposition 6.12.2. Let G be a group. Let S ⊂ G be a finite subset containing 1G . Then, the following conditions are equivalent: (2)

(a) MS 2→2 = 1; (2) (b) given ε > 0 there exists x ∈ R[G] such that x2 = 1 and MS x2 ≥ 1 − ε; (c) given ε > 0 there exists x ∈ R[G] such that x ≥ 0, x2 = 1 and (2) MS x2 ≥ 1 − ε; (d) given ε > 0 there exists x ∈ R[G] such that x ≥ 0, x2 = 1 and x − Ts(2) x2 ≤ ε (2)

for all s ∈ S;

(6.73)

(2)

(e) 1 belongs to the real spectrum σ(MS ) of MS . Proof. The implication (a) ⇒ (b) follows from the definition of 2 -norm and the density of R[G] in 2 (G) (cf. (6.70)). Let now x ∈ R[G]. We have    (2) 2 (2) MS x22 ≤ MS |x|  . 2

(6.74)

196

6 Finitely Generated Amenable Groups

Indeed, (2) MS x22

 2  1     = Ts(2) x  |S|  s∈S 2  2  1  = x(gs) |S| g∈G s∈S 2   1  ≤ |x(gs)| |S| g∈G s∈S  2  1     (2) = Ts |x|  |S|  s∈S 2  2  (2)  = MS |x|  . 2

Thus, replacing x by |x| gives the implication (b) ⇒ (c). Suppose that (d) fails to hold, that is, there exists ε0 > 0 such that for all (2) x ∈ R[G], x ≥ 0, x2 = 1, there exists s0 ∈ S such that x − Ts0 x2 ≥ ε0 . Since 2 (G) is uniformly convex (Lemma I.4.2) there exists δ0 > 0 such that    x + T (2) x  s0   (6.75)   ≤ 1 − δ0   2 2

for all x ∈ R[G] such that x2 = 1. It then follows    1    (2) (2)  MS x2 =  Ts x  |S|  s∈S 2       (2)   2  x + Ts0 x 1 (2)  + = T x s  |S|  2 |S|   s∈S\{1G ,s0 } 2   (2)    2  x + Ts0 x  1 ≤ Ts(2) x2   +  |S|  2 |S| 2

s∈S\{1G ,s0 }

2 1 (1 − δ0 ) + (|S| − 2) (by (6.75)) ≤ |S| |S| 2δ0 =1− |S| for all x ∈ R[G], such that x2 = 1 and x ≥ 0. This clearly contradicts (c). We have shown (c) ⇒ (d). Let us show (d) ⇒ (e). Suppose that (6.73) holds. Then, for every ε > 0 there exists x ∈ R[G] such that

6.12 The Theorems of Kesten and Day

     1   (2) x − MS x2 = x − Ts(2) x   |S| s∈S 2    1    (2)  = (x − Ts x)  |S|  s∈S

1  ≤ x − Ts(2) x2 |S|

197

2

s∈S

≤ ε. (2)

Therefore, by Corollary I.2.3, the linear map I − MS ∈ L(2 (G)) is not (2) bijective, that is, 1 ∈ σ(MS ). Finally, the implication (e) ⇒ (a) follow from (I.14) and the fact that (2)   MS 2→2 ≤ 1 (Proposition 6.12.1). Lemma 6.12.3. Let S ⊂ G be a non-empty finite subset. The following conditions are equivalent: (2)

(a) 1 ∈ σ(MS ); (2) (b) 1 ∈ σ(MS∪{1G } ). Proof. If S contains 1G there is nothing to prove. |S| Suppose that 1G ∈ / S and set α = |S|+1 . Note that 0 < α < 1 and that (2)

(2)

MS∪{1G } = (1 − α)I + αMS . (2)

(2)

(6.76) (2)

We then have I − MS∪{1G } = (1 − α)(I − MS ) so that I − MS∪{1G } is (2)

bijective if and only if I − MS

(2)

is bijective. In other words, 1 ∈ σ(MS∪{1G } )

(2)

if and only if 1 ∈ σ(MS ).

 

Before stating and proving the main result of this section we need a little more work. More precisely, we want to show the equivalence between the existence of an almost S-invariant positive element x ∈ R[G] of norm x2 = 1 (cf. condition (d) in Proposition 6.12.2) and the existence of an almost S-invariant non-empty finite set F ⊂ G (cf. condition (b) in Proposition 6.10.2). Let F ⊂ G be a finite set. Denote, as usual, by χF the characteristic map of F and observe that χF ∈ R[G]. Proposition 6.12.4. Let A, B ⊂ G be two finite sets. Then χA − χB 1 = χA − χB 22 = |A \ B| + |B \ A|.

(6.77)

198

6 Finitely Generated Amenable Groups

Proof. First note that the map χA − χB takes value 1 at each point of A \ B, value −1 at each point of B \ A, and value 0 everywhere else, that is, outside of the set (A \ B) ∪ (B \ A) = (G \ (A ∪ B)) ∪ (A ∩ B). We then have  |χA (g) − χB (g)|2 χA − χB 22 = g∈G

=



|χA (g) − χB (g)|

g∈G



=

|χA (g) − χB (g)|

g∈G: χA (g)−χB (g)=1



+

|χA (g) − χB (g)|

g∈G: χA (g)−χB (g)=−1

=



χA\B (g) +

g∈G



χB\A (g)

g∈G

= |A \ B| + |B \ A|.   Lemma 6.12.5. Let x, y ∈ 2 (G) such that x2 = y2 = 1. Then, the following holds: (i) x2 ∈ 1 (G) and x2 1 = 1; (ii) x2 − y 2 1 ≤ 2x − y2 ;

1 (iii) suppose that x, y ≥ 0. Then, x − y2 ≤ x2 − y 2 1 2 .  Proof. (i) We have x2 1 = g∈G |x2 (g)| = x22 = 1. (ii) We have  x2 − y 2 1 = |x2 (g) − y 2 (g)| g∈G

=



|x(g) − y(g)| · |x(g) + y(g)|

g∈G

= |x − y|, |x + y| ≤ x − y2 · x + y2 ≤ x − y2 · (x2 + y2 ) = 2x − y2 , where the first inequality follows from Cauchy-Schwarz.

6.12 The Theorems of Kesten and Day

199

(iii) We have x − y22 =



(x(g) − y(g))2

g∈G

=



|x(g) − y(g)| · |x(g) − y(g)|

g∈G





|x(g) − y(g)| · |x(g) + y(g)|

g∈G

=



|x2 (g) − y 2 (g)|

g∈G

= x2 − y 2 1 , where the inequality follows from the fact that x, y ≥ 0.

 

Lemma 6.12.6. Let F ⊂ G be a non-empty finite set and s ∈ G. Then the following holds: (p)

(i) Ts χF = χF s−1 , for p = 1, 2; (ii) χF 1 = χF 22 = |F |; (1) (2) (iii) χF − Ts χF 1 = χF − Ts χF 22 = 2|F \ F s|. (s)

Proof. For g ∈ G and p = 1, 2, one has that (Tp χF )(g) = χF (gs) equals 1 if and only if gs ∈ F , that is, if and only if g ∈ F s−1 . This shows (i). Moreover, recalling that χF (g) ∈ {0, 1}, for all g ∈ G, on has   χF 22 = χF (g)2 = χF (g) = χF 1 = |{g ∈ G : χF (g) = 1}| = |F |. g∈G

g∈G

This shows (ii). Finally, (iii) follows from (i), (6.77) (with A = F and B = F s−1 ) and (6.61).   Lemma 6.12.7. Let x ∈ R[G] such that x ≥ 0 and x1 = 1. Then there exist an integer n ≥ 1, nonempty finite subsets Ai ⊂ G and real numbers λi > 0, 1 ≤ i ≤ n, satisfying A1 ⊃ A2 ⊃ · · · ⊃ An and λ1 + λ2 + · · · + λn = 1 such that n  χA (6.78) x= λi i . |A i| i=1 Proof. Let 0 < α1 < α2 < · · · < αn be the values taken by x. For each 1 ≤ i ≤ n, let us set Ai = {g ∈ G : x(g) ≥ αi }.

200

6 Finitely Generated Amenable Groups

Clearly the sets Ai are nonempty finite subsets of G such that A1 ⊃ A2 ⊃ · · · ⊃ An . On the other hand, we have x = α1 χA1 + (α2 − α1 )χA2 + · · · + (αn − αn−1 )χAn = λ1

χA1 χA χA + λ2 2 + · · · + λ n n , |A1 | |A2 | |An |

by setting λ1 = α1 |A1 | and λi = (αi − αi−1 )|Ai | for 2 ≤ i ≤ n. Thus λi > 0 for 1 ≤ i ≤ n and n 

λi = α1 |A1 | + (α2 − α1 )|A2 | + · · · + (αn − αn−1 )|An |

i=1

= α1 (|A1 | − |A2 |) + α2 (|A2 | − |A3 |) + · · · + αn |An |  x(g) = 1. = g∈G

  Lemma 6.12.8. With the same notation and hypotheses as in Lemma 6.12.7, we have n  2|Ai \ Ai g| (6.79) λi x − Tg(1) x1 = |Ai | i=1 for every g ∈ G. Proof. Equality (6.78) gives us x−

Tg(1) x

=

n 

(1)

λi

χAi − Tg χAi |Ai |

λi

χAi − χAi g−1 . |Ai |

i=1

(by Lemma 6.12.6(a))

=

n  i=1

As we observed before (cf. the proof of Proposition 6.12.4), the map χAi − χAi g−1 takes value 1 at each point of Ai \ Ai g −1 , value −1 at each point of Ai g −1 \ Ai , and value 0 everywhere else. Let us set   B= (Ai \ Ai g −1 ) and C = (Ai g −1 \ Ai ). 1≤i≤n

1≤i≤n

Note that the sets B and C are disjoint. Indeed, for all 1 ≤ i, j ≤ n, we have (Ai \ Ai g −1 ) ∩ (Aj \ Aj g −1 ) = ∅ since either Ai ⊂ Aj or Aj ⊂ Ai (which implies Aj g −1 ⊂ Ai g −1 ). It follows that

6.12 The Theorems of Kesten and Day

x − Tg(1) x1 =



201

|(x − Tg(1) x)(a)|

a∈G

  n   (χAi − χAi g−1 )(a)  λi =     |Ai | a∈G i=1     n n   (χAi − χAi g−1 )(a)    (χAi − χAi g−1 )(a)  λi λi =   +      |Ai | |Ai | i=1 i=1 =

a∈B n  i=1

a∈C

λi

|Ai \ Ai g |Ai |

−1

|

+

n 

−1

λi

i=1

|Ai g \ Ai | |Ai |

n 

2|Ai \ Ai g| . (by (6.61)) = λi |Ai | i=1   Let G be a finitely generated group and let S be a finite (not necessarily symmetric) generating subset of G. Consider the combinatorial Laplacian ΔS : RG → RG (cf. Example 1.4.3(b)). For all x ∈ 2 (G) ⊂ RG and g ∈ G we have  (ΔS x)(g) = |S|x(g) − x(gs) s∈S

= |S|x(g) −

 (Ts(2) x)(g)

s∈S (2)

= |S|(I − MS )(x)(g). (2)

This shows that for the map ΔS = ΔS | 2 (G) one has   (2) (2) ΔS = |S| I − MS

(6.80)

(2)

and therefore ΔS ∈ L(2 (G)). Let λ ∈ R. From (6.80) we deduce that (2)

(2)

λI − ΔS = λI − |S|(I − MS ) = −|S| It follows that (2)

λ ∈ σ(ΔS ) ⇔ and therefore

(2)

   λ (2) I − MS . 1− |S|

  λ (2) ∈ σ(MS ) 1− |S| (2)

0 ∈ σ(ΔS ) ⇔ 1 ∈ σ(MS ).

(6.81)

Theorem 6.12.9 (Kesten-Day). Let G be a finitely generated group. Let S ⊂ G be a finite (not necessarily symmetric) generating subset of G. Then

202

6 Finitely Generated Amenable Groups

the following conditions are equivalent: (a) G is amenable; (2) (b) 0 ∈ σ(ΔS ). Proof. Suppose (a). By Theorem 4.9.1 we have that G satisfies the Følner conditions. Fix ε > 0. By Proposition 6.10.2 there exists a finite subset F ⊂ G 2 such that |F \ F s| < ε2 |F | for all s ∈ S. Set x = √1 χF and note that |F |

x2 = 1. We have

1  (2) Ts x2 |S| s∈S 1   = (x − Ts(2) x)2 |S| s∈S 1  ≤ x − Ts(2) x2 |S| s∈S  1  = χF − Ts(2) χF 2 |S| · |F | s∈S  1 1  (by Lemma 6.12.6(iii)) = 2|F \ F s| 2 |S| · |F | s∈S 1  1  |F \ F s| 2 = 2 |S| |F | (2)

(I − MS )x2 = x −

s∈S

=

2

|F | < ε.

1

1 2

ε2

(2)

From Corollary I.2.3 we deduce that I − MS is not bijective, that is, 1 ∈ (2) (2) σ(MS ). From (6.81) we deduce that 0 ∈ σ(ΔS ). Thus (a) implies (b). To prove the converse implication, first observe that, by virtue of Proposition 6.10.2, in order to prove (a), it suffices to show that given any ε > 0 there exists a finite subset F ⊂ G such that |F \ F s| < ε|F |

for all s ∈ S.

(6.82) (2)

Fix ε > 0. Suppose (b) and observe that by (6.81) we have 1 ∈ σ(MS ). Also note that, by Lemma 6.12.3, we can suppose that S  1G . Thus, by virtue of the implication (e) ⇒ (c) in Proposition 6.12.2, we can find a non(2) ε negative function xε ∈ R[G] such that xε 2 = 1 and xε − Ts xε 2 ≤ 2|S| for all s ∈ S. Setting x = x2ε , from Lemma 6.12.5 we deduce that x1 = 1 and ε for all s ∈ S. (6.83) x − Ts(1) x1 ≤ |S|

6.12 The Theorems of Kesten and Day

203

By Lemma 6.12.7, there exist an integer n ≥ 1, nonempty finite subsets A1 ⊃ A2 ⊃ · · · ⊃ An Ai ⊂ G and real numbers λi > 0, 1 ≤ i ≤ n, nsatisfying χ and λ1 + λ2 + · · · + λn = 1, such that x = i=1 λi |AAii| . Set Ω = {1, 2, . . . , n} and consider the unique probability measure μ on Ω such that μ({i}) = λi for every i ∈ Ω. Finally, for each g ∈ G, let Ωg denote the subset of Ω defined by

|Ai \ Ai g| ≥ε . Ωg = i ∈ Ω : |Ai | It follows from Lemma 6.12.8 that x − Tg(1) x1 =



λi

i∈Ω





2|Ai \ Ai g| |Ai |

λi

i∈Ωg

≥ 2ε

2|Ai \ Ai g| |Ai |



λi

i∈Ωg

= 2εμ(Ωg ). Therefore, we have (1)

μ(Ωg ) ≤

x − Tg x1 2ε

for all g ∈ G.

By using (6.83), we deduce 1 |S|

μ(Ωs ) < which implies

 μ



 Ωs



s∈S

Thus

for all s ∈ S,



μ(Ωs ) < 1.

s∈S



Ωs = Ω.

s∈S

This means that there is some i0 ∈ Ω such that |Ai0 \ Ai0 s| 0 such that for every h ∈ H there exists g ∈ G such that dH (ϕ(g), h) ≤ δ.

(6.99)

Proof. Suppose that ϕ is a quasi-isometric embedding. Let C ⊂ H be a finite subset such that H = ϕ(H)C and set δ = max{dH (1H , c) : c ∈ C}. Let h ∈ H, then there exist c ∈ C and g ∈ G such that h = ϕ(g)c, that is, ϕ(g)−1 h = c ∈ C. We deduce that dH (ϕ(g), h) ≤ δ and (6.99) follows. This shows (a) ⇒ (b). Conversely, suppose (b) and set C = BSG (1G , δ) ⊂ G. Let h ∈ H. By (6.99) there exists g ∈ G such that dH (ϕ(g), h) ≤ δ. It follows that ϕ(g)−1 h ∈ C, equivalently h ∈ ϕ(g)C. This shows that H = ϕ(G)C. Therefore ϕ is a quasiisometry and (a) follows.   Proposition 6.13.14. Let G and H be two finitely generated groups. Suppose that there is a quasi-isometric embedding ϕ : G → H. Then one has γ(G)  γ(H). Proof. By Remark 6.13.4, we can assume ϕ(1G ) = 1H . Denote by dG and dH the word metric associated with two finite symmetric generating subsets SG and SH for G and H respectively. By Proposition 6.13.12, there exist integers α ≥ 1 and β ≥ 0 such that 1 dG (g, g  ) − β ≤ dH (ϕ(g), ϕ(g  )) ≤ αdG (g, g  ) + β α

(6.100)

210

6 Finitely Generated Amenable Groups

for all g, g  ∈ G. Note that the left inequality in (6.100) implies that ϕ(g) = ϕ(g  ) whenever g, g  ∈ G satisfy dG (g, g  ) ≥ αβ+1. For n ∈ N, let BSGG (n) ⊂ G (resp. BSHH (n) ⊂ H) denote the ball of radius n centered at 1G (resp. 1H ). Choose a subset En ⊂ BSGG (n) of maximal cardinality such that dG (g, g  ) ≥ αβ + 1 for all g, g  ∈ En . Setting C = |BSGG (αβ + 1)|, we have |BSGG (n)| ≤ C|En |

(6.101)

for all n ∈ N . Indeed, the balls of radius C centered at the elements of En cover BSGG (n) by the maximality of En . Now observe that the images by ϕ of the elements of En are all distinct and belong to BSHH (αn + β) by the right inequality in (6.100). This implies that |En | ≤ |BSHH (αn + β)|. By using (6.101), we then deduce that |BSGG (n)| ≤ C|BSHH (αn + β)| ≤ C|BSHH (β)||BSHH (αn)| ≤ C  |BSHH (C  n)| for all n ≥ 1, where C  = Cα|BSHH (β)|. It follows that γ(G)  γ(H).

 

Corollary 6.13.15. If two finitely generated groups are quasi-isometric, then they have the same growth type. Proof. If G and H are quasi-isometric finitely generated groups, then there exist a quasi-isometric embedding from G into H and a quasi-isometric embedding from H into G. It follows that γ(G)  γ(H) and γ(H)  γ(G). Thus we have γ(G) = γ(H).   Corollary 6.13.16. Let G and H be two quasi-isometric finitely generated groups. Then G has exponential (resp. subexponential, resp. polynomial, resp. intermediate) growth if and only if H has exponential (resp. subexponential, resp. polynomial, resp. intermediate) growth.   An important property of quasi-isometric embeddings is the fact that every quasi-isometric embedding is uniformly finite-to-one: Proposition 6.13.17. Let G and H be two groups. Let ϕ : G → H be a quasi-isometric embedding. Then there exists an integer M ≥ 1 such that |ϕ−1 (h)| ≤ M for all h ∈ H. Proof. Since ϕ is a quasi-isometric embedding, we can find a finite set K ⊂ G such that g1−1 g2 ∈ K whenever g1 , g2 ∈ G satisfy ϕ(g1 )−1 ϕ(g2 ) ∈ {1H }. Let us set M = |K|. Let h ∈ H. Suppose that g0 ∈ ϕ−1 (h). Then, every g ∈ ϕ−1 (h) satisfies ϕ(g0 )−1 ϕ(g) = h−1 h = 1H and therefore g0−1 g ∈ K. Thus, we have ϕ−1 (h) ⊂ g0 K and hence |ϕ−1 (h)| ≤ |g0 K| = |K| = M .   Corollary 6.13.18. Let G and H be two groups. Suppose that there exists a quasi-isometric embedding ϕ : G → H and that H is finite. Then G is finite.

6.13 Quasi-Isometries

211

Proof. Taking M as in the preceding proposition, we have |G| ≤ M |H|.

 

Corollary 6.13.19. Let G and H be two groups. Suppose that H is finite. Then G is quasi-isometric to H if and only if G is finite. Proof. If G is finite then it is clear from the definition that any map from G to H is a quasi-isometry. On the other hand, if G is quasi-isometric to H then G is finite by Corollary 6.13.18.   Proposition 6.13.20. Let G and H be two groups. Suppose that there exists a quasi-isometric embedding ϕ : G → H and that H is locally finite. Then G is locally finite. Proof. Let K be a finitely generated subgroup of G and let S ⊂ K be a finite symmetric generating subset of K. Since ϕ is a quasi-isometric embedding, we can find a finite subset F ⊂ H such that ϕ(g1 )−1 ϕ(g2 ) ∈ F whenever g1 , g2 ∈ G satisfy g1−1 g2 ∈ S. Let L denote the subgroup of H generated by {ϕ(1G )} ∪ F . Suppose that k ∈ K. Since S is a symmetric generating subset of K, there exist an integer n ≥ 0 and elements s1 , s2 , . . . , sn ∈ S such that k = s1 s2 · · · sn . Consider the elements k0 , k1 , . . . , kn ∈ K defined by k0 = 1G and ki = ki−1 si for all 1 ≤ i ≤ n. Observe that kn = k and that −1 ϕ(ki−1 )−1 ϕ(ki ) ∈ F , for all 1 ≤ i ≤ n, since ki−1 ki = si ∈ S. Thus, we have ϕ(k) = ϕ(1G )(ϕ(k0 )−1 ϕ(k1 ))(ϕ(k1 )−1 ϕ(k2 )) · · · (ϕ(kn−1 )−1 ϕ(kn )) ∈ L. It follows that ϕ(K) ⊂ L. As H is locally finite, the subgroup L is finite. On the other hand, it follows from Proposition 6.13.17 that there exists an integer M ≥ 1 such that |ϕ−1 (h)| ≤ M for all h ∈ H. We deduce that |K| ≤ M |L| < ∞. This shows that G is locally finite.   Corollary 6.13.21. Let G and H be two groups. Suppose that G and H are quasi-isometric and that the group G is locally finite. Then H is locally finite.   Lemma 6.13.22. Let G be a group. Let E, Ω and C be three subsets of G. Then, for every c0 ∈ C one has ∂Ec0 (Ω) = ∂E (Ωc−1 0 ).

(6.102)

and (6.103) ∂EC (Ω) ⊃ ∂Ec0 (Ω).   −1 Proof. By (5.3) we have Ω +Ec0 = e∈E Ω(ec0 )−1 = e∈E Ωc−1 = 0 e  −1 +E −Ec0 −1 and from (5.2) we deduce that Ω = = (Ωc0 ) e∈E Ω(ec0 )  −1 −1 −1 −E +Ec0 −Ec0 Ωc e = (Ωc ) . It follows that ∂ (Ω) = Ω \ Ω = Ec 0 0 0 e∈E −1 −E −1 +E (Ωc−1 ) \ (Ωc ) = ∂ (Ωc ). Similarly, we have E 0 0 0

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6 Finitely Generated Amenable Groups

Ω +EC =



Ω(ec)−1 =

e∈E c∈C



Ωc−1 e−1 ⊃

e∈E c∈C



−1 +E Ωc−1 = (Ωc−1 0 e 0 )

e∈E

and Ω −EC =

 e∈E c∈C

Ω(ec)−1 =



Ωc−1 e−1 ⊂

e∈E c∈C



−1 +E Ωc−1 = (Ωc−1 . 0 e 0 )

e∈E

+E −E We deduce that ∂EC (Ω) = Ω +EC \ Ω −EC ⊃ (Ωc−1 \ (Ωc−1 = 0 ) 0 ) −1   ∂E (Ωc0 ) = ∂Ec0 (Ω), where the last equality follows from (6.102).

Theorem 6.13.23. Let G and H be two quasi-isometric groups. Suppose that H is amenable. Then G is amenable. Proof. Let ϕ : G → H be a quasi-isometry and let C ⊂ H be a finite set such that  H = ϕ(G)C = ϕ(G)c. (6.104) c∈C

Let EG ⊂ G be a finite set and ε > 0. Let us show that there exists a finite set FG ⊂ G such that |∂EG (FG )| < ε|FG |. (6.105)  ⊂ H such Since ϕ is a quasi-isometric embedding, we can find a finite set EH that  (6.106) g1−1 g2 ∈ EG ⇒ ϕ(g1 )−1 ϕ(g2 ) ∈ EH  C. Also, by Proposition 6.13.17 we for all g1 , g2 ∈ G. We then set EH = EH can find an integer M ≥ 1 such that

|ϕ−1 (F )| ≤ M |F |

(6.107)

for all finite sets F ⊂ H. Since H is amenable, it follows from Corollary 5.4.5  ⊂ H such that that we can find a finite subset FH  )| < |∂EH (FH

ε |F  |. M |C| H

(6.108)

By (6.104) we can find c0 ∈ C such that  −1 c0 ∩ ϕ(G) = ∅ FH

and

(6.109)

 −1  −1 |FH c ∩ ϕ(G)| ≤ |FH c0 ∩ ϕ(G)|  −1 FH c0

⊂ H and FG = ϕ for all c ∈ C. Set FH = FG = ∅ by (6.109). Then we have ϕ(FG ) = FH ∩ ϕ(G) ⊂ FH

−1

(6.110)

(FH ) ⊂ G. Note that (6.111)

6.13 Quasi-Isometries

213

and ϕ(G \ FG ) ⊂ H \ FH .

(6.112)

Moreover,

 |FH | ≤ |C| · |FG |. (6.113)    = c∈C (FH ∩ ϕ(G)c) so that Indeed, from (6.104) we deduce that FH  |FH |≤



 |FH ∩ ϕ(G)c|

c∈C

=



 −1 |FH c ∩ ϕ(G)|

c∈C





 −1 |FH c0 ∩ ϕ(G)| (by (6.110))

c∈C

= |C| · |FH ∩ ϕ(G)| ≤ |C| · |FG | (by the equality in (6.111)). Let us show that the set FG ⊂ G has the required property. Suppose that g ∈ ∂EG (FG ). This means that the set gEG meets both FG and G \ FG . Thus, there exist g1 ∈ FG and g2 ∈ G \ FG such that g −1 g1 ∈ EG and g −1 g2 ∈ EG .   This implies ϕ(g)−1 ϕ(g1 ) ∈ EH and ϕ(g)−1 ϕ(g2 ) ∈ EH by applying (6.106). As ϕ(g1 ) ∈ FH (by (6.111)) and ϕ(g2 ) ∈ H \ FH (by (6.112)), we deduce  meets both FH and H \ FH . In other words, we have that the set ϕ(g)EH  (FH ). This shows that ϕ(g) ∈ ∂EH  (FH )). ∂EG (FG ) ⊂ ϕ−1 (∂EH

(6.114)

By taking cardinalities, we finally get  (FH ))| (by (6.114)) |∂EG (FG )| ≤ |ϕ−1 (∂EH  (FH )| (by (6.107)) ≤ M |∂EH

  c (F = M |∂EH H )| (by (6.102)) 0  ≤ M |∂EH (FH )| (by (6.103))



ε |F  | (by (6.108)) |C| H

≤ ε|FG | (by (6.113)). This shows that FG satisfies (6.105). From Corollary 5.4.5 we deduce that G is amenable.  

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Notes The idea of looking at a finitely generated group with a geometer’s eye by investigating the properties of the family consisting of all its word metrics is due to M. Gromov ([Gro1, Gro3, Gro4]) and gave birth to the flourishing branch of mathematics which is commonly known as geometric group theory in the late 1970s. In the 1950s, the notion of growth of a finitely generated group arose in group theory in relation to volume growth in Riemannian manifolds. This ˇ line of study was initiated by V.A. Efremovich [Efr] and A.S. Svarc [Sva] in the USSR and, slightly later and completely independently, by J. Milnor [Mil1] and J.A. Wolf [Wol] in the USA. In [Mil1] Milnor proved that fundamental groups of closed Riemannian manifolds with negative sectional curvature have exponential growth. Wolf [Wol] proved that a polycyclic group has polynomial growth if it contains a nilpotent subgroup of finite index and has exponential growth otherwise. Then, Milnor [Mil2] proved that every finitely generated non-polycyclic solvable group has exponential growth. Finally, in 1972 H. Bass [Bas] showed that the growth of a nilpotent group G with a finite symmetric generating subset S is exactly polynomial in the sense that there are positive constants C1 and C2 such that C1 nd ≤ γS (n) ≤ C2 nd , for all n ≥ 1, where d = d(G) ≥ 0 is an integer which can be computed explicitly from the lower central series of G (see [Har1, page 201] for more information on the history and prehistory of these results). Note that the growth estimates of Milnor and Bass imply that a finitely generated solvable group has either polynomial or exponential growth. It was shown by J. Tits [Tits] (see also [Har1]) that every finitely generated linear group either is virtually nilpotent or contains a free subgroup of rank two. This last result, which is known as the Tits alternative for linear groups, implies that every finitely generated linear group has either polynomial growth or exponential growth. The problem of the characterization of finitely generated groups with polynomial growth remained open until Gromov proved in [Gro2] that a finitely generated group with polynomial growth contains a nilpotent subgroup of finite index. It follows from the above mentioned result of Bass and Proposition 6.6.6 that a group of polynomial growth has in fact exactly polynomial growth. Thus, for finitely generated groups, the notions of polynomial and exactly polynomial growth coincide. The (general) Burnside problem, posed by W. Burnside in 1902, asked whether a finitely generated periodic group is necessarily finite. It was answered in the negative in 1964 by E.S. Golod and I.R. Shafarevich [GolS], who gave an example of a finitely generated infinite p-group. The Grigorchuk group, also known as the first Grigorchuk group, was originally constructed by R. I. Grigorchuk in 1980 [Gri2] as a new example of a finitely generated infinite periodic group, thus providing another counterexample to the general Burnside problem. In 1984 Grigorchuk [Gri4] proved that this group has intermediate growth (this was announced by Grigorchuk in 1983 [Gri3]),

Notes

215

thus providing a positive answer to the Milnor problem, posed by Milnor in 1968, about the existence of finitely generated groups of intermediate growth. More precisely, in [Gri4] Grigorchuk proved, among other things, √ that exp( n)  γ(G)  exp(ns ), where s = log32 (31) ≈ 0.991. The Grigorchuk group also provides the first example of an amenable but not elementary amenable group (the class of elementary amenable groups is the smallest class of groups containing all finite and all abelian groups that is closed under taking subgroups, quotients, extensions, and directed unions), thus answering a question posed by Day in 1957 [Day1]. Among other interesting properties of the Grigorchuk group G, we mention the following (see [CMS2], [Har2], [Gri5], [GriP]): (a) G is not finitely presented (a recursive set of defining relations for G was found by I.G. Lys¨enok [Lys]), (b) G is just infinite (it is infinite but every proper quotient is finite), (c) G has solvable word problem, that is, there exists an algorithm that establishes whether, given s1 , s2 , . . . , sn ∈ {a, b, c, d}, one has s1 s2 · · · sn = 1G or not. Originally, the Grigorchuk group was defined as a group of Lebesgue measure-preserving transformations of the unit interval. By representing the elements of Σ ∗ as the vertices of an infinite binary rooted tree, the Grigorchik group may be also realized as a subgroup of the full automorphism group of the tree. Another description of this group was provided by regarding it as a group generated by a finite automaton (see [GriNS]). Simple random walks on groups were first considered by H. Kesten in [Kes1]. Given a finitely generated group G and a finite symmetric generating subset S ⊂ G, the simple random walk on G relative to S is the G-invariant Markov chain with state space G and transition probabilities given by  1 if g −1 h ∈ S p(g, h) = |S| 0 otherwise. This can be interpreted as follows: a “random walker” on G moves from a group element g with equal probability to one of its |S| neighbors gs, where s ∈ S. For g, h ∈ G, denote by p(n) (g, h) the probability of reaching h from g after exactly n steps. We then have p(0) (g, h) = δg,h where δg,h is the Kronecker symbol, p(1) (g, h) = p(g, h) and, more generally,  p(n) (g, h) = p(n−1) (g, k)p(k, h). k∈G

The quantity p(n) (g, g) does not depend on g ∈ G and is called the return probability after n steps. The number ρ(G, S) = lim sup

n

p(n) (g, g)

(6.115)

n→∞

is called the spectral radius of the simple random walk on G relative to S. Kesten [Kes1, Kes2] proved that one always has

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6 Finitely Generated Amenable Groups

 2 |S| − 1 ≤ ρ(G, S) ≤ 1 |S| with equality on the right if and only if G is amenable. Moreover, if S contains no involutions, equality on the left holds if and only if G is a free group and S is the symmetrization of a free base. Theorem 6.12.9 is just an 2 reformulation of the amenability criterion of Kesten’s theorem. Kesten also proved that if S is a finite symmetric generating subset of a group G and N ⊂ G is a normal subgroup, then, denoting by G = G/N (resp S ⊂ G) the corresponding quotient group (resp. generating subset of G) then ρ(G, S) ≤ ρ(G, S) with equality if and only if N is amenable. Day [Day3] extended Kesten’s amenability criterion to non symmetric random walks. The associated Markov chain is then determined by a probability density whose support generates the group. In this setting, the associated Markov operator on 2 (G) is no more self-adjoint, in general. The key ingredient of this new proof is the uniform convexity (uniform rotundity in Day’s terminology) of Hilbert spaces (cf. Lemma I.4.2) and more generally of p spaces with p > 1 (note that in fact Day considers, more generally, Markov operators on the Banach spaces p (G), for p > 1). For more on this we refer to the paper [KaV] by V.A. Kaimanovich and A.M. Vershik and to W. Woess’ review [Woe1] and monograph [Woe2]. Another important criterion for amenability of finitely generated groups has been obtained by R.I. Grigorchuk [Gri1]. Let G be a group with m generators. Then G is isomorphic to F/N where F is the free group on m generators and N ⊂ F is a normal subgroup. Let α = α(G; F, N ) be defined by setting  α = lim sup n w(n), n→∞

where w(n) equals the number of elements in N at distance at most n from the identity element 1F in the free group F . The non-negative number α is called the cogrowth of G relative to the presentation G = F ; N . Grigorchuk [Gri1] proved that either α = 1 (this holds if and only if N = {1F }) or √ 2m − 1 ≤ α ≤ 2m − 1. (6.116) He also showed that if ρ = ρ(G, S) is the spectral radius of the simple random walk on G relative to S (the symmetrization of the image of a free base of F under the canonical quotient homomorphism F → G = F/N ), then the following relation holds: ⎧√ √ ⎨ 2m−1 if 1 ≤ α ≤ 2m − 1 m √  (6.117) ρ= √ √ 2m−1 ⎩ 2m−1 √ α if 2m − 1 ≤ α ≤ 2m − 1. + 2m α 2m−1 Then, from (6.117) and Kesten’s criterion he deduced that in (6.116) equality holds on the right if and only if G is amenable. This is called the Grigorchuk

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criterion (or the cogrowth criterion) of amenability. Grigorchuk’s criterion was used by Ol’shanskii [Ols] to show the existence of non-amenable groups without non-abelian free subgroups and by Adyan [Ady] to show that the free Burnside groups B(m, n), with m ≥ 2 generators and exponent n ≥ 665 odd, are non-amenable. The program of the classification of all finitely generated groups up to quasi-isometries was posed and initiated by Gromov [Gro4]. The definition of quasi-isometry presented here is modeled after Y. Shalom [Sha].

Exercises 6.1. Let S = {s1 , s2 , . . . , sn } be a finite subset of Z. Show that S generates Z if and only if gcd(s1 , s2 , . . . , sn ) = 1. 6.2. Let G be a finitely generated group and let S be a finite symmetric generating subset of G. Let x ∈ G and denote by Lx , Rx : G → G the maps defined by Lx (g) = xg and Rx (g) = gx for all g ∈ G. (a) Show that the map g → dS (g, Rx (g)) is constant on G. (b) Show that if x is in the center of G, then Rx is an isometry of (G, dS ). (c) Show that supg∈G dS (g, Lx (g)) < ∞ if and only if the conjugacy class of x in G is finite. 6.3. Suppose that S and S  are two finite symmetric generating subsets of a group G with S ⊂ S  . Show that one has: (i) S (g) ≥ S  (g) and dS (g, h) ≥ dS  (g, h) for all g, h ∈ G; (ii) BS (n) ⊂ BS  (n) and γS (n) ≤ γS  (n) for all n ∈ N; (iii) λS ≤ λS  . 6.4. Let G1 and G2 be two finitely generated groups and let G = G1 ×G2 . Let S1 (resp. S2 ) be a finite symmetric generating subset of G1 (resp. G2 ). Show that S = (S1 × {1G2 }) ∪ ({1G1 } × S2 ) is a finite symmetric generating subset G1 G2 of G and that one has G S (g) = S1 (g1 ) + S2 (g2 ) for all g = (g1 , g2 ) ∈ G. 6.5. Let S1 and S2 be two sets. For i = 1, 2, let Qi = (Qi , Ei ) be an Si labeled graph. We define their direct product Q1 × Q2 as the S-labeled graph Q = (Q, E) with: ! ! (1) S = S1 S2 , where denotes the disjoint union; (2) Q = Q1 × Q2 ; (3) E = {((q1 , q2 ), s, (q1 , q2 )) : either q1 = q1 and (q2 , s, q2 ) ∈ E2 , or q2 = q2 and (q1 , s, q1 ) ∈ E1 }. Suppose that S1 (resp. S2 ) is endowed with an involution ι1 : S1 → S1 (resp. ι2 : S2 → S2 ) and that Q1 (resp. Q2 ) is edge-symmetric with respect to such involution. Denote by ι : S → S the map defined by ι(s) = ιi (s) if s ∈ Si , i = 1, 2, and observe that ι is an involution. Show that Q is edgesymmetric with respect to ι.

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6 Finitely Generated Amenable Groups

6.6. Let G1 and G2 be two finitely generated groups and let S1 ⊂ G1 and / S1 S2 ⊂ G2 be two finite and symmetric generating subsets such that 1G1 ∈ and 1G2 ∈ / S2 . Consider the direct product group G = G1 × G2 together with the (finite and symmetric) generating subset S = (S1 ×{1G2 })∪({1G1 }×S2 ). Denote by CS1 (G1 ), CS2 (G2 ) and CS (G) the corresponding Cayley graphs. If we identify S1 with S1 × {1G2 } (resp. S2 with {1G1 } × S2 ), we may regard CS1 (G1 ) (resp. CS2 (G2 )) as an (S1 ×{1G2 })-labeled graph (resp. ({1G1 }×S2 )labeled graph). Show that CS (G) = CS1 (G1 ) × CS2 (G2 ). 6.7. Let G = Z, S = {1, −1} and S  = {2, −2, 3, −3}. Find the best possible positive constants C1 and C2 such that C1 S (g) ≤ S  (g) ≤ C2 S (g) for all g ∈ G. Hint: Check that C1 = 1/3 and C2 = 2. 6.8. Let G = Zm , where m ≥ 1 is an integer. Consider the finite and symmetric generating subset S = {±(1, 0, 0, . . . , 0), ±(0, 1, 0, . . . , 0), . . . , ±(0, 0, . . . , 0, 1)} ⊂ Zm . (a) Show that if g = (a1 , a2 , . . . , am ) ∈ Zm then S (g) = |a1 | + |a2 | + · · · + |am |. (b) Let n ∈ N. Set P0 (n) = 1 and, for all integers t ≥ 1 denote by Pt (n) the number of distinct t-tuples (a1 , a2 , . . . , at ) of positive integers such that a1 + a2 + · · · + at ≤ n. Show that Pt (n) = nt for 1 ≤ t ≤ n. Hint: The map (a1 , a2 , . . . , at ) → {a1 , a1 + a2 , . . . , a1 + a2 + · · · + at } establishes a bijection between the set {(a1 , a2 , . . . , at ) ∈ Nt : ai ≥ 1 and a1 + a2 + · · · + at ≤ n} and the set of all subsets of cardinality t of the set {1, 2, . . . , n}. (c) For n, t ∈ N and t ≥  1 denote by Nt (n) the number of all m-tuples m (a1 , a2 , . . . , am ) ∈ Zm with  i=1 |ai | ≤ n and exactly t many of the ai ’s m m nonzero. Show that γSZ (n) = t=0 Nt (n). (d) Let 0 ≤ t ≤ n and let I be a subset of {1, 2, . . . , n} such that |I| = t. Show that there are precisely Pt (n) distinct elements g = (a1 , a2 , . . . , am ) ∈ Nm with I = {i : ai > 0} and such that S (g) ≤ n. n (e) Deduce from (d) that there are exactly m t t elements g = (a1 , a2 , . . . ,  (g) ≤ n. am ) ∈ Nm with |{i : ai > 0}| = t such that nS . (f) Deduce from (e) that Nt (n) = 2t m t t  n m m (g) Deduce from (f) and (c) that γSZ (n) = t=0 2t m t t . 6.9. Suppose that γ, γ  : N → [0,  +∞) are two growth  functions such that γ  γ  . Show that lim supn→∞ n γ(n) ≤ lim supn→∞ n γ  (n). 6.10. Let α = 21 11 ∈ SL2 (Z). Consider the metabelian group G = Z2 α Z, that is, the semidirect product of Z2 by the infinite cyclic subgroup of SL2 (Z) generated by α. Recall that   

x G= , z : x, y, z ∈ Z y

Exercises

219

with the multiplication defined by           x1 x2 x1 z1 x2 , z1 , z2 = +α , z1 + z2 y1 y2 y1 y2 for all x1 , x2 , y1 , y2 , z1 , z2 ∈ Z. for all n ∈ N, where (fk )k∈N is the (a) Show that αn 10 = f2n+1 f2n Fibonacci sequence which is inductively defined by f0 = 0, f1 = 1, and fk = fk−2 + fk−1 for all k ≥ 2. (b) Deduce from (a) that, for any integer n ≥ 1, the set   n     i−1 1 , 0 : ui ∈ {0, 1} for 1 ≤ i ≤ n ⊂ G ui α A(n) = 0 i=1

has cardinality |A(n)| = 2n . (c) Consider the subset S ⊂ G defined by S = {a, b, c, a−1 , b−1 , c−1 }, where          0 0 1 a= ,0 ,b = , 0 and c = ,1 . 0 1 0 Show that S is a finite symmetric generating subset of G and that one has A(n) ⊂ BSG (3n − 2) for all n ≥ 1. (d) Deduce from (b) and (c) that G has exponential growth. 6.11. Let n ≥ 2. Show that GLn (Z) is a finitely generated group of exponential growth. Hint: Use Exercise 2.18, Lemma 2.3.2 and Corollary 6.6.5. 6.12. Let F2 denote the free group of rank two. Show that the groups GL2 (Z) Use elementary row operations to show that and F2 are commensurable. Hint: the matrices 10 21 and 12 01 generate a finite index subgroup of GL2 (Z) and apply Lemma 2.3.2. 6.13. Growth of the Baumslag-Solitar group BS(1, m). Let m be an integer such that |m| ≥ 2. Prove that the metabelian group G = a, b : aba−1 = bm  studied in Exercises 2.7 and 4.21 has exponential growth. Hint: Use an argument similar to the one used for the case m = 2 in the proof of Proposition 6.7.1. More precisely, take S = {a, b, a−1 , b−1 } and prove that every element of the form g = bk , where 0 ≤ k ≤ |m|n − 1 and n ≥ 1, has word length S (g) ≤ |m|n + n − 2 by developing k in base |m|. 6.14. Affine representation of the Baumslag-Solitar group BS(1, m). Let m be an integer such that |m| ≥ 2. Consider the group G given by the presentation G = a, b : aba−1 = bm  (cf. Exercises 2.7, 4.21 and 6.13). (a) Let α, β : R → R be the maps respectively defined by α(x) = mx and β(x) = x + 1 fro all x ∈ R. Show that there is a unique homomorphism ϕ : G → Sym(R) satisfying ϕ(a) = α and ϕ(b) = β.

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6 Finitely Generated Amenable Groups

(b) Show that ϕ is injective. Hint: Use Exercise 2.7(a). (c) Let n0 be an integer such that |m|n0 ≥ 3. Consider the elements λ, μ ∈ Sym(R) respectively defined by λ = α−n0 and μ = βλβ −1 . Check that the open intervals i = (−1/2, 1/2) and J = (1/2, 3/2) satisfy λ(J) ⊂ I and μ(I) ⊂ J. (d) Let n ≥ 1 be an integer and let σ1 , σ2 , . . . , σn ∈ {λ, μ}. Prove that σ1 σ2 · · · σn = IdR . Hint: Use (c) and play ping-pong as in the proof of Theorem D.5.1. (e) Use (a), (b) and (d) to get another proof of the fact that G has exponential growth. 6.15. Growth of the lamplighter group. Let L = (Z/2Z)  Z denote the lamplighter group (cf. Exercise 4.19). Recall that L is the semidirect product of a normal subgroup H = ⊕n∈Z An , where each An is a subgroup of order 2, with an infinite cyclic subgroup N generated by an element t which satisfies tat−1 = (an−1 )n∈Z for all a = (an )n∈Z ∈ H. Let s denote the nontrivial element of A0 . (a) Show that S = {s, t, t−1 } is a symmetric generating subset of L. (b) For n ≥ 1, let Bn = ⊕n−1 k=0 Ak . Prove that Bn is a subgroup of H generated by the elements s, tst−1 , t2 st−2 , . . . , tn−1 st−n+1 and that |Bn | = 2n . (c) Deduce from (b) that 2n ≤ γSL (3n − 2) for all n ≥ 1. (d) Deduce from (c) that L has exponential growth. 6.16. Growth of the integral Heisenberg group. Let G = HZ denote the Heisenberg group over the ring of integers (cf. Example 4.6.5). Recall that G is the subgroup of SL3 (Z) consisting of all matrices of the form ⎛ ⎞ 1yz M (x, y, z) = ⎝0 1 x⎠ (x, y, z ∈ Z). 001 Let us set A = M (1, 0, 0), B = M (0, 1, 0), C = M (0, 0, 1), and S = {A, A−1 , B, B −1 , C, C −1 }. (a) Verify that M (x, y, z) = Ax B y C z for all x, y, z ∈ Z. (b) Show that S is a finite symmetric generating subset of G. (c) Show that C x Ay = Ay C x , C x B y = B y C x , and B x Ay = Ay B x C xy for all x, y, z ∈ Z. (d) Deduce from (c) that if P ∈ G satisfies S (P ) ≤ n, then there exist x, y, z ∈ Z with |x| ≤ n, |y| ≤ n, and |z| ≤ n2 + n, such that P = Ax B y C z . (e) Deduce from (d) that there exists a constant C1 > 0 such that γS (n) ≤ C1 n4 for all n ≥ 1. (f) Let n, x, y, z be integers such that 0 ≤ x ≤ n, 0 ≤ y ≤ n, and 0 ≤ z ≤ n2 . Show that there exist integers q, r with 0 ≤ q ≤ n and 0 ≤ r ≤ n − 1 such that

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Ax B y C z = Ax−q B n Aq B y−n C r . Hint: Use (c) and Euclidean division of z by n. (g) Deduce from (f) that γS (5n) ≥ (n + 1)2 (n2 + 1) for all n ≥ 0. (h) Deduce from (g) that there exists a constant C2 > 0 such that γS (n) ≥ C2 n4 for all n ≥ 0. (i) Deduce from (e) and (h) that γ(G) ∼ n4 . 6.17. Let G be the Grigorchuk group. (a) Show that G acts transitively on Σ n for all n ∈ N. Hint: Use induction on n. More precisely, let w1 , w2 ∈ Σ n and suppose first that w1 and w2 start with the same letter, that is, there exist x ∈ Σ and u1 , u2 ∈ Σ n−1 such that w1 = xu1 and w2 = xu2 . Use induction and Proposition 6.9.7(i). Otherwise, if w1 and w2 do not start with the same letter, observe that w1 and a(w2 ) do start with the same letter and reduce to the previous case. (b) Use (a) to recover the fact that G is infinite (cf. Theorem 6.9.8). 6.18. Let K1 = Z/2Z = {0, 1} and, for n ≥ 2, define by induction Kn = Kn−1  (Z/2Z). Recall that Kn−1  (Z/2Z) = (Kn−1 )Z/2Z  (Z/2Z), so that Kn consists of the elements (f, a) ∈ (Kn−1 )Z/2Z × (Z/2Z) with the multiplication defined by (f1 , a1 )(f2 , a2 ) = (f1 f2a1 , a1 + a2 ), for all f1 , f2 ∈ (Kn−1 )Z/2Z and a1 , a2 ∈ Z/2Z, where f a (a ) = f (a + a ) ∈ Kn−1 for all f ∈ (Kn−1 )Z/2Z and a, a ∈ Z/2Z. The group Kn is called the Kaloujnine 2-group of degree n. Set Σ = {0, 1}. For n = 1 and x ∈ Σ, we set 1(x) = 1 − x, and 0(x) = x. For n ≥ 2 let g = (f, a) ∈ Kn and w = xu ∈ Σ n , where x ∈ Σ and u ∈ Σ n−1 . We then set g(w) = a(x)f (x)(u) ∈ Σ n (note that a(x) ∈ Σ, f (x) ∈ Kn−1 , and f (x)(u) ∈ Σ n−1 is defined by induction). (a) Show that the map Kn × Σ n  (g, w) → g(w) ∈ Σ n defines an action of the group Kn on Σ n . (b) Show that this action is faithful. (c) By virtue of (b), we may regard Kn as a subgroup of Sym(Σ n ). Show, by simple counting arguments, that Kn is a Sylow 2-subgroup of Sym(Σ n ). (d) Consider the elements g1,n , g2,n , . . . , gn,n ∈ Kn defined by induction as follows. First define g1,1 = 1 ∈ K1 = Z/2Z and then, for 1 ≤ m ≤ n, set g1,m = (f1,m , 1) ∈ Km , where f1,m : Z/2Z → Km−1 is given by f1,m (a) = 1Km−1 for all a ∈ Z/2Z. Finally, for 2 ≤ k ≤ m ≤ n, set gk,m = (fk,m , 0) ∈ Km , where fk,m : Z/2Z → Km−1 is given by fk,m (0) = gk−1,m−1 and fk,m (1) = (f1,m−1 , 0). Show that g1,n , g2,n , . . . , gn,n generate Kn and verify that gi,n (uxv) = u(1 − x)v ∈ Σ n for all u ∈ Σ i−1 , x ∈ Σ and v ∈ Σ n−i . (e) Define a map πn : Kn → (Z/2Z)n by induction as follows. π1 : K1 = Z/2Z → Z/2Z  is the identity map, while, for n ≥ 2 and (f, a) ∈ Kn we set πn (f, a) = ( b∈(Z/2Z) πn−1 (f (b)), a) ∈ (Z/2Z)n−1 ×(Z/2Z) = (Z/2Z)n for all f ∈ (Kn−1 )Z/2Z and a ∈ Z/2Z. Show that πn is a surjective homomorphism. use induction on n and the fact Hint: To prove that πn is a homomorphism  that b∈(Z/2Z) πn−1 (f a (b)) = b∈(Z/2Z) πn−1 (f (b)) for all f ∈ (Kn−1 )Z/2Z

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and a ∈ Z/2Z. To show surjectivity, look at the πn -images of the elements g1,n , g2,n , . . . , gn,n ∈ Kn defined in (d). (f) Deduce from (e) that Kn /[Kn , Kn ] (the abelianization of Kn ) is isomorphic to (Z/2Z)n . (g) Consider the map Φn : Σ n → {1, 2, . . . , 2n } given by expansion in base two, that is, Φn (i1 , i2 , . . . , in ) = i1 + 2i2 + 4i3 + · · · + 2n−1 in , for all i1 , i2 , . . . , in ∈ Σ. Verify that, modulo the map Φ3 , one has g1,3 = (1 5)(2 6) × (3 7)(4 8), g2,3 = (1 3)(2 4), and g3,3 = (1 2). 6.19. Let G be the Grigorchuk group. (a) Consider the homomorphism Ψ3 : G → Sym(8) defined by Ψ3 (g) = g|Σ 3 (observe that the map Ψ3 is well defined since (g(w)) = (w) for all w ∈ Σ ∗ and g ∈ G). With the notation from Exercise 6.18(f), verify that Ψ3 (a) = (1 5)(2 6)(3 7)(4 8), Ψ3 (b) = (1 3)(2 4)(5 6), Ψ3 (c) = (1 3)(2 4) and Ψ3 (d) = (5 6). (b) Deduce from (a) and Exercise 6.18(e) that Ψ3 (G) = K3 , where K3 ⊂ Sym(8) is the Kaloujnine group (cf. Exercise 6.18). (c) By applying Proposition 6.9.3, deduce from (b) and Exercise 6.18(e) that G/[G, G] (the abelianization of G) is isomorphic to (Z/2Z) × (Z/2Z) × (Z/2Z). 6.20. The word problem for the Grigorchuk group. Let G be the Grigorchuk group. Describe an algorithm which, given any word w ∈ {a, b, c, d}∗ , determines whether w represents the identity element 1G or not. Hint: Given a word w, first count the number a (w) of occurrences of the letter a in w. Prove that if a (w) is odd then w does not represent 1G . If a (w) is even, use the maps φ0 , φ1 : H1 → G and apply induction on the length of the word w. 6.21. Let us say that a net (xi )i∈I in a set X converges to infinity if, for /F every finite subset F ⊂ X, there exists an element i0 ∈ I such that xi ∈ for all i ≥ i0 . Let ϕ : G → H be a map from a group G into a group H. Show that ϕ is a quasi-isometric embedding if and only if it satisfies the following condition: for any two nets (ui )i∈I and (vi )i∈I in G having the same index set, the net (u−1 i vi )i∈I converges to infinity in G if and only if the net (ϕ(ui )−1 ϕ(vi ))i∈I converges to infinity in H. 6.22. Let G be a group and let E(G) denote the set consisting of all quasiisometries ϕ : G → G. Define a binary relation ∼ in E(G) by declaring that ϕ1 and ϕ2 ∈ E(G) satisfy ϕ1 ∼ ϕ2 if and only if there exists a finite subset F ⊂ G such that ϕ1 (g)−1 ϕ2 (g) ∈ F for all g ∈ G. (a) Show that ∼ is an equivalence relation in E(G). (b) Show that if ϕ ∈ E(G) then there exists ψ ∈ E(G) such that ϕ ◦ ψ ∼ IdG and ψ ◦ ϕ ∼ IdG . (b) Show that the composition of maps in E(G) is compatible with ∼ and induces a group structure on the quotient set QI(G) = E(G)/ ∼. (c) Show that if G is a finite group then the group QI(G) is trivial.

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(d) Show that the group QI(Z) contains a subgroup isomorphic to R×Z/2Z and is therefore uncountable. Hint: Prove that the map ϕλ : Z → Z defined by ϕλ (x) = [λx], where λ ∈ R \ {0} and [α] denotes the integral part of α, that is, the largest integer n such that n ≤ α, is a quasi-isometry and that one has fλ1 ∼ fλ2 if and only if λ1 = λ2 . (e) Show that if G and H are quasi-isometric groups then the groups QI(G) and QI(H) are isomorphic. 6.23. Let A be a set and let G = (Q, E) be a finite A-labeled graph. Denote by λ : E → A the labeling map defined by λ(e) = a for every edge e = (q, a, q  ) ∈ E. A bi-infinite path in G is a sequence π = (en )n∈Z of edges en = (qn , an , qn ) ∈ E such that qn = qn+1 for all n ∈ Z. We define the label of a bi-infinite path π = (en )n∈Z as being the element λ(π) ∈ AZ given by λ(π)(n) = λ(en ) for all n ∈ Z. (a) Denote by X G the set of the labels λ(π) of all bi-infinite paths π in G. Show that X G is a subshift of AZ . It is called the subshift defined by the A-labeled graph G. (b) Show that if G is connected then the subshift X G is irreducible. 6.24. Let A = {0, 1} and consider the A-labeled graphs G1 and in G2 in Fig. 6.18. Check that the associated subshifts X G1 and X G2 are the even subshift (cf. Exercise 1.38) and the golden mean subshift (cf. Exercise 1.39) respectively.

Fig. 6.18 The A-labeled graphs G1 and G2

6.25. Let A be a set and let X ⊂ AZ be a subshift of finite type. Let M be a positive integer such that {1, 2, . . . , M } ⊂ Z is a memory set for X. Consider the A-labeled graph G = G(X, M ) = (Q, E) defined as follows: Q = LM −1 (X) is the set of all X-admissible words of length M − 1, and the edge set E consists of all triples e = (aw, a, wa ) ∈ Q × A × Q, where a, a ∈ A and w ∈ AM −2 are such that awa ∈ LM (X). (a) Check that X G = X. (b) Show that X is irreducible if and only if G is connected (cf. Exercise 6.23). 6.26. Let A be a set and let X ⊂ AZ be a subshift of finite type. Let also τ : AZ → AZ be a cellular automaton. Let M be a positive integer such that {1, 2, . . . , M } ⊂ Z is a memory set for both X and τ , and let μ : AM → A denote the corresponding local defining map for τ . Consider the A-labeled

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graph G = G(X, τ, M ) = (Q, E), where Q = LM −1 (X) is the set of all Xadmissible words of length M − 1 and the edge set E consists of all triples e = (aw, μ(awa ), wa ) ∈ Q × A × Q, where a, a ∈ A and w ∈ AM −2 are such that awa ∈ LM (X). Check that X G = τ (X). 6.27. Life on Z. Let A = {0, 1} and consider the cellular automaton τ : AZ → AZ defined in Exercise 5.3. Let G be the A-labeled graph in Fig. 6.19. Check that X G = τ (AG ).

Fig. 6.19 The A-labeled graph G = G(A, τ, 3)

6.28. Let A = {0, 1} and let τ : AZ → AZ be the majority action cellular automaton (cf. Example 1.4.3(c)) associated with the set S = {−1, 0, 1}. Let  G  be the A-labeled graph in Fig. 6.20. Check that X G = τ (AG ).

Fig. 6.20 The A-labeled graph G  = G(A, τ, 3)

6.29. Let A be a set and let G = (Q, E) be a finite A-labeled graph. (a) Suppose that for each pair (a, a ) ∈ A2 there exists at most one vertex q ∈ Q which is both the terminal vertex of an edge with label a and the

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initial vertex of an edge with label a . Show that the subshift X G ⊂ AZ is of finite type. Hint: Show that in fact {0, 1} ⊂ Z is a memory set for X G . (b) Show that the hypothesis in (a) is satisfied if the labeling map λ : E → A is injective. 6.30. Let A be a set and let G = (Q, E) be a finite A-labeled graph. We recall that given an edge e = (q, a, q  ) ∈ E we denote by α(e) = q ∈ Q (resp. ω(e) = q  ∈ Q) the initial (resp. terminal) vertex of e. Consider the E-labeled graph G  = (Q , E  ), where Q = Q and E  = {(α(e), e, ω(e)) ∈ Q × E × Q : e ∈ E}. Note that the labeling map λ : E  → E on G  is bijective. Identify the set {λ(e) : e ∈ E} ⊂ A formed by the labels of the edges of G with a subset of E in an arbitrary way and consider the cellular automaton τ : E Z → E Z with memory set S = {0} and local defining map μ : E S → E defined by μ(e) = λ(e) for all e ∈ E S = E. Observe that τ (x) ∈ AZ for all x ∈ E Z and  show that τ (X G ) = X G . 6.31. Let A be a set and X ⊂ AZ a subshift. Show that the following conditions are equivalent: (i) there exists a finite A-labeled graph G such that X = X G . (ii) there exists a set B containing A, a subshift Y ⊂ B Z of finite type and a cellular automaton τ : B Z → B Z such that X = τ (Y ). A subshift X ⊂ AZ is said to be sofic if it satisfies one of the two above equivalent conditions. Hint: For the implication (i) ⇒ (ii) use Exercise 6.30. For the converse implication, use Exercise 6.25. 6.32. Let A be a set. (a) Show that every subshift X ⊂ AZ of finite type is sofic. (b) Suppose that A has at least two distinct elements. Show that there exists a subshift X ⊂ AZ which is sofic but not of finite type. Hint: The even subshift is sofic (cf. Exercise 6.24) but not of finite type (cf. Exercise 1.38(c)). 6.33. Let A be a set. Let X ⊂ AG be a sofic subshift and let τ : AG → AG be a cellular automaton. Show that τ (X) ⊂ AG is a sofic subshift. 6.34. A subshift which is not sofic (cf. [LiM, Example 3.1.7]). Let A = {0, 1, 2} and consider the subset X ⊂ AZ consisting of all configurations x ∈ Z such that if x(n) = 0, x(n + 1) = x(n + 2) = · · · = x(n + h) = 1, x(n + h + 1) = x(n + h + 2) = · · · = x(n + h + k) = 2 and x(n + h + k + 1) = 0 for some n ∈ Z and h, k ∈ N, then necessarily h = k. (a) Show that X is a subshift of AZ . It is called the context-free subshift. (b) Show that X is not sofic. Hint: Suppose by contradiction that X = X G for some finite A-labeled graph G = (Q, E). Let r = |Q|. Observe that w = 01r+1 2r+1 0 ∈ L(X) so that there exists a path π in G such that λ(π) = w. Let π  denote the subpath of π such that λ(π  ) = 1r+1 . Since (π  ) = r + 1 > r = |Q|, we can write π  = π1 π2 π3 where π2 is a closed path of length s = (π2 ) > 0 (and π1 (resp. π3 ) is a possibly empty path). It follows that / L(X). π  = π1 π2 π2 π3 is a path in G and its label is λ(π  ) = 01r+1+s 2r+1 0 ∈

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6.35. Let A be a finite set and let X ⊂ AZ be a sofic subshift. A finite Alabeled graph G = (Q, E) is said to be a minimal presentation of X if (i) X = X G , and (ii) |Q| ≤ |Q | for all A-labeled graphs G  = (Q , E  ) such that  X = X G . Suppose that X is irreducible and that G = (Q, E) is a minimal presentation of X. (a) Show that for every vertex q ∈ Q there exists a word wq ∈ L(X) such that if a path π in G satisfies λ(π) = wq , then it passes through q. Hint: By contradiction, if this is not the case for some q ∈ Q then the A-labeled graph G  = (Q , E  ) where Q = Q\{q} and E  = E \{e ∈ E : α(e) = q or ω(e) = q}  satisfies X G = X and |Q | < |Q|, contradicting the minimality of G. (b) Deduce from (a) that G is connected. Hint: Let q, q  ∈ Q. Consider the words wq , wq ∈ L(X) described in (a). Then by irreducibility of X there exists u ∈ L(X) such that wq uwq ∈ L(X). Let π be a path in G such that λ(π) = wq uwq ; then π = π1 π2 π3 where λ(π1 ) = wq , λ(π2 ) = u and λ(π3 ) = wq . By definition of wq (resp. wq ) the path π1 (resp. π3 ) passes through q (resp. q  ), say π1 = π1 π1 with (π1 )+ = q = (π1 )− (resp. π3 = π3 π3 with (π3 )+ = q  = (π3 )− ). Then the path π1 π2 π3 connects q to q  . (c) Deduce from (b) that for every irreducible sofic subshift Y ⊂ AZ there exists a connected finite A-labeled graph G such that Y = X G . 6.36. Let A be a finite set and X ⊂ AZ an irreducible sofic subshift. (a) Show that the subset Xf consisting of all configurations in X whose Z-orbit is finite (cf. Example 1.3.1(c)) is dense in X. Hint: Let x ∈ X and let n ∈ N. By virtue of Exercise 6.35, we can find a connected A-labeled graph G = (Q, E) such that X = X G . Let π1 be a finite path in G such that λ(π1 ) = x(0)x(1) · · · x(n − 1) and set q = π1− and q  = π1+ . Since G is connected, we can find a finite path π2 in G connecting q  to q. Let m = (π2 ). It follows that the path π = π1 π2 is closed and t = (π) = n + m. If w = λ(π) we deduce that wk ∈ L(X) for all k ∈ N. Since X is closed, there exists a configuration y ∈ X such that y(ht)y(ht + 1) · · · y((h + 1)t − 1) = w for all h ∈ Z. In particular, y(0)y(1) · · · y(n − 1) = x(0)x(1) · · · x(n − 1) and y ∈ Xf . (b) Deduce from (a) that X is surjunctive. Hint: Cf. Exercise 3.29. 6.37. Show that the Morse subshift is not sofic. Hint: An infinite minimal subshift is irreducible (cf. Exercise 3.35(b)) and contains no configuration whose Z-orbit is finite (cf. Exercise 3.35(d)). On the other hand, by Exercise 6.36(a), every irreducible sofic subshift contains an abundance of configurations with finite Z-orbit. 6.38. Let A be a finite set and let X ⊂ AZ be an infinite Toeplitz subshift. Show that X is not sofic. Hint: The same arguments as for Exercise 6.37 apply. 6.39. Let A be a finite set. Show that there are at most countably many distinct sofic subshifts X ⊂ AZ . Hint: There are at most countably many finite A-labeled graphs up to isomorphism.

