Cellular Automata and Groups [2 ed.] 9783031433276, 9783031433283

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Springer Monographs in Mathematics

Tullio Ceccherini-Silberstein Michel Coornaert

Cellular Automata and Groups Second Edition

Springer Monographs in Mathematics Editors-in-Chief Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea International Centre for Mathematical Sciences, Edinburgh, UK Katrin Wendland, School of Mathematics, Trinity College Dublin, Dublin, Ireland Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NJ, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan Isabelle Gallagher, Département de Mathématiques et Applications, Ecole Normale Supérieure, Paris, France Sinan Güntürk, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France Alberto A. Pinto, Department of Mathematics, University of Porto, Porto, Portugal Gabriella Pinzari, Department of Mathematics, University of Padova, Padova, Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institute for Mathematical Stochastics, Technical University Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK Shmuel Weinberger, Department of Mathematics, University of Chicago, Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

Tullio Ceccherini-Silberstein • Michel Coornaert

Cellular Automata and Groups Second Edition

Tullio Ceccherini-Silberstein Dipartimento di Ingegneria Università degli Studi del Sannio Benevento, Italy

Michel Coornaert Institut de Recherche Mathématique Avancée Université de Strasbourg Strasbourg, France

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-031-43327-6 ISBN 978-3-031-43328-3 (eBook) https://doi.org/10.1007/978-3-031-43328-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2010, 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

To Katiuscia, Giacomo, and Tommaso To Martine and Nathalie

Preface to the Second Edition

This second edition of our book gave us the opportunity to correct some errors, update historical notes, and add a few sections and many new exercises. The interested reader may find detailed solutions to all the exercises contained in this new edition, as well as related historical notes and additional material, in the companion book Exercises in Cellular Automata and Groups published by Springer in 2023. We are very grateful to all the friends and colleagues who sent us comments on the first edition. In particular, we would like to thank Keino Hasawni Brown, Philip Dowerk, Yonatan Gutman, Xuan Kien Phung, Paul Schupp, and Zoran Šuni´c. We wish also to thank the staff of Springer Verlag, in particular Elena Griniari, Senior Editor, Mathematics, for inviting us to write this new edition and for her unwavering support, encouragement, and guidance throughout its development, and Francesca Ferrari, Assistant Editor, for her invaluable assistance with our manuscript finalization and related matters. Rome, Italy Strasbourg, France

Tullio Ceccherini-Silberstein Michel Coornaert

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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 20 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 non-amenable. It follows that a group which contains a subgroup isomorphic to .F2 is non-amenable. However, there are examples of groups which are non-amenable 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 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 ix

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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. .

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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 Machì, 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 spatial 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. Machì, 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 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

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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 Gábor Elek and Endre Szabó). 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 Szabó using different methods. Moreover, given a group G and a field K, the pre-injectivity 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

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cellular automaton with one-dimensional alphabet over a torsion-free group always pre-injective? .

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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. 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-Lyndon 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 nonexistence 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 maximal 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

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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 .l -spectrum of the associated Laplacian. Finally, we consider the notion of quasiisometry 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 GromovWeiss 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 Lsurjunctive. 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 André 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 Markov-Kakutani fixed point theorem is presented in Appendix G. In the subsequent appendix, of a pure graph-theoretical and combinatorial flavor, 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 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, Italy Strasbourg, France

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 2 3 4 6 14 15 17 20 22 25 28 30

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 64 66 67 69

Surjunctive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stability Properties of Surjunctive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Surjunctivity of Locally Residually Finite Groups . . . . . . . . . . . . . . . . . . . 3.4 Marked Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Expansive Actions on Uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 90 91 93 96

3

71 74 75 76

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3.6 Gromov’s Injectivity Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.7 Closedness of Marked Surjunctive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 112 117 117 120 122 127 129 133 134 138 140 142

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 152 154 155 162 163 165 170 173 175 176

Finitely Generated Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Word Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Labeled Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Cayley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Growth Functions and Growth Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Growth Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Growth of Subgroups and Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 A Finitely Generated Metabelian Group with Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 198 200 202 206 214 216

6

177 178 181 182

219

Contents

7

8

xvii

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 Uniform Embeddings and Quasi-Isometries . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Quasi-Isometric Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221 224 237 239 240 251 260 264 268

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 312 321 325 329 332

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 . . . . . . . . . . . . . . . . . . . . . . 8.17 A Characterization of Amenability in Terms of Linear Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

369 370 374 376 377 381 386 388 393 396 402 403 405 409 412 415

339 344 351 355 358

418 423 431 435

xviii

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

453 453 454 455 456 457 459

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

461 461 463 465 466 468

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

469 469 470 472 473

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

477 477 478 483 485 486

E Inductive Limits and Projective Limits of Groups . . . . . . . . . . . . . . . . . . . . . . . 489 E.1 Inductive Limits of Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 E.2 Projective Limits of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 F The Banach-Alaoglu Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2 The Weak-∗ Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.3 The Banach-Alaoglu Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

493 493 494 495

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

497 497 497 499

H The Hall Harem Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.1 Bipartite Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.2 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.3 The Hall Marriage Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.4 The Hall Harem Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

501 501 503 504 509 511

Contents

xix

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

513 513 514 516 517

J

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

519 519 522 525

Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

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.

xxi

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).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3_1

1

2

1 Cellular Automata

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 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 G by .gx ∈ A gx := x ◦ Lg −1 .

(1.2)

.

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

= (g1 g2 )x,

2

1

2

1

1.2 The Prodiscrete Topology

3

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 → A defined on some finite subset .Ω of G. The set .Ω is then called the support of p.

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 } =

n

.

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 Propositions A.4.1 and A.4.2, a product of Hausdorff (resp. totally disconnected) topological spaces is Hausdorff (resp. totally disconnected). u n 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.

4

1 Cellular Automata

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). u n

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

.

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 ) :=

n

.

{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 u n closed in .AG for each .g ∈ G. Therefore .Fix(H ) is closed in .AG by (1.4). Consider the set .H \G := {H g : g ∈ G} consisting of all right cosets of H in G and the canonical surjective map ρ : G → H \G

.

g |→ H g.

1.3 Periodic Configurations

5

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. u n 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 .O 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 = Ng 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 ).

u 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(γ x) = γ ϕ(x) for all .γ ∈ Γ and .x ∈ X.

6

1 Cellular Automata

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.

u n

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 .S ⊂ G and a map .μ : A → 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 )

.

by (1.2).

(1.6)

1.4 Cellular Automata

7

(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 S .μ : A → 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, 1.3, 1.4, and 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); • (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. 1.1 The cell c and its eight neighboring cells

8

1 Cellular Automata

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

Fig. 1.4 A live cell that has at most one live neighbor 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

1.4 Cellular Automata

9

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

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

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). Consider the map .μ : AS → A given by

μ(y) :=

.

for all .y ∈ AS .

⎧ ⎪ ⎪ ⎪ ⎪ ⎨1 ⎪ ⎪ ⎪ ⎪ ⎩

⎧Σ ⎪ ⎪ ⎨ s∈S y(s) = 3 if

or ⎪ ⎪ ⎩Σ

s∈S

0

otherwise

y(s) = 4 and y((0, 0)) = 1,

(1.8)

10

1 Cellular Automata

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 map .ΔS = G G ΔG S : K → K defined by Σ .ΔS (x)(g) := |S|x(g) − x(gs). s∈S

Then .ΔS is a cellular automaton over G with memory set .S ∪{1G } and local defining map .μ : K S∪{1G } → K given by Σ .μ(y) := |S|y(1G ) − y(s) for all y ∈ K S∪{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 ⎧ ⎪ ⎪ ⎨1 .τ (x)(g) := 0 ⎪ ⎪ ⎩x(g)

if if if

Σ Σ

s∈S

x(gs) >

Σ

s∈S

x(gs)


Σ

s∈S

y(s)
0, there exists g ∈ G and x ∈ X such that d(x, x1 ) < ε and d(gx, x2 ) < ε.

.

1.9 (Irreducible Actions) An action of a group G on a topological space X is said to be irreducible if, for every finite subset F ⊂ G and all non-empty open subsets U and V of X, there exists g ∈ G \ F such that U ∩ gV /= ∅. (a) Show that a finite group admits no irreducible action. (b) Show that every irreducible action is topologically transitive. (c) Suppose that a group G acts on a uniform space X. Show that the action of G on X is irreducible if and only if the following holds:

Exercises

33

(UI) for every finite subset F ⊂ G, for all x1 , x2 ∈ X, and for every entourage W of X, there exists g ∈ G \ F and x ∈ X such that (x, x1 ) ∈ W and (gx, x2 ) ∈ W.

.

(d) Suppose that a group G acts on a metric space (X, d). Show that the action of G on X is irreducible if and only if the following holds: (MI) for every finite subset F ⊂ G, for all x1 , x2 ∈ X, and for every ε > 0, there exists g ∈ G \ F and x ∈ X such that d(x, x1 ) < ε and d(gx, x2 ) < ε.

.

1.10 Let X be a topological space and let f : X → X be a homeomorphism. Recall that the action of Z on X generated by f is the continuous action α : Z × X → X defined by α(n, x) := f n (x) for all (n, x) ∈ Z × X. Show that the action of Z on X generated by f is irreducible if and only if f satisfies the following condition: (HI) for all non-empty open subsets U and V of X, there exists an integer k ≥ 1 such that U ∩ f k (V ) /= ∅. 1.11 (Topologically Mixing Actions) An action of a group G on a topological space X is said to be topologically mixing if, for all non-empty open subsets U and V of X, there exists a finite subset F ⊂ G such that U ∩gV /= ∅ for all g ∈ G\F . (a) Suppose that a finite group G acts on a topological space X. Show that the action of G on X is topologically mixing. (b) Suppose that an infinite group G acts on a topological space X. Show that if this action is topologically mixing then it is irreducible and topologically transitive. (c) Suppose that a group G acts on a uniform space X. Show that the action of G on X is topologically mixing if and only if the following holds: (UTM) for all x1 , x2 ∈ X and for every entourage W of X, there exists a finite subset F ⊂ G such that, for every g ∈ G \ F , there exists x ∈ X such that (x, x1 ) ∈ W and (gx, x2 ) ∈ W.

.

(d) Suppose that a group G acts on a metric space (X, d). Show that the action of G on X is topologically mixing if and only if the following holds: (MTM) for all x1 , x2 ∈ X and for every ε > 0, there exists a finite subset F ⊂ G such that for all g ∈ G \ F there exists x ∈ X such that d(x, x1 ) < ε and d(gx, x2 ) < ε.

.

34

1 Cellular Automata

1.12 Let G be a group and let A be a set. Equip AG with its prodiscrete topology. (a) Show that the shift action of G on AG is topologically mixing. (b) Suppose that the group G is infinite. Show that the shift action of G on AG is irreducible and topologically transitive. (c) Suppose that the group G is finite and that the set A has more than one element. Show that the shift action of G on AG is neither irreducible nor topologically transitive. 1.13 Let G be a group acting on two topological spaces X1 and X2 . Suppose that there exists a continuous G-equivariant surjective map f : X1 → X2 . Show that if the action of G on X1 is topologically transitive (resp. irreducible, resp. topologically mixing) then the action of G on X2 is topologically transitive (resp. irreducible, resp. topologically mixing). 1.14 (Nilpotent Cellular Automata) Let G be a group and let A be a set. One says that a cellular automaton τ : AG → AG is nilpotent if there exists an integer n ≥ 0 satisfying the following property: there exists a configuration y ∈ AG such that every configuration x ∈ AG satisfies τ n (x) = y. The smallest integer n ≥ 0 satisfying this property is then called the nilpotency class of the nilpotent cellular automaton τ . Let τ : AG → AG be a nilpotent cellular automaton with nilpotency class n. (a) Show that there exists a unique constant configuration cτ ∈ AG such that every x ∈ AG satisfies τ n (x) = cτ . (b) Show that Fix(τ ) = {cτ }. (c) Show that τ k = τ n for every integer k ≥ n. (d) Let σ : AG → AG be a cellular automaton commuting with τ . Show that τ ◦ σ is a nilpotent cellular automaton with nilpotency class ≤ n and that cτ ◦σ = cτ . (e) Let k ≥ 1 be an integer. Show that τ k is a nilpotent cellular automaton with nilpotency class ≤ n and that one has cτ k = cτ . (f) Let σ : AG → AG be a cellular automaton. Show that σ is nilpotent if and only if there is an integer k ≥ 1 such that σ k is nilpotent. 1.15 (Example of a Nilpotent Cellular Automaton of Nilpotency Class n) Let A := {0, 1} and let n be a positive integer. Consider the subset S ⊂ Z given by S := {−1, 0, 1, . . . , n − 2} and the map μ : AS → A defined by ⎧ ⎪ ⎪ ⎨0 .μ(p) := 0 ⎪ ⎪ ⎩p(0)

if p(−1) = p(0) = p(1) = · · · = p(n − 2) = 1, if p(−1) = 0 and p(0) = 1, otherwise,

for all p ∈ AS . Show that the cellular automaton τ : AZ → AZ admitting S as a memory set and μ as the associated local defining map is nilpotent of class n.

Exercises

35

1.16 Let G be a group and let A be a set. Let H be a subgroup of G and suppose that τ : AG → AG is a cellular automaton admitting a memory set contained in H . Show that τ is nilpotent of nilpotency class n if and only if the restriction cellular automaton τH : AH → AH is nilpotent of nilpotency class n. 1.17 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. 1.18 Let G be a group and let S be a finite subset of G of cardinality k. Let A and B be two finite sets with cardinality m := |A| and n := |B|. Show that there are k exactly nm cellular automata τ : AG → B G admitting S as a memory set. 1.19 Let G be a countable group and let A and B be two finite sets. Show that the set of all cellular automata τ : AG → B G is countable. 1.20 Let G := Z2 and A := {0, 1}. Let τ : AG → AG denote the cellular automaton associated with the Game of Life (see Example 1.4.3(a)). 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). 1.21 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 Map(A) 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 Sym(A) of A. 1.22 (Constant Cellular Automata) Let G be a group and let A be a set. A configuration x : G → A (resp. a cellular automaton τ : AG → AG ) is said to be constant if it is a constant map. An element u in a monoid M is called left-absorbing (resp. right-absorbing) if every v ∈ M satisfies uv = u (resp. vu = u). (a) Show that a configuration x ∈ AG is constant if and only if it is fixed by the shift action of G on AG . (b) Let τ : AG → AG be a cellular automaton. Show that the following conditions are equivalent: (i) τ is constant; a. there exists a constant configuration c ∈ AG such that every x ∈ AG satisfies τ (x) = c; (ii) the minimal memory set of τ is the empty set; (iii) τ is left-absorbing in the monoid CA(G; A). (c) Suppose that A has more than one element. Show that the monoid CA(G; A) contains no right-absorbing elements.

36

1 Cellular Automata

1.23 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. 1.24 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 denote by μ : AS → A the associated local defining map. Show that the following conditions are equivalent: (C1) S is the minimal memory set of τ ; (C2) for every s ∈ S, there exist two patterns p1 , p2 ∈ AS such that μ(p1 ) /= μ(p2 ) and p1 (t) = p2 (t) for all t ∈ S \ {s}; (C3) for every s ∈ S, there exist two configurations x1 , x2 ∈ AG such that τ (x1 )(1G ) /= τ (x2 )(1G ) and x1 (g) = x2 (g) for all g ∈ G \ {s}. 1.25 Prove that there are exactly 218 cellular automata τ : {0, 1}Z → {0, 1}Z whose minimal memory set is {−1, 0, 1}. 1.26 Let τ ∈ CA(Z2 ; {0, 1}) denote the cellular automaton associated with the Game of Life (see Example 1.4.3(a)). Show that the minimal memory set of τ is the set {−1, 0, 1}2 . 1.27 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. 1.28 Let G be a group and let A be a set. Let H be a subgroup of G. (a) Let τ ∈ CA(G, H ; A). Show that τ and τH ∈ CA(H ; A) have the same minimal memory set. (b) Let σ ∈ CA(H ; A). Show that σ and σ G ∈ CA(G, H ; A) have the same minimal memory set. 1.29 A map f : X → Y between sets X and Y is said to be countable-to-one if the preimage set f −1 (y) := {x ∈ X : f (x) = y} is countable (finite or infinite) for each y ∈ X. Let A be a set and let G be an uncountable group. Show that every countable-to-one cellular automaton τ : AG → AG is injective. 1.30 Let G be a group and let A be a set. Let F be a non-empty finite subset of G and set B := AF . Equip the sets AG and B G with their prodiscrete uniform structure and the G-shift action. Show that the map Φ : AG → B G , defined by Φ(x)(g) := (g −1 x)|F for all x ∈ AG and g ∈ G, is a G-equivariant uniform embedding. 1.31 Let G be a group and let A be a set. A configuration x ∈ AG is called almost constant if there exists an element a ∈ A such that x(g) = a for all but finitely many elements g ∈ G. (a) Show that the set of almost constant configurations is dense in AG for the prodiscrete topology.

Exercises

37

(b) Let τ : AG → AG be a cellular automaton and let x ∈ AG be an almost constant configuration. Show that the configuration τ (x) is almost constant. 1.32 Let G be a group and let H be a finite index subgroup of G. Let T ⊂ G be a complete set of representatives for the right cosets of H in G, so that G = ut∈T H t. Let A be a set and define B := AT . Equip AG (resp. B H ) with its prodiscrete uniform structure and with the G-shift (resp. H -shift) action. Show that the map Ψ : AG → B H defined by Ψ (x)(h)(t) := x(ht) for all x ∈ AG , h ∈ H , and t ∈ T is an H -equivariant uniform isomorphism. 1.33 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 and g ∈ G (cf. Example 1.4.3(e)). (a) Show that τs ∈ ICA(G; A) for every s ∈ G. (b) Show that the map Φ : G → ICA(G; A) defined by φ(s) := τs for all s ∈ G is a group homomorphism. (c) Show that if A has more than one element, then Φ is injective but not surjective. 1.34 Let G be a group and let A be a set with more than one element. Prove that the set I 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). 1.35 (Elementary Cellular Automata) Let A := {0, 1}. One says that a cellular automaton τ : AZ → AZ is an elementary cellular automaton if the set S := {−1, 0, 1} is a memory set for τ . This amounts to saying that the minimal memory set of τ is contained in S. Denote by E the set consisting of all elementary cellular automata. For n ∈ N, let In := {0, 1, . . . , n − 1} denote the set consisting of all non-negative integers that are smaller than n. (a) Show that the map ϕ : AS → I8 , defined by ϕ(p) := 4p(−1) + 2p(0) + p(1) for all p ∈ AS , is bijective. (b) For τ ∈ E , denote by μτ : AS → A the local defining map of τ associated with S. Show that the map W : E → I256 , defined by W (τ ) :=

Σ

.

μτ (p)2ϕ(p)

p∈AS

for all τ ∈ E , is bijective. (c) Compute W (τ ) for each of the following elementary cellular automata τ : AZ → AZ : (i) the constant cellular automaton given by τ (x)(n) := 0 for all x ∈ AZ and n ∈ Z; (ii) the constant cellular automaton given by τ (x)(n) := 1 for all x ∈ AZ and n ∈ Z; (iii) the identity cellular automaton given by τ (x) := x for all x ∈ AZ ;

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(iv) the right-shift given by τ (x)(n) := x(n − 1) for all x ∈ AZ and n ∈ Z; (v) the left-shift given by τ (x)(n) := x(n + 1) for all x ∈ AZ and n ∈ Z; (vi) the cellular automaton given by τ (x)(n) := min(x(n), x(n + 1)) for all x ∈ AZ and n ∈ Z; (vii) the cellular automaton given by τ (x)(n) := x(n) + x(n + 1) mod 2 for all x ∈ AZ and n ∈ Z; (viii) the cellular automaton given by τ (x)(n) := x(n − 1) + x(n + 1) mod 2 for all x ∈ AZ and n ∈ Z; (ix) the cellular automaton given by τ (x)(n) := x(n − 1) + x(n) + x(n + 1) mod 2 for all x ∈ AZ and n ∈ Z; (x) the cellular automaton given by τ (x)(n) := x(n − 1) × x(n + 1) for all x ∈ AZ and n ∈ Z; (xi) the majority rule cellular automaton given by τ (x)(n) := 0 if x(n1 ) + x(n) + x(n + 1) ≤ 1 and τ (x)(n) := 1 otherwise for all x ∈ AZ and n ∈ Z. 1.36 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 . 1.37 Let G := Z and let K be a field. Let A := K[[t]] denote the ring of all formal power series in one indeterminate t with coefficients in K. Consider the map τ : AG → AG defined by τ (x)(n) := x(n) − tx(n + 1)

.

for all x ∈ AG and n ∈ Z. Then τ is a bijective cellular automaton whose inverse map σ : AG → AG is given by ⎛ ⎞ i Σ Σ ⎝ .σ (y)(n) := yn+j,i−j ⎠ t i i∈N

j =0

Σ for all y ∈ AG , where y(n) := i∈N yn,i t i for all n ∈ G (cf. Example 1.10.3). Show that σ is discontinuous, with respect to the prodiscrete topology on AZ , at every configuration y ∈ AZ . 1.38 Let G be a group and let H ⊂ G be a subgroup. Let A and B be two sets. Suppose that τ : AG → B G is a cellular automaton over the group G admitting a memory set S ⊂ H and let τH : AH → B H denote the cellular automaton over

Exercises

39

the group H obtained by restriction of τ to H . Show that τ (AG ) is closed in B G for the prodiscrete topology on B G if and only if τH (AH ) is closed in B H for the prodiscrete topology on B H . 1.39 (Subshifts) Let G be a group and let A be a set. A subset X ⊂ AG is called a subshift of AG if X is G-invariant i.e., 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 . The subshift AG is called the full subshift. (b) Let Y ⊂ AG be a G-invariant subset and let Y denote the closure of Y in AG . Show that Y is a subshift of AG . (c) Let x ∈ AG . Show that the closure Gx of the G-orbit of x in AG is a subshift of AG . This subshift is called the orbit closure of the configuration x. n (d) Let (Xi )i∈I be a family of subshifts of AG . Show that i∈I Xi is a subshift of AG . U (e) Let (Xi )i∈I be a finite family of subshifts of AG . Show that i∈I Xi is a subshift of AG . (f) Let B be a set and let τ : AG → B G be a G-equivariant continuous map (e.g. a cellular automaton). Let Y ⊂ B G be a subshift. Show that τ −1 (Y ) is a subshift of AG . (g) Suppose now that A is finite and let B be a set. Let X be a subshift of AG and let τ : AG → B G be a cellular automaton. Show that τ (X) is a subshift of B G . 1.40 (Cellular Automata Between Subshifts) Let G be a group. Let A and B be sets. Let X ⊂ AG and Y ⊂ B G be two subshifts. One says that a map σ : X → Y is a cellular automaton if there exists a cellular automaton τ : AG → B G such that σ (x) = τ (x) for all x ∈ X. Let σ : X → Y be a map. Show that the following conditions are equivalent: (CA1) σ is a cellular automaton; (CA2) σ is G-equivariant (with respect to the actions of G on X and Y induced by the shift actions of G on AG and B G ) and uniformly continuous (with respect to the uniform structures induced on X and Y by the prodiscrete uniform structures on AG and B G ); (CA3) there exists a finite subset S ⊂ G and a map λ : XS → B, where XS := {x|S : x ∈ X}, such that σ (x)(g) = λ((g −1 x)|S )

.

for all x ∈ X and g ∈ G. 1.41 Let G be a group. Let A and B be two sets with A finite. Suppose that X ⊂ AG and Y ⊂ B G are subshifts and let σ : X → Y be a map. Show that the following conditions are equivalent: (C1) σ is a cellular automaton;

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(C2) σ is G-equivariant and continuous (with respect to the topologies induced on X and Y by the prodiscrete topologies on AG and B G ). 1.42 Let G be an infinite group and let A be a set. Show that if X ⊂ AG is a subshift with non-empty interior then X = AG . 1.43 Let G be an infinite group and let A be a set. Let τ : AG → AG be a cellular automaton. Show that the following conditions are equivalent: (i) τ is nilpotent; (ii) there exists a configuration y ∈ AG such that for every x ∈ AG , there is an integer nx ≥ 1 such that τ n (x) = y for all n ≥ nx . (iii) there exists a constant configuration y ∈ AG such that for every x ∈ AG , there is an integer nx ≥ 1 such that τ nx (x) = y. 1.44 (Patterns Appearing in a Subshift) Let G be a group and let A be a set. One says that a pattern p ∈ P(G, A) appears in a configuration x ∈ AG if p = x|supp(p) . Given a subset X ⊂ AG , one says that a pattern p ∈ P(G, A) appears in X if there exists a configuration x ∈ X such that p appears in x. Let P(X) ⊂ P(G, A) denote the set of all patterns appearing in a subset X. (a) Let X, Y ⊂ AG and suppose that Y is a subshift. Show that one has X ⊂ Y if and only if P(X) ⊂ P(Y ). (b) Let X ⊂ AG be a subshift and let x ∈ AG . Show that one has x ∈ X if and only if every pattern appearing in x is in P(X). (c) Let X, Y ⊂ AG be subshifts. Show that one has X = Y if and only if P(X) = P(Y ). (d) Let X ⊂ AG be a subshift and let P := P (X). Show that P satisfies the following properties: (P1) P is a G-invariant subset of P(G, A); (P2) if p ∈ P and Ω ⊂ supp(p), then p|Ω ∈ P ; (P3) if p ∈ P and Ω is a finite subset of G such that supp(p) ⊂ Ω, then there exists a pattern q ∈ P with support Ω such that q|supp(p) = p. (e) Suppose that the group G is countable. Show that if P ⊂ P(G, A) satisfies conditions (P1), (P2), and (P3) in (d), then there exists a unique subshift X ⊂ AG such that P = P (X). 1.45 Let G be a group. Let A and B be 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 f∗ : AG → B G is a cellular automaton. (b) Show that if A is finite and X is a subshift of AG then f∗ (X) is a subshift of B G. (c) 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. Show that X is a subshift of AZ but f∗ (X) is not a subshift of B Z .

Exercises

41

1.46 Let G be a group. Let A and B be two sets. Equip AG and B G with their prodiscrete topologies and the shift actions of G. Suppose that the group G acts continuously on a topological space Y . (a) Let Z ⊂ Y be a closed G-invariant subset and let f : AG → Y be a Gequivariant continuous map. Show that f −1 (Z) := {x ∈ AG : f (x) ∈ Z} is a subshift of AG . (b) Suppose that Y is Hausdorff and let f, g : AG → Y be G-equivariant continuous maps. Show that Eq(f, g) := {x ∈ AG : f (x) = g(x)} is a subshift of AG . (c) Let f : AG → B G be a G-equivariant continuous map (e.g. a cellular automaton over the group G and the alphabets A and B) and let Z ⊂ B G be a subshift. Show that f −1 (Z) is a subshift of AG . (d) Let f, g : AG → B G be G-equivariant continuous maps (e.g. cellular automata over the group G and the alphabets A and B). Show that Eq(f, g) := {x ∈ AG : f (x) = g(x)} is a subshift of AG . (e) Let f : AG → AG be a G-equivariant continuous map (e.g. a cellular automaton over the group G and the alphabet A). Show that Fix(f ) := {x ∈ AG : f (x) = x} is a subshift of AG . 1.47 Let G be a group and let A be a set. Given a set F ⊂ P(G, A) of (G, A)patterns, define the subset XF ⊂ AG as being the set consisting of all configurations x ∈ AG satisfying the following condition: for every g ∈ G, there is no pattern in F appearing in gx, i.e., XF := {x ∈ AG : (gx)|supp(p) /= p for all g ∈ G and p ∈ F }.

.

(a) Let F ⊂ P(G, A). Show that XF is a subshift of AG . (b) Let X ⊂ AG be a subshift and let P (X) ⊂ P(G, A) denote the set consisting of all (G, A)-patterns appearing in X. One says that a set F ⊂ AG of (G, A)patterns is a defining set of forbidden patterns for X if X = XF . Show that the set F (X) := P(G, A) \ P (X) is a defining set of forbidden patterns for X and that every set of forbidden patterns for X is contained in F (X). (c) Let (Xi )i∈I be an family of subshifts of AG and consider the subshift X ⊂ AG defined by X := i∈I Xi (cf. Exercise 1.39(d)). Suppose that (Fi )i∈I is a family of subsets of P(G, A) such that Fi is a defining set of forbidden patterns for Xi U for each i ∈ I . Show that the set F := i∈I Fi is a defining set of forbidden patterns for X. 1.48 (Subshifts of Finite Type) Let G be a group and let A be a finite set. One says that a subshift X ⊂ AG is of finite type if it admits a finite defining set of forbidden patterns. (a) Let X ⊂ AG be a subshift of finite type. Show that X admits a finite defining set of forbidden patterns having all the same support.

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(b) Let Ω ⊂ G be a finite subset and let A ⊂ AΩ . Define X(A ) ⊂ AG by X(A ) := {x ∈ AG : (gx)|Ω ∈ A for all g ∈ G}.

.

(c)

(d)

(e) (f) (g) (h)

(1.25)

Show that X(A ) is a subshift of finite type of AG . Let X ⊂ AG be a subshift of finite type. Show that there exist a finite subset Ω ⊂ G and a subset A ⊂ AΩ such that X = X(A ). Such a set A is called a defining set of admissible patterns and the subset Ω is called a memory set for the subshift of finite type X. Let X ⊂ AG be a subshift of finite type and let Ω ⊂ G be a memory set for X. Let g ∈ G and let Λ be a finite subset of G such that Ω ⊂ Λ. Show that gΩ and Λ are also memory sets for X. Show that the full subshift AG is a subshift of finite type of itself. Let X1 , X2 ⊂ AG be two subshifts of finite type. Show that X1 ∩X2 is a subshift of finite type. Show that the union of two subshifts of finite type may fail to be of finite type. Fix a ∈ A and consider the constant configuration xa ∈ AG defined by xa (g) := a for all g ∈ G. Show that the singleton X := {xa } is a subshift of finite type of AG .

1.49 Let G be a group and let A be a finite set. (a) Let X ⊂ AG be a subshift. Let Ω ⊂ G be a finite subset and let A ⊂ AΩ denote the set of all patterns with support Ω appearing in X. Let X(A ) ⊂ AG denote the subshift of finite type admitting A as a defining set of admissible patterns. Show that X ⊂ X(A ). (b) With the notation in (a), find an example showing that the inclusion X ⊂ X(A ) may be strict. (c) Find an example of a subshift of finite type X ⊂ AG with memory set Ω ⊂ G admitting two distinct defining sets A , A ' ⊂ AΩ of admissible patterns. (d) Let X ⊂ AG be a subshift of finite type. Let Ω ⊂ G be a memory set for X and let A1 , A2 ⊂ AΩ be two defining sets of admissible patterns for X. Show that A := A1 ∩ A2 is a set of admissible patterns for X. (e) Let X ⊂ AG be a subshift of finite type and let Ω ⊂ G be a memory set for X. Let A1 , A2 , . . . , An ⊂ AΩ denoten all the defining sets of admissible patterns with support Ω for X and set A := ni=1 Ai . Show that A is a set of admissible patterns for X and that it equals the set of all patterns with support Ω appearing in X. 1.50 Let G be a group and let A be a finite set. Let X ⊂ AG be a subset. Let x0 ∈ {0, 1}G denote the constant 0-configuration. Show that the following conditions are equivalent: (i) X is a subshift of finite type of AG ; (ii) there exists a cellular automaton τ : AG → {0, 1}G such that X = τ −1 (x0 ).

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1.51 Let G be a group and let A and B be two finite sets. Let τ : B G → AG be a cellular automaton and suppose that X ⊂ AG is a subshift of finite type. Show that Y := τ −1 (X) ⊂ B G is a subshift of finite type. 1.52 Let G be a group and let A be a finite set. Let X ⊂ AG be a subshift. Denote by F the set of all finite subsets of G. For Ω ∈ F , let πΩ : AG → AΩ denote the projection map and set XΩ := {x ∈ AG : πΩ (gx) ∈ πΩ (X) for all g ∈ G}.

.

(a) Let Ω ∈ F . Show that XΩ is a subshift of finite type of AG . (b) Show that one has X=

n

.

XΩ .

(1.26)

Ω∈F

(c) Show that X is of finite type if and only if there exists Ω ∈ F such that X = XΩ . (d) Let Ω, Ω ' ∈ F such that Ω ⊂ Ω ' . Show that XΩ ' ⊂ XΩ and deduce that if X = XΩ then X = XΩ ' . 1.53 Let G be a group. Let A and B be two finite sets and let X ⊂ AG be a subshift. We keep the notation from Exercise 1.52. Let τ : AG → B G be a cellular automaton and let T ⊂ G be a memory set for τ . Show that τ (X) is a subshift of B G and that one has τ (XΩT ) ⊂ (τ (X))Ω

.

(1.27)

for all Ω ∈ F . 1.54 Let G be a group. Let A (resp. B) be a finite set and let X ⊂ AG (resp. Y ⊂ B G ) be a subshift. Suppose that the subshifts X and Y are topologically conjugate, i.e., there exists a G-equivariant homeomorphism ϕ : X → Y . Show that X is of finite type if and only if Y is of finite type. 1.55 Let G be a countable group, let A be a finite set, and let X ⊂ AG be a subshift. Show that the following conditions are equivalent: (i) X is of finite type; (ii) every descending sequence of subshifts of AG X0 ⊃ X1 ⊃ X2 ⊃ · · · ⊃ Xn ⊃ Xn+1 ⊃ · · ·

.

(1.28)

n such that X = n∈N Xn eventually stabilizes, that is, there exists n0 ∈ N such that Xn = Xn0 for all n ≥ n0 ;

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(iii) there is no infinite strictly descending sequence of subshifts of AG X0 C X1 C X2 C · · · C Xn C Xn+1 C · · ·

.

such that X =

n

n∈N Xn .

1.56 Let G be a group and let A be a set. Let X ⊂ AG be a subshift. One says that the subshift X is topologically transitive (resp. irreducible, resp. topologically mixing) if the shift action of G on X is topologically transitive (resp. irreducible, resp. topologically mixing). (a) Show that X is topologically transitive if and only if it satisfies the following condition: (TT) for all configurations x1 , x2 ∈ X and for every finite subset Ω of G, there exist a configuration x ∈ X and an element g ∈ G such that x|Ω = x1 |Ω and (gx)|Ω = x2 |Ω . (b) Show that X is irreducible if and only if it satisfies the following condition: (I) for every finite subset F ⊂ G, for all configurations x1 , x2 ∈ X, and for every finite subset Ω of G, there exist an element g ∈ G \ F and a configuration x ∈ X such that x|Ω = x1 |Ω and (gx)|Ω = x2 |Ω . (c) Show that X is topologically mixing if and only if it satisfies the following condition: (TM) for all configurations x1 , x2 ∈ X and for every finite subset Ω of G, there exists a finite subset F ⊂ G such that, for every element g ∈ G\F , there exists a configuration x ∈ X satisfying x|Ω = x1 |Ω and (gx)|Ω = x2 |Ω . (d) Show that every irreducible subshift of AG is topologically transitive. (e) Show that if G is infinite, then every topologically mixing subshift of AG is irreducible and topologically transitive. 1.57 Let G be a group and let A be a set. Let X ⊂ AG denote the subset consisting of all constant configurations. (a) Show that X is a subshift of AG . (b) Find a defining set of forbidden patterns for X. (c) Suppose that A contains at least two distinct elements. Show that X is not topologically transitive. (d) Suppose that G is infinite and that A contains at least two distinct elements. Show that X is not topologically mixing. (e) Suppose that A is finite and contains more than one element. Show that X is of finite type if and only if G is finitely generated. 1.58 Let G be a finitely generated group and let A be a finite set. Show that every finite subshift X ⊂ AG is of finite type.

Exercises

45

1.59 Let G be a finitely generated group and let A be a finite set. Let B and C be disjoint subsets of A. Suppose that X ⊂ B G and Y ⊂ C G are two subshifts of finite type. Show that the subset Z ⊂ AG defined by Z := X ∪ Y ⊂ B G ∪ C G ⊂ (B ∪ C)G ⊂ AG

.

is a subshift of finite type of AG . 1.60 Let G be a countable group and let A be a finite set. Show that the set of all subshifts of finite type X ⊂ AG is countable. 1.61 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 Eq(τ1 , τ2 ) := {x ∈ AG : τ1 (x) = τ2 (x)} ⊂ AG is a subshift of finite type. (b) Let X ⊂ AG . Show that the following conditions are equivalent: (i) X is a subshift of finite type; (ii) there exists a cellular automaton τ : AG → AG such that X = Fix(τ ). Here Fix(τ ) := {x ∈ AG : τ (x) = x} ⊂ AG denotes the fixed-point set of τ . (c) Let τ : AG → AG be a cellular automaton that is idempotent, i.e., such that τ 2 = τ , and set X := τ (AG ). Show that X = Fix(τ ) and deduce that X is a subshift of finite type of AG . 1.62 Let G be a group and let A be a set. Let B ⊂ A. 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 topologically transitive. (c) Suppose that B has more than one element. Show that X is not topologically transitive. 1.63 We recall that a topological space X is called a Baire space if the intersection of every countable family of open dense subsets of X is dense in X. We also recall that the Baire category theorem asserts that every completely metrizable space is a Baire space (see e.g. Theorem I.1.1 and Remark I.1.2). (a) Let G be a group acting continuously on a topological space X. Show that if there exists a point in X whose G-orbit is dense in X then the action of G on X is topologically transitive. (b) Let G be a group acting continuously on a Baire space X. Suppose that X is nonempty and satisfies the second axiom of countability. Show that the following conditions are equivalent: (1) the action of G on X is topologically transitive; (2) there is a point x ∈ X whose G-orbit is dense in X; (3) there exists a nonempty G-invariant Gδ subset D ⊂ X such that the G-orbit of each point x ∈ D is dense in X.