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6.40. Let I be a finite set. Let B = (bij )i,j∈I be a matrix with real entries (n) bij ≥ 0 for all i, j ∈ I. For an integer n ≥ 1, we denote by B n = (bij )i,j∈I the n-th power of B. One says that the matrix B is irreducible if for every i, j ∈ I (n) there exists an integer n = n(i, j) ≥ 1 such that bij > 0. Suppose that B is irreducible. The period per(i) of i ∈ I is the greatest common divisor of the (n) integers n ≥ 1 such that bii > 0. (a) Show that per(i) = per(j) for all i, j ∈ I. The period per(B) of the matrix B is defined as the common value of the numbers per(i), i ∈ I. Hint: (r) (s) Let i, j ∈ I. We can find r, s ≥ 1 such that bij > 0 and bji > 0. It follows (r+s)

that bii

(r) (s)

(r+n+s)

≥ bij bji > 0 and bii

(r) (n) (s)

≥ bij bjj bji > 0 for all n ≥ 1 such

(n)

that bjj > 0. It follows from the definition that per(i) divides both r + s and r + n + s and therefore also divides their difference n. This shows that per(i) divides per(j). (b) Let n ≥ 1 be an integer. Show that B n is irreducible if and only if n and per(B) are relatively prime. 6.41. Let A be a set and let G = (Q, E) be a finite A-labeled graph. The adjacency matrix of G is the (Q × Q)-matrix BG = (bqq )q,q ∈Q where bqq is the number of edges in E with initial vertex q and terminal vertex q  . (a) Show that G is connected if and only if the matrix BG is irreducible. (b) Suppose that G is connected. The period of G is the positive integer defined by per(G) = per(BG ). Show that per(G) is the greatest common divisor of the lengths of all closed paths in G. 6.42. Let A be a set. Given a subshift X ⊂ AZ denote by pern (X) = | Fix(nZ) ∩ X| (cf. Example 1.3.1(c)) the number of nZ-periodic configurations in X. Let G be a finite connected A-labeled graph. (a) Show that for every integer N there exists an integer n ≥ N such that pern (X G ) > 0. (b) The period per(X G ) of X G is defined as the greatest common divisor of all integers n ≥ 1 for which pern (X) > 0. Show that per(X G ) = per(G). 6.43. The N th higher block subshift (cf. Exercise 1.15 and Exercise 1.34). Let A be a set and N a positive integer. Consider the map ΦN : AZ → (AN )Z defined by ΦN (x)(n) = (x(n), x(n + 1), x(n + 2), . . . , x(n + N − 1)) for all x ∈ AZ and n ∈ Z. Similarly, consider the map ϕN : A∗ → (AN )∗ defined as follows: ϕN (w) = ε (the empty word) if (w) < N and ϕ(w) = (a1 , a2 , . . . , aN )(a2 , a3 , . . . , aN +1 ) · · · (am+1 , am+2 , . . . , am+N ) if w = a1 a2 · · · am+N , with m ≥ 0. Let X ⊂ AZ be a subshift and set X [N ] = ΦN (X) ⊂ (AN )Z . Let L ⊂ A∗ be a subset and set L[N ] = ϕN (L) ⊂ (AN )∗ . (a) Show that X [N ] is a subshift of (AN )Z (it is called the N th higher block subshift of X) and that L(X [N ] ) = (L(X))[N ] .

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(b) Show that X is irreducible (resp. topologically mixing, resp. strongly irreducible) if and only if X [N ] is irreducible (resp. topologically mixing, resp. strongly irreducible). (c) Show that if X is of finite type then X [N ] is of finite type. (d) Let Fn = {0, 1, 2, . . . , n − 1} ⊂ Z and let F = (Fn )n∈N . Show that entF (X) = entF (X [N ] ). 6.44. Let A be a set and let G = (Q, E) be an A-labeled graph. For N a positive integer, the N -higher edge graph G [N ] associated with G is the AN labeled graph (Q[N ] , E [N ] ) defined as follows. For N = 1 one has G [1] = G and, for N ≥ 2, the vertex set Q[N ] is the set of all paths of length N − 1 in G and E [N ] = {(π,λ(e1 )λ(e2 ) · · · λ(eN ), π  ) ∈ Q[N ] × AN × Q[N ] : π = (e1 , e2 , . . . , eN −1 ), π  = (e2 , e3 , . . . , eN −1 , eN )}. (a) Show that X G = (X G )[N ] . (b) Deduce from (b) that if X ⊂ AZ is a sofic subshift, then the subshift [N ] ⊂ (AN )Z is also sofic. X [N ]

6.45. The N th higher power subshift (cf. Exercise 1.16 and Exercise 1.35). Let A be a set and N a positive integer. Consider the map ΨN : AZ → (AN )Z defined by ΨN (x)(n) = (x(nN ), x(nN +1), . . . , x((n+1)N −1)) for all x ∈ AZ and n ∈ Z. Similarly, consider the map ψN : A∗ → (AN )∗ defined as follows: ψN (w) = ε if w ∈ A∗ \ (∪n≥1 AnN ) and ψN (w) = (a1 , a2 , . . . , aN )(aN +1 , aN +2 , . . . , a2N ) · · · (a(n−1)N +1 , a(n−1)N +2 , . . . , anN ) if w = a1 a2 · · · anN for some n ≥ 1. Let X ⊂ AZ be a subshift and set X (N ) = ΨN (X) ⊂ (AN )Z . Let L ⊂ A∗ be a subset and set L(N ) = ψN (L) ⊂ (AN )∗ . (a) Show that X (N ) is a subshift of (AN )Z (it is called the N th higher power subshift of X) and that L(X (N ) ) = (L(X))(N ) . (b) Show that if X is of finite type then X (N ) is of finite type. (c) Suppose that X is irreducible. Show that X (N ) is irreducible if and only if N and per(X) are relatively prime. (d) Let Fn = {0, 1, 2, . . . , n − 1} ⊂ Z and let F = (Fn )n∈N . Show that entF (X) = entF (X (N ) ). 6.46. Let A be a set and let G = (Q, E) be an A-labeled graph. For N a positive integer, the N -higher power graph G (N ) associated with G is the AN labeled graph (Q(N ) , E (N ) ) defined as follows. The vertex set is Q(N ) = Q and the edge set is E (N ) = {(π − , λ(π), π + ) ∈ Q × AN × Q : π a path of length N in G},

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where π − ∈ Q (resp. π + ∈ Q) is the initial (resp. terminal) vertex of the path π. Recall that given a labeled graph H, we denote by BH its adjacency matrix (cf. Exercise 6.40). (a) Show that BG (N ) = (BG )N . (N ) (b) Show that X G = (X G )(N ) . (c) Deduce from (b) that if X ⊂ AZ is a sofic subshift, then the subshift (N ) ⊂ (AN )Z is also sofic. X 6.47. (cf. [Sca, Lemma]) Let G = (Q, E) be a finite labeled graph. Suppose that G is connected and let e ∈ E. Show that there exists a positive integer n0 such that if π is any path in G of length n0 , then there exists a path π  = (e1 , e2 , . . . , en0 ) of the same length n0 , with the same initial and terminal vertices as π, and such that ei = e for some 1 ≤ i ≤ n0 . Hint: Given a path π = (e1 , e2 , . . . , en ) in G, denote by πQ = (q0 , q1 , . . . , qn ) the associated sequence of visited vertices (cf. Sect. 6.2). We define a decomposition of π as follows. Let i1 be the largest index such that the vertices q0 , q1 , . . . , qi1 −1 are all distinct. Then qi1 = qj1 for a suitable j1 < i1 and we set r1 = (e1 , e2 , . . . , ej1 ) and c1 = (ej1 +1 , ej1 +2 , . . . , ei1 ). Continuing this way, we obtain a decomposition of the path π = r1 c1 r2 c2 · · · rk ck rk+1 where the c1 , c2 , . . . , ck are closed simple paths and r1 , r2 , . . . , rk+1 are simple (possibly empty) paths. With this notation, for 1 ≤ s ≤ r and a positive integer d, we say that the path πs = r1 c1 r2 c2 rs rs+1 · · · ck rk is obtained from π by collapsing the sth closed simple path cs . Similarly, if π  is a closed path such that (π  )− = π − and d is a positive integer, we say that the path (π  )d π is obtained from π by adding d copies of π  at the beginning of π. Now, since G is connected, we can find a closed path π  with initial (= terminal) vertex π − containing the edge e. If n0 is large enough then in the decomposition π = r1 c1 r2 c2 · · · rk ck rk+1 there exists a cycle c that is repeated many times. If the length of c is , the length of π  is m and the cycle c is repeated at least m times, then we may collapse the first m copies of c and then add  copies of π  at the beginning of π to obtain the desired path π  . 6.48. Let G = (Q, E) be a finite labeled graph. We denote by Pn (G) the set of all paths of length n in G and we define the entropy of G as ent(G) = lim

n→∞

log |Pn (G)| . n

Show that the above limit exists and is finite. Hint: Use Lemma 6.5.1. 6.49. (cf. [Sca, Theorem]) Let G = (Q, E) be a finite connected labeled graph. Let e ∈ E and denote by H = (Q , E  ) the labeled subgraph of G were Q = Q and E  = E \ {e}. Let n0 be the positive integer given by Exercise 6.47 and set α = |Pn0 (G)|−1 . (a) Show that |P(k−1)n0 (G)| ≥ α|Pkn0 (G)| for all k = 1, 2, . . .. (b) We express a path π ∈ Pkn0 (G) as the composition π = π1 π2 · · · πk , where πi ∈ Pn0 (G), 1 ≤ i ≤ k. Also, for i = 1, 2, . . . , k, we denote by ϕi the set

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of all paths π ∈ Pkn0 (G) such that πi contains the edge e. Show that |ϕ1 | ≥ |P(k−1)n0 (G)| and deduce from (a) that |Pkn0 (G)\ϕ1 | ≤ (1−α)|Pn0 (G)|. Hint: ˜ ∈ Pkn0 (G) By Exercise 6.47 for any π ˜ ∈ P(k−1)n0 (G) there exits a path π  π  such that π contains e. i−1 i i (G) = Pkn0 (G)\ t=1 ϕt , C h = {π ∈ Pkn (G) : (c) For 1 ≤ i ≤ k we set Pkn 0 0 h πi contains e} and denote by D the set of pairs (σ1 , σ2 ) ∈ P(i−1)n0 (G) × i P(k−i)n0 (G) such that there exists π = σ1 σσ2 ∈ Pkn (G). Show that |C h | ≥ 0 i (G)|. Hint: Use Exercise 6.47 to show that for every (σ1 , σ2 ) ∈ |Dh | ≥ α|Pkn 0 i D there exists π = σ1 π  σ2 ∈ Pkn0 (G) such that π  contains e. k (d) Deduce from (c) that |Pkn0 (G) \ i=1 ϕi | ≤ (1 − α)k |Pkn0 (G)|. k (e) Observe that Pkn0 (H) ⊂ Pkn0 (G) \ i=1 ϕi and deduce from (d) that ent(H) ≤ log(1−α) + ent(G). n0 (f) Deduce from (e) that ent(H) < ent(G). 6.50. Let A be a finite set. Let Fn = {0, 1, . . . , n} ⊂ Z and F = (Fn )n∈N . Let X ⊂ AZ be a subshift and let w ∈ L(X). Set Xw = {x ∈ X : x(n + 1)x(n + 2) · · · x(n + N ) = w for all n ∈ Z}, where N = (w). (a) Show that Xw ⊂ X is a subshift of AZ . (b) Show that (Xw )[N ] = (X [N ] )ΦN (w) (cf. Exercise 6.43). (c) Suppose from now on that X is irreducible of finite type. Let M be a positive integer such that {1, 2, . . . , M } is a memory set for X, and let G be the A-labeled graph such that X = X(G, M ) (cf. Exercise 6.25). Let H denote the labeled subgraph of G [N ] obtained by removing all edges labeled by ΦN (w). Show that (X [N ] )ΦN (w) = XH . (d) Show that entF (Xw ) < entF (X). Hint: Use Exercises 6.49 and 6.43(d). 6.51. Let A be a finite set. Let Fn = {0, 1, . . . , n} ⊂ Z and F = (Fn )n∈N . Let X ⊂ AZ be an irreducible subshift of finite type. Show that if Y ⊂ AZ is such that Y  X, then entF (Y ) < entF (X). Hint: Show that there exists w ∈ L(X) \ L(Y ) such that Y ⊂ Xw ⊂ X. Then apply Exercise 6.50(d) and Proposition 5.7.2(ii). 6.52. (cf. [Hed-3], [CovP], and [LiM, Theorem 8.1.16]). Let A be a finite set. Let F = (Fn )n∈N , where Fn = {0, 1, . . . , n} ⊂ Z. Also let X ⊂ AZ an irreducible subshift of finite type and τ : X → X a pre-injective cellular automaton. (a) Let M be a positive integer such that {1, 2, . . . , M } is a memory set for X. Set Y = τ (X) and consider the labeled graph G = G(X, τ, M ) (cf. Exercise 6.26). Note that for every w ∈ L(Y ) there exists a path π ∈ G such that w = λ(π). Show that, by pre-injectivity of τ , for all q, q  ∈ Q and w ∈ L(Y ) there exists at most one path π in G with initial (resp. terminal) vertex π − = q (resp. π + = q  ) such that w = λ(π). (b) Deduce from (a) that |Ln (Y )| ≤ |Ln (X)| ≤ |Ln (Y )| · |Q|2 . (c) Deduce from (b) that entF (τ (X)) = entF (X).

Exercises

231

6.53. Let A be a finite set. Let Fn = {0, 1, . . . , n} ⊂ Z and F = (Fn )n∈N . Let X, Y ⊂ AZ be two irreducible subshifts of finite type such that entF (X) = entF (Y ). Show that every pre-injective cellular automaton τ : X → Y is surjective. Hint: Use the results of Exercise 6.51 and Exercise 6.52. 6.54. Let A be a finite set. Let X ⊂ AG be an irreducible subshift of finite type. Show that every pre-injective cellular automaton τ : X → X is surjective. Hint: Use Exercise 6.53 with Y = X. 6.55. (cf. [Hed-3], [CovP], and [LiM, Theorem 8.1.16]). Let A be a finite set and X ⊂ AZ an irreducible subshift of finite type. Let Fn = {0, 1, . . . , n} ⊂ Z and F = (Fn )n∈N . Let τ : X → X be a cellular automaton. Suppose that τ is not pre-injective. (a) Show that there exist a positive integer N and two distinct configurations x1 , x2 ∈ X such that x1 |Z\{1,2,...,N } = x2 |Z\{1,2,...,N } and τ (x1 ) = τ (x2 ). (b) Up to enlarging N if necessary, we may suppose that {1, 2, . . . , N } is a memory set for both X and τ . Let G = G(X, τ, N ) = (Q, E) (cf. Exercise 6.26). Show that there exist two vertices q, q  ∈ Q and two paths π1 , π2 in G with initial (resp. terminal) vertices π1− = π2− = q (resp. π1+ = π2+ = q  ) such that λ(π1 ) = x1 |{1,2,...,N } and λ(π2 ) = x2 |{1,2,...,N } . One says that the two paths π1 , π2 constitute a diamond in G (see Fig. 6.21).

Fig. 6.21 A diamond in an A-labeled graph G

(c) Up to further enlarging N if necessary, we may suppose that N is relatively prime with per(X) (cf. Exercise 6.40 and Exercise 6.42). By Exercise 6.45(c) the N th power graph G (N ) is also irreducible and π1 and π2 are edges in G (N ) with the same initial vertex and same terminal vertex. Let H be the A-labeled subgraph of G (N ) obtained by removing the edge π1 . Deduce from Exercise 6.49 that entF (XH ) < entF (XG (N ) ). (d) Show that XH = Y (N ) , where Y = τ (X). (e) Deduce from (c) and (d) and from Exercise 6.45(d) that entF (τ (X)) < entF (X). 6.56. Let A be a finite set. Let Fn = {0, 1, . . . , n} ⊂ Z and F = (Fn )n∈N . Let X, Y ⊂ AZ be two subshifts such that X is irreducible of finite type and entF (X) = entF (Y ). Show that every surjective cellular automaton τ : X → Y is pre-injective. Hint: Use the result of Exercise 6.55. 6.57. The Garden of Eden theorem for irreducible subshifts of finite type over Z [Fio1, Corollary 2.19]. Let A be a finite set and X ⊂ AZ an irreducible

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subshift of finite type. Let τ : X → X be a cellular automaton. Show that τ is surjective if and only if it is pre-injective. Hint: Combine together the results from Exercise 6.56 with Y = X and Exercise 6.54. 6.58. Let A = {0, 1}. Consider the subset X ⊂ AZ consisting of all configurations x ∈ AZ such that {n ∈ Z : x(n) = 1} is an interval of Z. Let σ : AZ → AZ be the cellular automaton with memory set S = {0, 1} and local defining map μ : AS → A given by  1 if (y(0), y(1)) = (0, 1), μ(y) = y(0) otherwise. (a) Show that X is a sofic subshift. (b) Show that X is neither of finite type nor irreducible. (c) Check that τ (X) ⊂ X (d) Show that the cellular automaton τ = σ|X : X → X is injective (and therefore pre-injective) but not surjective. (e) Deduce from (d) that X is not surjunctive.

Chapter 7

Local Embeddability and Sofic Groups

In this chapter we study the notions of local embeddability and soficity for groups. Roughly speaking, a group is locally embeddable into a given class of groups provided that the multiplicative table of any finite subset of the group is the same as the multiplicative table of a subset of some group in the class (cf. Definition 7.1.3). In Sect. 7.1 we discuss several stability properties of local embeddability. Subgroups of locally embeddable groups are locally embeddable (Proposition 7.1.7). Moreover, as the name suggests, local embeddability is a local property, that is, a group is locally embeddable into a class of groups if and only if all its finitely generated subgroups are locally embeddable into the class (Proposition 7.1.8). When the class is closed under finite direct products, the class of groups which are locally embeddable in the class is closed under (possibly infinite) direct products (Proposition 7.1.10). We also show that a marked group which is a limit of groups belonging to a given class is locally embeddable into this class (Theorem 7.1.16). Conversely, we prove that if the given class is closed under taking subgroups and the marking group is free, then any marked group which is locally embeddable is a limit of marked groups which are in the class (Theorem 7.1.19). If C is a class of groups which is closed under finite direct products, then every group which is residually C is locally embeddable into C (Corollary 7.1.14). Conversely, under the hypothesis that C is closed under taking subgroups, every finitely presented group which is locally embeddable into C is residually C (Corollary 7.1.21). In Sect. 7.2 we present a characterization of local embeddability in terms of ultraproducts: a group is locally embeddable into C if and only if it can be embedded into an ultraproduct of a family of groups in C. Section 7.3 is devoted to LEF and LEA-groups. A group is called LEF (resp. LEA) if it is locally embeddable into the class of finite (resp. amenable) groups. As the class of finite (resp. amenable) groups is closed under taking subgroups and taking finite direct products, all the results obtained in the previous section can be applied. This implies in particular that every locally residually finite (resp. locally residually amenable) group is LEF (resp. LEA). T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 7, © Springer-Verlag Berlin Heidelberg 2010

233

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7 Local Embeddability and Sofic Groups

We give an example of a finitely generated amenable group which is LEF but not residually finite (Proposition 7.3.9). This group is not finitely presentable since every finitely presented LEF-group is residually finite. The Hamming metric, which is a bi-invariant metric on the symmetric group of a finite set, is introduced in Sect. 7.4. In Sect. 7.5 we define the class of sofic groups. These groups are the groups admitting local approximations by finite symmetric groups equipped with their Hamming metric. Subgroups and direct products of sofic groups are sofic. Every LEA-group is sofic (Corollary 7.5.11). In particular, residually amenable groups, and therefore amenable groups and residually finite groups, are sofic. A group is sofic if and only if it can be embedded into an ultraproduct of a family of finite symmetric groups equipped with their Hamming metrics (Theorem 7.6.6). In Sect. 7.7 we give a characterization of finitely generated sofic groups in terms of their Cayley graphs. More precisely, we show that a finitely generated group G with a finite symmetric generating subset S ⊂ G is sofic if and only if, for every integer r ≥ 0 and every ε > 0, there exists a finite S-labeled graph Q such that there is a proportion of at least 1 − ε of vertices q ∈ Q such that the ball of radius r centered at q in Q is isomorphic, as a labeled graph, to a ball of radius r in the Cayley graph of G associated with S (Theorem 7.7.1). The last section of this chapter is devoted to the proof of the surjunctivity of sofic groups (Theorem 7.8.1).

7.1 Local Embeddability Definition 7.1.1. Let G and C be two groups. Given a finite subset K ⊂ G, a map ϕ : G → C is called a K-almost-homomorphism of G into C if it satisfies the following conditions: (K-AH-1) ϕ(k1 k2 ) = ϕ(k1 )ϕ(k2 ) for all k1 , k2 ∈ K; (K-AH-2) the restriction of ϕ to K is injective. Note that in the preceding definition, the map ϕ is not required to be a homomorphism nor to be globally injective. Remark 7.1.2. If ϕ : G → C is a K-almost-homomorphism and ϕ : G → C is a map which coincides with ϕ on K ∪ K 2 , then ϕ is also a K-almosthomomorphism. Let now C be a class of groups, that is, a collection of groups satisfying the following condition: if C ∈ C and C  is a group which is isomorphic to C, then C  ∈ C. For example, C might be the class of finite groups, the class of nilpotent groups, the class of solvable groups, the class of amenable groups, or the class of free groups.

7.1 Local Embeddability

235

Definition 7.1.3. Let C be a class of groups. One says that a group G is locally embeddable into the class C if, for every finite subset K ⊂ G, there exist a group C ∈ C and a K-almost-homomorphism ϕ : G → C. Examples 7.1.4. (a) Let C be a class of groups and let G be a group in C. Then G is locally embeddable into C. Indeed, the identity map IdG : G → G is a K-almost-homomorphism of G into itself for every finite subset K ⊂ G. (b) The group Z is locally embeddable into the class of finite groups. Indeed, let K be a finite subset of Z. Choose an integer n ≥ 0 such that K ⊂ [−n, n]. Then, the quotient homomorphism ϕ : Z → Z/(2n + 1)Z is a K-almost-homomorphism. Remark 7.1.5. Let G be a group and let C be a class of groups which is closed under taking subgroups. Suppose that G is finite and it is locally embeddable into C. Then G ∈ C. Indeed, any G-almost homomorphism ϕ : G → C from G into a group C ∈ C is an injective homomorphism. The following characterization of local embeddability will not be used in the sequel but it presents some interest on its own. Proposition 7.1.6. Let G be a group and let C be a class of groups which is closed under taking subgroups. Then the following conditions are equivalent: (a) G is locally embeddable into C; (b) for every finite subset K ⊂ G, there exist a set L with K ⊂ L ⊂ G and a binary operation  : L × L → L such that (L, ) is a group in C and k1 k2 = k1  k2 for all k1 , k2 ∈ K. Proof. Suppose (a). Fix a finite subset K ⊂ G. If G is finite, then G ∈ C by Remark 7.1.5. Therefore, condition (b) is satisfied in this case since we can take as (L, ) the group G itself. So let us assume that G is infinite. Consider the finite subset K  ⊂ G defined by K  = K ∪ K 2 . Since G is locally embeddable into C, we can find a group C  ∈ C and a K  -almosthomomorphism ϕ : G → C  . Note that, as K  is finite, the subgroup C ⊂ C  generated by ϕ (K  ) is countable. Since G is infinite, it follows that there exists a surjective map σ : G → C which coincides with ϕ on K  . Let us define a map ψ : C → G as follows. If c ∈ σ(K  ), we set ψ(c) = k , where k  is the unique element in K  such that σ(k ) = c. For c ∈ C \ σ(K  ), we take as ψ(c) an arbitrary element in σ −1 (c). Denote by L = ψ(C) ⊂ G the image of ψ. Observe that K ⊂ K  ⊂ L and that ψ induces a bijection from C onto L. Let us use this bijection to transport the group structure from C to L. If we denote by  the corresponding group operation on L, this means that ψ induces a group isomorphism from C onto (L, ). It follows that we have 1  2 = ψ(σ(1 )σ(2 )) for all 1 , 2 ∈ L. As C is closed under taking subgroups, C and hence (L, ) belong to C. Moreover, for all k1 , k2 ∈ K, we have

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k1  k2 = ψ(σ(k1 )σ(k2 )) = ψ(ϕ (k1 )ϕ (k2 )) = ψ(ϕ (k1 k2 )) = ψ(σ(k1 k2 )). Since K 2 ⊂ K  , we deduce that k1  k2 = k1 k2 for all k1 , k2 ∈ K. This shows that condition (b) is satisfied. Conversely, suppose (b). Fix a finite subset K ⊂ G. Set C = (L, ), where L is as in (b). Let ϕ : G → C be any map extending the identity map ι : L → L. Then ϕ|K = ι|K is injective. On the other hand, for all k1 , k2 ∈ K, we have k1 k2 = k1  k2 ∈ L and therefore ϕ(k1 k2 ) = ι(k1 k2 ) = k1 k2 = k1  k2 = ϕ(k1 )  ϕ(k2 ) Thus, ϕ is a K-almost-homomorphism of G into the group C in C. This shows that G is locally embeddable into C.  Proposition 7.1.7. Let C be a class of groups. Every subgroup of a group which is locally embeddable into C is locally embeddable into C. Proof. Let G be a group which is locally embeddable into C and let H be a subgroup of G. Given a finite subset K ⊂ H, let ϕ : G → C be a Kalmost-homomorphism of G into a group C ∈ C. Then the restriction map ϕ|H : H → C is a K-almost-homomorphism of H into C. This shows that H is locally embeddable into C.  As the name suggests, local embeddability is a local property for groups: Proposition 7.1.8. Let G be a group and let C be a class of groups. Then G is locally embeddable into C if and only if every finitely generated subgroup of G is locally embeddable into C. Proof. The necessity of the condition follows from Proposition 7.1.7. Conversely, suppose that every finitely generated subgroup of G is locally embeddable into C. Let K ⊂ G be a finite subset and denote by H ⊂ G the subgroup generated by K. Then, as H is locally embeddable into C, there exists a K-almost-homomorphism ϕ : H → C of H into a group C ∈ C. Extend arbitrarily ϕ to G, for example by setting ϕ(g) = 1C for all g ∈ G \ H. Then ϕ : G → C is a K-almost-homomorphism of G into C. This shows that G is locally embeddable into C.  Let C be a class of groups. Denote by C the class consisting of all groups which are locally embeddable into C. Proposition 7.1.9. Let C be a class of groups. Then C = C.

7.1 Local Embeddability

237

Proof. Let G be a group in C. Then, given any finite subset K ⊂ G there exist a group C  ∈ C and a K-almost-homomorphism ϕ : G → C  . Consider the finite subset K  = ϕ(K) ⊂ C  and let ϕ : C  → C be a K  -almosthomomorphism of C  into a group C ∈ C. Then the composition map Φ = ϕ ◦ ϕ : G → C is a K-almost-homomorphism. Indeed, for k1 , k2 ∈ K one has Φ(k1 k2 ) = ϕ (ϕ(k1 k2 )) = ϕ (ϕ(k1 )ϕ(k2 )) = ϕ (ϕ(k1 ))ϕ (ϕ(k2 )) = Φ(k1 )Φ(k2 ). Moreover, as ϕ|K and ϕ |K  are injective, and K  = ϕ(K), one has that Φ|K is also injective. It follows that G is locally embeddable into C. This shows that C ⊂ C. The inclusion C ⊂ C follows from the observation in Example 7.1.4(a).  This shows that C = C. A class of groups C is said to be closed under finite direct products if one has G1 × G2 ∈ C whenever G1 ∈ C and G2 ∈ C. Proposition 7.1.10. Let C be a class of groups which is closed under finite direct products. Let (Gi )i∈I be a family of  groups which are locally embeddable into C. Then, their direct product G = i∈I Gi is locally embeddable into C. Proof. For each i ∈ I, let πi : G → Gi denote the projection homomorphism. Fix a finite subset K ⊂ G. Then there exists a finite subset J⊂ I such  that the projection homomorphism πJ = j∈J πj : G → GJ = j∈J Gj is injective on K. Since the group Gj is locally embeddable into C, we can find, for each j ∈ J, a πj (K)-almost homomorphism ϕj : Gj → Cj of Gj into a groupCj ∈ C. As the class C is closed under finite direct products, the group is also in C. Consider the map ϕ : G → C defined by ϕ = ϕJ ◦πJ , C = j∈J Cj  where ϕJ = j∈J ϕj : GJ → C. For all k = (ki )i∈I , k = (ki )i∈I ∈ K, we have ϕ(kk ) = ϕJ (πJ (kk )) = ϕJ ((kj kj )j∈J ) = (ϕj (kj kj ))j∈J = (ϕj (kj )ϕj (kj ))j∈J = (ϕj (kj ))j∈J (ϕj (kj ))j∈J = ϕ(k)ϕ(k ). Moreover, ϕ|K is injective. Indeed, given k = (ki )i∈I , k = (ki )i∈I ∈ K, if ϕ(k) = ϕ(k ) then ϕj (kj ) = ϕj (kj ) for all j ∈ J. By injectivity of ϕj |πj (K) , we deduce that kj = kj for all j ∈ J. This implies that k = k  since πJ |K is injective. This shows that ϕ is a K-almost-homomorphism of G into C. It follows that G is locally embeddable into C. 

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Corollary 7.1.11. Let C be a class of groups which is closed under finite direct products. Let (Gi )i∈I be a family  of groups which are locally embeddable into C. Then their direct sum G = i∈I Gi is locally embeddable into C. Proof. This follows immediately from Proposition 7.1.7  and Proposition 7.1.10, since G is the subgroup of the direct product P = i∈I Gi consisting of all  g = (gi )i∈I ∈ P for which gi = 1Gi for all but finitely many i ∈ I. Corollary 7.1.12. Let C be a class of groups which is closed under finite direct products. If a group G is the limit of a projective system of groups which are locally embeddable into C, then G is locally embeddable into C. Proof. Let (Gi )i∈I be a projective system of groups which are locally embedof a projective limit (see dable into C such that G = lim Gi . By construction  ←− Appendix E), G is a subgroup of the group i∈I Gi . We deduce that G is locally embeddable into C by using Proposition 7.1.10 and Proposition 7.1.7.  Recall that if C is a class of groups, then a group G is called residually C if for each element g ∈ G with g = 1G , there exist a group C ∈ C and a surjective homomorphism φ : G → C such that φ(g) = 1C . Proposition 7.1.13. Let C be a class of groups which is closed under finite direct products. Let G be a group which is residually C. Then, for every finite subset K ⊂ G, there exist a group C ∈ C and a homomorphism ϕ : G → C whose restriction to K is injective. Proof. Fix a finite subset K ⊂ G and consider the set L = {hk −1 : h, k ∈ K and h = k}. C and a Since G is residually C, we can find, for each g ∈ L, a group Cg ∈ homomorphism φg : G → Cg such that φg (g) = 1Cg . The group C = g∈L Cg is in the class C since L is finite and C is closed under finite direct products.  Consider the group homomorphism ϕ = g∈L φg : G → C. If h and k are distinct elements in K, then g = hk−1 ∈ L and φg (hk−1 ) = φg (g) = 1Cg . It follows that ϕ(hk−1 ) = 1C and hence ϕ(h) = ϕ(k). Thus, the restriction of ϕ to K is injective.  Corollary 7.1.14. Let C be a class of groups which is closed under finite direct products. Then every group which is residually C is locally embeddable into C. Proof. Let G be a group which is residually C and let K ⊂ G be a finite subset. By Proposition 7.1.13, there exist a group C ∈ C and a homomorphism ϕ : G → C whose restriction to K is injective. Such a ϕ is a K-almosthomomorphism. Consequently, G is locally embeddable into C. 

7.1 Local Embeddability

239

Corollary 7.1.15. Let C be a class of groups which is closed under finite direct products. Then every group which is locally residually C is locally embeddable into C. Proof. This immediately follows from Corollary 7.1.14 and Proposition 7.1.8.  Given a group Γ , let N (Γ ) denote the space of all Γ -marked groups (cf. Sect. 3.4). Recall that N (Γ ) may be identified with the set consisting of all normal subgroups of Γ and that N (Γ ) is a compact Hausdorff space for the topology induced by the prodiscrete topology on P(Γ ) = {0, 1}Γ . Theorem 7.1.16. Let Γ be a group and let C be a class of groups. Let N ∈ N (Γ ). Suppose that there exists a net (Ni )i∈I which converges to N in N (Γ ) such that Γ/Ni ∈ C for all i ∈ I. Then the group Γ/N is locally embeddable into C. Proof. Denote by ρ : Γ → Γ/N and ρi : Γ → Γ/Ni the quotient homomorphisms. Fix a finite subset K ⊂ Γ/N and let F ⊂ Γ be a finite symmetric subset such that 1Γ ∈ F and ρ(F ) = K ∪K −1 ∪{1Γ/N }. Since the net (Ni )i∈I converges to N , we can find i0 ∈ I such that N ∩ F 4 = N i0 ∩ F 4 . Set C = Γ/Ni0 and define a map ϕ : Γ/N → C by setting  ρi0 (f  ) if g = ρ(f  ) for some f  ∈ F 2 ϕ(g) = 1C otherwise.

(7.1)

(7.2)

Note that ϕ is well defined. Indeed, suppose that g = ρ(f1 ) = ρ(f2 ), for some f1 , f2 ∈ F 2 . Then, from 1Γ/N = ρ(f1 )−1 ρ(f2 ) = ρ((f1 )−1 f2 ), we deduce that (f1 )−1 f2 ∈ ker(ρ) = N. (7.3) As (f1 )−1 f2 ∈ F 4 , we deduce from (7.3) and (7.1) that (f1 )−1 f2 ∈ Ni0 = ker(ρi0 ). Therefore, we have ρi0 (f1 ) = ρi0 (f2 ). Let us check now that ϕ is a K-almost-homomorphism. Let k1 , k2 ∈ K. Then, there exist f1 , f2 ∈ F such that ρ(f1 ) = k1 and ρ(f2 ) = k2 . Observe that f1 , f2 ∈ F 2 since F ⊂ F 2 . Thus, we get ϕ(k1 ) = ρi0 (f1 ) and ϕ(k2 ) = ρi0 (f2 ) by applying (7.2). On the other hand, we also have f1 f2 ∈ F 2 and k1 k2 = ρ(f1 )ρ(f2 ) = ρ(f1 f2 ). Therefore, by applying again (7.2), we obtain ϕ(k1 k2 ) = ρi0 (f1 f2 ) = ρi0 (f1 )ρi0 (f2 ) = ϕ(k1 )ϕ(k2 ). Moreover, if ϕ(k1 ) = ϕ(k2 ), then ρi0 (f1 ) = ρi0 (f2 ) and hence f1−1 f2 ∈ ker(ρi0 ) = Ni0 . As f1−1 f2 ∈ F 2 ⊂ F 4 , we deduce from (7.1) that f1−1 f2 ∈

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N = ker(ρ). Therefore k1 = ρ(f1 ) = ρ(f2 ) = k2 . This shows that ϕ is injective on K. We have shown that ϕ is a K-almost-homomorphism of Γ/N into the group C ∈ C. Therefore, Γ/N is locally embeddable into C.  Corollary 7.1.17. Let Γ be a group and let C be a class of groups. Then the set of all N ∈ N (Γ ) such that Γ/N is locally embeddable into C is closed (and hence compact) in N (Γ ). Proof. Let (Ni )i∈I be a convergent net in N (Γ ) and suppose that the groups Γ/Ni are locally embeddable into C for all i ∈ I. Let N = limi∈I Ni . By applying Theorem 7.1.16, we deduce that Γ/N is locally embeddable into the class of groups which are locally embeddable into C. It follows from Proposition 7.1.9 that Γ/N is locally embeddable into C.  For the next theorem we need some (slightly technical) preliminaries. Let n ≥ 2 be an integer and G, C two groups. Given a finite subset K ⊂ G, we say that a map ϕ : G → C is an n-K-almost-homomorphism if it satisfies the following conditions: (n-K-AH-1) ϕ(k1ε1 k2ε2 · · · ktεt ) = ϕ(k1 )ε1 ϕ(k2 )ε2 · · · ϕ(kt )εt for all ki ∈ K ∪ {1G }, εi ∈ {−1, 1}, 1 ≤ i ≤ t ≤ n; (n-K-AH-2) ϕ(1G ) = 1C ; (n-K-AH-3) the restriction of ϕ to (K ∪ K −1 ∪ {1G })n is injective. Lemma 7.1.18. Let G be a group and let C be a class of groups. Let n ≥ 2 be an integer. Then, the following conditions are equivalent: (a) G is locally embeddable into C; (b) for every finite subset K ⊂ G, there exist a group C ∈ C and an n-Kalmost-homomorphism ϕ : G → C. Proof. It is clear that any n-K-almost-homomorphism is also a K-almosthomomorphism (take ε1 = ε2 = 1 and t = 2 in (n-K-AH-1), and observe that K ⊂ (K ∪ K −1 ∪ {1G })n ). Thus, (b) implies (a). Conversely, suppose (a). Given a finite subset K ⊂ G, set K  = (K ∪ −1 K ∪ {1G })n . As G is locally embeddable into C, there exists a K  -almosthomomorphism ϕ : G → C of G into a group C ∈ C. Let us show that ϕ is an n-K-almost-homomorphism. As the restriction of ϕ to K  is injective, Property (n-K-AH-3) is trivially satisfied. On the other hand, we have ϕ(k1 k2 ) = ϕ(k1 )ϕ(k2 ) for all k1 , k2 ∈ K  . For k1 = k2 = 1G , this gives us ϕ(1G ) = ϕ(1G )2 , so that Property (n-K-AH-2) also holds. By taking k1 ∈ K  and k2 = k1−1 , we get 1C = ϕ(1G ) = ϕ(k1 k1−1 ) = ϕ(k1 )ϕ(k1−1 ). This implies that ϕ(k1−1 ) = ϕ(k1 )−1 for all k1 ∈ K  . Then, Property (n-K-AH-1) immediately follows by induction on t. This shows that ϕ is an n-K-almosthomomorphism of G into C.  Theorem 7.1.19. Let F be a free group and let C be a class of groups which is closed under taking subgroups. Let N ∈ N (F ) and suppose that the group F/N is locally embeddable into C. Then there exists a net (Ni )i∈I which converges to N in N (F ) such that F/Ni ∈ C for all i ∈ I.

7.1 Local Embeddability

241

Proof. Let X ⊂ F be a free base of F and denote by ρ : F → F/N the quotient homomorphism. Let I denote the set of all finite subsets E of F partially ordered by inclusion. Given E ∈ I, we denote by UE ⊂ X the subset of X consisting of all elements which appear in the reduced form of some element in E. Since E is finite, UE is also finite and there exists nE ∈ N such that each w ∈ E may be written in the form w = xε11 xε22 · · · xεt t

(7.4)

for suitable xi ∈ UE , εi ∈ {−1, 1}, i = 1, 2, . . . , t, and 1 ≤ t ≤ nE . Set KE = ρ(BnE ) ⊂ F/N , where BnE denotes the ball of radius nE centered at the identity element 1F in the Cayley graph of the subgroup of F generated by UE . Since F/N is locally embeddable into C it follows from Lemma 7.1.18 that there exists a group CE ∈ C and an nE -KE -almost-homomorphism ϕE : F/N → CE . As F is a free group with base X, there exists a unique homomorphism ρE : F → CE such that ρE (x) = ϕE (ρ(x)) for all x ∈ X. Setting NE = ker(ρE ) we have NE ∈ N (F ). Moreover, since C is closed under taking subgroups, we have that the quotient group F/NE , being isomorphic to ρE (F ), which is a subgroup of CE , also belongs to C. Let us show that the net (NE )E∈I converges to N . Fix a finite subset E0 ⊂ F . Let E ∈ I be such that E0 ⊂ E. Let us show that N ∩ E0 = N E ∩ E 0 .

(7.5)

Given w ∈ E0 , we can write w as in (7.4). Using the fact that ϕE is an nE -KE -almost-homomorphism, we get ρE (w) = ρE (xε11 xε22 · · · xεt t ) = ρE (x1 )ε1 ρE (x2 )ε2 · · · ρE (xt )εt = ϕE (ρ(x1 ))ε1 ϕE (ρ(x2 ))ε2 · · · ϕE (ρ(xt ))εt = ϕE (ρ(x1 )ε1 ρ(x2 )ε2 · · · ρ(xt )εt ) = ϕE (ρ(xε11 xε22 · · · xεt t )) = ϕE (ρ(w)). As ϕE is injective on KE , and ρ(w) ∈ ρ(BE ) = KE , we deduce that ρE (w) = 1F/NE if and only if ρ(w) = 1F/N , equivalently, w ∈ NE if and only if w ∈ N . Thus (7.5) follows. This shows that N = limE NE .  From Theorem 7.1.16 (with Γ a free group F ) and Theorem 7.1.19 we immediately deduce the following. Corollary 7.1.20. Let F be a free group and let C be a class of groups which is closed under taking subgroups. Let N ∈ N (F ). Then the following conditions are equivalent. (a) the group F/N is locally embeddable into C;

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(b) there exists a net (Ni )i∈I which converges to N in N (F ) such that F/Ni ∈ C for all i ∈ I.  Corollary 7.1.21. Let C be a class of groups which is closed under taking subgroups. Then every finitely presented group which is locally embeddable into C is residually C. Proof. Let G be a finitely presented group which is locally embeddable into C. Fix g ∈ G \ {1G }. Since G is finitely presented, there exists a free group F of finite rank and a finite subset R ⊂ F such that G = F/N , where N ∈ N (F ) denotes the normal closure of R in F . By Theorem 7.1.19, there exists a net (Ni )i∈I in N (F ) converging to N such that F/Ni ∈ C for all i ∈ I. Let ρ : F → G denote the quotient homomorphism and choose an element f ∈ F such that ρ(f ) = g. Since the net (Ni )i∈I converges to N , we can find i0 ∈ I such that (7.6) N ∩ (R ∪ {f }) = Ni0 ∩ (R ∪ {f }). As R ⊂ N , this implies R ⊂ Ni0 and hence N ⊂ Ni0 . Thus, there is a /N canonical epimorphism φ : G = F/N → C = F/Ni0 . Observe now that f ∈ and therefore f ∈ / Ni0 by (7.6). It follows that φ(g) = 1C . As C ∈ C, this shows that G is residually C.  Corollary 7.1.22. Let C be a class of groups which is closed under taking subgroups and under finite direct products. Then a finitely presented group is locally embeddable into C if and only if it is residually C. Proof. The fact that every finitely presented group which is locally embeddable into C is residually C follows from the previous corollary. The converse implication follows from Corollary 7.1.14.  We end this section by producing an example showing that Theorem 7.1.19 becomes false if we omit the hypothesis that F is free. Let us first establish the following: Proposition 7.1.23. Let C be a class of groups. Let G be a group. Suppose that there exists a net (Ni )i∈I which converges to {1G } in N (G) such that G/Ni ∈ C for all i ∈ I. Then G is residually C. Proof. Let g ∈ G \ {1G }. Let us set F = {1G , g}. Since the net (Ni )i∈I converges to {1G } in N (G), there exists i0 ∈ I such that Ni0 ∩ F = {1G } ∩ F . / Ni0 . Thus, the quotient homomorphism As F ∩{1G } = {1G }, this implies g ∈ φ : G → G/Ni0 satisfies φ(g) = 1G/Ni0 . As G/Ni0 ∈ C, this shows that G is residually C.  Now, to exhibit the promised example showing the necessity of the freeness hypothesis on F in Theorem 7.1.19, we consider the additive group G = Q of rational numbers and take as C the class of finite groups. It follows from Example 2.1.9 that Q is not residually finite. Thus, by the preceding propo-

7.2 Local Embeddability and Ultraproducts

243

sition, there is no net (Ni )i∈I converging to {1G } in N (G) such that G/Ni is finite for all i ∈ I. On the other hand, G is locally residually finite since every finitely generated abelian group is residually finite by Corollary 2.2.4. Therefore, G is locally embeddable into C by Corollary 7.1.15.

7.2 Local Embeddability and Ultraproducts Suppose that we are given a family of groups (Gi )i∈I and a filter ω (cf. Sect. J.1) on the index set I. Consider the group  P = Gi . i∈I

Let α = (αi )i∈I and β = (βi )i∈I be elements of P . We write α ∼ω β if {i ∈ I : αi = βi } ∈ ω. Proposition 7.2.1. One has: (i) α ∼ω α; (ii) α ∼ω β if (iii) if α ∼ω β (iv) if α ∼ω β (v) α ∼ω β if

and and and and

only if β ∼ω α; β ∼ω γ, then α ∼ω γ; γ ∼ω δ, then αγ ∼ω βδ; only if α−1 ∼ω β −1 ;

for all α, β, γ, δ ∈ P . Proof. Let α = (αi )i∈I , β = (βi )i∈I , γ = (γi )i∈I and δ = (δi )i∈I ∈ P . We have α ∼ω α since {i ∈ I : αi = αi } = I belongs to ω (cf. (F-6) in Sect. J.1). This shows (i). Also, since {i ∈ I : αi = βi } = {i ∈ I : βi = αi } we deduce (ii). Suppose now that α ∼ω β and β ∼ω γ. We have {i ∈ I : αi = γi } ⊃ {i ∈ I : αi = βi } ∩ {i ∈ I : βi = γi }. Thus (iii) follows from the fact that ω, being a filter, is closed under finite intersections and taking supersets (cf. (F-2) and (F-3) in Sect. J.1). Suppose now that α ∼ω β and γ ∼ω δ, that is, {i ∈ I : αi = βi } and {i ∈ I : γi = δi } both belong to ω. We have {i ∈ I : αi γi = βi δi } ⊃ {i ∈ I : αi = βi } ∩ {i ∈ I : γi = δi }. As ω is closed under finite intersections and taking supersets, we deduce that {i ∈ I : αi γi = βi δi } ∈ ω, that is, αγ ∼ω βδ. This shows (iv). Finally, from  {i ∈ I : αi = βi } = {i ∈ I : αi−1 = βi−1 } we deduce (v). Note that it follows from (i), (ii) and (iii) in Proposition 7.2.1 that ∼ω is an equivalence relation in P . Consider now the subset Nω ⊂ P defined by Nω = {α ∈ P : α ∼ω 1P = (1Gi )i∈I }.

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Proposition 7.2.2. The set Nω is a normal subgroup of P . Proof. This is an easy consequence of the properties of ∼ω stated in Proposition 7.2.1. Indeed, we immediately deduce from Proposition 7.2.1(i) that 1P ∈ Nω . Suppose now that α, β ∈ Nω , that is, α ∼ω 1P and β ∼ω 1P . It follows from Proposition 7.2.1(iv) that αβ ∼ω 1P 1P = 1P , that is, αβ ∈ Nω . = 1P , Similarly, from Proposition 7.2.1(v) we deduce that α−1 ∼ω 1−1 P that is, α−1 ∈ Nω . Thus, Nω is a subgroup of P . Finally, let γ ∈ P . Proposition 7.2.1(iv) implies that γαγ −1 ∼ω γ1P γ −1 = 1P . It follows that γαγ −1 ∈ Nω for all α ∈ Nω and γ ∈ P . This shows that Nω is a normal subgroup of P .  Observe that, given α and β in P , one has αNω = βNω ⇐⇒ α ∼ω β.

(7.7)

Indeed, one has αNω = βNω if and only if αβ −1 ∈ Nω , that is, if and only if αβ −1 ∼ω 1P . This is equivalent to α ∼ω β by Proposition 7.2.1(iv). The quotient group Pω = P/Nω is called the reduced product of the family of groups (Gi )i∈I with respect to the filter ω. In the particular case when ω is an ultrafilter, one also says that Pω is the ultraproduct of the family of groups (Gi )i∈I with respect to the ultrafilter ω. Theorem 7.2.3. Let C be a class of groups, (Gi )i∈I a family of groups such that Gi ∈ C for all i ∈ I, and ω an ultrafilter on the index set I. Then the ultraproduct Pω of the family of groups (Gi )i∈I with respect to the ultrafilter ω is locally embeddable into C. Proof. Fix a finite subset K ⊂ Pω . We want to show that there exist a group C ∈ C and a K-almost-homomorphism ϕ : Pω → C. Choose a representative of each element g ∈ Pω , that is, an element g = ( gi )i∈I ∈ P such that g = gNω . If h and k are arbitrary elements of K, we  ω =  ∼ω  have hkN h kNω and therefore hk h k by (7.7). It follows that the set i =  hi  Ih,k = {i ∈ I : hk ki } belongs to ω. Thus, since ω is closed under finite intersections (cf.(F-5) in Sect. J.1), we have that  Ih,k IK = h,k∈K

also belongs to ω. On the other hand, if h and k are distinct elements of K, ki } does not belongs to ω. As ω is an ultrafilter, the subset {i ∈ I :  hi =   this implies that Ih,k = I \ {i ∈ I :  hi =  ki } = {i ∈ I :  hi =  ki } belongs to ω (cf. (UF) in Sect. J.1). Using again the fact that ω is closed under finite intersections, we deduce that    = Ih,k IK h,k∈K h=k

7.2 Local Embeddability and Ultraproducts

245

also belongs to ω. Since ω is closed under finite intersections and ∅ ∈ / ω (cf. (F-1) in Sect. J.1), we can find an index i0 ∈ I such that  . i0 ∈ IK ∩ IK

Consider the map ϕ : Pω → Gi0 defined by ϕ(g) = gi0 for all g ∈ Pω . The map ϕ satisfies the following properties: i =  (1) ϕ(hk) = hk hi 0  ki0 = ϕ(h)ϕ(k) for all h, k ∈ K (because i0 ∈ IK ); 0   (2) ϕ(h) = hi0 = ki0 = ϕ(k) for all h, k ∈ K such that h = k (because  ). i0 ∈ IK This shows that ϕ is a K-almost homomorphism. It follows that Pω is locally embeddable into the class C.  Remark 7.2.4. When the ultrafilter ω is principal, then the group Pω itself belongs to C. Indeed, in this case, there is an element i0 ∈ I such that ω consists of all subsets of I containing i0 . This implies that α ∼ω β if and only if αi0 = βi0 for all α = (αi )i∈I , β = (βi )i∈I ∈ P . Therefore, Nω consists of all α ∈ P such that αi0 = 1Gi0 . It follows that the group Pω is isomorphic to the group Gi0 and therefore that Pω ∈ C. Theorem 7.2.5. Let C be a class of groups and let G be a group. The following conditions are equivalent: (a) G is locally embeddable into C; (b) there exists a family of groups (Gi )i∈I such that Gi ∈ C for all i ∈ I and an ultrafilter ω on I such that G is isomorphic to a subgroup of the ultraproduct Pω of the family (Gi )i∈I with respect to the ultrafilter ω. Proof. If G satisfies (b), then G is locally embeddable into C since Pω is locally embeddable into C by Theorem 7.2.3 and every subgroup of a group which is locally embeddable into C is itself locally embeddable into C by Proposition 7.1.7. Conversely, suppose that G is locally embeddable into C. Consider the set I consisting of all finite subsets of G. For each K ∈ I we define the set IK = {K  ∈ I : K ⊂ K  }. Observe that IK = ∅ as K ∈ IK . Moreover, the family of nonempty subsets (IK )K∈I is closed under finite intersections, since IK1 ∩ IK2 = IK1 ∪K2 for all K1 , K2 ∈ I. It follows from Proposition J.1.3 and Theorem J.1.6 that there exists an ultrafilter ω on I such that IK ∈ ω for all K ∈ I. As G is locally embeddable into C, we can find, for each K ∈ I, a group GK in C and a K-almost-homomorphism ϕK : G → GK .

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Consider the ultraproduct Pω of the family (GK )K∈I with respect to ω. By into C. Let ϕ  : G → P = Theorem 7.2.3, Pω is locally embeddable K∈I GK  denote the product map ϕ  = K∈I ϕK . Thus, we have ϕ(g)  = (ϕK (g))K∈I for all g ∈ G. Let ρ : P → Pω = P/Nω denote the canonical epimorphism. Let us show that the composite map Φ = ρ ◦ ϕ  : G → Pω is an injective homomorphism. This will prove that G is isomorphic to a subgroup of Pω . Let g, h ∈ G. Consider the element K0 = {g, h} ∈ I. If K ∈ I satisfies K0 ⊂ K, then ϕK (gh) = ϕK (g)ϕK (h) since ϕK is a K-almost-homomorphism. Thus, the set {K ∈ I : ϕK (gh) = ϕK (g)ϕK (h)} contains IK0 and therefore belongs to ω. This implies that  ϕ(h).  ϕ(gh)  ∼ω ϕ(g) Therefore, we have Φ(gh) = Φ(g)Φ(h) by (7.7). This shows that Φ is a homomorphism. On the other hand, if g = h, we have ϕK (g) = ϕK (h) for all K ∈ I such that K0 ⊂ K. This implies that {K ∈ I : ϕK (g) = / ω, ϕK (h)} ∈ ω. As ω is a filter, it follows that {K ∈ I : ϕK (g) = ϕK (h)} ∈ that is, ϕ(g)  ∼ω ϕ(h).  Therefore we have Φ(g) = Φ(h), again by (7.7). Consequently, Φ is injective. This shows that (a) implies (b).



Remark 7.2.6. Suppose that G is a group and that C is a class of groups which is closed under taking subgroups. If G is locally embeddable into C and G ∈ /C then for any family of groups (Gi )i∈I and any ultrafilter ω on the index set I satisfying condition (b) in Theorem 7.2.5, we deduce from Remark 7.2.4 that the ultrafilter ω is necessarily non-principal.

7.3 LEF-Groups and LEA-Groups Let us rewrite the results obtained in Sect. 7.1 in the particular case when C is either the class of finite groups or the class of amenable groups. Note that these classes are closed under taking subgroups and taking finite direct

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247

products (this is trivial for finite groups and follows from Proposition 4.5.1 and Corollary 4.5.6 for amenable groups), so that all the results established in Sect. 7.1 apply to these two classes. We shall use the following terminology, which is very popular in the field. A group which is locally embeddable into the class of finite groups is briefly called an LEF -group. Similarly, a group which is locally embeddable into the class of amenable groups is called an LEA-group. Note that every LEF-group is LEA since the class of finite groups is contained in the class of amenable groups by Proposition 4.4.6. We deduce from Proposition 7.1.6 the following characterization of LEFgroups. Proposition 7.3.1. Let G be a group. Then the following conditions are equivalent: (a) G is an LEF-group; (b) for every finite subset K ⊂ G, there exist a finite set L with K ⊂ L ⊂ G and a binary operation  : L × L → L such that (L, ) is a group and k1 k2 = k1  k2 for all k1 , k2 ∈ K.  Analogously, we deduce from Proposition 7.1.6 the following characterization of LEA-groups. Proposition 7.3.2. Let G be a group. Then the following conditions are equivalent: (a) G is an LEA-group; (b) for every finite subset K ⊂ G, there exist a set L with K ⊂ L ⊂ G and a binary operation  : L × L → L such that (L, ) is an amenable group and k1 k2 = k1  k2 for all k1 , k2 ∈ K.  From Theorem 7.2.5 we deduce the following characterization of LEF (resp. LEA) groups in terms of ultraproducts. Corollary 7.3.3. Let G be a group. The following conditions are equivalent: (a) G is an LEF (resp. LEA) group; (b) there exists a family of finite (resp. amenable) groups (Gi )i∈I and an ultrafilter ω on I such that G is isomorphic to a subgroup of the ultraproduct Pω of the family (Gi )i∈I with respect to the ultrafilter ω.  From Corollary 7.1.15, we get: Proposition 7.3.4. Every locally residually finite group is LEF and therefore LEA. 

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Corollary 7.3.5. All finite groups, all residually finite groups, all profinite groups, all free groups, all abelian groups, and all locally finite groups are LEF and therefore LEA. Proof. In order to complete the proof, it suffices to recall that free groups (resp. profinite groups) are residually finite by Theorem 2.3.1 (resp. Corollary 2.2.8) and that abelian groups are locally residually finite by Corollary 2.2.4.  Similarly, by applying Corollary 7.1.15 to the class of amenable groups, we get: Proposition 7.3.6. Every locally residually amenable group is LEA.



From Proposition 7.1.7, Proposition 7.1.8, Proposition 7.1.10, Corollary 7.1.11, Corollary 7.1.12, Corollary 7.1.17, and Theorem 7.1.19, we get: Proposition 7.3.7. The following assertions hold: (i) every subgroup of an LEF-group (resp. LEA-group) is an LEF-group (resp. LEA-group); (ii) a group is LEF (resp. LEA) if and only if all its finitely generated subgroups are LEF (resp. LEA); a family of LEF-groups (resp. LEA-groups). Then their (iii) let (Gi )i∈I be  direct product i∈I Gi is an LEF-group (resp. LEA-group); be a family of LEF-groups (resp. LEA-groups). Then their (iv) let (Gi )i∈I  direct sum i∈I Gi is an LEF-group (resp. LEA-group); (v) let (Gi )i∈I be a projective system of LEF-groups (resp. LEA-groups). Then their projective limit G = lim Gi is an LEF-group (resp. LEA←− group); (vi) Let Γ be a group. Then the set of all N ∈ N (Γ ) such that Γ/N is LEF (resp. LEA) is closed (and hence compact) in N (Γ ). (vii) let F be a free group and let N ∈ N (F ). Then F/N is an LEF-group (resp. LEA-group) if and only if there exists a net (Ni )i∈I in N (F ) with F/Ni finite (resp. amenable) for all i ∈ I such that N = limi Ni .  Finally, we deduce from Corollary 7.1.21 the following: Proposition 7.3.8. A finitely presented group is LEF (resp. LEA) if and only if it is residually finite (resp. residually amenable).  As we have observed at the end of the previous section, the additive group Q is LEF but not residually finite. On the other hand, it follows from Proposition 7.3.8 that every finitely presented LEF-group is residually finite. The group Q is not finitely generated. However, there exist finitely generated LEF-groups which are not residually finite. An example of such a group is provided by the group G1 introduced in Sect. 2.6:

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Proposition 7.3.9. The group G1 of Sect. 2.6 is a finitely generated amenable LEF-group which is not residually finite. Proof. By construction, the group G1 is finitely generated since it is defined as being a subgroup of Sym(Z) generated by two elements, namely the transposition (0 1) and the translation n → n + 1. The group G1 is not residually finite by Proposition 2.6.1. Recall that, denoting by Sym0 (Z) the normal subgroup of Sym(Z) consisting of all permutations of Z with finite support, the group G1 is the semidirect product of Sym0 (Z) with the infinite cyclic group generated by the translation T : n → n + 1 (see Lemma 2.6.4). Therefore, each element g ∈ G1 can be uniquely written in the form g = T i(g) σ(g), where i(g) ∈ Z and σ(g) ∈ Sym0 (Z). Note that we have i(gh) = i(g) + i(h) and σ(gh) = T −i(h) σ(g)T i(h) σ(h) for all g, h ∈ G1 . To prove that G1 is an LEF-group, consider a finite subset K ⊂ G1 . Let us show that there exist a finite group F and a K-almost-homomorphism of G1 into F . Let  = maxk∈K |i(k)| and choose an integer r ≥  such that the supports of the elements T −j σ(k)T j ∈ Sym0 (Z) are contained in the interval [−r, r] for all − ≤ j ≤  and k ∈ K. Let us set R = 4r, X = {−R, −R + 1, . . . , −1, 0, 1, . . . , R − 1, R}, and F = Sym(X). Consider the (2R + 1)-cycle γ ∈ F given by γ = (−R − R + 1 − R + 2 · · · R − 1 R). For all σ ∈ Sym0 (Z) whose support is contained in X, let σ ∈ F denote the element defined by σ(x) = σ(x) for all x ∈ X. Observe that if σ, σ  ∈ Sym0 (Z) have both their support contained in X, then so does σσ  and that we have σσ  = σσ  .

(7.8)

Moreover, for all k ∈ K and j ∈ Z such that − ≤ j ≤ , the supports of σ(k) and T −j σ(k)T j are contained in X and we have T −j σ(k)T j = γ −j σ(k)γ j .

(7.9)

Indeed, suppose first that x ∈ [−2r, 2r]. Then x + j ∈ [−3r, 3r] ⊂ X and we have [T −j σ(k)T j ](x) = [T −j σ(k)](x + j) = T −j (σ(k)(x + j)) = σ(k)(x + j) − j, and

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7 Local Embeddability and Sofic Groups

[γ −j σ(k)γ j ](x) = [γ −j σ(k)](x + j)

= γ −j σ(k)(x + j) = γ −j (σ(k)(x + j)) = σ(k)(x + j) − j. Suppose now that x ∈ X \ [−2r, 2r]. Then [T −j σ(k)T j ](x) = x since the support of T −j σ(k)T j is contained in [− + r, r + ] ⊂ [−2r, 2r]. On the other hand, if we denote, for y ∈ Z, by y the unique element in {y + (2R + 1)n : n ∈ Z} ∩ [−R, R], we have x + j ∈ X \ [−r, r]

(7.10)

and + j) [γ −j σ(k)γ j ](x) = [γ −j σ(k)](x

+ j) = γ −j σ(k)(x + j) by (7.10) = γ −j (x = x. This shows (7.9). Define a map ϕ : G1 → F by setting ϕ(g) = γ i(g) σ(g) if the support of σ(g) is contained in X, and ϕ(g) = 1F otherwise. Let us show that ϕ is a K-almost-homomorphism. Let k1 , k2 ∈ K. We have ϕ(k1 k2 ) = ϕ[T i(k1 )+i(k2 ) (T −i(k2 ) σ(k1 )T i(k2 ) σ(k2 ))] = γ i(k1 )+i(k2 ) T −i(k2 ) σ(k1 )T i(k2 ) σ(k2 ) (by (7.8)) = γ i(k1 )+i(k2 ) · T −i(k2 ) σ(k1 )T i(k2 ) σ(k2 ) (by (7.9)) = γ i(k1 )+i(k2 ) · γ −i(k2 ) σ(k1 )γ i(k2 ) · σ(k2 ) = γ i(k1 ) σ(k1 )γ i(k2 ) σ(k2 ) = ϕ[T i(k1 ) σ(k1 )]ϕ[T i(k2 ) σ(k2 )] = ϕ(k1 )ϕ(k2 ). Let us show that ϕ|K is injective. Let k1 , k2 ∈ K and suppose that ϕ(k1 ) = ϕ(k2 ). This implies that γ i(k1 ) σ(k1 ) = γ i(k2 ) σ(k2 ), that is, γ i(k1 )−i(k2 ) = −1

−1

σ(k2 ) · σ(k1 ) . But the support of σ(k2 ) · σ(k1 ) is contained in [−r, r] while the support of γ i , i ∈ Z, is the whole set X if i is not a multiple of 2R + 1. As |i(k1 ) − i(k2 )| ≤ 2 < 2R + 1, we deduce that i(k1 ) − i(k2 ) = 0 −1

and σ(k2 ) · σ(k1 ) = 1F . This implies i(k1 ) = i(k2 ) and σ(k1 ) = σ(k2 ), and therefore k1 = k2 . This shows that the restriction of ϕ to K is injective. It

7.4 The Hamming Metric

251

follows that ϕ is a K-almost homomorphism and therefore that G1 is an LEF group. Finally, observe that the group Sym0 (Z) is locally finite and therefore amenable by Corollary 4.5.12. Now, G1 is the semidirect product of the amenable group Sym0 (Z) with an infinite cyclic (and therefore amenable) group so that, by Proposition 4.5.5, it is an amenable group as well.  Remarks 7.3.10. (a) From Proposition 7.3.4 and Proposition 7.3.9, we deduce that the class of locally residually finite groups is strictly contained in the class of LEF-groups. (b) Proposition 7.3.8 and Proposition 7.3.9 imply that the group G1 is not finitely presentable.

7.4 The Hamming Metric Let G be a group. A metric d on G is called left-invariant (resp. right-invariant) if d(hg1 , hg2 ) = d(g1 , g2 ) (resp. d(g1 h, g2 h) = d(g1 , g2 )) for all g1 , g2 , h ∈ G. A metric on G which is both left and right-invariant is called bi-invariant. Note that a left-invariant (resp. right-invariant) metric d on G is entirely determined by the map g → d(1G , g), g ∈ G, since d(h, k) = d(1G , h−1 k) (resp. d(h, k) = d(1G , kh−1 )) for all h, k ∈ G. Let now F be a nonempty finite set and consider the symmetric group Sym(F ). For α ∈ Sym(F ), we denote by Fix(α) the set {x ∈ F : α(x) = x} of fixed points of α. The support of α is the set {x ∈ F : α(x) = x} = F \Fix(α), so that we have |{x ∈ F : α(x) = x}| = |F | − | Fix(α)|.

(7.11)

Consider the map dF : Sym(F ) × Sym(F ) → R defined by dF (α1 , α2 ) =

|{x ∈ F : α1 (x) = α2 (x)}| |F |

(7.12)

for all α1 , α2 ∈ Sym(F ). Observe that the set {x ∈ F : α1 (x) = α2 (x)} = {x ∈ F : x = α1−1 α2 (x)} is the support of α1−1 α2 , so that (7.11) gives us dF (α1 , α2 ) = 1 −

| Fix(α1−1 α2 )| . |F |

(7.13)

Proposition 7.4.1. Let F be a nonempty finite set. Then dF is a bi-invariant metric on Sym(F ). Proof. It is immediate from the definition that dF (α1 , α2 ) ≥ 0 and dF (α1 , α2 ) = dF (α2 , α1 ) for all α1 , α2 ∈ Sym(F ). Moreover, the equality dF (α1 , α2 ) = 0 holds if and only if α1 (x) = α2 (x) for all x ∈ F , that is, if and only if α1 = α2 .

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7 Local Embeddability and Sofic Groups

Let now α1 , α2 , α3 ∈ Sym(F ). If x ∈ F satisfies α1 (x) = α2 (x), then α1 (x) = α3 (x) or α2 (x) = α3 (x). Thus, we have the inclusion {x ∈ F : α1 (x) = α2 (x)} ⊂ {x ∈ F : α1 (x) = α3 (x)} ∪ {x ∈ F : α2 (x) = α3 (x)}. This implies 1 |{x ∈ F : α1 (x) = α2 (x)}| |F | 1 |{x ∈ F : α1 (x) = α3 (x)} ∪ {x ∈ F : α2 (x) = α3 (x)}| ≤ |F | 1 (|{x ∈ F : α1 (x) = α3 (x)}| + |{x ∈ F : α2 (x) = α3 (x)}|) ≤ |F | = dF (α1 , α3 ) + dF (α2 , α3 ).

dF (α1 , α2 ) =

This shows that dF also satisfies the triangle inequality. Therefore, dF is a metric on Sym(F ). It remains to show that dF is bi-invariant. Let α1 , α2 , β ∈ Sym(F ). Since β is bijective, we have {x ∈ F : βα1 (x) = βα2 (x)} = {x ∈ F : α1 (x) = α2 (x)}. This implies that 1 |{x ∈ F : βα1 (x) = βα2 (x)}| |F | 1 |{x ∈ F : α1 (x) = α2 (x)}| = |F | = dF (α1 , α2 ).

dF (βα1 , βα2 ) =

Thus, dF is left-invariant. On the other hand, we have {x ∈ F : α1 β(x) = α2 β(x)} = β −1 ({x ∈ F : α1 (x) = α2 (x)}), which implies 1 |{x ∈ F : α1 β(x) = α2 β(x)}| |F | 1 −1 |β ({x ∈ F : α1 (x) = α2 (x)})| = |F | 1 |{x ∈ F : α1 (x) = α2 (x)}| = |F | = dF (α1 , α2 ).

dF (α1 β, α2 β) =

Consequently, dF is also right-invariant.



Definition 7.4.2. Let F be a nonempty finite set. The bi-invariant metric dF is called the (normalized) Hamming metric on Sym(F ). Suppose now that m is a positive integer and that F1 , F2 , . . . , Fm are nonempty finite sets. Consider the Cartesian product F = F1 × F2 × · · · × Fm

7.4 The Hamming Metric

253

m and the natural group homomorphism Φ : i=1 Sym(Fi ) → Sym(F ) defined by Φ(α)(x) = (α1 (x1 ), α2 (x2 ), . . . , αm (xm )) m for all α = (α1 , α2 , . . . , αm ) ∈ i=1 Sym(Fi ) and x = (x1 , x2 , . . . , xm ) ∈ F . Proposition 7.4.3. With the above notation, one has dF (Φ(α), Φ(β)) = 1 −

m 

(1 − dFi (αi , βi ))

(7.14)

i=1

for all α = (αi )1≤i≤m and β = (βi )1≤i≤m in

m

Proof. First observe that if α = (αi )1≤i≤m ∈ m Fix(Φ(α)) = i=1 Fix(αi ), and therefore

i=1 Sym(Fi ). m i=1 Sym(Fi ),

then we have

| Fix(α)| |F | m | Fix(αi )| m = 1 − i=1 i=1 |Fi | m  | Fix(αi )| =1− |Fi | i=1

dF (IdF , Φ(α)) = 1 −

=1−

m 

(1 − dFi (IdFi , αi )).

i=1

We deduce that, for all α = (αi )1≤i≤m , β = (βi )1≤i≤m ∈ have dF (Φ(α), Φ(β)) = d(IdF , Φ(α)−1 Φ(β))

m i=1

Sym(Fi ), we

(by left-invariance of dF )

−1

= dF (IdF , Φ(α β)) m  =1− (1 − dFi (IdFi , αi−1 βi ) =1−

i=1 m 

(1 − dFi (αi , βi ))

i=1

where the last equality follows from the left-invariance of dFi .



Corollary 7.4.4. Let m be a positive integer and let F be a nonempty finite set. Consider the homomorphism Ψ : Sym(F ) → Sym(F m ) defined by Ψ (α)(x1 , x2 , . . . , xm ) = (α(x1 ), α(x2 ), . . . , α(xm ))

(7.15)

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7 Local Embeddability and Sofic Groups

for all α ∈ Sym(F ) and x1 , x2 , . . . , xm ∈ F . Then one has dF m (Ψ (α), Ψ (β)) = 1 − (1 − dF (α, β))m for all α, β ∈ Sym(F ).

(7.16) 

7.5 Sofic Groups Definition 7.5.1. Let G be a group, K ⊂ G a finite subset, and ε > 0. Let F be a nonempty finite set. A map ϕ : G → Sym(F ) is called a (K, ε)-almosthomomorphism if it satisfies the following conditions: ((K, ε)-AH-1) for all k1 , k2 ∈ K, one has dF (ϕ(k1 k2 ), ϕ(k1 )ϕ(k2 )) ≤ ε; ((K, ε)-AH-2) for all k1 , k2 ∈ K, k1 = k2 , one has dF (ϕ(k1 ), ϕ(k2 )) ≥ 1 − ε, where dF denotes the normalized Hamming metric on Sym(F ). Definition 7.5.2. A group G is called sofic if it satisfies the following condition: for every finite subset K ⊂ G and every ε > 0, there exist a nonempty finite set F and a (K, ε)-almost-homomorphism ϕ : G → Sym(F ). Proposition 7.5.3. Every finite group is sofic. Proof. Let G be a finite group. Consider the map L : G → Sym(G) defined by L(g)(h) = gh for all g, h ∈ G. As L is a homomorphism (cf. the proof of Cayley’s theorem (Theorem C.1.2)), we have dG (L(g1 g2 ), L(g1 )L(g2 )) = 0

(7.17)

for all g1 , g2 ∈ G. Moreover, for all distinct g1 , g2 ∈ G we have L(g1 )(h) = g1 h = g2 h = L(g2 )(h) for all h ∈ G so that dG (L(g1 ), L(g2 )) = 1.

(7.18)

This shows that L is a (K, ε)-almost-homomorphism for all K ⊂ G and ε > 0. It follows that G is sofic.  Proposition 7.5.4. Every subgroup of a sofic group is sofic. Proof. Let G be a sofic group and let H be a subgroup of G. Fix a finite subset K ⊂ H and ε > 0. As G is sofic, there exists a nonempty finite set F and a (K, ε)-almost-homomorphism ϕ : G → Sym(F ). Then the restriction map ϕ|H : H → Sym(F ) is a (K, ε)-almost-homomorphism. This shows that H is sofic.  Proposition 7.5.5. Every locally sofic group is sofic.

7.5 Sofic Groups

255

Proof. Let G be a locally sofic group. Let K ⊂ G be a finite subset and ε > 0. Denote by H the subgroup of G generated by K. Then, as H is sofic, there exist a nonempty finite set F and a (K, ε)-almost-homomorphism ψ : H → Sym(F ). Extend arbitrarily ψ to a map ϕ : G → Sym(F ), for example by setting ϕ(g) = IdF for all g ∈ G \ H. It is clear that ϕ is a (K, ε)-almosthomomorphism. This shows that G is sofic.  From Proposition 7.5.3 and Proposition 7.5.5, we immediately deduce that every locally finite group is sofic. As any locally finite group is amenable by Corollary 4.5.12, this is actually covered by the following: Proposition 7.5.6. Every amenable group is sofic. Proof. Suppose that G is an amenable group. Let K ⊂ G be a finite subset and ε > 0. Set S = ({1G } ∪ K ∪ K −1 )2 . Since G is amenable, it follows from Theorem 4.9.1 and Proposition 4.7.1(a) that there exists a nonempty finite subset F ⊂ G such that ε |F | (7.19) |F \ sF | ≤ |S| for all s ∈ S. Consider the set E = s∈S sF . Observe that E ⊂ F since 1G ∈ S. In fact, as S = S −1 , we get sE ⊂ F

(7.20)

for all s ∈ S. Moreover, we have |F \ E| = |F \ =|



sF |

s∈S

(F \ sF )| (7.21)

s∈S





|F \ sF |

s∈S

≤ ε|F |

by (7.19).

This implies |E| ≥ (1 − ε)|F |.

(7.22)

For each g ∈ G, we have |F | = |gF | and hence |F \ gF | = |gF \ F |. Therefore, we can find a bijective map αg : gF \ F → F \ gF . Consider the map ϕ : G → Sym(F ) defined by setting  gf if gf ∈ F ϕ(g)(f ) = αg (gf ) otherwise for all g ∈ G and f ∈ F (see Fig. 7.1).

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7 Local Embeddability and Sofic Groups

Fig. 7.1 The maps αg : gF \ F → F \ gF and ϕ(g) ∈ Sym(F ). We have ϕ(g)(f1 ) = gf1 , ϕ(g)(f2 ) = αg (gf2 ) and ϕ(g)(f3 ) = αg (gf3 )

Now, suppose that k1 , k2 ∈ K and f ∈ E. Then we have k2 , k1 k2 ∈ S, so that k2 f, k1 k2 f ∈ F by (7.20). This implies ϕ(k1 k2 )(f ) = k1 k2 f and (ϕ(k1 )ϕ(k2 ))(f ) = ϕ(k1 )(ϕ(k2 )(f )) = ϕ(k1 )(k2 f ) = k1 k2 f . Therefore, the permutations ϕ(k1 k2 ) and ϕ(k1 )ϕ(k2 ) coincide on E. From (7.21), we deduce that |F \ E| ≤ε dF (ϕ(k1 k2 ), ϕ(k1 )ϕ(k2 )) ≤ |F | for all k1 , k2 ∈ K. On the other hand, if k1 , k2 ∈ K, k1 = k2 , and f ∈ E, then we have k1 f, k2 f ∈ F so that ϕ(k1 )(f ) = k1 f = k2 f = ϕ(k2 )(f ). By using (7.22), we deduce that |E| ≥1−ε dF (ϕ(k1 ), ϕ(k2 )) ≥ |F | for all k1 , k2 ∈ K with k1 = k2 . Thus, the map ϕ : G → Sym(F ) is a (K, ε)almost-homomorphism. This shows that G is a sofic group.  Observe that, when G is finite, the proof of Proposition 7.5.6 reduces to that of Proposition 7.5.3 by taking F = G. Proposition7.5.7. Let (Gi )i∈I be a family of sofic groups. Then, their direct product G = i∈I Gi is sofic. Proof. For each i ∈ I, let πi : G → Gi denote the projection homomorphism. Fix a finite subset K ⊂ G and ε > 0. Then there exists a finite subset J ⊂ I

7.5 Sofic Groups

257

  such that the projection πJ = j∈J πj : G → GJ = j∈J Gj is injective on K. Choose a constant 0 < η < 1 small enough so that 1 − (1 − η)|J| ≤ ε

(7.23)

η ≤ ε.

(7.24)

and Since the group Gj is sofic for each j ∈ J, we can find a nonempty finite set Fj and a (πj (K), η)-almost  homomorphism ϕj : Gj → Sym(Fj ). Consider the nonempty finite set F = j∈J Fj and the map ϕ : G → Sym(F ) defined by ϕ(g)(f ) = (ϕj (gj )(fj ))j∈J for all g = (gi )i∈I ∈ G, and f = (fj )j∈J ∈ F . Then, for all k, k  ∈ K, we have, by applying (7.14),   1 − dFj (ϕj (kj kj ), ϕj (kj )ϕj (kj ) dF (ϕ(kk ), ϕ(k)ϕ(k )) = 1 − j∈J

≤ 1 − (1 − η)|J| ≤ε

(by (7.23)).

On the other hand, if k and k are distinct elements in K, then there exists j0 ∈ J such that kj0 = kj 0 . This implies, again by using (7.14), dF (ϕ(k), ϕ(k )) = 1 −



(1 − dFj (ϕj (kj ), ϕj (kj ))

j∈J

≥ 1 − (1 − dFj0 (ϕj0 (kj0 ), ϕj0 (kj 0 )) ≥1−η ≥1−ε

(by (7.24)).

This shows that ϕ is a (K, ε)-almost-homomorphism of G. It follows that G is sofic.  Corollary7.5.8. Let (Gi )i∈I be a family of sofic groups. Then their direct sum G = i∈I Gi is sofic. Proof. This follows immediately from Proposition 7.5.4  and Proposition 7.5.7, since G is the subgroup of the direct product P = i∈I Gi consisting of all  g = (gi ) ∈ P for which gi = 1Gi for all but finitely many i ∈ I. Corollary 7.5.9. The limit of a projective system of sofic groups is sofic. Proof. Let (Gi )i∈I be a projective system of sofic groups such that G = lim Gi . By construction of a projective limit (see Appendix E), G is a sub←−

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7 Local Embeddability and Sofic Groups

 group of the group i∈I Gi . We deduce that G is sofic by using Proposition 7.5.7 and Proposition 7.5.4.  Proposition 7.5.10. Every group which is locally embeddable into the class of sofic groups is sofic. Proof. Let G be a group which is locally embeddable into the class of sofic groups. Let K ⊂ G be a finite subset and ε > 0. By definition of local embeddability, there exists a sofic group G and a K-almost-homomorphism ϕ : G → G . Set K  = ϕ(K). By soficity of G , there exists a nonempty finite set F and a (K  , ε)-almost-homomorphism ϕ : G → Sym(F ). Let us prove that the composite map Φ = ϕ ◦ ϕ : G → Sym(F ) is a (K, ε)-almosthomomorphism. Let k1 , k2 ∈ K. Then we have dF (Φ(k1 k2 ), Φ(k1 )Φ(k2 )) = dF (ϕ (ϕ(k1 k2 )), ϕ (ϕ(k1 ))ϕ (ϕ(k2 ))) = dF (ϕ (ϕ(k1 )ϕ(k2 )), ϕ (ϕ(k1 ))ϕ (ϕ(k2 ))) ≤ε

(as ϕ(k1 ), ϕ(k2 ) ∈ K  ).

Finally, let k1 , k2 ∈ K be such that k1 = k2 . Since ϕ|K is injective, we have ϕ(k1 ) = ϕ(k2 ) and therefore dF (Φ(k1 ), Φ(k2 )) = dF (ϕ (ϕ(k1 )), ϕ (ϕ(k2 ))) ≥ 1 − ε. Thus, Φ is a (K, ε)-almost-homomorphism. It follows that G is sofic.



Since every amenable group is sofic by Proposition 7.5.6, an immediate consequence of Proposition 7.5.10 is the following: Corollary 7.5.11. Every LEA-group is sofic. In particular, every LEFgroup, every locally residually amenable group, every locally residually finite group, every residually amenable group, and every residually finite group is sofic.  As the class of sofic groups is closed under direct products by Proposition 7.5.7, it follows from Corollary 7.1.15 that every locally residually sofic group is locally embeddable into the class of sofic groups. By applying Proposition 7.5.10, we get: Corollary 7.5.12. Every locally residually sofic group is sofic.



It follows from Proposition 7.5.10 that the class of groups which are locally embeddable into the class of sofic groups coincide with the class of sofic groups. By applying Corollary 7.1.17, we then deduce the following: Corollary 7.5.13. Let Γ be a group. Then the set of Γ -marked groups N ∈ N (Γ ) such that Γ/N is sofic is closed (and hence compact) for the prodiscrete topology on N (Γ ) ⊂ P(Γ ) = {0, 1}Γ . 

7.5 Sofic Groups

259

We end this section by showing that extensions of sofic groups by amenable groups are sofic. This is a generalization of Proposition 7.5.6 since every amenable group is an extension of the trivial group by itself. Proposition 7.5.14. Let G be a group. Suppose that G contains a normal subgroup N such that N is sofic and G/N is amenable. Then G is sofic. Proof. Let K ⊂ G be a finite subset and 0 < ε < 1. Denote by g the image of an element g ∈ G under the canonical epimorphism of G onto G/N . Fix a set T ⊂ G of representatives for the cosets of N in G and denote by σ : G/N → T the map which associates with each element in G/N its representative in T . √ Note that σ(g)−1 g ∈ N for all g ∈ G. Also set ε = 1 − 1 − ε, so that 0 < ε < ε and (1 − ε )2 = 1 − ε. Since G/N is amenable and hence sofic by Proposition 7.5.6, there exist a nonempty finite set F1 and a (K, ε )-almost-homomorphism ϕ1 : G/N → Sym(F1 ). In fact, in the proof of Proposition 7.5.6 it is shown that we can take F1 ⊂ G/N such that there exists a subset E1 ⊂ F1 with |E1 | ≥ (1 − ε )|F1 | satisfying ϕ1 (k)(f1 ) = kf1 ∈ F1 and ϕ1 (hk)(f1 ) = hkf1 ∈ F1 for all h, k ∈ K and f1 ∈ E1 . Set M = N ∩ (σ(F1 )−1 · K · σ(F1 )) ⊂ N . As N is sofic, we can find a finite set F2 and an (M, ε )-almost-homomorphism ϕ2 : N → Sym(F2 ). Thus, for all m, m ∈ M we can find a set E2 ⊂ F2 such that |E2 | ≥ (1 − ε)|F2 | and ϕ2 (mm )(f2 ) = ϕ2 (m)(ϕ2 (m )(f2 )) for all f2 ∈ E2 .

(7.25)

Set F = F1 × F2 and E = E1 × E2 and observe that |E| = |E1 | · |E2 | ≥ (1 − ε )2 |F1 | · |F2 | = (1 − ε)|F |.

(7.26)

Consider the map Φ : G → Sym(F ) defined by setting Φ(g)(f1 , f2 ) = (ϕ1 (g)(f1 ), ϕ2 (σ(gf1 )−1 gσ(f1 ))(f2 )) for all g ∈ G and (f1 , f2 ) ∈ F . Let us show that Φ is a (K, ε)-almosthomomorphism. Let h, k ∈ K and (f1 , f2 ) ∈ E. Recall that the elements kf1 = ϕ1 (k)(f1 ) and hkf1 = ϕ1 (hk)(f1 ) = ϕ1 (h)(ϕ1 (k)(f1 )) both belong to F1 for all f1 ∈ E1 . It follows that Φ(h)Φ(k)(f1 , f2 ) = Φ(h)(ϕ1 (k)(f1 ), ϕ2 (σ(kf1 )−1 kσ(f1 ))(f2 )) = (ϕ1 (h)(ϕ1 (k)(f1 )), ϕ2 (σ(hϕ1 (k)(f1 ))−1 hσ(ϕ1 (k)(f1 ))) (ϕ2 (σ(kf1 )−1 kσ(f1 ))(f2 ))) = (ϕ1 (hk)(f1 ), ϕ2 (σ(hkf1 )−1 hσ(kf1 )) (ϕ2 (σ(kf1 )−1 kσ(f1 ))(f2 ))) =∗ (ϕ1 (hk)f1 , ϕ2 (σ(hkf1 )−1 hkσ(f1 ))(f2 )) = Φ(hk)(f1 , f2 )

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7 Local Embeddability and Sofic Groups

where =∗ follows from (7.25) since m = σ(hkf1 )−1 hσ(kf1 ) and m = σ(kf1 )−1 kσ(f1 ) both belong to (σ(F1 )−1 Kσ(F1 )) ∩ N = M . It follows from (7.26) that dF (Φ(hk), Φ(h)Φ(k)) ≤ ε. Let now h, k ∈ K be such that h = k. We distinguish two cases. If h = k then, as ϕ1 is a (K, ε)-almost-homomorphism, we have dF1 (ϕ1 (h), ϕ1 (k)) ≥ 1 − ε. It follows that there exists a subset B ⊂ F1 such that |B| ≥ (1 − ε)|F1 | such that ϕ1 (h)(b) = ϕ1 (k)(b) for all b ∈ B. Setting B  = B × F2 we have that

and

|B  | = |B| · |F2 | ≥ (1 − ε)|F1 | · |F2 | = (1 − ε)|F |

(7.27)

Φ(h)(b ) = Φ(k)(b ) for all b ∈ B  .

(7.28)

Suppose now that h = k. As ϕ2 is an (M, ε)-almost-homomorphism one has dF2 (ϕ2 (m), ϕ2 (m )) > 1 − ε for all m, m ∈ M such that m = m . It follows that for all distinct m, m ∈ M there exists a subset D ⊂ F2 such that |D| ≥ (1 − ε)|F2 | and ϕ2 (m)(d) = ϕ2 (m )(d) for all d ∈ D. From h = k we deduce that for all f1 ∈ F1 one has that if m = σ(hf1 )−1 × hσ(f1 ) and m = σ(kf1 )−1 kσ(f1 ), then m, m ∈ M and m = σ(hf1 )−1 hσ(f1 ) = σ(kf1 )−1 hσ(f1 ) = σ(kf1 )−1 kσ(f1 ) = m . Thus ϕ2 (σ(hf1 )−1 hσ(f1 ))(d) = ϕ2 (σ(kf1 )−1 kσ(f1 ))(d) for all d ∈ D. Set D = D × F2 so that |D | = |D| · |F2 | ≥ (1 − ε)|F1 | · |F2 | = (1 − ε)|F |.

(7.29)

It follows that for all d ∈ D one has Φ(h)(d ) = Φ(k)(d ) for all d ∈ D .

(7.30)

From (7.28) and (7.30), and taking into account (7.27) and (7.29) respectively, we deduce that in either cases dF (Φ(h), Φ(k)) ≥ 1 − ε. This shows that Φ : G → Sym(F ) is a (K, ε)-almost-homomorphism. Thus G is sofic. 

7.6 Sofic Groups and Metric Ultraproducts of Finite Symmetric Groups Suppose that we are given a triple T = (I, ω, F) consisting of the following data: a set I, an ultrafilter ω on I, and a family F = (Fi )i∈I of nonempty finite sets indexed by I. Our first goal in this section is to associate with such a triple T a sofic group GT . We start by forming the direct product group  PT = Sym(Fi ). i∈I

7.6 Sofic Groups and Metric Ultraproducts of Finite Symmetric Groups

261

Let α = (αi )i∈I and β = (βi )i∈I be elements of PT . Since 0 ≤ dFi (αi , βi ) ≤ 1 for all i ∈ I, it follows from Corollary J.2.6 that the Hamming distances dFi (αi , βi ) have a limit δω (α, β) = lim dFi (αi , βi ) ∈ [0, 1] i→ω

along the ultrafilter ω. Proposition 7.6.1. One has: (i) δω (α, α) = 0; (ii) δω (β, α) = δω (α, β); (iii) δω (α, β) ≤ δω (α, γ) + δω (γ, β); (iv) δω (γα, γβ) = δω (α, β); (v) δω (αγ, βγ) = δω (α, β). for all α, β, γ ∈ PT . Proof. Let α = (αi )i∈I , β = (βi )i∈I , γ = (γi )i∈I ∈ PT . For each i ∈ I, we have dFi (αi , αi ) = 0, dFi (βi , αi ) = dFi (αi , βi ), dFi (αi , βi ) ≤ dFi (αi , γi ) + dFi (γi , βi ), dFi (γi αi , γi βi ) = dFi (αi , βi ), and dFi (αi γi , βi γi ) = dFi (αi , βi ) since dFi is a bi-invariant metric on Sym(Fi ). This gives us properties (i), (ii), (iii), (iv), and (v) for δω by taking limits along ω (cf. Corollary J.2.10).  Consider now the subset NT ⊂ PT defined by NT = {α ∈ PT : δω (1PT , α) = 0}. Proposition 7.6.2. The set NT is a normal subgroup of PT . Proof. This is an easy consequence of the properties of δω stated in Proposition 7.6.1. Indeed, we deduce from Property (i) that δω (1PT , 1PT ) = 0, that is, 1PT ∈ NT . On the other hand, by using successively (iv), (iii), and (ii), we get δω (1PT , α−1 β) = δω (α, β) ≤ δω (α, 1PT )+δω (1PT , β) = δω (1PT , α)+δω (1PT , β) for all α, β ∈ PT . This implies that α−1 β ∈ NT if α, β ∈ NT . Thus, NT is a subgroup of PT . Finally, Properties (iv) and (v) imply that δω (1PT , γαγ −1 ) = δω (γ −1 , αγ −1 ) = δω (γ −1 γ, α) = δω (1PT , α) for all α, γ ∈ PT . It follows that γαγ −1 ∈ NT for all α ∈ NT and γ ∈ PT .  This shows that NT is a normal subgroup of PT . Observe that, given α = (αi )i∈I and β = (βi )i∈I in PT , one has αNT = βNT

⇐⇒ δω (α, β) = 0.

(7.31)

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Indeed, one has αNT = βNT if and only if α−1 β ∈ NT , that is, if and only if δω (1PT , α−1 β) = 0. This is equivalent to δω (α, β) = 0 by the left-invariance of δω . Theorem 7.6.3. The group GT = PT /NT is sofic. Proof. Fix a finite subset K ⊂ GT and ε > 0. We want to show that there exist a nonempty finite set F and a (K, ε)-almost-homomorphism ϕ : GT → Sym(F ). Choose a representative of each element g ∈ GT , that is, an element g = ( gi )i∈I ∈ PT such that g = gNT . We first introduce some constants that will be used in the proof. If h and h,  k) > 0 by (7.31). Let us set k are distinct elements in K, then δω ( η = min

h,k∈K h=k

δω ( h,  k) . 2

(7.32)

Note that 0 < η ≤ 1/2. Now choose an integer m ≥ log ε/ log(1 − η), so that 1 − (1 − η)m ≥ 1 − ε.

(7.33)

Finally, choose a real number ξ with 0 < ξ < 1 sufficiently small to make 1 − (1 − ξ)m ≤ ε.

(7.34)

 T =  If h and k are arbitrary elements of K, we have hkN h kNT and    therefore δω (hk, hk) = 0 by (7.31). It follows that the set  i,  A(h, k) = {i ∈ I : dFi (hk hi  ki ) ≤ ξ}

(7.35)

belongs to ω. On the other hand, if h and k are distinct elements of K, we have δω ( h,  k) ≥ 2η by (7.32). As η > 0, this implies that the set C(h, k) = {i ∈ I : dFi ( hi ,  ki ) ≥ η}

(7.36)

belongs to ω. As any finite intersection of elements of ω is in ω and therefore nonempty, we deduce that there exists an index j ∈ I such that ⎞ ⎛ ⎞ ⎛ ⎜   ⎟ A(h, k)⎠ ⎜ C(h, k)⎟ j∈⎝ ⎠. ⎝ h,k∈K

h,k∈K h=k

Consider the map ψ : GT → Sym(Fj ) defined by ψ(g) = gj for all g ∈ GT . It immediately follows from (7.35) and (7.36) that the map ψ satisfies the following properties:

7.6 Sofic Groups and Metric Ultraproducts of Finite Symmetric Groups

263

(1) dFj (ψ(hk), ψ(h)ψ(k)) ≤ ξ for all h, k ∈ K; (2) dFj (ψ(h), ψ(k)) ≥ η for all h, k ∈ K such that h = k. Consider now the Cartesian product F = Fj × Fj × · · · × Fj    m times

and the homomorphism Ψ : Sym(Fj ) → Sym(F ) defined by Ψ (σ)(x1 , x2 , . . . , xm ) = (σ(x1 ), σ(x2 ), . . . , σ(xm )) for all σ ∈ Sym(Fj ) and (x1 , x2 , . . . , xm ) ∈ F . By Corollary 7.4.4 we have dF (Ψ (σ), Ψ (σ  )) = 1 − (1 − dFj (σ, σ  ))m for all σ, σ  ∈ Sym(Fj ). It follows that the composite map ϕ = Ψ ◦ ψ : G → Sym(F ) satisfies the following properties: (1’) dF (ϕ(hk), ϕ(h)ϕ(k)) ≤ 1 − (1 − ξ)m for all h, k ∈ K; (2’) dF (ϕ(h), ϕ(k)) ≥ 1 − (1 − η)m for all h, k ∈ K such that h = k. As 1−(1−ξ)m ≤ ε by (7.34) and 1−(1−η)m ≥ 1−ε by (7.33), we deduce that ϕ is a (K, ε)-almost-homomorphism. This shows that GT is a sofic group.  Remark 7.6.4. When the ultrafilter ω is principal, then the group GT is finite. Indeed, in this case, there is an element i0 ∈ I such that ω consists of all subsets of I containing i0 . This implies that δω (α, β) = dFi0 (αi0 , βi0 ) for all α = (αi )i∈I , β = (βi )i∈I ∈ PT . Therefore, NT consists of all α ∈ PT such that αi0 = 1Sym(Fi0 ) . It follows that the group GT is isomorphic to the group Sym(Fi0 ). Remark 7.6.5. Let α, α , β, β  ∈ PT such that αNT = α NT and βNT = β  NT . By applying Proposition 7.6.1(iii) and (7.31), we get dω (α, β) ≤ dω (α, α ) + dω (α , β  ) + dω (β  , β) = dω (α , β  ). By exchanging the roles of α and α and of β and β  , we obtain dω (α , β  ) ≤ dω (α, β). It follows that dω (α, β) = dω (α , β  ). Therefore, if g, h ∈ GT and α, β ∈ PT are such that g = αNT and h = βNT , the quantity Δω (g, h) = δω (α, β)

(7.37)

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is well defined. Moreover, the map Δω : GT × GT → [0, 1] given by (7.37) is a bi-invariant metric on GT . This follows immediately from Proposition 7.6.1 taking into account that NT is a normal subgroup. Theorem 7.6.6. Let G be a group. The following conditions are equivalent: (a) G is sofic; (b) there exists a triple T = (I, ω, F), where I is a set, ω is an ultrafilter on I, and F = (Fi )i∈I is a family of nonempty finite sets indexed by I, such that G is isomorphic to a subgroup of the group GT = PT /NT . Proof. If G satisfies (b), then G is sofic since GT is sofic by Theorem 7.6.3 and every subgroup of a sofic group is itself sofic by Proposition 7.5.4. Conversely, suppose that G is sofic. Consider the set I consisting of all pairs (K, ε), where K is a finite subset of G and ε > 0. We partially order the set I by setting (K, ε)  (K  , ε ) if K ⊂ K  and ε ≤ ε. For each i = (K, ε) ∈ I we define the set Ii = {j ∈ I : i  j} ⊂ I. Observe that Ii = ∅ as i ∈ Ii . Moreover, the family of nonempty subsets {Ii }i∈I is closed under finite intersections, since I(K1 ,ε1 ) ∩ I(K2 ,ε2 ) = I(K1 ∪K2 ,min{ε1 ,ε2 }) for all (K1 , ε1 ), (K2 , ε2 ) ∈ I. It follows from Proposition J.1.3 and Theorem J.1.6 that there exists an ultrafilter ω on I such that I(K,ε) ∈ ω for all (K, ε) ∈ I. As G is sofic, we can find, for each i = (K, ε) ∈ I, a nonempty finite set Fi and a (K, ε)-almost-homomorphism ϕi : G → Sym(Fi ). Consider the triple T = (I, ω, F), where F = (Fi )i∈I , and the associated  sofic group  : G → PT denote the product map ϕ  = i∈I ϕi . Thus, GT = PT /NT . Let ϕ we have ϕ(g)  = (ϕi (g))i∈I for all g ∈ G. Let ρ : PT → GT = PT /NT denote the canonical epimorphism. Let us show that the composite map Φ = ρ ◦ ϕ  : G → GT is an injective homomorphism. This will prove that G is isomorphic to a subgroup of GT . Let g, h ∈ G and let η > 0. Consider the element i0 = ({g, h}, η) ∈ I. If i = (K, ε) ∈ I satisfies i0  i, then dFi (ϕi (gh), ϕi (g)ϕi (h)) ≤ ε ≤ η since ϕi is a (K, ε)-almost-homomorphism. Thus, the set {i ∈ I : dFi (ϕi (gh), ϕi (g)ϕi (h)) ≤ η} contains Ii0 and therefore belongs to ω. This implies that

7.7 A Characterization of Finitely Generated Sofic Groups

265

δω (ϕ(gh),  ϕ(g)  ϕ(h))  = lim dFi (ϕi (gh), ϕi (g)ϕi (h)) = 0. i→ω

Therefore, we have Φ(gh) = Φ(g)Φ(h) by (7.31). This shows that Φ is a homomorphism. On the other hand, if g = h, we have dFi (ϕi (g), ϕi (h)) ≥ 1 − ε ≥ 1 − η for all i ∈ I such that i0  i. This implies that  ϕ(h))  = lim dFi (ϕi (g), ϕi (h)) = 1. δω (ϕ(g), i→ω

(7.38)

Therefore, we have Φ(g) = Φ(h) by (7.31). Consequently, Φ is injective. This shows that (a) implies (b).  Remarks 7.6.7. (a) If G is an infinite sofic group and T = (I, ω, F ) is a triple satisfying condition (b) in Theorem 7.6.6, then the ultrafilter ω is necessarily non-principal by Remark 7.6.4. (b) Let G be a sofic group and let Φ : G → GT be as in the proof of Theorem 7.6.6. Using the notation from Remark 7.6.5, we deduce from (7.38) that Δω (Φ(g), Φ(h)) = 1 for all g, h ∈ G such that g = h. It follows that the restriction of the bi-invariant metric Δω to the subgroup Φ(G) ⊂ GT is the discrete metric.

7.7 A Characterization of Finitely Generated Sofic Groups In this section we give a geometric characterization of finitely generated sofic groups in terms of a finiteness condition on their Cayley graphs. Let G be a finitely generated group and let S be a finite symmetric generating subset of G. Given r ∈ N, we denote by BS (r) the ball of radius r centered at the vertex corresponding to the identity element 1G of G in the Cayley graph CS (G) of G with respect to S, with the induced S-labeled graph structure (cf. Sects. 6.1, 6.2 and 6.3). Let also Q = (Q, E) be an S-labeled graph. Given q ∈ Q and r ∈ N, we denote by B(q, r) the ball of radius r centered at q with the induced S-labeled graph structure. Given r ∈ N we denote by Q(r) the set of all q ∈ Q such that there exists an S-labeled graph isomorphism ψq,r : BS (r) → B(q, r)

(7.39)

ψq,r (1G ) = q.

(7.40)

satisfying

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7 Local Embeddability and Sofic Groups

Observe that if such a map ψq,r exists it is unique. We have the inclusions Q = Q(0) ⊃ Q(1) ⊃ Q(2) ⊃ · · · ⊃ Q(r) ⊃ Q(r + 1) ⊃ · · ·

(7.41)

Fig. 7.2 The inclusions Q ⊃ Q(1) ⊃ Q(2) ⊃ Q(3)

Note also that since CS (G) (and therefore the induced S-labeled subgraph BS (r)) is edge-symmetric (with respect to the involution s → s−1 on S (cf. Sect. 6.3)) and ψq,r is an S-labeled graph isomorphism, then B(q, r) = ψq,r (BS (r)) is edge-symmetric as well. Theorem 7.7.1. Let G be a finitely generated group and let S be a finite symmetric generating subset of G. The following conditions are equivalent: (a) the group G is sofic; (b) for all ε > 0 and r ∈ N, there exists a finite S-labeled graph Q = (Q, E) such that |Q(r)| ≥ (1 − ε)|Q|, (7.42) where Q(r) ⊂ Q denotes the set consisting of all vertices q ∈ Q for which there exists an S-labeled graph isomorphism ψq,r : BS (r) → B(q, r) from the ball BS (r) in the Cayley graph CS (G) of G with respect to S onto the ball B(q, r) in Q satisfying ψq,r (1G ) = q. Before starting the proof of the theorem we present some preliminary results. Lemma 7.7.2. Let Q = (Q, E) be an S-labeled graph and r0 , i ∈ N. Suppose that q0 ∈ Q((i + 1)r0 ). Then B(q0 , r0 ) ⊂ Q(ir0 ). Proof. Let q  ∈ B(q, r0 ) and let us show that q  ∈ Q(ir0 ). It follows from the triangle inequality that the ball B(q  , ir0 ) is entirely contained in the ball B(q0 , (i + 1)r0 ). Moreover, since ψq0 ,(i+1)r0 is isometric, setting g =

7.7 A Characterization of Finitely Generated Sofic Groups

267

ψq−1 (q  ), we have g ∈ BS (r0 ) so that gh ∈ B((i+1)r0 ) for all h ∈ B(ir0 ). 0 ,(i+1)r0 It follows that the map ψq ,ir0 : BS (ir0 ) → B(q  , ir0 )

(7.43)

defined by ψq ,ir0 (h) = ψq0 ,(i+1)r0 (gh) for all h ∈ BS (ir0 ) yields an S-labeled graph isomorphism satisfying ψq ,ir0 (1G ) = ψq0 ,(i+1)r0 (g) = q  . This shows  that q  ∈ Q(ir0 ). We deduce that B(q0 , r0 ) ⊂ Q(ir0 ). Lemma 7.7.3. Let Q = (Q, E) be an S-labeled graph and r0 ∈ N. Let q1 , q2 ∈ Q(2r0 ) such that q1 = q2 and g ∈ BS (r0 ). Then we have ψq1 ,2r0 (g) = ψq2 ,2r0 (g).

(7.44)

Proof. If g = 1G we have ψq1 ,2r0 (g) = ψq1 ,2r0 (1G ) = q1 = q2 = ψq2 ,2r0 (1G ) = ψq2 ,2r0 (g). Suppose now that g = 1G . Suppose by contradiction that ψq1 ,2r0 (g) = ψq2 ,2r0 (g) = q0 . Since ψq1 ,2r0 is isometric we have q0 ∈ B(q1 , r0 ). It follows from Lemma 7.7.2 that q0 ∈ Q(r0 ). As g ∈ BS (r0 ), we can find 1 ≤ r ≤ r0 and s1 , s2 , . . . , sr ∈ S such that g = s1 s2 · · · sr . Consider the path π = ((1G , s1 , s1 ), (s1 , s2 , s1 s2 ), . . . , (s1 s2 · · · sr −1 , sr , g)) and observe that it is contained in BS (r0 ). Now, π is mapped by ψq1 ,2r0 and ψq2 ,2r0 into two paths π1 and π2 in Q with initial vertices π1− = q1 , π2− = q2 and same terminal vertex π1+ = ψq1 ,2r0 (g) = q0 = ψq2 ,2r0 (g) = π2+ . Note that since ψq1 ,2r0 and ψq2 ,2r0 are isometric π1 and π2 are both contained in B(q0 , r0 ). Moreover, since ψq1 ,2r0 and ψq2 ,2r0 are label-preserving, π1 and π2 have the same label s1 s2 · · · sr . The inverse images of π1 and π2 under the S-label preserving graph isomorphism ψq0 ,r0 : BS (r0 ) → B(q0 , r0 ) are both equal to π. Indeed in a Cayley graph there exists a unique path which ends at a given vertex and with a given label. It follows that π1 = π2 and therefore q1 = π1− = π2− = q2 . This contradicts our assumptions. We deduce that  ψq1 ,2r0 (g) = ψq2 ,2r0 (g). Lemma 7.7.4. Let Q = (Q, E) be an S-labeled graph and r0 ∈ N. Let h, k ∈ BS (r0 ) and q0 ∈ Q(2r0 ). We have ψq0 ,2r0 (h) ∈ Q(r0 )

(7.45)

ψq0 ,2r0 (hk) = ψψq0 ,2r0 (h),r0 (k),

(7.46)

and where ψq,r , q ∈ Q(r), r ∈ N, is as in (7.39) and (7.40). Proof. Since ψq0 ,2r0 is isometric we have ψq0 ,2r0 (h) ∈ B(q0 , r0 ). Then (7.45) follows from Lemma 7.7.2. Let us show (7.46). First note that (7.46) makes sense by virtue of (7.45). If k = 1G (7.46) follows from ψq0 ,2r0 (hk) =

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ψq0 ,2r0 (h) = ψψq0 ,2r0 (h),r0 (1G ) = ψψq0 ,2r0 (h),r0 (k). Now suppose that k = 1G . Then we can find 1 ≤ r ≤ r0 and s1 , s2 , . . . , sr ∈ S such that k = s1 s2 · · · sr . Consider the path π1 = ((h, s1 , hs1 ), (hs1 , s2 , hs1 s2 ), . . . , (hs1 s2 · · · sr −1 , sr , hk)) and observe that it is contained in BS (2r0 ), since h, k ∈ BS (r0 ). The path π1 is mapped by ψq0 ,2r0 into the path π 1 = ((ψq0 ,2r0 (h),s1 , ψq0 ,2r0 (hs1 )), (ψq0 ,2r0 (hs1 ), s2 , ψq0 ,2r0 (hs1 s2 )), . . . . . . , (ψq0 ,2r0 (hs1 s2 · · · sr −1 ), sr , ψq0 ,2r0 (hk))). As ψq0 ,2r0 is isometric, we have that q  = ψq0 ,2r0 (h) belongs to B(q0 , r0 ) and since q0 ∈ Q(2r0 ) we deduce from Lemma 7.7.2 that q  ∈ Q(r0 ). Consider the inverse image of the path π 1 under the S-labeled graph isomorphism ψq ,r0 . −1 −1 −  Since ψq−1  ,r ((π 1 ) ) = ψq  ,r (ψq0 ,2r0 (h)) = ψq  ,r (q ) = 1G and ψq  ,r0 preserves 0 0 0 the label, this inverse image is necessarily equal to the path π2 = ((1G , s1 , s1 ), (s1 , s2 , s1 s2 ), . . . , (s1 s2 · · · sr −1 , sr , k)), since in a Cayley graph there exists a unique path which starts at a given ver−1 + tex and with a given label. It follows that ψq−1  ,r (ψq0 ,2r0 (hk)) = ψq  ,r (π 1 ) = 0 0 π2+ = k, that is, ψq0 ,2r0 (hk) = ψq ,r0 (k). Since q  = ψq0 ,2r0 (h), we deduce (7.46).  We are now in position to prove Theorem 7.7.1. Proof of Theorem 7.7.1. Suppose that G is sofic. Fix ε > 0 and r ∈ N. Set K = BS (2r + 1) and ε = ε(1 + |BS (r)| · |S| + |BS (r)|2 )−1 .

(7.47)

Since G is sofic, there exists a nonempty finite set F and a (K, ε )-almosthomomorphism ϕ : G → Sym(F ). We construct an S-labeled graph Q = (Q, E) as follows. We take as vertex set Q = F . Then, as set of edges we take the set E ⊂ Q × S × Q consisting of all the triples (q, s, ϕ(s−1 )(q)), were q ∈ Q and s ∈ S. Note that Q may have loops and multiple edges and that Q is not necessarily edge-symmetric with respect to the involution s → s−1 on S. Observe however that, if q ∈ Q and s ∈ S are fixed, then there exists a unique edge in Q with initial vertex q and label s. For each q ∈ Q denote by ψq : G → Q the map defined by setting ψq (g) = ϕ(g −1 )(q) for all g ∈ G. Denote by Q0 the subset of Q consisting of all q ∈ Q satisfying the following conditions: (∗) ψq (1G ) = q, (∗∗) ψq (gs) = ψψq (g) (s) for all g ∈ BS (r) and s ∈ S, (∗ ∗ ∗) ψq (g) = ψq (h) for all g, h ∈ BS (r) with g = h.

7.7 A Characterization of Finitely Generated Sofic Groups

269

Suppose that q ∈ Q0 . Let g ∈ BS (r). If g = 1G we have ψq (g) = ψq (1G ) = q ∈ B(q, r), by (∗). If g = 1G , then there exist 1 ≤ r ≤ r and s1 , s2 , . . . , sr ∈ S such that g = s1 s2 · · · sr . Consider the sequence of edges e1 = (q, s1 , ϕ(s−1 1 )(q)) = (q, s1 , ψq (s1 )), e2 = (ψq (s1 ), s2 , ϕ(s−1 2 )ψq (s1 )) = (ψq (s1 ), s2 , ψψq (s1 ) (s2 )) = (ψq (s1 ), s2 , ψq (s1 s2 )) (by (∗∗)), ······ er = (ψq (s1 s2 · · · sr −1 ), sr , ϕ(s−1 r  )(ψq (s1 s2 · · · sr  −1 ))) = (ψq (s1 s2 · · · sr −1 ), sr , ψψq (s1 s2 ···sr −1 ) (sr )) = (ψq (s1 s2 · · · sr −1 ), sr , ψq (s1 s2 · · · sr −1 sr )) (by (∗∗)) = (ψq (s1 s2 · · · sr −1 ), sr , ψq (g)). The path π = (e1 , e2 , . . . , er ) connects q to ψq (g) and has length (π) ≤ r. This shows that the graph distance from q to ψq (g) in Q does not exceed r so that ψq (BS (r)) ⊂ B(q, r). Conversely, let q  ∈ B(q, r). If q  = q then by (∗) we have q  = q = ψq (1G ) ∈ ψq (BS (r)). If q = q  then there exist 1 ≤ r ≤ r and a sequence of edges (q, s1 , q1 ), (q1 , s2 , q2 ), . . . , (qr −1 , sr , q  ) ∈ E. Using (∗∗) as above, we get q  = ψq (g), where g = s1 s2 · · · sr ∈ BS (r). This shows that B(q, r) ⊂ ψq (BS (r)). Thus B(q, r) = ψq (BS (r)). Moreover, by condition (∗ ∗ ∗), the map ψq |BS (r) is injective. Finally, from (∗∗) we deduce that if g ∈ BS (r) and s ∈ S then we have (ψq (g), s, ψq (gs)) = (ψq (g), s, ψψq (g) (s)) = (ψq (g), s, ϕ(s−1 )ψq (g)) ∈ E. Thus, the map ψq,r = ψq |BS (r) : BS (r) → B(q, r) is an S-labeled graph isomorphism such that ψq,r (1G ) = ψq (1G ) = q, where the last equality follows from (∗). We deduce that Q0 ⊂ Q(r). Let’s now estimate from below the cardinality of Q0 . We denote by dQ the normalized Hamming metric on Sym(Q). In order to estimate the cardinality of the set of q ∈ Q for which condition (∗) is satisfied, let us first observe that dQ (IdQ , ϕ(1G )) ≤ ε .

(7.48)

Indeed, since ϕ is a (K, ε )-almost-homomorphism, taking k1 = k2 = 1G in property (i) of Definition 7.5.1, we deduce that dQ (ϕ(1G ), ϕ(1G )ϕ(1G )) ≤ ε which, by left-invariance of dQ , implies (7.48). From (7.48) we deduce the

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existence of a subset Q ⊂ Q of cardinality |Q | ≤ ε |Q| such that ϕ(1G )(q) = q

(7.49)

for all q ∈ Q \ Q . It follows that ψq (1G ) = ϕ(1−1 G )(q) = ϕ(1G )(q) = q, for all q ∈ Q \ Q so that condition (∗) is satisfied in Q \ Q . Let now estimate the cardinality of the set of q ∈ Q for which condition (∗∗) is satisfied. Let g ∈ BS (r) and s ∈ S. As ϕ is a (K, ε )-almosthomomorphism we have dQ (ϕ(s−1 g −1 ), ϕ(s−1 )ϕ(g −1 )) ≤ ε so that there exists a subset Q (g, s) ⊂ Q with |Q (g, s)| ≤ ε |Q| such that ψq (gs) = ϕ((gs)−1 )(q) = ϕ(s−1 g −1 )(q) = ϕ(s−1 )ϕ(g −1 )(q) = ϕ(s−1 )(ψq (g)) = ψψq (g) (s) for all q ∈ Q \ Q (g, s). Setting Q =

Q (g, s)

g∈BS (r) s∈S

we have |Q | ≤ |BS (r)| · |S|ε |Q| and condition (∗∗) holds for all q ∈ Q \ Q . Fix now two distinct elements g, h ∈ BS (r). Since ϕ is a (K, ε )-almosthomomorphism and g −1 , h−1 ∈ K, by property (ii) of Definition 7.5.1 we deduce that dQ (ϕ(g −1 ), ϕ(h−1 )) ≥ 1 − ε . Thus we can find a subset Q (g, h) ⊂ Q of cardinality |Q (g, h)| ≤ ε |Q| such that ψq (g) = ϕ(g −1 )(q) = ϕ(h−1 )(q) = ψq (h)

(7.50)

for all q ∈ Q \ Q (g, h). Let now g = h vary in BS (r) and set

Q = Q (g, h). g,h∈BS (r) g=h

Observe that |Q | ≤ |BS (r)|2 ε |Q|. It follows from (7.50) that condition (∗ ∗ ∗) holds for all q ∈ Q \ Q . In conclusion, conditions (∗), (∗∗) and (∗ ∗ ∗) are satisfied for all q ∈ Q outside of Q ∪ Q ∪ Q . We have |Q ∪ Q ∪ Q | ≤ (1 + |BS (r)| · |S| + |BS (r)|2 )ε |Q| = ε|Q|,

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271

where the equality follows from (7.47). We deduce that |Q(r)| ≥ |Q0 | ≥ (1 − ε)|Q|. Thus G satisfies condition (b). This shows (a) ⇒ (b). Conversely, suppose (b). Fix a finite set K ⊂ G and ε > 0. Let r0 ∈ N be such that K ∪ K 2 ⊂ BS (r0 ). Let Q = (Q, E) be the finite S-labeled graph given by condition (b) corresponding to r = 2r0 and ε. Let g ∈ BS (r0 ). Since the map from Q(2r0 ) into Q defined by q → ψq,2r0 (g) is injective (by Lemma 7.44), we have that |{ψq,2r0 (g) : q ∈ Q(2r0 )}| = |Q(2r0 )|

(7.51)

|Q \ {ψq,2r0 (g) : q ∈ Q(2r0 )}| = |Q \ Q(2r0 )|.

(7.52)

and therefore

As a consequence, there exists a bijection αg : Q \ Q(2r0 ) → Q \ {ψq,2r0 (g) : q ∈ Q(2r0 )}. Since ψq,2r0 (1G ) = q for all q ∈ Q(2r0 ), we have {ψq,2r0 (1G ) : q ∈ Q(2r0 )} = Q(2r0 ) and therefore we can take α1G = IdQ\Q(2r0 ) .

(7.53)

Consider now the map ϕ : G → Sym(Q) defined by ⎧ −1 ⎪ ⎨ψq,2r0 (g ) ϕ(g)(q) = αg−1 (q) ⎪ ⎩ q

if g ∈ BS (r0 ) and q ∈ Q(2r0 ); if g ∈ BS (r0 ) and q ∈ Q \ Q(2r0 );

(7.54)

otherwise.

Note that ϕ(g) ∈ Sym(Q) for all g ∈ G, by construction. Let us show that the map ϕ : G → Sym(Q) is a (K, ε)-almost-homomorphism. Let k1 , k2 ∈ K ⊂ BS (r0 ) and q ∈ Q(2r0 ). We have ϕ(k1 k2 )(q) = ψq,2r0 (k2−1 k1−1 ) = ψψq,2r

−1 0 (k2 ),r0

(k1−1 ) (by (7.46))

= ϕ(k1 )(ψq,2r0 (k2−1 )) = [ϕ(k1 )ϕ(k2 )](q). This shows that on Q(2r0 ) we have ϕ(k1 k2 ) = ϕ(k1 )ϕ(k2 ). As |Q(2r0 )| = |Q(r)| ≥ (1 − ε)|Q|

(7.55)

we deduce that dQ (ϕ(k1 k2 ), ϕ(k1 )ϕ(k2 )) ≤ ε. Finally, suppose that k1 = k2 . We have ϕ(k1 )(q) = ψq,2r0 (k1−1 ) = ψq,2r0 (k2−1 ) = ϕ(k2 )(q), since k1−1 , k2−1 ∈ BS (2r0 ) and ψq,2r0 is injective.

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From (7.55) we deduce that dQ (ϕ(k1 ), ϕ(k2 )) ≥ 1 − ε. It follows that ϕ is a (K, ε)-almost-homomorphism. Therefore G is sofic. This shows that (b) ⇒ (a). 

7.8 Surjunctivity of Sofic Groups In this section we prove that sofic groups are surjunctive. Note that this result covers the fact that locally residually amenable groups are surjunctive, which had been previously established in Corollary 5.9.3. Indeed, all locally residually amenable groups are sofic by Corollary 7.5.11. Theorem 7.8.1 (Gromov-Weiss). Every sofic group is surjunctive. Let us first establish the following: Lemma 7.8.2. Let G be a group, A a finite set, and equip AG with the prodiscrete topology. Let X ⊂ AG be a closed G-invariant subset and let f : X → AG be a continuous G-equivariant map. Then there exists a cellular automaton τ : AG → AG such that f = τ |X . Proof. From the continuity of f we deduce the existence of a finite set S ⊂ G such that if two configurations y, z ∈ X coincide on S then f (y)(1G ) = f (z)(1G ). Let a0 ∈ A and consider the map μ : AS → A defined by setting  f (x)(1G ) if there exists x ∈ X such that x|S = u μ(u) = otherwise a0 for all u ∈ AS . Then μ is well defined and if we denote by τ : AG → AG the cellular automaton with memory set S and local defining map μ, by G  equivariance of f we clearly have τ |X = f . Proof of Theorem 7.8.1. Let G be a sofic group. Let A be a finite set of cardinality |A| ≥ 2 and let τ : AG → AG be an injective cellular automaton. We want to show that τ is surjective. Every subgroup of a sofic group is sofic by Proposition 7.5.4. On the other hand it follows from Proposition 3.2.2 that a group is surjunctive if all its finitely generated subgroups are surjunctive. Thus we can assume that G is finitely generated. Let then S ⊂ G be a finite symmetric generating subset of G. As usual, for r ∈ N, we denote by BS (r) ⊂ G the ball of radius r centered at 1G in the Cayley graph of G with respect to S. We set Y = τ (AG ). Observe that Y is G-invariant and, by Lemma 3.3.2, it is closed in AG . The inverse map τ −1 : Y → AG is G-equivariant and, by compactness of G A , it is also continuous. By Lemma 7.8.2, there exists a cellular automaton σ : AG → AG such that σ|Y = τ −1 : Y → AG . Choose r0 large enough so

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273

that the ball BS (r0 ) is a memory set for both τ and σ. Let μ : ABS (r0 ) → A and ν : ABS (r0 ) → A denote the corresponding local defining maps for τ and σ respectively. We proceed by contradiction. Suppose that τ is not surjective, that is, Y  AG . Then, since Y is closed in AG , there exists a finite subset Ω ⊂ G such that πΩ (Y )  AΩ , where, for a subset E ⊂ G, we denote by πE : AG → AE the projection map. It is not restrictive, up to taking a larger r0 , again if necessary, to suppose that Ω ⊂ BS (r0 ). Thus, πBS (r0 ) (Y )  ABS (r0 ) . Fix ε > 0 such that ε 0 for ! all a ∈ A and a∈A pa = 1) and a countable group G, then, denoting by

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 μp = g∈G p the corresponding probability measure on the product space shift. The associated entropy AG , the triple (G, AG , μp ) is called a Bernoulli ! is the nonnegative number H(p) = − a∈A pa log pa . Two Bernoulli shifts (G, AG , μp ) and (G, B G , μq ) are said to be isomorphic (or measurably conjugate) if there exist two G-invariant subsets X ⊂ AG and Y ⊂ B G such that μp (AG \ X) = μq (B G \ Y ) = 0 and a G-equivariant bijective measurable map θ : X → Y with measurable inverse θ−1 : Y → X such that θ∗ μp = μq . Bowen [Bow, Theorem 1.1] proved that if G is a countable sofic group, then two isomorphic Bernoulli shifts (G, AG , μp ) and (G, B G , μq ) have the same entropy, i.e. H(p) = H(q). This result had been established when G = Z by N. Kolmogorov [Ko1, Ko2] in 1958–59 and then extended to countable amenable groups by D. Ornstein and B. Weiss [OrW] in 1987. L. Glebsky and L.M. Rivera [GlR] introduced the concept of a weaklysofic group. This is a natural extension of the definition of soficity where the Hamming metric on symmetric groups is replaced by general bi-invariant metrics on finite groups. Glebsky and Rivera showed that the existence of a non-weakly-sofic group is equivalent to a conjecture on the closure in the profinite topology of products of conjugacy classes in free groups of finite rank.