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(c) Let G be a countable group and let A be a countable (e.g. finite) set. Let X ⊂ AG be a non-empty subshift. Show that the following conditions are equivalent: (1) the subshift X is topologically transitive; (2) there is a configuration x ∈ X whose G-orbit is dense in X; (3) there exists a non-empty G-invariant Gδ subset D ⊂ X such that the G-orbit of each configuration x ∈ D is dense in X. (d) Let G be an infinite countable group and let A be a countable (e.g. finite) nonempty set. Show that there exists a configuration x ∈ AG whose orbit under the G-shift action is dense in AG . (e) Let G be an infinite countable group and let A be an uncountable set. Show that the G-shift action on AG is topologically transitive but that there is no configuration x ∈ AG whose orbit under the G-shift action is dense in AG . 1.64 (Strongly Irreducible Subshifts) 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 ∩ Ω2 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 1.62 is {1G }-irreducible and therefore strongly irreducible. (b) Show that every strongly irreducible subshift X ⊂ AG is topologically mixing. 1.65 Let G be a group and let A be a set. Let X ⊂ AG be a subshift. Show that the following conditions are equivalent: (i) X is strongly irreducible, i.e., there exists a finite subset Δ ⊂ G such that X is Δ-irreducible; (ii) there exists a finite symmetric subset Δ' ⊂ G such that the following holds: if Ω1 and Ω2 are two finite subsets of G such that Ω1 ∩ Ω2 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 . (iii) there exists a finite subset Δ'' ⊂ G such that the following holds: if Ω1 and Ω2 are two finite subsets of G such that Ω1 Δ'' ∩ Ω2 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 . (iv) there exists a finite symmetric subset Δ''' ⊂ G such that the following holds: if Ω1 and Ω2 are two finite subsets of G such that Ω1 Δ''' ∩ Ω2 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 . 1.66 (Weak Specification Property) Let X be a uniform space equipped with a uniformly continuous action of a group G. One says that the action of G on X has the weak specification property if it satisfies the following condition:

Exercises

47

(WSP) for every entourage U of X, there exists a finite subset Λ = Λ(U ) ⊂ G such that the following holds: for any finite family (Ωi )i∈I of finite subsets of G such that Ωj ∩ ΛΩk = ∅ for all distinct j, k ∈ I and for any family of points (xi )i∈I in X, there exists a point x ∈ X such that (gx, gxi ) ∈ U for all i ∈ I and g ∈ Ωi . Such a subset Λ ⊂ G is then called a specification subset for (X, G, U ). Suppose that the action of G on X has the weak specification property. Let Y be a uniform space equipped with a uniformly continuous action of G and suppose there exists a G-equivariant uniformly continuous surjective map f : X → Y . Show that the action of G on Y has the weak specification property. 1.67 Let (Xk )k∈K be a (possibly infinite) family of uniform spaces and let G be a group. Suppose that each Xk , k ∈ K, is equipped with a uniformly continuous action of G having the weak specification property. Show that the diagonal action || of G on the uniform product X := k∈K Xk has the weak specification property. 1.68 (Weak Specification Property for Ultra-Uniform Spaces) Let X be a ultrauniform space (cf. Exercise 1.3) equipped with a uniformly continuous action of a group G. Show that the action of G on X has the weak specification property, that is, it satisfies condition (WSP) in Exercise 1.66, if and only if the following condition is satisfied: (WSP’) for every entourage U of X, there exists a finite subset Λ ⊂ G such that the following holds: if Ω1 and Ω2 are two finite subsets of G such that Ω1 ∩ ΛΩ2 = ∅, then, given any two points x1 , x2 ∈ X, there exists a point x ∈ X such that (gx, gxk ) ∈ U for all k ∈ {1, 2} and g ∈ Ωk . 1.69 Let G be a group and let A be a set. Equip AG with its prodiscrete uniform structure and the shift action of G. Let X ⊂ AG be a subshift. Show that the following conditions are equivalent: (i) X is strongly irreducible; (ii) the shift action of G on X has the weak specification property. 1.70 Let G be a group. Let A and B be sets. Let X ⊂ AG and Y ⊂ B G be subshifts. Suppose that there exists a surjective cellular automaton τ : X → Y . Show that if X is topologically transitive (resp. irreducible, resp. topologically mixing, resp. strongly irreducible) then Y is topologically transitive (resp. irreducible, resp. topologically mixing, resp. strongly irreducible). 1.71 (The At-Most-One-One Subshift) Let G be a group and let A := {0, 1}. Consider the set X ⊂ AG consisting of all configurations x ∈ AG such that 1 appears at most once in x (i.e., such that the set {g ∈ G : x(g) = 1} is either empty or reduced to a single element). (a) Show that X is a subshift of AG . It is called the at-most-one-one subshift. (b) Describe the set of all cellular automata τ : X → X. (c) Show that X is topologically transitive if and only if G is infinite.

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(d) Show that X is not irreducible. (e) Show that if G is infinite then X is neither topologically mixing, nor strongly irreducible, nor of finite type. 1.72 Let G be a group, let H be a finite index subgroup of G, and let A be a finite set. Let T ⊂ G be a complete set of representatives for the right cosets of H in G and set B := AT . Consider the map Ψ : AG → B H defined in Exercise 1.32. 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(H,T ) is of finite type then X is of finite type. Conversely, show that if G is abelian and X is of finite type then X(H,T ) is of finite type. 1.73 Let G be a group and let H be a finite index subgroup of G. Let T ⊂ G be a complete set of representatives for the right cosets of H in G, so that G = ut∈T H t. G Let A1 , A2 be two sets and let τ : AG 1 → A2 be a cellular automaton. For i = 1, 2, H set Bi := ATi and consider the H -equivariant uniform isomorphism Ψi : AG i → Bi G defined by setting Ψi (x)(h)(t) := x(ht) for all x ∈ Ai , h ∈ H , and t ∈ T (cf. Exercise 1.32). Define a map τ (T ) : B1H → B2H by setting τ (T ) := Ψ2 ◦ τ ◦ Ψ1−1 .

.

(a) Show that τ (T ) is a cellular automaton. G (b) Let X ⊂ AG 1 be a subshift and let Y := τ (X) ⊂ A2 denote the corresponding image subshift under τ . Consider the subshifts X(H,T ) := Ψ1 (X) ⊂ B1H and Y (H,T ) := Ψ2 (Y ) ⊂ B2H (cf. Exercise 1.72). Show that τ (T ) (X(H,T ) ) = Y (H,T ) . 1.74 (Language of a Subshift Over Z) Let A be a set. Let A∗ denote the free monoid based on A, that is, the set consisting of all words on the alphabet A with the concatenation of words as the monoid operation (cf. Sect. D.1). One says that a word w ∈ A∗ appears in a configuration x ∈ AZ if there exist integers n, m ∈ Z with n ≤ m such that w = x(n)x(n + 1) · · · x(m).

.

By convention, the empty word ε ∈ A∗ appears in any configuration x ∈ AZ . The set consisting of all words appearing in a configuration x ∈ AZ is called the language of x and denoted by L(x). More generally, one says that a word w ∈ A∗ appears in a subset X ⊂ A∗ if there exists a configuration x ∈ X such that w appears in x. The set U .L(X) := L(x) ⊂ A∗ , (1.29) x∈X

consisting of all words appearing in X, is called the language of X.

Exercises

49

(a) Let X, Y ⊂ AZ and suppose that Y is a subshift. Show that one has X ⊂ Y if and only if L(X) ⊂ L(Y ). (b) Let X ⊂ AZ be a subshift and let x ∈ AZ . Show that one has x ∈ X if and only if L(x) ⊂ L(X). (c) Let X ⊂ AZ be a subshift and let x ∈ AZ . Show that one has L(x) = L(X) if and only if X is the orbit closure of x. (d) Let X and Y be two subshifts of AZ . Show that one has X = Y if and only if L(X) = L(Y ). (e) 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) denote the language of 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, b ∈ A such that awb ∈ L. (f) Conversely, show that if a subset L ⊂ A∗ satisfies conditions (i) and (ii) in (e), then there exists a unique subshift X ⊂ AZ such that L = L(X). 1.75 Let A be a set and let X, Y ⊂ AZ be subshifts. (a) Show that X ∪ Y is a subshift of AZ and that one has L(X ∪ Y ) = L(X) ∪ L(Y ). (b) Show that X ∩ Y is a subshift of AZ and that one has L(X ∩ Y ) ⊂ L(X) ∩ L(Y ). (c) Give an example showing that the inclusion in (b) may be strict. 1.76 Let A be a set and let X ⊂ AZ be a subshift. Let L(X) ⊂ A∗ denote the language of X. (a) Show that X is topologically transitive if and only if the following holds: for all words u and v in L(X), there exists a word w ∈ L(X) such that both u and v are subwords of w. (b) Show that X is irreducible if and only if the following holds: for all words u and v in L(X), there exists w ∈ A∗ such that uwv ∈ L(X). (c) Show that X is topologically mixing if and only if the following holds: for all words u and v in L(X), there exists an integer n0 = n0 (u, v) ≥ 0 such that for any integer n ≥ n0 there exists a word w ∈ A∗ of length l(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 all words u and v in L(X), and for any integer n ≥ n0 , there exists a word w ∈ A∗ of length l(w) = n such that uwv ∈ L(X). 1.77 Let A be a set. Given a set F ⊂ A∗ of words on the alphabet A, define the subset XF ⊂ AZ as being the set consisting of all configurations x ∈ AZ such that there is no word in F appearing in x. (a) Let F ⊂ A∗ . Show that XF is a subshift of AZ . (b) Let X ⊂ AZ be a subshift and let L(X) ⊂ A∗ denote the language of X. One says that a set F ⊂ A∗ is a defining set of forbidden words for X if X = XF .

50

1 Cellular Automata

Show that the set F (X) := A∗ \ L(X) is a defining set of forbidden words for X and that every set of forbidden words for X is contained in F (X). 1.78 Let A be a finite set. (a) Show that a subshift X ⊂ AZ is of finite type if and only if it admits a finite defining set of forbidden words. (b) Let n ≥ 0 be an integer and let W ⊂ A∗ be a set of words of length n. Show that the subset X(W ) ⊂ AZ , defined by X(W ) := {x ∈ AZ : x(i)x(i +1) · · · x(i +n−1) ∈ W for all i ∈ Z},

.

(1.30)

is a subshift of finite type. (c) Let X ⊂ AZ be a subshift of finite type. Show that there exists an integer n ≥ 0 and a set W ⊂ A∗ of words of length n such that X = X(W ). One then says that the subshift of finite type X has memory length n and that W is a defining set of admissible words for X. 1.79 Let A be a finite set and let X ⊂ AZ be a subshift. 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 uw, wv ∈ L(X) and w has length l(w) ≥ n0 , then one has uwv ∈ L(X). 1.80 Let A be a finite set and let X ⊂ AZ be a subshift of finite type. Show that X is topologically mixing if and only if X is strongly irreducible. 1.81 Let A := {0, 1}. Consider the subshift of finite type X ⊂ AZ admitting {00, 11} as a defining set of forbidden words. Show that the subshift X is irreducible and topologically transitive but neither topologically mixing nor strongly irreducible. 1.82 Let A be a set and let X ⊂ AZ be an irreducible subshift. Suppose that Y, Z ⊂ AZ are subshifts such that Y ∪ Z = X. Show that one has Y = X or Z = X. 1.83 Let A := {0, 1} and consider the subset X ⊂ AZ consisting of all x ∈ AZ such that (x(n), x(n + 1)) /= (0, 1) for all n ∈ Z. (a) (b) (c) (d)

Show that X is a subshift of finite type. Show that X is topologically transitive. Show that X is not irreducible. Show that X is neither topologically mixing nor strongly irreducible.

1.84 (The Golden Mean Subshift) Let A := {0, 1} and consider the subset X ⊂ AZ consisting of all configurations x ∈ AZ such that (x(n), x(n + 1)) /= (1, 1) for all n ∈ Z. (a) Show that X is a subshift of finite type. (b) Show that X is strongly irreducible. (c) Show that X is topologically mixing, topologically transitive, and irreducible. The subshift X is called the golden mean subshift.

Exercises

51

1.85 (The Even Subshift) Let A := {0, 1} and consider the set X ⊂ AZ consisting of all configurations x ∈ AZ such that the following holds: if x(m) = x(n) = 1 for some integers m, n ∈ Z with m < n, then the number of k ∈ Z such that x(k) = 0 and m < k < n is even. (a) (b) (c) (d)

Show that X is a subshift of AZ . Show that X is not of finite type. Show that X is strongly irreducible. Show that X is topologically mixing, topologically transitive, and irreducible. The subshift X is called the even subshift.

1.86 Let A := {0, 1} and let X ⊂ AZ be the subshift with defining set of forbidden words {01k 0h 1 : h, k ∈ N and 1 ≤ h ≤ k}.

.

Show that X is topologically mixing, topologically transitive, and irreducible but not strongly irreducible. 1.87 Let G be a group and let A be a set. Let H be a subgroup of G. Consider the subset X ⊂ AG consisting of all configurations x ∈ AG that are constant on each left coset of H in G, that is, such that x(gh1 ) = x(gh2 ) for all g ∈ G and h1 , h2 ∈ H . Show that X is a subshift of AG . 1.88 Let G be a group and let A be a set. The reverse of a configuration x ∈ AG is the configuration x R ∈ AG defined by x R (g) := x(g −1 ) for all g ∈ G. The reverse of a subset X ⊂ AG is the subset XR := {x R : x ∈ X} ⊂ AG . The reverse of a word w = a1 a2 · · · an ∈ A∗ is the word w R := an an−1 · · · a1 ∈ A∗ . The reverse of a language L ⊂ A∗ is the language LR := {w R : w ∈ L} ⊂ A∗ . (a) Show that a subset X ⊂ AG is closed (in the prodiscrete topology) if and only if XR is closed. (b) Give an example of a subshift X ⊂ AG such that XR fails to be a subshift. (c) Suppose that G is abelian. Let x ∈ AG and let g ∈ G. Show that g(x R ) = (g −1 x)R . (d) Suppose that G is abelian and let X ⊂ AG . Show that X is G-invariant if and only if XR is G-invariant. (e) Suppose that G is abelian and let X ⊂ AG . Show that X is a subshift if and only if XR is a subshift. (f) Suppose that G is abelian. Show that a subshift X ⊂ AG is topologically transitive (resp. irreducible, resp. topologically mixing, resp. strongly irreducible) if and only if the subshift XR is topologically transitive (resp. irreducible, resp. topologically mixing, resp. strongly irreducible). (g) Suppose that G is abelian and that the set A is finite. Show that a subshift X ⊂ AG is of finite type if and only if the subshift XR is of finite type.

52

1 Cellular Automata

(h) Suppose that G = Z and let X ⊂ AZ be a subshift. Let L(X) ⊂ A∗ be the language of X. Show that L(XR ) = L(X)R and give another proof of the fact that X is irreducible if and only if XR is irreducible. 1.89 Let A be a countable set. The forward-orbit (resp. backward-orbit) of a configuration x ∈ AZ is the subset {nx : n ∈ N} ⊂ AZ (resp. {(−n)x : n ∈ N} ⊂ AZ ). Let X ⊂ AZ be a non-empty subshift. Show that the following conditions are equivalent: (i) X is irreducible; (ii) there exists a configuration x ∈ X whose backward and forward orbits are both dense in X; (iii) there exists a configuration x ∈ X whose backward-orbit is dense in X; (iv) there exists a configuration x ∈ X whose forward-orbit is dense in X. 1.90 (Sofic Subshifts) Let G be a group and let A be a finite set. One says that a subshift X ⊂ AG is a sofic subshift if there exist a finite set B, a subshift of finite type Y ⊂ B G , and a cellular automaton τ : B G → AG such that X = τ (Y ). (a) Show that every subshift of finite type X ⊂ AG is a sofic subshift. One says that a subshift X ⊂ AG is strictly sofic if it is sofic but not of finite type. (b) Let A := {0, 1} and consider the at-most-one-one subshift X ⊂ AZ , i.e., the subshift consisting of all configurations x ∈ AZ such that 1 appears at most once in x (cf. Exercise 1.71). Show that the subshift X is strictly sofic. 1.91 Let G be a countable group and let A be a finite set. Show that the set of all sofic subshifts X ⊂ AG is countable. 1.92 (S-Gap Subshifts) Let A := {0, 1} and let S ⊂ N. Consider the subset X(S) ⊂ AZ consisting of all the configurations x ∈ AZ satisfying the following condition: if x(n) = 1, x(n + 1) = x(n + 2) = · · · = x(n + h) = 0, and x(n + h + 1) = 1 for some n ∈ Z and h ∈ N, then h ∈ S. (a) Show that X(S) is a subshift of AZ . It is called the S-gap subshift. (b) Determine X(S) in each of the following cases: (i) (ii) (iii) (iv) (v)

S S S S S

:= ∅; := {0}; := N; := N \ {0}; := 2N.

(c) Suppose that S1 , S2 ⊂ N are distinct. Show that X(S1 ) /= X(S2 ). (d) Let B be a set with more than one element. Deduce from (c) that there are uncountably many subshifts of B Z . 1.93 Let A be a finite set with more than one element. Show that there are uncountably many subshifts X ⊂ AZ that are non-sofic.

Exercises

53

1.94 Let G be a group. Let A and B be finite sets. (a) Suppose that τ : AG → B G is a cellular automaton and that X is a sofic subshift of AG . Show that τ (X) is a sofic subshift of B G . (b) Suppose that X ⊂ AG and Y ⊂ B G are topologically conjugate subshifts (cf. Exercise 1.54). Show that X is sofic if and only if Y is sofic. 1.95 Let G be a group and let A be a finite set. Let F be a non-empty finite subset of G and let B := AF . Consider the map Φ : AG → B G defined in Exercise 1.30. Let X ⊂ AG be a subshift and set X[F ] := Φ(X). (a) Show that X[F ] is a subshift of B G . (b) Show that the subshifts X and X[F ] are topologically conjugate. (c) Show that X is topologically transitive (resp. irreducible, resp. topologically mixing, resp. strongly irreducible, resp. of finite type, resp. sofic) if and only if X[F ] is topologically transitive (resp. irreducible, resp. topologically mixing, resp. strongly irreducible, resp. of finite type, resp. sofic). 1.96 Let A be a finite set. (a) Show that every non-empty subshift of finite type X ⊂ AZ contains a periodic configuration. (b) Deduce from (a) that every non-empty sofic subshift X ⊂ AZ contains a periodic configuration. 1.97 Let A := {0, 1}. Denote by X ⊂ AZ and Y ⊂ AZ the golden mean subshift and the even subshift, respectively. Consider the cellular automata σ : AZ → AZ and τ : AZ → AZ with memory set S := {0, 1} and local defining maps ν : AS → A and μ : AS → A given respectively by ν(p) :=

{ 1

if (p(0), p(1)) = (1, 0)

.

0

otherwise,

and { μ(p) :=

1

if p(0) = p(1) = 0

0

otherwise,

.

for all p ∈ AS . (a) (b) (c) (d) (e)

Show that σ is idempotent, i.e., σ ◦ σ = σ . Show that X = σ (AZ ) = Fix(σ ). Show that τ (X) = Y . Show that Y is strictly sofic. Show that the cellular automaton ϕ := τ ◦ σ : AZ → AZ satisfies Y = ϕ(AZ ) and give a memory set and the associated local defining map for ϕ.

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1 Cellular Automata

1.98 (Splicable Subshifts) Let G be a group and let A be a finite set. Given two configurations x, y ∈ AG and a subset Ω ⊂ G, the splice of x and y over Ω is the configuration z = spliceΩ (x, y) ∈ AG defined by z|Ω := x|Ω and z|G\Ω := y|G\Ω . Let Δ ⊂ G be a finite subset containing 1G . One says that a subshift X ⊂ AG is Δ-splicable if the following holds: (Splic-Δ) for every finite subset Ω ⊂ G and any two configurations x, y ∈ X such that x|ΩΔ\Ω = y|ΩΔ\Ω , one has spliceΩ (x, y) ∈ X. One says that a subshift X ⊂ AG is splicable if there exists a finite subset Δ ⊂ G containing 1G such that X is Δ-splicable. (a) Show that every subshift of finite type X ⊂ AG is splicable. (b) Suppose that G is an infinite group and let A := {0, 1}. Show that the at-mostone-one subshift X ⊂ AG is not splicable. (c) Let A := {0, 1}. Show that the even subshift X ⊂ AZ is not splicable. 1.99 (Characterization of Splicable Subshifts Over Z) Let A be a finite set and let X ⊂ AZ be a subshift. Let L(X) ⊂ A∗ denote the language of X. Given an integer n ≥ 0, one says that X is n-splicable if the following holds: (Splic-n) for all u, u' , v, v ' , w1 , w2 ∈ A∗ such that (i) l(v) = l(v ' ) = n and l(w1 ) = l(w2 ), (ii) uvw1 v ' u' ∈ L(X) and vw2 v ' ∈ L(X), then one has uvw2 v ' u' ∈ L(X). Show that X is splicable if and only if there exists an integer n ≥ 0 such that X is n-splicable. 1.100 Let A := {0, 1}. For a ∈ A, let ca ∈ AZ be the constant configuration defined by ca (k) := a for all k ∈ Z. For n ∈ Z, let xn , yn ∈ AZ be the configurations defined, for all k ∈ Z, by xn (k) := 0 and yn (k) := 1 if k < n, xn (k) := 1 and yn (k) := 0 if k ≥ n. Consider the subset X ⊂ AZ defined by X := {c0 , c1 } ∪ {xn , yn : n ∈ Z}.

.

(a) Show that X is a sofic subshift of AZ . (b) Show that the subshift X is not of finite type. (c) Show that the subshift X is splicable. 1.101 Let G be a finitely generated group and let A be a finite set. Let X, Y ⊂ AG be sofic subshifts. Show that X ∪ Y is a sofic subshift of AG . 1.102 Let A := {0, 1}. Let X ⊂ AZ (resp. Y ⊂ AZ ) denote the subshift of finite type admitting the singleton {11} (resp. {00}) as a defining set of forbidden words. Show that X ∪ Y is a strictly sofic subshift.

Exercises

55

1.103 Let G be a group and let A be a set. Let H be a subgroup of G and let X ⊂ AH be a subshift. For x ∈ AG and g ∈ G, denote by xgH ∈ AH the configuration defined by xgH := (gx)|H . (a) Show that the set X(G) defined by X(G) := {x ∈ AG : xgH ∈ X for all g ∈ G}

.

(b) (c)

(d)

(e)

is a subshift of AG . H = hx H for all x ∈ AG , h ∈ H , and g ∈ G. Check that xhg g From now on, we fix a complete set T ⊂ G of representatives for the right cosets of H in G, so that G = ut∈T H t. We suppose that 1G ∈ T . Let x ∈ AG . H Show that one has x ∈ X(G) if and || onlyHif xt ∈ X for all t ∈ T(. H ) G Show that the map ϕ : A → t∈T A defined by ϕ(x) := xt t∈T for all x ∈ AG is a uniform isomorphism with respect to the || prodiscrete uniform structure on AG and the product uniform structure on t∈T AH , where each copy of AH is equipped with the prodiscrete uniform||structure. || Deduce that ϕ|X(G) yields a uniform isomorphism from X(G) onto t∈T X ⊂ t∈T AH . Let x ∈ AH . Define x T ∈ AG by setting x T (t −1 h) := x(h) for all t ∈ T and h ∈ H . Show that x T ∈ X(G) if and only if x ∈ X. Finally observe that x T |H = x.

.

(1.31)

(f) Given g ∈ G, there exist unique η(g) ∈ H and θ (g) ∈ T such that g = η(g)θ (g).

.

(1.32)

Show that for every t ∈ T fixed, the map g |→ η(tg)is a surjection from G onto H,

.

(1.33)

while, for every g ∈ G fixed, the map t |→ θ (tg) is a permutation of T .

.

(1.34)

(g) Show that if x ∈ AG and g ∈ G then, for all t ∈ T , H (gx)H t = η(tg)xθ(tg) .

.

(1.35)

(h) Show that if the index of H in G is infinite, then the subshift X(G) ⊂ AG is topologically transitive. (i) Suppose A is finite. Show that X(G) is of finite type if and only if X is of finite type. (j) Show that if X(G) is topologically mixing then X is topologically mixing.

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(k) Show that X(G) is strongly irreducible if and only if X is strongly irreducible. (l) Let B be another alphabet set. Suppose that σ : B H → AH is a cellular G G G cellular automaton. Check automaton and ( Glet σ)H: B →H A be the induced that one has σ (x) g = σ (xg ) for all x ∈ B G and g ∈ G, and deduce that ( ) (σ (X))(G) = σ G X(G) .

.

(1.36)

(m) Suppose that A is finite. Show that if X is sofic then X(G) is sofic. 1.104 (Limit Set of a Cellular Automaton) Let G be a group and let A be a nonempty finite set. Let τ : AG → AG be a cellular automaton. The subset Ω(τ ) ⊂ AG defined by Ω(τ ) :=

n

.

τ n (AG )

(1.37)

n∈N

is called the limit set of τ . (a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

(k)

Show that Ω(τ ) is a non-empty subshift of AG . Show that every subset X ⊂ AG such that X ⊂ τ (X) satisfies X ⊂ Ω(τ ). Show that Fix(τ ) ⊂ Ω(τ ). Show that τ (Ω(τ )) = Ω(τ ). One says that x ∈ AG admits a backward orbit if there exists a sequence (xn )n∈N of elements of AG such that x0 = x and xn = τ (xn+1 ) for all n ∈ N. Show that Ω(τ ) is the set of configurations in AG admitting a backward orbit. Show that Ω(τ k ) = Ω(τ ) for every integer k ≥ 1. Show that Ω(τ ) contains at least one constant configuration. Show that τ is surjective if and only if Ω(τ ) = AG . Show that τ is nilpotent if and only if Ω(τ ) is a singleton. Let A := {0, 1}. Determine Ω(τ ) when τ : AZ → AZ is the cellular automaton defined, for all x ∈ AZ and n ∈ Z, by τ (x)(n) := 1 if (x(n), x(n + 1)) = (1, 0) and τ (x)(n) := 0 otherwise. Let A := {0, 1}. Determine Ω(τ ) when τ : AZ → AZ is the cellular automaton defined, for all x ∈ AZ and n ∈ Z, by τ (x)(n) := 1 if x(n) = x(n + 1) = 1 and τ (x)(n) := 0 otherwise.

1.105 (Stable Cellular Automata) Let G be a group and let A be a non-empty finite set. Suppose that τ : AG → AG is a cellular automaton and let Ω(τ ) denote the limit set of τ . One says that τ is stable if there exists k ∈ N such that τ k (AG ) = τ k+1 (AG ). One says that τ is unstable if it is not stable. (a) (b) (c) (d)

Show that if τ is surjective or nilpotent then it is stable. Show that if τ is stable then Ω(τ ) is sofic. Show that if Ω(τ ) is of finite type then τ is stable. Show that if Ω(τ ) is of finite type then it is topologically mixing.

Exercises

57

(e) Let A := {0, 1} and consider the cellular automaton τ : AZ → AZ defined, for all x ∈ AZ and n ∈ Z, by τ (x)(n) := 1 if x(n) = x(n + 1) = 1 and τ (x)(n) := 0 otherwise. Show that τ is unstable and that Ω(τ ) is strictly sofic and not topologically transitive. 1.106 Let G be an infinite group and let A be a non-empty finite set. Let τ : AG → AG be a cellular automaton. Show that the following conditions are equivalent: (i) τ is nilpotent; (ii) the limit set Ω(τ ) of τ is a singleton; (iii) the limit set Ω(τ ) of τ is finite. 1.107 Let G be a group and let A be a set. Let N ⊂ G be a normal subgroup and denote by ρ : G → K := G/N the canonical quotient homomorphism. Consider the map ρ ∗ : AK → AG defined by setting ρ ∗ (y) := y ◦ ρ for all y ∈ AK . (a) Show that ρ ∗ is uniformly continuous with respect to the prodiscrete uniform structures on AK and AG . (b) Let g ∈ G and y ∈ AK . Show that one has gρ ∗ (y) = ρ ∗ (ρ(g)y). 1.108 Let G be a group and let A be a finite set. Let N ⊂ G be a normal subgroup and denote by ρ : G → K := G/N the canonical quotient homomorphism. Let ρ ∗ : AK → AG be the map defined by setting ρ ∗ (y) := y ◦ ρ for all y ∈ AK . Let Y ⊂ AK be a subshift and set X := ρ ∗ (Y ) ⊂ AG . (a) Show that X is a subshift of AG . (b) Suppose that X is of finite type. Show that Y is of finite type. 1.109 (Group Subshifts) Let G be a group and let A be a finite group. Equip the configuration space AG with the product group structure (thus, given two configurations x, y ∈ AG , their product is defined as the configuration xy ∈ AG given by (xy)(g) := x(g)y(g) for all g ∈ G). A subshift X ⊂ AG which is also a subgroup of AG is called a group subshift. (a) Let X ⊂ AG be a group subshift of finite type. Let Ω ⊂ G be a memory set for X and equip AΩ with the product group structure. Show that there exists a subgroup A of AΩ which is a defining set of admissible patterns for X. (b) Let H ⊂ G be a subgroup. Let Y ⊂ AH be a group subshift. Show that the set X := Y (G) = {x ∈ AG : (gx)|H ∈ Y for all g ∈ G}

.

(cf. Exercise 1.103) is a group subshift of AG . (c) Let N ⊂ G be a normal subgroup and denote by ρ : G → K := G/N the canonical quotient homomorphism. Show that the map ρ ∗ : AK → AG defined by setting ρ ∗ (y) := y ◦ ρ for all y ∈ AK is a group homomorphism. (d) Show that if Y ⊂ AK is a group subshift, then the set X := ρ ∗ (Y ) is a group subshift of AG .

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1 Cellular Automata

1.110 Let G, A, and B be groups. Equip AG and B G with their product group structure and the shift action of G. Show that the following conditions are equivalent: (i) there exists a G-equivariant group isomorphism Φ : AG → B G ; (ii) the groups A and B are isomorphic. 1.111 (Groups of Finite Markov Type) One says that a group G is of finite Markov type if for any finite group A every group subshift X ⊂ AG is of finite type. (a) (b) (c) (d) (e) (f) (g) (h) (i)

Show that every finite group is of finite Markov type. Show that every group of finite Markov type is finitely generated. Show that every group of finite Markov type is countable. Show that every subgroup of a group of finite Markov type is of finite Markov type. Show that every subgroup of a group of finite Markov type is finitely generated. Let F be a nonabelian free group. Show that F is not of finite Markov type. Let G be a group containing a non-abelian free subgroup. Show that G is not of finite Markov type. Show that every quotient of a group of finite Markov type is of finite Markov type. Show that every group containing a finite index subgroup of finite Markov type is itself of finite Markov type.

1.112 Let G be a group. Show that the following conditions are equivalent: (i) G is of finite Markov type; (ii) G is countable and, for every finite group A, any descending sequence of group subshifts of AG X0 ⊃ X1 ⊃ X2 ⊃ · · · ⊃ Xn ⊃ Xn+1 ⊃ · · ·

.

(1.38)

eventually stabilizes, that is, there exists n0 ∈ N such that Xn = Xn0 for all n ≥ n0 . 1.113 Let G be a group. Suppose that there exists a normal subgroup H ⊂ G which is of finite Markov type and such that K := G/H is infinite cyclic. The purpose of this exercise is to show that G is itself of finite Markov type. Let A be a finite group and let X ⊂ AG be a group subshift. Let ρ : G → K denote the canonical quotient homomorphism and let t ∈ G such that ρ(t) generates K. (a) Show that for every n ∈ N the set Yn := {x|H : x ∈ X such that x(ht i )

.

= 1A for all h ∈ H and − n ≤ i ≤ −1} ⊂ AH

(1.39)

Exercises

59

is a group subshift of AH . (b) Show that Y0 ⊃ Y1 ⊃ · · · ⊃ Yn ⊃ Yn+1 ⊃ · · · and that the sequence (Yn )n∈N stabilizes, that is, there exists n0 ∈ N such that Yn = Yn0 for all n ≥ n0 . j (c) For i, j ∈ Z with i ≤ j , set Si := {t n : i ≤ n ≤ j } ⊂ G. Show that j +1

j

tH Si = H Si+1 .

.

(1.40) j

(d) Given i, j ∈ Z with i ≤ j , let XH S j := {x|H S j : x ∈ X} ⊂ AH Si . Let n0 ∈ N i i be the integer provided by (b) and, for m ∈ N such that m ≥ n0 , set m m X(m) := {x ∈ AG : (gx)|H S−n ∈ XH S−n for all g ∈ G} ⊂ AG .

.

0

0

(1.41)

Let m ∈ N such that m ≥ n0 and let z ∈ X(m). Prove successively the following results: m m ; (1) there exists x0 ∈ X such that x0 |H S−n = z|H S−n 0

0

m m ; (2) there exists x1 ∈ X such that x1 |H S−n = (t −1 z)|H S−n 0 0 := (3) the configuration x2 tx1 is in X and satisfies that x2 |H S m+1

−n0 +1

z|H S m+1 ;

=

−n0 +1

(4) the configuration x3 := x0−1 x2 is in X and satisfies that x3 (g) = 1A for all m g ∈ H S−n ; 0 +1 (5) the configuration y := (t −(m+1) x3 )|H ∈ AH is in Yn0 ; (6) the set L(y) := {x ∈ X : x|H = y and x(ht i ) = 1A for all h ∈ H and i ≤ −1} ⊂ X

.

is nonempty; (7) given x4 ∈ L(y), the configuration x5 := x0 · (t m+1 x4 )−1 is in X and satisfies that x5 |H S m+1 = z|H S m+1 ; −n0

a. X(m) = X(m + 1).

−n0

(e) Let z0 ∈ X(n0 ). Show that, given a finite subset F ⊂ G, there exists x ∈ X such that x|F = z0 |F . (f) Show that X = X(n0 ). n0 (g) Set S := S−n and consider the finite group B := AS . Show that the map 0 Θ : AG → B H defined by setting Θ(x)(h)(s) := x(hs) for all x ∈ AG , h ∈ H , and s ∈ S is an H -equivariant continuous group homomorphism. (h) Show that Z := Θ(X) ⊂ B H is a group subshift and that for x ∈ AG one has x ∈ X if and only if Θ(gx) ∈ Z for all g ∈ G. (i) Show that X is of finite type. (j) Show that the group G is of finite Markov type. 1.114 Show that every finitely generated abelian group is of finite Markov type.

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 .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3_2

61

62

2 Residually Finite Groups

Proposition 2.1.2 Let G be a group. Then the following conditions are equivalent: (a) G is residually finite; (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). u n 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 := n u 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. u n 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) . u n 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 . u n The preceding lemma immediately yields the following result. Proposition 2.1.8 A nontrivial divisible group cannot be residually finite.

O

2.1 Definition and First Examples

63

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 := n −1 g∈G gH g . Then K is a normal subgroup of G contained in H . Moreover, if H is of finite index in G, then Kis of finite index in G. Proof Since gH g −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 gH g −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. u n 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 gH g −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. u n

64

2 Residually Finite Groups

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. u n 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 ' := φ ◦ π , where π : G → Gi0 is the projection onto Gi0 . We have φ ' (g) = φ(gi0 ) /= 1F . Consequently, G is residually finite. u n Corollary 2.2.3 O 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||Propositions 2.2.1 and 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 . u n 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 Propositions 2.1.3 and 2.1.4, we deduce that G is residually finite. u n 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 Propositions 2.1.3, 2.2.2 and 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 H :=

""

.

g∈G\{1G }

Fg .

2.2 Stability Properties of Residually Finite Groups

65

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). n u 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 G = lim Gi . By construction of a projective limit (see Appendix E), G is a subgroup of ← − || the group i∈I Gi . We deduce that G is residually finite by using Propositions 2.2.2 and 2.2.1. u n 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. 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] < ∞. u n Proposition 2.2.12 Every virtually residually finite group is residually finite.

66

2 Residually Finite Groups

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. u n

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 .a := 01

)

(

and

10 b := 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 g11 t + g12 g21 t + g22

gt :=

.

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

.

a k 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].

One immediately checks that, for all .k ∈ Z \ {0}, one has a k 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 u n b generate a free group of rank 2.

2.4 Hopfian Groups

67

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

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 k appear in the reduced form of g for some .k ∈ Z \ {0}. .x ∈ X such that .x 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 u n finite.

2.4 Hopfian Groups Definition 2.4.1 A group G is called Hopfian if every surjective endomorphism of G is injective.

68

2 Residually Finite Groups

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. u n 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. u n 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. 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.

2.5 Automorphism Groups of Residually Finite Groups

69

(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. u n 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 n

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 (

)

n

α(H ) = α

Ker(ψ)

.

ψ∈Hom(G,F )

=

n

α(Ker(ψ))

ψ∈Hom(G,F )

=

n

ψ∈Hom(G,F )

Ker(ψ ◦ α −1 ).

70

2 Residually Finite Groups

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 ) =

n

.

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. u n 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.

O

.

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. .O 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 u n Proposition 2.2.1. Corollary 2.5.7 Let R be a ring. Suppose that R is finitely generated as a .Zmodule. 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 (R n ), where .R n is viewed as a left module over R. u n

.

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

2.6 Examples of Finitely Generated Groups Which Are Not Residually Finite

71

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). u n 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. 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. u n 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)

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2 Residually Finite Groups

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 . u n 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) (Theorem C.4.3) implies + that .Ker(φ) ∩ Sym+ (Z) is either equal to .Sym (Z) or reduced to the identity. 0 0 + + We cannot have .Ker(φ) ∩ Sym+ 0 (Z) = Sym0 (Z), that is, .Sym0 (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 (Z) = {IdZ }. If .Ker(φ) is not reduced to the identity, it follows that 0 .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. u n 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 O .H := Hi , i∈Z

2.6 Examples of Finitely Generated Groups Which Are Not Residually Finite

73

where each .Hi is a copy of the alternating group .Sym+ 5 . Let .ψ : Z → Aut(H ) be the homomorphism defined by .ψ(n) := α n , 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 xψ 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' ) := (hα n (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) = ht n

(2.2)

t n ht −n = α n (h)

(2.3)

.

and .

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 = t i 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. 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 to .H0 and hence to + m m = 1 . Thus, it follows from (2.4) .Sym . Let .m := |F |. We have .φ(t ) = φ(t) F 5 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. u n .

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.

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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 G .A ; (c) there exists a set A having at least two elements such that the set of points of G which have a finite G-orbit is dense in .AG ; .A (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 G is faithful. .x(1G ) = a. Consequently, the action of G on .A u n 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. 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 of finite index .Ns ⊂ G such that .s ∈ / Ns . The set .K := ∩s∈S Ns is a normal subgroup of finite index in G by Lemma 2.5.2. Let .ρ : G → G/K be the canonical homomorphism. / K and hence .ρ(g) /= ρ(h). n u If g and h are distinct elements in .Ω, then .g −1 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.

.