Exercises 7.1. Let G and C be two groups. Suppose that K is a finite symmetric subset of G such that 1G ∈ K. Show that a map ϕ : G → C is a K-almost homomorphism if and only if it satisfies ϕ(k1 k2 ) = ϕ(k1 )ϕ(k2 ) for all k1 , k2 ∈ K and ϕ(k) = 1C for all k ∈ K \ {1G }. 7.2. Show that every group which is locally embeddable into the class of abelian groups is itself abelian. 7.3. Show that every group which is locally embeddable into the class of metabelian groups is itself metabelian. 7.4. Show that every abelian group is locally embeddable into the class of finite cyclic groups. 7.5. Let C be a class of groups. Suppose that a group G contains " a family (Hi )i∈I of subgroups satisfying the following properties: (1) G = i∈I Hi ; (2) For all i, j ∈ I there exists k ∈ I such that Hi ∪ Hj ⊂ Hk ; (3) Hi is locally embeddable into C for all i ∈ I. Show that G is locally embeddable into C. 7.6. Let C be a class of groups which is closed under finite direct products. Show that every group which is residually locally embeddable into C is locally embeddable into C.

Exercises

279

7.7. Let C be a class of groups which is closed under taking subgroups and taking finite direct products. Let G be a group which is residually C. Show that there exists a net (Ni )i∈I which converges to {1G } in N (G) such that G/Ni ∈ C fro all i ∈ I. Hint: Take as I the set of finite subsets of G partially ordered by inclusion and use Proposition 7.1.13. 7.8. Show that if G is a finitely presented infinite simple group then G is not LEF. 7.9. Show that if G is a finitely presented non-amenable simple group then G is not LEA. 7.10. (cf. [VeG]) Let G1 and G2 be two groups. Recall that, by Proposition 7.3.1, Gi (i = 1, 2) is an LEF-group if and only if the following holds: (∗) for every finite subset Ki ⊂ Gi there exist a finite set Li such that Ki ⊂ Li ⊂ Gi and a binary operation i : Li × Li → Li such that (Li , i ) is a group and ki ki = ki  ki for all ki , ki ∈ Ki . Suppose that G1 and G2 are LEF-groups. An action of G2 on G1 by group automorphisms, i.e., a group homomorphism ϕ : G2 → Aut(G1 ), is said to be equivariantly approximable, if for all finite subsets K1 ⊂ G1 and K2 ⊂ G2 there exist finite groups (L1 , 1 ) and (L2 , 2 ) as in (∗) and an action of L2 on L1 by group automorphisms ψ : L2 → Aut(L1 ) such that if k1 ∈ K1 , k2 ∈ K2 and ϕ(k2 )(k1 ) ∈ K1 then ϕ(k2 )(k1 ) = ψ(k2 )(k1 ). Show that if G1 and G2 are LEF-groups and ϕ : G2 → Aut(G1 ) is an equivariantly approximable action, then the semidirect product G1 ϕ G2 is an LEF-group. 7.11. Use the previous exercise to show that the group G1 of Sect. 2.6 is an LEF-group (cf. Proposition 7.3.9). 7.12. Show that every residually free group is torsion-free. 7.13. A group G is called fully residually free if for any finite subset K ⊂ G, there exist a free group F and a group homomorphism φ : G → F whose restriction to K is injective. (a) Show that every fully residually free group is residually free. (b) Show that every fully residually free group is locally embeddable into the class of free groups. (c) A group G is called commutative-transitive if whenever a, b, c ∈ G\{1G } satisfy ab = ba and bc = cb, then ac = ca. Prove that every group which is locally embeddable into the class of free groups is commutative-transitive. Hint: First prove that if F is a free group and a, b ∈ F satisfy ab = ba, then there exist an element x ∈ F and integers m, n ∈ Z such that a = xm and b = xn . (d) Show that if F is a nonabelian free group, then the group F × Z is residually free but not locally embeddable into the class of free groups.

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7.14. Give an example of a finitely generated LEF-group which is neither residually finite nor amenable. Hint: Take for instance the group G = G1 ×F2 , where G1 is the group introduced in Sect. 2.6 and F2 denotes a free group of rank two. 7.15. Let F be a nonempty finite set. Denote by GL(RF ) the automorphism group of the real vector space RF = {x : F → R}. For α ∈ Sym(F ), define λ(α) : RF → RF by λ(α)(x) = x ◦ α−1 for all x ∈ RF . (a) Show that λ(α) ∈ GL(RF ) for all α ∈ Sym(F ). (b) Show that the map λ : Sym(F ) → GL(RF ) is an injective group homomorphism. (c) Show that the Hamming metric dF on Sym(F ) satisfies dF (α, β) =

1 Tr(λ(α−1 β)) |F |

for all α, β ∈ Sym(F ), where Tr(·) denotes the trace. 7.16. Let H be a real or complex Hilbert space of finite dimension n ≥ 1. Let L(H) denote the vector space consisting of all linear maps u : H → H. If u ∈ L(H), we denote by Tr(u) the trace of u and by u∗ its adjoint. For u, v ∈ L(H), we set 1 u, vHS = Tr(u ◦ v ∗ ). n (a) Show that ·, ·HS is a scalar product on L(H). Let  · HS denote the associated norm. (b) Let U(H) = {u ∈ L(H) : u ◦ u∗ = IdH }. Show that U(H) is a group for the composition of maps. (c) Show that the map dHS : U(H) × U(H) → R defined by dHS (u, v) = u − vHS for all u, v ∈ U(H) is a bi-invariant metric on U(H). (The group U(H) is called the unitary group of H and dHS is called the normalized Hilbert-Schmidt metric on U(H).) 7.17. Let G be a group, K ⊂ G a finite subset, C a finite group, and ϕ : G → C a K-almost-homomorphism. Denote by L : C → Sym(C) the Cayley homomorphism, that is, the map defined by L(g)(h) = gh for all g, h ∈ C and set Φ = L ◦ ϕ : G → Sym(C). Show that Φ is a (K, ε)-almosthomomorphism for all ε > 0. 7.18. Let G be a group and let K be a finite subset of G. Let F be a nonempty finite set. Show that if 0 < ε < 2/|F | then every (K, ε)-almosthomomorphism ϕ : G → Sym(F ) is a K-almost-homomorphism of G into the group Sym(F ). 7.19. Suppose that a group G contains " a family (Hi )i∈I of subgroups satisfying the following properties: (1) G = i∈I Hi ; (2) For all i, j ∈ I there exists k ∈ I such that Hi ∪ Hj ⊂ Hk ; (3) Hi is sofic for all i ∈ I. Show that G is sofic.

Exercises

281

7.20. Show that every virtually sofic group is sofic. Hint: Use Proposition 7.5.14. 7.21. By using Exercise 6.5 and Exercise 6.6, give a direct proof of the fact that the direct product of finitely many groups which satisfy condition (b) in Theorem 7.7.1 also satisfies it. 7.22. Let G be a finitely generated group and suppose that S and S  are two finite symmetric generating subsets of G. Show that the pair (G, S) satisfies condition (b) in Theorem 7.7.1 if and only if (G, S  ) does. 7.23. Give a direct proof of the fact that every finitely generated residually finite group satisfies the condition (b) in Theorem 7.7.1. 7.24. Give a direct proof of the fact that every finitely generated amenable group satisfies the condition (b) in Theorem 7.7.1.

Chapter 8

Linear Cellular Automata

In this chapter we study linear cellular automata, namely cellular automata whose alphabet is a vector space and which are linear with respect to the induced vector space structure on the set of configurations. If the alphabet vector space and the underlying group are fixed, the set of linear cellular automata is a subalgebra of the endomorphism algebra of the configuration space (Proposition 8.1.4). An important property of linear cellular automata is that the image of a finitely supported configuration by a linear cellular automaton also has finite support (Proposition 8.2.3). Moreover, a linear cellular automaton is entirely determined by its restriction to the space of finitely-supported configurations (Proposition 8.2.4) and it is pre-injective if and only if this restriction is injective (Proposition 8.2.5). The algebra of linear cellular automata is naturally isomorphic to the group algebra of the underlying group with coefficients in the endomorphism algebra of the alphabet vector space (Theorem 8.5.2). Linear cellular automata may be also regarded as endomorphisms of the space of finitely-supported configurations, viewed as a module over the group algebra of the underlying group with coefficients in the ground field (Proposition 8.7.5). This representation of linear cellular automata is always one-to-one and, when the alphabet vector space is finite-dimensional, it is also onto (Theorem 8.7.6). The image of a linear cellular automaton is closed in the space of configurations for the prodiscrete topology, provided that the alphabet is finite dimensional (Theorem 8.8.1). We exhibit an example showing that if one drops the finite dimensionality of the alphabet, then the image of a linear cellular automaton may fail to be closed. In Sect. 8.9 we prove a linear version of the Garden of Eden theorem. For the proof, we introduce the mean dimension of a vector subspace of the configuration space. We show that for a linear cellular automaton with finite-dimensional alphabet, both pre-injectivity and surjectivity are equivalent to the maximality of the mean dimension of the image of the cellular automaton (Theorem 8.9.6). We exhibit two examples of linear cellular automata with finite-dimensional alphabet over the free group of rank two, one which is pre-injective but not surjective, and one which is surjective but T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 8, © Springer-Verlag Berlin Heidelberg 2010

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not pre-injective. This shows that the linear version of the Garden of Eden theorem fails to hold for groups containing nonabelian free subgroups (see Sects. 8.10 and 8.11). Provided the alphabet is finite dimensional, the inverse of every bijective linear cellular automaton is also a linear cellular automaton (Corollary 8.12.2). In Sect. 8.13 we study the pre-injectivity and surjectivity of the discrete Laplacian over the real numbers and prove a Garden of Eden type theorem (Theorem 8.13.2) for such linear cellular automata with no amenability assumptions on the underlying group. As an application, we deduce a characterization of locally finite groups in terms of real linear cellular automata (Corollary 8.13.4). In Sect. 8.14 we define linear surjunctivity and prove that all sofic groups are linearly surjunctive (Theorem 8.14.4). The notion of stable finiteness for rings is introduced in Sect. 8.15. A stably finite ring is a ring for which one-sided invertible square matrices are also two-sided invertible. It is shown that linear surjunctivity is equivalent to stable finiteness of the associated group algebra (Corollary 8.15.6). As a consequence, we deduce that group algebras of sofic groups are stably finite for any ground field (Corollary 8.15.8). In the last section, we prove that the absence of zero-divisors in the group algebra of an arbitrary group is equivalent to the fact that every non-identically-zero linear cellular automaton with one-dimensional alphabet is pre-injective (Corollary 8.16.12). We recall that in this book all rings are assumed to be associative (but not necessarily commutative) with a unity element, and that a field is a nonzero commutative ring in which each nonzero element is invertible.

8.1 The Algebra of Linear Cellular Automata Let G be a group and let V be a vector space over a field K. The set V G consisting of all configurations x : G → V over the group G and the alphabet V has a natural structure of vector space over K in which addition and scalar multiplication are given by (x + x )(g) = x(g) + x (g)

and

(kx)(g) = kx(g)

for all x, x ∈ V G , k ∈ K, and g ∈ G. With the prodiscrete topology, V G becomes a topological vector space (cf. Sect. F.1). The G-shift (see Sect. 1.1) is then K-linear and continuous, that is, for each g ∈ G, the map x → gx is a continuous endomorphism of V G . A linear cellular automaton over the group G and the alphabet V is a cellular automaton τ : V G → V G which is K-linear, i.e., which satisfies τ (x + x ) = τ (x) + τ (x ) for all x, x ∈ V G and k ∈ K.

and τ (kx) = kτ (x)

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285

Proposition 8.1.1. Let G be a group and let V be a vector space over a field K. Let τ : V G → V G be a cellular automaton with memory set S ⊂ G and local defining map μ : V S → V . Then τ is linear if and only if μ is K-linear. Proof. Suppose first that τ is linear. Let y, y  ∈ V S and k ∈ K. Denote by x and x two configurations in V G extending y and y  respectively, i.e., such that x|S = y and x |S = y  . We then have μ(y + y  ) = τ (x + x )(1G ) = (τ (x) + τ (x ))(1G ) = τ (x)(1G ) + τ (x )(1G ) = μ(y) + μ(y  ). Similarly, as (kx)|S = ky, we have μ(ky) = τ (kx)(1G ) = kτ (x)(1G ) = kμ(y). This shows that μ is K-linear. Conversely, suppose that μ is K-linear. Then, for all x, x ∈ V G , k ∈ K and g ∈ G we have τ (x + x )(g) = μ((g −1 (x + x ))|S ) = μ((g −1 x)|S + (g −1 x )|S ) = μ((g −1 x)|S ) + μ((g −1 x )|S ) = τ (x)(g) + τ (x )(g) = (τ (x) + τ (x ))(g) and τ (kx)(g) = μ((g −1 (kx))|S ) = μ(k(g −1 x)|S ) = kμ((g −1 x)|S ) = kτ (x)(g). This shows that τ (x + x ) = τ (x) + τ (x ) and τ (kx) = kτ (x). It follows that τ is linear.   The following result is a linear analogue of the Curtis-Hedlund theorem (Theorem 1.8.1). Theorem 8.1.2. Let G be a group and let V be a vector space over a field K. Let τ : V G → V G be a G-equivariant and K-linear map. Then the following conditions are equivalent: (a) the map τ is a linear cellular automaton; (b) the map τ is uniformly continuous (with respect to the prodiscrete uniform structure on V G ); (c) the map τ is continuous (with respect to the prodiscrete topology on V G ); (d) the map τ is continuous (with respect to the prodiscrete topology on V G ) at the constant configuration x = 0.

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Proof. The implication (a) ⇒ (b) immediately follows from Theorem 1.9.1. Since the topology associated with the prodiscrete uniform structure is the prodiscrete topology (cf. Example B.1.4(a)) and every uniformly continuous map is continuous (cf. Proposition B.2.2), we also have (b) ⇒ (c). The implication (c) ⇒ (d) is trivial. Therefore, we are only left to show that (d) ⇒ (a). Suppose that τ is continuous at 0. Then, the map V G → V defined by x → τ (x)(1G ) is continuous at 0 since the projection maps V G → V defined by x → x(g) are continuous (for the prodiscrete topology) for all g ∈ G and the composition of continuous maps is continuous. We deduce that there exists a finite subset M ⊂ G such that if x ∈ V G satisfies x(m) = 0 for all m ∈ M , then τ (x)(1G ) = 0. By linearity, we have that if two configurations x and y coincide on M then τ (x)(1G ) = τ (y)(1G ). Thus there exists a linear map μ : V M → V such that τ (x)(1G ) = μ(x|M ) for all x ∈ V G . As τ is G-equivariant, we deduce that τ (x)(g) = τ (g −1 x)(1G ) = μ((g −1 x)|M ) for all x ∈ V G and g ∈ G. This shows that τ is the (linear) cellular automaton with memory set M and local defining map μ. Thus (d) implies (a).   Examples 8.1.3. (a) Let G be a group, let S be a nonempty finite subset of G, and let K be a field. The discrete Laplacian ΔS : KG → KG (cf. Example 1.4.3(b)) is a linear cellular automaton. (b) Let G be a group, V a vector space over a field K, and f ∈ EndK (V ). Then the map τ : V G → V G defined by τ (x) = f ◦ x for all x ∈ V G is a linear cellular automaton (cf. Example 1.4.3(d)). (c) Let G be a group, V a vector space over a field K, and s0 an element of G. Let Rs0 : G → G be the right multiplication by s0 in G, that is, the map defined by Rs0 (g) = gs0 for all g ∈ G. Then the map τ : V G → V G defined by τ (x) = x ◦ Rs0 is a linear cellular automaton (cf. Example 1.4.3(e)). (d) Let G = Z and K be a field. Consider the vector space V = K[t] of all polynomials in the indeterminate t with coefficients in K. A configuration x ∈ V G may be viewed as a sequence x = (xn )n∈Z , where xn = xn (t) is a polynomial for all n ∈ Z. Let S = {0, 1} and consider the K-linear map μ : V S → V defined by μ(p, q) = p − tq  for all (p, q) ∈ V S = V × V , where q  ∈ V denotes the derivative of the polynomial q. The linear cellular automaton τ : V Z → V Z with memory set S and local defining map μ is then given by τ (x) = y, where yn = xn − txn+1 ∈ V , n ∈ Z, for all x = (xn )n∈Z ∈ V Z. We recall that an algebra over a field K (or a K-algebra) is a vector space A over K endowed with a product A × A → A such that A is a ring with respect to the sum and the product and such that the following associative law holds for the product and the multiplication by scalars: (ha)(kb) = (hk)(ab) for all h, k ∈ K and a, b ∈ A.

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287

A subset B of a K-algebra A is called a subalgebra of A if B is both a vector subspace and a subring of A. If V is a vector space over a field K, the set EndK (V ) consisting of all endomorphisms of the vector space V has a natural structure of a K-algebra for which (f + f  )(v) = f (v) + f  (v), (f f  )(v) = (f ◦ f  )(v) = f (f  (v)) and (kf )(v) = kf (v) 

for all f, f ∈ EndK (V ), k ∈ K, and v ∈ V . The identity map IdV is the unity element of EndK (V ). Given a group G and a vector space V over a field K, we denote by LCA(G; V ) the set of all linear cellular automata over the group G and the alphabet V . It immediately follows from the definition of a linear cellular automaton that LCA(G; V ) ⊂ EndK (V G ). Proposition 8.1.4. Let G be a group and let V be a vector space over a field K. Then, LCA(G; V ) is a subalgebra of EndK (V G ). Proof. Let τ1 , τ2 ∈ LCA(G; V ). Let S1 and S2 be memory sets for τ1 and τ2 . Then, the set S = S1 ∪ S2 is also a memory set for τ1 and τ2 (cf. Sect. 1.5). Let μ1 : V S → V and μ2 : V S → V be the corresponding local defining maps and set μ = μ1 + μ2 . For all x ∈ V G and g ∈ G we have (τ1 + τ2 )(x)(g) = τ1 (x)(g) + τ2 (x)(g) = μ1 (g −1 x|S ) + μ2 (g −1 x|S ) = μ(g −1 x|S ). This shows that τ1 + τ2 is a cellular automaton with memory set S and local defining map μ. Since the map τ1 + τ2 is K-linear, we deduce that τ1 + τ2 ∈ LCA(G; V ). On the other hand, let k ∈ K and let τ ∈ LCA(G; V ) with memory set S and local defining map μ : V S → V . Then, for all x ∈ V G and g ∈ G, we have (kτ )(x)(g) = kτ (x)(g) = kμ(g −1 x|S ) = (kμ)(g −1 x|S ). Therefore the K-linear map kτ is a cellular automaton with memory set S and local defining map kμ. It follows that kτ ∈ LCA(G; V ). We clearly have IdV G ∈ LCA(G; V ) (cf. Example 1.4.3(d)). Finally, it follows from Proposition 1.4.9 that if τ1 , τ2 ∈ LCA(G; V ) then the K-linear map τ1 τ2 = τ1 ◦ τ2 is also a cellular automaton and hence τ1 τ2 ∈ LCA(G; V ).   This shows that LCA(G; V ) is a subalgebra of EndK (V G ).

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8 Linear Cellular Automata

8.2 Configurations with Finite Support Let G be a group and let V be a vector space over a field K. The support of a configuration x ∈ V G is the set {g ∈ G : x(g) = 0V }. We denote by V [G] the subset of V G consisting of all configurations with finite support. Proposition 8.2.1. Let G be a group and let V be a vector space over a field K. Then the set V [G] is a vector subspace of V G . Moreover, V [G] is dense in V G for the prodiscrete topology. Proof. If k1 , k2 ∈ K and x1 , x2 ∈ V G , then the support of k1 x1 + k2 x2 is contained in the union of the support of x1 and the support of x2 . Therefore, if x1 and x2 have finite support, so does k1 x1 + k2 x2 . Consequently, V [G] is a vector subspace of V G . Let x ∈ V G and let W ⊂ V G be a neighborhood of x for the prodiscrete topology. By definition of the prodiscrete topology, there exists a finite subset Ω ⊂ G such that W contains all configurations which coincide with x on Ω. It follows that the configuration y ∈ V [G] which coincides with x on Ω and is identically zero outside of Ω is in W . This shows that V [G] is   dense in V G . Proposition 8.2.2. Let G be a group and let V be a vector space over a field K. Let x, x ∈ V G . Then the configurations x and x are almost equal if and only if x − x ∈ V [G]. Proof. By definition, x and x are almost equal if and only if the set {g ∈ G : x(g) = x (g)} is finite. This is equivalent to x − x ∈ V [G] since {g ∈ G :   x(g) = x (g)} = {g ∈ G : (x − x )(g) = 0V }. Proposition 8.2.3. Let G be a group and let V be a vector space over a field K. Let τ ∈ LCA(G; V ). Then one has τ (V [G]) ⊂ V [G]. Proof. Denote by S ⊂ G a memory set for τ and let μ : V S → V be the corresponding local defining map. Let x ∈ V [G] and let T ⊂ G denote the support of x. For all g ∈ G, we have τ (x)(g) = μ((g −1 x)|S ). As μ is linear (Proposition 8.1.1) and the support of g −1 x is g −1 T , we deduce that τ (x)(g) = 0 if g −1 T ∩ S = ∅. It follows that the support of τ (x) is contained in the finite   set T S −1 ⊂ G. This shows that τ (x) ∈ V [G]. Observe that if τ ∈ LCA(G; V ), then the restriction map τ |V [G] : V [G] → V [G] is K-linear, that is, τ |V [G] ∈ EndK (V [G]).

8.3 Restriction and Induction of Linear Cellular Automata

289

Let A and B be two algebras over a field K. A map F : A → B is called a K-algebra homomorphism if F is both a vector space homomorphism (i.e., a K-linear map) and a ring homomorphism. This is equivalent to the fact that F satisfies F (a + a ) = F (a) + F (a ), F (aa ) = F (a)F (a ), F (ka) = kF (a) for all a, a ∈ A and k ∈ K, and F (1A ) = 1B . Proposition 8.2.4. Let G be a group and let V be a vector space over a field K. Then the map Λ : LCA(G; V ) → EndK (V [G]) defined by Λ(τ ) = τ |V [G] , where τ |V [G] : V [G] → V [G] is the restriction of τ to V [G], is an injective K-algebra homomorphism. Proof. The fact that Λ is an algebra homomorphism immediately follows from the definition of the algebra operations on LCA(G; V ) and EndK (V [G]). Suppose that τ ∈ LCA(G; V ) satisfies Λ(τ ) = 0. This means that τ (y) = 0 for all y ∈ V [G]. As τ : V G → V G is continuous by Proposition 1.4.8 and V [G] is dense in V G by Proposition 8.2.1 for the prodiscrete topology, we deduce that τ (x) = 0 for all x ∈ V G , that is, τ = 0. This shows that Λ is injective.   Proposition 8.2.5. Let G be a group and let V be a vector space over a field K. Let τ ∈ LCA(G; V ). Then the following conditions are equivalent: (a) τ is pre-injective; (b) τ |V [G] : V [G] → V [G] is injective. Proof. Suppose that τ is pre-injective. Let x ∈ ker(τ |V [G] ). Then x ∈ V [G] and τ (x) = τ |V [G] (x) = 0. As the trivial configuration 0 and the configuration x are almost equal and τ (0) = 0 by linearity of τ , the pre-injectivity of τ implies that x = 0. This shows that τ |V [G] is injective. Conversely, suppose that τ |V [G] is injective. Let x, x ∈ V G be two configurations which are almost equal and such that τ (x) = τ (x ). Then x−x ∈ V [G] by Proposition 8.2.2 and we have τ |V [G] (x−x ) = τ (x−x ) = τ (x)−τ (x ) = 0. As τ |V [G] is injective, this implies x − x = 0, that is, x = x . Therefore, τ is pre-injective.  

8.3 Restriction and Induction of Linear Cellular Automata In this section, we show that the operations of restriction and induction for cellular automata that were introduced in Sect. 1.7 preserve linearity. Let G be a group and let V be a vector space over a field K. Let H be a subgroup of G. We denote by LCA(G, H; V ) = LCA(G; V ) ∩ CA(G, H; V ) the set of all linear cellular automata τ : V G → V G admitting a memory set S such that S ⊂ H. Recall from Sect. 1.7, that, given a cellular automaton τ : V G → V G with memory set S ⊂ H, we denote by τH : V H → V H

290

8 Linear Cellular Automata

the restriction cellular automaton. Similarly, given a cellular automaton σ : V H → V H , we denote by σ G : V G → V G the induced cellular automaton. Proposition 8.3.1. Let τ ∈ CA(G, H; V ). Then, τ ∈ LCA(G, H; V ) if and only if τH ∈ LCA(H; V ). Proof. It follows immediately from the definitions of restriction and induction that a cellular automaton τ ∈ CA(G, H; V ) is linear if and only if τH ∈ CA(H; V ) is linear.   Let A and B be two algebras over a field K. A map F : A → B is called a K-algebra isomorphism if F is a bijective K-algebra homomorphism. It is clear that if F : A → B is a K-algebra isomorphism then its inverse map F −1 : B → A is also a K-algebra isomorphism. Proposition 8.3.2. The set LCA(G, H; V ) is a subalgebra of LCA(G; V ). Moreover, the map τ → τH is a K-algebra isomorphism from LCA(G, H; V ) onto LCA(H; V ) whose inverse is the map σ → σ G . Proof. Let τ1 , τ2 ∈ LCA(G, H; V ) with memory sets S1 , S2 ⊂ H respectively. Then the linear cellular automaton τ1 + τ2 admits S1 ∪ S2 as a memory set. As S1 ∪ S2 ⊂ H, we have τ1 + τ2 ∈ LCA(G, H; V ). If k ∈ K and τ ∈ LCA(G, H; V ), with memory set S ⊂ H, then S is also a memory set for kτ and therefore kτ ∈ LCA(G, H; V ). This shows that LCA(G, H; V ) is a vector subspace of LCA(G; V ). Since LCA(G, H; V ) is a submonoid of LCA(G; V ) by Proposition 1.7.1, we deduce that LCA(G, H; V ) is a subalgebra of LCA(G; V ). To simplify notation, denote by Φ : LCA(G, H; V ) → LCA(H; V ) and Ψ : LCA(H; V ) → LCA(G, H; V ) the maps defined by Φ(τ ) = τH and Ψ (σ) = σ G respectively. It is clear from the definitions that Ψ ◦ Φ : LCA(G, H; V ) → LCA(G, H; V ) and Φ ◦ Ψ : LCA(H; V ) → LCA(H; V ) are the identity maps. Therefore, Φ is bijective with inverse Ψ . It remains to show that Φ is a Kalgebra homomorphism.  ∈ VG Let τ1 , τ2 ∈ LCA(G, H; V ) and k1 , k2 ∈ K. Let x ∈ V H and let x extending x. By applying (1.13), we have x)(h) = k1 τ1 ( x)(h) + k2 τ2 ( x)(h) Φ(k1 τ1 + k2 τ2 )(x)(h) = (k1 τ1 + k2 τ2 )( for all h ∈ H. We deduce that Φ(k1 τ1 + k2 τ2 )(x) = (k1 Φ(τ1 ) + k2 Φ(τ2 ))(x) for all x ∈ V H , that is, Φ(k1 τ1 + k2 τ2 ) = k1 Φ(τ1 ) + k2 Φ(τ2 ). This shows that Φ is K-linear. Finally, it follows from Proposition 1.7.2 that Φ(IdV G ) = IdV H and Φ(τ1 τ2 ) = Φ(τ1 )Φ(τ2 ) for all τ1 , τ2 ∈ LCA(G, H; V ). We have shown that Φ is a K-algebra isomorphism.  

8.4 Group Rings and Group Algebras

291

8.4 Group Rings and Group Algebras Let G be a group and let R be a ring. We denote by 0R (resp. 1R ) the zero (resp. unity) element of R. We regard R as a left R-module over itself. Then RG = {α : G → R} has a natural structure of left R-module with addition and scalar multiplication given by (α + β)(g) = α(g) + β(g)

and (rα)(g) = rα(g)

for all α, β ∈ RG , r ∈ R, and g ∈ G. Define the support of an element α ∈ RG as being the set {g ∈ G : α(g) = 0R }. Let R[G] denote the set consisting of all elements α ∈ RG which have finite support. Then R[G] is a free submodule of RG . We have   R⊂ R = RG , R[G] = g∈G

g∈G

so that the elements δg : G → R, g ∈ G, defined by  1R if h = g δg (h) = 0R if h = g

(8.1)

freely generate R[G] as a left R-module. Note that the decomposition of an element α ∈ R[G] in this basis is simply given by the formula  α= α(g)δg . g∈G

Let now α and β be two elements of R[G] and denote their supports by S and T . The convolution product of α and β is the element αβ ∈ RG defined by   (αβ)(g) = α(h)β(h−1 g) = α(h)β(h−1 g) (8.2) h∈G

h∈S

for all g ∈ G. Note that αβ ∈ R[G] as the support of αβ is contained in ST = {st : s ∈ S, t ∈ T }. By using the change of variables h1 = h and h2 = h−1 g, the convolution product (8.2) may be also expressed as follows:  (αβ)(g) = α(h1 )β(h2 ). (8.3) h1 ,h2 ∈G h1 h2 =g

Proposition 8.4.1. Let G be a group and let R be a ring. Then the addition and the convolution product gives a ring structure to R[G]. Proof. We know that (R[G], +) is an abelian group. Let α, β, γ ∈ R[G] and g ∈ G. We have

292

[(αβ)γ](g) =



8 Linear Cellular Automata

(αβ)(h)γ(h−1 g)

h∈G

=



α(k)β(k−1 h)γ(h−1 g)

h∈G k∈G

(by setting s = k

−1

h) =



α(k)β(s)γ(s−1 k −1 g)

s∈G k∈G

=



α(k)(

=

β(s)γ(s−1 k −1 g))

s∈G

k∈G





α(k)(βγ)(k−1 g)

k∈G

= [α(βγ)](g). This shows that (αβ)γ = α(βγ). Thus, the convolution product is associative. On the other hand, given α ∈ R[G] we have, for all g ∈ G,   δ1G (h)α(h−1 g) = α(g) = α(k)δ1G (k−1 g) = [αδ1G ](g). [δ1G α](g) = h∈G

k∈G

Thus, δ1G α = αδ1G = α for all α ∈ R[G]. This shows that δ1G is an identity element for the convolution product. Finally,  (α + β)(h)γ(h−1 g) [(α + β)γ](g) = h∈G

=



[α(h) + β(h)]γ(h−1 g)

h∈G

=



α(h)γ(h−1 g) +

h∈G



β(k)γ(k−1 g)

k∈G

= (αγ)(g) + (βγ)(g). Thus, we have (α + β)γ = αγ + βγ. Similarly, one shows that α(β + γ) = αβ +αγ. It follows that the distributive laws also hold in R[G]. Consequently,   R[G] is a ring with unity element 1R[G] = δ1G . The ring R[G] is called the group ring of G with coefficients in R. Given a ring R, denote by U (R) the multiplicative group consisting of all invertible elements in R. Proposition 8.4.2. Let G be a group and let R be a ring. Then one has δg ∈ U(R[G]) for all g ∈ G. Moreover the map φ : G → U(R[G]) given by φ(g) = δg is a group homomorphism.  Proof. Let g1 , g2 , g ∈ G. Then (δg1 δg2 )(g) = h∈G δg1 (h)δg2 (h−1 g) equals 1 if g1−1 g = g2 , that is, if g = g1 g2 , and equals 0 otherwise. This shows that δ g1 δg2 = δg1 g2 .

(8.4)

8.4 Group Rings and Group Algebras

293

We deduce that δg δg−1 = δgg−1 = δ1G = 1R[G] and, similarly, δg−1 δg = 1R[G] . This shows that δg belongs to U(R[G]). The fact that φ is a group homomorphism follows from (8.4).   Let G be a group and let R be a ring. For α ∈ R[G] define α∗ : G → R by setting α∗ (g) = α(g −1 ) for all g ∈ G. If S is the support of α, then the support of α∗ is S −1 . Thus one has α∗ ∈ R[G]. Proposition 8.4.3. Let G be a group and let R be a ring. Let α, β ∈ R[G]. Then one has (i) (1R[G] )∗ = 1R[G] , (ii) (α∗ )∗ = α, (iii) (α + β)∗ = α∗ + β ∗ . Moreover, if the ring R is commutative, then one has (iv) (αβ)∗ = β ∗ α∗ . Proof. (i) We have (1R[G] )∗ = 1R[G] since the support of 1R[G] = δ1G is {1G }. (ii) For all g ∈ G, we have (α∗ )∗ (g) = α((g −1 )−1 ) = α(g). Therefore (α∗ )∗ = α. (iii) For all g ∈ G, we have (α+β)∗ (g) = (α+β)(g −1 ) = α(g −1 )+β(g −1 ) = α∗ (g)+β ∗ (g) = (α∗ +β ∗ )(g). Therefore (α + β)∗ = α∗ + β ∗ . (iv) Suppose that the ring R is commutative. For all g ∈ G, we have (αβ)∗ (g) = (αβ)(g −1 )  = α(h)β(h−1 g −1 ) h∈G

=



β(h−1 g −1 )α(h)

(since R is commutative)

h∈G

=



β ∗ (gh)α∗ (h−1 )

h∈G

=



β ∗ (k)α∗ (k−1 g)

k∈G

= (β ∗ α∗ )(g). Therefore (αβ)∗ = β ∗ α∗ .

 

If R = (R, +, ·) is a ring, its opposite ring is the ring Rop = (R, +, ∗) having the same underlying set and addition as R and with multiplication ∗ defined by r ∗ s = sr for all r, s ∈ R.

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8 Linear Cellular Automata

From Proposition 8.4.3, we deduce that, if G is a group and R is a commutative ring, then the map α → α∗ is a ring isomorphism between the ring R[G] and its opposite ring (R[G])op . Thus we have Corollary 8.4.4. Let G be a group and let R be a commutative ring. Then   the ring R[G] is isomorphic to its opposite ring (R[G])op . Remarks 8.4.5. Let G be a group and let R be a ring. (a) It is immediate to verify that the map R → R[G] defined by r → rδ1G is an injective ring homomorphism. This is often used to regard R as a subring of R[G]. (b) Observe that (rα)β = r(αβ) for all r ∈ R and α, β ∈ R[G]. Therefore, we can use the notation rαβ = (rα)β = r(αβ). If the ring R is commutative, then one has the additional property (r1 α)(r2 β) = r1 r2 αβ for all r1 , r2 ∈ R and α, β ∈ R[G]. (c) Suppose that the group G is abelian and that the ring R is commutative. Then the ring R[G] is commutative. Indeed, it immediately follows from (8.3) that we have αβ = βα for all α, β ∈ R[G] under these hypotheses. Suppose now that we are given an algebra A over a field K. Then A[G] is both a K-vector space and a ring; Moreover, we have (k1 α)(k2 β) = k1 k2 αβ for all k1 , k2 ∈ K and α, β ∈ A[G]. Therefore, A[G] is an algebra over K. The K-algebra A[G] is called the group algebra of G with coefficients in the K-algebra A. In the particular case A = K, this gives the K-algebra K[G].

8.5 Group Ring Representation of Linear Cellular Automata Let G be a group and let V be a vector space over a field K. Consider the K-algebra EndK (V )[G], that is, the group algebra of G with coefficients in the K-algebra EndK (V ) (see Sect. 8.4). For each α ∈ EndK (V )[G], we define a map τα : V G → V G by setting  τα (x)(g) = α(h)(x(gh)) (8.5) h∈G

for all x ∈ V G and g ∈ G. Observe that α(h) ∈ EndK (V ) and x(gh) ∈ V for all x ∈ V G and g, h ∈ G. Note also that there is only a finite number of nonzero terms in the sum appearing in the right hand side of (8.5) since α has finite support. It follows that τα is well defined. For each v ∈ V , define the configuration cv ∈ V [G] by  v if g = 1G , (8.6) cv (g) = 0 otherwise.

8.5 Group Ring Representation of Linear Cellular Automata

295

Note that the map from V to V [G] given by v → cv is injective and K-linear. Proposition 8.5.1. Let α ∈ EndK (V )[G]. Then one has: (i) τα ∈ LCA(G; V ); (ii) α(g)(v) = τα (cv )(g) for all v ∈ V and g ∈ G; (iii) the support of α is the minimal memory set of τα . Proof. Let S ⊂ G denote the support of α. Consider the map μ : V S → V given by  μ(y) = α(s)(y(s)) s∈S

for all y ∈ V . Then, for all x ∈ V S

and g ∈ G, we have  τα (x)(g) = α(h)(x(gh)) G

h∈G

=



α(h)(g −1 x(h)) (8.7)

h∈G

=



α(s)(g

−1

x(s))

s∈S

= μ((g −1 x)|S ). Thus τα is the cellular automaton with memory set S and local defining map μ. It is clear from (8.5) that τα is a K-linear map. Thus, we have τα ∈ LCA(G; V ). This shows (i). Let v ∈ V and let cv ∈ V [G] as in (8.6). By applying (8.5), we get, for all g ∈ G,  α(h)(cv (g −1 h)) = α(g)(v). τα (cv )(g −1 ) = h∈G

This gives us (ii). Finally, let S0 ⊂ G denote the minimal memory set of τα . We have seen in the proof of (i) that S is a memory set for τα . Therefore, we have S0 ⊂ S. On the other hand, we deduce from (ii) that, for all v ∈ V and g ∈ G, we have α(g)(v) = τα (cv )(g) = μ0 ((g −1 cv )|S0 ), where μ0 : V S0 → V denote the local defining map for τα associated with S0 . Since the configuration g −1 cv is identically zero on G \ {g}, it follows that α(g) = 0 if g ∈ / S0 . This implies S ⊂ S0 . Thus, we have S = S0 . This shows (iii).   Theorem 8.5.2. Let G be a group and let V be a vector space over a field K. Then the map Ψ : EndK (V )[G] → LCA(G; V ) defined by α → τα is a Kalgebra isomorphism. Proof. Let α, β ∈ EndK (V )[G] and k ∈ K. By applying (8.5), we get

296

τα+β (x)(g) =



8 Linear Cellular Automata

[(α + β)(h)](x(gh))

h∈G

=



(α(h) + β(h))(x(gh))

h∈G

=



[α(h)(x(gh)) + β(h)(x(gh))]

h∈G

=



α(h)(x(gh)) +

h∈G



β(h)(x(gh))

h∈G

= τα (x)(g) + τβ (x)(g) and, similarly, τkα (x)(g) =



[(kα)(h)](x(gh))

h∈G

=



k[α(h)](x(gh))

h∈G

=k



[α(h)](x(gh))

h∈G

= kτα (x)(g) for all x ∈ V G and g ∈ G. Thus τα+β = τα + τβ and τkα = kτα . This shows that Ψ is a K-linear map. Let us show that Ψ is a ring homomorphism. Let α, β ∈ EndK (V )[G]. Let x ∈ V G and set y = τβ (x). For all g, h ∈ G, we have  β(t)(x(ght)). (8.8) y(gh) = t∈G

It follows that τα (τβ (x))(g) = τα (y)(g)  = α(h)(y(gh)) h∈G

(by (8.8)) =





α(h)

t∈G

h∈G

(by setting z = ht) =





α(h)

h∈G

=

 

z∈G

=



β(t)(x(ght))

β(h−1 z)(x(gz))

z∈G

α(h)β(h−1 z) (x(gz))

h∈G

[αβ](z)(x(gz))

z∈G

= ταβ (x)(g).

8.5 Group Ring Representation of Linear Cellular Automata

297

Thus we have ταβ = τα ◦ τβ , that is, Ψ (αβ) = Ψ (α)Ψ (β). Observe that 1EndK (V )[G] = δ1G , where δ1G : G → EndK (V ) is given by δ1G (h) = IdV if h = 1G and δ1G (h) = 0 otherwise. Thus, if x ∈ V G we have Ψ (1EndK (V )[G] )(x)(g) = Ψ (δ1G )(x)(g)  = δ1G (h)(x(gh)) h∈G

= x(g) for all g ∈ G. It follows that Ψ (1EndK (V )[G] ) = IdV G . This proves that Ψ is a ring homomorphism. Let us show now that Ψ is injective. Let α ∈ ker(Ψ ). Then, τα (x) = 0

(8.9)

for all x ∈ V G . Taking x = cv in (8.9), where, for v ∈ V , the element cv ∈ V [G] is as in (8.6), we deduce from Proposition 8.5.1(ii) that α(g)(v) = 0 for all v ∈ V and g ∈ G. Thus, we have α(g) = 0 for all g ∈ G and therefore α = 0. It follows that ker(Ψ ) = {0}, that is, Ψ is injective. Let us show that Ψ is surjective. Suppose that τ ∈ LCA(G; V ) has memory set S and local defining map μ : V S → V . As μ is K-linear (cf. Proposition  8.1.1), there exist K-linear maps αs : V → V , s ∈ S, such that μ(y) = s∈S αs (y(s)) for all y ∈ V S . For all x ∈ V G and g ∈ G, we have τ (x)(g) = μ((g −1 x)|S ) =



 αs g −1 x(s) = αs (x(gs)) .

s∈S

s∈S

This shows that τ = τα , where α ∈ EndK (V )[G] is defined by  αs if h = s ∈ S, α(h) = 0 otherwise.  

Thus Ψ is surjective.

If the vector space V is one-dimensional over the field K, then each endomorphism of V is of the form v → kv, for some k ∈ K. It follows that the K-algebra EndK (V ) is canonically isomorphic to K in this case. Thus, we get: Corollary 8.5.3. Let G be a group and let V be a one-dimensional vector space over a field K. Then the map Ψ : K[G] → LCA(G; V ) defined by  Ψ (α)(x)(g) = α(h)x(gh) h∈G

for all α ∈ K[G], x ∈ V G , and g ∈ G, is a K-algebra isomorphism.

 

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8 Linear Cellular Automata

When G is an abelian group, then the K-algebra K[G] is commutative for any field K by Remark 8.4.5(c). Thus, as an immediate consequence of the preceding corollary, we get: Corollary 8.5.4. Let G be an abelian group and let V be a onedimensional vector space over a field K. Then the K-algebra LCA(G; V ) is commutative.   Examples 8.5.5. (a) Let G be a group, S a nonempty finite subset of G, and let K be a field. Consider the element  δs ∈ K[G], α = |S|δ1G − s∈S

Then, taking V = K, we have, for all x ∈ KG and g ∈ G,   Ψ (α)(x)(g) = α(h)x(gh) = |S|x(g) − x(gs). s∈S

h∈G

It follows that Ψ (α) is the discrete Laplacian ΔS : KG → KG associated with S (cf. Example 8.1.3(a)). (b) Let G be a group, V a vector space over a field K and f ∈ EndK (V ). Consider the element α ∈ EndK (V )[G] defined by α = f δ1G . Let x ∈ V G and g ∈ G. We have  α(h)(x(gh)) = f (x(g)). Ψ (α)(x)(g) = h∈G

It follows that Ψ (α) is the linear cellular automaton τ ∈ LCA(G; V ) defined by τ (x) = f ◦ x for all x ∈ V G (cf. Example 8.1.3(b)). (c) Let G be a group, V a vector space over a field K, and s0 an element of G. Consider the group algebra element α ∈ EndK (V )[G] defined by α = δs0 . For all x ∈ V G and g ∈ G, we have  Ψ (α)(x)(g) = α(h)(x(gh)) = x(gs0 ). h∈G

It follows that Ψ (α) is the linear cellular automaton τ ∈ LCA(G; V ) defined by τ (x) = x ◦ Rs0 , where Rs0 : G → G is the right multiplication by s0 (cf. Example 8.1.3(c)). (d) Let G = Z, let K be a field, and let V = K[t] be the vector space consisting of all polynomials in the indeterminate t with coefficients in K. Denote by D and U the elements in EndK (V ) defined respectively by D(p) = p and U (p) = tp, for all p ∈ V . Consider the element α = δ0 − (U ◦ D)δ1 ∈ EndK (V )[Z].

8.6 Modules over a Group Ring

299

For all x = (xn )n∈Z ∈ V Z , we have Ψ (α)(x) = (yn )n∈Z ∈ V Z , where  yn = α(m)(xn+m ) = xn − txn+1 m∈Z

for all n ∈ Z. It follows that Ψ (α) is the linear cellular automaton τ ∈ LCA(Z; K[t]) described in Example 8.1.3(d).

8.6 Modules over a Group Ring Let R be a ring and let M be a left R-module. We denote by EndR (M ) the endomorphism ring of M . Recall that EndR (M ) is the set consisting of all maps f : M → M satisfying f (x + x ) = f (x) + f (x ) and

f (rx) = rf (x)

for all r ∈ R and x, x ∈ M . The ring operations in EndR (M ) are given by (f + f  )(x) = f (x) + f  (x)

and

(f f  )(x) = (f ◦ f  )(x) = f (f  (x))

for all f, f  ∈ EndR (M ) and x ∈ M , and the unity element of EndR (M ) is the identity map IdM . Suppose that there is a group G which acts on M by endomorphisms. We then define a structure of left R[G]-module on M as follows. For α ∈ R[G] and x ∈ M , define the element αx ∈ M by  α(g)gx. (8.10) αx = g∈G

Note that this definition makes sense. Indeed, one has α(g) ∈ R and gx ∈ M for each g ∈ G. On the other hand, there is only finitely many nonzero terms in the right hand side of (8.10) since the support of α is finite. Proposition 8.6.1. One has: (i) 1R[G] x = x, (ii) α(x + x ) = αx + αx , (iii) (α + β)x = αx + βx, (iv) α(βx) = (αβ)x for all α, β ∈ R[G] and x, x ∈ M . Proof. (i) We have 1R[G] x = δ1G x = x.

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8 Linear Cellular Automata

(ii) We have α(x + x ) =



α(g)g(x + x ) =

g∈G

=



α(g)gx +

g∈G



(α(g)gx + α(g)gx )

g∈G



α(g)gx = αx + αx .

g∈G

(iii) We have (α + β)x =



(α + β)(g)gx =

g∈G

=



α(g)gx +

g∈G





(α(g) + β(g))gx

g∈G

β(g)gx = αx + βx.

g∈G

(iv) We have, by using (8.3),  α(βx) = α(h1 )h1 (βx) h1 ∈G

=



α(h1 )h1



h1 ∈G

=

 

h2 ∈G

α(h1 )β(h2 )h1 h2 x

h1 ∈G h2 ∈G

=

 

g∈G

=



β(h2 )h2 x

α(h1 )β(h2 ) gx

h1 ,h2 ∈G h1 h2 =g

(αβ)(g)gx

g∈G

= (αβ)x.   It follows from Proposition 8.6.1 that the addition on M and the multiplication (α, x) → αx defined by (8.10) gives us a left R[G]-module structure on M . Observe that this R[G]-module structure extends the R-module structure on M if we regard R as a subring of R[G] via the map r → rδ1G (cf. Remark 8.4.5(a)). Proposition 8.6.2. Let f : M → M be a map. Then the following conditions are equivalent: (a) f ∈ EndR[G] (M ); (b) f ∈ EndR (M ) and f is G-equivariant. Proof. Suppose that f satisfies (b). Then f (x + x ) = f (x) + f (x ) for all x, x ∈ M since f ∈ EndR (M ). On the other hand, if α ∈ R[G] and x ∈ M ,

8.7 Matrix Representation of Linear Cellular Automata

301

we have, by using the G-equivariance and the R-linearity of f ,

   α(g)gx = α(g)f (gx) = α(g)gf (x) = αf (x). f (αx) = f g∈G

g∈G

g∈G

It follows that f ∈ EndR[G] (M ). This shows that (b) implies (a). Conversely, suppose that f satisfies (a). Let x, x ∈ M and r ∈ R. Then we have f (x + x ) = f (x) + f (x ), f (rx) = f (rδ1G x) = rδ1G f (x) = rf (x), and f (gx) = f (δg x) = δg f (x) = gf (x) since f ∈ EndR[G] (M ). This shows   that f ∈ EndR (M ) and that f is G-equivariant. Thus, (a) implies (b). Remark 8.6.3. Conversely, suppose that we are given a left R[G]-module M . Then, by applying the above construction, we recover the initial R[G]-module structure on M if we start from the R-module structure on M obtained by restricting the scalars and the R-linear action of G on M defined by gx = δg x for all g ∈ G and x ∈ M .

8.7 Matrix Representation of Linear Cellular Automata Let G be a group and let V be a vector space over a field K. Since the G-shift action on V G is K-linear, it induces a structure of left K[G]-module on V G which extends the K-vector space structure on V G (see Sect. 8.6). Note that by applying (8.10) we get, for all α ∈ K[G] and x ∈ V G ,  αx = α(h)hx, (8.11) h∈G

that is, (αx)(g) =



α(h)x(h−1 g)

h∈G

for all g ∈ G. Thus, αx may be regarded as the “convolution product” of α and x (compare with (8.2)). Let τ : V G → V G be a linear cellular automaton. Then τ is K-linear by definition. On the other hand, τ is G-equivariant by Proposition 1.4.4. Thus, it follows from Proposition 8.6.2 that τ is an endomorphism of the K[G]-module V G . Consequently, we have LCA(G; V ) ⊂ EndK[G] (V G ). It is clear that EndK[G] (V G ) is a subalgebra of the K-algebra EndK (V G ). Since LCA(G; V ) is a subalgebra of EndK (V G ) by Proposition 8.1.4, we get: Proposition 8.7.1. The set LCA(G; V ) is a subalgebra of the K-algebra EndK[G] (V G ).   Remark 8.7.2. When G is a finite group, we have LCA(G; V ) = EndK[G] (V G ). Indeed, in this case, every u ∈ EndK[G] (V G ) is a linear cellular automaton

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8 Linear Cellular Automata

with memory set G and local defining map μ : V G → V given by μ(y) = u(y)(1G ), since u(x)(g) = g −1 u(x)(1G ) = u(g −1 x)(1G ) for all x ∈ V G and g ∈ G. Consider now the vector subspace V [G] ⊂ V G consisting of all configurations with finite support. Observe that if x ∈ V G has support Ω ⊂ G and g ∈ G, then the support of the configuration gx is gΩ. Therefore, V [G] is a submodule of the K[G]-module V G . Recall that, for v ∈ V , the configuration cv ∈ V [G] is defined by  v if g = 1G , cv (g) = 0 otherwise. Note that if g ∈ G and v ∈ V , then gcv = δg cv is the configuration which takes the value v at g and is identically 0 on G \ {g}. It follows that every x ∈ V [G] can be written as  x= gcx(g) . (8.12) g∈G

Proposition 8.7.3. Suppose that (ei )i∈I is a basis of the K-vector space V . Then the family of configurations (cei )i∈I is a free basis for the left K[G]module V [G]. Proof. Let x ∈ V [G]. As (ei )i∈I is a K-basis for V , we can find elements αi ∈ K[G], i ∈ I, such that  x(g) = αi (g)ei i∈I

for all g ∈ G. This gives us x=



gcx(g)

g∈G

=

  g αi (g)cei

g∈G

= = =

i∈I

 

g∈G

i∈I

i∈I

g∈G

   i∈I



αi (g)gcei

αi (g)gcei

αi cei ,

8.7 Matrix Representation of Linear Cellular Automata

303

where the last equality follows from (8.11). If x = 0 ∈ V [G], then x(g) = 0 for all g ∈ G and therefore αi = 0 for all i ∈ I. This shows that (cei )i∈I is a basis for the left K[G]-module V [G].   As every vector space admits a basis, we deduce the following Corollary 8.7.4. The left K[G]-module V [G] is free.

 

If τ : V → V is a linear cellular automaton, we have τ (V [G]) ⊂ V [G] by Proposition 8.2.3. Moreover, it follows from Proposition 8.2.4 that the map Λ : LCA(G; V ) → EndK (V [G]) which associates with each τ ∈ LCA(G; V ) its restriction τ |V [G] : V [G] → V [G] is an injective homomorphism of K-algebras. As we have τ |V [G] ∈ EndK[G] (V [G]) ⊂ EndK (V [G]) for all τ ∈ LCA(G; V ), we get: G

G

Proposition 8.7.5. Let G be a group and let V be a vector space over a field K. Then the map Φ : LCA(G; V ) → EndK[G] (V [G]) defined by Φ(τ ) = τ |V [G] , where τ |V [G] : V [G] → V [G] is the restriction of τ to V [G], is an injective K-algebra homomorphism.   When the alphabet is finite-dimensional, we have the following: Theorem 8.7.6. Let G be a group and let V be a finite-dimensional vector space over a field K. Then the map Φ : LCA(G; V ) → EndK[G] (V [G]) defined by Φ(τ ) = τ |V [G] , where τ |V [G] : V [G] → V [G] is the restriction of τ to V [G], is a K-algebra isomorphism. Proof. By the preceding proposition, it suffices to show that Φ is surjective. Let u ∈ EndK[G] (V [G]). Consider an element x ∈ V [G]. We have x=



gcx(g)

g∈G

by (8.12). This implies u(x) = u



gcx(g)

g∈G

=



gu(cx(g) )

g∈G

Since u is K[G]-linear. We deduce that  u(x)(1G ) = u(cx(g) )(g −1 ).

(8.13)

g∈G

Suppose now that dimK (V ) = d and let e1 , e2 , . . . , ed be a K-basis for V . Denote by Ti ⊂ G the support

of u(cei ), 1 ≤ i ≤ d, and consider the finite subset S ⊂ G defined by S = 1≤i≤d Ti−1 . d If v ∈ V , we can write v = i=1 ki ei with ki ∈ K, 1 ≤ i ≤ d. We then d have cv = i=1 ki cei and hence

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8 Linear Cellular Automata

u(cv ) = u

 d

ki cei

i=1

=

d 

ki u(cei ).

i=1

We deduce that u(cv )(g −1 ) = 0 if g ∈ / S. Thus, using (8.13), we get  u(x)(1G ) = u(cx(g) )(g −1 ). g∈S

This shows that u(x)(1G ) only depends on the restriction of x to S. More precisely, there is a K-linear map μ : V S → V such that u(x)(1G ) = μ(x|S )

for all x ∈ V [G].

Using the K[G]-linearity of u, we get u(x)(g) = (g −1 u(x))(1G ) = u(g −1 x)(1G ) = μ((g −1 x)|S ) for all x ∈ V [G] and g ∈ G. Therefore u is the restriction to V [G] of the linear cellular automaton τ : V G → V G with memory set S and local defining map μ. This shows that Φ is surjective.   Remarks 8.7.7. (a) When G is a finite group and V is a (not necessarily finite-dimensional) vector space over a field K, one has V [G] = V G and Φ : LCA(G; V ) → EndK[G] (V [G]) = EndK[G] (V G ) is the identity map by Remark 8.7.2. (b) Suppose now that G is an infinite group and that V is an infinitedimensional vector space over a field K. Then the map Φ : LCA(G; V ) → EndK[G] (V [G]) is not surjective. To see this, take a K-basis (ei )i∈I for V . As G and I are infinite, we can find a family (Ti )i∈I of finite subsets of G such that i∈I Ti is infinite. Choose, for each i ∈ I, a configuration zi ∈ V [G] whose support is Ti . It follows from Proposition 8.7.3 that V [G] is a free left K[G]-module admitting the family (cei )i∈I as a basis. Therefore, there is an element u ∈ EndK[G] (V [G]) such that u(cei ) = zi for all i ∈ I. Let us show that u is not in the image of Φ. We proceed by contradiction. Suppose that u is in the image of Φ. This means that there is a linear cellular automaton τ : V G → V G such that u = τ |V [G] . If S is a memory set for τ , this implies that, for x ∈ V [G], the value of u(x)(1G ) depends only on the restriction of x

to S. As S is finite and i∈I Ti−1 is infinite, we can find elements g0 ∈ G and / S and g0 ∈ Ti−1 . Consider the configuration x0 ∈ V [G] i0 ∈ I such that g0 ∈ 0 which takes the value ei0 at g0 and is identically 0 on G \ {g0 }. Observe that x0 = g0 cei0 , so that u(x0 ) = u(g0 cei0 ) = g0 u(cei0 ) = g0 zi0 .

8.8 The Closed Image Property

Thus, we have

305

u(x0 )(1G ) = (g0 zi0 )(1G ) = zi0 (g0−1 ).

We deduce that u(x0 )(1G ) = 0 as g0−1 is in the support Ti0 of zi0 . This gives us a contradiction since x0 coincides with the 0 configuration on S. This shows that Φ is not surjective. (c) By combining the two preceding remarks with Theorem 8.7.6, we deduce that the map Φ : LCA(G; V ) → EndK[G] (V [G]) is surjective if and only if G is finite or V is finite-dimensional. Let us recall the following basic facts from linear algebra. Let R be a ring and let d ≥ 1 be an integer. We denote by Matd (R) the ring consisting of all d × d matrices A = (aij )1≤i,j≤d with entries aij ∈ R. Recall that the addition and the multiplication in Matd (R) are given by A + B = (aij + bij )1≤i,j≤d

and AB =

 d k=1

aik bkj 1≤i,j≤d

for all A = (aij )1≤i,j≤d , B = (bij )1≤i,j≤d ∈ Matd (R). Suppose now that M is a free left R-module with basis e1 , e2 , . . . , ed . If u ∈ EndR (M ), then we d have u(ei ) = j=1 aij ej , where aij ∈ R for 1 ≤ i, j ≤ d. Then the map u → (aij )1≤i,j≤d is a ring isomorphism from EndR (M ) onto Matd (Rop ), where Rop denotes the opposite ring of R. Moreover, when R is an algebra over a field K, it is an isomorphism of K-algebras. Let G be a group and let V be a vector space of finite dimension d ≥ 1 over a field K. By Proposition 8.7.3, the left K[G]-module V [G] admits a free basis of cardinality d. As the K-algebras K[G] and (K[G])op are isomorphic by Corollary 8.4.4, we deduce from Theorem 8.7.6 the following: Corollary 8.7.8. Let G be a group and let V be a vector space over a field K of finite dimension dimK (V ) = d ≥ 1. Then the K-algebras LCA(G; V ) and   Matd (K[G]) are isomorphic. Remark 8.7.9. For d = 1, Corollary 8.7.8 tells us that if G is a group and V is a one-dimensional vector space over a field K, then the K-algebras LCA(G; V ) and K[G] are isomorphic. Note that this last result also follows from Corollary 8.5.3.

8.8 The Closed Image Property As we have seen in Lemma 3.3.2, the image of a cellular automaton τ : AG → AG is always closed in AG for the prodiscrete topology if the alphabet A is finite. When A is infinite, the image of τ may fail to be closed in AG (cf. Example 8.8.3). However, it turns out that if A = V is a finite-dimensional

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8 Linear Cellular Automata

vector space over a field K and τ : V G → V G is a linear cellular automaton, then the image of τ is closed in V G even when the field K is infinite. Theorem 8.8.1. Let G be a group and let V be a finite-dimensional vector space over a field K. Let τ : V G → V G be a linear cellular automaton. Then τ (V G ) is closed in V G for the prodiscrete topology. Proof. We split the proof into two steps. Suppose first that G is countable. Then we can find a sequence (An )n∈N of finite subsets of G such that G =

n∈N An and An ⊂ An+1 for all n ∈ N. Let S be a memory set for τ

and let Bn = A−S denote the S-interior of A (cf. Sect. 5.4). Note that G = n n n∈N Bn and Bn ⊂ Bn+1 for all n ∈ N. It follows from Proposition 5.4.3 that if x and x are elements in V G such that x and x coincide on An then the configurations τ (x) and τ (x ) coincide n ∈ V G a configuration on Bn . Therefore, given xn ∈ V An and denoting by x extending xn , the pattern xn )|Bn ∈ V Bn yn = τ ( does not depend on the particular choice of the extension x n . Thus we can define a map τn : V An → V Bn by setting τn (xn ) = yn for all xn ∈ V An . It is clear that τn is K-linear. Let y ∈ V G and suppose that y is in the closure of τ (V G ). Then, for all n ∈ N there exists zn ∈ V G such that y|Bn = τ (zn )|Bn .

(8.14)

Consider, for each n ∈ N, the affine subspace Ln ⊂ V An defined by Ln = τn−1 (y|Bn ). We have Ln = ∅ for all n by (8.14). For n ≤ m, the restriction map V Am → V An induces an affine map πn,m : Lm → Ln . Consider, for all n ≤ m, the affine subspace Kn,m ⊂ Ln defined by Kn,m = πn,m (Lm ). We have Kn,m ⊂ Kn,m for all n ≤ m ≤ m since πn,m = πn,m ◦ πm,m . As the sequence Kn,m (m = n, n + 1, . . . ) is a decreasing sequence of finitedimensional affine subspaces, it stabilizes, i.e., for each n ∈ N there exist a non-empty affine subspace Jn ⊂ Ln and an integer m0 = m0 (n) ≥ n such that Kn,m = Jn if m0 ≤ m. For all n ≤ n ≤ m, we have πn,n (Kn ,m ) ⊂ Kn,m since πn,n ◦ πn ,m = πn,m . Therefore, πn,n induces by restriction an affine map ρn,n : Jn → Jn for all n ≤ n . We claim that ρn,n is surjective. To see this, let u ∈ Jn . Let us choose m large enough so that Jn = Kn,m and Jn = Kn ,m . Then we can find v ∈ Lm such that u = πn,m (v). We have u = ρn,n (w), where w = πn ,m (v) ∈ Kn ,m = Jn . This proves the claim. Now, using the surjectivity of ρn,n+1 for all n, we construct by induction a sequence of elements xn ∈ Jn , n ∈ N, as follows. We start by choosing an arbitrary element x0 ∈ J0 . Then, assuming xn has been constructed, we take as xn+1 an arbitrary element in ρ−1 n,n+1 (xn ). Since xn+1 coincides with xn on An , there exists x ∈ V G such that x|An = xn for all n. We have

8.8 The Closed Image Property

307

τ (x)|Bn = τn (xn ) = yn = y|Bn for all n. Since G = ∪n∈N Bn , we deduce that τ (x) = y. This ends the proof in the case when G is a countable group. We now drop the countability assumption on G and prove the theorem in the general case. Let S ⊂ G be a memory set for τ and denote by H ⊂ G the subgroup of G generated by S. Then H is countable since it is finitely generated. Consider the restriction cellular automaton τH : V H → V H and observe that, by the first step of the proof, τH (V H ) is closed in V H

(8.15)

for the prodiscrete topology. With the notation from Sect. 1.7 we have, by virtue of (1.16), that  τ (V G ) = τc (V c ). (8.16) c∈G/H

Also, it follows from (1.18) that, for all c ∈ G/H and g ∈ c, τc (V c ) = (φ∗g )−1 τH (V H ) .

(8.17)

As the map φ∗g : V c → V H is a uniform isomorphism and therefore a homeomorphism, it follows from (8.15) and (8.17) that τc (V c ) is closed in V c for all c ∈ G/H. As the product of closed subspaces is closed in the product topology (cf. Proposition A.4.3), we deduce from (8.16) that τ (V G ) is closed   in V G . Corollary 8.8.2. Let G be a group and let V be a finite-dimensional vector space over a field K. Let τ : V G → V G be a linear cellular automaton. Suppose that every configuration with finite support y ∈ V [G] lies in the image of τ . Then τ is surjective. Proof. By our hypothesis, we have V [G] ⊂ τ (V G ) ⊂ V G . As V [G] is dense in V G by Proposition 8.2.1 and τ (V G ) is closed in V G by Theorem 8.8.1, we   deduce that τ (V G ) = V G . Thus, τ is surjective. In the example below we show that if we drop the finite dimensionality of the alphabet vector space V , then the image of a linear cellular automaton may fail to be closed in V G . Example 8.8.3. Let G = Z and let K be a field. Consider the K-algebra V = K[t] consisting of all polynomials in the indeterminate t with coefficients in K. A configuration in V Z is therefore a sequence x = (xn )n∈Z , where xn = xn (t) is a polynomial for all n ∈ Z. Consider the K-linear map τ : V Z → V Z defined by setting τ (x) = y where y = (yn )n∈Z is given by yn = xn+1 − txn for all n ∈ Z. Then τ is a linear cellular automaton with memory set {0, 1} and local defining map μ : V {0,1} → V given by μ(x0 , x1 ) = x1 − tx0 .

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8 Linear Cellular Automata

Consider the configuration z = (zn )n∈Z , where zn = 1 for all n ∈ Z. Let Ω be a finite subset of Z and choose an integer M ∈ Z such that Ω ⊂ [M, ∞). Consider the configuration x = (xn )n∈Z defined by  1 + t + · · · + tn−M if n ≥ M, xn = 0 if n < M. Observe that xn+1 = txn +1 for all n ≥ M , so that the configuration y = τ (x) coincides with z on [M, ∞) and hence on Ω. Thus z is in the closure of τ (V Z ) in V Z . On the other hand, the configuration z is not in the image of τ . Indeed, z = τ (x) for some x ∈ V Z would imply xn+1 = txn + 1 and hence deg(xn+1 ) > deg(xn ) for all n ∈ Z, which is clearly impossible. This shows that τ (V Z ) is not closed in V Z for the prodiscrete topology. Observe that every y ∈ V [Z] is in the image of τ . Indeed, if y = (yn )n∈Z ∈ V [Z] has support contained in [M, ∞) for some M ∈ Z, we can construct x = (xn )n∈Z ∈ V Z with τ (x) = y inductively by setting xn = 0 for all n ≤ M and xn+1 = txn + yn for all n ≥ M . This shows that we cannot omit the hypothesis saying that V is finite-dimensional in Corollary 8.8.2.

8.9 The Garden of Eden Theorem for Linear Cellular Automata In this section, we present a linear version of Theorem 5.8.1. We first introduce the notion of mean dimension, which plays here the role which was played by the entropy in the finite alphabet case, and we present some of its basic properties. In the definition of mean dimension, the dimension of finitedimensional vector spaces replaces the cardinality of finite sets. The proof of the properties of the mean dimension and the proof of the linear version of the Garden of Eden Theorem follow the same lines as their finite alphabet counterparts (see Sects. 5.7 and 5.8). From now on, in this section, G is an amenable group, F = (Fj )j∈J a right Følner net for G, and V a finite-dimensional vector space over a field K. For E ⊂ G, we denote by πE : V G → V E the canonical projection (restriction map). We thus have πE (x) = x|E for all x ∈ V G . Note that πE is K-linear so that, if X is a vector subspace of V G , then πE (X) is a vector subspace of V E . Definition 8.9.1. Let X be a vector subspace of V G . The mean dimension mdimF (X) of X with respect to the right Følner net F = (Fj )j∈J is defined by dim(πFj (X)) mdimF (X) = lim sup , (8.18) |Fj | j

8.9 The Garden of Eden Theorem for Linear Cellular Automata

309

where dim(πFj (X)) denotes the dimension of the K-vector subspace πFj (X) ⊂ V Fj . Remark 8.9.2. In the particular case when K is a finite field, every finitedimensional vector space W over K is finite of cardinality |W | = |K|dim(W ) . Therefore, in this case, we have mdimF (X) =

1 entF (X) log |K|

for every vector subspace X ⊂ V G . Here are some immediate properties of mean dimension. Proposition 8.9.3. One has: (i) mdimF (V G ) = dim(V ); (ii) mdimF (X) ≤ mdimF (Y ) if X ⊂ Y are vector subspaces of V G ; (iii) mdimF (X) ≤ dim V for all vector subspaces X ⊂ V G . Proof. (i) If X = V G , then, for every j, we have πFj (X) = V Fj and therefore dim(πFj (X)) dim(V Fj ) |Fj | dim(V ) = = = dim(V ). |Fj | |Fj | |Fj | This implies that mdimF (X) = dim(V ). (ii) If X ⊂ Y are vector subspaces of V G , then πFj (X) ⊂ πFj (Y ) are vector subspaces of V Fj and hence dim(πFj (X)) ≤ dim(πFj (Y )) for all j. This implies mdimF (X) ≤ mdimF (Y ). (iii) This follows immediately from (i) and (ii).   An important property of linear cellular automata is the fact that applying a linear cellular automaton to a vector subspace of configurations cannot increase the mean dimension of the subspace. More precisely, we have the following: Proposition 8.9.4. Let τ : V G → V G be a linear cellular automaton and let X be a vector subspace of V G . Then one has mdimF (τ (X)) ≤ mdimF (X). Proof. Let us set Y = τ (X) and observe that Y is a vector subspace of V G , by linearity of τ . Let S ⊂ G be a memory set for τ . After replacing S by S ∪ {1G }, we can assume that 1G ∈ S. Let Ω be a finite subset of G. Observe first that τ induces a map τΩ : πΩ (X) → πΩ −S (Y ) defined as follows. If u ∈ πΩ (X), then

310

8 Linear Cellular Automata

τΩ (u) = (τ (x))|Ω −S , where x is an element of X such that x|Ω = u. Note that the fact that τΩ (u) does not depend on the choice of such an x follows from Proposition 5.4.3. Clearly τΩ is surjective. Indeed, if v ∈ πΩ −S (Y ), then there exists x ∈ X such that (τ (x))|Ω −S = v. Then, setting u = πΩ (x) we have, by construction, τΩ (u) = v. Since τΩ is K-linear and surjective, we have dim(πΩ −S (Y )) ≤ dim(πΩ (X)).

(8.19)

Observe now that Ω −S ⊂ Ω, since 1G ∈ S (cf. Proposition 5.4.2(iv)). Thus −S πΩ (Y ) is a vector subspace of πΩ −S (Y ) × V Ω\Ω . This implies dim(πΩ (Y )) ≤ dim(πΩ −S (Y ) × V Ω\Ω

−S

)

= dim(πΩ −S (Y )) + dim(V Ω\Ω = dim(πΩ −S (Y )) + |Ω \ Ω ≤ dim(πΩ (X)) + |Ω \ Ω

−S

−S

−S

)

| dim(V )

| dim(V ),

by (8.19). As Ω \ Ω −S ⊂ ∂S (Ω), we deduce that dim(πΩ (Y )) ≤ dim(πΩ (X)) + |∂S (Ω)| dim(V ). By taking Ω = Fj , this gives us dim(πFj (Y )) dim(πFj (X)) |∂S (Fj )| ≤ + dim(V ). |Fj | |Fj | |Fj | Since lim j

|∂S (Fj )| =0 |Fj |

by Proposition 5.4.4, we finally get mdimF (Y ) = lim sup j

dim(πFj (Y )) dim(πFj (X)) ≤ lim sup = mdimF (X). |Fj | |Fj | j  

By Proposition 8.9.3, the maximal value for the mean dimension of a vector subspace X ⊂ V G is dim(V ). The following result gives a sufficient condition on X which guarantees that its mean dimension is strictly less than dim(V ). Proposition 8.9.5. Let X be a G-invariant vector subspace of V G . Suppose that there exists a finite subset E ⊂ G such that πE (X)  V E . Then one has mdimF (X) < dim(V ). Proof. Let E  = {g1 g2−1 : g1 , g2 ∈ E}. By Proposition 5.6.3, we may find an (E, E  )-tiling T ⊂ G.

8.9 The Garden of Eden Theorem for Linear Cellular Automata

311

For each j ∈ J, let us define, as in Proposition 5.6.4, the subset Tj ⊂ T by Tj = T ∩ Fj−E = {g ∈ T : gE ⊂ Fj } and set 

Fj∗ = Fj \

gE.

g∈Tj

Since πE (X)  V E and X is G-invariant, we have πgE (X)  V gE for all g ∈ G. We thus have dim(πgE (X)) ≤ dim(V gE ) − 1 = |gE| dim(V ) − 1 for all g ∈ T. As



πFj (X) ⊂ V Fj ×



(8.20)

πgE (X),

g∈Tj

we get ∗

dim(πFj (X)) ≤ dim(V Fj ×



πgE (X))

g∈Tj

= |Fj∗ | dim(V ) +



dim(πgE (X))

g∈Tj

≤ |Fj∗ | dim(V ) +



(|gE| dim(V ) − 1) (by (8.20))

g∈Tj

=



 |Fj∗ | + |gE| dim(V ) − |Tj | g∈Tj

= |Fj | dim(V ) − |Tj |,

since

|Fj | = |Fj∗ | +



|gE|

and

|gE| = |E|.

g∈Tj

Now, by Proposition 5.6.4, there exist α > 0 and j0 ∈ J such that |Tj | ≥ α|Fj | for all j ≥ j0 . Thus dim(πFj (X)) ≤ dim(V ) − α |Fj |

for all j ≥ j0 .

This implies that mdimF (X) = lim sup j

dim(πFj (X)) ≤ dim(V ) − α < dim(V ). |Fj |  

312

8 Linear Cellular Automata

We are now in position to state the linear version of the Garden of Eden theorem. Theorem 8.9.6. Let G be an amenable group and let V be a finite-dimensional vector space over a field K. Let F = (Fj )j∈J be a right Følner net for G. Let τ : V G → V G be a linear cellular automaton. Then the following conditions are equivalent: (a) τ is surjective; (b) mdimF (τ (V G )) = dim(V ); (c) τ is pre-injective. Since every injective cellular automaton is pre-injective, we immediately deduce the following: Corollary 8.9.7. Let G be an amenable group and let V be a finite-dimensional vector space over a field K. Then every injective linear cellular automaton τ : V G → V G is surjective.   We divide the proof of Theorem 8.9.6 into several lemmas. Lemma 8.9.8. Let τ : V G → V G be a linear cellular automaton. Suppose that τ is not surjective. Then mdimF (τ (V G )) < dim(V ). Proof. Let X = τ (V G ) and choose a configuration y ∈ V G \ X. Since V G \ X is an open subset of V G for the prodiscrete topology by Theorem 8.8.1, we / πΩ (X). Therefore we have can find a finite subset Ω ⊂ G such that πΩ (y) ∈ πΩ (X)  V Ω . Observe that X is a G-invariant vector subspace of V G , as τ is G-equivariant and linear. By applying Proposition 8.9.5, we deduce that mdimF (X) < dim(V ).   Lemma 8.9.9. Let τ : V G → V G be a linear cellular automaton. Suppose that (8.21) mdimF (τ (V G )) < dim(V ). Then τ is not pre-injective. Proof. Let us set X = τ (V G ). Let S be a memory set for τ such that 1G ∈ S. +S As πF +S (X) is a vector subspace of πFj (X) × V Fj \Fj , we have j

dim(πF +S (X)) ≤ dim(πFj (X)) + |Fj+S \ Fj | dim(V ) j

≤ dim(πFj (X)) + |∂S (Fj )| dim(V ). We thus have dim(πF +S (X)) j

|Fj |



dim(πFj (X)) |∂S (Fj )| + dim(V ). |Fj | |Fj |

8.9 The Garden of Eden Theorem for Linear Cellular Automata

313

It follows from (8.21) and Proposition 5.4.4 that we can find j0 ∈ J such that dim(πF +S (X)) < |Fj0 | dim(V ). Let Z denote the (finite-dimensional) j0

vector subspace of V G consisting of all configurations whose support is contained in Fj0 . Observe that τ (x) vanishes outside of Fj+S for every x ∈ Z by 0 Proposition 5.4.3. Thus we have dim(τ (Z)) = dim(πF +S (τ (Z))) j0

≤ dim(πF +S (X)) < |Fj0 | dim(V ) = dim(Z). j0

This implies that the restriction of τ to Z is not injective. As all configurations in Z have finite support, we deduce that τ is not pre-injective (cf. Proposition 8.2.5).   Lemma 8.9.10. Let τ : V G → V G be a linear cellular automaton. Suppose that τ is not pre-injective. Then mdimF (τ (V G )) < dim(V ). Proof. Since the linear cellular automaton τ is not pre-injective, we can find, by Proposition 8.2.5, an element x0 ∈ V G with nonempty finite support Ω ⊂ G such that τ (x0 ) = 0. Let S be a memory set for τ such that 1G ∈ S 2 and S = S −1 . Let E = Ω +S . By Proposition 5.6.3, we can find a finite subset F ⊂ G such that G contains a (E, F )-tiling T . Note that for each g ∈ G, the support of gx0 is gΩ ⊂ gE. Let us choose, for each g ∈ T , a hyperplane Hg ⊂ V gΩ which does not contain the restriction to gΩ of gx0 . Consider the vector subspace X ⊂ V G consisting of all x ∈ V G such that the restriction of x to gΩ belongs to Hg for each g ∈ T . We claim that τ (V G ) = τ (X). Indeed, let z ∈ V G . Then, for each g ∈ T , there exists a scalar kg ∈ K such that the restriction to gΩ of z + kg (gx0 ) belongs to Hg . Let z  ∈ V G be  (z + kg (gx0 )) for each g ∈ T and z  = z outside such  that πgΩ (z ) = πgΩ  of g∈T gΩ. We have z ∈ X by construction. On the other hand, since z    and z coincide outside g∈T gΩ, we have τ (z  ) = τ (z) outside g∈T gΩ +S . 2

Now, if h ∈ gΩ +S for some g ∈ T , then hS ⊂ gΩ +S = gE and therefore τ (z  )(h) = τ (z + kg (gx0 ))(h) = τ (z)(h) since gx0 lies in the kernel of τ . Thus τ (z) = τ (z  ) and the claim follows. We deduce that mdimF (τ (V G )) = mdimF (τ (X)) ≤ mdimF (X) < dim(V ) where the first inequality follows from Proposition 8.9.4 and the second one from Proposition 8.9.5.   Proof of Theorem 8.9.6. Condition (a) implies (b) since we have mdimF (V G ) = dim(V ) by Proposition 8.9.3(i). The converse implication follows from Lemma 8.9.8. On the other hand, condition (c) implies (b) by Lemma 8.9.9. Finally, (b) implies (c) by Lemma 8.9.10.  

314

8 Linear Cellular Automata

We end this section by showing that Corollary 8.9.7 as well as both implications (a) ⇒ (c) and (c) ⇒ (a) in Theorem 8.9.6 fail to hold when the vector space V is infinite-dimensional. Example 8.9.11. Let G be any group and let V be an infinite-dimensional vector space over a field K. Let us choose a basis subset B for V . Every map ϕ : B → B uniquely extends to a K-linear map ϕ  : V → V . The product G : V G → V G is a linear cellular automaton with memory set map τϕ = ϕ  Since B is infinite, we can find a map S = {1G } and local defining map ϕ. ϕ1 : B → B which is surjective but not injective and a map ϕ2 : B → B which is injective but not surjective. Consider first the cellular automaton τ1 = τϕ1 . Let us show that τ1 is surjective but not pre-injective. Observe that ϕ 1 is surjective. As a product of surjective maps is a surjective map, it follows that τ1 is surjective. On the 1 ) \ {0}. Consider the other hand, as ϕ 1 is not injective, there exists v ∈ ker(ϕ configuration x ∈ V G defined by x(1G ) = v and x(g) = 0 for all g ∈ G \ {1G }. Then x ∈ V [G] ∩ ker(τ1 ) and x = 0. This shows that τ1 is not pre-injective. Consider now the cellular automaton τ2 = τϕ2 . Let us show that τ2 is injective (and therefore pre-injective) but not surjective. Suppose that x ∈ 2 (x(g)) for all g ∈ G. As ϕ 2 is injective, ker(τ2 ), that is, 0 = τ2 (x)(g) = ϕ we deduce that x(g) = 0 for all g ∈ G, in other words, x = 0. This shows that τ2 is injective. On the other hand, as ϕ 2 is not surjective, there exists v ∈ V \ϕ 2 (V ). Consider the constant configuration x ∈ V G where x(g) = v for all g ∈ G. It is clear that x ∈ V G \ τ2 (V G ). This shows that τ2 is not surjective.

8.10 Pre-injective but not Surjective Linear Cellular Automata In this section, we give examples of linear cellular automata with finitedimensional alphabet which are pre-injective but not surjective. We recall that the underlying groups for such automata cannot be amenable by Theorem 8.9.6. Proposition 8.10.1. Let G = F2 be the free group of rank two and let V be a two-dimensional vector space over a field K. Then there exists a linear cellular automaton τ : V G → V G which is pre-injective but not surjective. Proof. We may assume V = K2 . Let a and b denote the canonical generators of G = F2 . Let p1 and p2 be the elements of EndK (V ) defined respectively by p1 (v) = (k1 , 0) and p2 (v) = (k2 , 0) for all v = (k1 , k2 ) ∈ V . Consider the linear cellular automaton τ : V G → V G given by τ (x)(g) = p1 (x(ga)) + p2 (x(gb)) + p1 (x(ga−1 )) + p2 (x(gb−1 ))

8.11 Surjective but not Pre-injective Linear Cellular Automata

315

for all x ∈ V G and g ∈ G. This cellular automaton is the one described in Sect. 5.11 for H = K. By Proposition 5.11.1, τ is pre-injective but not surjective.   If H is a subgroup of a group G and τ : V H → V H is a linear cellular automaton over H which is pre-injective and not surjective, then the induced cellular automaton τ G : V G → V G is also linear, pre-injective and not surjective (see Proposition 1.7.4 and Proposition 5.2.2). Therefore, an immediate consequence of Proposition 8.10.1 is the following: Corollary 8.10.2. Let G be a group containing a free subgroup of rank two and let V be a two-dimensional vector space over a field K. Then there exists a linear cellular automaton τ : V G → V G which is pre-injective but not surjective.   This shows that implication (c) ⇒ (a) in Theorem 8.9.6 becomes false if G is a group containing a free subgroup of rank two.

8.11 Surjective but not Pre-injective Linear Cellular Automata We now present examples of linear cellular automata with finite-dimensional alphabet which are surjective but not pre-injective. We recall that, by Theorem 8.9.6, the underlying groups for such examples are necessarily nonamenable. Proposition 8.11.1. Let G = F2 be the free group of rank two and let V be a two-dimensional vector space over a field K. Then there exists a linear cellular automaton τ : V G → V G which is surjective but not pre-injective. Proof. Let a, b denote the canonical generators of G. We can assume that V = K2 . Consider the elements q1 , q2 ∈ EndK (V ) respectively defined by q1 (v) = (k1 , 0) and q2 (v) = (0, k1 ) for all v = (k1 , k2 ) ∈ V . Let τ : V G → V G be the linear cellular automaton given by τ (x)(g) = q1 (x(ga)) + q1 (x(ga−1 )) + q2 (x(gb)) + q2 (x(gb−1 )) for all x ∈ V G , g ∈ G. A memory set for τ is the set S = {a, b, a−1 , b−1 }. Let k0 be any nonzero element in K. Consider the configuration x0 ∈ V G defined by  (0, k0 ) if g = 1G x0 (g) = (0, 0) otherwise. Then, x0 is almost equal to 0 but x0 = 0. As τ (x0 ) = 0, we deduce that τ is not pre-injective.

316

8 Linear Cellular Automata

However, τ is surjective. To see this, let z = (z1 , z2 ) ∈ KG × KG = V G . Let us show that there exists x ∈ V G such that τ (x) = z. We define x(g) by induction on the graph distance (cf. Sect. 6.2), which we denote by |g|, of g ∈ G from 1G in the Cayley graph of G. We first set x(1G ) = (0, 0). Then, for s ∈ S we set ⎧ ⎨ (z1 (1G ), 0) if s = a x(s) = (z2 (1G ), 0) if s = b (8.22) ⎩ (0, 0) otherwise. Suppose that x(g) has been defined for all g ∈ G with |g| ≤ n, for some n ≥ 1. For g ∈ G with |g| = n, let g  ∈ G and s ∈ S be the unique elements such that |g  | = n − 1 and g = g  s . Then, for s ∈ S with s s = 1G , we set ⎧ (z1 (g) − x1 (g  ), 0) if s ∈ {a, a−1 } and s = s ⎪ ⎪ ⎪ ⎪ if s ∈ {a, a−1 } and s = b ⎨ (z2 (g), 0) if s ∈ {b, b−1 } and s = a x(gs) = (z1 (g), 0) (8.23) ⎪   −1  ⎪ (z (g) − x (g ), 0) if s ∈ {b, b } and s = s ⎪ 2 2 ⎪ ⎩ (0, 0) otherwise. Let us check that τ (x) = z. We have τ (x)(1G ) = q1 (x(a)) + q1 (x(a−1 )) + q2 (x(b)) + q2 (x(b−1 )) (by (8.22)) = q1 (z1 (1G ), 0) + q1 (0, 0) + q2 (z2 (1G ), 0) + q2 (0, 0) = (z1 (1G ), 0) + (0, 0) + (0, z2 (1G )) + (0, 0) = (z1 (1G ), z2 (1G )) = z(1G ). Let now g = g  s ∈ G with |g| = |g  | + 1 > 2. Suppose, for instance, that s = a. We then have: τ (x)(g) = τ (x)(g  a) (by (8.23)) = q1 (x(g  a2 )) + q1 (x(g  )) + q2 (x(g  ab)) + q2 (x(g  ab−1 )) = q1 (z1 (g) − x1 (g  ), 0) + q1 (x1 (g  ), x2 (g  )) + q2 (z2 (g), 0) + q2 (0, 0) = (z1 (g) − x1 (g  ), 0) + (x1 (g  ), 0) + (0, z2 (g)) + (0, 0) = (z1 (g), z2 (g)) = z(g). The cases when s = a−1 , b, b−1 are similar. It follows that τ (x) = z. This shows that τ is surjective.   If H is a subgroup of a group G, and τ : V H → V H is a linear cellular automaton over H which is surjective and not pre-injective, then the induced cellular automaton τ G : V G → V G is also linear, surjective and not

8.12 Invertible Linear Cellular Automata

317

pre-injective (see Proposition 1.7.4 and Proposition 5.2.2). Therefore, an immediate consequence of Proposition 8.11.1 is the following: Corollary 8.11.2. Let G be a group containing a free subgroup of rank two and let V be a two-dimensional vector space over a field K. Then there exists a linear cellular automaton τ : V G → V G which is surjective but not preinjective.   As a consequence, we deduce that implication (a) ⇒ (c) in Theorem 8.9.6 becomes false if the group G contains a free subgroup of rank two.

8.12 Invertible Linear Cellular Automata Theorem 8.12.1. Let G be a group and let V be a finite-dimensional vector space over a field K. Let τ : V G → V G be an injective linear cellular automaton. Then there exists a linear cellular automaton σ : V G → V G such that σ ◦ τ = IdV G . Proof. We split the proof into two steps. Suppose first that G is countable. Since τ is injective, it induces a bijective map τ : V G → Y where Y = τ (V G ) is the image of τ . The fact that τ is K-linear and G-equivariant implies that Y is a G-invariant vector subspace of V G and that the inverse map τ−1 : Y → V G is K-linear and G-equivariant. Let us show that the following local property is satisfied by τ−1 : there exists a finite subset T ⊂ G such that (∗) for y ∈ Y , the element τ−1 (y)(1G ) only depends on the restriction of y to T . Let us assume by contradiction that there exists no such T . Let S be a memory set for τ such that 1G ∈ S. Since G is countable, we

can find a sequence (An )n∈N of finite subsets of G such that G = n∈N An , −S S ⊂ A0 and An ⊂ An+1 for all n ∈ N.

Let Bn = An denote the S-interior of An (cf. Sect. 5.4). Note that G = n∈N Bn and Bn ⊂ Bn+1 for all n ∈ N. Since there exists no finite subset T ⊂ G satisfying condition (∗), we can find, for each n ∈ N, two configurations yn , yn ∈ Y such that yn |Bn = yn |Bn and τ−1 (yn )(1G ) = τ−1 (yn )(1G ). By linearity of τ−1 , the configuration yn = yn − yn ∈ Y satisfies yn |Bn = 0 and τ−1 (yn )(1G ) = 0. It follows from Proposition 5.4.3 that if x and x are elements in V G such that x and x coincide on An then the configurations τ (x) and τ (x ) coincide n ∈ V G a configuration on Bn . Therefore, given xn ∈ V An and denoting by x extending xn , the pattern xn )|Bn ∈ V Bn un = τ ( does not depend on the particular choice of the extension x n of xn . Thus we can define a map τn : V An → V Bn by setting τn (xn ) = un . It is clear that τn is K-linear.

318

8 Linear Cellular Automata

Consider, for each n ∈ N, the vector subspace Ln ⊂ V An defined by Ln = Ker(τn ). where for n ≤ m, the restriction map V Am → V An induces a K-linear map πn,m : Lm → Ln . Indeed, if u ∈ Lm , then we have u|An ∈ Ln since τm (u) = 0 and therefore τn (u|An ) = (τm (u))|Bn = 0. Consider now, for all n ≤ m, the vector subspace Kn,m ⊂ Ln defined by Kn,m = πn,m (Lm ). We have Kn,m ⊂ Kn,m for all n ≤ m ≤ m since πn,m = πn,m ◦πm,m . Therefore, if we fix n, the sequence Kn,m , where m = n, n+1, . . ., is a decreasing sequence of vector subspaces of Ln . As Ln ⊂ V An is finite-dimensional, this sequence stabilizes, i.e., there exist a vector subspace Jn ⊂ Ln and an integer kn ≥ n such that Kn,m = Jn for all m ≥ kn . For all n ≤ n ≤ m, we have πn,n (Kn ,m ) ⊂ Kn,m since πn,n ◦ πn ,m = πn,m . Therefore, πn,n induces by restriction a linear map ρn,n : Jn → Jn for all n ≤ n . We claim that ρn,n is surjective. To see this, let u ∈ Jn . Let us choose m large enough so that Jn = Kn,m and Jn = Kn ,m . Then we can find v ∈ Lm such that u = πn,m (v). We have u = ρn,n (w), where w = πn ,m (v) ∈ Kn ,m = Jn . This proves the claim. Now, using the surjectivity of ρn,n+1 for all n, we construct by induction a sequence of elements zn ∈ Jn , n ∈ N, as follows. We start by taking as z0 the restriction of xk0 to A0 . Observe that xk0 ∈ Lk0 and hence z0 = π0,k0 (xk0 ) ∈ J0 . Then, assuming that zn has been constructed, we take as zn+1 an arbitrary element in ρ−1 n,n+1 (zn ). Since zn+1 coincides with zn on An , there exists a unique element z ∈ V G such that z|An = zn for all n. We have z ∈ Y , since zn ∈ πAn (Y ) for all n and X is closed in V G . As z(1G ) = z0 (1G ) = xk0 (1G ) = 0, we have z = 0. On the other hand, τ (z) = 0 since τ (z)|Bn = τn (zn ) = 0 for all n by construction. This contradicts the injectivity of τ . Thus, there exists a finite subset T ⊂ G satisfying (∗). z )(1G ), where Consider the linear map ν : V T → V defined by ν(z) = τ−1 ( z ∈ V G is any configuration extending the pattern z ∈ V T . Then the linear cellular automaton σ : V G → V G with memory set T and local defining map ν clearly satisfies σ ◦ τ = IdV G . Indeed, if x ∈ V G , then denoting by y = τ (x) ∈ Y its image by τ , we have x = τ−1 (y) and (σ ◦ τ )(x)(1G ) = σ(y)(1G ) = ν(y|T ) = τ−1 (y)(1G ) = x(1G ) showing that (σ ◦ τ )(x) = x, by G-equivariance of σ ◦ τ . It follows that σ ◦ τ = IdV G and this proves the statement when G is a countable group. We now drop the countability assumption on G and prove the theorem in the general case. Let S ⊂ G be a memory set for τ and denote by μ : V S → V the corresponding local defining map. Let H ⊂ G be the subgroup generated by S. Note that H is countable. Consider the set G/H of all left cosets of H in G. For c ∈ G/H denote by

8.12 Invertible Linear Cellular Automata

319

πc : V G → V c =



V

g∈c

 the projection map. We have V G = c∈G/H V c and, for every x ∈ V G we write x = (xc )c∈G/H , where xc = πc (x) ∈ V c . For c ∈ G/H and g ∈ c denote by φg : H → c the linear map defined by φg (h) = gh for all h ∈ H. Consider the map φ∗g : V c → V H defined by φ∗g (z) = z ◦ φg . Then, if x = (xc )c∈G/H ∈ V G and c ∈ G/H, we have (φg ∗ (xc ))(h) = xc (gh) = x(gh) = (g −1 x)(h) = (g −1 x)H (h) for all h ∈ H, that is,

φ∗g (xc ) = (g −1 x)H .

(8.24)

For c ∈ G/H, define the map τc : V c → V c by setting τc (z)(g) = μ((φ∗g (z))|S ) for all z ∈ V c and g ∈ c. Note that τH : V H → V H is the restriction of τ to the subgroup H. We then have  τ= τc . (8.25) c∈G/H

 From (8.25) we immediately deduce that Y = c∈G/H Yc , where Yc = πc (Y ) = τc (V c ), and that τc is injective for all c ∈ G/H. By the first part of the present proof, there exists a linear cellular automaton σH : V H → V H such that σH ◦ τH = IdV H .

(8.26)

Let T ⊂ H be a memory set for σH and let ν : V T → V be the corresponding local defining map. Consider the linear cellular automaton σ : V G → V G defined by setting σ(y)(g) = ν((g −1 y)|T ) for all y ∈ V G and g ∈ G. Note that σ = (σH )G is the induced cellular automaton of σH from H to G, so that, if σc : V c → V c is the map defined by setting σc (z)(g) = ν((φ∗g (z))|T ) for all z ∈ V c and g ∈ c then σ=



σc .

c∈G/H

Given c ∈ G/H, z ∈ V c and g ∈ c, we have

(8.27)

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8 Linear Cellular Automata

(σc ◦ τc )(z)(g) = σc (τc (z))(g) = [φ∗g σc (τc (z))](1G ) (by (8.24)) = σH (φ∗g (τc (z))(1G ) (again by (8.24)) = σH (τH (φ∗g (z)))(1G ) (by (8.26)) = (φ∗g (z))(1G ) = z(g). This shows that (σc ◦ τc )(z) = z for all z ∈ V c . We deduce that σc ◦ τc = IdV c for all c ∈ C. It follows from (8.25) and (8.27) that σ ◦ τ = IdV G .   We recall that given a group G and a set A, a cellular automaton τ : AG → A is said to be invertible if τ is bijective and the inverse map τ −1 : AG → AG is also a cellular automaton. When A is a finite set, every bijective cellular automaton τ : AG → AG is invertible by Theorem 1.10.2. The following is a linear analogue. G

Corollary 8.12.2. Let G be a group and let V be a finite-dimensional vector space over a field K. Then every bijective linear cellular automaton τ : V G → V G is invertible. Proof. Let τ : V G → V G be a bijective linear cellular automaton. It follows from Theorem 8.12.1 that there exists a linear cellular automaton σ : V G → V G such that σ◦τ = IdV G . This implies that τ −1 = σ is a cellular automaton. Thus τ is an invertible cellular automaton.   Remark 8.12.3. Let G = Z and let K be a field. Consider the infinite dimensional K-vector space V = K[[t]] consisting of all formal power series in one indeterminate t with coefficients in K. Thus, an element of V is just a sequence v = (ki )i∈N of elements of K written in the form  v = k0 + k1 t + k2 t2 + k3 t3 + · · · = ki ti . i∈N

Consider the map τ : V Z → V Z defined by τ (x)(n) = x(n) − tx(n + 1) for all x ∈ V Z , n ∈ Z. In Example 1.10.3 we have showed that τ is a bijective cellular automaton whose inverse map τ −1 : V Z → V Z is not a cellular automaton. It is clear that τ is K-linear. This shows that Corollary 8.12.2 becomes false if we omit the finite-dimensionality of the alphabet V . Let G be a group and let V be a vector space over a field K. We have seen in Sect. 1.10 that the set ICA(G; V ) consisting of all invertible cellular automata τ : V G → V G is a group for the composition of maps. The subset of ICA(G; V ) consisting of all invertible linear cellular automata is a subgroup

8.13 Pre-injectivity and Surjectivity of the Discrete Laplacian

321

of ICA(G; V ) since it is the intersection of ICA(G; V ) with the automorphism group of the K-vector space V G . Given a ring R and an integer d ≥ 1, we denote by GLd (R) the group of invertible elements of the matrix ring Matd (R). An immediate consequence of Corollary 8.12.2 and Corollary 8.7.8 is the following Corollary 8.12.4. Let G be a group and let V be a vector space over a field K of finite dimension dimK (V ) = d ≥ 1. Then the set consisting of all bijective linear cellular automata τ : V G → V G is a subgroup of ICA(G; V ) isomorphic   to GLd (K[G]).

8.13 Pre-injectivity and Surjectivity of the Discrete Laplacian Let G be a group and let K be a field. Given a nonempty finite subset S of G, we recall that the discrete Laplacian associated with G and S is the linear cellular automaton ΔS : KG → KG (with memory set S ∪ {1G }) defined by  ΔS (f )(g) = |S|f (g) − f (gs) s∈S

for all f ∈ KG and g ∈ G, where |S| denotes the cardinality of S (cf. Example 1.4.3(b) and Example 8.1.3(a)). Let us observe that ΔS is never injective since all constant maps f : G → K are in the kernel of ΔS . Proposition 8.13.1. Let G be a group and let K be a field. Let S ⊂ G be a non-empty finite subset and suppose that the subgroup H ⊂ G generated by S is finite. Then ΔS is neither pre-injective nor surjective. Proof. Let (ΔS )H : KH → KH denote the restriction cellular automaton of ΔS to H. Observe that (ΔS )H is the discrete Laplacian on KH associated with H and S. As we have seen above, (ΔS )H is not injective. As H is finite, it follows that (ΔS )H is not pre-injective. On the other hand, the non-injectivity of (ΔS )H implies its non-surjectivity since (ΔS )H is K-linear and KH is finite-dimensional. By applying Proposition 5.2.2 (resp. Proposition 1.7.4),   we deduce that ΔS is not pre-injective (resp. not surjective). In the remaining of this section, we assume that the field K is the field R of real numbers. The following result is a Garden of Eden type theorem for the discrete laplacian. Note that there is no amenability hypothesis for the underlying group.

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Theorem 8.13.2. Let G be a group and let S be a nonempty finite subset of G. Let ΔS : RG → RG denote the associated discrete real laplacian. Then the following conditions are equivalent: (a) ΔS is surjective; (b) the subgroup of G generated by S is infinite; (c) ΔS is pre-injective. Let us first establish the following simple fact. Recall that given a group G, the subsemigroup generated by a subset S ⊂ G is the smallest subset P of G containing S which is closed under the group operation, i.e., such that p1 p2 ∈ P for all p1 , p2 ∈ P . Clearly, P consists of all elements of the form p = s1 s2 · · · sn where n ≥ 1 and si ∈ S for 1 ≤ i ≤ n. Lemma 8.13.3. Let G be a group and let S be a subset of G. Then the following conditions are equivalent: (a) the subsemigroup of G generated by S is infinite; (b) the subgroup of G generated by S is infinite. Proof. Let P (resp. H) denote the subsemigroup (resp. subgroup) of G generated by S. The implication (a) ⇒ (b) is obvious since P ⊂ H. Suppose that S is nonempty and that P is finite. Then, for each s ∈ S, the set {sn : n ≥ 1} ⊂ P is also finite. This implies that every element of S has finite order. Therefore, for each s ∈ S, we can find an integer n ≥ 2 such that sn = 1G . It follows that s−1 = sn−1 ∈ S. Consequently, we have P −1 = P and 1G ∈ P . This implies H = P , so that H is finite. This shows (b) ⇒ (a).   Proof of Theorem 8.13.2. The implication (a) ⇒ (b) immediately follows from Proposition 8.13.1. Suppose that the subgroup generated by S is infinite. Let f be an element of R[G] such that ΔS (f ) = 0. Let us show that f = 0. Let g0 ∈ G such that |f (g0 )| = maxg∈G |f (g)|. Note that such a g0 exists since f takes only finitely many values. As  0 = ΔS (f )(g0 ) = |S|f (g0 ) − f (g0 s), s∈S

we get |f (g0 )| ≤

1  |f (g0 s)| |S|

(8.28)

s∈S

by applying the triangle inequality. We deduce from (8.28) and the definition of g0 that |f (g0 s)| = |f (g0 )| for all s ∈ S. By iterating the previous argument, we get |f (g0 p)| = |f (g0 )| for all p ∈ P , where P denotes the subsemigroup of G generated by S. As P is infinite (cf. Lemma 8.13.3), this implies that |f (h)| = |f (g0 )| = maxg∈G |f (g)| for infinitely many h ∈ G. Since f has finite

8.13 Pre-injectivity and Surjectivity of the Discrete Laplacian

323

support, it follows that f = 0. Therefore, ΔS is pre-injective. This proves the implication (b) ⇒ (c). To complete the proof, it suffices to prove that (c) implies (a). Suppose first that G is an amenable group. It follows from the implication (c) ⇒ (a) in Theorem 8.9.6 that if ΔS is pre-injective, then it is surjective. This shows the implication (c) ⇒ (a) for G amenable. Suppose now that G is non-amenable (and hence infinite) and that S is a generating subset for G. This implies that G is countable. Consider the Hilbert space 2 (G) ⊂ RG consisting of all square-summable real functions on (2) G and the continuous linear map ΔS : 2 (G) → 2 (G) obtained by restriction 2 of ΔS to  (G) (cf. Sect. 6.12). By the Kesten-Day amenability criterion (Theorem 6.12.9), the nona(2) menability of G implies that 0 does not belong to the spectrum σ(ΔS ) (2) (2) of ΔS . Thus, ΔS is bijective and hence (2)

R[G] ⊂ 2 (G) = ΔS (2 (G)) = ΔS (2 (G)) ⊂ ΔS (RG ). By applying Corollary 8.8.2, we deduce that ΔS (RG ) = RG . This shows that ΔS is surjective in the case when G is non-amenable and S generates G. Finally, suppose now that G is an arbitrary non-amenable group and that ΔS is pre-injective. Denote by H the subgroup of G generated by S. Then the restriction cellular automaton (ΔS )H : RH → RH , which is the discrete laplacian associated with H and S, is pre-injective by Proposition 5.2.2. The subgroup H may be amenable or not. However, it follows from the two preceding cases that (ΔS )H is surjective. By applying Proposition 1.7.4, we deduce that ΔS is surjective as well. This completes the proof that (a) implies (c).   As a consequence of Theorem 8.13.2, we obtain the following characterization of locally finite groups in terms of real linear cellular automata: Corollary 8.13.4. Let G be a group and let V be a real vector space of finite dimension d ≥ 1. Then the following conditions are equivalent: (a) G is locally finite; (b) every surjective linear cellular automaton τ : V G → V G is injective. Proof. Suppose (a). Let τ : V G → V G be a surjective linear cellular automaton with memory set S ⊂ G. As G is locally finite, the subgroup H generated by S is finite. Consider the linear cellular automaton τH : V H → V H obtained from τ by restriction. Observe that τH is surjective by Proposition 1.7.4(ii). Since V H is finite-dimensional, it follows that τH is injective. By applying Proposition 1.7.4(i), we deduce that τ is also injective. This shows that (a) implies (b). Now, suppose that G is not locally finite. Let us show that there exists a linear cellular automaton τ : V G → V G which is surjective but not injective.

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We can assume that V = Rd . Since G is not locally finite, we can find a finite subset S ⊂ G such that the subgroup H generated by S is infinite. By Theorem 8.13.2, the discrete Laplacian ΔS : RG → RG is surjective. As mentioned above ΔS is not injective since all constant configurations are in its kernel. Consider now the product map τ = (ΔS )d : (Rd )G → (Rd )G , where we use the natural identification (Rd )G = (RG )d . Clearly, τ is a linear cellular automaton admitting S ∪ {1G } as a memory set. On the other hand, τ is surjective but not injective since any product of surjective (resp. noninjective) maps is a surjective (resp. non-injective) map. This shows that (b) implies (a).  

8.14 Linear Surjunctivity In analogy with the finite alphabet case, we introduce the following definition. Definition 8.14.1. A group G is said to be L-surjunctive if, for any field K and any finite-dimensional vector space V over K, every injective linear cellular automaton τ : V G → V G is surjective. Proposition 8.14.2. Every subgroup of an L-surjunctive group is L-surjunctive. Proof. Suppose that H is a subgroup of a L-surjunctive group G. Let V be a finite-dimensional vector space over a field K and let τ : V H → V H be an injective linear cellular automaton over H. Consider the cellular automaton τ G : V G → V G over G obtained from τ by induction (see Sect. 1.7). The fact that τ is injective implies that τ G is injective by Proposition 1.7.4(i). Also, τ G is linear by Proposition 8.3.1. Since G is L-surjunctive, it follows that τ G is surjective. By applying Proposition 1.7.4(ii), we deduce that τ is surjective. This shows that H is L-surjunctive.   Proposition 8.14.3. Let G be a group. Then the following conditions are equivalent: (a) G is L-surjunctive; (b) every finitely generated subgroup of G is L-surjunctive. Proof. The fact that (a) implies (b) follows from Proposition 8.14.2. Conversely, let G be a group all of whose finitely generated subgroups are Lsurjunctive. Let V be a finite dimensional vector space over a field K and let τ : V G → V G be an injective linear cellular automaton with memory set S. Let H denote the subgroup of G generated by S and consider the linear cellular automaton τH : V H → V H obtained by restriction of τ (see Sect. 1.7 and Proposition 8.3.1). The fact that τ is injective implies that τH is injective by Proposition 1.7.4(i). As H is finitely generated, it is L-surjunctive by

8.14 Linear Surjunctivity

325

our hypothesis on G. It follows that τH is surjective. By applying Proposition 1.7.4(ii), we deduce that τ is also surjective. This shows that (b) implies (a).   Note that the preceding proposition may be stated by saying that a group is L-surjunctive if and only if it is locally L-surjunctive. Every finite group is obviously L-surjunctive. Indeed, if G is a finite group and V is a finite-dimensional vector space, then the vector space V G is also finite-dimensional and therefore every injective endomorphism of V G is surjective. More generally, it follows from Corollary 8.9.7 that every amenable group is L-surjunctive. In fact, we have the following result, which is a linear analogue of Theorem 7.8.1, Theorem 8.14.4. Every sofic group is L-surjunctive. Proof. Let G be a sofic group. Let V be a finite-dimensional vector space over a field K of dimension dimK (V ) = d ≥ 1 and let τ : V G → V G be an injective linear cellular automaton. We want to show that τ is surjective. Every subgroup of a sofic group is sofic by Proposition 7.5.4. On the other hand, it follows from Proposition 3.2.2 that a group is L-surjunctive if all its finitely generated subgroups are L-surjunctive. Thus we can assume that G is finitely generated. Let then S ⊂ G be a finite symmetric generating subset of G. As usual, for r ∈ N, we denote by BS (r) ⊂ G the ball of radius r centered at 1G in the Cayley graph associated with (G, S). We set Y = τ (V G ). Observe that Y is a G-invariant vector subspace of V G . On the other hand, it follows from Theorem 8.8.1 that Y is closed in V G with respect to the prodiscrete topology. By Theorem 8.12.1, there exists a linear cellular automaton σ : V G → V G such that σ ◦ τ = IdV G . Choose r0 large enough so that the ball BS (r0 ) is a memory set for both τ and σ. Let μ : V BS (r0 ) → V and ν : V BS (r0 ) → V denote the corresponding local defining maps for τ and σ respectively. We proceed by contradiction. Suppose that τ is not surjective, that is, Y  V G . Then, since Y is closed in V G , there exists a finite subset Ω ⊂ G such that Y |Ω  V Ω . It is not restrictive, up to taking a larger r0 , again if necessary, to suppose that Ω ⊂ BS (r0 ). Thus, Y |BS (r0 )  V BS (r0 ) . Let ε > 0 be such that ε
1 − (1 − ε)−1 < 1 +

1 d|B(2r0 )|+1

1 . d|B(2r0 )|

(8.29) which yields (8.30)

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Since G is sofic, we can find a finite S-labeled graph (Q, E, λ) such that |Q(3r0 )| ≥ (1 − ε)|Q|,

(8.31)

where we recall that Q(r), r ∈ N, denotes the set of all q ∈ Q such that there exists an S-labeled graph isomorphism ψq,r : BS (r) → B(q, r) satisfying ψq,r (1G ) = q (cf. Theorem 7.7.1). Note the inclusions Q(r0 ) ⊃ Q(2r0 ) ⊃ · · · ⊃ Q(ir0 ) ⊃ Q((i + 1)r0 ) ⊃ · · · . (cf. (7.41); see also Fig. 7.2). Also recall from Lemma 7.7.2 that B(q, r0 ) ⊂ Q(ir0 ) for all Q((i + 1)r0 ) and i ≥ 0. For each integer i ≥ 1, we define the map μi : V Q(ir0 ) → V Q((i+1)r0 ) by setting, for all u ∈ V Q(ir0 ) and q ∈ Q((i + 1)r0 ), μi (u)(q) = μ u|B(q,r0 ) ◦ ψq,r0 (1G ), where ψq,2r0 is the unique isomorphism of S-labeled graphs from BS (r0 ) ⊂ G to B(q, r0 ) ⊂ Q sending 1G to q (cf. 7.39 and 7.40). Similarly, we define the map νi : V Q(ir0 ) → V Q((i+1)r0 ) by setting, for all u ∈ V Q(ir0 ) and q ∈ Q((i + 1)r0 ), νi (u)(q) = ν u|B(q,r0 ) ◦ ψq,r0 (1G ). From the fact that τ −1 ◦ τ is the identity map on V G , we deduce that the composite νi+1 ◦μi : V Q(ir0 ) → V Q((i+2)r0 ) is the identity on V Q((i+2)r0 ) . More precisely, denoting by ρi : V Q(ir0 ) → V Q((i+2)r0 ) the restriction map, we have that νi+1 ◦ μi = ρi for all i ≥ 1. In particular, we have ν2 ◦ μ1 = ρ1 . Thus, setting Z = μ1 (V Q(r0 ) ) ⊂ V Q(2r0 ) , we deduce that ν2 (Z) = ρ1 (V Q(r0 ) ) = V Q(3r0 ) . It follows that (8.32) dim(Z) ≥ d|Q(3r0 )|.  Let Q ⊂ Q(3r0 ) be as in (7.59) and set Q = q ∈Q B(q  , r0 ). Note that Q ⊆ Q(2r0 ) so that |Q(2r0 )| = |Q | · |BS (r0 )| + |Q(2r0 ) \ Q |.

(8.33)

Now observe that, for all q ∈ Q(2r0 ), we have a natural isomorphism of vector spaces Z|B(q,r0 ) → Y |BS (r0 ) given by u → u◦ψq,r0 , where ψq,r0 denotes as above the unique isomorphism of S-labeled graphs from BS (r0 ) to B(q, r0 ) such that ψq,r0 (1G ) = q. Since Y |BS (r0 )  V BS (r0 ) , this implies that dim Z|B(q,r0 ) = dim(Y |BS (r0 ) ) ≤ d · |BS (r0 )| − 1, (8.34) for all q ∈ Q . Thus we have

8.15 Stable Finiteness of Group Algebras

dim(Z) ≤ dim Z|Q + dim Z|Q(2r0 )\Q

327

≤ |Q | · (d · |BS (r0 )| − 1) + d · |Q(2r0 ) \ Q |  |Q |  = d |Q(2r0 )| − d where the last equality follows from (8.33). Comparing this with (8.32) we obtain |Q | . |Q(3r0 )| ≤ |Q(2r0 )| − d Thus, |Q| ≥ |Q(2r0 )| ≥ |Q(3r0 )| +

|Q | d

|Q(3r0 )| by (7.59), d|B(2r0 )|   1 = |Q(3r0 )| 1 + d|B(2r0 )|

≥ |Q(3r0 )| +

> |Q(3r0 )|(1 − ε)−1 where the last inequality follows from (8.30). This yields |Q(3r0 )| < (1 − ε)|Q| which contradicts (8.31). This shows that τ (V G ) = Y = V G , that is, τ is surjective. It follows that the group G is L-surjunctive.  

8.15 Stable Finiteness of Group Algebras Let R be a ring and denote by 1R its unity element. If a, b ∈ R satisfy ab = 1R , then one says that b is a right-inverse of a and that a is a left-inverse of b. An element a ∈ R is said to be right-invertible (resp. left-invertible) if it admits a right-inverse (resp. a left-inverse). Every invertible element in R is both right-invertible and left-invertible. Conversely, suppose that an element a ∈ R admits a right-inverse b and a left-inverse b . Then, we have ab = 1R and b a = 1R , so that b = (b a)b = b (ab ) = b . This shows that a is invertible with inverse a−1 = b = b . Thus, if an element is both right-invertible and left-invertible, then it is invertible. One says that the ring R is directly finite if every right-invertible element (or, equivalently, every left-invertible element) is invertible. This is equivalent to saying that if any two elements a, b ∈ R satisfy ab = 1R , then they also satisfy ba = 1R .

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The ring R is said to be stably finite if the matrix ring Matd (R) is directly finite for any d ≥ 1. Observe that every stably finite ring R is directly finite since Mat1 (R) = R. Proposition 8.15.1. Every finite ring is stably finite. Proof. Let R be a finite ring. Suppose that a, b ∈ R satisfy ab = 1R . Consider the map f : R → R defined by f (r) = ar. We have f (br) = a(br) = (ab)r = 1R r = r for all r ∈ R. Therefore, the map f is surjective. As R is finite, this implies that f is also injective. Since f (ba) = a(ba) = (ab)a = a = f (1R ), we deduce that ba = 1R . This shows that every finite ring is directly finite. If the ring R is finite, then the ring Matd (R) is also finite for every d ≥ 1. Consequently, every finite ring is stably finite.   Proposition 8.15.2. Every commutative ring is stably finite. Proof. Let d ≥ 1 and suppose that a ∈ Matd (R) is right-invertible. Then there exists b ∈ Matd (R) such that ab = Id . This implies that det(a) det(b) = det(ab) = det(Id ) = 1R . It follows that det(a) is an invertible element in R. Therefore a is invertible in Matd (R). This shows that Matd (R) is directly finite for any d ≥ 1. Consequently, R is stably finite.   Let us give an example of a ring which is not directly finite. Example 8.15.3. Let R be a nonzero ring and consider the free left R-module M = ⊕n∈N R. Every element in M can be represented in the form m = (mn )n∈N where mn ∈ R for all n ∈ N and mn = 0R for all but finitely many n ∈ N. Consider the maps a : M → M and b : M → M defined by setting a(m) = m (resp. b(m) = m ) where mn = mn−1 for n ≥ 1 and m0 = 0R (resp. mn = mn+1 for all n ∈ N). Then a and b are obviously R-linear, in other words, a, b ∈ EndR (M ) and ab = IdM = 1EndR (M ) . Consider the element m ∈ M such that m0 = 1R and mn = 0R for all n ≥ 1. As a(m) = 0 we have ba(m) = 0. This shows that ba = 1EndR (M ) . It follows that the ring EndR (M ) is not directly finite. Let G be a group and let V be a vector space over a field K. Recall that the set LCA(G; V ) consisting of all linear cellular automata τ : V G → V G has a natural structure of K-algebra in which the multiplication is given by the composition of maps. Proposition 8.15.4. Let G be a group and let V be a vector space over a field K. Let τ : V G → V G be a linear cellular automaton. Then the following hold: (i) if τ is left-invertible in LCA(G; V ), then τ is injective;

8.15 Stable Finiteness of Group Algebras

329

(ii) if τ is right-invertible in LCA(G; V ), then τ is surjective; (iii) if τ is invertible in LCA(G; V ), then τ is bijective. If, in addition, the vector space V is finite-dimensional, then: (iv) τ is left-invertible in LCA(G; V ) if and only if τ is injective; (v) τ is invertible in LCA(G; V ) if and only if τ is bijective. Proof. (i) Suppose that τ ∈ LCA(G; V ) is left-invertible. This means that there exists σ ∈ LCA(G; V ) such that σ ◦ τ = IdV G . As IdV G is injective, this implies that τ is injective. (ii) Suppose that τ ∈ LCA(G; V ) is right-invertible. This means that there exists σ ∈ LCA(G; V ) such that τ ◦ σ = IdV G . As IdV G is surjective, this implies that τ is surjective. (iii) This immediately follows from (i) and (ii). (iv) This immediately follows from (i) and Theorem 8.12.1. (v) This immediately follows from (iii) and Corollary 8.12.2.   Remarks 8.15.5. (a) If V is infinite-dimensional, it may happen that a bijective linear cellular automaton τ : V G → V G is not left-invertible in LCA(G; V ). Therefore, the converses of assertions (i) and (iii) in Proposition 8.15.4 are false if we do not add the hypothesis that V is finitedimensional. To see this, consider the bijective linear cellular automaton τ : V Z → V Z described in Remark 8.12.3. Then, there is no σ ∈ LCA(G; V ) such that σ ◦ τ = IdV G since otherwise the inverse map of τ would coincide with σ, which is impossible as τ is not an invertible cellular automaton. (b) The converse of assertion (ii) in Proposition 8.15.4 does not hold even under the additional hypothesis that V is finite-dimensional. For instance, take an arbitrary field K and consider the linear cellular automaton τ : KZ → KZ defined by τ (x)(n) = x(n + 1) − x(n) for all x ∈ KZ and n ∈ Z. Observe that τ is surjective. Indeed, given y ∈ KZ , the configuration x ∈ KZ defined by ⎧ ⎪ if n = 0, ⎨0 x(n) = y(0) + y(1) + · · · + y(n − 1) if n > 0, ⎪ ⎩ y(n) + y(n + 1) + · · · + y(−1) if n < 0, clearly satisfies τ (x) = y. However, τ is not right-invertible in LCA(Z; K). To see this, observe that, as the K-algebra LCA(Z; K) is commutative by Corollary 8.5.4, the right-invertibility of τ would imply the bijectivity of τ by Proposition 8.15.4(iii). But τ is not injective as all constant configurations are mapped to 0. Corollary 8.15.6. Let G be a group and let K be a field. Let V be a vector space over K of finite dimension dimK (V ) = d ≥ 1. Then the following conditions are equivalent: (a) every injective linear cellular automaton τ : V G → V G is surjective; (b) the K-algebra LCA(G; V ) is directly finite;

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8 Linear Cellular Automata

(c) the K-algebra Matd (K[G]) is directly finite. Proof. The equivalence between conditions (b) and (c) follow from the fact that the K-algebras LCA(G; V ) and Matd (K[G]) are isomorphic by Corollary 8.7.8. Suppose (a). Let τ ∈ LCA(G; V ) be a left-invertible element in LCA(G; V ). Then τ is injective by Proposition 8.15.4(i). Condition (a) implies then that τ is surjective and hence bijective. By applying Proposition 8.15.4(v), we deduce that τ is invertible in LCA(G; V ). This shows that (a) implies (b). Conversely, suppose (b). Let τ : V G → V G be an injective linear cellular automaton. Then τ is left-invertible in LCA(G; V ) by Proposition 8.15.4(iv). It follows from condition (b) that τ is invertible in LCA(G; V ). By using Proposition 8.15.4(ii), we deduce that τ is surjective. This shows that (b) implies (a).   Corollary 8.15.7. Let G be a group. Then the following conditions are equivalent: (a) the group G is L-surjunctive; (b) for any field K, the group algebra K[G] is stably finite.

 

From Theorem 8.14.4 and Corollary 8.15.7, we deduce the following: Corollary 8.15.8. Let G be a sofic group and let K be a field. Then the group algebra K[G] is stably finite.  

8.16 Zero-Divisors in Group Algebras and Pre-injectivity of One-Dimensional Linear Cellular Automata Let R be a ring. A nonzero element a ∈ R is said to be a left zero-divisor (resp. a right zero-divisor ) if there exists a nonzero element b in R such that ab = 0 (resp. ba = 0). Note that the existence of a left zero-divisor in R is equivalent to the existence of a right zero-divisor. One says that the ring R has no zero-divisors if R admits no left (or, equivalently, no right) zero-divisors. Examples 8.16.1. (a) Let R = Z/nZ, n ≥ 2, be the (commutative) ring of integers modulo n. Then R has no zero-divisors if and only if n is a prime number. (b) Let G be a group and let R be a nonzero ring. Suppose that G contains an element g0 of finite order n ≥ 2 and let H = {1G , g0 , g02 , . . . , g0n−1 } denote the subgroup of G generated by g0 . Consider the elements α, β ∈ R[G] defined respectively by

8.16 Zero-Divisors in Group Algebras

⎧ ⎪ ⎨1 α(g) = −1 ⎪ ⎩ 0

if g = 1G , if g = g0 , if g ∈ / {1G , g0 }.

331

 1 and β(g) = 0

if g ∈ H, otherwise.

One easily checks that α = 0, β = 0, and αβ = βα = 0. Thus, α and β are both left and right zero-divisors in R[G]. Let R be a ring. An element x ∈ R is called an idempotent if it satisfies 1. x2 = x. An idempotent x ∈ R is said to be proper if x = 0 and x = Proposition 8.16.2. Let R be a ring. Then the following hold: (i) if R has no zero-divisors, then R has no proper idempotents; (ii) if R has no proper idempotents, then R is directly finite. Proof. (i) Every idempotent x ∈ R satisfies x(x − 1) = x2 − x = 0. If R has no zero-divisors, this implies x = 0 or x = 1. (ii) Suppose that R has no proper idempotents. Let a, b ∈ R such that ab = 1. Then we have (ba)2 = b(ab)a = ba so that ba is an idempotent. Therefore, ba = 0 or ba = 1. If ba = 0, then a = (ab)a = a(ba) = 0 so that 0 = 1 and R is reduced to 0. Therefore, we have ba = 1 in all cases. This shows that R is directly finite.   Remarks 8.16.3. (a) A ring without proper idempotents may admit zerodivisors. For example, the ring Z/8Z has no proper idempotents. However, the classes of 2, 4, and 6 are zero-divisors in Z/8Z. (b) A directly finite ring may admit proper idempotents. For example, take a nonzero commutative ring R. Then the ring 2 (R) is directly fi Mat nite by Proposition 8.15.2. However, the matrices 10 00 and 00 01 are proper idempotents in Mat2 (R). A group G is called a unique-product group if, given any two non-empty finite subsets A, B ⊂ G, there exists an element g ∈ G which can be uniquely expressed as a product g = ab with a ∈ A and b ∈ B. Remark 8.16.4. A unique-product group is necessarily torsion-free. Indeed, suppose that G is a group containing an element g0 of order n ≥ 2. Take A = B = {1G , g0 , g02 , . . . , g0n−1 }. Then AB = A and there is no g ∈ AB which can be uniquely written in the form g = ab with a ∈ A and b ∈ B. Recall that a total ordering on a set X is a binary relation ≤ on X which is reflexive, antisymmetric, transitive, and such that one has x ≤ y or y ≤ x for all x, y ∈ X. A group G is called orderable if it admits a left-invariant total ordering, that is, a total ordering ≤ such that g1 ≤ g2 implies gg1 ≤ gg2 for all g, g1 , g2 ∈ G.