Notes

75

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 Propositions 1.2.1, 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 u n 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. An example of a finitely presented group which is residually finite but not linear was recently given by C. Dru¸tu and M. Sapir [DruS]. 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) := , that is, the quotient of the free group .F2 based on two generators x and y by the smallest normal subgroup

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of .F2 containing the element .x −3 yx 2 y −1 (see [BauS], [MaKS], [LS]). On the other hand, the Baumslag-Solitar group .BS(2, 4) := is an example of a finitely presented Hopfian group which is not residually finite (the incorrect statement in [BauS] 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 xψ 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]). In [CeC20], it is shown that if M is a monoid and A is a finite set with more than one element then the residual finiteness of M is equivalent to that of the monoid .CA(M; A) consisting of all cellular automata with universe M and alphabet A.

Exercises 2.1 (The Residual Subgroup) Let G be a group and let G0 ⊂ G denote its residual subgroup. Recall that G0 is defined as being the intersection of all the finite index subgroups of G and that G0 is a normal subgroup of G since it is also equal to the intersection of all the normal subgroups of finite index of G (cf. Proposition 2.1.11.(ii)). (a) Show that the group G/G0 is residually finite. (b) Let φ : G → G' be a group homomorphism from G to a residually finite group G' . Show that G0 ⊂ ker(φ). (c) Suppose that an element x ∈ G satisfies the following condition: for every integer n ≥ 1, there exists y ∈ G such that x = y n . Show that x belongs to the residual subgroup of G. (d) Show that if G is a divisible group then G0 = G. (e) Show that the following conditions are equivalent: (1) G0 = G, (2) G contains no proper subgroup of finite index, (3) G contains no proper normal subgroup of finite index, (4) all finite quotients of G are trivial, (5) all residually finite quotients of G are trivial.

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77

2.2 (Maximal Normal Subgroups) One says that a subgroup H of a group G is a maximal normal subgroup of G if H is a proper normal subgroup of G and every proper normal subgroup K of G such that H ⊂ K satisfies H = K. (a) Let G be a group and let H be a maximal normal subgroup of G. Show that the quotient G/H is a non-trivial simple group. (b) Show that every non-trivial finitely generated group admits a maximal normal subgroup. (c) Let G be a non-trivial finitely generated group admitting no proper subgroups of finite index. Show that G admits an infinite simple quotient group. (d) Show that the additive group Q has no maximal normal subgroups. 2.3 Show that every quotient of a divisible group is a divisible group. 2.4 Recall that a group G is said to be torsion-free if 1G is the unique element of finite order in G. Show that every torsion-free divisible abelian group G is isomorphic to a direct sum of copies of Q. 2.5 (Profinite Topology) 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 satisfying 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) Let g ∈ G and let H be a finite index subgroup of G. Show that the set gH is both open and closed in G for the profinite topology. (c) Show that G with its profinite topology is a topological group. (d) Show that the group G is finite if and only if the profinite topology on G is discrete. (e) Show that G is residually finite if and only if the profinite topology on G is Hausdorff. 2.6 Let α, β ∈ C such that |α|, |β| ≥ 2. Consider the matrices A, B ∈ SL2 (C) defined by ( A :=

.

1α 01

)

( and

B :=

) 10 . β1

Show that A and B generate a free subgroup of rank 2 of SL2 (C). 2.7 (Residual Finiteness of Free Groups) The goal of this exercise is to present an alternative proof of the fact that all free groups are residually finite (cf. Theorem 2.3.1). Let X be a set and let G denote the free group based on X. Let g ∈ G with g /= 1G and let g = an · · · a2 a1

.

denote the reduced form of G. Recall that one has n ≥ 1, ai ∈ X ∪ X−1 for all i ∈ {1, . . . , n}, and ai ai−1 /= 1G for all i ∈ {2, . . . , n}.

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For each x ∈ X, define the subsets Ex+ , Ex− ⊂ {1, . . . , n} by Ex+ := {i : ai = x} and Ex− := {i : ai = x −1 }.

.

Define also the subsets Fx+ , Fx− ⊂ {2, . . . , n + 1} by Fx+ := {i + 1 : i ∈ Ex+ } and Fx− := {i + 1 : i ∈ Ex− }.

.

(a) Show that Ex+ ∩ Fx− = Fx+ ∩ Ex− = ∅.

.

for all x ∈ X. (b) Let x ∈ X. Show that there exists a permutation σx ∈ Symn+1 such that σx (i) = i + 1 for all i ∈ Ex+ and σx−1 (i) = i + 1 for all i ∈ Ex− . (c) Show that there exists a group homomorphism φ : G → Symn+1 such that φ(x) = σx for all x ∈ X. (d) Compute φ(g)(1) and conclude. 2.8 Show that the class of residually finite groups is not closed under taking quotients. 2.9 Let X be an infinite set. Show that the symmetric group Sym(X) is not residually finite. 2.10 Let G be a residually finite group and let A be a finite set with more than one element. Let ICA(G; A) denote the group consisting of all reversible cellular automata τ : AG → AG (cf. Sect. 1.10). Show that the group ICA(G; A) is residually finite if and only if G is residually finite. 2.11 (Residually Finite Rings) A ring R is said to be residually finite if for every non-zero element r ∈ R, there exist a finite ring F and a ring homomorphism φ : R → F such that φ(r) /= 0F . (a) (b) (c) (d) (e)

Show that every finite ring is residually finite. Show that the ring Z is residually finite. Show that any subring of a residually finite ring is itself residually finite. Show that every product of residually finite rings is itself residually finite. Show that every ring which is the limit of a projective system of residually finite rings is residually finite. (f) Let p be a prime number. Show that the ring Zp of p-adic integers is residually finite. (g) Let R be a ring. Show that the following conditions are equivalent: (1) the ring R is residually finite; (2) given any two distinct elements r, s ∈ R, there exist a finite ring F and a ring homomorphism φ : R → F such that φ(r) /= φ(s); (3) given any finite subset Ω ⊂ R, there exist a finite ring F and a ring

Exercises

79

homomorphism φ : R → F such that the restriction of φ to Ω is injective; (4) the intersection of all the finite index two-sided ideals of R is reduced to the zero element of R. (h) Let R be a residually finite ring. Let U (R) denote the set consisting of all invertible elements of R. Show that the additive group R and the multiplicative group U (R) are both residually finite. (i) Let R be a residually finite ring and let n ≥ 1 be an integer. Show that the matrix ring Matn (R) is residually finite. (j) Let R be a residually finite ring and let n ≥ 1 be an integer. Show that the group GLn (R) is residually finite. 2.12 (Residual Finiteness of the Ring of n-Adic Rationals) Let n ≥ 2 be an integer. Let Z[1/n] denote the ring of n-adic rationals, i.e., the subring of Q consisting of all r ∈ Q that can be written in the form r = knm for some k, m ∈ Z. (a) Show that the ring Z[1/n] is residually finite. (b) Show that the additive group Z[1/n] is residually finite and Hopfian but not finitely generated. 2.13 (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 := . The group G is the Baumslag-Solitar group BS(1, m). (a) Show that every element g ∈ G may be written in the form g = a i bj a k for some i, j, k ∈ Z such that i ≤ 0 and k ≥ 0. (b) Show that there is a unique group homomorphism ρ : G → GL2 (Q) that satisfies ( ) ( ) m0 11 .ρ(a) = and ρ(b) = . 0 1 01 (c) Show that ρ is injective and that the image of ρ is the subgroup H ⊂ GL2 (Q) given by {( H :−

.

(d) (e) (f) (g) (h) (i)

) } mn r : n ∈ Z, r ∈ Z[1/m] , 0 1

(2.5)

where Z[1/m] denotes the ring of m-adic rationals, i.e., the subring of Q consisting of all rationals r ∈ Q which can be written in the form r = umv for some u, v ∈ Z (cf. Exercise 2.12). Show that G is not abelian. Show that G is torsion-free. Show that there is a short exact sequence of groups 0 → Z[1/m] → G → Z → 0. Show that G is residually finite. Show that G is Hopfian. Show that G is not of finite Markov type.

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2 Residually Finite Groups

2.14 (The Baumslag-Solitar Group BS(2, 3)) Consider the group G given by the presentation G := . Thus, G is the quotient group G = F /N, where F is the free group based on two generators x, y and N is the normal closure of xy 2 x −1 y −3 in F . Let ρ : F → G denote the quotient group homomorphism. The group G is the Baumslag-Solitar group BS(2, 3). (a) Consider the elements a, b ∈ G defined by a := ρ(x) and b := ρ(y). Show that every element g ∈ G may be uniquely written in the form g = bη0 a ε1 bη1 a ε2 bη2 · · · a εn bηn ,

.

(2.6)

where n ∈ N, η0 ∈ Z, εi ∈ {−1, 1} and ηi ∈ {−1, 0, 1}, i = 1, 2, . . . , n, are subject to the conditions (i) if εi = 1 for some 1 ≤ i ≤ n, then ηi /= −1; (ii) if ηi = 0 for some 1 ≤ i ≤ n − 1, then εi /= −εi+1 . (b) Show that there is a unique group endomorphism φ : G → G such that φ(a) = a and φ(b) = b2 . (c) Show that φ is surjective. (d) Show that the element g ∈ G given by g := b−1 (aba −1 b−1 )2 is in the kernel of φ. (e) Show that G is not Hopfian. (f) Show that G is not residually finite. 2.15 Let G be a group and let H be a normal subgroup of G. (a) Suppose that there exists a surjective group homomorphism ϕ : G → G such that ϕ(H ) ⊂ H and (G \ H ) ∩ ϕ −1 (H ) /= ∅. Show that the group G/H is not Hopfian. (b) Suppose that there exists a group automorphism α ∈ Aut(G) such that α(H ) C H . Show that the group G/H is not Hopfian. 2.16 Let G be a Hopfian group and let H be a non-trivial group. Show that the groups G and G × H are not isomorphic. 2.17 Show that all elements of the additive group Q/Z have finite order but that Q/Z is not residually finite. 2.18 Show that the multiplicative group Q∗ of nonzero rational numbers is residually finite. 2.19 Show that the automorphism group Aut(Q) of the additive group Q is isomorphic to the multiplicative group Q∗ . 2.20 Let G be a group with trivial center and suppose that the automorphism group of G is residually finite. Show that G is residually finite. 2.21 Let E be an infinite-dimensional vector space over a field K. Show that the additive group E is not Hopfian.

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81

2.22 Let K be a field of characteristic p > 0. (a) Show that the additive group K is residually finite. (b) Show that the additive group K is not Hopfian unless K is finite. 2.23 Show that the additive group R is neither residually finite nor Hopfian. 2.24 Show that the multiplicative group R∗ of nonzero real numbers is neither residually finite nor Hopfian. 2.25 Let p be a prime number. Show that the additive group Zp of p-adic integers is Hopfian. 2.26 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 to prove that |S| = n and that S is a base for F . 2.27 Let F be a free group of finite rank n. Suppose that S is a finite generating subset of F . Show that S has cardinality at least n. 2.28 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. 2.29 (Characteristic Subgroups) Let G be a group. A subgroup C ⊂ G is said to be characteristic if it is invariant under all automorphisms of G, that is, α(C) = C for all α ∈ Aut(G). (a) Show that every characteristic subgroup of G is normal in G. (b) Find an example of a group G containing a subgroup H that is normal but not characteristic in G. (c) Let G be a group. Show that the center, the derived subgroup, and the residual subgroup of G are all characteristic subgroups of G. (d) Let G be a finitely generated group and let H ⊂ G be a finite index subgroup. Show that there exists a finite index characteristic subgroup C of G such that C ⊂ H. (e) Find an example of a countable group G containing a finite index subgroup H such that there is no finite index characteristic subgroup C of G satisfying C ⊂ H. 2.30 (External Semidirect Product) Let H and K be two groups and let α : K → Aut(H ) be a group homomorphism from K into the automorphism group of H . Equip the set G := H × K with the binary operation defined by (h, k) · (h' , k ' ) := (hα(k)(h' ), kk ' )

.

for all h, h' ∈ H and k, k ' ∈ K. (a) Show that G is a group. (b) Show that the maps ι : H → G and t : K → G, given by ι(h) := (h, 1K ) for all h ∈ H and t (k) := (1H , k) for all k ∈ K, are group monomorphisms and

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that the map π : G → K, given by π(h, k) := k for all h ∈ H and k ∈ K, is a group epimorphism. (c) Show that ι

π

1 → H→ G → K → 1

.

(2.7)

is a short exact sequence of groups and that the group homomorphism t : K → G satisfies π ◦ t = IdK . (d) Show that the following holds: (i) ι(H ) is a normal subgroup of G and the quotient group G/ι(H ) is canonically isomorphic to K, (ii) G = ι(H )t (K), (iii) ι(H ) ∩ t (K) = {1G }. (e) Check that t (k)ι(h)t (k)−1 = ι(α(k)(h)) for all h ∈ H and k ∈ K. 2.31 (Split Exact Sequences) An exact sequence of groups ι

π

1 → H→ G → K → 1

.

is said to be split provided that there exists a group homomorphism t : K → G such that π ◦ t = IdK . Two exact sequences of groups ι

π

ι'

π'

1 → H→ G → K → 1

.

(2.8)

and 1 → H → G' → K → 1

.

(2.9)

are said to be equivalent provided that there exists a group isomorphism ϕ : G → G' such that ι' = ϕ ◦ ι

and

.

π = π ' ◦ ϕ,

that is, the diagram G ι

H

is commutative.

K

ϕ ι

.

π

G

π

(2.10)

Exercises

83

(a) Suppose that two short exact sequences of groups are equivalent. Show that if one is split so is the other. (b) Let H xα K be the semidirect product of two groups H and K with respect to a group homomorphism α : K → Aut(H ). Consider the maps ια : H → H xα K defined by ια (h) := (h, 1K ) for all h ∈ H and the map πα : H xα K → K defined by πα (h, k) := k for all (h, k) ∈ H xα K. Show that ια

πα

1 → H → H xα K → K → 1

.

(2.11)

is a split exact sequence of groups. (c) Let 1 → H → G → K → 1 be an exact sequence of groups. Show that the following conditions are equivalent: (i) the exact sequence 1 → H → G → K → 1 is split; (ii) there exists a group homomorphism α : K → Aut(H ) such that the exact sequence 1 → H → G → K → 1 is equivalent to the exact sequence (2.11). (d) Let K be a free group. Show that every exact sequence of groups 1 → H → G → K → 1 is split. 2.32 (Internal Semidirect Product) Let G be a group. Let H and K be subgroups of G and denote by ι : H → G and t : K → G the inclusion maps. (a) Show that the following conditions are equivalent. (i) H is normal in G, G = H K, and H ∩ K = {1G }; (ii) H is normal in G and every element g ∈ G can be uniquely written in the form g = hk with h ∈ H and k ∈ K; (iii) there is a short exact sequence of groups ι

π

1 → H → G → K → 1,

.

(2.12)

such that π ◦ t = IdK . (b) Suppose that one (and hence any) of the equivalent conditions in (a) is satisfied. Let α : K → Aut(H ) denote the group homomorphism associated with the action of K on H through conjugation, i.e., α(k)(h) = khk −1 for all k ∈ K and h ∈ H (this is well defined since H is normal in G). Show that the map ϕ : H xα K → G defined by ϕ((h, k)) := hk for all h ∈ H and k ∈ K, is a group isomorphism. 2.33 (Residual Finiteness of Semidirect Products) Let H and K be two groups and let α : K → Aut(H ) be a group homomorphism from H into the automorphism group of K. Consider the semidirect product G := H xα K. One can regard H and K as subgroups of G via the group embeddings ι : H → G and t : K → G introduced in Exercise 2.30. Every element g ∈ G can be uniquely written in the form g = hk with h ∈ H and k ∈ K, and the map π : G → K defined by π(g) := k is a group

84

2 Residually Finite Groups

epimorphism with kernel H . Moreover, one has khk −1 = α(k)(h) for all h ∈ H and k ∈ K. Suppose that the groups H and K are residually finite and that the group H is finitely generated. The goal of the exercise is to show that G is residually finite. (a) Let g ∈ G \ H . Show that there is a finite group F and a group homomorphism φ : G → F such that φ(g) /= 1F . (b) Let L be a finite index subgroup of H . Show that there exists a characteristic subgroup C of H which is of finite index in H such that C ⊂ L. (c) Show that α induces a group homomorphism α : K → Aut(H /C). (d) Let N denote the kernel of α . Show that N is a finite index normal subgroup of K and that α induces a group homomorphism β : K/N → Aut(H /C). (e) Consider the semidirect product F := (H /C) xβ (K/N). Show that the group F is finite and that the map φ : G → F , defined by φ(g) := (hC, kN) for all g = hk ∈ G, h ∈ H and k ∈ K, is a group homomorphism. (f) Use (a)–(e) to show that the group G is residually finite. 2.34 Let G be a group. Suppose that G contains a finitely generated normal subgroup H such that H is residually finite and G/H is free. Show that G is residually finite. 2.35 Let G be a group. Suppose that G contains a finitely generated normal subgroup H such that H is residually finite and G/H is (finite or infinite) cyclic. Show that G is residually finite. 2.36 (Wreath Product) Let K and H be two groups. Consider the group D := O of K indexed by H . In h∈H Kh , the direct sum of a family (Kh )h∈H of copies || other words, D is the subgroup of the product group h∈H K = K H = {x : H → K} consisting of all d : H → K such that d(h) = 1K for all but finitely many h ∈ H . Let σ : H → Aut(D) denote the group homomorphism associated with the restriction to D ⊂ K H of the shift action of H on K H . This means that, given h ∈ H and d ∈ D, the element d ' := σ (h)(d) ∈ D is given by the formula d ' (h' ) = d(h−1 h' ) for all h' ∈ H . The semidirect product G := D xσ H associated with σ (cf. Exercise 2.33) is called the wreath product of the groups K and H and is denoted by K | H . Since G is a semidirect product of D and H , the groups D and H can be regarded as subgroups of G. We also regard K as a subgroup of D (and hence of G) by using the identification of K1H with K. (a) Show that hKh−1 = Kh for all h ∈ H . (b) Let d ∈ D. Show that d can be uniquely written in the form d=

""

.

hkh h−1 ,

h∈S

where S ⊂ H is a finite subset and 1K /= kh ∈ K for all h ∈ S. (c) Show that G is generated by K ∪ H . (d) Show that G is finitely generated whenever K and H are both finitely generated.

Exercises

85

(e) Suppose that K is residually finite and that H is finite. Show that G is residually finite. (f) Suppose that K is abelian. Let H ' be a group and let ϕ : H → H ' be a group homomorphism. Let G' := K | H ' . Show that ϕ uniquely extends to a group homomorphism Φ : G → G' fixing every element of K. (g) Suppose that K and H are residually finite and that K is abelian. Show that G is residually finite. (h) Suppose that K is non-abelian and that H is infinite. Show that G is not residually finite. (i) Show that the group G is residually finite if and only if the following two conditions are both satisfied: (1) the groups K and H are both residually finite, (2) the group K is abelian or the group H is finite. (j) Suppose that H is infinite. Show that if the group K is non-trivial then G is not of finite Markov type. 2.37 Let n ≥ 1 be an integer. All matrices considered here are n × n matrices with entries in Z. An elementary row operation on matrices is a transformation of one of the following types: Type 1: Type 2: Type 3: Type 4:

multiplication of row i by −1, operation denoted by Ri ← −Ri ; interchange of rows i and j , operation denoted by Ri ↔ Rj ; addition of row j to row i, operation denoted by Ri ← Ri + Rj ; subtraction of row j to row i, operation denoted by Ri ← Ri − Rj .

Here 1 ≤ i, j ≤ n and i /= j . The elementary column operations Ci ← −Ci , Ci ↔ Cj , Ci ← Ci + Cj , and Ci ← Ci − Cj are defined analogously. A matrix is called an elementary matrix if it can be obtained from the identity matrix In by applying an elementary row operation. One says that a matrix A is equivalent to a matrix B, and one writes A ∼ B, if B can be obtained from A by performing a finite sequence of elementary row and column operations. (a) Show that the transpose of an elementary matrix is an elementary matrix. (b) Show that every elementary matrix is in GLn (Z) and that its inverse is an elementary matrix. (c) Show that the matrix obtained from a matrix A by applying an elementary row (resp. column) operation T is of the form T (A) = EA (resp. T (A) = AE), where E is an elementary matrix. (d) Show that ∼ is an equivalence relation. (e) Let A ∈ GLn (Z). Show that every matrix that is equivalent to A is in GLn (Z). (f) Let A ∈ GLn (Z) and let (a1 a2 · · · an ) be the first row of A. Show that gcd(a1 , a2 , . . . , an ) = 1. (g) Let A ∈ GLn (Z). Show that A is equivalent to a matrix whose first row is (1 0 0 · · · 0). (h) Let A be an n × n matrix with entries in Z. Show that the following conditions are equivalent:

86

2 Residually Finite Groups

(C1) A ∈ GLn (Z); (C2) A is equivalent to In ; (C3) A is a finite product of elementary matrices. (i) Show that the group GLn (Z) is finitely generated. (j) Show that the group SLn (Z) is finitely generated. 2.38 Let n ≥ 1 be an integer. Show that the group GLn (Z) is Hopfian. 2.39 Let A be a finitely generated abelian group and let G := Aut(A) denote its automorphism group. (a) Prove that G is residually finite. (b) Prove that G is finitely generated. (c) Prove that G is Hopfian. 2.40 Let G denote the subgroup of Sym(Z) generated by the translation T : n |→ n + 1 and the transposition S = (0 1) (thus G is the group G1 considered in Sect. 2.6). Let H be a non-trivial normal subgroup of G. Show that the group H is not residually finite. 2.41 Show that the wreath product G := Sym+ 5 |Z is a Hopfian group. 2.42 (Profinite Completion of a Group) 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 forms 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 kernel (c) Show that there is a canonical homomorphism η : G → G of η is the residual subgroup of G. (d) Prove that G is residually finite if and only if the canonical homomorphism - is injective. η: G → G 2.43 (The Transfer Homomorphism) Let G be a group and let H := Z(G) denote its center. Suppose that H has finite index n in G. Let T ⊂ G be a complete set of representatives for the cosets of H in G. Thus, for each g ∈ G, there exists a unique pair (θ (g), η(g)) ∈ T × H such that g = θ (g)η(g). (a) Let g ∈ G. Show that the map σg : T → T , defined by σg (t) := θ (gt) for all t ∈ T , is bijective. (b) Define a map ϕ : G → H by setting ϕ(g) :=

""

.

t∈T

η(gt)

(2.13)

Exercises

87

for all g ∈ G (observe that ϕ(g) does not depend on the order of the factors in the right-hand side of (2.13) since H is abelian). Show that ϕ is a group homomorphism. (c) Show that ϕ(g) = g n for all g ∈ G. (d) Show that (xy)n = x n y n for all x, y ∈ G. 2.44 (FC-Groups) Let G be a group. Denote by Z(G) its center and by D(G) its derived subgroup. One says that G is an FC-group if every conjugacy class of G is finite. (a) (b) (c) (d) (e)

Show that if D(G) is finite then G is an FC-group. Show that if Z(G) has finite index in G then G is an FC-group. Show that if G is an FC-group then G/Z(G) is residually finite. Show that if G is a finitely generated FC-group then G/Z(G) is finite. Show that if G is a finitely generated FC-group then there exists an integer d ≥ 0 such that G contains a finite index subgroup isomorphic to Zd .

2.45 Let G be an FC-group. (a) Let x, y ∈ G. Let H denote the subgroup of G generated by x and y. Show that the center of H has finite index in H . (b) Deduce from (a) and Exercise 2.43 that every torsion-free FC-group is abelian. (c) Show that the subset T ⊂ G, consisting of all elements of G with finite order, is a characteristic (and therefore normal) subgroup of G. (d) Show that G/T is a torsion-free abelian group. 2.46 Let G be a group. Denote by Δ the subset of G consisting of all elements whose conjugacy class is finite. (a) Show that Δ is a characteristic (and hence normal) subgroup of G. (b) Show that Δ is an FC-group. (c) Show that if G is torsion-free then Δ is abelian. 2.47 (ICC-Groups) One says that a group G has the infinite conjugacy class property (ICC-property for short) or that G is an ICC-group for short, if every conjugacy class of G except {1G } is infinite. (a) Let G be an ICC-group. Show that the center of G is trivial. (b) Let G be an ICC-group. Show that every non-trivial normal subgroup of G is infinite. (c) Let X be an infinite set and let Sym0 (X) denote the group consisting of all permutations of X with finite support. Show that Sym0 (X) is an ICC-group. (d) Show that every non-abelian free group is an ICC-group. 2.48 Let G be a finitely generated group and let φ : G → G be a group endomorphism of G. Let F be a finite group and, for n ∈ N, denote by Rn the set consisting of all group homomorphisms f : φ n (G) → F . (a) Show that the set Rn is finite.

88

2 Residually Finite Groups

(b) Let ψn : φ n (G) → φ n+1 (G) denote the group homomorphism obtained by restriction of φ. Show that the map ρn : Rn+1 → Rn , defined by ρn (f ) := f ◦ψn for all f ∈ Rn+1 , is injective. (c) Show that the sequence (|Rn |)n∈N is eventually constant. (d) Show that ρn is bijective for n large enough. 2.49 Let G be a finitely generated residually finite group and let φ : G → G be a group endomorphism such that φ(G) has finite index in G. (a) For n ∈ N, n let Hn := φ n (G). Show that Hn+1 has finite index in Hn . (b) Let Nn := g∈Hn gHn+1 g −1 . Show that Nn is a finite index normal subgroup of Hn . (c) Let ψn : Hn → Hn+1 denote the group homomorphism obtained by restriction of α. Show that ψn (Nn ) ⊂ Nn+1 . (d) Show that ψn induces a surjective quotient group homomorphism θn : Hn /Nn → Hn+1 /Nn+1 . (e) Show that there exists n0 ∈ N such that θn is a group isomorphism for all n ≥ n0 . (f) Let Kn := Hn ∩ ker(φ). Show that Kn = Kn0 for all n ≥ n0 . (g) Show that the restriction of φ to Hn0 is injective. (h) Show that the kernel of φ is finite. (i) Suppose that G contains no finite non-trivial normal subgroups (e.g., G is torsion-free). Show that φ is injective.

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 Gequivariant continuous map .f : AG → AG is surjective. Proposition 3.1.3 Every finite group is surjunctive. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3_3

89

90

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. u n

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. u n 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). u n 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 u n 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 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

3.3 Surjunctivity of Locally Residually Finite Groups

91

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. u n 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: 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.

92

3 Surjunctive Groups

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 .

u n

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. u n Corollary 3.3.5 Every free group is surjunctive. Proof Free groups are residually finite by Theorem 2.3.1.

u n

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. u n 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. u n 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.

3.4 Marked Groups

93

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

Γ

φ∼ = 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 , 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 },

.

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3 Surjunctive Groups

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) .γ αγ −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)(πγ αγ −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 ). u n

.

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 . 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 || .1Γ partially ordered by inclusion. For .i ∈ I we denote by .φi := γ ∈i φγ : Γ → || ' ' := (φγ (γ ))γ ∈I for all .γ ' ∈ Γ . Set γ ∈i Γγ the homomorphism defined by .φi (γ ) := .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 .

3.4 Marked Groups

95

iF ⊂ i. Then .φi (f ) /= 1Γi since .φf (f ) /= 1Γf . Thus

.

f ∈ / Ni and f ∈ / {1Γ } ∩ F.

.

(3.1)

It follows that if .1Γ ∈ / F then .Ni ∩ F = ∅ = {1Γ } ∩ F . On the other hand, if 1Γ ∈ F , from (3.1) we deduce .Ni ∩ F = {1Γ } = {1Γ } ∩ F . In either cases we have .Ni ∩ F = {1Γ } ∩ F . We deduce that .limi Ni = {1Γ }. Finally, by our 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 .Γ /NiF satisfy .P. / NiF . In other words, if .φF : Γ → Γ /NiF As .{1Γ } ∩ F = ∅ we have that .γ ∈ denotes the canonical epimorphism, we have .φF (γ ) /= 1Γ /NiF . We deduce that .Γ is residually .P. This shows (b) .⇒ (a). u n .

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 : γ x = 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 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)

96

3 Surjunctive Groups

where .πF : AΓ → AF is the projection map. Consider now the finite subset .E ⊂ Γ defined by E := F · F −1 = {γ η−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 −1 x(1 ) = x(1 ) = a. This implies .N ⊂ N . By symmetry, we also .x(ν) = ν Γ Γ 2 1 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. u n .

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) := (γ x, γ y) 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 |→ γ x, is uniformly continuous for each .γ ∈ Γ . This is 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 n . 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 .(γ x, γ y) ∈ 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.

3.6 Gromov’s Injectivity Lemma

97

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(γ x, γ y) < 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 )}.

.

−1 W if and only if .x(g −1 ) = y(g −1 ). Thus Given .g ∈ G, we have .(x, y) ∈ g 0 n −1 G G . u n g∈G g W0 is equal to the diagonal in .A × A .

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 . 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 .)

.

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3 Surjunctive Groups

Proof By expansivity of the action of .Γ , there is an entourage .W0 of X such that n .

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 f (γ z' ) = f (γ z'' )

.

(3.11)

for all .γ ∈ Γ . As the points .γ z' and .γ z'' stay in Z, the fact that .Z ⊂ V [Y ] implies that there are points .yγ' and .yγ'' in Y such that .(γ z' , yγ' ) ∈ V and .(γ z'' , yγ'' ) ∈ V . Since .V ⊂ E, it follows from (3.10) that .(f (γ z' ), f (yγ' )) and .(f (γ z'' ), f (yγ'' )) are in U . As U is symmetric, we also have .(f (yγ' ), f (γ z' )) ∈ 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 .(γ z , yγ ) ∈ S and .(yγ , γ z ) ∈ S since .V ⊂ S and S is symmetric. It follows that (γ z' , γ z'' ) ∈ S ◦ S ◦ S ⊂ W0

.

3.7 Closedness of Marked Surjunctive Groups

99

by (3.7). This gives us (z' , z'' ) ∈

n

.

γ −1 W0 ,

γ ∈Γ

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

u n

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 π : AS → AS Γ denote the bijective map induced by ψ. Consider the cellular automaton τ: A → AΓ over Γ with memory set S and local defining map μ := μ ◦ π −1 : AS → A. To τ , Y := Fix(N ) and Zi := Fix(Ni ). simplify notation, let us set X := AΓ , f := 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 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. u n

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3 Surjunctive Groups

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). The topology of the space .N (Γ ) of .Γ -marked groups is often called the Cayley topology. It was first used by Grigorchuk [Gri4] to show the existence of an uncountable family of groups with pairwise inequivalent growth functions. See also [Cha]. 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 Section 4 of [Gro5] (see also [CeC10]). There is another proof based on techniques from model theory (see [Gro5] and [GleG]). A topological dynamical system is a pair .(X, G) consisting of a topological space X, called the phase space, and a group G acting by homeomorphisms of X. An endomorphism of a topological dynamical system .(X, G) is a continuous Gequivariant self-mapping .f : X → X. One says that a topological dynamical system .(X, G) is surjunctive if every injective endomorphism of .(X, G) is surjective. Given a group G and a set A, the pair .(AG , G), where .AG is equipped with the prodiscrete topology and the G-shift, is a topological dynamical system, called the shift dynamical system (associated with G and A). When A is finite, it follows from the Curtis-Hedlund-Lyndon theorem that the endomorphisms of .(AG , G) are exactly the cellular automata .τ : AG → AG . Thus, the group G is surjunctive if and only if the shift dynamical system .(AG , G) is surjunctive for every finite set A. In recent years, several surjunctivity results have been obtained for various classes of subshifts and other topological dynamical systems naturally arising from geometry or algebra (see [BhCC] in collaboration with S. Bhattacharya; [CKT] by S. Capobianco, J. Kari, and S. Taati; [CeC1], [CeC2], [CeC3], [CeC4], [CeC5], [CeC9], [CeC10], [CeC12], [CeC13], [CeC14], [CeC15], [CeC16], [CeC18], [CeC19], [CeC21], [CeC22], [CeC23], and [CeC24]; [CFS], [CCFS1], and

Exercises

101

[CCFS2] in collaboration with F. Fiorenzi, F. Scarabotti, and Z. Šuni´c; [CCL1] and [CCL2] in collaboration with H. Li; [CCP1], [CCP2], and [CCP3] in collaboration with X.K. Phung; [Dou] and [DouG] by M. Doucha and J. Gismatullian; [Fio1] and [Fio2] by Fiorenzi; [Li] by Li; and [Phu1], [Phu5], [Phu6], [Phu8], and [Phu9] by Phung). See the recent survey [CCP6, Section 9] for more details.

Exercises 3.1 Let X be a set and let f : X → X be an injective map. Suppose that there exists n ∈ N such that f n (X) = f n+1 (X). Show that f is surjective. 3.2 Let K be an algebraically closed field. Show that every injective polynomial map f : K → K is surjective. 3.3 Show that every injective polynomial map f : R → R is surjective. 3.4 Show that the polynomial map f : Q → Q defined by f (x) = x 3 is injective but not surjective. 3.5 Show that every injective holomorphic map f : C → C is surjective. 3.6 Give an example of a real analytic map f : R → R which is injective but not surjective. 3.7 Let K be a field and let V be a vector space over K. Show that V is finitedimensional if and only if every injective endomorphism f : V → V is surjective. 3.8 (Artinian Modules) 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 of submodules of M 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. 3.9 Let U := {z ∈ C : |z| = 1}. Show that every injective continuous map f : U → U is surjective. 3.10 Let n ≥ 0 be an integer. An n-dimensional topological manifold is a non-empty 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. 3.11 Let P be a property of groups. Show that every subgroup of a locally P group is itself locally P.

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3 Surjunctive Groups

3.12 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. 3.13 Show that the additive group Q is locally cyclic. 3.14 Show that every element of a locally finite group has finite order. 3.15 Let G be an abelian group. Show that G is locally finite if and only if every element of G has finite order. 3.16 Show that every subgroup and every quotient of a locally finite group is a locally finite group. 3.17 Let G be a group. Suppose that G contains a normal subgroup N such that both N and G/N are locally finite. Show that G is locally finite. 3.18 Let G be a group which is the limit of an inductive system of locally finite groups. Show that G is locally finite. 3.19 Show that the direct sum of any family of locally finite groups is a locally finite group. 3.20 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 U and K ∈ S then H K ∈ 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. 3.21 Let G be an infinite locally finite group and let A be a finite set. Let τ : AG → AG be a cellular automaton. Show that the following conditions are all equivalent: (i) (ii) (iii) (iv) (iv)

τ τ τ τ τ

is surjective; is injective; is finite-to-one; is countable-to-one; is bijective.

3.22 Let G be a locally finite group and let A be a set. Show that every bijective cellular automaton τ : AG → AG is invertible. 3.23 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. 3.24 Let G and A be two groups. Suppose that there is an element s0 ∈ G of infinite order and that the group A is non-trivial. Show that the map τ : AG → AG defined by τ (x)(g) := x(g)−1 x(gs0 ), for all x ∈ AG and g ∈ G, is a cellular automaton over the group G which is surjective but not injective. 3.25 Let X be an infinite set. Show that the symmetric group Sym(X) is not locally residually finite.

Exercises

103

3.26 Show that every virtually surjunctive group is surjunctive. 3.27 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 n 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. 3.28 Let X be a topological space and let f : X → X be a map such that f (X) is closedUin X. Suppose that there is a family (Xi )i∈I of subsets of X such that the set Y := i∈I Xi is dense in X and Xi ⊂ f (Xi ) for all i ∈ I . Show that f is surjective. 3.29 Let X be a set. Let (Ai )i∈I be a net of subsets of X. (a) Show that Un .

Aj ⊂

i j ≥i

nU

Aj .

(3.12)

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=

Un

.

i j ≥i

Aj =

nU

Aj .

(3.13)

i j ≥i

3.30 Let G be a group and let A be a set. Let H be a subgroup of G. In Exercise 1.103, we associated with each subshift X ⊂ AH the subshift X(G) ⊂ AG defined by X(G) := {x ∈ AG : xgH ∈ X for all g ∈ G},

.

where xgH := (gx)|H for all x ∈ AG and g ∈ G. Let σ : AH → AH 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). 3.31 (Surjunctive Subshifts) Let G be a group and let A be a set. One says that a subshift X ⊂ AG is surjunctive if every injective cellular automaton σ : X → X is surjective. (a) Show that every finite subshift X ⊂ AG is surjunctive. (b) Let A and B be finite sets. Suppose that X ⊂ AG and Y ⊂ B G are topologically conjugate subshifts. Show that X is surjunctive if and only if Y is surjunctive.

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3 Surjunctive Groups

3.32 Let G be a group and let A be a finite set. Let X ⊂ AG be a subshift and let Per(X) denote the set of all periodic configurations in X. (a) Let H be a subgroup of finite index of G and let XH := Fix(H ) ∩ X denote the set consisting of all configurations in X that are fixed by H . Show that XH is a finite set and that σ (XH ) ⊂ XH for every cellular automaton σ : X → X. (b) Deduce from (a) that if Per(X) is dense in X then X is surjunctive. 3.33 Let G be a group and let A be a set. Let X ⊂ AG be a subshift. Denote by P(X) ⊂ P(G, A) the set of (G, A)-patterns appearing in X and by Per(X) ⊂ X the set of its periodic configurations. Show that Per(X) is dense in X if and only if the following condition is satisfied: for every pattern p ∈ P(X), there exists a finite index subgroup H of G and an element y ∈ AH \G such that, denoting by ρH : G → H \G the canonical surjection g |→ H g, the configuration y ◦ ρH is in X and the pattern p appears in y ◦ ρH . 3.34 Let A be a set. Given a word w ∈ A∗ of length p ≥ 1, denote by w ∞ ∈ AZ the unique periodic configuration with period p such that w ∞ (0)w ∞ (1) · · · w ∞ (p − 1) = w.

.

(3.14)

Let X ⊂ AZ be a subshift. Denote by L(X) the language of X and by Per(X) the set of periodic configurations in X. (a) Let w ∈ A∗ . Show that one has w ∞ ∈ X if and only if w n ∈ L(X) for all n ∈ N. (b) Show that Per(X) is dense in X if and only if the following holds: (DP) for every word u ∈ L(X), there exists a non-empty word w ∈ A∗ such that u is a subword of w and w∞ ∈ X. 3.35 Let A be a finite set and let X ⊂ AZ be an irreducible subshift of finite type. Let Per(X) denote the set of periodic configurations in X. (a) Show that Per(X) is dense in X. (b) Show that X is surjunctive. (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 Y is topologically transitive and that Per(Y ) is not dense in Y . (d) Show that the subshift Y is surjunctive. 3.36 Show that the golden mean subshift is surjunctive. 3.37 Let X ⊂ {0, 1}Z denote the even subshift. (a) Show that X contains a dense set of periodic configurations. (b) Show that X is surjunctive.

Exercises

105

3.38 Let A := {0, 1, 2} and consider the subshift of finite type X ⊂ AZ with defining set of forbidden words {02, 10, 20, 21}. (a) Show that X is neither strongly irreducible, nor topologically mixing, nor irreducible, nor topologically transitive. (b) Show that X is not surjunctive. 3.39 Let A := {0, 1} and consider the subshift of finite type X ⊂ AZ with defining set of forbidden words {100, 110}. (a) Show that X is neither topologically transitive, nor irreducible, nor topologically mixing, nor strongly irreducible. (b) Show that X is not surjunctive. 3.40 Let G be a group and let A be a finite set. Suppose that G contains an element of infinite order and that A has more than one element. (a) Show that there exists a subshift of finite type X ⊂ AG which is not surjunctive. (b) Show that if, in addition, G is not virtually cyclic, then there exists a topologically transitive subshift of finite type X ⊂ AG which is not surjunctive. 3.41 (W-Subshifts) Let A be a finite set and let X ⊂ AZ be a subshift. One says that X is a W-subshift if there exists an integer n0 ≥ 0 such that the following holds: for all words u, v ∈ L(X), there exists a word w ∈ A∗ .

with length l(w) ≤ n0 such that uwv ∈ L(X).

(3.15)

(a) Show that every W-subshift X ⊂ AZ is irreducible. (b) Show that every strongly irreducible subshift X ⊂ AZ is a W-subshift. (c) Give an example of a W-subshift X ⊂ AZ that is of finite type but not strongly irreducible. (d) Let X ⊂ AZ be a W-subshift. Show that Per(X) is dense in X. (e) Show that every W-subshift X ⊂ AZ is surjunctive. 3.42 Let G be a group and let A := {0, 1}. Show that the at-most-one-one subshift X ⊂ AG is surjunctive. 3.43 (Minimal Subshifts) Let G be a group and let A be a set. A subshift X ⊂ AG is called a minimal subshift if X is non-empty and the G-orbit of every configuration x ∈ X is dense in X. (a) Show that a subshift X ⊂ AG is minimal if and only if X /= ∅ and there is no subshift Y of AG such that ∅ /= Y C X. (b) Show that the finite minimal subshifts in AG are precisely the finite G-orbits. (c) Show that every minimal subshift X ⊂ AG is topologically transitive. (d) Give an example of a minimal subshift that is not topologically mixing. (e) Show that if X ⊂ AG is an infinite minimal subshift then the G-orbit of every configuration x ∈ X is infinite.

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3 Surjunctive Groups

(f) Show that if X and Y are minimal subshifts of AG then either X = Y or X∩Y = ∅. 3.44 Let G be a group. Let A and B be sets with A finite. Let X ⊂ AG and Y ⊂ B G be subshifts. Suppose that X is non-empty and Y is minimal. Show that every cellular automaton σ : X → Y is surjective. 3.45 Let G be a group and let A be a finite set. Show that every minimal subshift X ⊂ AG is surjunctive. 3.46 Let G be a group and let A be a finite set. Show that every non-empty subshift X ⊂ AG contains a minimal subshift. 3.47 Let A be a finite set and let X ⊂ AZ be an infinite minimal subshift. Show that X is not sofic. 3.48 (Almost Periodic Configurations) 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 (or syndetically recurrent) if for every neighborhood U of x in AG the set Rx (U ) := {g ∈ G : gx ∈ U } is syndetic in G. (a) Show that a configuration x ∈ AG is almost periodic if and only if it satisfies the following condition: for every finite subset Ω ⊂ G the set R(x, Ω) := {g ∈ G : (gx)|Ω = x|Ω } is syndetic in G. (b) Let H be a subgroup of G. Show that H is syndetic in G if and only if H is of finite index in G. (c) Show that every periodic configuration x ∈ AG is almost periodic. (d) Let x ∈ AG be an almost periodic configuration. Show that its orbit closure X := Gx ⊂ AG is a minimal subshift. (e) Suppose that the set A is finite and let X ⊂ AG be a minimal subshift. Show that every configuration x ∈ X is almost periodic. (f) Suppose that the set A is finite and let X ⊂ AG be an infinite minimal subshift. Show that every configuration x ∈ X is almost periodic but not periodic. (g) Suppose that the set A is finite. Show that a non-empty subshift X ⊂ AG is minimal if and only if it satisfies the following two conditions: (1) X is topologically transitive; (2) every configuration x ∈ X is almost periodic. 3.49 Let A be a set. Given a configuration x ∈ AZ (resp. a subshift X ⊂ AZ ), denote by L(x) (resp. L(X)) its language. (a) Show that a non-empty subshift X ⊂ AZ is minimal if and only if one has L(x) = L(X) for all x ∈ X. (b) Let x ∈ AZ . Show that the following conditions are equivalent: (AP1) x is almost periodic; (AP2) for every word u ∈ L(x) of length l(u), the set consisting of all n ∈ Z such that x(n)x(n + 1) · · · x(n + l(u) − 1) = u is a syndetic subset of Z;

Exercises

107

(AP3) for every word u ∈ L(x), there exists an integer r = r(u) ≥ 1 such that u is a subword of any word v ∈ L(x) of length r. 3.50 Let G be an infinite group and let A be a finite set. Show that every minimal subshift X ⊂ AG is irreducible. 3.51 (The Thue-Morse Sequence) Let A := {0, 1}. The Thue-Morse sequence is the sequence T : N → A defined by T (n) := 0 if the number of 1s in the binary expansion of n is even and T (n) := 1 otherwise. (a) Check that T (0)T (1) · · · T (15) = 0110100110010110.

.

(b) Show that T (2n) = T (n) and T (2n + 1) = 1 − T (n) for all n ∈ N. (c) Let k, n ∈ N such that k ≤ 2n − 1. Show that T (k + 2n ) = 1 − T (k). (d) Consider the sequences (an )n∈N and (bn )n∈N of elements of A∗ defined by the initial conditions a0 := 0 and b0 := 1, and the recurrence relations an+1 := an bn and bn+1 := bn an for all n ∈ N. Let ι : A∗ → A∗ denote the unique monoid automorphism such that ι(0) := 1 and ι(1) := 0. Show that one has bn = ι(an )

(3.16)

an = T (0)T (1) · · · T (2n − 1)

(3.17)

.

and .

for all n ∈ N. (e) Let k, n ∈ N. Show that one has { T (k2 )T (k2 + 1) · · · T (k2 + 2 − 1) =

.

n

n

n

n

an

if T (k) = 0,

bn

if T (k) = 1.

(3.18)

(f) Consider the unique monoid homomorphism ϕ : A∗ → A∗ that satisfies ϕ(0) := 01

and

ϕ(1) := 10.

an+1 = ϕ(an )

and

bn+1 = ϕ(bn )

.

Show that one has .

for all n ∈ N. (g) Show that an = ϕ n (0) and bn = ϕ n (1) for all n ∈ N.

(3.19)

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3 Surjunctive Groups

(h) For u ∈ A∗ , let ρ(u) := uR (cf. Exercise 1.88) denote the word in A∗ obtained by reading the letters of u from right to left. Thus, ρ : A∗ → A∗ is the unique anti-automorphism of the monoid A∗ that fixes 0 and 1. Show that one has { (ρ(an ), ρ(bn )) =

.

(an , bn )

if n is even,

(bn , an )

if n is odd,

(3.20)

for all n ∈ N. 3.52 (The Morse Subshift) Let A := {0, 1}. Keeping the notation introduced in Exercise 3.51, let x ∈ AZ be the configuration defined by x(n) := T (n) if n ≥ 0 and x(n) := T (−n − 1) if n ≤ −1. (a) Let k, n ∈ Z with n ≥ 0. Let wk,n ∈ A∗ be the word of length 2n defined by wk,n := x(k2n )x(k2n + 1) · · · x(k2n + 2n − 1).

.

(b) (c) (d) (e) (f)

Show that wk,n ∈ {an , bn } (here an and bn are the words of length 2n in A∗ introduced in Exercise 3.51). Show that the configuration x is almost periodic. Show that the configuration x is not periodic. Let X ⊂ AZ denote the orbit closure of x. Show that X is an infinite minimal subshift of AZ . Show that the subshift X is surjunctive and irreducible. Show that the subshift X is not sofic.

3.53 (Toeplitz Subshifts) Let G be a group and let A be a finite set. A configuration x ∈ AG is called a Toeplitz configuration if for every neighborhood U of x in AG , there exists a finite index subgroup H of G such that the set Rx (U ) := {g ∈ G : gx ∈ U } contains H . One says that a subshift X ⊂ AG is a Toeplitz subshift if it is the orbit closure of some Toeplitz configuration x ∈ AG . (a) (b) (c) (d) (e)

Show that every periodic configuration x ∈ AG is Toeplitz. Show that every Toeplitz configuration x ∈ AG is almost periodic. Show that every Toeplitz subshift X ⊂ AG is minimal. Show that every Toeplitz subshift X ⊂ AG is surjunctive. Let x ∈ AG . Show that the following conditions are equivalent: (TO1) x is Toeplitz; (TO2) for every finite subset Ω ⊂ G, there exists a finite index subgroup H of G such that (hx)|Ω = x|Ω for all h ∈ H ; (TO3) for every g ∈ G, there exists a finite index subgroup H of G such that (hx)(g) = x(g) for all h ∈ H .

3.54 Let A be a finite set. (a) Show that every Toeplitz subshift X ⊂ AZ is irreducible.

Exercises

109

(b) Let x ∈ AZ . Show that x is Toeplitz if and only if it satisfies the following condition: (To) for every n ∈ Z, there exists an integer r ≥ 1 such that x(n) = x(n + kr) for all k ∈ Z; (c) Given a Toeplitz configuration x ∈ AZ and n ∈ Z we define the integer p(x, n) ≥ 1 by p(x, n) := min{r ≥ 1 : x(n) = x(n + kr) for all k ∈ Z}

.

and the subset P (x) ⊂ N \ {0} by P (x) := {p(x, n) : n ∈ Z}. Let x ∈ AZ be a Toeplitz configuration. Show that x is periodic if and only if the set P (x) is finite. (d) Let A := {0, 1}. Consider the bijective map ι : Z → N defined by ι(n) := 2n if n ≥ 0 and ι(n) := −(2n + 1) if n ≤ −1. Show that the following inductive procedure yields a well defined non-periodic Toeplitz configuration x ∈ AZ : (1) Step 1 consists in setting x(n) := 1 for all n ∈ 2Z. (2) Suppose that Step k − 1 has been performed for some integer k ≥ 2 and let Jk denote the set consisting of all n ∈ Z for which x(n) has not yet been defined. Denoting by nk the integer in Jk whose image under ι is minimal, Step k consists in setting x(n) := k mod 2 for all n ∈ nk + 2k Z. 3.55 Let A be a finite set and let X ⊂ AZ be an infinite Toeplitz subshift. Show that X is not sofic. 3.56 (Strongly Aperiodic Subshifts) Let G be a group and let A be a set. A configuration x ∈ AG is called strongly aperiodic if the stabilizer of x in G is trivial, i.e., gx = x implies g = 1G for all g ∈ G. A subshift X ⊂ AG is called strongly aperiodic if X is non-empty and every configuration x ∈ X is strongly aperiodic. (a) Let G be a finite group and let A be a set with more than one element. Show that there exists a strongly aperiodic subshift X ⊂ AG . (b) Let A be a finite set. Show that there exists no strongly aperiodic sofic subshift X ⊂ AZ . (c) Let A be a set. Show that every infinite minimal subshift X ⊂ AZ is strongly aperiodic. (d) Show that the Morse subshift X ⊂ {0, 1}Z is strongly aperiodic. (e) Let A be a finite set. Show that every infinite Toeplitz subshift X ⊂ AZ is strongly aperiodic. (f) Let A and B be two sets and let τ : B G → AG be a G-equivariant continuous map (e.g., a cellular automaton). Suppose that X ⊂ AG is a strongly aperiodic subshift. Show that τ −1 (X) is a strongly aperiodic subshift of B G .

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, taking direct sums, 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).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3_4

111

112

4 Amenable Groups

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 = ∅. 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) (ii) (iii) (iv) (v)

μ(∅) = 0; μ(A ∪ B) = μ(A) + μ(B) − μ(A ∩ B); .μ(A ∪ B) ≤ μ(A) + μ(B); .A ⊂ B ⇒ μ(B \ A) = μ(B) − μ(A); .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. u n .

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

.

||x||∞ = sup |x(a)|.

.

a∈E

We equip .l∞ (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 .l∞ (E) which is identically equal to .λ on E.

4.1 Measures and Means

113

Definition 4.1.4 A mean on E is a linear map .m : l∞ (E) → R such that: (1) .m(1) = 1, (2) .x ≥ 0 ⇒ m(x) ≥ 0 for all .x ∈ l∞ (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 Σ(s) > 0 for all .s ∈ S; (C2) . s∈S f (s) = 1. Then the map .mf : l∞ (E) → R defined by mf (x) :=

Σ

.

f (s)x(s)

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 : l∞ (E) → R be a mean on E. Then one has: (i) (ii) (iii) (iv)

m(λ) = λ, x ≤ y ⇒ m(x) ≤ m(y), .infE x ≤ m(x) ≤ supE x, .|m(x)| ≤ ||x||∞ , . .

for all .λ ∈ R and .x, y ∈ l∞ (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 .infE x ≤ x ≤ supE x and therefore .infE x ≤ m(x) ≤ supE x by applying (i) and (ii). (iv) From (iii) we get .−||x||∞ ≤ m(x) ≤ ||x||∞ , that is, .|m(x)| ≤ ||x||∞ . u n Consider the topological dual of .l∞ (E), that is, the vector space .(l∞ (E))∗ consisting of all continuous linear maps .u : l∞ (E) → R. Recall that .(l∞ (E))∗ is a Banach space for the operator norm .|| · || defined by ||u|| := sup |u(x)|

.

||x||∞ ≤1

for all .u ∈ (l∞ (E))∗ .

(4.1)

114

4 Amenable Groups

Proposition 4.1.7 Let .m : l∞ (E) → R be a mean on E. Then .m ∈ (l∞ (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. u n 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 .(l∞ (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. Let .m ∈ M (E). Consider the map .m - : P(E) → R given by m -(A) := m(χA ).

.

Observe that .0 ≤ χA ≤ 1 and therefore .m -(A) ∈ [0, 1] by Proposition 4.1.6(ii). 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 - ∈ PM (E). linearity of m. It follows that .m 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 .l∞ (E) spanned by the set of characteristic maps .{χA : A ⊂ E}. Lemma 4.1.9 The vector subspace .E (E) is dense in .l∞ (E). Proof Let .x ∈ l∞ (E) and .ε > 0. Set .α := infE 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 ∞ .l (E). u n Let .μ ∈ PM (E). Let us set, for all .x ∈ E (E), μ(x) :=

Σ

.

λ∈R

μ(x −1 (λ))λ.

(4.2)

4.1 Measures and Means

115

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 Σ μ(Ai )αi . .μ(x) = i∈I

Proof Since .μ is finitely additive, we have Σ .

i∈I

μ(Ai )αi =

Σ

⎛ ⎝

⎞

Σ

μ(Ai )⎠ λ =

αi =λ

λ∈R

Σ

μ(x −1 (λ))λ = μ(x).

λ∈R

u n 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 Σ

μ(ξ x + ηy) =

μ(x −1 (α) ∩ y −1 (β))(ξ α + ηβ)

.

(α,β)∈V ×W

⎛

Σ

=ξ⎝

⎞ μ(x −1 (α) ∩ y −1 (β))α ⎠

(α,β)∈V ×W

⎛

+ ⎝η

Σ

⎞ μ(x −1 (α) ∩ y −1 (β))β ⎠

(α,β)∈V ×W

= ξ μ(x) + ημ(y). Consequently, .μ is linear. Formula (4.2) implies .infE x ≤ μ(x) ≤ supE x since Σ .

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. u n

116

4 Amenable Groups

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 .(yn ) the constant sequence .yn = x in this case. Thus .ϕ extends .ϕ. u n Proof of Theorem 4.1.8 Let .μ ∈ PM (E). By Lemmas 4.1.9, 4.1.11, and 4.1.12, the map .μ : E (E) → R can be extended to a continuous linear map .μ : l∞ (E) → R. We have 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 ∈ l∞ (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 .l∞ (E) by Lemma 4.1.9, we deduce that .m1 = m2 by continuity of .m1 and .m2 . This shows that .Φ is injective. u n

4.3 Measures and Means on Groups

117

4.2 Properties of the Set of Means Let E be a set. The topology on .(l∞ (E))∗ associated with the operator norm .||·|| defined in (4.1) is called the strong topology on .(l∞ (E))∗ . Another topology that is commonly used is the weak-.∗ topology on .(l∞ (E))∗ . Recall that the weak-.∗ topology on .(l∞ (E))∗ is by definition the smallest topology for which the evaluation map ψx : (l∞ (E))∗ → R

.

u |→ u(x) is continuous for each .x ∈ l∞ (E) (see Sect. F.2). By Proposition 4.1.7, the set .M (E) is contained in the unit sphere {u : ||u|| = 1} ⊂ (l∞ (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 .(l∞ (E))∗ with respect to the weak-.∗ topology. 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 ∈ l∞ (E) such that .x ≥ 0. Thus .tm1 + (1 − t)m2 ∈ M (E). This shows that .M (E) is convex. Equip now .(l∞ (E))∗ with the weak-.∗ topology and suppose that .(mi )i∈I is a net in .M (E) converging to .u ∈ (l∞ (E))∗ . Then, for every .i ∈ I , we have .ψ1 (mi ) = mi (1) = 1 and .ψx (mi ) = mi (x) ≥ 0 for all .x ∈ l∞ (E) such that .x ≥ 0. By taking limits, we get .u(1) = ψ1 (u) = 1 and .u(x) = ψx (u) ≥ 0 for all .x ∈ l∞ (E) such that .x ≥ 0. Thus .u ∈ M (E). This shows that .M (E) is closed in .(l∞ (E))∗ (Proposition A.2.1). As .M (E) is contained in the unit ball of .(l∞ (E))∗ , which is compact for the weak-.∗ topology by the Banach-Alaoglu Theorem (Theorem F.3.1), it follows that .M (E) is compact. u n

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.

118

4 Amenable Groups

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 .l∞ (G) ⊂ RG is left invariant by both actions. Observe that .||gx||∞ = ||xg||∞ = ||x||∞ for all .g ∈ G and .x ∈ l∞ (G). Thus the left (resp. right) action of G on .l∞ (G) is isometric (and therefore continuous). By duality, we also get a left and a right action of G on .(l∞ (G))∗ . More precisely, for .g ∈ G and .u ∈ (l∞ (G))∗ , we define the elements gu and ug by gu(x) := u(g −1 x)

.

and

ug(x) := u(xg −1 ),

for all .x ∈ l∞ (G). We have .gu ∈ (l∞ (G))∗ and .ug ∈ (l∞ (G))∗ since the left and right actions of G on .l∞ (G) are linear and continuous. Note that the set .M (G) is invariant under both actions of G on .(l∞ (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 .(l∞ (G))∗ is linear. On the other hand, these actions are continuous if we equip .(l∞ (G))∗ with the weak-.∗ topology. Indeed, if we fix .g ∈ G, the map .u |→ gu is continuous on .(l∞ (G))∗ since, for each .x ∈ l∞ (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 .(l∞ (G))∗ , we deduce that and continuous. u n Given .μ ∈ PM (G), we define the map .μ∗ : P(G) → [0, 1] by μ∗ (A) := μ(A−1 ) for all A ∈ P(G).

.

4.3 Measures and Means on Groups

119

It is clear that .μ∗ ∈ PM (G) and that the map .μ |→ μ∗ is an involution of .PM (G). For .x ∈ l∞ (G), define .x ∗ ∈ l∞ (G) by x ∗ (g) := x(g −1 ) for all g ∈ G.

.

The map .x |→ x ∗ is an isometric involution of .l∞ (G). By duality, it gives an isometric involution .u |→ u∗ of .(l∞ (G))∗ defined by u∗ (x) := u(x ∗ )

.

for all x ∈ l∞ (G).

Note that .m∗ ∈ M (G) for all .m ∈ M (G). Proposition 4.3.2 Let .g ∈ G, .x ∈ l∞ (G), .μ ∈ PM (G) and .u ∈ (l∞ (G))∗ . Then one has (i) (ii) (iii) (iv) (v) (vi)

(gx)∗ = x ∗ g −1 ; ∗ −1 x ∗ ; .(xg) = g ∗ ∗ −1 ; .(gμ) = μ g ∗ −1 .(μg) = g μ∗ ; ∗ ∗ −1 .(gu) = u g ; ∗ −1 u∗ . .(ug) = g .

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.

u n

Proposition 4.3.3 Let .g ∈ G and .m ∈ M (G). Then one has: (i) .gm - = gm; (ii) .mg - =m -g; -∗ = 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 −1 h) = 1 ⇔ g −1 h ∈ A ⇔ h ∈ gA ⇔ χ (h) = 1. This shows .(gχA )(h) = χA (g gA that .gχA = χgA . u n

120

4 Amenable Groups

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) = gm(A).

.

Thus .gm - = gm. Similarly, we get .mg - =m -g. On the other hand, for every .A ∈ P(G), using Lemma 4.3.4(i) one obtains -∗ (A) m -∗ (A) = m -(A−1 ) = m(χA−1 ) = m((χA )∗ ) = m∗ (χA ) = m

.

-∗ . and this shows that .m -∗ = m

u n

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. 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).

u n

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

u n

Proposition 4.4.3 Let .m ∈ M (G). Then m is left-invariant (resp. right-invariant, 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. u n 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;

4.4 Definition of Amenability

121

(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 : l∞ (G) → R on G; (e) there exists a right-invariant mean .m : l∞ (G) → R on G; (f) there exists a bi-invariant mean .m : l∞ (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 leftx: G → R invariant mean .m : l∞ (G) → R. For each .x ∈ l∞ (G), define the map .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 ∈ l∞ (G) for all .x ∈ l∞ (G). Consider now the map .M : l∞ (G) → R defined by M(x) := m(x ) for all x ∈ l∞ (G).

.

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

.

- =since m is left-invariant. We deduce that .hx x . 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 x (g). xh(g) = m((xh)g) = m(x(hg)) = x (hg) = h−1-

.

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

.

Therefore the mean M is also right-invariant. This shows that (d) implies (f).

u n

122

4 Amenable Groups

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.

u n

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 2 .μ(F2 ) ≥ μ(A) + μ(bA) + μ(b A) = 3μ(A). This gives μ(A) ≤

.

1 3 u n

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 ( μ(A) := μ

U

.

) Ar

for all A ∈ P(H )

r∈R

is a left-invariant finitely additive probability measure on H .

4.5 Stability Properties of Amenable Groups

123

Firstly, we have ( μ(H ) = μ

)

U

= μ(G) = 1.

Hr

.

r∈R

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

U

.

) (A ∪ B)r

r∈R

(( =μ

U

)

(

Ar ∪

r∈R

( =μ

U

U

)) Br

r∈R

)

(

Ar + μ

r∈R

U

) Br

r∈R

=μ(A) + μ(B), since the sets U .

and

Ar

r∈R

U

Br

r∈R

are disjoint. Finally, for all h ∈ H and A ∈ P(H ), we have ( μ(hA) = μ

U

.

) hAr

r∈R

( =μ h

U r∈R

) Ar

( =μ

U

) Ar

=μ(A).

r∈R

This shows that H is amenable.

u n

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. O 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 SL2 (Z) is not amenable. Indeed, as we ) seen ( in )Lemma 2.3.2, the ( have subgroup of SL2 (Z) generated by the matrices 10 21 and 12 01 is isomorphic to F2 .

124

4 Amenable Groups

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))

.

since ρ −1 (A) ∩ ρ −1 (B) = ∅. Thus we have μ(A ∪ B) = μ(A) + μ(B). Finally, if g ∈ G and A ⊂ G/H , we have μ(ρ(g)A) = μ(ρ −1 (ρ(g)A)) = μ(gρ −1 (A)) = μ(ρ −1 (A)) = μ(A).

.

As ρ is surjective, it follows that μ is left-invariant with respect to G/H . This shows that G/H is amenable. u n In the next proposition, we show that the class of amenable groups is closed under taking extensions. 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 : l∞ (H ) → R. Let x ∈ l∞ (G). Consider the map x : G/H → R defined by x (g) := m0 ((g −1 x)|H )

.

for all g ∈ G,

where g := gH = H g ∈ G/H denote the class of g modulo H . The map x is 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 )

.

4.5 Stability Properties of Amenable Groups

125

since m0 is H -left-invariant. Observe also that x ∈ l∞ (G/H ) since, by Proposition 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 : l∞ (G/H ) → R. Let us set, for each x ∈ l∞ (G), m(x) := m1 (x ).

.

Clearly m is a mean on G. Let us show that it is G-left-invariant. Let g ∈ G and x ∈ l∞ (G). For all g ' ∈ G, we have gx(g ' ) = m0 ((g ' .

−1

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

= gx (g ' ).

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

.

which shows that m is G-left-invariant.Therefore G is amenable.

u n

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 isomorphic u n 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 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 gH g −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. u n

126

4 Amenable Groups

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 (l∞ (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 ∈ l∞ (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. u n 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 : l∞ (Hi ) → R. ∞ Consider the family mi : l (G) → R of means on G defined by mi (x) := m -i (x|Hi )

.

for all x ∈ l∞ (G).

Let g ∈ G. Since G = lim Gi , there exists i0 (g) ∈ I such that g ∈ Hi for all − → i ≥ i0 (g). For x ∈ l∞ (G) and i ≥ i0 (g), we have (gmi − mi )(x) = gmi (x) − mi (x) = gmi (x|Hi ) − m -i (x|Hi ) = 0,

.

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. u n Every group is the inductive limit of its finitely generated subgroups. Therefore we have: Corollary 4.5.11 Every locally amenable group is amenable.

O

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

O

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.

4.6 Solvable Groups

127

Corollary O 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. u n

4.6 Solvable Groups Theorem 4.6.1 Every abelian group is amenable. Proof Let G be an abelian group. Equip (l∞ (G))∗ with the weak-∗ topology. By Theorem 4.2.1, the set M (G) is a nonempty convex compact subset of (l∞ (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. u n 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 (D i (G))i≥0 of subgroups of G inductively defined by D 0 (G) := G and D i+1 (G) := D(D i (G)) for all i ≥ 0. One has G = D 0 (G) ⊃ D 1 (G) ⊃ D 2 (G) ⊃ . . .

.

with D i+1 (G) normal in D i (G) and D i (G)/D i+1 (G) abelian for all i ≥ 0. The group G is said to be solvable if there is an integer i ≥ 0 such that D i (G) = {1G }. The smallest integer i ≥ 0 such that D i (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.

128

4 Amenable 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. In the case when K has two elements, i.e., K := {0, 1}, then Aff(K) is cyclic of order 2 and therefore it is abelian. Suppose now that K has more than two elements. Then the group D 1 (Aff(K)) is the group of translations x |→ x + b, b ∈ K. Since D 1 (Aff(K)) is abelian, we have D i (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. u n Remark 4.6.4 The alternating group Sym+ 5 is amenable since it is finite. However, is not solvable. An example of an infinite amenable as mentioned above, Sym+ 5 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. 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 ⎛

⎞ 1y z .M(x, y, z) := ⎝0 1 x ⎠ 001

(x, y, z ∈ R).

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.

4.7 The Følner Conditions

129

Proposition 4.6.6 Every nilpotent group is solvable. Proof An easy induction yields D i (G) ⊂ C i (G) for all i ≥ 0.

u n

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.

O

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)),

4.9 The Theorems of Tarski and Følner

137

which gives |K1 F | ≥ (1 + ε0 )|F |.

.

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 −1 F respectively. Therefore, we have .NL (F ) := K .

|NR (F )| = |KF | ≥ 2|F |,

.

by applying (4.17). On the other hand, if .k ∈ K, then .NL (F ) ⊃ k −1 F so that |NL (F )| ≥ |k −1 F | = |F | ≥

.

1 |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, −1 ∈ K for all .g ∈ G. This shows that (c) implies (e). .g(ϕ(g)) (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} and Bk := {g ∈ G : θ2 (g) = k}.

.

138

4 Amenable Groups

We have G = uk∈K Ak = uk∈K Bk .

.

(4.19)

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

.

(4.20)

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 ((uk∈K kAk ) u (uk∈K kBk )) Σ Σ μ(kAk ) + μ(kBk ) = k∈K .

=

Σ

k∈K

μ(Ak ) +

k∈K

Σ

μ(Bk )

k∈K

= μ(uk∈K Ak ) + μ(uk∈K Bk ) = μ(G) + μ(G) = 2, which is clearly absurd. Therefore G is not amenable. This shows that (g) implies (a). u n

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.

4.10 The Fixed Point Property

139

Choose an arbitrary point .x ∈ C and set xj :=

.

1 Σ hx. |Fj | h∈Fj

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 ⎞ Σ 1 1 Σ hx ⎠ − hx .gxj − xj = g ⎝ |Fj | |Fj | ⎛

h∈Fj

=

h∈Fj

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

h∈Fj

=

1 Σ 1 Σ hx, hx − |Fj | |Fj | h∈Fj

h∈gFj

which yields, after simplification, 1 |Fj |

gxj − xj =

.

Σ

hx −

h∈gFj \Fj

1 |Fj |

Σ

(4.21)

hx.

h∈Fj \gFj

Consider the points yj :=

.

1 |gFj \ Fj |

Σ

hx,

and

zj :=

h∈gFj \Fj

1 |Fj \ gFj |

Σ

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

140

4 Amenable Groups

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

.

j

This shows that c is fixed by G.

u n

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 .(l∞ (G))∗ is a locally convex Hausdorff topological vector space for the weak-.∗ topology (see Sect. F.2). u n

.

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 non-amenable 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 n .Fm generated by all n-powers, that is, all elements of the form .w 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

Notes

141

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 which contain no subgroup isomorphic to .F2 were given by A.Yu Ol’shanskii and M. Sapir [OlsS]. 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 [HallP2] 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 Sects. 6.10 and 6.12) and Adyan’s cogrowth estimates [Ady] for the free Burnside groups .B(m, n) (see (6.111) 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].

142

4 Amenable Groups

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. 4.2 Let X be an infinite set. Show that the symmetric group Sym(X) is not amenable. 4.3 Show that every FC-group is amenable. 4.4 Show that the wreath product of two amenable groups is an amenable group. 4.5 Suppose that G is a non-amenable group and that N is an amenable normal subgroup of G. Show that the group G/N is non-amenable. 4.6 Let G1 (resp. G2 ) denote the finitely generated and non residually finite group described in Exercise 2.40 (resp. Exercise 2.41). Show that the groups G1 and G2 are amenable. 4.7 Let G be a group. (a) Suppose that H and K are normal subgroups of G. Show that H K is a normal subgroup of G and that the groups H K/K and H /(H ∩ K) are isomorphic. (b) Suppose that H and K are normal amenable subgroups of G. Show that H K is itself a normal amenable subgroup of G. (c) Show that the set of all normal amenable subgroups of G has a unique maximal element for inclusion. 4.8 Let G be a group and let H be a subgroup of G. Show that the following conditions are equivalent: (i) [G, G] ⊂ H ; (ii) H is normal in G and G/H is abelian. 4.9 Let G be a group. Show that G is metabelian if and only if it contains an abelian normal subgroup N such that the group G/N is abelian. 4.10 Let G be a group. Let x, y, z ∈ G. (a) Show that [x, yz] = [x, y] · [y, [x, z]] · [x, z].

(4.22)

[xy, z] = [y, z] · [[z, y], x] · [x, z].

(4.23)

.

(b) Show that .

4.11 Let G be a group and let Z(G) denote the center of G. Suppose that H is a subgroup of G such that [H, G] ⊂ Z(G).

Exercises

143

(a) Let x, x ' ∈ H and y, y ' ∈ G. Show that one has [xx ' , yy ' ] = [x, y] · [x, y ' ] · [x ' , y] · [x ' , y ' ], .

.

[x, y]−1 = [x −1 , y] = [x, y −1 ], . [x, y] = [x , y] = [x, y ]. n

n

n

(4.24) (4.25) (4.26)

(b) Let x ∈ H and y ∈ G. Show that if x or y has finite order then [x, y] has finite order. (c) Suppose that S is a generating subset of H and that T is a generating subset of G. Show that the set C := {[s, t] : s ∈ S, t ∈ T } is a generating subset of [H, G]. (d) Suppose that the groups H and G are finitely generated. Show that the group [H, G] is finitely generated. 4.12 (The Second Center of a Group) Let G be a group and let Z(G) denote its center. Let Z2 (G) denote the subset of G consisting of all x ∈ G such that [x, y] ∈ Z(G) for all y ∈ G. (a) Show that Z(G) ⊂ Z2 (G). (b) Show that Z2 (G) is a normal subgroup of G and that Z2 (G)/Z(G) is the center of G/Z(G). (c) Show that if Z(G) is torsion-free then Z2 (G)/Z(G) is torsion-free. 4.13 Let G and H be two groups. Let ϕ : G → H be a group homomorphism. (a) Show that ϕ([x, y]) = [ϕ(x), ϕ(y)] for all x, y ∈ G. (b) Let A and B be subgroups of G. Show that ϕ([A, B]) = [ϕ(A), ϕ(B)].

.

(4.27)

(c) Show that ϕ(D i (G)) ⊂ D i (H ) and ϕ(C i (G)) ⊂ C i (H ) for all i ≥ 0. (d) Suppose that ϕ is surjective. Show that ϕ(D i (G)) = D i (H ) and ϕ(C i (G)) = C i (H )

.

for all i ≥ 0. 4.14 Show that every subgroup of a solvable (resp. nilpotent) group of solvability (resp. nilpotency) degree d is solvable (resp. nilpotent) with solvability (resp. nilpotency) degree at most d. 4.15 Show that every quotient of a solvable (resp. nilpotent) group of solvability (resp. nilpotency) degree d is solvable (resp. nilpotent) with solvability (resp. nilpotency) degree at most d.

144

4 Amenable Groups

4.16 Let G be a group and let G' = [G, G] denote the derived subgroup of G. (a) Show that if G' and G/G' are Hopfian then G is Hopfian. (b) Show that if G is finitely generated and G' is Hopfian then G is Hopfian. 4.17 Let G be a group and let i ≥ 0 be an integer. (a) Show that the group G/D i 4(G) (resp. G/C i (G)) is solvable (resp. nilpotent) and has solvability (resp. nilpotency) degree at most i. (b) Suppose that the group G is solvable (resp. nilpotent) with solvability (resp. nilpotency) degree d. Show that the group G/D i (G) (resp. G/C i (G)) is solvable (resp. nilpotent) with solvability (resp. nilpotency) degree min(d, i). (c) Suppose that the group G is nilpotent with nilpotency degree d ≥ 1 and let Z(G) denote the center of G. Show that the group G/Z(G) is nilpotent with nilpotency degree at most d − 1. 4.18 Show that the class of torsion-free groups is closed under forming extensions. 4.19 Let G be a nilpotent group and let Z(G) denote its center. Show that the following conditions are equivalent (i) the group G is torsion-free; (ii) the group Z(G) is torsion-free; (iii) the groups Z(G) and G/Z(G) are torsion-free. 4.20 Let G be a group and let H be a normal subgroup of G such that G/H is solvable (resp. nilpotent). Let d ≥ 0 denote the solvability (resp. nilpotency) degree of G/H . Show that D d (G) ⊂ H (resp. C d (G) ⊂ H ). 4.21 Let G be a group. Show that G is solvable (resp. nilpotent) if and only if there exist an integer n ≥ 0 and a decreasing sequence (Hi )0≤i≤n of subgroups of G such that H0 = G, Hn = {1G } and [Hi , Hi ] ⊂ Hi+1 (resp. [Hi , G] ⊂ Hi+1 ) for all 0 ≤ i ≤ n − 1. 4.22 Let G be a group. Show that G is solvable if and only if there exist an integer n ≥ 0 and a decreasing sequence (Hi )0≤i≤n of subgroups of G such that H0 = G, Hn = {1G }, Hi+1 is normal in Hi and Hi /Hi+1 is abelian for all 0 ≤ i ≤ n − 1. 4.23 Show that a group is nilpotent with nilpotency degree ≤ 2 if and only if its derived subgroup is contained in its center. 4.24 Let G be a group such that G/Z(G) is cyclic. Show that G is abelian (so that G = Z(G)). 4.25 Let G be a group and let H be a subgroup of G. Suppose that H is contained in the center of G and that the group G/H is nilpotent. Show that G is nilpotent. 4.26 (Torsion Groups) One says that a group G is a torsion group if all elements of G have finite order. Show that every finitely generated solvable torsion group is finite.

Exercises

145

4.27 Let G be a group containing a normal subgroup H such that both G/H and H are solvable. Let d (resp. d ' ) denote the solvability degree of G/H (resp. H ). Show that G is solvable with solvability degree at most d + d ' . 4.28 Show that the direct product of two nilpotent groups is a nilpotent group. 4.29 Show that a semidirect product of two nilpotent groups may fail to be nilpotent. 4.30 Let G be a nilpotent group. (a) Let H be a non-trivial normal subgroup of G. Show that H ∩ Z(G) /= {1G }. (b) Show that every non-trivial nilpotent group has a non-trivial center. (c) Let G' be a group and let ϕ : G → G' be a group homomorphism. Let ψ : Z(G) → G' denote the restriction of ϕ to Z(G). Show that ϕ is injective if and only if ψ is injective. 4.31 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). 4.32 Let G be a finite group whose cardinality is a power of a prime number. Show that G is nilpotent. 4.33 Let p be a prime number. (a) Show that every group with cardinality p or p2 is abelian (and hence nilpotent). (b) Give an example of a group with cardinality p3 that is not abelian. 4.34 Let G be a group. Given x, y ∈ G, we write x y := yxy −1 . (a) Let x, y, z ∈ G. Show that [[x −1 , y], z]x · [[z−1 , x], y]z · [[y −1 , z], x]y = 1G .

.

(4.28)

(b) Let X, Y , and Z be subgroups of G. Suppose that [[X, Y ], Z] = [[Y, Z], X] = {1G }. Show that [[Z, X], Y ] = {1G }. (c) Let X, Y , and Z be subgroups of G and let N be a normal subgroup of G. Suppose that [[X, Y ], Z] ⊂ N and [[Y, Z], X] ⊂ N. Show that [[Z, X], Y ] ⊂ N. (d) Deduce from (c) that [C i (G), C j (G)] ⊂ C i+j +1 (G)

.

for all i, j ∈ N.