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Examples 8.16.5. (a) Every subgroup of an orderable group is orderable. Indeed, if H is a subgroup of a group G and ≤ is a left-invariant total ordering on G, then the restriction of ≤ to H is a left-invariant total ordering on H. (b) The additive groups Z, Q, and R are orderable since the usual ordering on R is translation-invariant. (c) The direct product of two orderable groups is an orderable group. Indeed, let G1 and G2 be two groups and suppose that ≤1 and ≤2 are leftinvariant total orderings on G1 and G2 respectively. Then the lexicographic ordering ≤ on G1 × G2 defined by ⎧ ⎪ ⎨g1 ≤1 g2 (g1 , g2 ) ≤ (h1 , h2 ) ⇐⇒ or ⎪ ⎩ g1 = g2 and h1 ≤2 h2 is a left-invariant total ordering on G1 × G2 . (d) More generally, the direct product of any family (finite or infinite) of orderable groups is an orderable group. Indeed, let (Gi )i∈I be a family of orderable groups. Let ≤i be a left-invariant total ordering on Gi for each i ∈ I. Let us fix a well-ordering on I, i.e., a total ordering such that every nonempty subset of I admits a minimal element (the fact that any set can be well-ordered is a basic fact in set theory which may be deduced from the  Axiom of Choice). Then the lexicographic ordering on G = i∈I Gi , which is defined by setting g ≤ h for g = (gi ), h = (hi ) ∈ G if and only if g = h or gi0 < hi0 where i0 = min{i ∈ I  : gi = hi }, is a left-invariant total ordering on G. As ⊕i∈I Gi is a subgroup of i∈I Gi , it follows that the direct sum of any family of orderable groups is an orderable group. Since a free abelian group is isomorphic to a direct sum of copies of Z, we deduce in particular that every free abelian group is orderable. (e) In fact, every torsion-free abelian group is orderable. Indeed, if G is a torsion-free abelian group then G embeds in the Q-vector space G ⊗Z Q via the map g → g ⊗ 1. On the other hand, the additive group underlying a Q-vector space V is isomorphic to a direct sum of copies of Q (since V admits a Q-basis) and hence orderable. (f) Suppose that G is a group containing a normal subgroup N such that both N and G/N are orderable groups. Then the group G is orderable. Indeed, let ≤1 (resp. ≤2 ) be a left-invariant total ordering on N (resp. G/N ). Then one easily checks that the binary relation ≤ on G defined by setting ⎧ ⎪ ⎨ρ(g) ≤2 ρ(h) g ≤ h ⇐⇒ or ⎪ ⎩ ρ(g) = ρ(h) and 1G ≤1 g −1 h is a left-invariant total ordering on G. (g) Let R be any ring. Consider the Heisenberg group

8.16 Zero-Divisors in Group Algebras

⎧ ⎛ ⎞ 1yz ⎨ HR = M (x, y, z) = ⎝0 1 x⎠ ⎩ 001

333

:

x, y, z ∈ R

⎫ ⎬ ⎭

.

(cf. Example 4.6.5). The kernel of the group homomorphism of HR onto R2 given by M (x, y, z) → (x, y) is the normal subgroup ⎧⎛ ⎫ ⎞ ⎨ 10z ⎬ N = ⎝0 1 0⎠ : z ∈ R . ⎩ ⎭ 001 As the groups N and HR /N are both abelian, it follows from Examples (e) and (f) above that HR is orderable. Given a set X equipped with a total ordering ≤, we denote by Sym(X, ≤) the group of order-preserving permutations of X, that is, the subgroup of Sym(X) consisting of all bijective maps f : X → X such that x ≤ y implies f (x) ≤ f (y) for all x, y ∈ X. Proposition 8.16.6. Let G be a group. Then the following conditions are equivalent: (a) G is orderable; (b) there exists a set X equipped with a total ordering ≤ such that G is isomorphic to a subgroup of Sym(X, ≤). Proof. Suppose that ≤ is a left invariant total ordering on G. Then the action of G on itself given by left multiplication is order-preserving. Thus G is isomorphic to a subgroup of Sym(G, ≤). This shows that (a) implies (b). Conversely, suppose that X is a set equipped with a total ordering ≤. Choose a well ordering ≤W on X. Then we can define a lexicographic ordering ≤L on Sym(X, ≤) by setting f ≤L g for f, g ∈ Sym(X, ≤) if and only if f = g or f (x0 ) ≤ g(x0 ) where x0 = min{x ∈ X : f (x) = g(x)}. Clearly ≤L is a left invariant total ordering on Sym(X, ≤). It follows that Sym(X, ≤) is an orderable group. As any subgroup of an orderable group is orderable, we conclude that (b) implies (a).   Corollary 8.16.7. The group Homeo+ (R) of orientation-preserving homeomorphisms of R is orderable.   Proposition 8.16.8. Every orderable group is a unique-product group. Proof. Let G be an orderable group and let ≤ be a left-invariant total ordering on G. Let A and B be nonempty finite subsets of G. Set bm = min B and let am ∈ A be the unique element such that am bm = min Abm . Consider the element g = am bm . For all a ∈ A and b ∈ B, we have am bm ≤ abm ≤ ab. Thus, if g = ab for some a ∈ A and b ∈ B, then am bm = abm = ab, so that we get am = a and bm = b after right and left cancellation. This shows that G is a unique-product group.  

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Proposition 8.16.9. Let G be a unique-product group and let R be a ring with no zero-divisors. Then the group ring R[G] has no zero-divisors. Proof. Let α and β be nonzero elements in R[G], and denote by A and B their supports. As G is a unique-product group, there is an element g ∈ AB such that there exists a unique element (a, b) ∈ A × B such that g = ab. Then we have  α(h1 )β(h2 ) = α(a)β(b). (αβ)(g) = h1 ∈A,h2 ∈B h1 h2 =g

As R has no zero-divisors, this implies (αβ)(g) = 0. Thus, we have αβ = 0. This shows that R[G] has no zero-divisors.   Corollary 8.16.10. Let G be a unique-product group and let K be a field. Then the group algebra K[G] has no zero-divisors.   Let G be a group and let K be a field. We have seen in Corollary 8.5.3 that the map Ψ : K[G] → LCA(G; K) defined by  Ψ (α)(x)(g) = α(h)x(gh) h∈G

for all α ∈ K[G], x ∈ KG , and g ∈ G, is a K-algebra isomorphism. Proposition 8.16.11. Let G be a group and let K be a field. Let α be a nonzero element in K[G]. Then the following conditions are equivalent: (a) α is not a left zero-divisor in K[G]; (b) the linear cellular automaton Ψ (α) : KG → KG is pre-injective. Proof. By Proposition 8.2.5, the linear cellular automaton Ψ (α) is not preinjective if and only if there exists a nonzero element β ∈ K[G] such that Ψ (α)(β) = 0. As   Ψ (α)(β)(g) = α(h)β(gh) = α(h)β ∗ (h−1 g −1 ) = (αβ ∗ )(g −1 ), h∈G

h∈G

for all β ∈ K[G] and g ∈ G, we deduce that Ψ (α) is not pre-injective if and only if there exists a nonzero element β ∈ K[G] such that αβ ∗ = 0, that is, if and only if α is a left zero-divisor in K[G]. This shows that conditions (a) and (b) are equivalent.   Corollary 8.16.12. Let G be a group and let K be a field. Then the following conditions are equivalent: (a) the group algebra K[G] has no zero-divisors; (b) every non-identically-zero linear cellular automaton τ : KG → KG is preinjective.  

Notes

335

From Corollary 8.16.12 and Theorem 8.9.6 we deduce the following. Corollary 8.16.13. Let G be an amenable group and let K be a field such that K[G] has no zero-divisors. Then every non-identically-zero linear cellular automaton τ : KG → KG is surjective.   Observe that it follows from Theorem 4.6.1, Example 8.16.5(e), Proposition 8.16.8, and Corollary 8.16.10 that torsion-free abelian groups satisfy the hypotheses of Corollary 8.16.13 for any field K. This implies in particular that if G is a torsion-free abelian group and S is a nonempty subset of G which is not reduced to the identity element, then the discrete Laplacian ΔS : KG → KG is surjective for any field K.

Notes In the literature, the term linear is used by some authors with a different meaning, namely to designate a cellular automaton τ : AG → AG for which the alphabet A is a finite abelian group and τ is a group endomorphism of AG . Such cellular automata are also called additive cellular automata. Linear cellular automata with vector spaces as alphabets were considered by the authors in a series of papers starting with [CeC1]. The fact that the algebra of linear cellular automata over a group G whose alphabet is a ddimensional vector space over a field K is isomorphic to the algebra of d × d matrices with coefficients in K[G] (Corollary 8.7.8) was proved in Sect. 6 of [CeC1] for d = 1 and in Sect. 4 of [CeC2] for all d ≥ 1. The representations of linear cellular automata both as elements in EndK (V )[G] and as elements in EndK[G] (V [G]) were given in Sect. 4 of [CeC5]. Recall that a Laurent polynomial over a field K is a polynomial in the variable t and its inverse t−1 with coefficients in K. The K-algebra of Laurent polynomials is thus denoted by K[t, t−1 ]. Also, a Laurent polynomial matrix is a matrix whose entries are Laurent polynomials. For d ≥ 1, denote by Matd (K[t, t−1 ]) the K-algebra of d × d Laurent polynomial matrices over K. When G = Z, there are canonical isomorphisms of K-algebras K[Z] ∼ = Matd (K[t, t−1 ]). In [LiM, Sect. 1.6] = K(t, t−1 ] and LCA(Z; Kd ) ∼ d cellular automata τ ∈ LCA(Z; K ) are called (d × d) convolutional encoders. The notion of mean dimension for vector subspaces of V G , where G is an amenable group and V is a finite-dimensional vector space, was introduced by Gromov in [Gro5] and [Gro6]. Mean dimension was used by Elek [Ele] to prove that, given any amenable group G and any field K, there exists a non-trivial homomorphism from the Grothendieck group of finitely generated modules over the group algebra K[G] into the additive group of real numbers. As for entropy, it can be shown, as an application of the Ornstein-Weiss convergence

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theorem (see [OrW], [LiW], [Gro6], [Ele], [Kri]) that the lim sup in (8.18) is in fact a true limit and does not depend on the Følner net F . The fact that the image of a linear cellular automaton with finite dimensional alphabet is closed (Theorem 8.8.1) was proved in Sect. 3 of [CeC1]. The linear version of the Garden of Eden theorem (Theorem 8.9.6) was first proved for countable amenable groups in [CeC1] and then extended to all amenable groups in [CeC8] using induction and restriction for linear cellular automata. The linear version of the Garden of Eden theorem was generalized to R-linear cellular automata with coefficients in semisimple left R-modules of finite length over a ring R and over amenable groups in [CeC4]. Invertibility of linear cellular automata with finite dimensional alphabet V was proved in Sect. 3 of [CeC2] for bijective linear cellular automata τ : X → Y between closed linear subshifts X, Y ⊂ V G over countable groups and in [CeC8] for bijective linear cellular automata τ : V G → V G over any group G. In [CeC11] it is shown that if G is a non-periodic group, then for every infinite-dimensional vector space V over a field K there exist a bijective cellular automaton τ : V G → V G which is not invertible (cf. Theorem 8.12.1 and Remark 8.12.3) and a cellular automaton τ  : V G → V G whose image τ  (V G ) is not closed in V G with respect to the prodiscrete topology (cf. Theorem 8.8.1 and Example 8.8.3). Theorem 8.13.2 and Corollary 8.13.4 were proved in [CeC6] and [CeC9]. Directly finite rings are sometimes called Dedekind finite rings or von Neumann finite rings. In [Coh], P.M. Cohn constructed, for each integer d ≥ 1, a ring R such that Matd (R) is directly finite but Matd+1 (R) is not directly finite. I. Kaplansky [Kap2, p. 122], [Kap3, Problem 23] observed that techniques from the theory of operator algebras could be used to prove that, for any group G and any field K of characteristic 0, the group algebra K[G] is stably finite and asked whether this property remains true for fields of characteristic p > 0. The stable finiteness of K[G] in arbitrary characteristic was established for free-by-abelian groups G by P. Ara, K.C. O’Meara and F. Perera in [AOP]. This was extended to all sofic groups by Elek and Szab´ o [ES1], using the notion of von Neumann dimension for continuous regular rings. The proof of Elek and Szab´ o’s result via linear cellular automata which is presented in this chapter (cf. Corollary 8.15.8) was given in [CeC2]. The notion of L-surjunctivity was also introduced in [CeC2] and the equivalence between L-surjunctivity and stable finiteness was established in Corollary 4.3 therein. In [CeC3], the authors proved that if R is a ring and G is a residually finite group, then every injective R-linear cellular automaton over G whose alphabet is an Artinian left R-module is surjunctive. In [CeC5], it was shown that if R is a ring, then R-linear cellular automata with coefficients in left R-modules of finite length (thus a stronger condition than being Artinian) over sofic groups (thus a weaker condition than being residually finite) are surjunctive. This last result was used to show in [CeC5] that the group ring R[G] is stably finite whenever R is a left (or right) Artinian ring and G is a

Notes

337

sofic group. This yields an extension of the Elek and Szab´ o stable finiteness result since any division ring is Artinian. Unique-product groups were introduced by W. Rudin and H. Schneider in [RuS] under the name of Ω-groups. The question of the existence of a torsionfree group which is not unique-product was raised by Rudin and Schneider [RuS, p. 592]. This question was answered in the affirmative by E. Rips and Y. Segev [RiS] (see also [Pro]). More information on orderable groups may be found for example in [BoR], [Pas], and [Gla]. A group G is said to be locally indicable if any nontrivial finitely generated subgroup of G admits an infinite cyclic quotient. Every locally indicable group is orderable (see for example [BoR, Theorem 7.3.1] or [Gla, Lemma 6.9.1]). This implies in particular that locally nilpotent torsion-free groups, free groups, and fundamental groups of surfaces not homeomorphic to the real projective plane are all orderable groups. It was observed by G.M. Bergman [Ber] that the  universal covering group SL 2 (R) of SL2 (R) is orderable but not locally indi cable. The group SL 2 (R) is orderable since it has a natural faithful action by orientation-preserving homeomorphisms of the real line but it is not locally indicable since it contains nontrivial finitely generated perfect groups. By a recent result due to D.W. Morris [Morr, Theorem B], every amenable orderable group is locally indicable. The orderability of the braid groups Bn was established independently by P. Dehornoy [Deh] and W. Thurston. It follows from a result of E.A. Gorin and V.Ja. Lin [GoL] that the group Bn is not locally indicable for n ≥ 5. A group is called bi-orderable if it admits a total ordering which is both left and right invariant. All free groups and all locally nilpotent torsion-free groups are bi-orderable. For n ≥ 3, the braid group Bn is not bi-orderable. However, the pure braid groups Pn (i.e., the kernel of the natural epimorphism of Bn onto Symn ) is bi-orderable for all n. A theorem due to A.I. Mal’cev [Mal2] and B.H. Neumann [Neu1] says that, given a group G equipped with a total ordering which is both left and right invariant and a field K, the vector subspace of KG consisting of all maps x : G → K whose support is a wellordered subset of G is a division K-algebra for the convolution product (see [Pas, Theorem 2.11 in Chap. 13]). When G = Z, this division algebra is the field of Laurent series with coefficients in K. The Mal’cev-Neumann theorem implies in particular that, for any bi-orderable group G and any field K, the group algebra K[G] embeds in a division K-algebra and is therefore stably finite. A famous conjecture attributed to Kaplansky is the zero-divisor conjecture which states that if G is a torsion-free group then the group algebra K[G] has no zero-divisors for any field K (see [Kap1], [Kap2, p. 122], [Pas, Chap. 13]). The observation that unique-product groups (and hence orderable groups) satisfy the Kaplansky zero-divisor conjecture (see Corollary 8.16.10) was made by Rudin and Schneider [RuS, Theorem 3.2]. The class of elementary amenable groups is the smallest class of groups containing all finite

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and all abelian groups that is closed under taking subgroups, quotients, extensions, and directed unions. It is known (see [KLM, Theorem 1.4]) that torsion-free elementary amenable groups satisfy the Kaplansky zero-divisors conjecture. This implies in particular that all torsion-free virtually solvable groups satisfy the Kaplansky zero-divisors conjecture. According to Corollary 8.16.12, the zero-divisor conjecture is equivalent to saying that if G is a torsion-free group and K is a field then every non-identically-zero linear cellular automaton τ : KG → KG is pre-injective. This reformulation of the Kaplansky zero-divisor conjecture in terms of linear cellular automata was given in Sect. 6 of [CeC1].

Exercises 8.1. Let G be a group and let R be a nonzero ring. Show that the ring R[G] is commutative if and only if both G and R are commutative. 8.2. Recall that the center of a ring A is the subring B of A consisting of the elements x ∈ A which satisfy ax = xa for all a ∈ A. Let G be a group and let R be a commutative ring. Let α ∈ R[G]. Show that α is in the center of R[G] if and only if α is constant on each conjugacy class of G. 8.3. One says that a group G has the ICC-property if every conjugacy class of G except {1G } is infinite. (a) Show that if a group G has the ICC-property then the center of G is reduced to the identity element. (b) Show that if a group G has the ICC-property then every normal subgroup of G which is not reduced to the identity element is infinite. (c) Let X be an infinite set. Show that the group Sym0 (X), which consists of all permutations of X with finite support, has the ICC-property. (d) Show that every nonabelian free group has the ICC-property. (e) Let G be a group with the ICC-property and let R be a commutative ring. Show that the center of R[G] consists of the maps x : G → R which satisfy x(g) = 0R for all g ∈ G \ {1G }. Hint: Use the result of Exercise 8.2. (f) Let G be a group and let R be a nonzero commutative ring. Show that G has the ICC-property if and only if the center of R[G] is isomorphic to the ring R. 8.4. Let G be a group, K a field, and d ≥ 1 an integer. Show that the Kalgebras Matd (K)[G] and Matd (K[G]) are isomorphic. 8.5. Show that Theorem 8.13.2 remains valid if the field R is replaced by the field C of complex numbers. Hint: Write every f ∈ CG in the form f = f0 +if1 , where f0 , f1 ∈ RG .

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8.6. Let G be a group and let S ⊂ G be a non-empty finite subset. Denote by ·, · the standard scalar product on R[G] ⊂ 2 (G) and let  ·  denote the associated norm. (a) Show that for all x ∈ R[G] and λ ∈ R, one has x, ΔS (x) + λx =

1  |x(g) − x(gs)|2 + λx2 . 2 g∈G s∈S

(b) Show that if the subgroup generated by S is infinite then for any non–empty finite subset F ⊂ G there exists s ∈ S such that F s ⊂ F . (c) Deduce from (a) and (b) that if λ ≥ 0 and the subgroup generated by S is infinite, then the linear cellular automaton ΔS + λ IdRG : RG → RG is pre-injective. 8.7. Showthat if (Ri )i∈I is a family of directly finite rings then the product ring P = i∈I Ri is directly finite. 8.8. Show that every subring of a directly finite (resp. stably finite) ring is directly finite (resp. stably finite). 8.9. Let K be a field and let V be an infinite-dimensional vector space over K. Show that the K-algebra EndK (V ) is not directly finite. 8.10. Let R be a ring and let M be a left R-module. One says that the module M is Hopfian if every surjective endomorphism of M is injective. Show that if M is Hopfian, then the ring EndR (M ) is directly finite. 8.11. Let R be a ring and let M be a left R-module. One says that the module M is Noetherian if its submodules satisfy the ascending chain condition, i.e., every increasing sequence N 1 ⊂ N2 ⊂ . . . of submodules of M stabilizes (there is an integer i0 ≥ 1 such that Ni = Ni0 for all i ≥ i0 ). Show that if M is Noetherian, then M is Hopfian. Hint: Suppose that f is a surjective endomorphism of M and consider the sequence of submodules Ni = Ker(f i ), i ≥ 1. 8.12. Let R be a ring and let P be a left R-module. One says that the module P is projective if for every homomorphism f : P → M and every surjective  → M of left R-modules, there exists a homomorphism homomorphism g : M  such that f = g ◦ h. Show that if the module P is projective, then h: P → M P is Hopfian if and only if the ring EndR (P ) is directly finite. 8.13. Let R be a ring and let d ≥ 1 be an integer. Equip Rd with its natural structure of left R-module. Show that Rd is Hopfian if and only if the ring Matd (R) is directly finite.

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8.14. One says that a ring R is left Noetherian if R is Noetherian as a left module over itself. (a) Let R be a left Noetherian ring. Show by induction that Rd is Noetherian as a left R-module for each integer d ≥ 1. (b) Show that every left Noetherian ring is stably finite. Hint: Use Exercises 8.11 and 8.13. 8.15. One says that a ring R has the unique rank property if it satisfies the following condition: if m and n are positive integers such that Rm and Rn are isomorphic as left R-modules, then one has m = n. Show that every stably finite ring has the unique rank property. 8.16. Let K be a field and let V be an infinite-dimensional vector space over K. Show that the ring R = EndK (V ) does not have the unique rank property. Hint: Observe that the vector spaces V and V ⊕ V are isomorphic and then prove that the set consisting of all K-linear maps f : V ⊕ V → V , with its natural structure of left R-module, is isomorphic to both R and R2 . 8.17. Show that every division ring is stably finite. 8.18. A ring R is said to be unit-regular if for any a ∈ R there exists an invertible element u ∈ R such that a = aua. (a) Show that every division ring is unit-regular. (b) Show that if K is a field then the ring Matd (K) is unit-regular for every d ≥ 1. Hint: Prove that if A ∈ Matd (K) has rank r then there exist invertible matrices U, V ∈ GLd (K) such that A = U Dr V , where Dr ∈ Matd (K) is the diagonal matrix defined by δij = 1 if 1 ≤ i = j ≤ r and δij = 0 otherwise, and then observe that A = AXA, where X = (U V )−1 . (c) Show that every unit-regular ring is directly finite. (d) Prove that the ring Z is not unit-regular. (e) A ring R is called a Boolean ring if a2 = a for all a ∈ R. Show that every Boolean ring is commutative and unit-regular. 8.19. One says that a ring R is a right Ore ring if R has no zero-divisors and if for any pair a, b of nonzero elements in R there exist nonzero elements u, v ∈ R such that au = bv. Left Ore rings are defined similarly. Show that any right (or left) Ore ring is directly finite. Hint: Prove that ab = 1R implies ba = 1R by a direct argument, or show that R can be embedded as a subring of a division ring by adapting the construction of the field of fractions of an integral domain. 8.20. Let K be a field. Show that every finite-dimensional K-algebra is stably finite. 8.21. Let G be an amenable group and let F be a Følner net for G. Let V be a finite-dimensional vector space. Suppose that X (resp. Y ) is a vector subspace of V G . Show that mdimF (X ∪ Y ) ≤ mdimF (X) + mdimF (Y ).

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8.22. Use Corollary 8.15.7 and the result of Exercise 8.20 to recover the fact that every finite group is L-surjunctive. 8.23. Use Proposition 8.15.2 and Corollary 8.15.7 to recover the fact that every abelian group is L-surjunctive. 8.24. It follows from Theorem 8.14.4 that every residually finite group is Lsurjunctive. The goal of this exercise is to present an alternative proof of this result. Let G be a residually finite group and let V be a finite-dimensional vector space over a field K. Let τ : V G → V G be an injective linear cellular automaton. Fix a family (Γi )i∈I of subgroups of finite index of G such that  i∈I Γi = {1G } (the existence of such a family follows from the residual finiteness of G). (a) Show that, for each i ∈ I, the set Fix(Γi ) = {x ∈ V G : gx = x for all g ∈ Γi } is a finite-dimensional vector subspace of V G . (b) Show that τ (Fix(Γi )) = Fix(Γi ) for all i ∈ I. (c) Prove that i∈I Fix(Γi ) is dense in V G and conclude. 8.25. Let R be a ring and let x ∈ R. Show that x is an idempotent if and only if 1R − x is an idempotent. 8.26. Show that any subgroup of a unique-product group is a unique-product group. 8.27. Let G be a group. Suppose that G contains a normal subgroup N such that N and G/N are both unique-product groups. Show that G is a unique-product group. Hint: See for example [RuS, Theorem 6.1]. 8.28. Show that every residually orderable group is orderable. 8.29. Show that the limit of a projective system of orderable groups is orderable. 8.30. A group G is called bi-orderable if it admits a total ordering ≤ which is both left and right invariant, i.e., such that g1 ≤ g2 implies gg1 ≤ gg2 and g1 g ≤ g2 g for all g, g1 , g2 ∈ G. Let G be a bi-orderable group. Suppose that an element g ∈ G satisfies the following property: there exist an integer n ≥ 1 and elements h1 , h2 , · · · , hn ∈ G such that −1 −1 h1 gh−1 1 h2 gh2 · · · hn ghn = 1G .

Show that g = 1G . 8.31. Let G be a bi-orderable group. Show that if g, h ∈ G are such that g n = hn for some integer n ≥ 1, then g = h. 8.32. The Klein bottle group is the group K given by the presentation K = x, y; xyx−1 y. Thus, K is the quotient group K = F/N , where F is the free

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group based on x and y, and N is the normal closure of xyx−1 y in F . Let ρ : F → K denote the quotient homomorphism. (a) Let H denote the subgroup of K generated by ρ(y). Show that H is normal in K and that H and K/H are both infinite cyclic. (b) Show that K is an orderable group. (c) Show that the group K is not bi-orderable. 8.33. Let G be an amenable group, F a right Følner net for G, and V a finitedimensional vector space over some field K. Let X be a vector subspace of V G and let X denote the closure of X in V G for the prodiscrete topology. Show that X is a vector subspace of V G and that one has mdimF (X) = mdimF (X).

Appendix A

Nets and the Tychonoff Product Theorem

A.1 Directed Sets Recall that a partially ordered set is a set I equipped with a binary relation ≤ which is both reflexive (i ≤ i for all i ∈ I) and transitive (i ≤ j and j ≤ k implies i ≤ k for all i, j, k ∈ I). A directed set is a partially ordered set I which satisfies the following condition: for all i, j ∈ I, there exists an element k ∈ I such that i ≤ k and j ≤ k. Examples A.1.1. (a) The set Z equipped with the relation ≤ defined by i ≤ j ⇐⇒ i divides j is a directed set. (b) If E is an arbitrary set, then the set P(E) of all subsets of E is a directed set for inclusion. (c)) If X is a topological space and x is a point of X, then the set of neighborhoods of x, equipped with the relation ≤ defined by V ≤ W ⇐⇒ W ⊂ V, is a directed set. Indeed, if V and W are neighborhoods of x, then V ∩ W is a neighborhood of x satisfying V ≤ V ∩ W and W ≤ V ∩ W .

A.2 Nets in Topological Spaces Let X be a set. A net of points of X (or net in X) is a family (xi )i∈I of points of X indexed by some directed set I.

T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 9, © Springer-Verlag Berlin Heidelberg 2010

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Let (xi )i∈I and (yj )j∈J be nets in a set X indexed by directed sets I and J respectively. One says that the net (yj ) is a subnet of the net (xi ) if there is a map ϕ : J → I which satisfies the following conditions: (SN-1) yj = xϕ(j) for all j ∈ J; (SN-2) for each i ∈ I, there exists j ∈ J such that if k ∈ J and j ≤ k then i ≤ ϕ(k). Let X be a topological space. Let (xi )i∈I be a net in X and let a ∈. One says that the net (xi ) converges to a, or that a is a limit point of the net (xi ), if the following condition is satisfied: for each neighborhood V of a in X, there exists an element i0 ∈ I such that xi ∈ V for all i ≥ i0 . One says that the net (xi ) is convergent if there exists a point a in X such that (xi ) converges to a. It is clear that if the net (xi ) converges to a, then every subnet of (xi ) also converges to a. Proposition A.2.1. Let X be a topological space, Y ⊂ X and a ∈ X. Let Y denote the closure of Y in X. Then the following conditions are equivalent: (a) a ∈ Y ; (b) there exists a net (yi )i∈I of points of Y which converges to a in X. Proof. Suppose that (yi )i∈I is a net of points of Y which converges to a. Then, for each neighborhood V of a, there is an element i0 ∈ I such that yi ∈ V for all i ≥ i0 . Thus, every neighborhood of a meets Y . This shows that a ∈ Y . Conversely, suppose that a ∈ Y . Let I denote the directed set consisting of all neighborhoods of a in X partially ordered by reverse inclusion, that is, V ≤ W ⇐⇒ W ⊂ V. Since a is in the closure of Y , we can find for each neighborhood V ∈ I a point yV in Y such that yV ∈ V . It is clear that the net (yV )V ∈I converges to a.

Proposition A.2.2. A topological space X is Hausdorff if and only if every convergent net in X admits a unique limit. Proof. Suppose that X is Hausdorff. Consider a net (xi )i∈I in X which converges to some point a ∈ X. Let b be a point in X with a = b. Since X is Hausdorff, there exist a neighborhood V of a and a neighborhood W of b such that V ∩ W . For i large enough, the point xi is in V and therefore not in W . Therefore the net (xi )i∈I does not converge to b. This shows that every convergent net in X has a unique limit. Suppose now that X is not Hausdorff. Then there exist distinct points a and b in X such that each neighborhood of a meets each neighborhood of b. Consider the set I consisting of all pairs (V, W ), where V is a neighborhood of

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a and W is a neighborhood of b. partially ordered by declaring that (V  , W  ) ≤ (V, W ) if and only if V ⊂ V  and W ⊂ W  . Clearly I is a directed set. If we choose, for each i = (V, W ) ∈ I a point xi ∈ V ∩ W , then the net (xi )i∈I admits both points a and b as limits. This proves the converse implication.

Let (xi )i∈I be a net in a topological space X. One says that a point a ∈ X is a cluster point of the net (xi ) if it satisfies the following condition: for each neighborhood V of a in X and each i ∈ I, there exists an element j ∈ I such that i ≤ j and xj ∈ V . Proposition A.2.3. Let X be a topological space, (xi )i∈I a net in X, and a ∈ X. Then the following conditions are equivalent: (a) the point a is a cluster point of the net (xi ); (b) the net (xi ) admits a subnet converging to a. Proof. Suppose that (yj )j∈J is a subnet of the net (xi )i∈I converging to a. Let ϕ : J → I be a map satisfying conditions (SN-1) and (SN-2) above. Consider a neighborhood V of a and an element i0 ∈ I. By (SN-2), we may find j0 ∈ J such that i0 ≤ ϕ(k) for all k ∈ J satisfying j0 ≤ k. Since the net (yj ) converges to a, there exists k0 ∈ J such that j0 ≤ k0 and yk0 ∈ V . Then we have i0 ≤ ϕ(k0 ) and xϕ(k0 ) = yk0 ∈ V . This shows that a is a cluster point for the net (xi ). Thus (b) implies (a). Conversely, suppose that a is a cluster point for the net (xi ). Denote by Na the set of all neighborhoods of a, partially ordered by reverse inclusion. Let J be the subset of the Cartesian product I × Na consisting of all pairs (i, V ) ∈ I × Na such that xi ∈ V . The fact that a is a cluster point of the net (xi ) implies that J is a directed set for the partial ordering ≤ defined by def

(i, V ) ≤ (i , V  ) ⇐⇒ (i ≤ i and V ≤ V  ). Consider the non decreasing map ϕ : J → I given by ϕ((i, V )) = i. If we set yj = xϕ(j) for all j ∈ J, it is clear that ϕ satisfies conditions (SN-1) and (SN-2) above and that the net (yj )j∈J converges to a. This shows that (a) implies (b).

Note that if f : X → Y is a continuous map between topological spaces and a ∈ X is a limit (resp. cluster) point of the net (xi ), then f (a) is a limit (resp. cluster) point of the net (f (xi ). This immediately follows from the definition and the fact that f −1 (W ) is a neighborhood of a for each neighborhood W of f (a).

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A.3 Initial Topology Let X be a set and let (Yλ )λ∈Λ be a family of topological spaces indexed by an arbitrary set Λ. Suppose that we are given, for each λ ∈ Λ, a map fλ : X → Yλ . Then one constructs a topology on X in the following way. Let F denote the set of all subsets of X of the form fλ−1 (Uλ ), where λ ∈ Λ and Uλ is an open subset of Yλ . Let B be the set of all subsets of X which may be written as a finite intersection of elements of F. Finally, let T denote the set of all subsets of X which may be written as a (finite or infinite) union of elements of B. It is straightforward to verify that the set T is the set of open sets of a topology on X which admits B as a base, and that this topology is the smallest topology on X for which all maps fλ : X → Yλ are continuous. The topology on X whose open sets are the elements of T is called the initial topology on X associated with the topological spaces (Yλ )λ∈Λ and the maps (fλ )λ∈Λ . If Z is a topological space and g : Z → X is a map, then g is continuous with respect to the initial topology on X if and only if all composite maps fλ ◦g : Z → Yλ , λ ∈ Λ, are continuous. If (xi )i∈I is a net in X and a is a point in X, then the net (xi )i∈I converges to a if and only if the net (fλ (xi ))i∈I converges to fλ (a) for every λ ∈ Λ. Similarly, a is a cluster point of the net (xi )i∈I if and only if fλ (a) is a cluster point of the net (fλ (xi ))i∈I for every λ ∈ Λ.

A.4 Product Topology Let (Xλ )λ∈Λ be a family of topological spaces  indexed by a set Λ. The initial topology on the cartesian product X = λ∈Λ Xλ associated with the projection maps πλ : X → Xλ is called the product topology on X. Abase for the product topology on X consists of all subsets of the form V = λ∈Λ Uλ , where Uλ is an open subset of Xλ for each λ ∈ Λ and Uλ = Xλ for all but finitely many λ ∈ Λ. In the case when each  Xλ is endowed with the discrete topology, the product topology on X = λ∈Λ Xλ is called the prodiscrete topology. Proposition  A.4.1. Let (Xλ )λ∈Λ be a family of Hausdorff topological spaces. Then X = λ∈Λ Xλ is Hausdorff for the product topology. Proof. Let x = (xλ ) and y = (yλ ) be distinct points of X. Then there exists λ0 ∈ Λ such that xλ0 = yλ0 . Since Xλ0 is Hausdorff, we may find disjoint open subsets U and V of Xλ0 containing xλ0 and yλ0 respectively. The pullbacks of U and V by the projection map πλ0 : X → Xλ0 are disjoint open subsets of X containing x and y respectively. Therefore X is Hausdorff.

Recall that a topological space X is called totally disconnected if any nonempty connected subset of X is reduced to a single point.

A.5 The Tychonoff Product Theorem

347

Proposition A.4.2. Let (X λ )λ∈Λ be a family of totally disconnected topological spaces. Then X = λ∈Λ Xλ is totally disconnected for the product topology. Proof. Let C be a nonempty connected subset of X. Then, for each λ ∈ Λ, the image of C by the projection map πλ : X → Xλ is a nonempty connected subset of Xλ . Since Xλ is totally disconnected, the set πλ (C) is reduced to a single point for every λ ∈ Λ. This implies that C is reduced to a single point. Therefore, X is totally disconnected.

Suppose Proposition A.4.3. Let (Xλ )λ∈Λ be a family of topological spaces.  of X for each λ ∈ Λ. Then F = that Fλ is a closed subset λ λ∈Λ Fλ is a  closed subset of X = λ∈Λ Xλ for the product topology. Proof. We have F =



πλ−1 (Fλ ).

λ∈Λ

Thus F is closed in X as it is the intersection of a family of closed subsets of X.

A.5 The Tychonoff Product Theorem Recall that a topological space X is called compact if every open cover of X admits a finite subcover. This means that if (Uα )α∈A is a family of open  subsets of X with X = α∈A Uα , then there exists a finite subset B ⊂ A such  that X = α∈B Uα . By taking complements, one sees that the compactness of X is equivalent to the fact that every family (F α )α∈A of closed subsets of X with the finite intersection property,, that is, α∈B Fα = ∅ for every finite subset B ⊂ A, has a nonempty intersection. It is well known that a metric space X is compact if and only if every sequence in X admits a convergent subsequence. There is an analogous characterization of compactness for general topological spaces using nets: Theorem A.5.1. Let X be a topological space. Then the following conditions are equivalent: (a) X is compact; (b) every net in X admits a cluster point; (c) every net in X admits a convergent subnet. Proof. The equivalence between conditions (b) and (c) follows from Proposition A.2.3. Thus, it suffices to prove that conditions (a) and (b) are equivalent. Suppose first that X is compact. Let (xi )i∈I be a net in X. For each i ∈ I, define the subset Yi ⊂ X by

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Yi = {xj : j ∈ I and i ≤ j}. Let J be a finite subset of I. Since I is a directed set, we may find an element k∈I such that  i ≤ k for all i ∈ J. This implies xk ∈ Yi for all i ∈ J and hence i∈J Yi ⊃ i∈J Yi = ∅. It follows that the family of closed sets (Yi )i∈I has  By compactness of X, we deduce that  the finite intersection property. Y = ∅. Let a be a point in i i∈I i∈I Yi . Then, any neighborhood of a meets Yi for all i ∈ I. This means that a is a cluster point of the net (xi ). Therefore, (a) implies (b). Conversely, suppose that every net in X admits a cluster point. Let (Fα )α∈A be a family of closed subsets of X with the finite intersection property. Consider the directed set E consisting of all finite subsets of  A partially ordered by inclusion. Choose, for each E ∈ E, an element xE ∈ α∈E Fα . By our hypothesis, the net (xE )E∈E admits a cluster point a ∈ X. Let α0 ∈ A. If V is a neighborhood of a, then there exists  a finite set E0 ⊂ A such that {α0 } ⊂ E0 and xE0 ∈ V . Since xE0 ∈ α∈E0 Fα ⊂ Fα0 , it follows that that any neighborhood of a meets Fα0 . Since the set Fα0 is closed, we deduce  an arbitrary element in A, we conclude that a ∈ a ∈ Fα0 . As α0 was α∈A Fα .  This shows that i∈I Fi = ∅. Consequently, the space X is compact. This proves that (b) implies (a).

Theorem A.5.2 (Tychonoff theorem). Let (Xλ )λ∈Λ be a family of com pact topological spaces. Then X = λ∈Λ Xλ is compact for the product topology. Proof. By Theorem A.5.1, it suffices to prove that every net in X admits a cluster point. Let (xi ) be net in X. We shall prove that (xi ) admits a cluster point by applying Zorn’s Lemma.Let us first introduce some notation. Given a subset A ⊂ Λ, we set X(A) = λ∈A Xλ and equip X(A) with the product B topology. If A and B are subsets of Λ such that A ⊂ B, we denote by πA the projection map X(B) → X(A). Consider the set E consisting of all pairs (A, a), where A is a subset of Λ and a ∈ X(A) is a cluster point of the net Λ (xi ))i∈I . We partially order E by declaring that two elements (A, a) and (πA B (B, b) satisfy (A, a) ≤ (B, b) if and only if A ⊂ B and πA (b) = a. The set E is not empty since it contains the pair (A, a), where A = ∅ and a is the unique element of X(∅). On the other hand, E isinductive. Indeed, suppose that F is a totally ordered subset of E. Let B = (A,a)∈F A and consider the unique B element b ∈ X(B) such that πA (b) = a for all (A, a) ∈ F. Clearly (B, b) ∈ E and (B, b) is an upper bound for F. By applying Zorn’s Lemma, we deduce that E admits a maximal element (M, m). To prove that the net (xi ) admits a cluster point, it suffices to show that M = Λ. Suppose not and choose an element λ0 ∈ Λ\M . Since the space Xλ0 is compact, the net (πλ0 (xi )) admits a cluster point a0 ∈ Xλ0 . Let us set M  = M ∪ {λ0 } and consider the element M M (m ) = m and π{λ (m ) = a0 . Clearly m is a m ∈ X(M  ) defined by πM 0} Λ   cluster point of the net (πM  (xi )) and (M, m) ≤ (M , m ). This contradicts the maximality of (M, m) and completes the proof.

Notes

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It follows from the definition of compactness that if a topological space X has only finitely many open subsets, then X is compact. In particular, every finite topological space is compact. Therefore, an immediate consequence of Tychonoff theorem is the following: Corollary A.5.3 (Tychonofftheorem). Let (Xλ )λ∈Λ be a family of finite topological spaces. Then X = λ∈Λ Xλ is compact for the product topology.

Notes The original Tychonoff theorem was only stated for product of compact intervals. The proof of the general Tychonoff theorem we have presented here is based on the one given by P. Chernoff in [Che]. Three other proofs may be found in Kelley’s book [Kel]: a proof using Alexander’s subbase theorem, Bourbaki’s proof [Bou] using ultrafilters, and a proof based on universal nets. There is also another proof using non-standard analysis in the book of A. Robinson [RobA].

Appendix B

Uniform Structures

B.1 Uniform Spaces Let X be a set. We shall use the following notation. We denote by ΔX the diagonal in X × X, that is, ΔX = {(x, x) : x ∈ X} ⊂ X × X. Suppose that R is a subset of X × X (in other words, R is a binary relation on X). For y ∈ X, we define the set R[y] ⊂ X by R[y] = {x ∈ X : (x, y) ∈ R}. −1

The inverse R ⊂ X × X of R is defined by −1

R = {(x, y) : (y, x) ∈ R}. −1

One says that R is symmetric if it satisfies R = R. If R and S are subsets of X × X, we define their composite R ◦ S ⊂ X × X by R ◦ S = {(x, y) : there exists z ∈ X such that (x, z) ∈ R and (z, y) ∈ S}. Definition B.1.1. Let X be a set. A uniform structure on X is a non–empty set U of subsets of X × X satisfying the following conditions: (UN-1) if V ∈ U, then ΔX ⊂ V ; (UN-2) if V ∈ U and V ⊂ V  ⊂ X × X, then V  ∈ U; (UN-3) if V ∈ U and W ∈ U, then V ∩ W ∈ U; −1

(UN-4) if V ∈ U, then V ∈ U; (UN-5) if V ∈ U, then there exists W ∈ U such that W ◦ W ⊂ V . T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 10, © Springer-Verlag Berlin Heidelberg 2010

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A set X equipped with a uniform structure U is called a uniform space and the elements of U are called the entourages of X (see Fig. B.1).

Fig. B.1 An entourage in a uniform space X

Examples B.1.2. (a) Let X be a set. Then U = {X ×X} is a uniform structure on X. This uniform structure is called the trivial uniform structure on X. It is the smallest uniform structure on X. (b) The discrete uniform structure on a set X is the uniform structure whose entourages consist of all subsets of X × X containing ΔX . This is the largest uniform structure on X. It follows from (UN-2) that the discrete uniform structure on X is the only uniform structure on X admitting the diagonal ΔX ⊂ X × X as an entourage. (c) Suppose that d is a metric on X. For each ε > 0 let Vε ⊂ X × X denote the set of pairs (x, y) such that d(x, y) < ε. Let U be the set of all subsets W ⊂ X × X such that one can find ε > 0 for which Vε ⊂ W . Then U is a uniform structure on X which is called the uniform structure associated with the metric d. A uniform structure U on a set X is said to be metrizable if U is the uniform structure on X associated with some metric on X. Example B.1.3. If d is the discrete metric on a set X, that is, the metric given by d(x, y) = 0 if x = y and d(x, y) = 1 otherwise, then the uniform structure defined by d is the discrete uniform structure on X. Thus the discrete uniform structure on X is metrizable. Let X be a uniform space. One easily verifies that is possible to define a topology on X by taking as open sets the subsets Ω ⊂ X which satisfy the following property: for each x ∈ Ω, there exists an entourage V ⊂ X × X such that V [x] = Ω. One says that this topology is the topology associated with the uniform structure on X. A subset N ⊂ X is a neighborhood of a point x ∈ X for this topology if and only if there exists an entourage V such

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that N = V [x]. This topology is Hausdorff if and only if the intersection of the entourages of X coincides with the diagonal ΔX ⊂ X × X. Examples B.1.4. (a) The topology associated with the discrete uniform structure on a set X is the discrete topology on X (every subset is open). (b) If U is the uniform structure associated with a metric d on a set X, then the topology defined by U coincides with the topology defined by d. Let U be a uniform structure on a set X. If Y is a subset of X, then UY = {V ∩ (Y × Y ) : V ∈ U} is a uniform structure on Y , which is said to be induced by U. The topology on Y associated with UY is the topology induced by the topology on X associated with U. A subset B ⊂ U is called a base of U if for each W ∈ U there exists V ∈ B such that V ⊂ W . Example B.1.5. If d is a metric on X, then B = {Vε : ε > 0}, where Vε = {(x, y) ∈ X × X : d(x, y) < ε}, is a base for the uniform structure on X associated with d. The proof of the following statement is straightforward. Proposition B.1.6. Let X be a set and let B be a nonempty set of subsets of X × X. Then B is a base for some (necessarily unique) uniform structure on X if and only if it satisfies the following properties: (BU-1) if V ∈ B, then ΔX ⊂ V ; (BU-2) if V ∈ B and W ∈ B, then there exists U ∈ B such that U ⊂ V ∩ W ; −1

(BU-3) if V ∈ B, then there exists W ∈ B such that W ⊂ V ; (BU-4) if V ∈ B, then there exists W ∈ B such that W ◦ W ⊂ V .

 

B.2 Uniformly Continuous Maps Let X and Y be uniform spaces. A map f : X → Y is called uniformly continuous if it satisfies the following condition: for each entourage W of Y , there exists an entourage V of X such that (f ×f )(V ) ⊂ W . Here f ×f denotes the map from X × X into Y × Y defined by (f × f )(x1 , x2 ) = (f (x1 ), f (x2 )) for all (x1 , x2 ) ∈ X × X. If B (resp. B  ) is a base of the uniform structure on X (resp. Y ), then a map f : X → Y is uniformly continuous if and only if it satisfies the following condition: for each W ∈ B  , there exists V ∈ B such that (f × f )(V ) ⊂ W . Note that this condition is equivalent to the fact that (f × f )−1 (W ) is an entourage of Y for each entourage W of X.

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Example B.2.1. Let (X, dX ) and (Y, dY ) be metric spaces. Then a map f : X → Y is uniformly continuous if and only if it satisfies the following condition: for each ε > 0, there exists δ > 0 such that dX (x1 , x2 ) < δ implies dY (f (x1 ), f (x2 )) < ε. Proposition B.2.2. Let X and Y be uniform spaces. Then every uniformly continuous map f : X → Y is continuous (with respect to the topologies on X and Y associated with the uniform structures). Proof. Suppose that f : X → Y is uniformly continuous. Let x ∈ X and let N ⊂ Y be a neighborhood of f (x). Then there exists an entourage W of Y such that W [f (x)] = N . Since f is uniformly continuous, the set V = (f × f )−1 (W ) is an entourage of X. The set V [x] is a neighborhood of x and satisfies f (V [x]) ⊂ W [f (x)] = N . This shows that f is continuous.   A continuous map between uniform spaces may fail to be uniformly continuous. For example, the map x → x2 is not uniformly continuous on R (equipped with the uniform structure associated with its usual metric). However, this is true when the source space is compact: Theorem B.2.3. Let X and Y be uniform spaces and suppose that X is compact. Then every continuous map f : X → Y is uniformly continuous. Let us first establish the following: Lemma B.2.4 (Lebesgue lemma). Let (Ωi )i∈I be an open cover of a compact uniform space X. Then there exists an entourage Λ of X satisfying the following property: for each x ∈ X, there exists an index i ∈ I such that Λ[x] ⊂ Ωi . Proof. Let us choose, for each x ∈ X, an index i(x) ∈ I such that x ∈ Ωi(x) . Since Ωi(x) is a neighborhood of x, there is an entourage Vx such that Vx [x] = Ωi(x) . By (UN-5), we may find an entourage Wx such that Wx ◦ Wx ⊂ Vx . The set Wx [x] is a neighborhood of x for each x ∈ X.  By compactness of X, there exists a finite subset A ⊂ X such that X = a∈A Wa [a]. Let us show that the entourage  Wa Λ= a∈A

has the required property. Let x ∈ X. Choose a point a ∈ A such that x ∈ Wa [a]. Suppose that y ∈ Λ[x]. Since (x, a) ∈ Wa and (y, x) ∈ Λ ⊂ Wa , we have (y, a) ∈ Wa ◦ Wa ⊂ Va . Thus y ∈ Va [a]. Since Va [a] ⊂ Ωi(a) , this   shows that Λ[x] ⊂ Ωi(a) . Proof of Theorem B.2.3. Let f : X → Y be a continuous map and let W be an entourage of Y . By (UN-3), (UN-4), and (UN-5), we may find a symmetric entourage S of Y such that S ◦ S ⊂ W . Since f is continuous, there exists, for each x ∈ X, an open neighborhood Ωx of X such that f (Ωx ) ⊂ S[f (x)].

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By Lemma B.2.4, we may find an entourage Λ of X such that, for each y ∈ X, there exists x ∈ X such that Λ[y] ⊂ Ωx . Suppose that (x1 , x2 ) ∈ Λ. Choose a ∈ X such that Λ[x2 ] ⊂ Ωa . Since x1 and x2 are in Λ[x2 ], we deduce that the points f (x1 ) and f (x2 ) are in S[f (a)]. It follows that (f (x1 ), f (a)) and (f (a), f (x2 )) are in S, and hence (f (x1 ), f (x2 )) ∈ S ◦ S ⊂ W . Thus (f × f )(Λ) ⊂ W . This shows that f is uniformly continuous.   Let X and Y be uniform spaces. One says that a map f : X → Y is a uniform isomorphism if f is bijective and both f and f −1 are uniformly continuous. One says that a map f : X → Y is a uniform embedding if f is injective and induces a uniform isomorphism between X and f (X) ⊂ Y . Proposition B.2.5. Let X and Y be uniform spaces with X compact and Y Hausdorff. Suppose that f : X → Y is a continuous injective map. Then f is a uniform embedding. Proof. As X is compact and Y is Hausdorff, f induces a homeomorphism from X onto f (X). This homeomorphism is a uniform isomorphism by Theorem B.2.3.  

B.3 Product of Uniform Spaces Let X be a set. Suppose that we are given a family (Xλ )λ∈Λ of uniform spaces and a family (fλ )λ∈Λ of maps fλ : X → Xλ . Then the initial uniform structure associated with these data is the smallest uniform structure on X such that all maps fλ : X → Xλ , λ ∈ Λ, are uniformly continuous. In the particular case when X = λ∈Λ Xλ and fλ : X → Xλ is the projection map, the associated initial uniform structure on X is called the product uniform structure. A base of entourages for the product uniform structure on X is obtained by taking all subsets of X × X which are of the form   Vλ ⊂ Xλ × Xλ λ∈Λ

λ∈Λ

 =



 Xλ

λ∈Λ

 ×



 Xλ

λ∈Λ

= X × X, where Vλ ⊂ Xλ × Xλ is an entourage of Xλ and Vλ = Xλ × Xλ for all but finitely many λ ∈ Λ. When each Xλ is endowed with  the discrete uniform structure, the product uniform structure on X = λ∈Λ Xλ is called the prodiscrete uniform structure.

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B.4 The Hausdorff-Bourbaki Uniform Structure on Subsets Let X be a uniform space with uniform structure U. In this section, we construct a uniform structure on the set P(X) of all subsets of X. We shall use the following notation. Suppose that R is a subset of X × X. Given a subset Y ⊂ X, we set  R[y] = {x ∈ X : (x, y) ∈ R for some y ∈ Y }, (B.1) R[Y ] = y∈Y

⊂ P(X) × P(X) by and we define the subset R = {(Y, Z) ∈ P(X) × P(X) : Y ⊂ R[Z] and Z ⊂ R[Y ]} . R

(B.2)

Proposition B.4.1. The set {V : V ∈ U} is a base for a uniform structure on P(X). Proof. Let us check that the conditions of Proposition B.1.6 are satisfied. Property (BU-1) follows from the fact that we have Y ⊂ V [Y ] for all Y ∈ P(X) and V ∈ U since ΔX ⊂ V . Then observe that (V ∩ W )[Y ] = {x ∈ X : (x, y) ∈ V ∩ W for some y ∈ Y } ⊂ {x ∈ X : (x, yV ) ∈ V and (x, yW ) ∈ W for some yV , yW ∈ Y } = V [Y ] ∩ W [Y ],

for all for all V, W ∈ U and Y ⊂ X. Therefore we have V ∩ W ⊂ V ∩ W V, W ∈ U, so that (BU-2) is satisfied. Property (BU-3) follows from the fact that the set V is a symmetric subset of P(X)×P(X) for each V ∈ U. Finally, let us verify (BU-4). Let V ∈ U and take W ∈ U such that W ◦ W ⊂ V . We

◦W

⊂ V . To see this, let (Y, Z) ∈ W

◦W

. This means that claim that W

. In particular there exists T ∈ P(X) such that (Y, T ) ∈ W and (T, Z) ∈ W we have Y ⊂ W [T ] and T ⊂ W [Z]. Thus, given y ∈ Y , there exist t ∈ T and z ∈ Z such that (y, t) ∈ W and (t, z) ∈ W . We have (y, z) ∈ W ◦W ⊂ V . This shows that Y ⊂ V [Z]. Similarly, we get Z ⊂ V [Y ] by using Z ⊂ W [T ] and T ⊂ W [Y ]. We deduce that (Y, Z) ∈ V . This proves the claim. Consequently, (BU-4) is satisfied.   The uniform structure on P(X) admitting {V : V ∈ U} as a base is called the Hausdorff-Bourbaki uniform structure on P(X) and the topology associated with this uniform structure is called the Hausdorff-Bourbaki topology on P(X). Remarks B.4.2. (a) The empty set is an isolated point in P(X). (b) One easily checks that the map i : X → P(X) defined by i(x) = {x} is a uniform embedding.

B.4 The Hausdorff-Bourbaki Uniform Structure on Subsets

357

Proposition B.4.3. Let X be a uniform space. Let Y and Z be closed subsets of X. Suppose that there is a net (Ti )i∈I of subsets of X which converges to both Y and Z with respect to the Hausdorff-Bourbaki topology on P(X). Then one has Y = Z. Proof. Let y ∈ Y and let Ω be a neighborhood of y in X. Then there is a symmetric entourage V of X such that V [y] ⊂ Ω. Choose an entourage W of X such that W ◦ W ⊂ V . Since the net (Ti )i∈I converges to both Y and Z, we can find an element i0 ∈ I such that ⊂ W [Ti0 ] and Ti0 ⊂ W [Z]. Thus, there exist t ∈ Ti0 and z ∈ Z such that (y, t) ∈ W and (t, z) ∈ W . This implies (y, z) ∈ W ◦ W ⊂ V . As V is symmetric, it follows that (z, y) ∈ V and hence z ∈ V [y] ⊂ Ω. This shows that y is in the closure of Z. Since Z is closed in X, we deduce that Y ⊂ Z. By symmetry, we also have Z ⊂ Y . Consequently, Y = Z.   By using Proposition A.2.2, we immediately deduce from Proposition B.4.3 the following: Corollary B.4.4. Let X be a uniform space. Then the topology induced by the Hausdorff-Bourbaki topology on the set of closed subsets of X is Hausdorff.   Remark B.4.5. Suppose that (X, d) is a metric space and let Cb (X) denote the set consisting of all closed bounded subsets of X. For x ∈ X and r > 0, denote by B(x, r) the open ball of radius r centered at x. Then it is not difficult to verify that the map δ : Cb (X) × Cb (X) → R defined by   δ(Y, Z) = inf{r > 0 : Z ⊂ B(y, r) and Y ⊂ B(z, r)} y∈Y

z∈Z

is a metric on Cb (X) and that the uniform structure associated with δ is the uniform structure induced by the Hausdorff-Bourbaki structure on the set of subsets of X. The metric δ is called the Hausdorff metric on Cb (X). Proposition B.4.6. Let X and Y be uniform spaces and let f : X → Y be a uniformly continuous map. Then the map f∗ : P(X) → P(Y ) which sends each subset A ⊂ X to its image f (A) ⊂ Y is uniformly continuous with respect to the Hausdorff-Bourbaki uniform structures on P(X) and P(Y ). Proof. Let W be an entourage of Y and let

= {(B1 , B2 ) ∈ P(Y ) × P(Y ) : B2 ⊂ W [B1 ] and B1 ⊂ W [B2 ]} W be the associated entourage of P(Y ). Since f is uniformly continuous, there is an entourage V of X such that (f × f )(V ) ⊂ W.

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Suppose that (A1 , A2 ) ∈ V , that is, A2 ⊂ V [A1 ] and A1 ⊂ V [A2 ]. If a1 ∈ A1 , then there exists a2 ∈ A2 such that (a1 , a2 ) ∈ V and hence (f (a1 ), f (a2 )) ∈ W . Therefore, we have f∗ (A1 ) ⊂ W [f∗ (A2 )]. Similarly, we get f∗ (A2 ) ⊂

. This shows that (f∗ × W [f∗ (A1 )]. It follows that (f∗ (A1 ), f∗ (A2 )) ∈ W

. Consequently, f∗ is uniformly continuous. f∗ )(V ) ⊂ W  

Notes Uniform structures were introduced by Andr´e Weil [Weil]. The reader is referred to [Bou, Ch. 2], [Kel, Ch. 6], and [Jam] for a detailed exposition of the general theory of uniform spaces. The Hausdorff-Bourbaki uniform structure on the set of subsets of a uniform space was introduced in exercises by Bourbaki (see [Bou, ch. II exerc. 5 p. 34 and exerc. 6 p. 36]).

Appendix C

Symmetric Groups

C.1 The Symmetric Group Let X be a set. A permutation of X is a bijective map σ : X → X. Let Sym(X) denote the set of all permutations of X. We equip Sym(X) with a group structure by defining the product σ1 σ2 of two elements σ1 , σ2 ∈ Sym(X) as the composite map σ1 ◦ σ2 . The associative property follows from the associativity of the composition of maps. The identity map IdX : X → X is the identity element and the inverse of σ ∈ Sym(X) is the inverse map σ −1 . The group Sym(X) is called the symmetric group on X. Remark C.1.1. Suppose that f : X → Y is a bijection from a set X onto a set Y . Then the map f∗ : Sym(X) → Sym(Y ) defined by f∗ (σ) = f ◦ σ ◦ f −1 is a group isomorphism. As a consequence, symmetric groups on equipotent sets are isomorphic. Theorem C.1.2 (Cayley’s theorem). Every group G is isomorphic to a subgroup of Sym(G). Proof. Let G be a group. Given g ∈ G denote by Lg : G → G the left multiplication by g, that is, the map defined by Lg (h) = gh for all h ∈ G. Observe that Lg ∈ Sym(G). Indeed, Lg is bijective since, given h, h ∈ G, we have Lg (h) = h if and only if h = g −1 h . Let us show that the map L : G → Sym(G) g → Lg is a group homomorphism. Given g, g  , h ∈ G we have Lgg (h) = gg  h = Lg (g  h) = Lg (Lg (h)) which shows that Lgg = Lg Lg . Moreover, L is injective. Indeed, if g ∈ ker(L), that is, Lg = IdG , we have g = g · 1G = Lg (1G ) = 1G . This shows that ker(L) = {1G }, and therefore L is injective. It follows that G is isomorphic to L(G) ⊂ Sym(G).   T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 11, © Springer-Verlag Berlin Heidelberg 2010

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C.2 Permutations with Finite Support Let X be a set. The support of a permutation σ ∈ Sym(X) is the set S(σ) ⊂ X consisting of all x ∈ X such that σ(x) = x. Proposition C.2.1. Let σ, τ ∈ Sym(X). Then (i) σ(S(σ)) = S(σ); (ii) S(σ) = S(σ −1 ); (iii) S(στ ) ⊂ S(σ) ∪ S(τ ); (iv) if S(σ) ∩ S(τ ) = ∅, then στ = τ σ; (v) S(τ στ −1 ) = τ (S(σ)). Proof. (i) Let x ∈ X. Then, x ∈ S(σ), that is, σ(x) = x, if and only if σ(σ(x)) = σ(x), that is, σ(x) ∈ S(σ). (ii) This follows from the fact that σ(x) = x if and only if x = σ −1 (x). (iii) Suppose that x ∈ X \ (S(σ) ∪ S(τ )). Then σ(x) = x = τ (x) and therefore (στ )(x) = σ(τ (x)) = σ(x) = x. It follows that x ∈ / S(στ ). (iv) Let x ∈ X. Suppose first that x ∈ X \ (S(σ) ∪ S(τ )). Then, by (iii), x∈ / S(στ ) and x ∈ / S(τ σ), so that σ(τ (x)) = x = τ (σ(x)). Suppose now that x is in the support of one of the two permutations, say x ∈ S(σ). It then follows from our assumptions that x ∈ / S(τ ) and therefore τ (x) = x. Also, by (i), σ(x) ∈ S(σ) and therefore, again by our assumptions, σ(x) ∈ / S(τ ), so that τ (σ(x)) = σ(x). We thus have τ (σ(x)) = σ(x) = σ(τ (x)). It follows that στ = τ σ. (v) Let x ∈ X. Then, x ∈ S(σ), that is σ(x) = x, if and only if, (τ στ −1 )(τ (x)) = τ (σ(x)) is not equal to τ (x). Thus, x ∈ S(σ) if and only if   τ (x) ∈ S(τ στ −1 ). Let Sym0 (X) denote the subset of Sym(X) consisting of all permutations of X with finite support. Proposition C.2.2. Let X be a set. Then the set Sym0 (X) is a normal subgroup of Sym(X). Proof. The support of the identity map IdX is the empty set and therefore IdX ∈ Sym0 (X). By Proposition C.2.1(ii), if σ ∈ Sym0 (X) then σ −1 ∈ Sym0 (X). On the other hand, by Proposition C.2.1(iii), the set Sym0 (X) is closed under multiplication. Thus Sym0 (X) is a subgroup of Sym(X). Finally, from Proposition C.2.1(v) we deduce that if σ ∈ Sym0 (X) then τ στ −1 ∈ Sym0 (X) for all τ ∈ Sym(X). It follows that Sym0 (X) is a normal subgroup of Sym(X).   Let r ≥ 2 be an integer and let x1 , x2 , . . . , xr be distinct elements in X. We denote by γ = (x1 x2 · · · xr )

C.2 Permutations with Finite Support

361

the permutation of X that maps x1 to x2 , x2 to x3 , . . . , xr−1 to xr , xr to x1 , and maps each element of X \ {x1 , x2 , . . . , xr } to itself. Thus, the support of γ is the set {x1 , x2 , . . . , xr }. One says that γ is a cycle of length r, or an r-cycle. A 2-cycle is called a transposition. Observe that γ = (x γ(x) γ 2 (x) · · · γ r−1 (x)) for all x ∈ {x1 , x2 , . . . , xr } and that the inverse of γ is the r-cycle γ −1 = (xr xr−1 · · · x2 x1 ). Proposition C.2.3. Let X be a set and let σ ∈ Sym0 (X). Then there exists an integer n ≥ 0 and cycles γ1 , γ2 , . . . , γn with pairwise disjoint supports such that σ = γ1 γ 2 · · · γ n . (C.1) Moreover, such a factorization is unique up to a permutation of the factors. Proof. If σ = IdX then n = 0 and there is nothing to prove. Suppose now that σ = IdX . Let S = S(σ) ⊂ X be the support of σ. We introduce an equivalence relation on S by setting ∼ y ifand only if  x  there exists k ∈ Z such that y = σ k (x). Let S = X1 X2 · · · Xn be the partition of S into the equivalence classes of ∼ and let us set ri = |Xi | for 1 ≤ i ≤ n. Note that ri ≥ 2 for all i. For each i = 1, 2, . . . , n, choose a representative xi ∈ Xi . Observe that the elements xi , σ(xi ), σ 2 (xi ), . . . , σ ri −1 (xi ) are all distinct since otherwise the class of xi would have less that ri elements. Consider the cycle γi = (xi σ(xi ) σ 2 (xi ) · · · σ ri −1 (xi )). The support of γi is Xi . Thus, the cycles γ1 , γ2 , . . . , γn have pairwise disjoint supports. Clearly σ = γ1 γ2 · · · γn . Suppose now that σ = δ1 δ2 · · · δs , where δ1 , δ2 , . . . , δs are cycles with pairwise disjoint supports. The supports of the cycles δi are the equivalence classes of ∼. We deduce that s = n. Moreover, up to a permutation of the factors, we may suppose that the support of γi equals the support of δi for all i = 1, 2, . . . , n. We have γi = (xi γi (xi ) γi2 (xi ) · · · γiri −1 (xi )) = (xi σ(xi ) σ 2 (xi ) · · · σ ri −1 (xi )) = (xi δi (xi ) δi2 (xi ) · · · δiri −1 (xi )) = δi and this completes the proof.