(4.29)

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4 Amenable Groups

4.35 (Polycyclic Groups) A group G is called polycyclic if it admits a finite sequence of subgroups {1G } = G0 ⊂ G1 ⊂ G2 ⊂ · · · ⊂ Gn = G

.

such that Gi is normal in Gi+1 and Gi+1 /Gi is a (finite or infinite) cyclic group for all 0 ≤ i ≤ n − 1. Such a sequence of subgroups is then called a polycyclic series of length n for G. (a) (b) (c) (d) (e) (f) (g) (h) (i)

Show that every polycyclic group is solvable. Show that every subgroup of a polycyclic group is polycyclic. Show that every quotient of a polycyclic group is polycyclic. Show that the class of polycyclic groups is closed under forming extensions. Show that every polycyclic group is finitely generated. Show that every polycyclic group is residually finite. Show that an abelian group is polycyclic if and only if it is finitely generated. Show that every subgroup of a polycyclic group is finitely generated. One says that a group G satisfies the ascending condition for subgroups if every increasing sequence (Hn )n∈N of subgroups of G eventually stabilizes, i.e., there exists n0 ∈ N such that Hn0 = Hn for all n ≥ n0 . Show that every polycyclic group satisfies the ascending chain condition for subgroups. (j) Show that a solvable group is polycyclic if and only if it satisfies the ascending chain condition for subgroups.

4.36 Show that every finite solvable group is polycyclic. 4.37 Show that every virtually polycyclic group is of finite Markov type. 4.38 Let G be a group. For each integer i ≥ 0, denote by ρi : C i (G) → C i (G)/C i+1 (G) the canonical group epimorphism. (a) Let i ≥ 1 be an integer. Let x ∈ G. Show that [x, y] ∈ C i (G) for all y ∈ C i−1 (G) and that the map ϕxi : C i−1 (G) → C i (G)/C i+1 (G) defined by ϕix (y) := ρi ([x, y])

.

for all y ∈ C i−1 (G) is a group homomorphism whose kernel contains C i (G). (b) Let y ∈ C i−1 (G). Show that [y, x] ∈ C i (G) for all x ∈ G and that the map y ψi : G → C i (G)/C i+1 (G) defined by y

ψi (x) := ρi ([y, x])

.

for all x ∈ G is a group homomorphism whose kernel contains C 1 (G). (c) Suppose that the abelianization G/[G, G] = C 0 (G)/C 1 (G) of the group G is finitely generated. Show that the group C i (G)/C i+1 (G) is finitely generated for every i ≥ 0.

Exercises

147

4.39 Let G be a nilpotent group. Show that the following conditions are equivalent: (i) G is finitely generated; (ii) G/[G, G] is finitely generated; (iii) G is polycyclic. 4.40 (Residual Finiteness of Finitely Generated Nilpotent Groups) Show that every finitely generated nilpotent group is residually finite. 4.41 (A Finitely Generated Solvable Group that Is not Residually Finite) Let n ≥ 2 be an integer and let R := Z[1/n] denote the ring of n-adic rationals. Consider the set G ⊂ GL3 (R) consisting of all matrices of the form ⎛

⎞ 1 y z k .M(k, x, y, z) := ⎝0 n x ⎠ 0 0 1 with k ∈ Z and x, y, z ∈ R. (a) (b) (c) (d)

Show that G is a subgroup of GL3 (R). Show that the group G is finitely generated, residually finite, and Hopfian. Show that G is a solvable group of solvability degree 3. Let ⎛ n0 .D := ⎝ 0 1 00

⎞ 0 0⎠ ∈ GL3 (R) 1

and let α denote the inner automorphism of GL3 (R) associated with D so that α(X) = DXD −1 for all X ∈ GL3 (R). Show that α(G) = G. (e) Let H denote the subgroup of G generated by the matrix M(0, 0, 0, 1). Show that H is normal in G and that α(H ) C H . (f) Show that the quotient group G/H is not Hopfian. (g) Show that G/H is a finitely generated solvable group with solvability degree 3 that is not residually finite. 4.42 Let G be a nilpotent group. Suppose that G admits a finite generating subset S all of whose elements have finite order. Show that G is finite. 4.43 Let G be a nilpotent group and let T denote the subset of G consisting of all finite order elements in G. (a) Show that T is a normal subgroup of G. (b) Show that if G is finitely generated then T is finite.

148

4 Amenable Groups

4.44 (The Lamplighter Group) The lamplighter group is the wreath O 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) (b) (c) (d) (e) (f)

Show that the group L is metabelian. Show that the group L is residually finite. Show that L is finitely generated. Show that L is not of finite Markov type. Show that L is not virtually polycyclic. Show that L is solvable but not nilpotent.

4.45 Let m be an integer such that |m| ≥ 2. Let G be the group given by the presentation G := . (a) Show that G is solvable with solvability degree 2. (b) Show that G is not polycyclic. (c) Show that G is not nilpotent. 4.46 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. 4.47 (Følner Sequences in a Finite Cartesian Product) Let G1 (resp. G2 ) be a (1) (2) countable amenable group. Let F1 = (Fn )n∈N (resp. F2 = (Fn )n∈N ) be a left Følner sequence for G1 (resp. for G2 ). (1)

(2)

(a) Show that F = (Fn )n∈N , where Fn := Fn × Fn for all n ∈ N, is a left Følner sequence for G := G1 × G2 . (b) Let r ≥ 1 be an integer. Show that F (r) := (Fnr )n∈N , where Fnr := {0, 1, . . . , n}r ⊂ Zr for all n ∈ N, is a Følner sequence for Zr . 4.48 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. 4.49 Let G := HZ denote the integral Heisenberg group (cf. Example 4.6.5). Recall that G is the subgroup of SL3 (Z) consisting of all upper triangular matrices with only 1s on the diagonal. For each integer n ≥ 1, define the subset Fn ⊂ G by ⎧⎛ ⎫ ⎞ ⎨ 1x z ⎬ ⎝0 1 y ⎠ ∈ G : 1 ≤ x ≤ n, 1 ≤ y ≤ n, 1 ≤ z ≤ n2 . .Fn := ⎩ ⎭ 001 Show that the sequence (Fn )n≥1 is a Følner sequence for G. 4.50 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 |.

Exercises

149

4.51 Let G be a finite group. Let I be a directed set and let F = (Fi )i∈I be a net of non-empty finite subsets of G. Show that F is a left Følner net for G if and only if there exists i0 ∈ I such that Fi = G for all i ≥ i0 . 4.52 Let G be an infinite amenable group and let F = (Fi )i∈I be a Følner net for G. Show that limi |Fi | = ∞. 4.53 Let G be a group and let H be a finite index subgroup of G. Suppose that F H = (FjH )j ∈J is a left Følner net for H and that T ⊂ G is a complete set of representatives for the left cosets of H in G, so that G = ut∈T tH . Show that the family F G = (FjG )j ∈J , where FjG := T FjH for all j ∈ J , is a left Følner net for G. 4.54 Let G be an amenable group and let F = (Fi )i∈I be a left Følner net for G. Let A ⊂ G be a finite subset. Show that the net (Fi ∪ A)i∈I is a left Følner net for G. 4.55 Let G be a countable amenable group. Show that G admits a left Følner sequence U (Fn )n∈N which exhausts G, i.e., such that Fn ⊂ Fn+1 for all n ∈ N and G = n∈N Fn . 4.56 Let G be a group and let Ω be a subset of G. (a) Let A and B be two subsets of G. Show that AΩ \ BΩ ⊂ (A \ B)Ω.

.

(4.30)

(b) Give an example showing that the inclusion in (4.30) may be strict. (c) Suppose that (Fj )j ∈J is a left Følner net for G and that the subset Ω ⊂ G is finite and non-empty. Show that the family (Fj Ω)j ∈J is a left Følner net for G. 4.57 Let (K, (Ak )k∈K , (Bk )k∈K ) be a left paradoxical decomposition of a group G. Show that |K| ≥ 3. 4.58 Let G be a group and let H be a subgroup of G. Suppose that (K, (Ak )k∈K , (Bk )k∈K ) is a left paradoxical decomposition of H and let T ⊂ G be a complete set of representatives for the right cosets of H in G. For each k ∈ K, set A'k := ut∈T Ak t and Bk' := ut∈T Bk t. Show that (K, (A'k )k∈K , (Bk' )k∈K ) is a left paradoxical decomposition of G. 4.59 Let G be a group and let N be a normal subgroup of G. Suppose that (K, (Ak )k∈K , (Bk )k∈K ) is a left paradoxical decomposition of G/N. Let π : G → G/N denote the canonical quotient group homomorphism and let K ' ⊂ G such that π(K ' ) = K and |K ' | = |K|. For each k ∈ K, denote by k ' the unique element in K ' such that π(k ' ) = k and set Ak ' := π −1 (Ak ) (resp. Bk ' := π −1 (Bk )). Show that (K ' , (Ak ' )k ' ∈K ' , (Bk ' )k ' ∈K ' ) is a paradoxical decomposition of G. 4.60 (The Tarski Number of a Group) Let G be a group. Given a (left) paradoxical decomposition P = (K, (Ak )k∈K , (Bk )k∈K ) of G, the integer c(P) := m + n, where m = m(P) := |{k ∈ K : Ak /= ∅}| and n = n(P) := |{k ∈ K : Bk /= ∅}|, is called the complexity of P. The quantity T (G) := inf c(P), where the infimum

150

4 Amenable Groups

is taken over all 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. (a) Let H be a subgroup of G. Show that T (G) ≤ T (H ). (b) Let N be a normal subgroup of G. Show that T (G) ≤ T (G/N). (c) Show that there exists a finitely generated subgroup H of G such that T (H ) = T (G). (d) Show that one always has T (G) ≥ 4. (e) Show that T (G) = 4 if and only if G contains a subgroup isomorphic to F2 . (f) Suppose that all elements of G have finite order. Show that T (G) ≥ 6. 4.61 Let G be a group. A strong (left) paradoxical decomposition of G is a tuple Q = (I, (A'i )i∈I , (gi )i∈I ; J, (Bj' )j ∈J , (hj )j ∈J ), where I (resp. J ) is a finite index set, A'i ⊂ G (resp. Bj' ⊂ G) and gi ∈ G (resp. hj ∈ G) for all i ∈ I (resp. j ∈ J ), such that ) ) ( ( ' ' = ui∈I gi A'i = uj ∈J hj Bj' . .G = ui∈I Ai u uj ∈J Bj The number c' (Q) := |I |+|J | is called the complexity of Q. Let T ' (G) := inf c(Q), where the infimum is taken over all strong paradoxical decompositions Q of G and the convention that T ' (G) = +∞ if G admits no strong paradoxical decompositions is used. (a) Show that G admits a paradoxical decomposition if and only if it admits a strong paradoxical decomposition. (b) Show that G is amenable if and only if it does not admit any strong paradoxical decomposition. (c) Show that T (G) = T ' (G). (d) Suppose that G is non-amenable. Let Q = (I, (A'i )i∈I , (gi )i∈I ; J, (Bj' )j ∈J , (hj )j ∈J ) be a strong paradoxical decomposition of G such that c' (Q) = T (G) and denote by H (Q) ⊂ G the subgroup generated by the elements gi , hj , i ∈ I, j ∈ J . Show that one has T (H (Q)) = T (G). (e) Let Q = (I, (A'i )i∈I , (gi )i∈I ; J, (Bj' )j ∈J , (hj )j ∈J ) be a strong paradoxical decomposition of G. Show that there exists a strong paradoxical decomposition Q ' = (I, (A'i )i∈I , (gi )i∈I ; J, (Bj' )j ∈J , (hj )j ∈J ) such that gi = hj = 1G for some i ∈ I and j ∈ J . (f) Let m ≥ 1 and suppose that every subgroup H ⊂ G generated by m elements is amenable. Show that T (G) ≥ m + 3. (g) Recover from (f) the fact already established in Exercise 4.60.(d) that one always has T (G) ≥ 4.

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 pre-injective. 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 pre-injectivity 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 Sects. 5.10 and 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 non-existence of Garden of Eden configurations. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3_5

151

152

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 → A 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 . n U

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 pre-injective 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 pre-injective map .f : AG → Z may fail to be injective when G is infinite.

5.2 Pre-injective Maps

153

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 .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 → AG 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 G be the configuration defined by .x (1 ) := 1 and .x (g) := 0 if .g /= 1 . .x1 ∈ A 1 G 1 G 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 preinjective. Proof First recall from (1.16) the factorizations AG =

||

.

c∈G/H

Ac

and

τ=

||

τc ,

c∈G/H

where .τc : Ac → Ac satisfies .τc (x |c ) = (τ (x ))|c for all .x ∈ Ac .

(5.1)

154

5 The Garden of Eden Theorem

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) :=

x(g)

if g ∈ H,

a0

otherwise

.

{ and

y (g) :=

y(g)

if g ∈ H,

a0

otherwise

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

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 (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 +

5.4 Interiors, Closures, and Boundaries

155

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 5.7). (b) Let .G := Z2 and .A := {0, 1}. Consider the cellular automaton .τ : AG → AG 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). (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}

.

and

Ω +E := {g ∈ G : gE ∩ Ω /= ∅} (see Figs. 5.1 and 5.2). Observe that Ω −E =

n

Ωe−1

(5.2)

Ωe−1 = ΩE −1 .

(5.3)

.

e∈E

and Ω +E =

U

.

e∈E

The E-boundary of .Ω is the subset .∂E (Ω) of G defined by ∂E (Ω) := Ω +E \ Ω −E

.

(see Fig. 5.3). Observe that G \ Ω −E = {g ∈ G : gE /⊂ Ω} = {g ∈ G : gE ∩ (G \ Ω) /= ∅},

.

156

5 The Garden of Eden Theorem

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)

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

.f

so that .∂E (Ω) consists of all .g ∈ G such that the set .gE meets both .Ω and .G \ Ω, in formulæ ∂E (Ω) = {g ∈ G : gE ∩ Ω /= ∅ /= gE ∩ (G \ Ω)}.

.

5.4 Interiors, Closures, and Boundaries

157

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 Eboundary .∂E (Ω) = (Ω ∪ Ωa −1 ) \ (Ω ∩ Ωa −1 ) of a set .Ω when .E := {1G , a}

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)

.

is the symmetric difference between the sets .Ω and .Ωa −1 (see Fig. 5.4).

(5.4)

158

5 The Garden of Eden Theorem

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

.Ω

(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) (ii) (iii) (iv) (v) (vi) (vii) (viii)

(G \ Ω)−E = G \ Ω +E ; +E = G \ Ω −E ; .(G \ Ω) if .a ∈ E, then .Ω −E ⊂ Ωa −1 ⊂ Ω +E ; if .1G ∈ E, then .Ω −E ⊂ Ω ⊂ Ω +E ; if E is nonempty and .Ω is finite, then .Ω −E is finite; if E and .Ω are both finite, then .Ω +E and .∂E (Ω) are finite; if .E1 ⊂ E2 , then .Ω −E2 ⊂ Ω −E1 , .Ω +E1 ⊂ Ω +E2 and .∂E1 (Ω) ⊂ ∂E2 (Ω); 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).

5.4 Interiors, Closures, and Boundaries

159

(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).

.

U n 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 −S = G \ Ω +S by the first part of the proof and Proposition 5.4.2(i). .(G \ Ω) U n 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: (a) the net .(Fj )j ∈J is a right Følner net for G; (b) one has .

lim j

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

Proof (b) .⇒ (a). Suppose (b). Let .g ∈ G and take .E := {1G , g −1 }. By (5.4), we have Fj \ Fj g ⊂ ∂E (Fj ),

.

160

5 The Garden of Eden Theorem

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 ( ∂E (Fj ) =

) (

U

Fj a −1 \

.

a∈E

( =

) Fj b−1

b∈E

)

U

n

Fj a

−1

n

( G\

a∈E

( =

Fj a

−1

( n U

a∈E

=

U

) Fj b

−1

b∈E

)

U

n

) (G \ Fj b

−1

)

b∈E

(Fj a −1 \ Fj b−1 ).

a,b∈E

This implies |∂E (Fj )| ≤

Σ

.

|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 .

|∂E (Fj )| |Fj \ Fj g| ≤ |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). U n

5.4 Interiors, Closures, and Boundaries

161

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 Følner net .(Fj )j ∈J |∂ (F )| 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, there exists .j0 ∈ J such that |∂E (Fj )| . |Fj | < 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 Propositions 5.4.4 and 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). U n

162

5 The Garden of Eden Theorem

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 G → AG associated with the Game of Life (see Example 1.4.3(a)). Let .Ω := .τ : A {−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). Suppos 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: 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. Fig. 5.6 Two mutually erasable patterns for the Game of Life (recall that open circle denotes a dead cell, while filled circle denotes a live cell)

5.6 Tilings

163

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 finite sets .S 2 = {s1 s2 : s1 , s2 ∈ S} and +S 2 . Let us show that the patterns .p = x | and .p = x | are mutually .Ω := Σ 1 1 Ω 2 2 Ω 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 −S . Now observe that .Σ +S ⊂ Ω −S . Indeed, if .g ∈ .τ (y1 ) and .τ (y2 ) coincide on .Ω +S Σ , 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 U n 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 ∩Ug2 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 .E are subsets of G such that .E1 ⊂ E and .E ⊂ 1 ' ' E1 , then it is clear that T is also an .(E1 , E1 )-tiling of G.

164

5 The Garden of Eden Theorem

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, U 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. U n .

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 | ≤ |T | · |E |, which gives j

.

|Tj+ | |Fj |

≥

1 |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 )

5.7 Entropy

165

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

.

Using (5.10), we then deduce

.

|Tj+ | |∂E ' (Fj )| |∂E ' (Fj )| |Tj | 1 ≥ − ≥ ' − . |Fj | |Fj | |Fj | |E | |Fj |

Hence we have .

|Tj | 1 ≥α= 2|E ' | |Fj | U n

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

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)| |Fj |

=

|Fj | log |A| log |A||Fj | = = log |A|. |Fj | |Fj |

Thus we have .entF (X) = log |A|.

166

5 The Garden of Eden Theorem

Fig. 5.7 The set .T7 := T ∩ F7−E ⊂ Z2 , where .E := {(0, 0), (1, 0)}, .E ' := {0, 1} × {0, 1} and |T | 12 3 2 1 2 ' 2 .T := (2Z + 1) × (2Z + 1) ⊂ Z is an .(E, E )-tiling in .Z . Note that . 7 = |F7 | 64 = 16 ≥ 16 = 8 = 1 2|E ' |

=α

(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). U n 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:

5.7 Entropy

167

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. 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 Ω\Ω −S . This implies .πΩ (Y ) ⊂ πΩ −S (Y ) × A −S

.

log |πΩ (Y )| ≤ log |πΩ −S (Y ) × AΩ\Ω | −S

= log |πΩ −S (Y )| + log |AΩ\Ω | = 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 (Y )| |Fj |

≤

log |πFj (X)| |Fj |

+

|∂S (Fj )| log |A|. |Fj |

168

5 The Garden of Eden Theorem

Since .

lim j

|∂S (Fj )| =0 |Fj |

by Proposition 5.4.4, we finally get .

entF (Y ) = lim sup

log |πFj (Y )|

≤ lim sup

|Fj |

j

log |πFj (X)|

j

|Fj |

= entF (X). U n

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) C 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

.

Fj∗ := Fj \ Ug∈Tj gE

.

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

.

As ∗

πFj (X) ⊂ AFj ×

||

.

πgE (X),

g∈Tj

we get ∗

.

log |πFj (X)| ≤ log |AFj ×

||

πgE (X)|

g∈Tj

= |Fj∗ | log |A| +

Σ

log |πgE (X)|

g∈Tj

≤ |Fj∗ | log |A| +

Σ

g∈Tj

log(|A||gE| − 1)

(by (5.12))

(5.12)

5.7 Entropy

169

Fig. 5.8 The set .F7∗ := F7 \ Ug∈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)

= |Fj∗ | log |A| +

Σ

|gE| log |A| +

g∈Tj

Σ

log(1 − |A|−|gE| )

g∈Tj

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

Σ

.

|gE|

and

|gE| = |E|.

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.

170

5 The Garden of Eden Theorem

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)| |Fj |

≤ log |A| − cα

for all j ≥ j0 .

This implies that .

entF X = lim sup j

log |πFj (X)| |Fj |

≤ log |A| − cα < log |A|. U n

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

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) C 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, E )-tiling .T ⊂ G. Since .πE (X) C A and X is G-invariant, we have .πgE (X) C AgE for all .g ∈ G. This implies .entF (X) < log |A| by Proposition 5.7.4. U n

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. 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|.

5.8 Proof of the Garden of Eden Theorem

171

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 )) C AE . The set .τ (AG ) is G-invariant since .τ is G-equivariant by Proposition 1.4.4. We deduce that G .entF (τ (A )) < log |A| by applying Corollary 5.7.5. U n 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 −S .F ⊂ Fj ⊂ Fj+S by Proposition 5.4.2(iv) and therefore .Fj+S \ Fj ⊂ ∂S (Fj ). As j +S

πF +S (Y ) ⊂ πFj (Y ) × AFj

.

\Fj

j

.

, it follows that

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 )| |Fj |

+

|∂S (Fj )| log |A|. |Fj |

(5.14)

As .

ent(Y ) = lim sup

log |πFj (Y )|

j

|Fj |

< log |A|

by hypothesis, and .

lim j

|∂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

.

|Fj0 |

< log |A|.

(5.15)

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 |πF +S (Y )| < |A||Fj0 | = |Z|.

.

j0

172

5 The Garden of Eden Theorem

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

|τ (Z)| = |πF

.

j0+S

(τ (Z))| ≤ |πF

j0+S

(Y )| < |Z|.

This implies that we may find distinct configurations .z1 , z2 ∈ Z such that .τ (z1 ) = τ (z2 ). Since .z1 and .z2 coincide outside the finite set .Fj0 , this shows that .τ is not pre-injective. U n 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) C 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

173

Let .z ∈ AG be the configuration defined by { z(g) :=

.

hx2 (g)

if there is h ∈ T ' such that g ∈ hE,

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 −1 s = gs ∈ hΩ. As .s −1 s ∈ R, this implies .gss 0 0 0 gs ∈ (hΩ)+R = hΩ +R .

(by Proposition 5.4.2(viii))

= hE.

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. U n Proof of Theorem 5.8.1 If .τ is surjective, then .τ (AG ) = AG and hence G G .entF (τ (A )) = entF (A ) = 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. U n

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. 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

174

5 The Garden of Eden Theorem

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 Γ :=

||

.

Λs

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 −1 h ∈ S and .φ (g) /= φ (h) since .(φ (g))−1 φ (h) = φ (g −1 h) = .Ω. Then .s := g s s s s s φs (s) /= 1Λs . This implies .ρ(g) /= ρ(h). Therefore, the restriction of .ρ to .Ω is injective. U n Proof of Theorem 5.9.1 Since every injective cellular automaton is pre-injective, the Garden of Eden theorem (Theorem 5.3.1) implies that every amenable group is surjunctive. By applying Lemmas 3.3.4 and 5.9.2, it follows that every residually U n 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.

U n

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

175

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 pre-injective 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 ⎧ ⎪ ⎪ ⎨1 .τ (x)(g) := 0 ⎪ ⎪ ⎩x(g)

if if if

Σ Σ

s∈S

x(gs) > 2,

Σ

s∈S

x(gs) < 2,

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 G be the configuration defined by .x(g) := y(ψ(g)) .ψ(g) = s1 s2 . . . sn−1 . Let .x ∈ A 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.

176

5 The Garden of Eden Theorem

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 pre-injective cellular automaton with finite alphabet over an amenable group is 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 preinjective but not surjective. Proof The image of .τ is contained in .(H × {0})G C (H × H )G = AG . Thus .τ is not surjective. Let us show that .τ is pre-injective. Suppose not. Then there exist configurations G satisfying .τ (x ) = τ (x ) such that the set .Ω := {g ∈ G : x (g) /= .x1 , x2 ∈ A 1 2 1 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 \ {s0−1 }, 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.

U n

5.12 A Characterization of Amenability in Terms of Cellular Automata

177

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 2 .F2 whose alphabet is finite of cardinality .n . From Proposition 1.7.4 we deduce the following: .

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 preinjective but not surjective. U n

5.12 A Characterization of Amenability in Terms of Cellular Automata In Sect. 5.10, it was shown that if G is a group containing a free subgroup of rank 2, then there exists a cellular automaton with finite alphabet over G which is surjective but not pre-injective. In fact, the following theorem shows that the existence of such a cellular automaton extends to 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 (ϕ(g))−1 g ∈ S for all g ∈ G.

.

(5.19)

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

.

μ(y) :=

.

⎧ ' ' ⎪ ⎪ ⎨(s , t )

if there exists a unique element (s, t) ∈ S × S with s < t

⎪ ⎪ ⎩(s , s ) 0 0

otherwise,

such that y(s) = (s, s ' ) and y(t) = (t, t ' ), where s ' , t ' ∈ S, (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 G .x0 , x1 ∈ A , 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.

178

5 The Garden of Eden Theorem

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)). 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. U n 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 pre-injective. U n

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 (Fig. 5.9). 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 Z2 satisfies .x| .x ∈ {0, 1} Cn = u (the fact that .(τ (x))|Dn does not depend of the

5.13 Garden of Eden Patterns for Life

179

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

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). 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)

2

)n = 2(25+log2 (1−2

Therefore we have .|τn (Xn )| < |Yn | if (25 + log2 (1 − 2−25 ))n2 < 25n2 − 20n + 4.

.

−25 ))n2

.

180

5 The Garden of Eden Theorem

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

This inequality is equivalent to .

− n2 log2 (1 − 2−25 ) − 20n + 4 > 0,

which is verified if and only if n>

.

2 (5 + − log2 (1 − 2−25 )

/

25 + log2 (1 − 2−25 )),

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

.

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.

.

U n

Notes

181

Notes According to M. Gardner (see [Gar2, 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 MooreMyhill theorem. The next step in the proof of the Garden of Eden theorem was done by A. Machì 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 all finitely generated amenable groups by A. Machì, 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], and [CCK]) 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 [Mul] (unpublished class notes). The underlying group was the modular group .G := PSL(2, Z) = (Z/2Z) ∗ (Z/3Z) (note that G contains the free group .F2 ). Theorem 5.12.1 is due to L. Bartholdi [Bar1]. The computation in Sect. 5.13 is taken from [BerCG, p. 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. It was proved that there exist no Garden of Eden patterns for Life with support contained in a rectangle of size .6 × 5. In the internet site https://conwaylife.com/wiki/Garden_of_Eden, the interested reader may find a comprehensive and updated list of “records” for “minimal” Garden of Eden patterns for Life. Several Garden of Eden type theorems have been established for certain classes of subshifts and other topological dynamical systems (see [CeC1, CeC4, CeC13, CeC22], and [CeC23]; [CCL1] in collaboration with H. Li; [CCP2] in collaboration with X.K. Phung; [Dou] by M. Doucha; [Fio1] and [Fio2] by F. Fiorenzi; [Li]

182

5 The Garden of Eden Theorem

by Li; and [Phu1, Phu5], and [Phu10] by Phung). See the recent survey [CCP6, Section 9] for more details.

Exercises 5.1 (Homoclinic Points) Let X be a uniform space equipped with an action of a group G. One says that two points x, y ∈ X are homoclinic if they satisfy the following condition: for every entourage U of X, there exists a finite subset Ω = Ω(U ) ⊂ G such that (gx, gy) ∈ U for all g ∈ G \ Ω. (a) Show that homoclinicity is an equivalence relation on X. (b) Suppose that the uniform structure on X is induced by a metric d. Show that x, y ∈ X are homoclinic if and only if the following holds: (MH) for every ε > 0 there exists a finite subset Ω = Ω d (ε) ⊂ G such that d(gx, gy) < ε for all g ∈ G \ Ω. (c) Let Y be a uniform space equipped with an action of G and suppose that there exists a uniformly continuous G-equivariant map f : X → Y . Show that if x, y ∈ X are homoclinic then f (x), f (y) ∈ Y are homoclinic. 5.2 Let G be a group and let A be a set. Equip AG with its prodiscrete uniform structure and the shift action of G. (a) Show that two configurations in AG are homoclinic if and only if they are almost equal. (b) Let X ⊂ AG be a subset, let Z be a set, and let f : X → Z be a map. Show that f is pre-injective if and only if its restriction to each homoclinicity class in X is injective. 5.3 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. Let μ : AM → A denote the associated local defining map. Show that the following conditions are equivalent: (i) (ii) (iii) (iv) (v)

τ is pre-injective; τ is injective; μ is injective μ is surjective; τ is surjective.

5.4 Let G be an infinite group and let A be a finite set. Let τ : AG → AG be a non-surjective cellular automaton. Show that there are uncountably many Garden of Eden configurations in AG .

Exercises

183

5.5 (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 { μ(y) :=

Σ

y(s) = 2

1

if

0

otherwise

.

s∈S

for all y ∈ AS . (a) Show that τ is not pre-injective. (b) Deduce from (a) that τ is not surjective either. (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 . 5.6 Let G := Z, let A := {0, 1}, let S := {−1, 0, 1}, and let τ : AG → AG denote the associated majority action cellular automaton (cf. Example 1.4.3(c)). Thus, one has { Σ 1 if s∈S x(n + s) ≥ 2 .τ (x)(n) := Σ 0 if s∈S x(n + s) ≤ 1, for all x ∈ AG and n ∈ Z. As observed in Example 5.2.1(c), the celluar automaton τ is not pre-injective so that, by amenability of Z and the Garden of Eden Theorem, τ is not surjective either (cf. Example 5.3.2(c)). It then 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 . 5.7 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 and pre-injective. 5.8 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) + x(n + 1) 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. 5.9 Let G be a group. Let A and B be sets. (a) Let τ : AG → B G be a cellular automaton. Suppose that the configurations x, x ' ∈ AG are almost equal. Show that the configurations τ (x), τ (x ' ) ∈ B G are almost equal.

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5 The Garden of Eden Theorem

(b) Let X ⊂ AG , Y ⊂ B G be subshifts and let σ : X → Y be a cellular automaton. Suppose that the configurations x, x ' ∈ X are almost equal. Show that the configurations τ (x), τ (x ' ) ∈ Y are almost equal. 5.10 Let G be a group and let A, B be sets. Let X ⊂ AG , Y ⊂ B G be subshifts and let Z be a set. Suppose that τ : X → Y is a pre-injective cellular automaton and that f : Y → Z is a pre-injective map. Show that the composite map f ◦ τ : X → Z is pre-injective. 5.11 Let G be a group and let A, B, C be sets. Let X ⊂ AG , Y ⊂ B G , and Z ⊂ C G be subshifts. Suppose that τ : X → Y and σ : Y → Z are pre-injective cellular automata. Show that the composite cellular automaton σ ◦ τ : X → Z is pre-injective. 5.12 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. 5.13 (The Garden of Eden Theorem for Locally Finite Groups) Let G be a locally finite group and let A be a finite set. Let τ : AG → AG be a cellular automaton. Show that the following conditions are all equivalent: (C1) τ is pre-injective; (C2) τ is injective; (C3) τ is surjective. 5.14 (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

μ(p) :=

.

⎧ ⎪ ⎪ ⎪ ⎪ ⎨1 ⎪ ⎪ ⎪ ⎪ ⎩

⎧Σ ⎪ ⎪ ⎨ s∈S p(s) = 3 if

or ⎪ ⎪ ⎩Σ

s∈S

0

p(s) = 4 and p(1G ) = 1,

otherwise

for all p ∈ AS (cf. Example 1.4.3(a)). Show that τ is surjective but not pre-injective. 5.15 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}, F := {1, 2, 3}, and Ω := {−2, −1, 1}. Determine the three sets ∂F (∂E (Ω)), ∂E (∂F (Ω), and ∂E+F (Ω), and check that these sets are all distinct. 5.16 Let G be a group and let E ⊂ G. Determine the sets ∂E (∅) and ∂E (G).

Exercises

185

5.17 Let G be a group. Let E and Ω be subsets of G. Show that ∂E (G \ Ω) = ∂E (Ω). 5.18 Let G be a group. Let E, Ω1 and Ω2 be subsets of G. (a) Show that ∂E (Ω1 ∪ Ω2 ) ⊂ ∂E (Ω1 ) ∪ ∂E (Ω2 ). (b) Give an example showing that one may have ∂E (Ω1 ∪Ω2 ) /= ∂E (Ω1 )∪∂E (Ω2 ). 5.19 Let G be a group. Let E, Ω1 and Ω2 be subsets of G. (a) Show that ∂E (Ω1 \ Ω2 ) ⊂ ∂E (Ω1 ) ∪ ∂E (Ω2 ). (b) Give an example showing that one may have ∂E (Ω1 \ Ω2 ) /= ∂E (Ω1 ) ∪ ∂E (Ω2 ). 5.20 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. 5.21 Let G be a group. Show that a subset R ⊂ G is syndetic (cf. Exercise 3.48) if and only if there exists a finite subset S ⊂ G such that the set Ω := R −1 satisfies Ω +S = G. −1

5.22 Let G be a group. Let E and Ω be subsets of G. Show that Ω ⊂ (Ω +E )−E . 5.23 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 configurations −1 x1 , x2 ∈ AG coincide on Ω +M then the configurations τ (x1 ) and τ (x2 ) coincide on Ω. 5.24 Let (A, +) be an abelian group, let k be a positive integer, and let ϕ : Ak → A be a map. Consider the map τ : AZ → AZ defined by τ (x)(n) := x(n) + ϕ(x(n + 1), x(n + 2), . . . , x(n + k))

.

for all n ∈ Z and x ∈ AZ . (a) (b) (c) (d) (e)

Show that τ is a cellular automaton over the group Z and the alphabet A. Show that τ is pre-injective. Show that the image of τ is dense in AZ for the prodiscrete topology. Show that if A is finite then τ is surjective. Take A := Z, k := 1, and ϕ : A → A given by ϕ(a) := −3a for all a ∈ A, so that τ (x)(n) = x(n) − 3x(n + 1) for all n ∈ Z and x ∈ AZ . Show that the cellular automaton τ is injective but not surjective and that the image of τ is not closed in AZ for the prodiscrete topology.

5.25 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.

186

5 The Garden of Eden Theorem

5.26 Let G be an amenable group, let A be a finite set, let F be a right Følner net for G, and let X ⊂ AG be a subset. Let Ω be a finite subset of G such that 1G ∈ Ω. Show that .

lim sup

log |πFj Ω (X)| |Fj |

j

= entF (X).

5.27 (Topological Entropy) Let G be an amenable group, let F = (Fj )j ∈J be a right Følner net for G, and let X be a compact topological space equipped with a continuous action of G. Suppose that U = (Ui )i∈I is an open cover of X. By compactness, U admits a finite subcover. We denote by N (U ) the minimal cardinality of a finite subcover of U . Thus, N(U ) is the smallest U integer n ≥ 0 such that there is a finite subset I0 ⊂ I of cardinality |I0 | = n with i∈I0 Ui = X. (a) Let U = (Ui )i∈I and V = (Vk )k∈K be two open covers of X. Suppose that V is finer than U , that is, for each k ∈ K, there exists i ∈ I such that Vk ⊂ Ui . Show that N(U ) ≤ N(V ). (b) Suppose that U = (Ui )i∈I is 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 ). Let I0 := {i ∈ I : Ui /= ∅}. Show that N(U ) = |I0 |. (c) Let U = (Ui )i∈I be an open cover of X and let F be a non-empty finite subset of G. Consider the family U (F ) = (Wα )α∈I F indexed by the set I F = {α : F → I } defined by Wα =

n

gUα(g)

.

g∈F

for all α ∈ I F . Show that U (F ) is an open cover of X and that N (U (F ) ) ≤ N (U )|F | . (d) Let U be an open cover of X. Define hF (U ) := lim sup

.

j

log N(U (Fj ) ) . |Fj |

Show that hF (U ) ≤ log N(U ). (e) The topological entropy of the dynamical system (X, G) is the quantity hF (X, G) defined by hF (X, G) = sup hF (U ),

.

U

where U ranges over all open covers of X. Suppose that G acts continuously on another compact space Y and that the dynamical systems (X, G) and (Y, G) are topologically conjugate, i.e., there is a G equivariant homeomorphism ϕ : X → Y . Show that hF (X, G) = hF (Y, G).

Exercises

187

(f) 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).

.

5.28 Let G be an amenable group and let F = (Fj )j ∈J be a right Følner net for G. Let A and B be two finite sets and let X ⊂ AG and Y ⊂ B G be two subshifts. Suppose that the subshifts X and Y are topologically conjugate, i.e., there exists a G-equivariant homeomorphism ϕ : X → Y . Show that entF (X) = entF (Y ). 5.29 Let G be an amenable group, let F = (Fj )j ∈J be a right Følner net for G, and let A be 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). 5.30 Let G be a finite group, let F = (Fj )j ∈J be a right Følner net for G, and let A be a finite set. Let X ⊂ AG be a subset. Show that .

entF (X) =

log |X| . |G|

5.31 Let G be an infinite amenable group, let F = (Fj )j ∈J be a right Folner net for G, and let A be a finite set. Let X ⊂ AG be a non-empty finite subset. Show that entF (X) = 0. 5.32 Let G be an infinite amenable group and let F = (Fj )j ∈J be a right Følner net for G. Let A := {0, 1} and let X ⊂ AG denote the at-most-one-one subshift (cf. Exercise 1.71). Show that entF (X) = 0. 5.33 Let G be an amenable group, let F = (Fj )j ∈J be a right Følner net for G, and let A be a finite set. Let X and Y be two non-empty subshifts of AG . (a) Suppose that G is infinite or that either X or Y has more than one configuration. Show that entF (X ∪ Y ) ≤ entF (X) + entF (Y ). (b) Suppose that G is finite. Let A := {0, 1} and set X := {x0 } and Y := {x1 }, where xi : G → A is the constant configuration xi (g) = i for all g ∈ G, i = 0, 1. Check that entF (X ∪ Y ) /≤ entF (X) + entF (Y ). 5.34 Let G be an amenable group, let A be a finite set, let F be a right Følner net for G, and let X ⊂ AG be a subshift. Let F be a non-empty finite subset of G and consider the subshift X[F ] ⊂ B G , where B := AF , introduced in Exercise 1.95. Show that entF (X[F ] ) = entF (X). 5.35 Let G be an amenable group and let H be a finite index subgroup of G. Let T ⊂ G be a complete set of representatives for the right cosets of H in G. Let FH = (Fj )j ∈J be a right Følner net for H .

188

5 The Garden of Eden Theorem

(a) Show that FG := (Fj T )j ∈J is a right Følner net for G. (b) Let A be a finite set and set B := AT . Let X ⊂ AG be a subshift and consider the subshift X(H,T ) ⊂ B H (cf. Exercise 1.72). Show that entFH (X(H,T ) ) = [G : H ] entFG (X). 5.36 A sequence (un )n≥1 of real numbers is said to be subadditive (resp. submultiplicative) if it satisfies un+m ≤ un + um (resp. un+m ≤ un um ) for all n, m ≥ 1. (a) Let (un )n≥1 be a subadditive sequence ( u ) of real numbers such that un ≥ 0 for all n n ≥ 1. Show that the sequence is convergent and that one has n n≥1 .

lim

n→∞

un un = inf . n≥1 n n

(b) Let (vn )n≥1 be a submultiplicative sequence of)real numbers such that vn ≥ 1 ( log vn for all n ≥ 1. Show that the sequence is convergent and that one n n≥1 has .

lim

n→∞

log vn log vn = inf . n≥1 n n

(c) Let A be a finite set and let X ⊂ AZ be a non-empty subshift. Consider the Følner sequence F for Z defined by F = (Fn )n≥1 , where Fn := {0, 1, . . . , n − 1} for all n ≥ 1. For n ≥ 1, let Ln (X) ⊂ A(∗ denote the set ) of words of length n log |Ln (X)| that appear in X. Show that the sequence is convergent and n n≥1 that one has .

entF (X) = lim

n→∞

log |Ln (X)| log |Ln (X)| = inf . n≥1 n n

(5.21)

5.37 (Entropy of the Golden Mean and Even Subshifts) Let A := {0, 1}. Let X ⊂ AZ and Y ⊂ AZ denote the golden mean subshift and the even subshift, respectively. Consider the Følner sequence F = (Fn )n≥1 for Z given by Fn := {0, 1, . . . , n − 1}. The goal of the exercise is to show that entF (X) = entF (Y ) = log ϕ, where √ 1+ 5 .ϕ := 2 is the golden mean.

Exercises

189

(a) A sequence (fn )n≥1 of real numbers is called a Fibonacci sequence if it satisfies the recurrence relation fn+2 = fn+1 + fn

.

(5.22)

for all n ≥ 1. Let (fn )n≥1 be a Fibonacci sequence and write a :=

.

f2 (ϕ − 1) + f1 (2 − ϕ) f2 ϕ − f1 (ϕ + 1) and b := . √ √ 5 5

(5.23)

Show that fn = aϕ n + b(1 − ϕ)n

.

(5.24)

for all n ∈ N. (b) Let (fn )n≥1 be a Fibonacci sequence such that f1 , f2 > 0. Show that .

lim

n→∞

log fn = log ϕ. n

(5.25)

(c) For n ≥ 1, let Ln (X) = πFn (X) ⊂ A∗ denote the set of words of length n that appear in X. Show that the sequence (|Ln (X)|)n≥1 is a Fibonacci sequence and deduce that entF (X) = log ϕ. (d) For n ≥ 1, let Ln (Y ) = πFn (Y ) ⊂ A∗ denote the set of words of length n that appear in Y . Show that the sequence (1 + |Ln (Y )|)n≥1 is a Fibonacci sequence and deduce that entF (Y ) = log ϕ. 5.38 (Entropy of the Morse Subshift) Let A := {0, 1} and let X ⊂ AZ denote the Morse subshift (cf. Exercise 3.52). For n ≥ 1 let Ln (X) ⊂ A∗ denote the set of words of length n that appear in X. (a) Show that |Ln (X)| ≤ 8n for all n ≥ 1. (b) Let F := (Fn )n≥1 denote the Følner sequence for Z defined by Fn := {0, 1, . . . , n − 1}. Show that entF (X) = 0. 5.39 Let A := {0, 1, 2}. Let 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 C X. (b) Show that X is not topologically transitive. (c) Show that X is not irreducible. (d) Check that X and Y have the same entropy entF (X) = entF (Y ) = log 2 with respect to the Følner sequence F = (Fn )n≥1 for Z given by Fn := {0, 1, . . . , n − 1}.

190

5 The Garden of Eden Theorem

5.40 Let G be an amenable group, let A be a finite set, and let F = (Fj )j ∈J be a right Følner net for G. Suppose that X ⊂ AG is a strongly irreducible subshift containing at least two distinct configurations. Show that one has entF (X) > 0. 5.41 Let G be an amenable group, let F = (Fj )j ∈J be a right Følner net for G, and let A be a finite set. Let X ⊂ AG be a strongly irreducible subshift. Given a subset Ω ⊂ G, denote by πΩ : AG → AΩ the projection map. Let Δ be a finite subset of G such that 1G ∈ Δ and X is Δ-irreducible. Let D be a finite subset of G. Set E := D +Δ and E ' := EE −1 = {ab−1 : a, b ∈ E}. Let T ⊂ G be an (E, E ' )-tiling. Suppose now that Z is a subset of X such that πgD (Z) C πgD (X)

.

for all g ∈ T .

(5.26)

(a) Write χ := |πE (X)| and Tj := {g ∈ T : gE ⊂ Fj }. Show that |πFj (Z)| ≤ (1 − χ −1 )|Tj | |πFj (X)|.

.

for all j ∈ J . (b) Deduce from (a) that entF (Z) < entF (X). 5.42 Let G be an amenable group, let F = (Fj )j ∈J be a right Følner net for G, and let A be a finite set. Suppose that X and Y are subshifts of AG with X strongly irreducible and Y C X. Show that entF (Y ) < entF (X). 5.43 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 family of (possibly infinite) subsets of G, indexed by a (possibly infinite) set I , such that Ωj+Δ ∩ Ωk = ∅

.

for all distinct j, k ∈ I.

(5.27)

Suppose also that (xi )i∈I is 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 a closed non-empty subset of X for each Λ ∈ Pf (G). (b) Let n be a positive integer and let Λ1 , Λ2 , . . . , Λn ∈ Pf (G). Show that X(Λ1 ) ∩ X(Λ2 ) ∩ · · · ∩ X(Λn ) /= ∅. (c) Show that there exists a configuration x ∈ X such that x coincides with xi on Ωi for all i ∈ I . 5.44 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 .

Exercises

191

5.45 Let G be a group and let A be a finite set. Let X ⊂ AG be a strongly irreducible subshift. Let x ∈ X and let CX (x) denote the set consisting of all y ∈ X such that y is almost equal to x. Show that CX (x) is dense in X. 5.46 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 subshifts with X strongly irreducible and entF (Y ) < entF (X). Let τ : X → Y be a cellular automaton. For Ω ⊂ G, let πΩ : AG → AΩ and ηΩ : B G → B Ω denote the projection maps. (a) Let S be a finite subset of G that is a memory set for τ . Up to enlarging the subset S if necessary, one can also suppose that 1G ∈ S and X is S-irreducible. Show that there exists j0 ∈ J such that |η +S 2 (Y )| < |πFj0 (X)|. Fj

0

(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 \ Fj+S . 0 Show that |Z| ≥ |πFj0 (X)|. (c) Show that τ (z) coincides with τ (x0 ) on G \ Fj+S for all z ∈ Z. 0 (d) Show that |τ (Z)| = |η +S 2 (τ (Z))|. 2

Fj

0

(e) Show that there exist distinct configurations z1 , z2 ∈ Z such that τ (z1 ) = τ (z2 ). (f) Show that τ is not pre-injective. 5.47 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 strongly irreducible subshifts such that entF (X) = entF (Y ). Show that every pre-injective cellular automaton τ : X → Y is surjective. 5.48 (The Moore and the Myhill Properties) Let G be an infinite group and let A be a set. One says that a subshift X ⊂ AG has the Moore property (resp. the Myhill property) if every surjective (resp. pre-injective) cellular automaton τ : X → X is pre-injective (resp. surjective). (a) Show that every subshift X ⊂ AG that has the Myhill property is surjunctive. (b) Let A := {0, 1} and let X ⊂ AG denote the subshift consisting of the two constant configurations. Show that the subshift X is surjunctive and that it has the Moore property but not the Myhill property. (c) Let A := {0, 1} and let X ⊂ AG denote the at-most-one-one subshift (cf. Exercise 1.71). Show that the subshift X is surjunctive and that it has the Moore property but not the Myhill property. (d) Let G be an amenable group and let A be a finite set. Show that the subshift X := AG has both the Moore and the Myhill property. (e) Let G be a free group of rank 2 and let A := {0, 1, 2, 3}. Show that the subshift X := AG has neither the Moore property nor the Myhill property. 5.49 Let G be a group. Let A and B be finite sets. Suppose that X ⊂ AG and Y ⊂ B G are topologically conjugate subshifts. Show that X has the Myhill property

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5 The Garden of Eden Theorem

(resp. the Moore property) if and only if Y has the Myhill property (resp. the Moore property). 5.50 Let G be an amenable group and let A be a finite set. Show that every strongly irreducible subshift X ⊂ AG has the Myhill property. 5.51 Let G be an amenable group and let A be a finite set. Show that every strongly irreducible subshift X ⊂ AG is surjunctive. 5.52 Let G be an amenable group, let F = (Fj )j ∈J be a right Følner net for G, and let A be a finite set. Let X ⊂ AG be a strongly irreducible subshift of finite type and let τ : X → AG be a cellular automaton. Suppose that τ is not pre-injective. Thus, there exist configurations x1 , x2 ∈ X such that τ (x1 ) = τ (x2 ) and the set D := {g ∈ G : x1 (g) /= x2 (g)} is a non-empty 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 −1 suppose that 1G ∈ S and that X is S-irreducible. Set E := D +(S S) = {ds −1 s ' : d ∈ D and s, s ' ∈ S} and E ' := EE −1 = {ab−1 : a, b, ∈ E}. Let T ⊂ G be an (E, E ' )-tiling. Finally, let Z ⊂ X be the set consisting of all configurations z in X such that z|tD /= (tx1 )|tD for all t ∈ T . (a) Show that entF (Z) < entF (X). (b) Show that entF (τ (Z)) ≤ entF (Z). (c) Let x ∈ X. Show that the configuration z ∈ AG defined by { z(g) :=

.

tx2 (g)

if there is t ∈ T such that g ∈ tE and x|tE = tx1 |tE ,

x(g)

otherwise.

satisfies z ∈ Z and τ (z) = τ (x). (d) Show that τ (Z) = τ (X). (e) Show that entF (τ (X)) < entF (X). 5.53 Let G be an amenable group, let F = (Fj )j ∈J be a right Følner net for G, and let A be a finite set. Suppose that X, Y ⊂ AG are 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. 5.54 (The Garden of Eden Theorem for Strongly Irreducible Subshifts of Finite Type) Let G be an amenable group and let A be a finite set. Let X ⊂ AG be a strongly irreducible subshift of finite type and let τ : X → X be a cellular automaton. Show that τ is surjective if and only if it is pre-injective. 5.55 Let G be a group. Let F be a non-trivial 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 the infinite cyclic subgroup of G 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 that satisfy

Exercises

193

x2 (gh0 ) = x2 (g) for all 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 of AG . (b) Show that the map τ : X → X, defined by τ (x) := (x1 + x2 , 0) for all x = (x1 , x2 ) ∈ X, is a cellular automaton. (c) Show that the cellular automaton τ : X → X 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 topologically transitive. 5.56 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. Let X ⊂ AH be a subshift and denote by X(G) ⊂ AG the associated subshift defined in Exercise 1.103.(a). Suppose that σ (X) ⊂ X. (a) Show that σ G (X(G) ) ⊂ X(G) . (b) 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. 5.57 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) (b) (c) (d) (e)

Show that X := {x0 , x1 } is a subshift of finite type. Describe all the cellular automata τ : X → X. Show that X has the Moore property. Show that X does not have the Myhill property. Show that X is surjunctive.

5.58 Let A := {0, 1}, let G := Z2 , and let H := Z × {0} ⊂ G. Consider the subset X := Fix(H ) ⊂ AG consisting of all the configurations x ∈ AG that are fixed by each element of H . (a) Show that X is a subshift of finite type of AG . (b) Show that the subshift X is topologically transitive and irreducible but neither topologically mixing nor strongly irreducible. (c) Show that two configurations in X which are almost equal are equal. (d) Show that X has the Moore property. (e) 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. (f) Show that X does not have the Myhill property. (g) Show that X is surjunctive. 5.59 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 topologically transitive, not irreducible, and not topologically mixing.

194

5 The Garden of Eden Theorem

(b) Consider the cellular automaton σ : AZ → AZ with memory set S := {0, 1} and local defining map μ : AS → A defined by { μ(p) :=

p(0)

if p(1) /= 0

0

otherwise.

.

Show that σ (X) ⊂ X. (c) Show that the cellular automaton τ : X → X, defined by τ (x) := σ (x) for all x ∈ X, is surjective but not pre-injective. (d) Show that X does not have the Moore property. (e) Show that X is not surjunctive. (f) Show that X does not have the Myhill property. 5.60 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 and suppose that X ⊂ AG is a splicable subshift. Let τ : X → B G be a pre-injective cellular automaton. Show that one has entF (τ (X)) = entF (X). 5.61 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 and suppose that X ⊂ AG is a splicable subshift and that Y ⊂ B G is a strongly irreducible subshift such that entF (X) = entF (Y ). Show that every pre-injective cellular automaton τ : X → Y is surjective. 5.62 Let A := {0, 1, 2} and let X ⊂ AZ and σ : AZ → AZ be as in Exercise 5.59. 2 2 Let H := Z = Z × {0} ⊂ Z2 , let X(Z ) be the associated subshift in AZ defined as 2 2 2 in Exercise 1.103, and let σ Z : AZ → AZ be the induced cellular automaton. 2

(a) Show that the subshift X(Z ) is topologically transitive. 2 (b) Show that X(Z ) is of finite type. 2 2 2 (c) Show that the cellular automaton σ Z |X(Z2 ) : X(Z ) → X(Z ) is surjective but not pre-injective. 5.63 (Post-surjective Cellular Automata) Let G be a group and let A be a set. Let X ⊂ AG be a subshift. For x ∈ X, let CX (x) ⊂ X denote the homoclinicity class of x in X, i.e., the set consisting of all configurations y ∈ X such that y is almost equal to x. One says that a cellular automaton τ : X → X is post-surjective if it satisfies the following condition: if x ∈ X and z ∈ CX (τ (x)), then there exists y ∈ CX (x) such that z = τ (y). (a) Let τ : X → X be a cellular automaton and let x ∈ X. Show that τ induces by restriction a map τx : CX (x) → CX (τ (x)). (b) Let τ : X → X be a cellular automaton. Show that τ is pre-injective (resp. postsurjective) if and only if it satisfies the following condition: for every x ∈ X, the map τx : CX (x) → CX (τ (x)) is injective (resp. surjective).

Exercises

195

(c) Show that if a cellular automaton τ : X → X is invertible, i.e., τ is bijective and τ −1 : X → X is a cellular automaton, then τ is post-surjective. (d) Let K be a field. Take G := Z, the additive group of integers, and A := K[[t]], the ring of formal power series with coefficients in K. Show that the map τ : AG → AG defined by τ (x)(n) = x(n) − tx(n + 1) for all x ∈ AG and n ∈ Z, is a bijective cellular automaton which is not post-surjective. (e) Show that if G is finite then a cellular automaton τ : X → X is post-surjective if and only if it is surjective. (f) Suppose that the set A is finite and that the subshift X is strongly irreducible (e.g. X = AG ). Show that if τ : X → X is a post-surjective cellular automaton then τ is surjective. (g) Give an example showing that the hypothesis that X is strongly irreducible cannot be removed in (f). (h) Take G := Z, A := {0, 1} = Z/2Z, and the cellular automaton τ : AG → AG given by τ (x)(n) := x(n) + x(n + 1) for all x ∈ AG and n ∈ G. Show that τ is surjective but not post-surjective. 5.64 Consider the map τ : RZ → RZ defined by τ (x)(n) := x(n + 1) − x(n)2

.

(a) (b) (c) (d) (e) (f)

for all x ∈ RZ and n ∈ Z.

Show that τ is a cellular automaton over the group Z and the alphabet R. Show that τ is not injective. Show that τ is pre-injective. Show that τ (RZ ) is dense in RZ for the prodiscrete topology. Show that τ is not surjective. Show that τ (RZ ) is not closed in RZ for the prodiscrete topology.

Chapter 6

Finitely Generated 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 .l2 -spectrum of the associated Laplacian. In Sect. 6.13 we consider the notions of uniform embedding and quasi-isometry for (possibly uncountable) groups and we show that amenability is a quasi-isometry invariant (Theorem 6.13.19). Finally, in Sect. 6.14, we introduce the notion of a quasi-isometric embedding between finitely generated groups. We show that every quasi-isometric embedding between finitely generated groups is a uniform embedding (Proposition 6.14.5) and that the converse is false in general (Example 6.14.6).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3_6

197

198

6 Finitely Generated Groups

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 .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 = s1ε1 s2ε2 · · · snεn .

.

(6.1)

A subset .S ⊂ G is called symmetric if .S = S −1 , that is, .s −1 ∈ S whenever −1 is a .s ∈ S. Observe that if .S ⊂ G is a generating subset of G, then .S ∪ S 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 .lS (g) = lG 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, lS (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 lS (g) = 0 if an only if g = 1G .

.

(6.3)

Proposition 6.1.1 One has lS (g −1 ) = lS (g)

(6.4)

lS (gh) ≤ lS (g) + lS (h)

(6.5)

.

and .

for all .g, h ∈ G. Proof Let .g, h ∈ G. Set .m := lS (g) and .n := lS (h). Then there exist .s1 , s2 , . . . , sm and .t1 , t2 , . . . , tn in S such that .g = s1 s2 · · · sm and .h = t1 t2 · · · tn . We have .g −1 = −1 · · · s −1 s −1 so that .l (g −1 ) ≤ m = l (g). Exchanging the roles of g and .g −1 sm S S 2 1 we get .lS (g) ≤ lS (g −1 ) and therefore .lS (g −1 ) = lS (g). On the other hand, we have .gh = s1 s2 · · · sm t1 t2 · · · tn so that .lS (gh) ≤ m + n = lS (g) + lS (h). u n

6.1 The Word Metric

199

Consider the map .dS = dSG : G × G → N defined by dS (g, h) := lS (g −1 h)

.

(6.6)

for all .g, h ∈ G. Proposition 6.1.2 The map .dS is a metric on G. 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) = lS (g −1 k) + lS (k −1 h) ≥ lS ((g −1 k)(k −1 h)) .

= lS (g −1 h) = dS (g, h). u n

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 ) = lS ((gg1 )−1 gg2 ) = lS (g1−1 g −1 gg2 ) = lS (g1−1 g2 ) = dS (g1 , g2 ).

.

u n 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 : lS (h) ≤ n} and we simply write .BSG (n) instead of G .B (1G , n). Also, when there is no ambiguity on the group G, we omit the supscript S “G” and we simply write .BS (g, n) and .BS (n) instead of .BSG (g, n) and .BSG (n).

200

6 Finitely Generated Groups

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. 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 −1 : Q → Q is .φ : Q1 → Q2 which is bijective and such that the inverse map .φ 2 1 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 .l(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 .ε. Let + − ' ' ' .π1 = (e1 , e2 , . . . , en ) and .π2 = (e , e , . . . , em ) be two paths with .π 1 = π2 . Then 1 2 the path ' π1 π2 := (e1 , e2 , . . . , en , e1' , e2' , . . . , em )

.

is called the composition of the paths .π1 and .π2 . Note that .l(π1 π2 ) = l(π1 ) + l(π2 ) and that .λ(π1 π2 ) = λ(π1 )λ(π2 ).

6.2 Labeled Graphs

201

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{l(π ) : π a path connecting q to q ' }.

.

(6.7)

A path .π connecting q to .q ' with minimal length, that is, such that .l(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: (i) (ii) (iii) (iv)

dQ (q, q ' ) ∈ N ∪ {∞}; ' ' .dQ (q, q ) = 0 if and only if .q = q ; ' ' .dQ (q, q ) < ∞ if and only if there exists a path which connects q to .q ; '' ' ' '' .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 '' ) ≤ l(π1 π2 ) = l(π1 ) + l(π2 ) = dQ (q, q ' ) + dQ (q ' , q '' ).

.

u n 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 −1 ) = l(π ). .π . Note that .l(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.

.

202

6 Finitely Generated Groups

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. u n

.

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 n+1 defined by .q := α(e .πQ = (q0 , q1 , . . . , qn ) ∈ Q i i+1 ) for all .i = 0, 1, . . . , n and .qn = ω(en ). The vertices .q0 , q1 , . . . , qn are called the vertices visited by .π . 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 −1 .(i, j ) /= (0, n). One says that .π is proper, or with no back–tracking, if .ei+1 /= e i for .i = 1, 2, . . . , n − 1. 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}. 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

6.3 Cayley Graphs

203

that .dS (g, h) = lS (g −1 h) = lS (s1 s2 · · · sn ) ≤ n = l(π ) = dCS (G) (g, h). ' ∈ S Conversely, suppose that .dS (g, h) =: m. Then we can find .s1' , s2' , . . . , sm ' . Then the unique path .π ' = (e' , e' , . . . , e' ) with such that .g −1 h = s1' s2' · · · sm m 1 2 ' − = g and label .λ(π ' ) = s ' s ' · · · s ' clearly satisfies .(π ' )+ = h, that is, it .(π ) m 1 2 connects g to h. We deduce that .dCS (G) (g, h) ≤ l(π ' ) = m = dS (g, h). Then (6.8) follows. u n 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. (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). (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.

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

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

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

204

6 Finitely Generated Groups

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)}

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

(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 .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 .r 2 = s 2 = 1G . Taking .S := {r, s}, the Cayley graph CS (G) is as in Fig. 6.6. (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 2 .CS (Z ) is given in Fig. 6.8. .

6.3 Cayley Graphs

205

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)}

(h) Let .G := Z2 and take 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 −1 . Then, the Cayley graph .C (F ) is a regular tree of degree .S := X ∪ X S X .|S| = 2|X| (see Fig. 6.10). Indeed, since .1FX ∈ / S, the Cayley graph .CS (FX ) 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 + = π − h /= π − , that is, .π is .h := s1 s2 · · · sn ∈ FX are distinct and we have .π not closed. This shows that .CS (FX ) is a tree.

206

6 Finitely Generated 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 : lS (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 n .(S ∪ {1G }) to .BS (n), one has γS (n) ≤ |S ∪ {1G }|n

.

for all .n ∈ N.

(6.10)

6.4 Growth Functions and Growth Types

207

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

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{lS ' (s) : s ∈ S}. Then the following holds. (i) (ii) (iii) (iv)

lS ' (g) ≤ clS (g) for all .g ∈ G; .dS ' (g, h) ≤ cdS (g, h) for all .g, h ∈ G; .BS (n) ⊂ BS ' (cn) for all .n ∈ N; .γS (n) ≤ γS ' (cn) for all .n ∈ N. .

Proof (i) Let .g ∈ G. Suppose that .lS (g) = n. Then there exist .s1 , s2 , . . . , sn ∈ S such that .g = s1 s2 · · · sn . Using (6.5) we get lS ' (g) = lS ' (s1 s2 · · · sn ) ≤

n Σ

.

i=1

lS ' (si ) ≤ cn.

208

6 Finitely Generated Groups

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

.

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.

u n and .d '

Two metrics 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 Lipschitzequivalent. u n 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 - γ and that .γ1 < γ2 . Then .γ < γ . 2 1 2 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 .-.

6.4 Growth Functions and Growth Types

209

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' . u n 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). 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 a n - bn .

(6.11)

.

Indeed, suppose for instance that .a ≤ b. On the one hand we have .a n ≤ bn for all .n ≥ 1, so that trivially .a n < bn . On the other, setting .c := [loga b] + 1 > 1 (here .[ · ] denotes the integer part), one has bn = (a loga b )n = a (loga b)n ≤ a cn ≤ ca cn

.

for all .n ≥ 1, so that .bn < a n . We deduce that .a n - bn . In particular, we have a n - 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 , then .γ < exp(n) nd nd = 0, the sequence .( exp(n) )n≥1 is and .γ /- exp(n). Indeed, since .limn→∞ exp(n) .

d

n ≤ c for all .n ≥ 1. bounded and we can find an integer .c ≥ 1 such that . exp(n) d It follows that .n ≤ 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 nd

for all .n ≥ 1 contradicting the fact that .limn→∞ exp(n) = +∞. Thus .nd /- exp(n). nd 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). u n

210

6 Finitely Generated Groups

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). 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 all .n ≥ 1. Thus 1 .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. u n .

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. u n 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). u n

.

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 d .γ (G) < n . Proposition 6.4.10 Every finitely generated group of polynomial growth has subexponential growth.

6.4 Growth Functions and Growth Types

211

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). u 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. (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 language 2 of cellular automata, the ball .BSZ (g, 1) is commonly called the von Neumann Σ neighborhood of g. We have .γS (n) = 1 + nk=1 4k = 2n2 + 2n + 1. It follows that .γ (Z2 ) - γS (n) - n2 . In particular, .Z2 has polynomial growth.

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

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

212

6 Finitely Generated Groups

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

(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 Z2 Z 2 2 2 .γ S (n) = 4n + 4n + 1 = (2n + 1) = (γS (n)) . Note that this yields again 2 2 2 .γ (Z ) - γS (n) - n and the polynomial growth of .Z (cf. Example 6.4.11(b) above). ¯ (−1, 0), ¯ (1, 1), ¯ (−1, 1)}. ¯ Then the ball (d) Let .G := Z × (Z/2Z) and .S := {(1, 0), ¯ 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

213

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

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

(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∞ . (g) Let .G := Fk be the free group of rank .k ≥ 2. Let .{a1 , a2 , . . . , ak } be a free basis and set .S := {a1 , a1−1 , a2 , a2−1 , . . . , ak , ak−1 }. 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 Σ k(2k − 1)n − 1 . (2k − 1)j = k−1 j =0

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

214

6 Finitely Generated 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 .

exists and equals infn≥1

√ n

lim

n→∞

√ n

an

an .

Proof Fix an integer t ≥ 1 and, for all n ≥ 1 write n = qt + r, with 0 ≤ r < t. √ q/n √ q Then an ≤ aqt ar ≤ at 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 ≤ inft≥1 t at ≤ lim inft→∞ t at completing the proof. n u

6.5 The Growth Rate

215

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 λS := lim

.

n→∞

√ n

(6.13)

γS (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. u n 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
1. By Lemma 6.5.1 we have

√ n γS (n) ≥ λS so that

γS (n) ≥ λnS .

.

This shows that exp(n) - λnS < γS (n). By Corollary 6.4.5 we have γS (n) < exp(n) u 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. u n 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). u n

216

6 Finitely Generated 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. u n 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 = rn−1 sn = rn−1 sn 1G ∈ RSR −1 . Thus also hn ∈ S ' . This shows that S ' generates H . u n

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).

6.6 Growth of Subgroups and Quotients

217

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). u n Corollary 6.6.4 Every finitely generated group which contains a finitely generated subgroup of exponential growth has exponential growth. u n 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. u n 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 ).

u n

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 u n generated, then G1 is finitely generated and one has γ (G1 ) = γ (G2 ). 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).

218

6 Finitely Generated Groups

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 π −1 (BS ' (n)) and therefore G/N

γSG (n) = |BSG (n)| ≤ |N||BS '

.

G/N

(n)| = |N |γS '

G/N

(n) ≤ |N |γS '

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

⊂

(|N |n). u 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' .

u n

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 ) ( ) ( S := S1 × {1G2 } ∪ {1G1 } × S2

.

6.7 A Finitely Generated Metabelian Group with Exponential Growth

219

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) ≤ γSG1 1 (n)γSG2 2 (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 γSG1 1 (n)γSG2 2 (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 ). u n

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 .

u n

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, Corollaries 6.6.11, Corollary 6.4.7 that γ (G) = γ (Zd )γ (F ) - nd . This shows that G has polynomial growth. u n

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.

220

6 Finitely Generated Groups

Proof The group G is finitely generated by definition. We have {( k ) } 2 r .G = : 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 .

( k ) 2 r = Am B n Ak−m ∈ G. 0 1

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 q=

n Σ

.

ui 2i−1 ,

i=1

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

.

11 01

)u1 (

12 01

)u2 (

1 22 0 1

)u3 ···

( n−1 )un 12 . 0 1

As we have ( i−1 ) 12 = Ai−1 BA−(i−1) . 0 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

221

This shows that lS (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 λS = lim

.

n→∞

√ |BS (3n − 2)| ≥ lim

3n−2

n→∞

√

3n−2

2n =

√ 3 2 > 1. u 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 . u 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).

.

222

6 Finitely Generated Groups (i)

If .S ⊂ G and .i ≥ 2 we then denote by .SG the set consisting of all elements of the form [s1 , s2 , . . . , si ], s1 , s2 , . . . , si ∈ S.

(6.18)

.

(i)

The elements (6.18) are called simple S-commutators of weight i. Note that .SG ⊂ C i−1 (G). Lemma 6.8.3 Let G be any group. Let .SG ⊂ G be a generating subset of G. Then (i+1) for all .i ≥ 1, the subgroup .C i (G) is the normal closure in G of the set .SG . Proof Let us prove the statement by induction on i. We have that .C 1 (G) = [G, G] (2) 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 (i) for .i = 1. Suppose by induction that .C i−1 (G) is the normal closure in G of .SG , (i+1) denote by N the normal closure in G of .SG , and let us show that .C i (G) = N. (i+1) ⊂ C i (G) and .C i (G) is a normal subgroup, we deduce that Since .SG N ⊂ C i (G).

(6.19)

.

(i)

Denote by .π : G → G/N the quotient map. Let .w ∈ SG and .s ∈ SG . (i+1) and therefore .[w, s] ∈ N. It follows that Then, by definition, .[w, s] ∈ SG .[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 −1 , s]) = [π(hwh−1 ), π(s)] = [π(w), π(s)] = 1 .π([hwh G/N . It follows that [hwh−1 , s] ∈ N.

.

(6.20)

By the inductive hypothesis, every element in .C i−1 (G) can be expressed as a (i) −1 −1 product .(h1 w1 h−1 1 )(h2 w2 h2 ) · · · (hm wm hm ) with .wk ∈ SG and .hk ∈ G, .k = i−1 1, 2, . . . , m, for some .m ∈ N. By taking .H := C (G), .K := G, .S := {w h : w ∈ (i) SG , h ∈ G} and .T := SG in Lemma 6.8.2, we deduce that .C i (G) = [C i−1 (G), G] (i) is the normal closure in G of the set .{[hwh−1 , s] : 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. u 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 (i+1) Lemma 6.8.3 we have that .C i (G) is the normal closure in G of the finite set .SG . i+1 Therefore, if .π : G → G/C (G) denotes the quotient homomorphism, we have

6.8 Growth of Finitely Generated Nilpotent Groups

223

that .C i (G)/C i+1 (G) = π(C i (G)) is the normal closure in .G/C i+1 (G) of the set (i+1) (i+1) .π(S ). But .π(SG ) = (π(SG ))(i+1) ⊂ Z(C i (G)/C i+1 (G)) and therefore G (i+1) .π(S ) generates .C i (G)/C i+1 (G). G 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. u n 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 i .≤ d − 1. Indeed, as one immediately checks by induction, we have .C (H ) ⊂ i+1 d−1 d C (G) for all .i = 0, 1, . . . , d − 1, so that .C (H ) ⊂ C (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 ∈ G and suppose that .m := l (g) ≤ n. Then there exist .1 ≤ i1 , i2 , . . . , im ≤ k S 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 = s j1 s j2 · · · s jm h

.

(6.23)

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

224

6 Finitely Generated Groups

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, 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 lH 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 .s1n1 s2n2 · · · sknk , 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. u n 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. u n

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 .ε (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 =: l(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 ∗ , · · · 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 .lα (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 (αγ α) := δ, Φ1 (γ ) := δ, Φ1 (αβα) := α, Φ0 (δ) := ε, Φ0 (αδα) := β, Φ1 (β) := δ, Φ1 (αβα) := ε. 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 l(w0 ) + l(w1 ) ≤ l(w) + 1.

.

(6.51)

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 .l(w) = 2k + 1. Then .wσ = Φσ (w) and .Φσ (w) = Φσ (αλ1 α)Φσ (λ2 ) · · · Φσ (λk−1 )Φσ (αλk α) and therefore l(wσ ) = l(Φσ (w)) ≤ l(Φσ (w)) ≤ l(Φσ (αλ1 α)) + l(Φσ (λ2 )) + · · · + l(Φσ (λk−1 )) + l(Φσ (αλk α)) .

≤k= =

(2k + 1) − 1 2

l(w) − 1 . 2

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 .l(w) = 2k − 1. Then .wσ = Φσ (w) and .Φσ (w) = Φσ (λ1 )Φσ (αλ2 α) · · · Φσ (αλk−1 α)Φσ (λk )

6.9 The Grigorchuk Group and Its Growth

235

and therefore l(wσ ) = l(Φσ (w)) ≤ l(Φσ (w)) ≤ l(Φσ (λ1 )) + l(Φσ (αλ2 α)) + · · · + l(Φσ (αλk−1 α)) + l(Φσ (λk )) .

≤k= =

(2k − 1) + 1 2

l(w) + 1 . 2

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 .l(w) = 2k. Then .wσ = Φσ (w) and .Φσ (w) = Φσ (αλ1 α)Φσ (λ2 ) · · · Φσ (αλk−1 α)Φσ (λk ) and therefore l(wσ ) = l(Φσ (w)) ≤ l(Φσ (w)) ≤ l(Φσ (αλ1 α)) + l(Φσ (λ2 )) + · · · + l(Φσ (αλk−1 α)) + l(Φσ (λk )) .

≤k= =

2k 2

l(w) . 2

This shows (6.51). u n Lemma 6.9.16 For all .g ∈ H3 one has 1 Σ

lS (gij k ) ≤

.

i,j,k=0

5 lS (g) + 8. 6

(6.52)

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

l(wij ) ≤

i,j =0

1 Σ (l(wi ) + 1) i=0

.

=

1 Σ

l(wi ) + 2

i=0

≤ l(w) + 3

(6.53)

236

6 Finitely Generated Groups

and therefore 1 Σ

l(wij k ) ≤

1 Σ

(l(wij ) + 1)

i,j =0

i,j,k=0

≤

1 Σ

l(wij ) + 4

(6.54)

i,j =0

.

(by (6.53)) ≤

(Σ 1

) 1 Σ l(wi ) + 2 + 4 = l(wi ) + 6

i=0

i=0

(by (6.51)) ≤ l(w) + 1 + 6 = l(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. l(w0 ) + l(w1 ) ≤ l(w) + 1 − lδ (w),

.

(6.55)

where .lδ (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) = ε, while in the second one, .Φ1 (αδα) = ε. 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, l(w00 ) + l(w01 ) + l(w10 ) + l(w11 ) ≤ l(w) + 3 − lγ (w),

.

(6.56)

where .lγ (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, 1 Σ

l(wij k ) ≤ l(w) + 7 − lβ (w),

.

(6.57)

i,j,k=0

where .lβ (w) denotes the number of occurrences of the letter .β in the word w. Since .l(w) = lα (w) + lβ (w) + lγ (w) + lδ (w) and w is reduced, we necessarily and therefore have .lβ (w) + lγ (w) + lδ (w) ≥ l(w)−1 2 .

max lλ (w) >

λ∈{β,γ ,δ}

l(w) − 1. 6

(6.58)

6.10 The Følner Condition for Finitely Generated Groups

237

Taking into account the inequalities in (6.54) we obtain 1 Σ .

l(wij k ) ≤ min

i,j,k=0

⎧ 1 ⎨Σ ⎩

i=0

l(wi ) + 6,

1 Σ i,j =0

⎫ ⎬ l(wij ) + 4 . ⎭

(6.59)

Observe that .π(wij k ) = gij k so that .lS (gij k ) ≤ l(wij k ) 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 Σ

lS (gij k ) ≤

i,j,k=0

1 Σ

l(wij k )

i,j,k=0

≤ min{l(w) + 7 − lβ (w), l(w) + 7 − lγ (w), l(w) + 7 − lδ (w)} ≤ l(w) + 7 − max{lβ (w), lγ (w), lδ (w)} ) ( l(w) −1 ≤ l(w) + 7 − 6

.

5 l(w) + 8 6 5 = l(g) + 8. 6 ≤

u n

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. u n 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 Theorems 6.9.9 and 6.9.12 we then get: Theorem 6.9.17 The Grigorchuk group G has intermediate growth.

u n

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,

.

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

.

(6.60) (6.61)

238

6 Finitely Generated Groups

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. u n 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. 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

6.11 Amenability of Groups of Subexponential Growth

239

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 \ F s| = |F s \ F | ≤ |F S \ F | ≤

Σ

.

|F s ' \ F | =

s ' ∈S

Σ

|F \ F s ' |.

s ' ∈S

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

u n

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 infn→∞ an+1 an . If .α = 0 there is nothing to prove. Otherwise, let an+1 .0 < β < α. Then, there exists an integer .N ≥ 1 such that . an ≥ β for all .n ≥ N. Thus, for all .p = 1, 2 . . . one has .

aN +p−1 aN +p aN +p aN +1 · ··· ≥ βp. = aN +p−1 aN +p−2 aN aN

This gives .aN +p ≥ β p aN . It follows that setting .n := N + p, one has, for all .n ≥ N an ≥ β n−N aN = β n (β −N aN ).

.

240

6 Finitely Generated Groups

√ √ 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 infn→∞

(6.68) √ n

an ≥ α = lim infn→∞

an+1 an .

u n

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 previous √ lemma we have .1 ≤ lim infn→∞ γSγ(n+1) ≤ limn→∞ n γS (n) = 1, so that S (n) lim infn→∞

.

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 u n that G is amenable. From Theorems 6.9.12 and 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 { } Σ p G .l (G) := x ∈ R : |x(g)|p < ∞ g∈G

u n

6.12 The Theorems of Kesten and Day

241

consisting of all p-summable real functions on G. For .x ∈ lp (G), the nonnegative )1 (Σ p p is called the .lp -norm of x. number .||x||p := 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 .lp (G). When .p = 2, it is possible to endow .l2 (G) with a scalar product . defined by setting :=

Σ

.

x(g)y(g)

g∈G

for all .x,√y ∈ l2 (G). Then, the .l2 -norm of an element .x ∈ l2 (G) is given by .||x||2 := . The space .(l2 (G), ) is a real Hilbert space. p The .l -norm of a linear map .T : lp (G) → lp (G) is defined by ||T ||p→p :=

sup ||T x||p =

.

x∈lp (G) ||x||p ≤1

||T x||p . x∈lp (G) ||x||p sup

x/=0

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

||T ||p→p = sup ||T x||p = sup

.

x∈R[G] ||x||p ≤1

x/=0

for all .T ∈ L (lp (G)). Let .p ∈ [1, +∞) and .s ∈ G. For all .x ∈ lp (G) and .g ∈ G we set (p)

(Ts

.

x)(g) := x(gs).