 

Corollary C.2.4. Every permutation in Sym0 (X) can be expressed as a product of transpositions.

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C Symmetric Groups

Proof. First observe that every cycle is a product of transpositions. Indeed, for all distinct x1 , x2 , . . . , xr ∈ X, we have (x1 x2 · · · xr ) = (x1 xr )(x1 xr−1 ) · · · (x1 x3 )(x1 x2 ).

(C.2)

By applying Proposition C.2.3 we deduce that every σ ∈ Sym0 (X) is a product of transpositions.  

C.3 Conjugacy Classes in Sym0 (X) Let G be a group and let H ⊂ G be a subgroup. We recall that two elements h and h in H are said to be conjugate in G (resp. in H) if there exists an element g ∈ G (resp. g ∈ H) such that h = ghg −1 . Clearly conjugacy in G (resp. in H) defines an equivalence relation on H. Proposition C.3.1. Let X be a set. Let γ ∈ Sym0 (X) be a cycle of length r and let σ ∈ Sym(X). Then σγσ −1 is also a cycle of length r. More precisely, if γ = (x1 x2 · · · xr ), then σγσ −1 equals the cycle (σ(x1 ) σ(x2 ) · · · σ(xr )).

(C.3)

Proof. First observe that, by Proposition C.2.1(v) the support of σγσ −1 is the set {σ(x1 ), σ(x2 ), . . . , σ(xn )}. Given 1 ≤ i ≤ r, we have (σγσ −1 )(σ(xi )) = (σγ)(xi ) = σ(xi+1 ), where r + 1 = 1. It follows that σγσ −1 = (σ(x1 ) σ(x2 ) · · · σ(xr )).

 

Let σ ∈ Sym0 (X). The type of σ is the sequence t(σ) = (tr )r≥2 where tr is the number of cycles of length r in the factorization of σ as a product of cycles with pairwise disjoint supports (cf. Proposition C.2.3). Proposition C.3.2. Let X be a set and let σ and σ  in Sym0 (X). Then the following conditions are equivalent: (a) σ and σ  are conjugate in Sym0 (X); (b) σ and σ  are conjugate in Sym(X); (c) σ and σ  have the same type. Proof. The implication (a) ⇒ (b) is obvious. Suppose (b). let σ = γ 1 γ2 · · · γ r

(C.4)

be the factorization of σ as a product of cycles with disjoint supports and let α ∈ Sym(X) be a permutation such that σ  = ασα−1 . From (C.4) we deduce that σ  = (αγ1 α−1 )(αγ2 α−1 ) · · · (αγr α−1 ). It follows from Proposition C.3.1 that t(σ  ) = t(σ). This shows (b) ⇒ (c).

C.4 The Alternating Group

363

Finally, suppose that σ = (x1 x2 · · · xr1 )(y1 y2 · · · yr2 ) · · · (z1 z2 · · · zr ) and

σ  = (x1 x2 · · · xr1 )(y1 y2 · · · yr 2 ) · · · (z1 z2 · · · zr  )

are two permutations of the same type. Consider a permutation α, with support the union of the supports of σ and σ  , which maps xi to xi for all i = 1, 2, . . . , r1 , yj to yj , for all j = 1, 2, . . . , r2 , . . ., and zk to zk for all k = 1, 2, . . . , r . Note that α ∈ Sym0 (X). Then (cf. the proof of Proposition C.3.1)   ασα−1 = α . It follows that σ and σ  are conjugate in Sym0 (X).

C.4 The Alternating Group Proposition C.4.1. Let X be a set and let σ ∈ Sym0 (X). Suppose that σ can be expressed as a product of n transpositions. Then the parity of n only depends on σ. Proof. Suppose that σ can be expressed both as a product of an even and as a product of an odd number of transpositions. Then, the same holds for σ −1 . It follows that choosing an even writing for σ and an odd one for σ −1 , we can write the identity element IdX = σσ −1 as a product of an odd number of transpositions, say (C.5) IdX = τ1 τ2 · · · τ2m+1 . Let x be an element in X appearing in the support of one of the transpositions τi in (C.5). As transpositions with disjoint support commute (cf. Proposition C.2.1(iv)) and (y z)(x z) = (x y)(y z) for all distinct elements y, z in X \ {x}, we can move all transpositions of the form (x y) to the left in (C.5). In other words, we can write the identity IdX as a product of 2m + 1 transpositions   · · · τ2m+1 IdX = τ1 τ2 · · · τr τr+1

(C.6)

where 1 ≤ r ≤ 2m + 1 and x belongs to the support of τi if and only if 1 ≤ i ≤ r. Let τr = (x y) and observe that it cannot appear only once in the product (C.6), otherwise the element x would be mapped onto y, while it has to remain fixed, since that product is the identity. It follows that there exists 1 ≤ j ≤ r − 1 such that τj = τr and τi = τr for all i = j + 1, j + 2, . . . , r − 1. Now, for all j + 1 ≤ i ≤ r − 1, if τi = (x z), we have τi τr = (x z)(x y) = (x y)(y z) = τr (y z). Thus, we can move to the left the transposition τr next to τj and cancel them out, without changing the value of the product. Repeating this operation, we reduce every time by 2 the

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number of transpositions in (C.6). Eventually, we reach a single transposition. This clearly yields a contradiction.   Consider the map ε : Sym0 (X) → {−1, 1}

(C.7)

defined by setting ε(σ) = 1 if σ is a product of an even number of permutations and ε(σ) = −1 otherwise. Note that ε is well defined by virtue of Proposition C.4.1. It is obvious that ε is a group homomorphism. Moreover, ε is surjective if X has at least two elements. The normal subgroup ker(ε) ⊂ Sym0 (X) is called the alternating group on X and it is denoted by Sym+ 0 (X). Proposition C.4.2. Let X be a set. An r-cycle is in Sym+ 0 (X) if and only if r is odd. In particular, every 3-cycle belongs to Sym+ 0 (X). Proof. We have seen (cf. (C.2)) that and r-cycle is a product of r − 1 transpositions.   Recall that a nontrivial group G is simple if the only normal subgroups of G are the trivial subgroup {1G } and G itself. Theorem C.4.3. Let X be a set having at least five distinct elements. Then the group Sym+ 0 (X) is simple. Proof. Every element of Sym+ 0 (X) is a product of permutations of the form (s t)(u v) or (s t)(s u), where s, t, u and v are distinct elements of X. Since (s t)(u v) = (s u t)(s u v) and (s t)(s u) = (s u t), it follows that Sym+ 0 (X) is generated by the set of all 3-cycles. Let x and y be two distinct elements in X. Clearly, any 3-cycle is of one of the forms (x y s), (x s y), (x s t), (y t u), or (s t u), where s, t, u are distinct elements in X \ {x, y}. We have (x s y) = (x y s)2 , (x s t) = (x y t)(x s y) = (x y t)(x y s)2 , (y s t) = (x t y)(x y s) = (x y t)2 (x y s), (s t u) = (x s y)(x y u)(x t y)(x y s) = (x y s)2 (x y u)(x y t)2 (x y s). This shows that Sym+ 0 (X) is generated by the 3-cycles (x y z), where z ∈ X \ {x, y}. Let now N ⊂ Sym+ 0 (X) be a nontrivial normal subgroup. Let us show that N = Sym+ 0 (X). We distinguish a few cases (corresponding to the different possible cycle structures of a nontrivial element in N ). Case 1. N contains a 3-cycle (x y s). Then, for any z ∈ X \ {x, y, s} we have that the 3-cycle

C.4 The Alternating Group

365

(x y z) = (x y)(s z)(x y s)2 (s z)(x y) = [(x y)(s z)]−1 (x y z)2 [(s z)(x y)] belongs to N . From the preceding part of the proof, we deduce that N = Sym+ 0 (x). Case 2. N contains an element σ whose factorization as product of cycles with disjoint supports contains a cycle (x1 x2 · · · xr ) of length r ≥ 4. Write σ = (x1 x2 · · · xr )ρ, where ρ ∈ Sym0 (X) is the product of the remaining cycles. Consider the cycle γ = (x1 x2 x3 ). Then, σ(γσ −1 γ −1 ) ∈ N , as N is a normal subgroup. By Proposition C.3.1, we have σ(γσ −1 γ −1 ) = (σγσ −1 )γ −1 = (x2 x3 x4 )(x3 x2 x1 ) = (x1 x4 x2 ). Thus, N contains a 3-cycle and, by Case 1, N = Sym+ 0 (X). Case 3. N contains an element σ whose factorization as a product of cycles with disjoint supports contains at least two cycles (x1 x2 x3 ) and (x4 x5 x6 ) of length 3. Write σ = (x1 x2 x3 )(x4 x5 x6 )ρ, where ρ ∈ Sym0 (X) is the product of the remaining cycles. Consider the cycle γ = (x1 x2 x4 ). Then, as above, σγσ −1 γ −1 ∈ N . But, again by Proposition C.3.1, we have σγσ −1 γ −1 = (x2 x3 x5 )(x4 x2 x1 ) = (x1 x4 x3 x5 x2 ). Thus, N contains a 5-cycle and by Case 2, N = Sym+ 0 (X). Case 4. N contains an element σ which factorizes as a product of one single 3-cycle (x1 x2 x3 ) and transpositions with disjoint supports. Write σ = (x1 x2 x3 )ρ, where ρ ∈ Sym0 (X) is the product of the transpositions. Note that ρ2 = IdX . Then σ 2 ∈ N and σ 2 = (x1 x2 x3 )ρ(x1 x2 x3 )ρ = (x1 x2 x3 )2 ρ2 = (x1 x3 x2 ). Thus, again as in Case 1, N = Sym+ 0 (X). Case 5. N contains an element σ which is the product of an (even) number of transpositions with disjoint supports. We can write σ = (x1 x2 )(x3 x4 )ρ where ρ ∈ Sym0 (X) satisfies ρ2 = IdX . Consider the cycle γ = (x1 x2 x3 ). Then, the element π = σγσ −1 γ −1 belongs to N . By Proposition C.3.1, π = σγσ −1 γ −1 = (x2 x1 x4 )(x1 x3 x2 ) = (x1 x3 )(x2 x4 ). Since X has at least five distinct elements, there exists an element y ∈ X \ {x1 , x2 , x3 , x4 }. Set δ = (x1 x3 y). Then, π(δπ −1 δ −1 ) ∈ N . But, one more time by Proposition C.3.1, π(δπ −1 δ −1 ) = (πδπ −1 )δ −1 = (x3 x1 y)(y x3 x1 ) = (x1 x3 y). We are again in Case 1, and therefore N = Sym+ 0 (X). + In all cases, N = Sym+ 0 (X), and this shows that Sym0 (X) is simple.

 

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Note that if X is a finite set, then Sym0 (X) = Sym(X). Given an integer n ≥ 1, we denote by Symn the symmetric group of the set {1, 2, . . . , n}. The group Symn is called the symmetric group of degree n. By Remark C.1.1, if X is a finite set with |X| = n, we have Symn ∼ = Sym(X). The subgroup Sym+ 0 ({1, 2, . . . , n}) is called the alternating group of degree n and it is denoted by Sym+ n. + Remark C.4.4. The groups Sym+ 1 (= Sym1 ) and Sym2 are trivial groups. + The group Sym3 = {Id{1,2,3} , (1 2 3), (1 3 2)} is cyclic of order 3 and therefore it is a simple group. On the other hand, the subgroup

K = {Id{1,2,3,4} , (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} ⊂ Sym+ 4 + has index two in Sym+ 4 . It follows that K is a normal subgroup of Sym4 . + Therefore, Sym4 is not a simple group. From Theorem C.4.3 we deduce that the group Sym+ n is simple for all n ≥ 5.

Appendix D

Free Groups

D.1 Concatenation of Words Let A be a set. A word on the alphabet set A is an element of the set  An , A∗ = n∈N

where An is the Cartesian product of A with itself n times, that is, the set consisting of all n-tuples (a1 , a2 , . . . , an ) with ak ∈ A for 1 ≤ k ≤ n. The unique element of A0 is denoted by  and is called the empty word . The concatenation of two words w = (a1 , a2 , . . . , am ) ∈ Am and w =  (a1 , a2 , . . . , an ) ∈ An is the word ww ∈ Am+n defined by ww = (a1 , a2 , . . . , am , a1 , a2 , . . . , an ). We have w = w = w and (ww )w = w(w w ) for all w, w , w ∈ A∗ . Thus, A∗ is a monoid for the concatenation product whose identity element is the empty word . Observe that each word w = (a1 , a2 , . . . , an ) ∈ An may be uniquely written as a product of elements of A = A1 , namely w = a1 a2 · · · an .

D.2 Definition and Construction of Free Groups Definition D.2.1. A based free group is a triple (F, X, i), where F is a group, X is a set, and i : X → F is a map from X to F satisfying the following universal property: for every group G and any map f : X → G, there exists a unique homomorphism φ : F → G such that f = φ ◦ i.

T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 12, © Springer-Verlag Berlin Heidelberg 2010

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F i

X

φ

f

G

A group F is called free if there exist a set X and a map i : X → F such that the triple (F, X, i) is a based free group. One then says that (X, i) is a free base for F and that F is a free group based on (X, i). Remarks D.2.2. (a) If (F, X, i) is a based free group and α : Y → X is a bijective map from a set Y onto X, then the triple (F, Y, i ◦ α) is also a based free group. Indeed, if f : Y → G is a map from Y into a group G, then there is a unique homomorphism φ : F → G such that f = φ ◦ i ◦ α, namely the unique homomorphism φ : F → G satisfying f ◦ α−1 = φ ◦ i. (b) If (F, X, i) is a based free group and ψ : F → F  is an isomorphism from F onto a group F  , then the triple (F  , X, ψ ◦ i) is a based free group. Indeed, if f : X → G is a map from X into a group G, then there exists a unique homomorphism φ : F  → G such that f = φ ◦ ψ ◦ i, namely the homomorphism given by φ = φ ◦ ψ −1 , where φ : F → G is the unique homomorphism satisfying f = φ ◦ i. Proposition D.2.3. Let (F, X, i) be a based free group. Then the following hold: (i) the map i is injective; (ii) the set i(X) generates the group F ; (iii) the triple (F, X  , i ), where X  = i(X) and i : X  → F is the inclusion map, is also a based free group. Proof. (i) Let x1 and x2 be two distinct elements in X. Consider the map f : X → Z/2Z defined by f (x) = 0 if x = x2 and f (x2 ) = 1. Since (F, X, i) is a based free group, there exists a homomorphism φ : F → Z/2Z such that f = φ ◦ i. As f (x1 ) = f (x2 ), this implies i(x1 ) = i(x2 ). Therefore, the map i is injective. (ii) Denote by H the subgroup of F generated by i(X). Consider the map i∗ : X → H defined by i∗ (x) = i(x) for all x ∈ X. Since (F, X, i) is a based free group, there exists a homomorphism φ : F → H such that i∗ = φ ◦ i. Consider now the inclusion map ρ : H → F . The homomorphisms IdF and ρ ◦ φ satisfy IdF ◦i = ρ ◦ φ ◦ i. By uniqueness, we get IdF = ρ ◦ φ. This implies that ρ is surjective, that is, H = F . Therefore, i(X) generates F . (iii) The fact that (F, X  , i ) is a based free group immediately follows from Remark D.2.2(a) since i = i ◦ j −1 where j : X → X  is the bijective map defined by j(x) = i(x) for all x ∈ X.   From Proposition D.2.3(iii), we deduce that if F is a free group then there exists a subset X ⊂ F such that the triple (F, X, i), where i : X → F is the inclusion map, is a based free group. Such a subset X ⊂ F is then called a free base subset, or simply a base, for F .

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369

Proposition D.2.4. Let (F, X, i) be a based free group and let Y ⊂ X. Let K denote the subgroup of F generated by Y and let j : Y → K be the map defined by j(y) = i(y) for all y ∈ Y . Then (K, Y, j) is a based free group. Proof. Let G be a group and let f : Y → G be a map. Let us show that there exists a unique homomorphism φ : K → G satisfying f = φ ◦ j. Uniqueness follows from the fact that j(Y ) = i(Y ) generates K. Choose a map f  : X → G extending f . As (F, X, i) is a based free group, there exists a homomorphism φ : F → G such thatf  = φ ◦ i. Then φ = φ |K : K → G satisfies f = φ ◦ j. This proves that (K, Y, j) is a based free group.   In the case when i is an inclusion map, this gives us the following: Corollary D.2.5. Let F be a free group with base X ⊂ F . Let Y ⊂ X and let K denote the subgroup of F generated by Y . Then K is a free group with base Y .   Proposition D.2.6. Let (F1 , X1 , i1 ) and (F2 , X2 , i2 ) be two based free groups. Suppose that there is a bijective map u : X1 → X2 . Then there exists a unique group isomorphism ϕ : F1 → F2 satisfying ϕ ◦ i1 = i2 ◦ u. ϕ

F1 −−−−→  ⏐ i1 ⏐

F2  ⏐i ⏐2

X1 −−−−→ X2 u

Proof. Since (F1 , X1 , i1 ) is a based free group, there exists a unique homomorphism ϕ : F1 → F2 such that i2 ◦ u = ϕ ◦ i1 . It suffices to show that ϕ is bijective. To see this, we now use the fact that (F2 , X2 , i2 ) is a based free group. This implies that there exists a homomorphism ϕ : F2 → F1 such that i1 ◦u−1 = ϕ ◦i2 . The maps IdF1 and ϕ ◦ϕ are endomorphisms of F1 satisfying IdF1 ◦i1 = i1 and (ϕ ◦ ϕ) ◦ i1 = i1 . Since (F1 , X1 , i1 ) is a based free group, it follows that IdF1 = ϕ ◦ ϕ by uniqueness. Similarly, we get ϕ ◦ ϕ = IdF2 . This shows that ϕ is bijective.   Theorem D.2.7. Let X be a set. Then there exist a group F and a map i : X → F such that the triple (F, X, i) is a based free group. Proof. Let X  be a disjoint copy of X, that is, a set X  such that X ∩ X  = ∅ together with a bijective map γ : X → X  . Let A = X ∪ X  . For each a ∈ A, define the element  a ∈ A by setting  γ(a) if a ∈ X,  a= −1 γ (a) if a ∈ X  . Observe that the map a →  a is an involution of A, that is, it satisfies   a=a for all a ∈ A.

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Consider now the set A∗ consisting of all words on the alphabet set A (see Sect. D.1). Recall that A∗ is a monoid for the concatenation product whose identity element is the empty word . We say that a word w ∈ A∗ may be obtained from a word w ∈ A∗ by an elementary reduction if there exist an element a ∈ A and words u, v ∈ A∗ such that w = uv and w = ua av. Given words w, w ∈ A∗ , we write w ∼ w if either w may be obtained from w by an elementary reduction or w may be obtained from w by an elementary reduction. Finally, we define a relation ≡ on A∗ by writing w ≡ w for w, w ∈ A∗ if and only if there exist an integer n ≥ 1 and a sequence of words w1 , w2 , . . . , wn ∈ A∗ with w1 = w and wn = w such that wi ∼ wi+1 for each i = 1, 2, . . . , n − 1. It is immediate to check that ≡ is an equivalence relation on A∗ . Denote by [w] the equivalence class of an element w ∈ A∗ and consider the quotient set F = A∗ / ≡, that is, the set consisting of all equivalence classes [w], w ∈ A∗ . Observe that if u, u , v ∈ A∗ and u ∼ u then uv ∼ u v and vu ∼ vu . It follows from this observation that if the words u, v, u , v  ∈ A∗ satisfy u ≡ u and v ≡ v  then uv ≡ u v  , so that we can define the product of [u] and [v] in F by setting [u][v] = [uv]. Let us show that this product gives a group structure on F . The associativity immediately follows from the associativity of the concatenation product in A∗ . Indeed, for all u, v, w ∈ A∗ , we have ([u][v])[w] = [uv][w] = [(uv)w] = [u(vw)] = [u][vw] = [u]([v][w]). On the other hand, for all w ∈ A∗ , we have [][w] = [w] = [w] and [w][] = [w] = [w] which shows that [] is an identity element. Finally, let us show that every element in F admits an inverse. For w = a1 a2 · · · an ∈ A∗ , where n ≥ 0 and  ∈ A∗ by ak ∈ A for 1 ≤ k ≤ n, define the word w  w  = a na n−1 · · · a1 . Observe that ww  = a1 a2 · · · an a n · · · a2 a1 ∼ a1 · · · an−1 a n−1 · · · a1 .. . ∼ a1 a2 a2 a1 ∼ a1 a1 ∼ .

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371

Thus, we have ww  ≡ . This gives us [w][w]  = [ww]  = []. Similarly, we get [w][w]  = []. It follows that [w]  is an inverse of [w] for the product operation in F . This proves that F is a group. Consider the map i : X → F defined by i(x) = [x] for all x ∈ X. If w = a1 a2 · · · an ∈ A∗ , where ak ∈ A for 1 ≤ k ≤ n, then [w] = [a1 a2 · · · an ] = [a1 ][a2 ] · · · [an ]. a)−1 if a ∈ X  , we deduce that Since [a] = i(a) if a ∈ X and [a] = [ a]−1 = i( i(X) generates the group F . Let us show that the triple (F, X, i) is a based free group. Suppose that f : X → G is a map from X into a group G. We have to prove that there exists a unique homomorphism φ : F → G such that f = φ ◦ i. Uniqueness follows from the fact that i(X) generates F . To construct φ, we first extend f to a map g : A → G by setting  f (a) if a ∈ X, g(a) = −1 if a ∈ X  . f ( a) Observe that g( a) = g(a)−1 for all a ∈ A. Define now a map Φ : A∗ → G by setting Φ(w) = g(a1 )g(a2 ) · · · g(an ) for all w = a1 a2 · · · an ∈ A∗ . Note that we have Φ(ww ) = Φ(w)Φ(w )

(D.1)

for all w, w ∈ A∗ . Moreover, if w may be obtained from w by an elementary av for some a ∈ A and u, v ∈ A∗ , then reduction, that is, w = uv and w = ua a)Φ(v) = Φ(u)g(a)g(a)−1 Φ(v) = Φ(u)Φ(v) = Φ(w). Φ(w ) = Φ(u)g(a)g( It follows that Φ(w) = Φ(w ) whenever w, w ∈ A∗ satisfy w ∼ w . By induction, we deduce that Φ(w) = Φ(w ) for all w, w ∈ A∗ such that w ≡ w . Thus, we can define a map φ : F → G by setting φ([w]) = Φ(w) for all w ∈ A∗ . By using (D.1), we get φ([w][w ]) = φ([ww ]) = Φ(ww ) = Φ(w)Φ(w ) = φ([w])φ([w ]). Therefore, φ is a group homomorphism. On the other hand, for all x ∈ X, we have φ ◦ i(x) = φ([x]) = g(x) = f (x), which shows that φ ◦ i = f . Consequently, the triple (F, X, i) is a based free group.   Given an arbitrary set X, it follows from Theorem D.2.7 that there exist a based free group (F, X, i). Then one often says that F = F (X) is the free

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group based on X (this is a minor abuse of language since if (F  , X, i ) is another based free group then there is a unique isomorphism ϕ : F → F  such that ϕ ◦ i = i , by Proposition D.2.6). For k ∈ N, the free group based on {1, 2, . . . , k} is denoted by Fk . We recall that one says that two sets X1 and X2 are equipotent, or that they have the same cardinality, if there exists a bijective map u : X1 → X2 . Theorem D.2.8. Let F1 and F2 be two free groups based on X1 ⊂ F1 and X2 ⊂ F2 . Then the following conditions are equivalent: (a) the groups F1 and F2 are isomorphic; (b) the sets X1 and X2 are equipotent. For the proof, we shall need a few preliminary results. We use the following notation. If G1 and G2 are groups, we denote by Hom(G1 , G2 ) the set consisting of all homomorphisms φ : G1 → G2 . We denote by P(X) the set of all subsets of a set X. Lemma D.2.9. Let F be a free group based on X ⊂ F . Then the sets Hom(F, Z/2Z) and P(X) are equipotent. Proof. Since F is free with base X, each map f : X → Z/2Z can be uniquely extended to a homomorphism φ : F → Z/2Z. Therefore, the restriction map yields a bijection from Hom(F, Z/2Z) onto the set of maps from X to Z/2Z. Consequently, the sets Hom(F, Z/2Z) and P(X) are equipotent.   Lemma D.2.10. Let F be a free group based on X ⊂ F . Suppose that the set X is infinite. Then the sets X and F are equipotent. Proof. Let A = X ∪ X −1 . Since X generates F , the map ρ : A∗ → F defined by ρ(a1 , a2 , . . . , an ) = a1 a2 . . . an is surjective. As X is infinite, the sets X, A, and A∗ are all equipotent. It follows that there is a surjective map from X onto F and therefore an injective map from F into X. By applying the Cantor-Bernstein theorem (cf. Corollary H.3.5), we deduce that X and F are equipotent.   Proof of Theorem D.2.8. The fact that (b) implies (a) immediately follows from Proposition D.2.6. To prove the converse implication, suppose that there exists an isomorphism α : F1 → F2 . Then α induces a bijective map between the sets Hom(F1 , Z/2Z) and Hom(F2 , Z/2Z). It follows that P(X1 ) and P(X2 ) are equipotent by Lemma D.2.9. If X1 is infinite, this implies that X2 is also infinite, and we conclude that X1 and X2 are equipotent by applying Lemma D.2.10. On the other hand, if X1 is finite, the fact that P(X1 ) and P(X2 ) are equipotent implies that X2 is also finite and that 2|X1 | = 2|X2 | . This gives us |X1 | = |X2 | and we conclude that X1 and X2 are also equipotent in this case.  

D.3 Reduced Forms

373

Corollary D.2.11. Let F be a free group and let X1 , X2 ⊂ F be two bases   of F . Then X1 and X2 are equipotent. Let F be a free group. The cardinality of a base X ⊂ F depends only on F by Corollary D.2.11. This cardinality is called the rank of F . Two free groups are isomorphic if and only if they have the same rank by Theorem D.2.8. A group is free of finite rank k ∈ N if and only if it is isomorphic to Fk .

D.3 Reduced Forms In order to state the first result of this section, we use the notation introduced in the proof of Theorem D.2.7. A word w ∈ A∗ is said to be reduced if it contains no subword of the form a a with a ∈ A, that is, if there is no word w ∈ A∗ which can be obtained from w by applying an elementary reduction. Note that the empty word  is reduced and that every subword of a reduced word is itself reduced. Theorem D.3.1. Every equivalence class for ≡ contains a unique reduced word. Proof. It is clear that any word w ∈ A∗ can be transformed into a reduced word by applying a suitable finite sequence of elementary reductions. This shows the existence of a reduced word in any equivalence class for ≡. Let us prove uniqueness. Consider the subset R ⊂ A∗ consisting of all reduced words. For a ∈ A and r ∈ R, define the word αa (r) by  w if r =  aw for some w ∈ A∗ , αa (r) = ar otherwise. Note that the word αa (r) is always reduced. Thus, we get a map αa : R → R defined for each a ∈ A. Let us show that αa ◦ αea = αea ◦ αa = IdR .

(D.2)

Let a ∈ A and consider an arbitrary element r ∈ R. If r = aw for some w ∈ A∗ , then αea (r) = w and hence αa (αea (r)) = αa (w) = aw = r (observe ar that w cannot start by  a since r is reduced). Otherwise, we have αea (r) =  ar) = r. It follows that αa ◦ αAe = IdR . By and therefore αa (αea (r)) = αa ( replacing a by  a in this equality, we get αea ◦ αa = IdR since · is an involution on A. This proves (D.2). From (D.2), we deduce that αa ∈ Sym(R) for all a ∈ A. By setting ρ(w) = αa1 ◦ αa2 ◦ · · · ◦ αan

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for every word w = a1 a2 · · · an ∈ A∗ , we get a monoid homomorphism ρ : A∗ → Sym(R). Note that ρ(r)() = r

for all r ∈ R.

(D.3)

On the other hand, it immediately follows from (D.2) that ρ(w) = ρ(w ) whenever w, w ∈ A∗ satisfy w ≡ w . Thus, if r1 an r2 are reduced words in the same equivalence class for ≡, we have ρ(r1 ) = ρ(r2 ) and therefore r1 = r2 by applying (D.3).   Corollary D.3.2. Let (F, X, i) be a based free group. Then every element f ∈ F can be uniquely written in the form f = i(x1 )h1 i(x2 )h2 · · · i(xn )hn

(D.4)

with n ≥ 0, xi ∈ X and hi ∈ Z \ {0} for 1 ≤ i ≤ n, and xi = xi+1 for 1 ≤ i ≤ n − 1. Definition D.3.3. The expression (D.4) is called the reduced form of the element f in the based free group (F, X, i). Proof of Corollary D.3.2. We can assume that (F, X, i) is the based free group constructed in the proof of Theorem D.2.7. By construction, the element f ∈ F is an equivalence class for ≡. This equivalence class contains a unique reduced word r by Theorem D.3.1. The word r can be uniquely written in the form r = ak11 ak22 · · · aknn , where n ≥ 0, ai ∈ A and ki ∈ N \ {0} for 1 ≤ i ≤ n, and ai = ai+1 for 1 ≤ i ≤ n − 1. This gives us an expression of the form (D.4) with xi = ai and hi = ki if ai ∈ X, and xi = ai and hi = −ki otherwise. Uniqueness of such an expression for f follows from the uniqueness of the reduced word r ∈ f .   Corollary D.3.4. Let G be a group. Let U ⊂ G be a subset such that uk11 uk22 · · · uknn = 1G ,

(D.5)

 ui+1 for for all u1 , u2 , . . . , un ∈ U and k1 , k2 , . . . , kn ∈ Z \ {0} with ui = 1 ≤ i ≤ n − 1 and n ≥ 1. Then the subgroup of G generated by U is free with base U . Proof. Denote by H the subgroup of G generated by U and by ι : U → H the inclusion map. Let (F, U, i) be a based free group (cf. Theorem D.2.7). Then there exists a unique homomorphism φ : F → H such that ι = φ ◦ i. Note that φ is surjective since φ(i(U )) = ι(U ) = U generates H. Let us show that φ is also injective. Consider an element f ∈ F written in reduced form, that is, in the form f = i(u1 )h1 i(u2 )h2 · · · i(un )hn with n ≥ 0, ui ∈ U and hi ∈ Z \ {0} for 1 ≤ i ≤ n, and ui = ui+1 for 1 ≤ i ≤ n − 1. Then we

D.4 Presentations of Groups

375

have φ(f ) = uh1 1 uh2 2 · · · uhnn since φ(i(ui )) = ι(ui ) = ui for 1 ≤ i ≤ n. It follows from (D.5) that φ(f ) = 1H if and only if n = 0, that is, if and only if f = 1F . This shows that φ is an isomorphism. It follows from Remark D.2.2 that (H, U, ι) is a based free group.  

D.4 Presentations of Groups Proposition D.4.1. Let G be a group and let S be a generating subset of G. Let F denote the free group based on S. Then, the group G is isomorphic to a quotient of F . Proof. Let i : S → F and f : S → G denote the inclusion maps. Since F is free with base S, there exists a homomorphism φ : F → G such that f = φ ◦ i. This implies that S is contained in the image of φ. Since S generates G, we deduce that φ is surjective. Therefore, the group G is isomorphic to the quotient group F/ Ker(φ).   As every group admits a generating subset (e.g., the group itself), we deduce the following: Corollary D.4.2. Every group is isomorphic to a quotient of a free group.   A group is said to be finitely generated if it admits a finite generating subset. Proposition D.4.1 gives us: Corollary D.4.3. Every finitely generated group is isomorphic to a quotient of a free group of finite rank.   Let A be a subset of a group G. The intersection of all normal subgroups of G containing A is a normal subgroup of G which is called the normal closure of A in G. Let G be a group. By Corollary D.4.2, there exist a free group F and an epimorphism φ : F → G. Let X be a free base for F . If R is a subset of F whose normal closure is the kernel of φ, then one says that G admits the presentation G = X; R. (D.6) Note that as X generates F (cf. Proposition D.2.3(ii)), we have that φ(X) generates G. The elements x ∈ X (rather than the φ(x) ∈ G, x ∈ X) are called, by abuse of language, the generators of the presentation (D.6). The elements r ∈ R are called the relators of the presentation (D.6). For every r ∈ R let ur , vr be elements in F such that r = ur (vr )−1 . Then (D.6) is often expressed as G = X : r = 1, r ∈ R or G = X : ur = vr , r ∈ R. Note that G admits a presentation (D.6) with X finite if and only if G is finitely generated (cf. Corollary D.4.3).

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If a group G admits a presentation (D.6) with both X and R finite, then G is said to be finitely presented.

D.5 The Klein Ping-Pong Theorem The following theorem is often used to prove that a group is free: Theorem D.5.1. Let G be a group. Let X be a generating subset of G having at least two distinct elements. Suppose that G acts on a set E and that there is a family (Ax )x∈X of nonempty pairwise disjoint subsets of E such that ⎛ ⎞  xk ⎝ Ay ⎠ ⊂ Ax for all x ∈ X and k ∈ Z \ {0}. (D.7) y∈X\{x}

Then G is a free group with base X. Proof. Consider an element g ∈ G written as a nontrivial reduced word on the generating subset X, that is, in the form g = xk11 xk22 . . . xknn , where n ≥ 1, xi ∈ X and ki ∈ Z \ {0} for 1 ≤ i ≤ n, and xi = xi+1 for 1 ≤ i ≤ n − 1. By Corollary D.3.4, we have to show that g = 1G . Suppose first that either X contains at least three distinct elements or X has exactly two elements and x1 = xn . In this case, we can find an element y ∈ X such that y = x1 and y = xn . By successive applications of D.7, we get k

k

k

k

n−2 n−1 kn xn−1 xn Ay gAy = xk11 xk22 . . . xn−2 n−2 n−1 ⊂ xk11 xk22 . . . xn−2 xn−1 Axn

k

n−2 ⊂ xk11 xk22 . . . xn−2 Axn−1

⊂ xk11 xk22 . . . Axn−2 ... ⊂ xk11 xk22 Ax3 ⊂ xk11 Ax2 ⊂ Ax1 . As the sets Ay and Ax1 are disjoint and Ay = ∅ by our hypotheses, we deduce that g = 1G .

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377

It remains to treat the case when X has exactly two elements and x1 = xn . Then kn −k1 1 1 k2 k3 = x2k xk11 gx−k 1 1 x2 x3 . . . xn x1 1 1 is a reduced form of xk11 gx−k . We have xk11 gx−k = 1G by the first case. We 1 1   deduce that we also have g = 1G in this case.

Remark D.5.2. The proof of Theorem D.5.1 shows that the hypotheses on the family (Ax )x∈X may be relaxed: in fact, it suffices that the subsets Ax ⊂ X, x ∈ X, satisfy D.7 and Ay ⊂ Ax for all distinct elements x, y ∈ X. Corollary D.5.3. Let F be a free group of rank 2 and let n ≥ 2 be an integer. Then F contains a free subgroup of rank n. Proof. Suppose that F is based on the elements a and b. Let G be the subgroup of F generated by the subset X = {ai ba−i : 0 ≤ i ≤ n − 1}. Consider the action of G on F given by left multiplication. Define, for each x = ai ba−i ∈ X, the subset Ax ⊂ F as being the set of elements of F whose reduced form starts by ai bk for some k ∈ Z \ {0}. The subsets Ax , x ∈ X, clearly satisfy the hypotheses of Theorem D.5.1. Therefore G is free with base X.  

Appendix E

Inductive Limits and Projective Limits of Groups

E.1 Inductive Limits of Groups Let I be a directed set. An inductive system of groups over I consists of the following data: (1) a family of groups (Gi )i∈I indexed by I, (2) for each pair i, j ∈ I such that i ≤ j, a homomorphism ψji : Gi → Gj satisfying the following conditions: ψii = IdGi (identity map on Gi ) for all i ∈ I, ψkj ◦ ψji = ψki for all i, j, k ∈ I such that i ≤ j ≤ k. Then one speaks of the inductive system (Gi , ψji ) or simply of the inductive system (Gi ) if the homomorphisms ψji are understood. of groups over I. Consider the relation Let (Gi , ψji ) be an inductive system  ∼ on the disjoint union E = i∈I Gi defined as follows. If xi ∈ Gi and xj ∈ Gj are elements of E, then xi ∼ xj if and only if there is an element k ∈ I such that i ≤ k, j ≤ k and ψki (xi ) = ψkj (xj ). It easy to check that ∼ is a equivalence relation on the set E. Let G = E/ ∼ be the set of equivalence classes of ∼. For xi ∈ Gi , let [xi ] ∈ G denote the class of xi . One defines a binary operation on G in the following way. Given xi ∈ Gi and xj ∈ Gj , one defines the class [xi ][xj ] by setting [xi ][xj ] = [ψki (xi )ψkj (xj )], where k ∈ I is such that i ≤ k and j ≤ k. One checks that [xi ][xj ] depends neither on the choice of the representatives xi and xj , nor on the choice of k. Moreover, this operation gives a group structure on G. The group G is called the inductive limit (or the direct limit) of the inductive system (Gi ) and one writes G = lim Gi . For each i ∈ I, there is a canonical homomorphism −→ hi : Gi → G defined by hi (xi ) =[xi ]. Note that it immediately follows from the construction of G that G = i∈I hi (Gi ). Moreover, one has T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 13, © Springer-Verlag Berlin Heidelberg 2010

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E Inductive Limits and Projective Limits of Groups

hj ◦ ψji = hi for all i, j ∈ I such that i ≤ j. Examples E.1.1. (a) Let (Gi )i∈I be a family of groups and, for i, j ∈ I such that i ≤ j, let ψji : Gi → Gj be the trivial homomorphism, namely, ψji (g) = 1Gj for all g ∈ Gi . Then, (Gi , ψji ) is an inductive system whose inductive limit is a trivial group. (b) Let G be a group and denote by H the setof all finitely generated subgroups of G. Then (H, ⊂) is a directed set and H∈H H = G. Moreover, the set H together with the inclusion maps ψK,H : H → K, for all H, K ∈ H with H ⊂ K, is an inductive system whose limit is canonically isomorphic to G.

E.2 Projective Limits of Groups Let I be a directed set. A projective system of groups over I consists of the following data: (1) a family of groups (Gi )i∈I indexed by I, (2) for each pair i, j ∈ I such that i ≤ j, a homomorphism ϕij : Gj → Gi satisfying the following conditions: ϕii = IdGi (identity map on Gi ) for all i ∈ I, ϕij ◦ ϕjk = ϕik for all i, j, k ∈ I such that i ≤ j ≤ k. Then one speaks of the projective system (Gi , ϕij ) or simply of the projective system (Gi ) if the homomorphisms ϕij are understood.  Let (Gi , ϕij ) be a projective system of groups over I. Let P = i∈I Gi denote the direct product of the groups Gi . One immediately checks that G = {(xi ) ∈ P : ϕij (xj ) = xi for all i, j ∈ I such that i ≤ j} is a subgroup of P . The group G is called the projective limit of the projective system (Gi ), and one writes G = lim Gi . For each i ∈ I, there is a canonical ←− homomorphism fi : G → Gi obtained by restriction of the projection map πi : P → Gi . One has ϕij ◦ fj = fi for all i, j ∈ I such that i ≤ j. Examples E.2.1. (a) Let (Gi )i∈I be a family of groups and, for i, j ∈ I such that i ≤ j, let ϕij : Gj → Gi be the trivial homomorphism. Then, (Gi , ϕij ) is a projective system whose projective limit is a trivial group. (b) Let G be a group. Denote by N the set of all normal subgroups of G. Then (N , ⊃) is a directed set. The family of groups (G/H)H∈N , when

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equipped with the canonical quotient homomorphisms ϕH,K : G/K → G/H for all H, K ∈ N with H ⊃ K, gives rise to a projective system whose projective limit is canonically isomorphic to G.

Appendix F

The Banach-Alaoglu Theorem

All vector spaces considered in this appendix are vector spaces over the field R of real numbers.

F.1 Topological Vector Spaces A topological vector space is a real vector space X endowed with a topology such that the maps X ×X →X (x, y) → x + y and R×X →X (λ, x) → λx are continuous. Example F.1.1. Let  ·  be a norm on a real vector space X and let d denote the metric on X defined by d(x, y) = x − y for all x, y ∈ X. Then the topology defined by d yields a structure of topological vector space on X. Recall that a subset C of a real vector space X is said to be convex if (1 − λ)x + λy ∈ C for all λ ∈ [0, 1] and x, y ∈ C. A topological real vector space X is said to be locally convex if there is a base of neighborhoods of 0 consisting of convex subsets of X.

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F.2 The Weak-∗ Topology Let X be a real vector space equipped with a norm  · . Recall that a linear map u : X → R is continuous if and only if u is bounded on the unit ball B(X) = {x ∈ X : x ≤ 1}, that is, if and only if there is a constant C ≥ 0 such that |u(x)| ≤ C for all x ∈ B(X). The topological dual of X is the vector space X ∗ consisting of all continuous linear maps u : X → R. The operator norm on X ∗ is the norm defined by u =

sup |u(x)|. x∈B(X)

The topology defined by the operator norm is called the strong topology on X ∗ . Given x ∈ X, let ψx : X ∗ → R denote the evaluation map u → u(x) at x. The weak-∗ topology on X ∗ is the initial topology associated with the family of all evaluation maps ψx : X ∗ → R, x ∈ X. Thus, the weak-∗ topology is the smallest topology on X ∗ for which all evaluation maps ψx are continuous. Observe that every subset of X ∗ which is open for the weak-∗ topology is also open for the strong topology since all the evaluation maps are continuous for the strong topology. It follows that convergence with respect to the strong topology implies convergence with respect to the weak-∗ topology. The weak-∗ topology provides a topological vector space structure on X ∗ . A base of open neighborhoods of 0 for the weak-∗ topology is given by all subsets of the form V (F, ε) = {u ∈ X ∗ : |u(x)| < ε for all x ∈ F }, where F is a finite subset of X and ε > 0. Since all the sets V (F, ε) are convex, the topological vector space structure associated with the weak-∗ topology on X ∗ is locally convex. The weak-∗ topology on X ∗ is Hausdorff. Indeed, if u1 and u2 are two distinct elements in X ∗ , then there exists x ∈ X such that u1 (x) = u2 (x). If U1 and U2 are disjoint open subsets of R containing u1 (x) and u2 (x) respectively, then ψx−1 (U1 ) and ψx−1 (U2 ) are disjoint open subsets of X ∗ containing u1 and u2 respectively.

F.3 The Banach-Alaoglu Theorem In general, the unit ball in X ∗ is not compact for the strong topology (this follows from the fact that the unit ball in a normed space is never compact unless the space is finite-dimensional). However, this ball is always compact for the weak-∗ topology. This result, which is known as the Banach-Alaoglu

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theorem is one of the central results of classical functional analysis and may be easily deduced from the Tychonoff theorem. Theorem F.3.1 (Banach-Alaoglu theorem). Let X be a real normed vector space and let  ·  denote the operator norm on its topological dual X ∗ . Then the unit ball B(X ∗ ) = {u ∈ X ∗ : u ≤ 1} is compact for the weak-∗ topology on X ∗ . Proof. Observe first that X ∗ is a vector subspace of the vector space RX consisting of all real-valued functions on X. On the  other hand, setting Ix = [−x, x] ⊂ R for each x ∈ X. We haveB(X ∗ ) ⊂ x∈X Ix by definition of the operator norm. Let us equip RX = x∈X  R with the product topology. Then it is clear that the topology induced on x∈X Ix is the product topology and that the topology induced on X ∗ is the weak-∗ topology. Let f be an element of RX which is the limit of a net (ui ) of elements of B(x∗ ). For all x, y ∈ X and λ ∈ R, we have ui (λx) = λui (x), ui (x + y) = ui (x) + ui (y) and |ui (x)| ≤ x since ui ∈ B(X ∗ ). By taking limits, we get f (λx) = λf (x), ∗ f (x + y) = f (x) + f (y) and |f (x)| ≤ x.  This shows that f ∈ B(X ). ∗ X Consequently, B(X ) is closed in R . Since x∈X Ix is compact by Tychonoff theorem (Theorem A.5.2), we deduce that B(X ∗ ) is compact.

Appendix G

The Markov-Kakutani Fixed Point Theorem

All vector spaces considered in this appendix are vector spaces over the field R of real numbers.

G.1 Statement of the Theorem Let C be a convex subset of a real vector space X. A map f : C → C is called affine if f ((1 − λ)x + λy) = (1 − λ)f (x) + λf (y) for all λ ∈ [0, 1] and x, y ∈ C. Theorem G.1.1 (Markov-Kakutani). Let K be a nonempty convex compact subset of a Hausdorff topological vector space X. Let F be a set of continuous affine maps f : K → K. Suppose that all elements of F commute, that is, f1 ◦ f2 = f2 ◦ f1 for all f1 , f2 ∈ F. Then there exists a point in K which is fixed by all the elements of F.

G.2 Proof of the Theorem In the proof of the Markov-Kakutani theorem, we shall use the following lemmas. Lemma G.2.1. Let K be a compact subset of a topological vector space X and let V be a neighborhood of 0 in X. Then there exists a real number α > 0 such that λK ⊂ V for every real number λ such that |λ| < α. Proof. Since the multiplication by a scalar R × X → X is continuous, we can find, for each x ∈ X, a real number αx and an open neighborhood Ωx ⊂ X of x such that T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 15, © Springer-Verlag Berlin Heidelberg 2010

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|λ| < αx ⇒ λx Ωx ⊂ V.

(G.1)

The sets Ωx , x ∈ K, form an open cover of K. As K is compact, there is a finite subset F ⊂ K such that  Ωx . (G.2) K⊂ x∈F

If we take α = minx∈F αx , then α > 0 and |λ| < α ⇒ λK ⊂ V  

by (G.1) and (G.2).

Lemma G.2.2. Let K be a compact subset of a topological vector space X. Let (xi )i∈I be a net of points in K and let (λi )i∈I be a net of real numbers converging to 0 in R. Then the net (λi xi )i∈I converges to 0 in X. Proof. Let V be a neighborhood of 0 in X. By Lemma G.2.1, we can find α > 0 such that λK ⊂ V for every λ such that |λ| < α. As the net (λi ) converges to 0, there exists i0 ∈ I such that i ≥ i0 implies |λi | < α. Thus we have λi xi ∈ V for all i ≥ i0 . This shows that the net (λi xi ) converges to 0.   The following lemma is the theorem of Markov-Kakutani in the particular case when the set F is reduced to a single element. Lemma G.2.3. Let K be a nonempty convex compact subset of a Hausdorff topological vector space X and let f : K → K be an affine continuous map. Then f has a fixed point in K. Proof. Let us set C = {y − f (y) : y ∈ K}. The fact that f admits a fixed point in K is equivalent to the fact that 0 ∈ C. Choose an arbitrary point x ∈ K and consider the sequence (xn )n≥1 of points of X defined by xn =

n−1 1 k (f (x) − f k+1 (x)). n k=0

We have f k (x) − f k+1 (x) = f k (x) − f (f k (x)) ∈ C for 0 ≤ k ≤ n − 1. On the other hand, the set C is convex, since K is convex and f is affine. Thus xn ∈ C for every n ≥ 1. As 1 1 xn = x − f n (x). n n and f n (x) ∈ K for every n ≥ 1, it follows from Lemma G.2.2 that the sequence (xn )n≥1 converges to 0. The set C is compact since it is the image of the compact set K by the continuous map y → y − f (y). As every compact subset of a Hausdorff space is closed, we deduce that C is closed in X. Thus 0 ∈ C.  

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Proof of Theorem G.1.1. Let f ∈ F and consider the set Fix(f ) = {x ∈ K : f (x) = x} of its fixed points. The set Fix(f ) is not empty by Lemma G.2.3 and it is compact since it is a closed subset of the compact set K. On the other hand, Fix(f ) is convex since K is convex and f is affine. If g ∈ F and x ∈ Fix(f ), then the fact that f and g commute implies that g(x) ∈ Fix(f ) since f (g(x)) = g(f (x)) = g(x). Therefore we can apply Lemma G.2.3 to the restriction of g to Fix(f ). It follows that g fixes a point in Fix(f ), that is, Fix(f ) ∩ Fix(g) = ∅. By induction on n, we get Fix(f1 ) ∩ Fix(f2 ) ∩ · · · ∩ Fix(fn ) = ∅ for all f1 , f2 , . . . , fn ∈ F. Since K is compact, from the finite intersection property (see Sect. A.5) we deduce that  Fix(f ) = ∅. f ∈F

This shows that there is a point in K which is fixed by all elements of F .  

Notes The proof presented here is due to S. Kakutani [Kak] (see [Jac]). In the proof of A. Markov [Mar], the local convexity of X is needed.

Appendix H

The Hall Harem Theorem

H.1 Bipartite Graphs A bipartite graph is a triple G = (X, Y, E), where X and Y are arbitrary sets, and E is a subset of the Cartesian product X × Y . The set X (resp. Y ) is called the set of left (resp. right) vertices and E is called the set of edges of the bipartite graph G (see Fig. H.1).

Fig. H.1 The bipartite graph G = (X, Y, E) with X = {x1 , x2 , x3 , x4 , x5 }, Y = {y1 , y2 , y3 , y4 } and E = {(x1 , y1 ), (x2 , y1 ), (x2 , y2 ), (x2 , y3 ), (x3 , y4 ), (x5 , y3 ), (x5 , y4 )}

A bipartite subgraph of a bipartite graph G = (X, Y, E) is a bipartite graph G  = (X  , Y  , E  ) with X  ⊂ X, Y  ⊂ Y and E  ⊂ E (see Fig. H.2). Let G = (X, Y, E) be a bipartite graph. Two edges (x, y), (x , y  ) ∈ E are said to be adjacent if x = x or y = y  . Given a vertex x ∈ X (resp. y ∈ Y ) the right-neighborhood of x (resp. the left-neighborhood of y) is the subset NR (x) ⊂ Y (resp. NL (y) ⊂ X) defined by (see Figs. H.3–H.4):

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Fig. H.2 The bipartite subgraph G  = (X  , Y  , E  ) of the bipartite graph G in Fig. H.1 with X  = {x1 , x2 , x4 , x5 }, Y  = {y1 , y2 , y3 } and E  = {(x2 , y1 ), (x2 , y2 ), (x2 , y3 ), (x5 , y3 )}

Fig. H.3 The right-neighborhood NR (x2 ) ⊂ Y of the vertex x2 ∈ X in the bipartite graph G of Fig. H.1

NR (x) = NRG (x) = {y ∈ Y : (x, y) ∈ E} (resp. NL (y) = NLG (y) = {x ∈ X : (x, y) ∈ E}). For subsets A ⊂ X and B ⊂ Y , we define the right-neighborhood NR (A) of A and the left-neighborhood NL (B) of B by   NR (A) = NRG (A) = NR (a) and NL (B) = NLG (B) = NL (b). a∈A

b∈B

One says that the bipartite graph G = (X, Y, E) is finite if the sets X and Y are finite. One says that G is locally finite if the sets NR (x) and NR (y) are finite for all x ∈ X and y ∈ Y . Note that if G is locally finite then the sets NR (A) and NL (B) are finite for all finite subsets A ⊂ X and B ⊂ Y .

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Fig. H.4 The left-neighborhood NL (y1 ) ⊂ X of the vertex y1 ∈ Y in the bipartite graph G of Fig. H.1

H.2 Matchings Let G = (X, Y, E) be a bipartite graph. A matching in G is a subset M ⊂ E of pairwise nonadjacent edges. In other words, a subset M ⊂ E is a matching if and only if both projection maps p : M → X and q : M → Y are injective. A matching M is called left-perfect (resp. right-perfect) if for each x ∈ X (resp. y ∈ Y ), there exists y ∈ Y (resp. x ∈ X) such that (x, y) ∈ M (see Fig. H.5). Thus, a matching M is left-perfect (resp. right-perfect) if and only if the projection map p : M → X (resp. q : M → Y ) is surjective (and therefore bijective). A matching M is called perfect if it is both left-perfect and right-perfect.

Fig. H.5 A right-perfect matching M ⊂ E in the bipartite graph G of Fig. H.1. Note that there is no left-perfect matching (and therefore no perfect matching) in G since |X| > |Y |

Remarks H.2.1. (a) A subset M ⊂ E is a left-perfect (resp. right-perfect) matching if and only if there is an injective map ϕ : X → Y (resp. an injective

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map ψ : Y → X) such that M = {(x, ϕ(x)) : x ∈ X} (resp. M = {(ψ(y), y) : y ∈ Y }). (b) Similarly, a subset M ⊂ E is a perfect matching if and only if there is a bijective map ϕ : X → Y such that M = {(x, ϕ(x)) : x ∈ X}. Given a bipartite graph G = (X, Y, E), one often regards the set X as a set of boys and Y as a set of girls. One interprets (x, y) ∈ E as the condition that x and y know each other. In this context, a matching M ⊂ E is a process of getting boys and girls that know each other married (no polygamy is allowed here). The matching is left-perfect (resp. right-perfect) if and only if every boy (resp. girl) gets married. Finally, M is perfect if and only if there remain no singles.

H.3 The Hall Marriage Theorem We use the notation | · | to denote cardinality of sets. Definition H.3.1 (Hall conditions). Let G = (X, Y, E) be a bipartite graph. One says that G satisfies the left (resp. right) Hall condition if |NR (A)| ≥ |A|

(H.1)

(resp. |NL (B)| ≥ |B|) for every finite subset A ⊂ X (resp. for every finite subset B ⊂ Y ). One says that G satisfies the Hall marriage conditions if G satisfies both the left and the right Hall conditions. Theorem H.3.2. Let G = (X, Y, E) be a locally finite bipartite graph. Then the following conditions are equivalent. (a) G satisfies the left (resp. right) Hall condition; (b) G admits a left (resp. right) perfect matching. Proof. It is obvious that (b) implies (a). Let us prove that (a) implies (b). By symmetry, it suffices to show that if G satisfies the left Hall condition then it admits a left perfect matching. We fist treat the case when the set X is finite by induction on n = |X|. In the case |X| = 1, the statement is trivially satisfied. Suppose that we have proved the statement whenever |X| ≤ n − 1 and let us prove it for |X| = n. We distinguish two cases. Case (i): Suppose that |NR (A)| ≥ |A| + 1

(H.2)

for all nonempty proper subsets A ⊂ X. Then fix x0 ∈ X and y0 ∈ Y such that (x0 , y0 ) ∈ E. Consider the bipartite subgraph G  = (X  , Y  , E  ),

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where X  = X \ {x0 }, Y  = Y \ {y0 }, and E  = E ∩ (X  × Y  ). Then, for every nonempty subset A ⊂ X  , we have |NR (A )| ≥ |A | + 1 by (H.2), and   hence |NRG (A )| ≥ |A | since NRG (A ) = NR (A ) \ {y0 }. As |X  | = n − 1, it follows from our induction hypothesis that G  admits a left-perfect matching M  ⊂ E  . Then M = M  ∪ {(x0 , y0 )} is a left-perfect matching for G. Case (ii): Suppose that we are not in Case (i). This means that there exists a nonempty proper subset X  ⊂ X such that |NR (X  )| = |X  |.

(H.3)

Then the bipartite subgraph G  = (X  , Y  , E  ), where Y  = NR (X  ) and E  = E ∩ (X  × Y  ) clearly satisfies the left Hall condition. As |X  | ≤ n − 1, there exists, by our induction hypothesis, a left-perfect matching M  ⊂ E  for G  . Consider now the bipartite subgraph G  = (X  , Y  , E  ), where X  = X \ X  , Y  = Y \ Y  , and E  = E ∩ (X  × Y  ). We claim that the left Hall condition also holds for G  . Otherwise, there would be some subset A ⊂ X  such that, 

|NRG (A )| < |A |,

(H.4)

and then the left Hall condition for G would be violated by A = X  ∪ A ⊂ X since |NR (A)| = |NR (X  ∪ A )| = |NR (X  ) ∪ NR (A )| 

= |NR (X  ) ∪ NRG (A )| 

≤ |NR (X  )| + |NRG (A )| < |X  | + |A | (by (H.3) and (H.4)) = |X  ∪ A | = |A|. Therefore, as |X  | < |X| = n, induction applies again yielding a left-perfect matching M  ⊂ E  for G  . It then follows that M = M  ∪ M  is a leftperfect matching for G. This completes the proof that (a) implies (b) in the case when X is finite. To treat the general case, we shall apply the Tychonoff product theorem. Suppose that G = (X, Y, E) is a (possibly infinite) locally finite bipartite graph satisfying the left Hall condition, that is, |NR (A)| ≥ |A|for every finite subset A ⊂ X. Let us equip the Cartesian product K = x∈X NR (x) with its prodiscrete topology, that is, with the product topology obtained by taking the discrete topology on each factor NR (x). As each set NR (x) is finite, the space K is compact by the Tychonoff product theorem (Corollary A.5.3).

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For each x ∈ X, let πx : K → NR (x) denote the projection map. Let F be the set consisting of all nonempty finite subsets of X. Consider, for each F ∈ F, the set C(F ) ⊂ K consisting of all z ∈ K which satisfy πx1 (z) = πx2 (z) for all distinct elements x1 and x2 in F . It follows from this definition and the continuity of the projection maps πx , that C(F ) is an intersection of closed subsets of X and hence closed in K. On the other hand, C(F ) is not empty. Indeed, choose, for each x ∈ X, an element ψ(x) ∈ NR (x) (observe that NR (x) is not empty by the left Hall condition). Also set G  = (X  , Y  , E  ) where X  = F , Y  = NR (F ) and E  = E ∩ (X  × Y  ). Then, the left Hall condition for G implies the left Hall condition for the finite bipartite graph G  and therefore, by Theorem H.3.2, there exists a perfect matching for G  . By Remark H.2.1, there exists an injective mapping ϕ : F = X  → Y  = NR (F ). Then, the element (Φ(x))x∈X ∈ K, defined by Φ(x) = ϕ(x) if x ∈ F and Φ(x) = ψ(x) if x ∈ X \ F , clearly belongs to C(F ). Since C(F1 ) ∩ C(F2 ) ∩ · · · ∩ C(Fn ) ⊃ C(F1 ∪ F2 ∪ · · · ∪ Fn ) for all F1 , F2 , . . . , Fn ∈ F, we deduce that the family {C(F ) : F ∈ F} of subsets of K hasthe finite intersection property. By compactness of K, there is a point z0 ∈ F ∈F F . Then M = {(x, πx (z0 )) : x ∈ X} is a left-perfect matching for G. Remark H.3.3. The hypothesis of local finiteness for the bipartite graph can not be removed from the statement of the previous theorem. To see this, consider the bipartite graph G = (X, Y, E), where X = Y = N and E = {(0, n + 1) : n ∈ N} ∪ {(n + 1, n) : n ∈ N}. Observe that G is not locally finite as |NR (0)| = ∞. Also, it satisfies the left Hall condition. Indeed, given a finite subset A ⊂ X, the set NR (A) is infinite if 0 ∈ A, while |NR (A)| = |{n − 1 : n ∈ A}| = |A| if 0 ∈ / A. However, G admits no left perfect matching. Indeed, for any matching M ⊂ E, either 0 ∈ X remains unmatched, or, if (0, n) ∈ M for some n ∈ N, then n + 1 ∈ X remains unmatched (see Fig. H.6). Theorem H.3.4. Let G = (X, Y, E) be a bipartite graph. Suppose that G admits both a left perfect matching and a right perfect matching. Then G admits a perfect matching. Proof. Let MX (resp. MY ) be a left perfect (resp. right perfect) matching for G. Consider the equivalence relation in M = MX ∪ MY defined by declaring two edges e and e in relation if there exists a finite sequence e = e0 , e1 , . . . , en = e in M such that ei and ei+1 are adjacent for all 0 ≤ i ≤ n − 1. Then, each equivalence class consists of either (see Fig. H.7): – a single edge, or – a cycle of even length 2n ≥ 4, or – an infinite chain which can be bi-infinite, or infinite only in one direction.