We then have (p)

||Ts

p

x||p =

Σ

(p)

|Ts

x(g)|p

g∈G

=

Σ

|x(gs)|p

g∈G

.

=

Σ

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

h∈G p

= ||x||p

(6.70)

242

6 Finitely Generated Groups (p)

for all .x ∈ lp (G). We deduce that .Ts (p) .Ts : lp (G) → lp (G) has .lp -norm (p)

||Ts

.

x ∈ lp (G) and that the linear map

||p→p = 1.

(6.71)

(p)

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

(MS x)(g) :=

.

1 Σ x(gs) |S| s∈S

(p)

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

is called the .lp -Markov operator on

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

.

(p)

||MS ||p→p ≤ 1.

.

(p)

Proof Since .Ts have

(p)

∈ L (lp (G)), we deduce that .MS

(p)

||MS ||p→p = ||

.

(6.72) ∈ L (lp (G)). Finally, we

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

s∈S

u n

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: (a) .||MS(2) ||2→2 = 1; (b) given .ε > 0 there exists .x ∈ R[G] such that .||x||2 = 1 and .||MS(2) x||2 ≥ 1 − ε; (c) given .ε > 0 there exists .x ∈ R[G] such that .x ≥ 0, .||x||2 = 1 and .||MS(2) x||2 ≥ 1 − ε; (d) given .ε > 0 there exists .x ∈ R[G] such that .x ≥ 0, .||x||2 = 1 and ||x − Ts(2) x||2 ≤ 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 .l2 -norm and the density of .R[G] in .l2 (G) (cf. (6.70)).

6.12 The Theorems of Kesten and Day

243

Let now .x ∈ R[G]. We have || || || (2) ||2 (2) ||MS x||22 ≤ ||MS |x| || .

(6.74)

.

2

Indeed, (2) ||MS x||22

|| ||2 || || 1 Σ || || Ts(2) x || = || || || |S| s∈S

=

Σ

(

≤

Σ g∈G

)2 1 Σ x(gs) |S| s∈S

g∈G .

2

(

1 Σ |x(gs)| |S|

)2

s∈S

|| ||2 || || 1 Σ || || (2) Ts |x||| = || || |S| || s∈S

|| || || (2) ||2 = ||MS |x| || .

2

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 .x ∈ (2) R[G], .x ≥ 0, .||x||2 = 1, there exists .s0 ∈ S such that .||x − Ts0 x||2 ≥ ε0 . Since 2 .l (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 .||x||2 = 1. It then follows || || || || 1 Σ || || (2) Ts(2) x || ||MS x||2 = || || || |S| s∈S 2 || || ) ( || || Σ || || 2 x + Ts(2) x 1 0 (2) || || = || Ts x || + 2 |S| || || |S| s∈S\{1G ,s0 } 2 || || (2) || || . Σ 2 || x + Ts0 x || 1 ≤ ||Ts(2) x||2 || || + || |S| || 2 |S| 2

s∈S\{1G ,s0 }

1 2 (by (6.75)) ≤ (|S| − 2) (1 − δ0 ) + |S| |S| 2δ0 =1− |S|

244

6 Finitely Generated Groups

for all .x ∈ R[G], such that .||x||2 = 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 || || || || Σ 1 || || Ts(2) x || ||x − MS(2) x||2 = ||x − || || |S| s∈S

2

|| || || || 1 Σ || (2) || (x − Ts x)|| = || || || |S|

.

s∈S

2

1 Σ ||x − Ts(2) x||2 ≤ |S| s∈S

≤ ε. (2)

Therefore, by Corollary I.2.3, the linear map .I − MS ∈ L (l2 (G)) is not bijective, (2) that is, .1 ∈ σ (MS ). Finally, the implication (e) .⇒ (a) follow from (I.14) and the fact that (2) .||M u n S ||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: (a) .1 ∈ σ (MS(2) ); (2) (b) .1 ∈ σ (MS∪{1 ). G} Proof If S contains .1G there is nothing to prove. |S| Suppose that .1G ∈ . Note that .0 < α < 1 and that / S and set .α := |S|+1 (2)

(2)

MS∪{1G } = (1 − α)I + αMS .

.

(6.76)

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

(2)

if and only if .I − MS is bijective. In other words, .1 ∈ σ (MS∪{1G } ) if and only if

1 ∈ σ (MS(2) ).

.

u n

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 .||x||2 = 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].

6.12 The Theorems of Kesten and Day

245

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)

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 − χB ||22 =

Σ

|χA (g) − χB (g)|2

g∈G

=

Σ

|χA (g) − χB (g)|

g∈G

Σ

=

|χA (g) − χB (g)|

g∈G: χA (g)−χB (g)=1

.

Σ

+

|χA (g) − χB (g)|

g∈G: χA (g)−χB (g)=−1

=

Σ

χA\B (g) +

g∈G

Σ

χB\A (g)

g∈G

= |A \ B| + |B \ A|. u n Lemma 6.12.5 Let .x, y ∈ l2 (G) such that .||x||2 = ||y||2 = 1. Then, the following holds: (i) .x 2 ∈ l1 (G) and .||x 2 ||1 = 1; (ii) .||x 2 − y 2 ||1 ≤ 2||x − y||2 ;

( )1 (iii) suppose that .x, y ≥ 0. Then, .||x − y||2 ≤ ||x 2 − y 2 ||1 2 . Proof (i) We have .||x 2 ||1 = (ii) We have

Σ

g∈G |x

2 (g)|

||x 2 − y 2 ||1 =

= ||x||22 = 1.

Σ

.

|x 2 (g) − y 2 (g)|

g∈G

=

Σ

|x(g) − y(g)| · |x(g) + y(g)|

g∈G

=

246

6 Finitely Generated Groups

≤ ||x − y||2 · ||x + y||2 ≤ ||x − y||2 · (||x||2 + ||y||2 ) = 2||x − y||2 , where the first inequality follows from Cauchy-Schwarz. (iii) We have Σ ||x − y||22 = (x(g) − y(g))2 g∈G

Σ

=

|x(g) − y(g)| · |x(g) − y(g)|

g∈G

Σ

≤

.

|x(g) − y(g)| · |x(g) + y(g)|

g∈G

Σ

=

|x 2 (g) − y 2 (g)|

g∈G

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

u n

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 Σ Σ 2 .||χF ||2 = 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).

u n

Lemma 6.12.7 Let .x ∈ R[G] such that .x ≥ 0 and .||x||1 = 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 x=

n Σ

.

i=1

λi

χAi . |Ai |

(6.78)

6.12 The Theorems of Kesten and Day

247

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 }.

.

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

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

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

.

n Σ .

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

i=1

= α1 (|A1 | − |A2 |) + α2 (|A2 | − |A3 |) + · · · + αn |An | =

Σ

x(g) = 1.

g∈G

u n Lemma 6.12.8 With the same notation and hypotheses as in Lemma 6.12.7, we have ||x − Tg(1) x||1 =

n Σ

.

i=1

λi

2|Ai \ Ai g| |Ai |

(6.79)

for every .g ∈ G. Proof Equality (6.78) gives us x − Tg(1) x =

n Σ

.

λi

i=1

(by Lemma 6.12.6(a))

=

n Σ i=1

λi

χAi − Tg(1) χAi |Ai | χAi − χAi g −1 |Ai |

.

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

248

6 Finitely Generated Groups

value 0 everywhere else. Let us set B :=

U

(Ai \ Ai g −1 )

.

C :=

and

1≤i≤n

U

(Ai g −1 \ Ai ).

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 −1 ⊂ A g −1 ). .Aj g i It follows that Σ (1) .||x − Tg x||1 = |(x − Tg(1) x)(a)| .

a∈G

| n | Σ ||Σ (χAi − χAi g −1 )(a) || λi = | | | | |Ai | a∈G i=1 | | | n | n | Σ ||Σ (χAi − χA g −1 )(a) || Σ ||Σ (χ − χ −1 )(a) | A A g i i i = λi λi |+ | | | | | | | |Ai | |Ai | a∈C i=1

a∈B i=1

=

n Σ

λi

i=1

(by (6.61)) =

n Σ

λi

i=1

|Ai \ Ai g −1 | + |Ai |

n Σ

λi

i=1

|Ai g −1 \ Ai | |Ai |

2|Ai \ Ai g| . |Ai | u n

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 ∈ l2 (G) ⊂ RG and .g ∈ G we have (ΔS x)(g) = |S|x(g) −

Σ

x(gs)

s∈S

Σ = |S|x(g) − (Ts(2) x)(g)

.

s∈S (2)

= |S|(I − MS )(x)(g). (2)

This shows that for the map .ΔS = ΔS |l2 (G) one has ) ( (2) (2) ΔS = |S| I − MS

.

2 and therefore .Δ(2) S ∈ L (l (G)).

(6.80)

6.12 The Theorems of Kesten and Day

249

Let .λ ∈ R. From (6.80) we deduce that λI

.

− Δ(2) S

= λI − |S|(I

− MS(2) )

It follows that λ∈

.

(2) σ (ΔS )

) ) (( λ (2) . I − MS = −|S| 1− |S|

) ( λ (2) ∈ σ (MS ) ⇔ 1− |S|

and therefore (2)

(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 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 such that ε2 √1 .|F \ F s| < 2 |F | for all .s ∈ S. Set .x := |F | χF and note that .||x||2 = 1. We have (2)

||(I − MS )x||2 = ||x −

1 Σ (2) Ts x||2 |S| s∈S

=

1 || |S|

Σ

(x − Ts(2) x)||2

s∈S

1 Σ ≤ ||x − Ts(2) x||2 |S| s∈S

= .

(by Lemma 6.12.6(iii)) =

=

Σ 1 ||χF − Ts(2) χF ||2 √ |S| · |F | s∈S Σ 1 1 2|F \ F s| 2 √ |S| · |F | s∈S )1 ( 1 Σ |F \ F s| 2 2 |S| |F | s∈S

=

2 |F |

< ε.

1

1 2

ε2

250

6 Finitely Generated Groups (2)

(2)

From Corollary I.2.3 we deduce that .I − MS is not bijective, that is, .1 ∈ σ (MS ). (2) 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)

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

.

ε |S|

for all s ∈ S.

(6.83)

By Lemma 6.12.7, there exist an integer .n ≥ 1, nonempty finite subsets .Ai ⊂ G and real numbers .λi > 0, .1 ≤ i ≤ n, .A1 ⊃ A2 ⊃ · · · ⊃ An and Σnsatisfying χ Ai .λ1 + λ2 + · · · + λn = 1, such that .x = λ . i=1 i |Ai | 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) x||1 =

Σ

.

λi

i∈Ω

≥

Σ

2|Ai \ Ai g| |Ai |

λi

i∈Ωg

≥ 2ε

Σ

2|Ai \ Ai g| |Ai | λi

i∈Ωg

= 2εμ(Ωg ). Therefore, we have (1)

μ(Ωg ) ≤

.

||x − Tg x||1 2ε

for all g ∈ G.

6.13 Uniform Embeddings and Quasi-Isometries

251

By using (6.83), we deduce μ(Ωs )
1). For more on this we refer to the paper [KaiV] 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 = . Grigorchuk [Gri1] proved that either .α = 1 (this holds if and only if .N = {1F }) or √ .

2m − 1 ≤ α ≤ 2m − 1.

(6.110)

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 ) (√ √ .ρ = (6.111) √ 2m−1 α ⎩ 2m−1 if 2m − 1 ≤ α ≤ 2m − 1. + √2m−1 2m α Then, from (6.111) and Kesten’s criterion he deduced that in (6.110) equality holds on the right if and only if G is amenable. This is called the Grigorchuk 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-

268

6 Finitely Generated Groups

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 quasiisometries was posed and initiated by Gromov [Gro4]. The definition of quasiisometry 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 one has 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 ' . (a) (b) (c) (d) (e)

Show that one has lS ' (g) ≤ lS (g) for all g ∈ G. Show that one has dS ' (g, h) ≤ dS (g, h) for all g, h ∈ G. Show that BS (n) ⊂ BS ' (n) for all n ∈ N. Show that γS (n) ≤ γS ' (n) for all n ∈ N. Show that 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 of G and G1 G2 that one has lG S (g) = lS1 (g1 ) + lS2 (g2 ) for all g = (g1 , g2 ) ∈ G. 6.5 (Direct Product of Labeled Graphs) Let S1 and S2 be two sets. For i = 1, 2, let Gi = (Qi , Ei ) be an Si -labeled graph. We define their direct product G1 × G2 as the S-labeled graph G = (Q, E) with (1) S := S1 u S2 , where u denotes a disjoint union; (2) Q := Q(1 × 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 G1 (resp. G2 ) is edge-symmetric with respect to ι1 (resp. ι2 ). 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 G is edge-symmetric with respect to ι.

Exercises

269

6.6 Let G1 and G2 be two finitely generated groups and let S1 ⊂ G1 and S2 ⊂ G2 be two finite and symmetric generating subsets such that 1G1 ∈ / S1 and 1G2 ∈ / S2 . Consider the direct product group G := G1 × G2 together with the finite 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, let S := {1, −1}, and let S ' := {2, −2, 3, −3}. Find the best possible positive constants C1 and C2 such that C1 lS (g) ≤ lS ' (g) ≤ C2 lS (g) for all g ∈ G. 6.8 (Growth of Zd ) Let G := Zd , where d ≥ 1 is an integer. Consider the finite symmetric generating subset S ⊂ G defined by S := {±(1, 0, 0, . . . , 0), ±(0, 1, 0, . . . , 0), . . . , ±(0, 0, . . . , 0, 1)} ⊂ Zd .

.

(a) Show that if g = (a1 , a2 , . . . , ad ) ∈ Zd then lS (g) = |a1 | + |a2 | + · · · + |ad |. (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 , . . . (, a)t ) of positive integers such that a1 + a2 + · · · + at ≤ n. Show that Pt (n) = nt for 1 ≤ t ≤ n. (c) For n, t ∈ N and t ≥ 1Σ denote by Nt (n) the number of all d-tuples (a1 , a2 , . . . , ad ) ∈ Zd with di=1 |ai | ≤ n and exactly t many of the ai ’s Σ d nonzero. Show that γSZ (n) = dt=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 , . . . , ad ) ∈ Nm with I = {i : ai > 0} and such that lS (g) ≤ n. ( )( ) (e) Deduce from (d) that there are exactly dt nt elements g = (a1 , a2 , . . . , ad ) ∈ Nm with |{i : ai > 0}| = t (such )( )that lS (g) ≤ n. (f) Deduce from (e) that Nt (n) = 2t dt nt . ( )( ) Σ d (g) Deduce from (f) and (c) that γSZ (n) = dt=0 2t dt nt . 6.9 (Growth Series) Let G be a finitely generated group and let S ⊂ G be a finite symmetric generating subset. For n ∈ N we denote by ΣS (n) := {g ∈ G : lS (g) = n} = {g ∈ G : dS (g, 1G ) = n}

.

the sphere of radius n centered at 1G . The function σS : N → N, defined by σS (n) := |ΣS (n)| for all n ∈ N, is called the spherical growth function of G relative to S. The formal power series BS (z) :=

∞ Σ

.

n=0

γS (n)zn

270

6 Finitely Generated Groups

and AS (z) :=

∞ Σ

.

σS (n)zn

n=0

are called the growth series and the spherical growth series of G corresponding to the generating subset S, respectively. Σ (a) Show that γS (n) = Σni=0 σS (i) for all n ∈ N. (b) Show that AS (z) = g∈G zlS (g) . (c) Show that AS (z) = (1 − z)BS (z). (d) Let G := Z and let S := {−1, 1}. Compute the growth series AS (z) and BS (z). (e) Let G1 (resp. G2 ) be a finitely generated group and let S1 ⊂ G1 (resp. S2 ⊂ G2 ) denote a finite symmetric generating subset. Let G := G1 × G2 and let S := (S1 × {1G2 }) ∪ ({1G1 } × G2 ). Recall (cf. Proposition 6.6.10) that S is a finite symmetric generating subset of G. Show that AS (z) = AS1 (z)AS2 (z)

.

and

BS (z) = (1 − z)BS1 (z)BS2 (z).

(f) Let d ≥ 1 be an integer. Let G := Zd and let S := {e1 , −e1 , e2 , −e2 , . . . , ed , −ed }, where ei := (0, 0, . . . , 0, 1, 0, 0 . . . , 0) ∈ Zd with 1 being placed in the ith coordinate position, for i = 1, 2, . . . , d. Compute the growth series AS (z) and BS (z). (g) Suppose that G is a free group of finite rank r ≥ 1 and let X ⊂ G be a free base for G. Set S := X ∪ X−1 . Compute the growth series AS (z) and BS (z). 6.10 Suppose that the maps γ , γ ' : N → [0, √ +∞) are such that γ < γ ' . Show that √ n if lim supn→∞ γ (n) > 1 then lim supn→∞ n γ ' (n) > 1. ( ) 21 6.11 Let M := ∈ SL2 (Z). Consider the semidirect product G := Z2 xα 11 ( ) x 2 := Z, where α : Z → Aut(Z ) is the group homomorphism defined by α(z) y ( ) x for all x, y, z ∈ Z. Recall (cf. Exercise 2.30) that Mz y G=

.

{( ( ) ) } x , z : x, y, z ∈ Z y

with the multiplication defined by (( ) ) (( ) ) (( ) ( ) ) x1 x2 x1 x , z1 · , z2 = + M z1 2 , z1 + z2 y1 y2 y1 y2

.

for all x1 , x2 , y1 , y2 , z1 , z2 ∈ Z.

Exercises

271

(a) Show that Mn

.

( ) ( ) 1 f = 2n+1 f2n 0

for all n ∈ N, where (fk )k∈N is the Fibonacci sequence which is inductively defined by f0 := 0, f1 := 1, and fk := fk−2 + fk−1 for all k ≥ 2. (b) Show that n−1 Σ .

f2i+1 = f2n

(6.112)

i=0

for every integer n ≥ 1. (c) Deduce from (a) and (b) that, for any integer n ≥ 1, the set A(n) ⊂ G defined by A(n) :=

{( Σ n−1

.

ui M i

i=0

( ) ) } 1 , 0 : ui ∈ {0, 1} for 1 ≤ i ≤ n 0

has cardinality |A(n)| = 2n . (d) Consider the subset S ⊂ G defined by S := {s, t, z, s −1 , t −1 , z−1 }, where s :=

.

(( ) ) (( ) ) (( ) ) 1 0 0 , 0 , t := , 0 , and z = ,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. (e) Deduce from (c) and (d) that G has exponential growth. 6.12 Show that any two of the matrices S, T , U ∈ SL2 (Z) defined by ( ) 0 −1 .S := 1 0

( ) 11 T := 01

(

and

10 U := 11

)

generate the group SL2 (Z). 6.13 (Elements of Finite Order in SL2 (Z)) (a) Show that for every n ∈ {1, 2, 3, 4, 6} there exists an element of order n in the group SL2 (Z). (b) Suppose that M ∈ SL2 (Z) has finite order n. Show that one has n ∈ {1, 2, 3, 4, 6} and that n = 1 (resp. 2, resp. 3, resp. 4, resp. 6) if and only if Tr(M) = 2 (resp. −2, resp. 1, resp. 0, resp. 1).

272

6 Finitely Generated Groups

(c) Let m ≥ 3 be an integer and let ρ : SL2 (Z) → SL2 (Z/mZ) denote the canonical group homomorphism given by reduction modulo m. Show that the kernel of ρ is a finitely generated torsion-free group. 6.14 Consider the matrices S, V ∈ SL2 (Z) defined by ( ) 0 −1 .S := 1 0

and

( ) 0 −1 V := . 1 1

(a) Show that S has order 4 and that V has order 6. (b) Show that S and V generate SL2 (Z). 6.15 Show that every group homomorphism SL2 (Z) → Z is trivial. 6.16 (Indecomposable Groups) A group G is called indecomposable if G is nontrivial and there do not exist non-trivial groups H and K such that the group H × K is isomorphic to G. Show that the group SL2 (Z) is indecomposable. 6.17 Show that the matrices R, T ∈ GL2 (Z) defined by R :=

.

( ) 01 10

and

T :=

( ) 11 01

generate the group GL2 (Z). 6.18 (Free Groups Are Torsion-Free) Let F be a free group and let X ⊂ F be a base for F . Recall that every element g ∈ G can be uniquely written in its Xreduced form, i.e., in the form g = x1 x2 · · · xn

.

(6.113)

with n ≥ 0, xi ∈ X ∪ X−1 for 1 ≤ i ≤ n, and xi+1 /= xi−1 for 1 ≤ i ≤ n − 1. One says that g is cyclically reduced (with respect to X) if its reduced form (6.113) satisfies xn /= x1−1 . (a) Show that every element in F is conjugate to some cyclically reduced element. (b) Deduce from (a) that F is torsion-free. 6.19 (Schreier Lemma) Let G be a group and let H be a subgroup of G. Let S ⊂ G be a complete set of representatives for the right cosets of H in G such that 1G ∈ S. For g ∈ G, let g ∈ S denote the representative of the right coset H g. (a) (b) (c) (d)

Show that hg = g for all h ∈ H and g ∈ G. Show that gg −1 and g(g)−1 are both in H for all g ∈ G. Show that g1 g2 = g1 g2 for all g1 , g2 ∈ G. Let s ∈ S and x ∈ G. Show that the element t ∈ S defined by t := sx −1 satisfies sx −1 (sx −1 )−1 = (tx(tx)−1 )−1 .

.

(6.114)

Exercises

273

(e) Let X ⊂ G be a generating subset of G. Show that the set Y = Y (X, S) := {sx(sx)−1 : s ∈ S and x ∈ X}

.

(6.115)

is a generating subset of H . A complete set of representatives for the right cosets of a subgroup H of a group G is called a right transversal of H in G. The elements in the set Y (X, S) are called the Schreier generators of H (relative to the right transversal S and the generating subset X of G). 6.20 (The Sanov Subgroup) Let F denote the subgroup of SL2 (Z) generated by the matrices A, B ∈ SL2 (Z) defined by ( A :=

.

) 12 , 01

( B :=

) 10 . 21

(a) Show that F is a free group of rank 2 which is freely generated by A and B. (b) Show that F is a normal subgroup of index 12 of SL2 (Z). (c) Show that F consists of all matrices .

( ) ab ∈ SL2 (Z) cd

such that a ≡ d ≡ 1 mod 4 and b ≡ c ≡ 0 mod 2. 6.21 Let G := SL2 (Z) and let n ≥ 1 be an integer. Consider the group homomorphism ρn : G → SL2 (Z/nZ) given by reduction of entries modulo n. Let Kn ⊂ G denote the kernel of ρn . (a) Show that ρn is surjective. (b) Show that the index of Kn in G is given by the formula ) || ( 1 3 1− 2 , .[G : Kn ] = n p p

(6.116)

where p runs over all prime factors of n. 6.22 (The Projective Modular Group) Consider the modular group G := SL2 (Z) and let Z(G) denote its center. (a) Show that Z(G) = {I, −I }, where I denotes the identity matrix in G. (b) Let Γ := G/Z(G) = SL2 (Z)/{I, −I } and denote by π : G → Γ the canonical group epimorphism. Consider the matrices S, V ∈ SL2 (Z) defined by .S :=

( ) 0 −1 1 0

and

V :=

( ) 0 −1 . 1 1

274

(c) (d) (e)

(f) (g)

6 Finitely Generated Groups

Show that the elements α := π(S) and β := π(V ) have order 2 and 3, respectively, in Γ . Show that α and β generate Γ . Show that Γ is residually finite. Show that the subgroup Φ ⊂ Γ generated by the elements γ1 := βαβα and γ2 := β −1 αβ −1 α is a normal subgroup of index 6 of Γ and that Φ is free of rank 2. Show that Γ is non-amenable. Consider the finite symmetric generating subset X ⊂ Γ defined by X := {α, β, β −1 }. Given an integer n ≥ 0, one says that a sequence (xi )1≤i≤n of elements of X is admissible if, for all 1 ≤ i ≤ n − 1, one has xi = α if and only if xi+1 ∈ {β, β −1 }. Suppose that (xi )1≤i≤n is an admissible seqence of elements of X such that n ≥ 1. By playing ping-pong on the real projective line, show that one has x1 x2 · · · xn /= 1Γ .

.

(h) Let F (x, y) denote the free group of rank 2 based on the symbols x and y. Show that there is a unique group homomorphism ψ : F (x, y) → Γ such that ψ(x) = α and ψ(y) = β. (i) Show that ψ is surjective and that the kernel of ψ is the normal closure in F (x, y) of the set {x 2 , y 3 }. (j) Show that Γ admits the presentation Γ = .

.

(k) Let γ ∈ Γ and let lX (γ ) denote the word length of γ with respect to X. Show that there exist a unique integer n ≥ 0 and a unique admissible sequence (xi )1≤i≤n of elements of X such that γ = x1 x2 · · · xn , and that one has n = lX (γ ). This sequence (xi )1≤i≤n is called the normal form of γ . (l) Draw the Cayley graph of Γ with respect to X. 6.23 (Growth of the Projective Modular Group) Let G := PSL2 (Z) denote the projective modular group and consider the finite symmetric generating subset S := {α, β, β −1 } ⊂ G, where α and β are as in Exercise 6.22(g). Compute the spherical growth series AS . Compute the growth series BS . Compute the growth function γS (n). √ Compute the growth rate λS := limn→∞ n γS (n). Show that there exist constants C1 , C2 > 0 such that C1 λnS ≤ γS (n) ≤ C2 λnS for all n ∈ N. (f) Show that the sequence (γS (n)/λnS )n∈N is not convergent.

(a) (b) (c) (d) (e)

Exercises

275

6.24 (Presentation of SL2 (Z)) Let G := SL2 (Z). Consider the matrices S, V ∈ G defined by S :=

.

( ) 0 −1 1 0

and

V :=

( ) 0 −1 1 1

- := {S, V , V −1 }. Given an integer n ≥ 0, one says that a sequence and let X - is admissible if, for all 1 ≤ i ≤ n − 1, one has xi = S if (xi )1≤i≤n of elements of X −1 and only if xi+1 ∈ {V , V }. (a) Let M ∈ G. Show that there exist an unique integer n ≥ 0, an unique integer - such ε ∈ {1, −1}, and a unique admissible sequence (xi )1≤i≤n of elements of X that M = εx1 x2 · · · xn .

.

This expression is called the normal form of M. (b) Let F (x, y) denote the free group of rank 2 based on the symbols x and y. - : F (x, y) → G such that Show that there is a unique group homomorphism ψ ψ (x) = S and ψ (y) = V . - is surjective and that the kernel of ψ - is the normal closure in (c) Show that ψ 4 −3 2 F (x, y) of the set {x , y x }. (d) Show that G admits the presentation G = .

.

6.25 (Commutator Subgroups of PSL2 (Z) and SL2 (Z)) Recall the notation introduced in Exercise 6.22. The modular group is G := SL2 (Z) and the projective modular group is Γ := PSL2 (Z). The matrices S, V , T ∈ G are defined by S :=

.

( ) 0 −1 , 1 0

( V :=

) 0 −1 , 1 1

T :=

( ) 11 , 01

and the elements α, β ∈ Γ are given by α := π(S) and β := π(V ), where π : G → Γ is the canonical group epimorphism. (a) Show that there exists a surjective group homomorphism η : Γ → Z/6Z. (b) Show that ker(η) is generated by the elements c1 , c2 ∈ Γ defined by c1 := αβαβ −1 and c2 := αβ −1 αβ. (c) Show that ker(η) = [Γ, Γ ]. (d) Show that [Γ, Γ ] is a free group of rank 2 based on {c1 , c2 }. (e) Show that Γ /[Γ, Γ ] is a cyclic group of order 6 generated by the class of αβ. (f) Show that there is a surjective group homomorphism η : G → Z/12Z.

276

6 Finitely Generated Groups

(g) Consider the matrices C1 , C2 ∈ G defined by ( C1 := −

.

) 21 11

( and

C2 := −

) 11 . 12

Show that the commutator subgroup [G, G] is a free group of rank 2 based on {C1 , C2 } and that one has [G, G] = ker(η). (h) Show that G/[G, G] is a cyclic group of order 12 generated by the class of T . 6.26 (Short-Lex Ordering) Let A be a set. Recall that it follows from the Axiom of Choice that A admits a well-ordering, that is, a total ordering ≤ with the property that every non-empty subset of A has a minimal element. Consider the monoid A∗ consisting of all words on the alphabet A and denote by |u| the length of a word u ∈ A∗ . Define a relation < on A∗ by setting, for all u, v ∈ A∗ , u ≺ v if either |u| < |v| or there exist w, u' , v ' ∈ A∗ with |u' | = |v ' | and x, y ∈ A with x < y, such that u = u' xw and v = v ' yw. Show that < is a well-ordering on A∗ . 6.27 (Schreier Systems) Let A be a set. One says that a word u ∈ A∗ is a prefix of a word w ∈ A∗ if there exists a word v ∈ A∗ such that w = uv. A subset L ⊂ A∗ is said to be prefix-closed provided that, given any word w ∈ L, every prefix of w is itself in L. Let F be a free group based on X ⊂ F . Setting A := X ∪ X−1 , one can regard F as the set of all reduced words in A∗ . Let H be a subgroup of F . Show that there exists a complete set of representatives for the right cosets of H in F which is prefixclosed. Such a prefix-closed complete set of representatives for the right cosets of H in the free group F equipped with the base X is called a Schreier system. 6.28 (The Nielsen-Schreier Theorem) Let F be a free group and let H ⊂ F be a subgroup. Let X ⊂ F be a base of F . Let also S ⊂ F be a Schreier system of representatives of the right cosets of H in F with respect to X (cf. Exercise 6.27). Recall that one can identify each element g of F with its reduced form, which is a word w on the alphabet X ∪ X−1 . We denote by l(g) ∈ N the length of w, and, provided that g /= 1F , we denote by α(g) ∈ X∪X−1 (resp. ω(g) ∈ X∪X−1 ) the first letter (resp. the last letter) of w. For g ∈ F , let g denote the representative in S of the right coset H g. Recall (cf. Exercise 6.19) that the set U := Y (X, S) \ {1F } ⊂ H (cf. (6.115)) is a generating subset of H . (a) Let g1 , g2 ∈ F \{1F } such that ω(g1 ) /= α(g2 )−1 . Show that l(g1 g2 ) = l(g1 )+ l(g2 ). (b) Let s ∈ S and a ∈ X ∪ X−1 . Suppose that u := sa(sa)−1 /= 1F . Show that l(u) = l(s) + 1 + l(sa). (c) Let s1 , s2 ∈ S and a1 , a2 ∈ X ∪ X−1 . Set t1 := s1 a1 , t2 := s2 a2 , u1 := s1 a1 (s1 a1 )−1 = s1 a1 t1−1 , and u2 := s2 a2 (s2 a2 )−1 = s2 a2 t2−1 . Suppose that u1 , u2 /= 1F and u2 /= u−1 1 . Show that:

Exercises

277

(i) if l(t1 ) < l(s2 ), then t1 a1−1 is not a prefix of s2 ; (ii) if l(t1 ) > l(s2 ), then s2 a2 is not a prefix of t1 ; (iii) if l(t1 ) = l(s2 ), then a1 t1−1 s2 a2 /= 1F . Deduce that setting w := t1−1 s2 ∈ F one has u1 u2 = s1 a1 wa2 t2−1 and l(u1 u2 ) = l(s1 ) + 1 + l(w) + 1 + l(t1 ). (d) Let u1 , u2 , . . . un ∈ U ∪ U −1 . Suppose that ui /= u−1 i+1 for i = 1, 2, . . . , n − 1. Show that l(u1 u2 · · · un ) ≥ n − 1. (e) Show that H is a free group based on U . 6.29 (Schreier Index-Rank Formula) Let F be a free group of finite rank and let H be a finite index subgroup of F . (a) Show that H is a free group of finite rank. (b) Let X ⊂ F be a free base for F and let S ⊂ F be a Schreier system of representatives of the right cosets of H in F with respect to X (cf. Exercise 6.27). Recall that one can identify each element of F with its reduced form, which is a word on the alphabet X ∪ X−1 . For g ∈ F , let g denote the representative in S of the right coset H g. Set Z := {(s, x) ∈ S × X : sx ∈ / S}. Show that the map from Z into H , given by (s, x) |→ sx(sx)−1 , is injective. (c) Let n (resp. r) denote the rank of F (resp. H ). Show that one has r = 1 + [F : H ](n − 1).

.

(6.117)

6.30 (Cofinitely Hopfian Groups) A group G is said to be cofinitely Hopfian if every endomorphism of G whose image has finite index in G is an automorphism of G. (a) (b) (c) (d)

Show that every cofinitely Hopfian group is Hopfian. Show that every non-trivial finite group is Hopfian but not cofinitely Hopfian. Show that the group Z is Hopfian but not cofinitely Hopfian. Show that every free group of finite rank n ≥ 2 is cofinitely Hopfian.

6.31 (Rank of a finitely generated group) Let G be a finitely generated group. The minimal cardinality |X| of a (not necessarily symmetric) generating subset X ⊂ G is called the rank of G and it is denoted by rk(G). (a) Show that rk(G) = 0 (resp. rk(G) = 1) if and only if G is a trivial group (resp. is a nontrivial cyclic group). (b) Let n ≥ 0 be an integer. Show that if F is a free group of rank n, then rk(F ) = n. (c) Let π : G → K be a group epimorphism of G onto a group K. Show that rk(K) ≤ rk(G). (d) Show that rk(G) is equal to the minimal n ∈ N such that there exist a free group F of rank n and an group epimorphism π : F → G. (e) Let d ≥ 1 be an integer. Show that rk(Zd ) = d.

278

6 Finitely Generated Groups

(f) Let G1 and G2 be two finitely generated groups. Show that rk(G1 × G2 ) ≤ rk(G1 ) + rk(G2 ) and provide an example showing that one may have a strict inequality. (g) Let H be a finite index subgroup of G. Show that H is finitely generated and that one has rk(H ) ≤ 1 + [G : H ](rk(G) − 1). 6.32 (A Theorem of Ralph Strebel [Str]) Let G be a finitely generated residually finite group. Suppose that for every finite index subgroup H of G, the inequality established in Exercise 6.31(g) is actually an equality, i.e., one has .

rk(H ) = 1 + [G : H ](rk(G) − 1).

(6.118)

The goal of this exercise is to show that G is free. (a) Let A be a set. Recall that it follows from the Axiom of Choice that A admits a well-ordering, that is, a total ordering ≤ with the property that every non-empty subset of A has a minimal element. Let (≤i )i≥1 be a sequence of well-orderings on A. Consider the monoid A∗ consisting of all words on the alphabet A and denote by |u| the length of a word u ∈ A∗ . Define a relation < on A∗ by setting, for all u, v ∈ A∗ , u ≺ v if either |u| < |v| or there exist w, u' , v ' ∈ A∗ and x, y ∈ A such that u = u' xw and v = v ' yw with |u' | = |v ' | and x 0 such that γS (n) ≤ C1 n4 for all n ≥ 1. (f) Let n, x, y, z be integers such that n ≥ 1, 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

Exercises

281

such that Ax B y C z = Ax−q B n Aq B y−n C r .

.

(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) Show that γ (G) - n4 . (j) Show that the map ϕ : Z → G, defined by ϕ(k) := C k for all k ∈ Z, is a uniform embedding but not a quasi-isometric embedding. 6.42 Show that the Grigorchuk group is Hopfian. 6.43 Let G be the Grigorchuk group and let Σ := {0, 1}. Recall that G naturally acts on Σ ∗ . (a) Show that G acts transitively on Σ n for all n ∈ N. (b) Use (a) to recover the fact that G is infinite. 6.44 (The Automorphism Group of a Regular Rooted Tree) Let Σ be a set. Let Σ ∗ denote the free monoid based on Σ, i.e., the set consisting of all words on the alphabet Σ with the concatenation of words as the monoid operation. Recall that one says that a word u ∈ Σ ∗ is a prefix of a word w ∈ Σ ∗ , and one writes u < w, if there exists v ∈ Σ ∗ such that uv = w. The relation < is a partial order on Σ ∗ and the empty word ε ∈ Σ ∗ is the least element of (Σ ∗ , 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)-almost-homomorphism. This shows that G is sofic. u n From Propositions 7.5.3 and 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 |F \ sF | ≤

.

ε |F | |S|

(7.19)

334

7 Local Embeddability and Sofic Groups

for all s ∈ S. Consider the set E := In fact, as S = S −1 , we get

n

sF . Observe that E ⊂ F since 1G ∈ S.

s∈S

sE ⊂ F

(7.20)

.

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

n

sF |

s∈S

=|

U

(F \ sF )|

s∈S

.

≤

Σ

(7.21)

|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 ϕ(g)(f ) :=

{ gf

.

αg (gf )

if gf ∈ F otherwise

for all g ∈ G and f ∈ F (see Fig. 7.1). 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 dF (ϕ(k1 k2 ), ϕ(k1 )ϕ(k2 )) ≤

.

|F \ E| ≤ε |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 dF (ϕ(k1 ), ϕ(k2 )) ≥

.

|E| ≥1−ε |F |

7.5 Sofic Groups

335

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 )

for all k1 , k2 ∈ K with k1 /= k2 . Thus, the map ϕ : G → Sym(F ) is a (K, ε)-almostu n 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. Proposition 7.5.7 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 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

.

336

7 Local Embeddability and Sofic Groups

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), dF (ϕ(kk ' ), ϕ(k)ϕ(k ' )) = 1 −

) || ( 1 − dFj (ϕj (kj kj' ), ϕj (kj )ϕj (kj' ) 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. u n Corollary 7.5.8 Let (Gi )i∈I be a family of sofic groups. Then their direct sum G := O i∈I Gi is sofic. Proof This follows immediately from||Propositions 7.5.4 and 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 . u n 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 and set G := lim Gi . By ← − construction of a projective limit (see Appendix E), G is a subgroup of the group || u n i∈I Gi . We deduce that G is sofic by using Propositions 7.5.7 and 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

7.5 Sofic Groups

337

Φ := ϕ ' ◦ ϕ : G → Sym(F ) is a (K, ε)-almost-homomorphism. 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.

u n

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 LEF-group, every locally residually amenable group, every locally residually finite group, every residually amenable group, and every residually finite group is sofic. O 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.