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Fig. H.6 The bipartite graph G = (X, Y, E), where X = Y = N and E = {(0, n + 1) : n ∈ N} ∪ {(n + 1, n) : n ∈ N}

Note that an equivalence class is reduced to a single edge (x, y) if and only if (x, y) ∈ MX ∩ MY . On the other hand, an equivalence class is a cycle of length 2n if and only if it is of the form C ={(x1 , y1 ), (x2 , y2 ), . . . , (xn , yn )} ∪ {(x1 , y2 ), (x2 , y3 ), . . . , (xn−1 , yn ), (xn , y1 )} with xi ∈ X, all distinct, and yj ∈ Y , all distinct, 1 ≤ i, j ≤ n. In this case, we then set M (C) = {(x1 , y1 ), (x2 , y2 ), . . . , (xn , yn )}. Also, a bi-infinite chain is an equivalence class of the form C = {(xn , yn ) : n ∈ Z} ∪ {(xn , yn+1 ) : n ∈ Z} with xi ∈ X, all distinct, and yj ∈ Y , all distinct, i, j ∈ Z. In this case, we then set M (C) = {(xn , yn ) : n ∈ Z}.

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Fig. H.7 In the bipartite graph G = (X, Y, E), where X = {x0 , x1 , x2 , x3 , x4 , x5 }, Y = {y0 , y1 , y2 , y3 , y4 , y5 } and E = {(xi , yi ) : i = 0, 1, . . . , 5} ∪ {(xi , yi+1 ) : i = 1, 2, . . . , 4} ∪ {(x5 , y1 )}, we have a left-perfect matching MX = {(xi , yi ) : i = 0, 1, . . . , 5} (which is indeed perfect) and a right-perfect matching MY = {(x0 , y0 )} ∪ {(xi , yi+1 ) : i = 1, 2, . . . , 4} ∪ {(x5 , y1 )} (which is also perfect). There are two equivalence classes in M = MX ∪ MY , namely, MX ∩ MY = {(x0 , y0 )}, which consists of a single edge, and C = {(xi , yi ) : i = 1, 2 . . . , 5} ∪ {(xi , yi+1 ) : i = 1, 2, . . . , 4} ∪ {(x5 , y1 )}, which is a cycle of length 10

Finally, if the equivalence class is an infinite chain, which is infinite only in one direction, then it is either of the form C = {(xn , yn ) : n ∈ N} ∪ {(xn , yn+1 ) : n ∈ N} or C = {(xn , yn ) : n ∈ N} ∪ {(xn+1 , yn ) : n ∈ N} with xi ∈ X, all distinct, and yj ∈ Y , all distinct, i, j ∈ N. In both cases, we then set M (C) = {(xn , yn ) : n ∈ N}. It is then clear that the set M ⊂ E defined by  M (C), M= C

where C runs over all equivalence classes in M , is the required perfect matching for G. Corollary H.3.5 (Cantor–Bernstein Theorem). Let X and Y be two sets. Suppose that there exist injective maps f : X → Y and g : Y → X. Then there exists a bijective map h : X → Y . Proof. Consider the bipartite graph G = (X, Y, E), where E = {(x, f (x)) : x ∈ X} ∪ {(g(y), y) : y ∈ Y }. Now, MX = {(x, f (x)) : x ∈ X} and

H.4 The Hall Harem Theorem

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MY = {(g(y), y) : y ∈ Y } are left perfect and right perfect matchings in G, respectively (cf. Remark H.2.1(a)). By the previous theorem, there exists a perfect matching M ⊂ E for G. Then M = {(x, h(x)) : x ∈ X}, where h : X → Y is bijective (cf. Remark H.2.1(b)). Theorem H.3.6 (The Hall marriage Theorem). Let G = (X, Y, E) be a locally finite bipartite graph. Then the following conditions are equivalent: (a) G satisfies the Hall-marriage conditions; (b) G admits a perfect matching. Proof. The left (resp. right) Hall condition implies, by Theorem H.3.2, the existence of a left- (resp. right-) perfect matching for G. Then, Theorem H.3.4 guarantees the existence of a perfect matching for G. The converse implication is trivial.

H.4 The Hall Harem Theorem Let G = (X, Y, E) be a bipartite graph and let k ≥ 1 be an integer. A subset M ⊂ E is called a perfect (1, k)-matching if it satisfies the following conditions: (1) for each x ∈ X, there are exactly k elements in y ∈ Y such that (x, y) ∈ M , (2) for each y ∈ Y , there is a unique element x ∈ X such that (x, y) ∈ M (see Fig. H.8). Thus, a subset M ⊂ E is a perfect (1, k)matching if and only if there exists a k-to-one surjective map ψ : Y → X such that M = {(ψ(y), y) : y ∈ Y } (recall that a surjective map f : S → T from a set S onto a set T is said to be k-to-one if each element in T has exactly k preimages in S). Note that when k = 1, a perfect (1, k)-matching is the same thing as a perfect matching. In the language of boys, girls, and marriages, a perfect (1, k)-matching is a process for marrying each boy with exactly k girls (among the girls he knows) in such a way that each girl is married with exactly one boy (among the boys she knows). The girls that are married with a given boy constitute his harem. Definition H.4.1. Let G = (X, Y, E) be a locally finite bipartite graph and let k ≥ 1 be an integer. One says that G satisfies the Hall k-harem conditions if |NR (A)| ≥ k|A| and (H.5) |NL (B)| ≥ k1 |B| for all finite subsets A ⊂ X and B ⊂ Y . Theorem H.4.2 (The Hall harem Theorem). Let G = (X, Y, E) be a locally finite bipartite graph and let k ≥ 1 be an integer. Then, the following conditions are equivalent. (a) G satisfies the Hall k-harem conditions; (b) G admits a perfect (1, k)-matching.

400

H The Hall Harem Theorem

Fig. H.8 The bipartite graph G = (X, Y, E), where X = {x1 , x2 }, Y = {y1 , y2 , y3 , y4 } and E = {(x1 , yi ) : i = 1, 2, 3, 5} ∪ {(x2 , yj ) : j = 1, 2, 4, 5, 6}, with a perfect (1, 3)-matching M = {(x1 , yi ) : i = 1, 2, 3} ∪ {(x2 , yj ) : j = 4, 5, 6} and the harems H(x1 ) and H(x2 ) of x1 and x2 respectively

Proof. The fact that (b) implies (a) is trivial. Let us prove that (a) implies (b). Suppose that the Hall k-harem conditions are satisfied by G. Let X1 , X2 , . . . , Xk be disjoint copies of X and let φi : X → Xi , i = 1, 2, . . . , k, denote the copy maps. It will be helpful to think of φi (x) as the clone of x ∈ X in Xi . k Consider the new bipartite graph G  = (X  , Y  , E  ), where X  = i=1 Xi , Y  = Y , and E  = {(φi (x), y) : (x, y) ∈ E, i = 1, 2, . . . , k} ⊂ X  × Y  . that Let A be a finite subset of X  and denote by A the set of x ∈ X such  some clone of x belongs to A . Observe that |A | ≤ k|A | and NRG (A ) = NRG (A ). Thus, using (a), we get 

|NRG (A )| = |NRG (A )| ≥ k|A | ≥ |A |.

(H.6) 

On the other hand, if B  is a finite subset of Y  = Y , then NLG (B  ) is the set consisting of all clones of elements of NLG (B  ) so that   1  G  G  |B | = |B  | (H.7) |NL (B )| = k|NL (B )| ≥ k k by (a). Inequalities H.6 and H.7 say that G  satisfies the Hall marriage conditions. Therefore, G  admits a perfect matching M  ⊂ E  by Theorem H.3.6. Then the set M , consisting of all pairs (x, y) ∈ E such that (x , y) ∈ M  for some clone x of x, is clearly a perfect (1, k)-matching for G.

Notes

401

Notes The Hall marriage theorem was first established for finite bipartite graphs by P. Hall [Hall1] and then extended to infinite locally finite bipartite graphs by M. Hall [Hall-M-1]. The proof of Theorem H.3.2 which is given in this appendix is based on [Halm].

Appendix I

Complements of Functional Analysis

All vector spaces considered in this appendix are vector spaces over the field R of real numbers.

I.1 The Baire Theorem Let (X, d) be a metric space. Given x ∈ X and r > 0, we denote by BX (x, r) = {y ∈ X : d(x, y) ≤ r} the closed ball of radius r centered at x and by OB X (x, r) = {y ∈ X : d(x, y) < r} the open ball of radius r centered at x. For a subset A ⊂ X we denote by A (resp. Int A) the closure (resp. the interior) of A. Theorem I.1.1 (Baire’s Theorem). Let (X, d) be a complete metric space. Let (Xn )n≥1 be a sequence of closed subsets such that Int Xn = ∅ for all n ≥ 1. Then

⎛ Int ⎝



(I.1)

⎞ Xn ⎠ = ∅.

(I.2)

n≥1

Proof. Let us set  An = X \ Xn , so that An is an open dense subset of X. Let us show that n≥1 An is dense in X. Let x0 ∈ X and r0 > 0. We have to show that ⎞ ⎛   An ⎠ = ∅. BX (x0 , r0 ) ⎝ (I.3) n≥1

T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 17, © Springer-Verlag Berlin Heidelberg 2010

403

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 As A1 is dense and open we can find x1 ∈ BX (x0 , r0 ) A1 and r1 > 0 such that  BX (x1 , r1 ) ⊂ OB X (x0 , r0 ) A1 0 < r1
0 there exists r > 0 such that

I.2 The Open Mapping Theorem

405

T (OB X (0, r )) ⊃ OB Y (0, r ). Finally, since T (OB X (0, r )) = r T (OB X (0, 1)) and OB Y (0, r ) = r OB Y (0, 1), we are only left to show that there exists r > 0 such that T (OB X (0, 1)) ⊃ OB Y (0, r). (I.6) For all n ≥ 1 set Yn = nT (OB X (0, 1)) = T (OB X (0, n)). As T is surjective we have n≥1 Yn = Y . It follows from Theorem I.1.1 that there exists n0 ≥ 1 such that Int Yn0 = ∅. Since Int(n0 T (OB X (0, 1))) = n0 Int(T (OB X (0, 1))), we deduce that Int(T (OB X (0, 1))) = ∅. Thus we can find r > 0 and y ∈ Y such that OB Y (y, 4r) ⊂ T (OB X (0, 1)). (I.7) It follows that y ∈ T (OB X (0, 1)) and, by symmetry, − y ∈ T (OB X (0, 1)).

(I.8)

Summing (I.7) and (I.8) we obtain 2OB Y (0, 2r) = OB Y (0, 4r) = OB Y (y, 4r) − y ⊂ T (OB X (0, 1)) + T (OB X (0, 1)) ⊂ 2T (OB X (0, 1)). We deduce that OB Y (0, 2r) ⊂ T (OB X (0, 1)) so that, for all n ≥ 0, r OB Y 0, n ⊂ T (OB X (0, 1/2n+1 )). 2

(I.9)

Let y0 ∈ OB Y (0, r) and let us show that there exists x ∈ OB X (0, 1) such that y0 = T (x ). We deduce from (I.9) (with n = 0) that there exists x0 ∈ OB X (0, 12 ) such that y0 − T (x0 ) < 2r . Set y1 = y0 − T (x0 ) and observe that y1 ∈ OB Y (0, 2r ). Continuing this way, we find a sequence (yn )n≥1 in Y and a sequence (xn )n≥1 in X such that r yn+1 = yn − T (xn ) ∈ OB Y 0, n+1 , (I.10) 2 1 (I.11) xn ∈ OB X 0, n+1 2 and yn − T (xn )
0 there exists x ∈ X such that x = 1 and T (x) < ε. Then T is not bijective. Proof. Suppose by contradiction that T is bijective. Then, by Corollary I.2.2, the inverse linear map T −1 : Y → X is continuous. Thus we can find a constant M > 0 such that T −1 (y) ≤ M y for all y ∈ Y . Since T is bijective, this is equivalent to x ≤ M T (x) for all x ∈ X. This clearly contradicts the hypotheses. Thus, T is not bijective.



I.3 Spectra of Linear Maps Let (X, · ) be a Banach space. We denote by L(X) the space of all continuous linear maps T : X → X endowed with the norm T = sup x∈X x=0

T (x) . x

We denote by IdX : X → X the identity map. Definition I.3.1. Let T ∈ L(X). The set σ(T ) = {λ ∈ R : (T − λ IdX ) is not bijective}

(I.13)

is called the real spectrum of T . Proposition I.3.2. Let T ∈ L(X). Then the spectrum σ(T ) is a compact set and σ(T ) ⊂ [− T , T ]. (I.14) Proof. Let λ ∈ R and suppose that |λ| > T . For y ∈ X the equation (T − λ IdX )(x) = y

(I.15)

I.4 Uniform Convexity

407

admits a unique solution x ∈ X. Indeed (I.15) is equivalent to x=

1 (T (x) − y). λ

(I.16)

1 Moreover, λ1 (T (x1 )−y)− λ1 (T (x2 )−y) ≤ |λ| T · x1 −x2 for all x1 , x2 ∈ X 1 and |λ| T < 1. It then follows from the Banach fixed point theorem that there exists a unique x ∈ X satisfying (I.16). It follows that T − λ IdX is bijective and therefore λ ∈ / σ(T ). This shows (I.14). Let us show that R \ σ(T ) is open. Let λ0 ∈ R \ σ(T ) so that T − λ0 IdX is bijective. By Corollary I.2.2, we have that the inverse map (T − λ0 IdX )−1 is also continuous so that 0 = (T − λ0 IdX )−1 < ∞. Let λ ∈ R such that |λ − λ0 | < (T −λ0 1IdX )−1  . For y ∈ X we have that the linear equation (I.15) can be written as T (x) − λ0 x = y + (λ − λ0 )x, that is,

x = (T − λ0 IdX )−1 [y + (λ − λ0 )x].

(I.17)

By applying again the Banach fixed point theorem, we deduce that there exists a unique x ∈ X satisfying (I.17). This shows that T −λ IdX is bijective, that is, λ ∈ R\σ(T ). It follows that R\σ(T ) is open. Therefore σ(T ) is closed.



I.4 Uniform Convexity Definition I.4.1. A normed space (X, · ) is said to be uniformly convex if, for every ε > 0, there exists δ > 0 such that

x + y

ε implies 2 for all x, y ∈ X with x , y ≤ 1. Let Z be a nonempty set not reduced to a single point. Then the Banach space 1 (Z) is not uniformly convex. For instance, if z, z  ∈ Z are distinct, setting x = δz and y = δz , one has x 1 = y 1 = 1, x − y 1 = 2, but x+y 2 1 = 1. On the other hand, we have the following: Proposition I.4.2. Let X be a vector space equipped with a scalar product

·, · and denote by · the associated norm. Then the normed space (X, · ) is uniformly convex. In particular, every Hilbert space is uniformly convex. Proof. Let x, y ∈ X such that x , y ≤ 1. Then, we have

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I Complements of Functional Analysis



x + y 2 1 1 2

2 + 4 x − y = 4 ( x + y, x + y + x − y, x − y) 1 = ( x, x + y, y + 2 x, y + x, x + y, y − 2 x, y) 4  1 2 x 2 + 2 y 2 = 4 ≤ 1.

Therefore,



x + y 2 1 2

(I.18)

2 ≤ 1 − 4 x − y . √ Let now ε > 0 and set δ = 1 − 12 4 − ε2 . Suppose that x − y > ε. This 2 implies 1 − 14 x − y 2 < 1 − ε4 = (1 − δ)2 . From (I.18) we then deduce that x+y 2 < 1 − δ. This shows that X is uniformly convex.



Appendix J

Ultrafilters

J.1 Filters and Ultrafilters Let X be a set. We denote by P(X) the set of all subsets of X. Definition J.1.1. A filter on X is a nonempty set F ⊂ P(X) satisfying the following conditions: (F-1) ∅ ∈ / F; (F-2) if A ∈ F and A ⊂ B ⊂ X, then B ∈ F; (F-3) if A ∈ F and B ∈ F, then A ∩ B ∈ F. Examples J.1.2. (a) Let X be a set and let A0 be a nonempty subset of X. Then the set {A ∈ P(X) : A0 ⊂ A} is a filter on X. A filter F on X is said to be a principal filter if there exists a nonempty subset A0 of X such that F = {A ∈ P(X) : A0 ⊂ A}. One then says that F is the principal filter based on A0 . (b) Let X be a set and Ω ⊂ P(X). Suppose that ∅ ∈ / Ω and that, given A1 , A2 ∈ Ω, there exists A ∈ Ω such that A ⊂ A1 ∩ A2 . Then the set F(Ω) = {A ∈ P(X) : there exists A ∈ Ω such that A ⊂ A}

(J.1)

is a filter on X and one has Ω ⊂ F(Ω). The filter F(Ω) is called the filter generated by Ω. (c) Let (I, ≤) be a nonempty directed set. A subset A ⊂ I is called residual in I if there exists i ∈ I such that A ⊃ {j ∈ I : i ≤ j}. Clearly, the set Fr (I) of all residual subsets of I is a filter on I. It is the filter generated by the sets {j ∈ I : i ≤ j}, i ∈ I. The filter Fr (I) is called the residual filter on I. (d) Let X be an infinite set. Then the set F = {A ∈ P(X) : X \A is finite} is a filter. It is called the Fr´echet filter on X. If we consider the directed set (N, ≤), a subset A ⊂ N is residual if and only if its complement N \ A is finite. Therefore, the residual filter on N equals the Fr´echet filter on N. T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1 18, © Springer-Verlag Berlin Heidelberg 2010

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(e) Let X be a topological space and let x be a point in X. Then, the set Fx of all neighborhoods of x is a filter on X. (f) Let X be a nonempty uniform space (cf. Appendix B). Then the set U ⊂ P(X × X) of all entourages of X is a filter on X × X. Note that for a filter F ⊂ P(X) the following also holds true: (F-4) if A ⊂ X, then A and X \ A cannotboth belong to F; n (F-5) if A1 , A2 , . . . , An ∈ F, n ≥ 1, then i=1 Ai ∈ F; (F-6) X ∈ F. Indeed, (F-4) follows from (F-1) and (F-3) with B = X \ A. From (F-3) we deduce by induction condition (F-5). Finally, (F-6) follows from the fact that F = ∅ and (F-2). Proposition J.1.3. Let X be a set and Ω0 ⊂ P(X). Then, there exists a filter on X containing Ω0 if an only if Ω0 has the finite intersection property, that is, every finite family of elements in Ω0 has nonempty intersection. Proof. Let Ω0 ⊂ P(X) be a set and suppose that there exists a filter F such that Ω0 ⊂ F. By (F-1) and (F-5) we have that Ω0 has the finite intersection property. Conversely, suppose that Ω0 has the finite intersection property. Consider the set Ω consisting of all finite intersections of elements of Ω0 . Then, Ω satisfies the conditions in Example J.1.2(b), and therefore the filter

generated by Ω is a filter containing Ω0 . Definition J.1.4. A filter ω ⊂ P(X) is called an ultrafilter on X if it satisfies the condition (UF) if A ⊂ X, then A ∈ ω or (X \ A) ∈ ω. Example J.1.5. Let X be a set and x ∈ X. Then the principal filter based on {x} is an ultrafilter. An ultrafilter ω on X is called principal if there exists x ∈ X such that ω is the principal filter based on {x}. Note that a principal filter is an ultrafilter if and only if it is based on a singleton. If X is finite, then every ultrafilter is, clearly, principal. An ultrafilter which is not principal is called non-principal (or free). Theorem J.1.6. Let X be a nonempty set. Let F0 be a filter on X. Then there exists an ultrafilter ω on X containing F0 . Let us first prove the following: Lemma J.1.7. Let X be a set. Let F be a filter on X. Suppose there exists a set A0 ∈ P(X) such that A0 ∈ / F and (X \ A0 ) ∈ / F. Then there exists a filter F  on X such that

J.1 Filters and Ultrafilters

411

(1) F ⊂ F  ; (2) A0 ∈ F  . Proof. Consider the principal filter F0 based on A0 and set F  = {B ∈ P(X) : B ⊃ A ∩ A for some A ∈ F and A ∈ F0 }.

(J.2)

Let us show that F  satisfies the required conditions. First of all, taking A = X we have A = A ∩ X ∈ F  for all A ∈ F. This shows (1). On the other hand, taking A = X and A = A0 we have A0 = X ∩ A0 ∈ F, and this shows (2). We are only left to show that F  is a filter. Let B ∈ F  and denote by A ∈ F and A ∈ F0 two sets such that A ∩ A ⊂ B. Let us first show that A ∩ A = ∅. Suppose the contrary. Then A ⊂ (X \ A ) ⊂ (X \ A0 ). As A belongs to the filter F, it follows from (F-2) that (X \ A0 ) ∈ F, contradicting our assumptions. It follows that A ∩ A = ∅ and therefore B = ∅. This shows (F-1). Suppose now that B  ∈ P(X) contains B. Then (A ∩ A ) ⊂ B  so that B  ∈ F  . This shows (F-2). Finally, let B1 , B2 ∈ F  . For i = 1, 2 denote by Ai ∈ F and Ai ∈ F0 two sets such that Ai ∩ Ai ⊂ Bi . We have   B1 ∩ B2 ⊃ (A1 ∩ A1 ) (A2 ∩ A2 ) = (A1 ∩ A2 ) (A1 ∩ A2 ). But A1 ∩ A2 belongs to the filter F, and A1 ∩ A2 ∈ F0 as both A1 and A2 contain A0 . It follows that B1 ∩ B2 ∈ F  . This shows (F-3). It follows that

F  is a filter. Proof of Theorem J.1.6. Consider the set Φ0 consisting of all filters on X containing F0 . This is a nonempty set, partially orderedby inclusion. Let Φ be a totally ordered subset of Φ0 . We claim that F = F ∈Φ F is an upper bound for Φ. We only have to show that F belongs to Φ0 . As ∅ ∈ / F for all  This shows that F satisfies condition (F-1). Let F ∈ Φ we also have ∅ ∈ / F. now A ∈ F and B ⊂ X be such that A ⊂ B. Then there exists F ∈ Φ such that A ∈ F. As F is a filter, we have B ∈ F, by (F-2). It then follows that  This shows that F satisfies condition (F-2). Finally, suppose that B ∈ F.  Then there exist FA , FB ∈ Φ such that A ∈ FA and B ∈ FB . As Φ A, B ∈ F. is totally ordered, up to exchanging A and B we can suppose that FA ⊂ FB . We then have A, B ∈ FB and therefore A ∩ B ∈ FB , by (F-3). It follows that  Therefore F satisfies condition (F-3) as well. This shows that Φ0 A ∩ B ∈ F. is inductive. By Zorn’s lemma, Φ0 contains a maximal element ω. Let us show that ω is an ultrafilter on X. Let A0 ⊂ X. Suppose that A0 , (X \ A0 ) ∈ / ω. Then, by Lemma J.1.7, there exists a filter ω  containing A0 and ω. As ω  is in Φ0 and properly contains ω, this contradicts the maximality of ω. It follows that either A0 or X \ A0 belongs to ω. This shows that ω satisfies condition (UF), and therefore it is an ultrafilter.



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Corollary J.1.8. Let X be a set. Let F be a filter on X. Then the following conditions are equivalent. (a) F is an ultrafilter; (b) F is a maximal filter, that is, if F  is a filter containing F, then F  = F. Proof. Suppose (a) and let F  be a filter properly containing F. Let A ∈ F  \ F. As F is an ultrafilter, (X \ A) ∈ F ⊂ F  . Thus, both A and (X \ A) belong to F  contradicting (F-4). This shows that F  = F and the implication (a) ⇒ (b) follows. Suppose now that F is a maximal filter. By Theorem J.1.6 there exists an ultrafilter ω containing F. By maximality we have F = ω, that is, F is an ultrafilter. This shows (b) ⇒ (a).



J.2 Limits Along Filters Definition J.2.1. Let X be a topological space. A filter F on X is said to be convergent if there exists a point x0 ∈ X such that all neighborhoods of x0 belong to F. One then says that x0 is a limit of F and that F converges to x0 . Remark J.2.2. A topological space X is Hausdorff if and only if every convergent filter on X has a unique limit point in X. Indeed, suppose that X is Hausdorff and let F be a filter converging to two distinct points x, y ∈ X. Let U ∈ Fx ⊂ F and V ∈ Fy ⊂ F be such that U ∩ V = ∅. As U, V ∈ F we also have U ∩ V ∈ F and this contradicts (F-1). Conversely, suppose that X is not Hausdorff. Then there exist two distinct points x, y ∈ X such that for all U ∈ Fx and V ∈ Fy one has U ∩ V = ∅. It follows that the set Ω = Fx ∪ Fy ⊂ P(X) has the finite intersection property. Thus, by Proposition J.1.3, there exists a filter F containing both Fx and Fy . It follows that F converges to both x and y. Theorem J.2.3. Let X be a topological space. Then the following conditions are equivalent: (a) X is compact; (b) every ultrafilter on X is convergent. Proof. Suppose (a) and let ω be an ultrafilter on X. Let A1 , A2 , . . . , An , n elements in ω and denote by A1 , A2 , . . . , An their closures. As n≥ 1, be  n A ⊃ i=1 i i=1 Ai = ∅ (by (F-1) and (F-5)), we have that the family (A)A∈ω in X has the finite intersection property andtherefore, by compactness of X, it has a nonempty intersection. Let x ∈ A∈ω A. As x ∈ A for all A ∈ ω, it follows that for every V ∈ Fx we have A ∩ V = ∅. Consider the set Ω = {A ∩ V : A ∈ ω, V ∈ Fx } ⊂ P(X). Observe that Ω has the finite intersection property and denote by F(Ω) the filter generated by Ω

J.2 Limits Along Filters

413

(cf. Example J.1.2(b)). We have ω ⊂ Ω ⊂ F(Ω) and, as ω is an ultrafilter, it follows from Corollary J.1.8 that ω = F(Ω). As Fx ⊂ Ω, we deduce that Fx ⊂ ω, that is, ω converges to x. Conversely, suppose (b). Let (Ci )i∈I be a family of closed subsets of X with the finite intersection property. To prove that X is compact we have to show that  Ci = ∅. (J.3) i∈I

By Proposition J.1.3 there exists a filter F such that Ci ∈ F for all i ∈ I. By Theorem J.1.6 there exists an ultrafilter ω such that F ⊂ ω. By our assumptions, there exists x ∈ X such that ω converges to x, equivalently, have Ci ∩ V = ∅ for all V ∈ Fx . Fx ⊂ ω. Let i ∈ I. by (F-1) and (F-3), we  As Ci is closed, we have x ∈ Ci . Thus x ∈ i∈I Ci , and (J.3) follows.

Definition J.2.4. Let X be a set, Y a topological space, y0 a point of Y , f : X → Y a map and F a filter on X. One says that y0 is a limit of f along F (or that f (x) converges to y0 along F), and one writes f (x) −→ y0 , x→F

if f −1 (V ) = {x ∈ X : f (x) ∈ V } belongs to F for all neighborhoods V of y0 . If such a limit point y0 is unique one writes lim f (x) = y0 .

x→F

Examples J.2.5. (a) Let X and Y be two topological spaces, f : X → Y a map, x0 ∈ X and y0 ∈ Y . One has that f (x) converges to y0 in Y for x tending to x0 if and only if f (x) −→ y0 . x→Fx0

(b) Let X be a topological space and f : N → X a map. The sequence (f (n))n∈N converges to x ∈ X, if and only if x is a limit of f along the Fr´echet filter on N. (c) Let (I, ≤) be a directed set and let F be the residual filter on I. Let Y be a topological space and (yi )i∈I a net in Y . Then (yi )i∈I converges to a point y0 ∈ Y if and only if yi −→ y0 . Note that (b) is a particular case of i→F

the present example. Corollary J.2.6. Let X be a set, Y a compact topological space, f : X → Y a map, and ω an ultrafilter on X. Then there exists y0 ∈ Y such that f (x) −→ y0 . Moreover, if X is Hausdorff such an y0 is unique. x→F

Proof. The set f (ω) = {f (A) : A ∈ ω} ⊂ P(Y ) has the finite intersection property since f (A)∩f (B) ⊃ f (A∩B) = ∅ for all A, B ∈ ω. Let F denote the filter generated by f (ω). By Theorem J.1.6, there exists an ultrafilter ω  on Y which contains F. As Y is compact, by Theorem J.2.3 there exists y0 ∈ Y such that ω  converges to y0 . Let us show that if V is a neighborhood of y0 , then

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f −1 (V ) belongs to ω. Suppose the contrary. As ω is an ultrafilter, by (UF) / V } ∈ ω. Setting U = f (X \ f −1 (V )), we have X \ f −1 (V ) = {x ∈ X : f (x) ∈  we have U ∈ f (ω) ⊂ F ⊂ ω . But V ∈ Fy0 ⊂ ω  and, by construction, U ∩ V = ∅. As U ∩ V ∈ ω  , this contradicts (F-1). This shows that y0 is a limit of f along ω. If X is Hausdorff, then uniqueness of y0 follows from Remark J.2.2.

Proposition J.2.7. Let X be a set and let F be a filter on X. Let Y , Z be two topological spaces and f : X → Y and g : X → Z be two maps. Equip Y × Z with the product topology and let F : X → Y × Z be the map defined by F (x) = (f (x), g(x)) for all x ∈ X. Suppose that there exist y0 ∈ Y and z0 ∈ Z such that f (x) −→ y0 and g(x) −→ z0 . Then F (x) −→ (y0 , z0 ). x→F

x→F

x→F

Proof. Let W ⊂ Y × Z be a neighborhood of the point (y0 , z0 ). By definition of the product topology, there exist neighborhoods U ⊂ Y of y0 and V ⊂ Z of z0 such that U × V ⊂ W . As f (x) −→ y0 , we have f −1 (U ) ∈ F. Similarly, as x→F

g(x) −→ z0 , we have g −1 (V ) ∈ F. It follows that f −1 (U ) ∩ g −1 (V ) belongs x→F

to F and as F −1 (W ) ⊃ F −1 (U × V ) = f −1 (U ) ∩ g −1 (V ), we deduce from

(F-2) that F −1 (W ) ∈ F. This shows that F (x) −→ (y0 , z0 ). x→F

Proposition J.2.8. Let X be a set and let F be a filter on X. Let Y , Z be two topological spaces and f : X → Y and g : Y → Z be two maps. Suppose that there exists y0 ∈ Y such that f (x) −→ y0 and that g is continuous at y0 . Then (g ◦ f )(x) −→ g(y0 ).

x→F

x→F

Proof. Let W ⊂ Z be a neighborhood of g(y0 ). By continuity of g there exists a neighborhood V ⊂ Y of y0 such that g −1 (W ) ⊃ V . By definition of limit, we have f −1 (V ) ∈ F. As (g ◦ f )−1 (W ) = f −1 (g −1 (W )) ⊃ f −1 (V ), it follows

from (F-2) that (g ◦ f )−1 (W ) ∈ F. This shows that (g ◦ f )(x) −→ g(y0 ). x→F

Proposition J.2.9. Let X be a set and let F be a filter on X. Let f1 , f2 : X → R be two maps such that f1 ≤ f2 (i.e. f1 (x) ≤ f2 (x) for all x ∈ X). Suppose that there exists y1 ∈ R (resp. y2 ∈ R) such that y1 = limx→F f1 (x) (resp. y2 = limx→F f2 (x)). Then y1 ≤ y2 . 2 and let V1 = Proof. Suppose, by contradiction, that y1 > y2 . Set r = y1 −y 3 (y1 −r, y1 +r) and V2 = (y2 −r, y2 +r). Note that V1 ∩V2 = ∅. By definition of limit we have f1−1 (V1 ) ∈ F and f2−1 (V2 ) ∈ F. As F is a filter we deduce that f1−1 (V1 ) ∩ f2−1 (V2 ) ∈ F. On the other hand we have f1−1 (V1 ) ∩ f2−1 (V2 ) = ∅

since f1 ≤ f2 . This contradicts (F-2). It follows that y1 ≤ y2 .

Recall that given a set X, we denote by ∞ (X) the Banach space consisting of all bounded real maps f : X → R equipped with the norm f ∞ = sup{|f (x)| : x ∈ X}. Moreover, a linear map m : ∞ (X) → R satisfying m(1) = 1 and m(x) ≥ 0 for all x ∈ ∞ (E) such that x ≥ 0, is called a mean on X (cf. Definition 4.1.4).

Notes

415

Corollary J.2.10. Let X be a set and let ω be an ultrafilter on X. For every f ∈ ∞ (X) there exists a unique y0 ∈ R such that f (x) −→ y0 . Moreover, x→ω

the map mω : ∞ (X) → R defined by mω (f ) = limx→ω f (x) is a mean on X, in particular mω is continuous and mω  = 1. Proof. Let f ∈ ∞ (X). Using the fact that f (x) ∈ Y = [−f ∞ , f ∞ ] for all x ∈ X and that Y is compact Hausdorff, we deduce from Corollary J.2.6 that there exists a unique y0 ∈ R such that f (x) −→ y0 . x→ω Let us show that the map mω is linear. Let a ∈ R. Consider the continuous map g : R → R defined by g(y) = ay for all y ∈ R. Then af = g ◦ f and from Proposition J.2.8 we deduce mω (af ) = lim (af )(x) = a lim f (x) = amω (f ). x→ω

i→ω

Let now f, g ∈ ∞ (X). Consider the map F : X → R2 defined by setting F (x) = (f (x), g(x)) for all x ∈ X and the continuous map G : R2 → R defined by G(y1 , y2 ) = y1 + y2 . Then f + g = G ◦ F and from Proposition J.2.7 and Proposition J.2.8 we deduce     mω (f + g) = lim (f + g)(x) = lim f (x) + lim g(x) = mω (f ) + mω (g). x→ω

x→ω

x→ω

This shows that mω is linear. Let now f (x) = 1 for all x ∈ X. For every neighborhood V of 1 ∈ R we have f −1 (V ) = {x ∈ X : f (x) ∈ V } = X ∈ ω, by (F-6). This shows that mω (1) = mω (f ) = limx→ω f (x) = 1. Finally, it follows immediately from Proposition J.2.9 that if f ≥ 0 then mω (f ) = limx→ω f (x) ≥ 0. We have shown that mω is a mean. The last properties of mω follow from Proposition 4.1.7.



Notes The definition of filter is due to H. Cartan (1937). The full treatment of convergence along filters is given in Bourbaki [Bou] as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. L. Smith. Given a Hausdorff topological space X the set βX of all ultrafilters on X can be given the structure of a compact Hausdorff space, called the Stoneˇ Cech compactification of X. This is the largest compact Hausdorff space “generated” by X, in the sense that any map from X to a compact Hausdorff space factors through βX in a unique way. The elements x of X correspond to the principal ultrafilters ω({x}) on X. Let now X be any set. With every ultrafilter ω on X one associates the {0, 1}-valued finitely additive probability measure μω on X defined

416

J Ultrafilters

by μω (A) = 1 if A ∈ ω and μω (A) = 0 if A ∈ P(X) \ ω. Conversely, given a {0, 1}-valued finitely additive probability measure μ on X, the set ωμ = {A ∈ P(X) : μ(A) = 1} is an ultrafilter on X. This establishes a oneto-one correspondence between the ultrafilters ω on X and the {0, 1}-valued finitely additive probability measures on X. In Sect. 4.1 we considered the set MP(X) (resp. M(X)) of all finitely additive probability measures (resp. of all means) on X and we showed that there exists a natural bijective map Φ : MP(X) → M(X). Then one has Φ−1 (μω ) = mω for all ultrafilters ω on X.

Open Problems

In the list below we collect some open problems related to the topics treated in this book. (OP-1) Let G be an amenable periodic group which is not locally finite. Does there exist a finite set A and a cellular automaton τ : AG → AG which is surjective but not injective? (OP-2) Let G be a periodic group which is not locally finite and let A be an infinite set. Does there exist a bijective cellular automaton τ : AG → AG which is not invertible? (OP-3) Let G be a periodic group which is not locally finite and let V be an infinite-dimensional vector space over a field K. Does there exist a bijective linear cellular automaton τ : V G → V G which is not invertible? (OP-4) Is every Gromov-hyperbolic group residually finite (resp. residually amenable, resp. sofic, resp. surjunctive)? (OP-5) (Gottschalk’s conjecture) Is every group surjunctive? (OP-6) Let G be a periodic group which is not locally finite and let A be an infinite set. Does there exist a cellular automaton τ : AG → AG whose image τ (AG ) is not closed in AG with respect to the prodiscrete topology? (OP-7) Let G be a periodic group which is not locally finite and let V be an infinite-dimensional vector space over a field K. Does there exist a linear cellular automaton τ : V G → V G whose image τ (V G ) is not closed in V G with respect to the prodiscrete topology? (OP-8) Let G and H be two quasi-isometric groups. Suppose that G is surjunctive. Is it true that H is surjunctive? (OP-9) Let G be a non-amenable group. Does there exist a finite set A and a cellular automaton τ : AG → AG which is pre-injective but not surjective? (OP-10) Does there exist a non-sofic group? (OP-11) Does there exist a surjunctive group which is non-sofic? T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1, © Springer-Verlag Berlin Heidelberg 2010

417

418

Open Problems

(OP-12) Let G and H be two quasi-isometric groups. Suppose that G is sofic. Is it true that H is sofic? (OP-13) Let G be a non-amenable group and let K be a field. Does there exist a finite-dimensional K-vector space V and a linear cellular automaton τ : V G → V G which is pre-injective but not surjective? (OP-14) Let G be a non-amenable group and let K be a field. Does there exist a finite-dimensional K-vector space V and a linear cellular automaton τ : V G → V G which is surjective but not pre-injective? (OP-15) (Kaplanski’s stable finiteness conjecture) Is the group algebra K[G] stably finite for any group G and any field K? Equivalently, is every group L-surjunctive, that is, is it true that, for any group G, any field K, and any finite-dimensional K-vector space V , every injective linear cellular a automaton τ : V G → V G is surjective? (OP-16) (Kaplanski’s zero-divisors conjecture) Is it true that the group algebra K[G] has no zero-divisors for any torsion-free group G and any field K? Equivalently, is it true that, for any torsion-free group G and any field K, every non-identically-zero linear cellular automaton τ : KG → KG is pre-injective? (OP-17) Is every unique-product group orderable?

Comments (OP-1) The answer to this question is affirmative if G is non-periodic, i.e., it contains an element of infinite order (see Exercise 3.23), or if G is nonamenable (Theorem 5.12.1). On the other hand, if G is a locally finite group and A is a finite set, then every surjective cellular automaton τ : AG → AG is injective (see Exercise 3.21). An example of an amenable periodic group which is not locally finite is provided by the Grigorchuck group described in Sect. 6.9. (OP-2) The answer is affirmative if G is not periodic (cf. [CeC11, Corollary 1.2]). On the other hand, if G is locally finite and A is an arbitrary set, then every bijective cellular automaton τ : AG → AG is invertible (cf. Exercise 3.20 or [CeC11, Proposition 4.1]). (OP-3) The answer is affirmative if G is not periodic (cf. [CeC11, Theorem 1.1]). On the other hand, if G is locally finite and V is an arbitrary vector space, then every bijective linear cellular automaton τ : V G → V G is invertible (cf. [CeC11, Proposition 4.1]). (OP-5) Every sofic group is surjunctive (cf. Theorem 7.8.1). (OP-6) When A is a finite set and G is an arbitrary group, it follows from Lemma 3.3.2 that the image of every cellular automaton τ : AG → AG is closed in AG . When A is an infinite set and G is a non-periodic group, it is shown in [CeC11, Corollary 1.4] that there exists a cellular automaton τ : AG → AG whose image is not closed in AG . On the other hand, when G

Comments

419

is locally finite, then, for any set A, the image of every cellular automaton τ : AG → AG is closed in AG (cf. Exercise 3.22 or [CeC11, Proposition 4.1]). (OP-6) When V is a finite-dimensional vector space over a field K and G is an arbitrary group, it follows from Theorem 8.8.1 that the image of every linear cellular automaton τ : V G → V G is closed in V G . When V is an infinite-dimensional vector space and G is a non-periodic group, it is shown in [CeC11, Theorem 1.3] that there exists a linear cellular automaton τ : V G → V G whose image is not closed in V G . On the other hand, when G is locally finite, then, for any vector space V , the image of every linear cellular automaton τ : V G → V G is closed in V G (cf. [CeC11, Proposition 4.1]). (OP-9) The answer is affirmative if G contains a nonabelian free subgroup (cf. Proposition 5.11.1). (OP-13) The answer is affirmative if G contains a nonabelian free subgroup (cf. Corollary 8.10.2). (OP-14) The answer is affirmative if G contains a nonabelian free subgroup (cf. Corollary 8.11.2). (OP-15) See the discussion in the notes at the end of Chap. 8. (OP-16) See the discussion in the notes at the end of Chap. 8.

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List of Symbols

Symbol   ∼ T p→p 0R , 0 1G 1R , 1 γSG , γS ΔG S , ΔS (2)

ΔS

∂E (Ω) ιS (G) λG S , λS λ(e) λ(π) π−

Definition Page the empty word 367 the dominance relation in the set of growth functions γ : N → [0, +∞) 162 the equivalence relation in the set of growth functions γ : N → [0, +∞) 162 the p -norm of a linear map 194 T : p (E) → p (E) the zero element of the ring R 291 the identity element of the group G 2 the unity element of the ring R 291 the growth function of the group G relative to the finite symmetric generating subset S⊂G 160 the discrete laplacian on the group G associated with the subset S ⊂ G 9 the restriction of ΔS to the Hilbert space 201 2 (G) the E-boundary of the subset Ω ⊂ G 116 the isoperimetric constant of the group G with respect to the finite symmetric generating subset S ⊂ G 191 the growth rate of the group G with respect to the finite symmetric generating subset S⊂G 169 the label of the edge e ∈ E in a labeled graph G = (Q, E) 153 the label of the path π in a labeled graph G = (Q, E) 154, 223 the initial vertex of the path π in an S-labeled graph 154

T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1, © Springer-Verlag Berlin Heidelberg 2010

429

430

Symbol π+ σ(T ) ψq,r Ω −E Ω +E A∗ AG B(q, n) BSG (g, n), BS (g, n) BSG (n), BS (n) CA(G; A) CA(G, H; A)

(C i (G))i≥0 CS (G)

dF dG S , dS dQ D(G) (Di (G))i≥0 entF (X) F (X) Fn Fix(α) Fix(H) G = X; R

List of Symbols

Definition the terminal vertex of the path π in an S-labeled graph the real spectrum of T ∈ L(X) the S-labeled graph isomorphism from BS (r) onto B(q, r) such that ψq,r (1G ) = q the E-interior of the subset Ω ⊂ G the E-closure of the subset Ω ⊂ G the monoid consisting of all words on the alphabet A the set of all configurations x : G → A the ball of radius n in an S-labeled graph Q = (Q, E) centered at the vertex q ∈ Q the ball of radius n in G centered at the element g ∈ G with respect to the word metric the ball of radius n in G centered at the identity element 1G ∈ G with respect to the word metric the monoid consisting of all cellular automata τ : AG → AG the submonoid of CA(G; A) consisting of all cellular automata τ : AG → AG admitting a memory set S such that S ⊂ H the lower central series of the group G the Cayley graph of the group G with respect to the finite symmetric generating subset S ⊂ G the normalized Hamming distance on Sym(F ) the word metric on G with respect to the finite symmetric generating subset S ⊂ G the graph metric in the edge-symmetric S-labeled graph Q the derived subgroup of the group G the derived series of the group G the entropy of the subset X ⊂ AG with respect to the right Følner net F the free group based on the set X the free group of rank n the set of fixed points of the permutation α the set of configurations x ∈ AG fixed by H the presentation of the group G given by the generating subset X and the set of relators R

Page 154 406 265 115 115 367 2 265

153

153 13

16 93

156 251 152 155 92 92 125 371 372 251 4

375

List of Symbols

Symbol gx G = (X, Y, E) HR ICA(G; A) IdX (w) p (E) ∞ (E) G S (g), S (g) LCA(G; V ) LCA(G, H; V )

L(X) Ln (X) L(X) (p) MS

Matd (R) mdimF (X) M(E) PM(E) N (Γ ) NL (B) ⊂ X NL (y) ⊂ X

431

Definition the configuration defined by gx(h) = x(g −1 h) the bipartite graph with left (resp. right) vertex set X (resp. Y ) and set of edges E the Heisenberg group with coefficients in the ring R the group consisting of all invertible cellular automata τ : AG → AG the identity map on the set X the length of the word w ∈ A∗ the Banach space of all p-summable functions x : E → R the Banach space of all bounded functions x: E → R the word-length of the element g ∈ G with respect to the finite symmetric generating subset S ⊂ G the algebra of all linear cellular automata τ: VG →VG the subalgebra of LCA(G; V ) consisting of all linear cellular automata τ : V G → V G admitting a memory set S such that S ⊂ H the language associated with the subshift X the set of admissible words of length n of the subshift X the space of all continous endomorphisms of the Banach space X the p -Markov operator associated with the finite subset S ⊂ G the ring consisting of all d × d matrices with entries in the ring R the mean dimension of the vector subspace X ⊂ V G with respect to the right Følner net F the set of all means on the set E the set of all finitely additive probability measures on the set E the space of all normal subgroups of the group Γ or, equivalently, the space of all Γ -marked groups the left-neighborhood of the subset B ⊂ Y in the bipartite graph (X, Y, E) the left-neighborhood of the vertex y ∈ Y in the bipartite graph (X, Y, E)

Page 2 391 94 24 2 35 193 78

152 287

289 35 35 406 195 305

308 79 79

61 392 391

432

Symbol NR (A) ⊂ Y NR (x) ⊂ Y P(E) per(B) per(G) per(X) pern (X) Q(r)

Q = (Q, E) R[G] Rop Sym(X) Sym0 (X) Sym+ 0 (X) Symn Sym+ n Sym(X, ) U (R) V [G] x|Ω X∗ X(A) Xf XP XG Z(G)

List of Symbols

Definition Page the right-neighborhood of the subset A ⊂ X in the bipartite graph (X, Y, E) 392 the right-neighborhood of the vertex x ∈ X in the bipartite graph (X, Y, E) 391 the set of all subsets of the set E 77 the period of the matrix B 227 the period of the labeled graph G 227 the period of the irreducible sofic subshift X 227 the number of nZ-periodic configurations in the subshift X 227 the set of all vertices of the S-labeled graph Q = (Q, E) for which there exists an S-labeled graph isomorphism ψq,r : BS (r) → B(q, r) satisfying ψq,r (1G ) = q 265 the S-labeled graph with vertex set Q and edge set E ⊂ Q × S × Q 153 the group ring of the group G with coefficients in the ring R 292 the opposite ring of the ring R 293 the symmetric group of the set X 359 the subgroup of Sym(X) consisting of all permutations with finite support 360 the alternating group on X 364 the symmetric group of degree n 366 the alternating group of degree n 366 the subgroup of Sym(X) that preserve the partial order  of the set X 179, 333 the multiplicative group consisting of all invertible elements in the ring R, 292 the vector subspace of V G consisting of all configurations x : V → G with finite support 288 the restriction of the configuration x ∈ AG to the subset Ω ⊂ G 3 the topological dual of the real normed space X 384 the subshift of finite type defined by the set of admissible patterns A 32 the set of all configurations in the subshift X whose G-orbit is finite 71 the subshift defined by the set of forbidden patterns P 32 the subshift defined by the labeled graph G 223 the center of the group G 94

Index

Δ-irreducible subshift, 34 action continuous —, 3 equivariantly approximable —, 279 expansive —, 65 faithful —, 50 topologically mixing —, 29 topologically transitive —, 32 uniformly continuous —, 64 additive cellular automaton, 335 adjacency matrix of a labeled graph, 227 admissible pattern, 32 admissible word, 35 affine — group, 93 — map, 387 algebra, 286 — homomorphism, 289 — isomorphism, 290 almost — -homomorphism, 234, 254 — equal configurations, 112 — perfect group, 55 — periodic configuration, 72 alphabet, 2 alternating group, 364 — of rank n, 366 amenable — group, 87 elementary — group, 215, 338 Artinian module, 69 automaton additive cellular —, 335 cellular —, 6 linear cellular —, 284 automorphism group, 45

back-tracking, 156 Baire theorem, 403 Banach-Alaoglu theorem, 385 base — of a uniform structure, 353 free —, 368 based free group, 367 bi-invariant metric, 251 bi-orderable group, 341 bipartite — graph, 391 — subgraph, 391 finite — graph, 392 locally finite — graph, 392 Boolean ring, 340 boundary, 116 Burnside problem, 214 Cantor-Bernstein theorem, 398 Cayley graph, 156 cellular automaton, 6 additive —, 335 induced —, 17 invertible —, 24 linear —, 284, 335 reversible —, 24 characteristic map, 79 closed path, 155 closure, 115 cluster point of a net, 345 color, 2 commensurable groups, 171 commutative-transitive group, 279 commutator — of two group elements, 92 — subgroup, 92 simple —, 176

T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1, © Springer-Verlag Berlin Heidelberg 2010

433

434 compact topological space, 347 completion proamenable —, 108 pronilpotent —, 108 prosolvable —, 108 complexity of a paradoxical decomposition, 106 composition of paths, 154 concatenation, 367 configuration, 2 H-periodic —, 3 almost periodic —, 72 Garden of Eden —, 111 language of a —, 73 Toeplitz —, 74 conjugate elements, 362 connected labeled graph, 155 Connes embedding conjecture, 277 context-free subshift, 225 convergent net, 344 convex subset, 383 convolution product, 291 convolutional encoders, 335 Curtis-Hedlund theorem, 20 cycle, 361 Day’s problem, 105 Dedekind finite ring, 336 degree — of a graph, 155 — of a symmetric group, 366 — of a vertex, 155 — of an alternating group, 366 derived — series, 92 — subgroup, 92 directed set, 343 directly finite ring, 327 discrete uniform structure, 352 divisible group, 38 dominance of growth functions, 162 edge — -symmetric labeled graph, 155 — of a bipartite graph, 391 inverse —, 155 of a labeled graph, 153 elementary — amenable group, 215, 338 — reduction, 370 empty — path, 154 — word, 367

Index entourage, 352 entropy topological —, 142 equipotent sets, 372 equivalence of growth functions, 162 equivariant map, 5 equivariantly approximable action, 279 even subshift, 35 expansive action, 65 expansivity constant, 65 entourage, 65 exponential growth, 164 Følner — conditions, 96 — theorem, 99 left — net, 96 left — sequence, 96 right — net, 96 right — sequence, 96 faithful action, 50 Fibonacci sequence, 219 field, 284 filter, 409 — generated, 409 convergent —, 412 Fr´ echet —, 409 limit of a —, 412 principal —, 409 residual —, 409 ultra—, 410 finite intersection property, 347 finitely — additive probability measure, 77 — generated group, 152, 375 — presented group, 376 bi-invariant — additive probability measure, 85 left-invariant — additive probability measure, 85 right-invariant — additive probability measure, 85 forbidden — pattern, 32 — word, 35 Fr´ echet filter, 409 free — base, 368 — base subset, 368 — group, 368 — group of rank k, 372 — ultrafilter, 410 based — group, 367

Index rank of a — group, 373 fully residually free group, 279 Garden of Eden — configuration, 111 — pattern, 112 — theorem, 114, 128 — theorem for linear cellular automata, 312 generating subset, 151 generator of a presentation, 375 golden mean subshift, 35 graph — metric, 155 bipartite —, 391 Cayley —, 156 degree of a regular labeled —, 155 finite labeled —, 154 labeled —, 153 loop in a labeled —, 154 regular labeled —, 155 tree, 156 Grigorchuk group, 179 abelianization of the —, 222 group — algebra, 294 — of p-adic integers, 41 — of intermediate growth, 190 — ring, 292 affine —, 93 almost perfect —, 55 alternating —, 364 alternating — of rank n, 366 amenable —, 87 automorphism —, 45 bi-orderable —, 341 Cayley graph of a —, 156 commutative-transitive —, 279 divisible —, 38 elementary amenable —, 215, 338 finitely generated —, 152, 375 finitely presented —, 376 free —, 368 free — of rank k, 372 fully residually free —, 279 Grigorchuk —, 179 Heisenberg —, 94 Hopfian —, 44 hyperlinear —, 277 Kaloujnine —, 221 Klein bottle —, 341 L-surjunctive —, 324 lamplighter —, 108 LEA —, 247

435 LEF —, 247 linear —, 51 locally P —, 58 locally indicable —, 337 marked —, 61 metabelian —, 92 nilpotent —, 93 orderable —, 331 periodic —, 29, 105 polycyclic —, 106, 108 presentation of a —, 375 profinite —, 41 residually C —, 238 residually P —, 62, 131 residually amenable —, 132 residually finite —, 37 simple —, 44 sofic —, 254 solvable —, 93 surjunctive —, 57 symmetric —, 359 symmetric — of rank n, 366 unique-product —, 331 virtually P —, 41 growth — function, 160, 162 — rate, 169 — type of a group, 163 equivalence class of — functions, 163 equivalence of — functions, 162 exponential —, 164 intermediate —, 190 polynomial —, 164 subexponential —, 164 Hall — k-harem conditions, 399 — condition, 394 — harem theorem, 399 — marriage theorem, 399 Hamming metric, 252 Hausdorff metric, 357 Hausdorff-Bourbaki — topology, 356 uniform structure, 356 Heisenberg group, 94 homomorphism almost- —, 234, 254 labeled graph —, 154 Hopfian — group, 44 — module, 339 hyperlinear group, 277

436 ICC-property, 338 idempotent, 331 proper —, 331 induced — cellular automaton, 17 — labeled subgraph, 154 inductive — limit, 379 — system of groups, 379 initial — topology, 346 — uniform structure, 355 interior, 115 intermediate growth, 190 inverse — edge, 155 — path, 155 invertible cellular automaton, 24 irreducible — matrix, 227 — subshift, 32 isomorphism labeled graph —, 154 isoperimetric constant, 191 Kaloujnine group, 221 abelianization of the —, 222 Kesten-Day theorem, 201 Klein bottle group, 341 Klein Ping-Pong theorem, 376 L-surjunctive group, 324 labeled graph, 153 — homomorphism, 154 — isomorphism, 154 adjacency matrix of a —, 227 connected —, 155 edge-symmetric —, 155 finite —, 154 locally finite —, 155 path in a —, 154 subgraph, 154 subshift defined by a —, 223 labelling map, 153 lamplighter group, 108 language — of a configuration, 73 — of a subshift over Z, 35 Laplacian, 10 lattice, 95 LEA-group, 247 LEF-group, 247 left-invariant

Index — finitely additive probability measure, 85 — metric, 251 length — of a cycle, 361 — of a word, 35, 152 letter, 2 limit — along an ultrafilter, 413 — of a filter, 412 — point of a net, 344 inductive —, 379 projective —, 380 linear — cellular automaton, 284, 335 — group, 51 Lipschitz-equivalence, 162 local defining map, 6 locally — P group, 58 — convex topological vector space, 383 — embeddable, 235 — finite bipartite graph, 392 — finite labeled graph, 155 — indicable group, 337 loop, 154 lower central series, 93 majority action, 10 marked group, 61 Markov operator, 195 Markov-Kakutani theorem, 387 matching, 393 left-perfect —, 393 perfect —, 393 right-perfect—, 393 mean, 78 — dimension, 308 bi-invariant —, 86 left-invariant —, 86 right-invariant —, 86 memory set, 6 minimal —, 15 metabelian group, 92 metric bi-invariant —, 251 graph —, 155 Hamming —, 252 word —, 153 metrizable uniform structure, 352 Milnor problem, 215 minimal — memory set, 15 — set, 72

Index — subshift, 72 module Artinian —, 69 Hopfian —, 339 Noetherian —, 339 projective —, 339 monoid, 13 Moore neighborhood, 166 Morse subshift, 74 entropy of the —, 144 neighbor, 155 net, 343 nilpotency degree, 93 nilpotent group, 93 Noetherian — module, 339 — ring, 340 non-principal ultrafilter, 410 normal closure, 375 Open mapping theorem, 404 operator norm, 384 opposite ring, 293 orderable group, 331 Ore ring, 340 paradoxical decomposition left —, 98 right —, 98 partially ordered set, 343 path — in a labeled graph, 154 closed —, 155 closed simple —, 156 composition, 154 empty —, 154 inverse —, 155 label of a —, 154 proper —, 156 simple —, 156 pattern, 2 admissible —, 32 forbidden —, 32 periodic group, 29, 105 permutation, 359 support of a —, 360 polycyclic group, 106, 108 polynomial, 164 pre-injective map, 112 presentation — of a group, 375 generator of a —, 375 relator of a —, 375

437 principal — filter, 409 — ultrafilter, 410 proamenable completion, 108 prodiscrete — topology, 3, 346 — uniform structure, 22, 355 product — topology, 346 — uniform structure, 355 profinite — completion, 55 — group, 41 — kernel, 39 — topology, 53 projective — limit, 380 — module, 339 — system of groups, 380 pronilpotent completion, 108 proper — idempotent, 331 — path, 156 prosolvable completion, 108 quasi-isometric — embedding, 204 — groups, 206 quasi-isometry, 204 rank of a free group, 373 reduced — form, 374 — product, 244 — word, 373 regular labeled graph, 155 relator of a presentation, 375 residual — filter, 409 — set, 409 — subgroup, 39 residually — C group, 238 — P group, 62, 131 — amenable group, 132 — finite group, 37 restriction, 17 reversible cellular automaton, 24 right-invariant — finitely additive probability measure, 85 — metric, 251 ring Boolean —, 340 Dedekind finite —, 336

438 directly finite —, 327 group —, 292 Noetherian —, 340 opposite —, 293 Ore —, 340 stably finite —, 328 unit-regular—, 340 von Neumann finite —, 336 set directed —, 343 partially ordered —, 343 shift, 2 simple — group, 44 — path, 156 closed — path, 156 sofic — subshift, 225 — group, 254 solvable group, 93 spectrum real —, 406 stably finite ring, 328 state, 2 strong topology, 82, 384 strongly irreducible subshift, 34 subalgebra, 287 subexponential growth, 164 subgraph induced labeled —, 154 labeled —, 154 submonoid, 17 subnet, 344 subsemigroup, 322 subshift, 31 N -power —, 228 N th higher block —, 227 Δ-irreducible —, 34 — defined by a labeled graph, 223 — of finite type, 32 context-free —, 225 even —, 35 golden mean —, 35 irreducible —, 32 language of a — over Z, 35 minimal —, 72 Morse —, 74, 144 sofic —, 225 strongly irreducible —, 34 surjunctive —, 71 Toeplitz —, 74 topologically mixing —, 33 subword, 35

Index support — of a configuration, 288 — of a pattern, 2 — of a permutation, 360 surjunctive — group, 57 —subshift, 71 symbol, 2 symmetric — group, 359 — subset, 152 syndetic subset, 72 Tarski — alternative, 99 — number of a group, 106 Tarski-Følner theorem, 99 theorem Baire —, 403 Banach-Alaoglu —, 385 Cantor-Bernstein —, 398 Curtis-Hedlund —, 20 Garden of Eden —, 114, 128 Garden of Eden — for linear cellular automata, 312 Gromov-Weiss —, 272 Hall harem —, 399 Hall marriage —, 399 Kesten-Day —, 201 Klein Ping-Pong —, 376 Markov-Kakutani —, 387 open mapping —, 404 Tarski-Følner —, 99 Tychonoff —, 348 Thue-Morse sequence, 73 tiling, 122 Toeplitz — configuration, 74 — subshift, 74 topological — dual, 384 — entropy, 142 — manifold, 69 — vector space, 383 topologically mixing — action, 29 — subshift, 33 topologically transitive action, 32 topology Hausdorff-Bourbaki —, 356 initial —, 346 prodiscrete —, 3, 346 product —, 346 profinite —, 53

Index strong —, 384 weak-∗ —, 384 total ordering, 331 totally disconnected topological space, 346 transposition, 361 tree, 156 trivial uniform structure, 352 Tychonoff theorem, 348 ultrafilter, 410 free —, 410 limit along an — , 413 non-principal —, 410 principal —, 410 ultraproduct, 244 uniform — convexity, 407 — embedding, 355 — isomorphism, 355 — structure, 351 Hausdorff-Bourbaki — structure, 356 induced — structure, 353 prodiscrete — structure, 22, 355 uniformly continuous — action, 64 — map, 353 unique — -product group, 331 — rank property, 340 unit-regular ring, 340 universe, 2

439 valence of a vertex, 155 vertex — of a bipartite graph, 391 — of a labeled graph, 153 degree of a —, 155 neighbor, 155 valence of a —, 155 virtually P group, 41 von Neumann — conjecture, 105 — finite ring, 336 — neighborhood, 165 weak-∗ topology, 384 word, 367 — length, 152 — metric, 153 admissible —, 35 empty —, 367 forbidden —, 35 length of a —, 35 reduced —, 373 subword of a —, 35 wreath product, 52 zero-divisor, 330 — conjecture, 337 left —, 330 right —, 330