O

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 . O 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.

338

7 Local Embeddability and Sofic Groups

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 )

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

7.6 Sofic Groups and Metric Ultraproducts of Finite Symmetric Groups

339

have that |B ' | = |B| · |F2 | ≥ (1 − ε)|F1 | · |F2 | = (1 − ε)|F |

.

(7.27)

and 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 u n 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 :=

||

.

i∈I

Sym(Fi ).

340

7 Local Embeddability and Sofic Groups

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) (ii) (iii) (iv) (v)

δω (α, α) = 0; δω (β, α) = δω (α, β); .δω (α, β) ≤ δω (α, γ ) + δω (γ , β); .δω (γ α, γβ) = δω (α, β); .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 biinvariant 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). u n .

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 , γ αγ −1 ) = δω (γ −1 , αγ −1 ) = δω (γ −1 γ , α) = δω (1PT , α)

.

for all .α, γ ∈ PT . It follows that .γ αγ −1 ∈ NT for all .α ∈ NT and .γ ∈ PT . This shows that .NT is a normal subgroup of .PT . u n Observe that, given .α = (αi )i∈I and .β = (βi )i∈I in .PT , one has αNT = βNT

.

⇐⇒ δω (α, β) = 0.

(7.31)

7.6 Sofic Groups and Metric Ultraproducts of Finite Symmetric Groups

341

Indeed, one has .αNT = βNT if and only if .α −1 β ∈ NT , that is, if and only if −1 β) = 0. This is equivalent to .δ (α, β) = 0 by the left-invariance of .δ . .δω (1PT , α ω ω 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 = g NT . (gi )i∈I ∈ PT such that .g = We first introduce some constants that will be used in the proof. If h and k are distinct elements in K, then .δω (h, k) > 0 by (7.31). Let us set 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 hkNT 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 ⎛ j ∈⎝

n

.

h,k∈K

⎞ A(h, k)⎠

⎛

⎞

n⎜ n ⎟ ⎜ C(h, k)⎟ ⎝ ⎠. h,k∈K

h/=k

342

7 Local Embeddability and Sofic Groups

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: (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. u n 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 ' ).

.

7.6 Sofic Groups and Metric Ultraproducts of Finite Symmetric Groups

343

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)

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 .GT := PT /NT . || Let .ϕ : G → PT denote the product map .ϕ := i∈I ϕi . Thus, 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 .

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7 Local Embeddability and Sofic Groups

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 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 δω (ϕ (g), ϕ (h)) = lim dFi (ϕi (g), ϕi (h)) = 1.

.

i→ω

(7.38)

Therefore, we have .Φ(g) /= Φ(h) by (7.31). Consequently, .Φ is injective. This shows that (a) implies (b). u n 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

7.7 A Characterization of Finitely Generated Sofic Groups

345

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. 6.1–6.3). 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 .

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)

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 edgesymmetric 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 := ψq−1 (q ' ), we have 0 ,(i+1)r0 .g ∈ BS (r0 ) so that .gh ∈ B((i + 1)r0 ) for all .h ∈ B(ir0 ). It follows that the map ψq ' ,ir0 : BS (ir0 ) → B(q ' , ir0 )

.

(7.43)

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7 Local Embeddability and Sofic Groups

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 ). u n 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). u n 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) = ψq0 ,2r0 (h) = ψψq0 ,2r0 (h),r0 (1G ) = ψψq0 ,2r0 (h),r0 (k). Now suppose that .k /= 1G . Then we can find

7.7 A Characterization of Finitely Generated Sofic Groups

347

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 . Since −1 −1 −1 − ' ' .ψ ' q ,r0 ((π 1 ) ) = ψq ' ,r0 (ψq0 ,2r0 (h)) = ψq ' ,r0 (q ) = 1G and .ψq ,r0 preserves 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 vertex and −1 + + with a given label. It follows that .ψq−1 ' ,r (ψq0 ,2r0 (hk)) = ψq ' ,r (π 1 ) = π2 = k, that 0 0 u n 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 −1

ε' := ε(1 + |BS (r)| · |S| + |BS (r)|2 )

.

.

(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 −1 )(q)), were .q ∈ Q and .E ⊂ Q × S × Q consisting of all the triples .(q, s, ϕ(s .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.

348

7 Local Embeddability and Sofic Groups

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(s1−1 )(q)) = (q, s1 , ψq (s1 )), e2 := (ψq (s1 ), s2 , ϕ(s2−1 )ψq (s1 )) = (ψq (s1 ), s2 , ψψq (s1 ) (s2 )) = (ψq (s1 ), s2 , ψq (s1 s2 )) (by (∗∗)), .

······ er ' := (ψq (s1 s2 · · · sr ' −1 ), sr ' , ϕ(sr−1 ' )(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 .l(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)

7.7 A Characterization of Finitely Generated Sofic Groups

349

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 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' )-almost-homomorphism 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'' :=

U

.

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)

.

for all .q ∈ Q \ Q''' (g, h). Let now .g /= h vary in .BS (r) and set Q''' :=

U

.

g,h∈BS (r) g/=h

Q''' (g, h).

(7.50)

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7 Local Embeddability and Sofic Groups

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|,

.

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 (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

ϕ(g)(q) :=

.

⎧ −1 ⎪ ⎪ ⎨ψq,2r0 (g ) ⎪ ⎪ ⎩

if g ∈ BS (r0 ) and q ∈ Q(2r0 );

αg −1 (q)

if g ∈ BS (r0 ) and q ∈ Q \ Q(2r0 );

q

otherwise.

(7.54)

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

0

−1 (k2−1 ),r0 (k1 )

(by (7.46))

7.8 Surjunctivity of Sofic Groups

351

= 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. 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). u n

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 S .a0 ∈ A and consider the map .μ : A → A defined by setting μ(u) :=

.

{ f (x)(1G ) a0

if there exists x ∈ X such that x|S = u otherwise

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 . u n 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

.

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7 Local Embeddability and Sofic Groups

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 .AG , 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 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 C AG . Then, since Y is closed in .AG , there exists a finite subset .Ω ⊂ G such that .πΩ (Y ) C 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 B (r ) .Ω ⊂ BS (r0 ). Thus, .πBS (r0 ) (Y ) C A S 0 . Fix .ε > 0 such that ε 0. 7.47 Let G be a group and let K be a finite subset of G. Let F be a non-empty finite set. Show that if 0 < ε < 2/|F | then every (K, ε)-almost-homomorphism ϕ : G → Sym(F ) is a K-almost-homomorphism of G into the group Sym(F ). 7.48 Let G be a group. Show that G is residually finite if and only if the following condition is satisfied: there exist a directed set I and two families (Fi )i∈I and (ϕi )i∈I indexed by I , where, for each i ∈ I , Fi is a non-empty finite subset of G and ϕi : G → Sym(Fi ) is a group homomorphism, such that, for every g ∈ G \ {1G }, the Hamming distance between ϕi (g) and IdFi is eventually equal to 1. 7.49 (The Hilbert-Schmidt Metric) Let H be a complex Hilbert space of finite dimension n ≥ 1. Let L(H ) denote the C-algebra consisting of all linear maps u : H → H . If u ∈ L(H ), we denote by Tr(u) ∈ C the trace of u and by u∗ ∈ L(H ) its adjoint. For u, v ∈ L(H ), we set H S :=

.

1 Tr(uv ∗ ). n

(a) Show that H S is a scalar product on L(H ). Denote by || · ||H S the associated norm and by dH S the associated metric. Thus, one has ||u||H S =

.

for all u, v ∈ L(H ).

√

H S

and

dH S (u, v) = ||u − v||H S

Exercises

367

(b) Let U(H ) := {u ∈ L(H ) : uu∗ = IdH }. Show that U(H ) is a group for the composition of maps. (c) Show that one has / dH S (u, v) =

.

2(1 −

1 Re(Tr(v −1 u))). n

(7.62)

for all u, v ∈ U(H ). (d) Show that dH S induces by restriction a bi-invariant metric on U(H ). 7.50 Let F be a non-empty finite set and let dF denote the Hamming metric on Sym(F ). We consider the |F |-dimensional Hilbert space H := CF = {x : F → C}. We denote by U(H ) the unitary group of H and by dH S the Hilbert-Schmidt metric on U(H ) (cf. Exercise 7.49). For α ∈ Sym(F ), define λ(α) : H → H by λ(α)(x) := x ◦ α −1 for all x ∈ H . (a) Show that λ(α) ∈ U(H ) for all α ∈ Sym(F ). (b) Show that the map λ : Sym(F ) → U(H ) is an injective group homomorphism. (c) Show that one has dF (α, β) = 1 −

.

1 Tr(λ(β −1 α)) |F |

(7.63)

for all α, β ∈ Sym(F ). (d) Show that one has dH S (λ(α), λ(β)) =

.

√ 2dF (α, β)

(7.64)

for all α, β ∈ Sym(F ). 7.51 (Hyperlinear Groups) Let G be a group. Given a finite subset K ⊂ G, a real number ε > 0, and a complex Hilbert space H with finite positive dimension, a map ψ : G → U(H ) is called a (K, ε)- almost-homomorphism if it satisfies the following conditions: ((K, ε)-AHU-1) for all k1 , k2 ∈ K, one has dH S (ψ(k1 k2 ), ψ(k1 )ψ(k2 )) ≤ ε; ((K, ε)-AHU-2) for all distinct k1 , k2 ∈ K, one has dH S (ψ(k1 ), ψ(k2 )) ≥ 1 − ε, where dH S denotes the Hilbert-Schmidt metric on U(H ) (cf. Exercise 7.49). A group G is called hyperlinear if it satisfies the following condition: for every finite subset K ⊂ G and every real number ε > 0, there exist a complex Hilbert space H with finite positive dimension and a (K, ε)-almost-homomorphism ψ : G → U(H ). Show that every sofic group is hyperlinear. 7.52 Suppose that a group G contains a family (Hi )i∈I of subgroups satisfying the U 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.

368

7 Local Embeddability and Sofic Groups

7.53 Show that every virtually sofic group is sofic. 7.54 (The Weiss Condition) Let G be a finitely generated group and let S be a finite symmetric generating subset of G. Consider the following condition (W(G, S)) for all ε > 0 and n ∈ N, there exists a finite S-labeled graph Q = (Q, E) such that there is a proportion at least 1 − ε of vertices q ∈ Q for which there exists an S-labeled graph isomorphism BS (n) → BQ (q, n) sending 1G to q. It follows from Theorem 7.7.1 that condition (W(G, S)) is equivalent to the soficity of the group G. This implies in particular that the fact that condition (W(G, S)) is satisfied or not is actually independent of the choice of S. The purpose of this exercise is to give a direct proof of this independence (without using the notion of soficity). (a) Let S ' ⊂ G be another finite symmetric generating subset of G such that S ' ⊂ S. Suppose that condition (W(G, S)) is satisfied. Show that condition (W(G, S ' )) is also satisfied. (b) Let m ≥ 1 be an integer and let S '' := BS (m) denote the ball of radius m centered at 1G in the Cayley graph of (G, S). Suppose that condition (W(G, S)) is satisfied. Show that S '' is a finite symmetric generating subset of G and that condition (W(G, S '' )) is also satisfied. (c) Deduce that if S ''' ⊂ G is another finite symmetric generating subset of G, then condition (W(G, S)) is satisfied if and only if condition (W(G, S ''' )) is satisfied. 7.55 Let G1 and G2 be two finitely generated groups and let S1 and S2 be finite symmetric generated subsets of G1 and G2 , respectively. Set G := G1 × G2 . Then (cf. Proposition 6.6.10) S := (S1 × {1G2 }) ∪ ({1G1 } × S2 ) is a finite symmetric generating subset of G. Suppose that conditions (W(G1 , S1 )) and (W(G2 , S2 )) hold. Give a direct proof of the fact that condition (W(G, S)) holds as well. 7.56 Let G be a finitely generated residually finite group and let S be a finite symmetric generating subset of G. Give a direct proof of the fact that condition (W(G, S)) is satisfied. 7.57 Let G be a finitely generated amenable group and let S be a finite symmetric generating subset of G. Give a direct proof of the fact that condition (W(G, S)) is satisfied.

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 preinjective 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 finitedimensional alphabet over the free group of rank two, one which is pre-injective but not surjective, and one which is surjective but not pre-injective. This shows that the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3_8

369

370

8 Linear Cellular Automata

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 onesided 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 Sect. 8.16, 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). In Sect. 8.17, it is shown that, given a group G, the following conditions are equivalent: (1) G is amenable; (2) every pre-injective cellular automaton with finite alphabet over G is surjective; (3) for any (or, equivalently, some) field K, every preinjective linear cellular automaton over G whose alphabet is a finite-dimensional vector space over K is surjective (Corollary 8.17.7). This yields characterizations of amenability in terms of cellular automata. 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 Klinear and continuous, that is, for each .g ∈ G, the map .x |→ gx is a continuous endomorphism of .V G .

8.1 The Algebra of Linear Cellular Automata

371

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 ' )

.

and

τ (kx) = kτ (x)

for all .x, x ' ∈ V G and .k ∈ K. 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 Klinear. Proof Suppose first that .τ is linear. Let .y, y ' ∈ V S and .k ∈ K. Denote by x and ' G extending y and .y ' respectively, i.e., such that .x| = y .x two configurations in .V S ' ' 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. u n The following result is a linear analogue of the Curtis-Hedlund-Lyndon theorem (Theorem 1.8.1).

372

8 Linear Cellular Automata

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. 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 −1 x)(1 ) = μ((g −1 x)| ) for .τ is G-equivariant, we deduce that .τ (x)(g) = τ (g G M G all .x ∈ V 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). u n 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 : K G → K G (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 G may be viewed as a sequence .x = (x ) .x ∈ V n n∈Z , where .xn = xn (t) is a polynomial for all .n ∈ Z. Let .S := {0, 1} and consider the K-linear map S → V defined by .μ(p, q) := p − tq ' for all .(p, q) ∈ V S = V × V , .μ : 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

8.1 The Algebra of Linear Cellular Automata

373

' then given by .τ (x) = y, where .yn := xn − txn+1 ∈ V , .n ∈ Z, for all Z .x = (xn )n∈Z ∈ V .

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 . 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),

.

(ff ' )(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 G .LCA(G; V ) ⊂ EndK (V ). 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 ).

374

8 Linear Cellular Automata

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 G .LCA(G; V ) is a subalgebra of .EndK (V ). u n

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 . n u 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 }. u n 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].

.

8.2 Configurations with Finite Support

375

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 −1 ⊂ G. This shows that .τ (x) ∈ V [G]. .T S u n 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]). 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 Klinear 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 G .x ∈ V , that is, .τ = 0. This shows that .Λ is injective. u n 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 preinjective. u n .

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8 Linear Cellular Automata

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 G → V G with memory set .S ⊂ H , we denote by .τ : V H → .τ : V H H V the restriction cellular automaton. Similarly, given a cellular automaton H .σ : V → 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. u n 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 G .LCA(H ; V ) whose inverse is the map .σ |→ σ . 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 K-algebra homomorphism. Let .τ1 , τ2 ∈ LCA(G, H ; V ) and .k1 , k2 ∈ K. Let .x ∈ V H and let .x ∈ VG extending x. By applying (1.13), we have Φ(k1 τ1 + k2 τ2 )(x)(h) = (k1 τ1 + k2 τ2 )(x )(h) = k1 τ1 (x )(h) + k2 τ2 (x )(h)

.

8.4 Group Rings and Group Algebras

377

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. u n

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 G := {α : G → R} has a natural structure of left R-module with addition and .R scalar multiplication given by (α + β)(g) = α(g) + β(g)

.

and (rα)(g) = rα(g)

for all .α, β ∈ R G , .r ∈ R, and .g ∈ G. Define the support of an element .α ∈ R G as being the set .{g ∈ G : α(g) /= 0R }. Let .R[G] denote the set consisting of all elements .α ∈ R G which have finite support. Then .R[G] is a free submodule of .R G . We have O || .R[G] = R⊂ R = RG, g∈G

g∈G

so that the elements .δg : G → R, .g ∈ G, defined by { δg (h) :=

.

1R

if h = g

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=

Σ

.

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 ∈ R G defined by (αβ)(g) :=

Σ

.

h∈G

α(h)β(h−1 g) =

Σ h∈S

α(h)β(h−1 g)

(8.2)

378

8 Linear Cellular Automata

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: Σ

(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 Σ [(αβ)γ ](g) = (αβ)(h)γ (h−1 g) h∈G

=

ΣΣ

α(k)β(k −1 h)γ (h−1 g)

h∈G k∈G

(by setting s = k −1 h) =

ΣΣ

α(k)β(s)γ (s −1 k −1 g)

s∈G k∈G

.

=

Σ

α(k)(

k∈G

=

Σ

Σ

β(s)γ (s −1 k −1 g))

s∈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 α](g) =

Σ

.

δ1G (h)α(h−1 g) = α(g) =

h∈G

Σ

α(k)δ1G (k −1 g) = [αδ1G ](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)γ ](g) =

Σ

.

(α + β)(h)γ (h−1 g)

h∈G

=

Σ

[α(h) + β(h)]γ (h−1 g)

h∈G

=

Σ

α(h)γ (h−1 g) +

h∈G

= (αγ )(g) + (βγ )(g).

Σ k∈G

β(k)γ (k −1 g)

8.4 Group Rings and Group Algebras

379

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 . u n 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. Σ Proof Let .g1 , g2 , g ∈ G. Then .(δg1 δg2 )(g) = h∈G δg1 (h)δg2 (h−1 g) equals 1 if −1 .g 1 g = g2 , that is, if .g = g1 g2 , and equals 0 otherwise. This shows that δg1 δg2 = δg1 g2 .

.

(8.4)

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). u n 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 ∗ −1 . Thus one has .α ∗ ∈ R[G]. .α is .S 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.

380

8 Linear Cellular Automata

(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 ) Σ α(h)β(h−1 g −1 ) =

.

h∈G

=

Σ

β(h−1 g −1 )α(h)

(since R is commutative)

h∈G

=

Σ

β ∗ (gh)α ∗ (h−1 )

h∈G

=

Σ

β ∗ (k)α ∗ (k −1 g)

k∈G

= (β ∗ α ∗ )(g). Therefore .(αβ)∗ = β ∗ α ∗ .

u n

If .R = (R, +, ·) is a ring, its opposite ring is the ring .R op := (R, +, ∗) having the same underlying set and addition as R and with multiplication .∗ defined by .r ∗ s := sr for all .r, s ∈ R. 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 . .O

.

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].

8.5 Group Ring Representation of Linear Cellular Automata

381

(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 Kalgebra .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) :=

Σ

.

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 { cv (g) :=

.

v if g = 1G , 0 otherwise.

(8.6)

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 −1 ) 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) :=

Σ

.

s∈S

α(s)(y(s))

382

8 Linear Cellular Automata

for all .y ∈ V S . Then, for all .x ∈ V G and .g ∈ G, we have τα (x)(g) =

Σ

α(h)(x(gh))

h∈G

=

Σ

α(h)(g −1 x(h)) (8.7)

h∈G

.

=

Σ

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, τα (cv )(g −1 ) =

Σ

.

α(h)(cv (g −1 h)) = α(g)(v).

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). u n 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 K-algebra isomorphism. Proof Let .α, β ∈ EndK (V )[G] and .k ∈ K. By applying (8.5), we get τα+β (x)(g) =

Σ

.

[(α + β)(h)](x(gh))

h∈G

=

Σ

(α(h) + β(h))(x(gh))

h∈G

=

Σ

[α(h)(x(gh)) + β(h)(x(gh))]

h∈G

8.5 Group Ring Representation of Linear Cellular Automata

=

Σ

α(h)(x(gh)) +

h∈G

Σ

383

β(h)(x(gh))

h∈G

= τα (x)(g) + τβ (x)(g) and, similarly, τkα (x)(g) =

Σ

[(kα)(h)](x(gh))

h∈G

=

Σ

k[α(h)](x(gh))

h∈G

.

=k

Σ

[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 y(gh) =

Σ

.

(8.8)

β(t)(x(ght)).

t∈G

It follows that τα (τβ (x))(g) = τα (y)(g) Σ α(h)(y(gh)) = h∈G

(by (8.8)) =

Σ

α(h)

h∈G . (by setting z = ht) =

Σ

(Σ t∈G

α(h)

(Σ

h∈G

=

) β(t)(x(ght))

Σ( Σ

β(h−1 z)(x(gz))

z∈G

) α(h)β(h−1 z) (x(gz))

z∈G h∈G

=

Σ

)

[αβ](z)(x(gz))

z∈G

= ταβ (x)(g).

384

8 Linear Cellular Automata

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) Σ 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), Σ 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 ) =

Σ

.

( ) Σ 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 { α(h) :=

.

αs if h = s ∈ S, 0 otherwise. u n

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) :=

Σ

.

α(h)x(gh)

h∈G

for all .α ∈ K[G], .x ∈ V G , and .g ∈ G, is a K-algebra isomorphism.

O

.

8.5 Group Ring Representation of Linear Cellular Automata

385

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. .O 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 Σ .α := |S|δ1G − δs ∈ K[G]. s∈S

Then, taking .V := K, we have, for all .x ∈ K G and .g ∈ G, Ψ (α)(x)(g) =

Σ

.

α(h)x(gh) = |S|x(g) −

h∈G

Σ

x(gs).

s∈S

It follows that .Ψ (α) is the discrete Laplacian .ΔS : K G → K G 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 Σ .Ψ (α)(x)(g) = α(h)(x(gh)) = f (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) =

Σ

.

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)).

386

8 Linear Cellular Automata

(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].

.

For all .x = (xn )n∈Z ∈ V Z , we have .Ψ (α)(x) = (yn )n∈Z ∈ V Z , where yn :=

Σ

.

' 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

(ff ' )(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 αx :=

Σ

.

α(g)gx.

(8.10)

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 ' ,

8.6 Modules Over a Group Ring

387

(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.

.

(ii) We have α(x + x ' ) =

Σ

.

α(g)g(x + x ' ) =

g∈G

=

Σ

Σ

α(g)gx +

g∈G

Σ

(α(g)gx + α(g)gx ' )

g∈G

α(g)gx ' = αx + αx ' .

g∈G

(iii) We have Σ

(α + β)x =

.

(α + β)(g)gx =

g∈G

Σ

=

α(g)gx +

g∈G

Σ

Σ

(α(g) + β(g))gx

g∈G

β(g)gx = αx + βx.

g∈G

(iv) We have, by using (8.3), α(βx) =

Σ

.

α(h1 )h1 (βx)

h1 ∈G

=

Σ

α(h1 )h1

(Σ

h1 ∈G

=

Σ Σ

α(h1 )β(h2 )h1 h2 x

Σ( Σ

g∈G

=

Σ

β(h2 )h2 x

h2 ∈G

h1 ∈G h2 ∈G

=

)

) α(h1 )β(h2 ) gx

h1 ,h2 ∈G h1 h2 =g

(αβ)(g)gx

g∈G

= (αβ)x. u n

388

8 Linear Cellular Automata

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, we have, by using the G-equivariance and the R-linearity of f , f (αx) = f

(Σ

.

) α(g)gx

=

g∈G

Σ

α(g)f (gx) =

g∈G

Σ

α(g)gf (x) = αf (x).

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). u n 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 =

Σ

.

α(h)hx,

h∈G

that is, (αx)(g) =

Σ

.

h∈G

α(h)x(h−1 g)

(8.11)

8.7 Matrix Representation of Linear Cellular Automata

389

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 G .EndK (V ) 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 ). .O

.

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 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 G .K[G]-module .V . Recall that, for .v ∈ V , the configuration .cv ∈ V [G] is defined by { cv (g) :=

v

if g = 1G ,

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 Σ .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) =

Σ

.

i∈I

αi (g)ei

390

8 Linear Cellular Automata

for all .g ∈ G. This gives us x=

Σ

.

gcx(g)

g∈G

=

) Σ (Σ g αi (g)cei

g∈G

=

g∈G

=

Σ

)

αi (g)gcei

i∈I

Σ( Σ i∈I

=

i∈I

Σ (Σ

) αi (g)gcei

g∈G

αi cei ,

i∈I

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 u n 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.

O

.

If .τ : V G → V G 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: 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 Kalgebra homomorphism. .O 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.

8.7 Matrix Representation of Linear Cellular Automata

391

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

.

Σ

x=

.

gcx(g)

g∈G

by (8.12). This implies ) Σ (Σ u(x) = u gcx(g) = gu(cx(g) )

.

g∈G

g∈G

since u is .K[G]-linear. We deduce that u(x)(1G ) =

Σ

.

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 U −1 .S ⊂ G defined by .S := 1≤i≤d Ti . Σd If .v ∈ V , we can write .v = i=1 ki ei with .ki ∈ K, .1 ≤ i ≤ d. We then have Σd .cv = i=1 ki cei and hence ) Σ (Σ d d .u(cv ) = u ki cei = ki u(cei ). i=1

i=1

We deduce that .u(cv )(g −1 ) = 0 if .g ∈ / S. Thus, using (8.13), we get u(x)(1G ) =

Σ

.

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 )

392

8 Linear Cellular Automata

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. u n 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 infinite-dimensional 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 U 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 U −1 . is infinite, we can find elements .g0 ∈ G and .i0 ∈ I such that .g0 ∈ /S i∈I Ti and .g0 ∈ Ti−1 . Consider the configuration .x0 ∈ V [G] which takes the value .ei0 0 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 .

.

Thus, we have 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 :=

(Σ d k=1

) aik bkj

1≤i,j ≤d

8.8 The Closed Image Property

393

for all .A = (aij )1≤i,j ≤d , B = (bij )1≤i,j ≤d ∈ Matd (R). Suppose now that M is a free Σd left R-module with basis .e1 , e2 , . . . , ed . If .u ∈ EndR (M), then we 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 (R op ), where .R op 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. .O 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 vector space over a field K and G → V G is a linear cellular automaton, then the image of .τ is closed in .V G .τ : V 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. U Then we can find a sequence .(An )n∈N of finite subsets of G such that .G = n∈N An and −S denote .An ⊂ An+1 for all .n ∈ N. Let S be a memory set for .τ and let .Bn := An U the S-interior of .An (cf. Sect. 5.4). Note that .G = 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 on .Bn . Therefore, given .xn ∈ V An and denoting by .xn ∈ V G a configuration extending .xn , the pattern yn := τ (xn )|Bn ∈ V Bn

.

394

8 Linear Cellular Automata

does not depend on the particular choice of the extension .xn . 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 finite-dimensional 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 −1 element in .ρn,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 .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 ) =

||

.

c∈G/H

τc (V c ).

(8.16)

8.8 The Closed Image Property

395

Also, it follows from (1.18) that, for all .c ∈ G/H and .g ∈ c, ( ) τc (V c ) = (φg∗ )−1 τH (V H ) .

.

(8.17)

As the map .φg∗ : 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 . u n 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 .O 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 G G .τ (V ) = V . Thus, .τ is surjective. u n 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 . 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 xn :=

.

{ 1 + t + · · · + t n−M 0

if n ≥ M, 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 Z .V . 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.

396

8 Linear Cellular Automata

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 finitedimensional 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 finite-dimensional 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 .

mdimF (X) := lim sup j

dim(πFj (X)) |Fj |

,

(8.18)

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) =

for every vector subspace .X ⊂ V G .

1 entF (X) log |K|

8.9 The Garden of Eden Theorem for Linear Cellular Automata

397

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))

=

|Fj |

|Fj | dim(V ) dim(V Fj ) = = dim(V ). |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). u n 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 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

398

8 Linear Cellular Automata

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 .πΩ (Y ) −S is a vector subspace of .πΩ −S (Y ) × V Ω\Ω . This implies −S

.

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 −S | dim(V ) ≤ dim(πΩ (X)) + |Ω \ Ω −S | 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 )) |Fj |

dim(πFj (X))

≤

|Fj |

+

|∂S (Fj )| dim(V ). |Fj |

Since .

lim j

|∂S (Fj )| =0 |Fj |

by Proposition 5.4.4, we finally get .

mdimF (Y ) = lim sup j

dim(πFj (Y )) |Fj |

≤ lim sup j

dim(πFj (X)) |Fj |

= mdimF (X). u n

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 vector subspace of .V G . Suppose that there exist finite subsets E and .E ' of G and an .(E, E ' )-tiling .T ⊂ G such that .πgE C V gE for all .g ∈ T . Then one has .mdimF (X) < dim(V )

8.9 The Garden of Eden Theorem for Linear Cellular Automata

399

Proof 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 \ ug∈Tj gE.

.

We thus have .

dim(πgE (X)) ≤ dim(V gE ) − 1 = |gE| dim(V ) − 1

for all g ∈ T .

(8.20)

As ||

∗

πFj (X) ⊂ V Fj ×

.

πgE (X),

g∈Tj

we get ∗

.

dim(πFj (X)) ≤ dim(V Fj ×

||

πgE (X))

g∈Tj

= |Fj∗ | dim(V ) +

Σ

dim(πgE (X))

g∈Tj

≤ |Fj∗ | dim(V ) +

Σ

(|gE| dim(V ) − 1)

(by (8.20))

g∈Tj

) ( Σ |gE| dim(V ) − |Tj | = |Fj∗ | + g∈Tj

= |Fj | dim(V ) − |Tj |, since |Fj | = |Fj∗ | +

Σ

.

|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)) |Fj |

≤ dim(V ) − α

for all j ≥ j0 .

400

8 Linear Cellular Automata

This implies that .

mdimF (X) = lim sup

dim(πFj (X))

j

|Fj |

≤ dim(V ) − α < dim(V ). u n

We are now in position to state the linear version of the Garden of Eden theorem. Theorem 8.9.6 (Garden of Eden Theorem for Linear Cellular Automata) 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 G → V G is surjective. .τ : V .O 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 can find a finite subset .Ω ⊂ G such that .πΩ (y) ∈ / πΩ (X). Therefore we have .πΩ (X) C 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 ). u n Lemma 8.9.9 Let .τ : V G → V G be a linear cellular automaton. Suppose that .

mdimF (τ (V G )) < dim(V ).

(8.21)

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 j

.

\Fj

, we have

dim(πF +S (X)) ≤ dim(πFj (X)) + |Fj+S \ Fj | dim(V ) j

≤ dim(πFj (X)) + |∂S (Fj )| dim(V ).

8.9 The Garden of Eden Theorem for Linear Cellular Automata

401

We thus have dim(πF +S (X)) .

j

|Fj |

≤

dim(πFj (X)) |Fj |

+

|∂S (Fj )| dim(V ). |Fj |

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) vector j0

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 Proposition 5.4.3. 0 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). n u 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 and .S = S −1 . 2 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 which does .gΩ ⊂ gE. Let us choose, for each .g ∈ T , a hyperplane .Hg ⊂ V 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 such that .πgΩ (z' ) = πgΩ (z + kg (gx0 )) for each ' ' .g ∈ T and .z = z outside of .ug∈T gΩ. We have .z ∈ X by construction. On the ' other hand, since .z and z coincide outside .ug∈T gΩ, we have .τ (z' ) = τ (z) outside +S . Now, if .h ∈ gΩ +S for some .g ∈ T , then .hS ⊂ gΩ +S 2 = gE and .ug∈T gΩ 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 u n Proposition 8.9.5.

402

8 Linear Cellular Automata

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. u n 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 map .τϕ = ϕG : V G → G V is a linear cellular automaton with memory set .S := {1G } and local defining map .ϕ . Since B is infinite, we can find a map .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 .ϕ1 is surjective. As a product of surjective maps is a surjective map, it follows that .τ1 is surjective. On the other hand, as .ϕ1 is not injective, there exists .v ∈ ker(ϕ1 ) \ {0}. Consider the 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 ∈ ker(τ2 ), that is, .0 = τ2 (x)(g) = ϕ2 (x(g)) for all .g ∈ G. As .ϕ2 is injective, 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 .ϕ2 is not surjective, there exists .v ∈ V \ ϕ2 (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 finite-dimensional 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 := K 2 . 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

8.11 Surjective But Not Pre-injective Linear Cellular Automata

403

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 ))

.

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. u n 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 Propositions 1.7.4 and 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. .O 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 non-amenable. 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 := K 2 . 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 { x0 (g) :=

.

(0, k0 )

if g = 1G

(0, 0)

otherwise.

404

8 Linear Cellular Automata

Then, .x0 is almost equal to 0 but .x0 /= 0. As .τ (x0 ) = 0, we deduce that .τ is not pre-injective. However, .τ is surjective. To see this, let .z = (z1 , z2 ) ∈ K G × K G = 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) .x(s) := (z (1 ), 0) ⎩ 2 G (0, 0)

if s = a if s = b otherwise.

(8.22)

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) ⎪ ⎪ ⎪ ⎪ ⎨ (z2 (g), 0) .x(gs) := (z1 (g), 0) ⎪ ⎪ ⎪ (z2 (g) − x2 (g ' ), 0) ⎪ ⎪ ⎩ (0, 0)

if s ' ∈ {a, a −1 } and s = s ' if s ' ∈ {a, a −1 } and s = b if s ' ∈ {b, b−1 } and s = a if s ' ∈ {b, b−1 } and s = s ' otherwise.

(8.23)

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 ' a 2 )) + 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).

8.12 Invertible Linear Cellular Automata

405

The cases when .s ' = a −1 , b, b−1 are similar. It follows that .τ (x) = z. This shows that .τ is surjective. u n 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 pre-injective (see Propositions 1.7.4 and 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 pre-injective. .O 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 Gequivariant. 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 isUcountable, we can find a sequence (An )n∈N of finite subsets of G such that G = n∈N An , S ⊂ A0 and An ⊂ An+1 forU all n ∈ N. Let Bn := A−S n 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'' ∈ −1 Y satisfies yn |Bn = 0 and τ (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 on Bn . Therefore, given xn ∈ V An and denoting by xn ∈ V G a configuration extending xn , the pattern un := τ (xn )|Bn ∈ V Bn

.

406

8 Linear Cellular Automata

does not depend on the particular choice of the extension xn 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. 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 −1 (zn ). Since zn+1 coincides with zn on An , there exists a unique element in ρn,n+1 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 (∗). Let Y ' ⊂ V T be a supplementary vector space of YT in V T , so that V T = YT ⊕ Y ' , and denote by π : V T → YT the corresponding linear projection. Consider the linear map ν : V T → V defined by ν(z) := τ −1 (z)(1G ), where z ∈ Y is any configuration extending the pattern π(z) ∈ YT , for all 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

8.12 Invertible Linear Cellular Automata

407

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 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 φg∗ : V c → V H defined by φg∗ (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, φg∗ (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((φg∗ (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

408

8 Linear Cellular Automata

σc (z)(g) := ν((φg∗ (z))|T ) for all z ∈ V c and g ∈ c then σ =

||

.

(8.27)

σc .

c∈G/H

Given c ∈ G/H , z ∈ V c and g ∈ c, we have (σc ◦ τc )(z)(g) = σc (τc (z))(g) = [φg∗ σc (τc (z))](1G ) (by (8.24)) = σH (φg∗ (τc (z))(1G ) .

(again by (8.24)) = σH (τH (φg∗ (z)))(1G ) (by (8.26)) = (φg∗ (z))(1G ) = z(g).

This shows that (σc ◦ τc )(z) = z for all z ∈ V c . We deduce that σc ◦ τc = IdV c for u n 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 → AG 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. 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 u n 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 Σ 2 3 .v = k0 + k1 t + k2 t + k3 t + · · · = ki t i . i∈N

Consider the map τ : V Z → V Z defined by τ (x)(n) := x(n) − tx(n + 1)

.

8.13 Pre-injectivity and Surjectivity of the Discrete Laplacian

409

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 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 Corollaries 8.12.2 and 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]). O

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 : K G → K G (with memory set .S ∪ {1G }) defined by ΔS (f )(g) := |S|f (g) −

Σ

.

f (gs)

s∈S

for all .f ∈ K G and .g ∈ G, where .|S| denotes the cardinality of S (cf. Examples 1.4.3(b) and 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 nonempty 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 : K H → K H denote the restriction cellular automaton of .ΔS to H . Observe that .(ΔS )H is the discrete Laplacian on .K H 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 nonsurjectivity since .(ΔS )H is K-linear and .K H is finite-dimensional. By applying

410

8 Linear Cellular Automata

Proposition 5.2.2 (resp. Proposition 1.7.4), we deduce that .ΔS is not pre-injective (resp. not surjective). u n 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. Theorem 8.13.2 (Garden of Eden Theorem for Discrete Real Laplacians) 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 n : n ≥ 1} ⊂ P is also finite. This implies that every element of S has finite .{s order. Therefore, for each .s ∈ S, we can find an integer .n ≥ 2 such that .s n = 1G . It follows that .s −1 = s n−1 ∈ S. Consequently, we have .P −1 = P and .1G ∈ P . This implies .H = P , so that H is finite. This shows (b) .⇒ (a). u n 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 Σ .0 = ΔS (f )(g0 ) = |S|f (g0 ) − f (g0 s), s∈S

8.13 Pre-injectivity and Surjectivity of the Discrete Laplacian

411

we get |f (g0 )| ≤

.

1 Σ |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 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 .l2 (G) ⊂ RG consisting of all square-summable real functions on G and (2) the continuous linear map .ΔS : l2 (G) → l2 (G) obtained by restriction of .ΔS to 2 .l (G) (cf. Sect. 6.12). By the Kesten-Day amenability criterion (Theorem 6.12.9), the nonamenability (2) (2) (2) of G implies that 0 does not belong to the spectrum .σ (ΔS ) of .ΔS . Thus, .ΔS is bijective and hence .

(2)

R[G] ⊂ l2 (G) = ΔS (l2 (G)) = ΔS (l2 (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). u n 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.

412

8 Linear Cellular Automata

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. 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 d G = (RG )d . Clearly, .τ is a linear cellular automaton admitting .S ∪ {1 } as a .(R ) G memory set. On the other hand, .τ is surjective but not injective since any product of surjective (resp. non-injective) maps is a surjective (resp. non-injective) map. This u n 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 G → V G is surjective. .τ : V 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 finitedimensional 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 G is injective by Proposition 1.7.4(i). Also, .τ G is linear by Proposition 8.3.1. Since .τ 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. u n 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 L-surjunctive. Let V be a finite dimensional vector space over a field K and let .τ : V G → V G be an

8.14 Linear Surjunctivity

413

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 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). u n Note that the preceding proposition may be stated by saying that a group is Lsurjunctive 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 Lsurjunctive. 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 C V G . Then, since Y is closed in .V G , there exists a finite subset .Ω ⊂ G such that .Y |Ω C V Ω . It is not restrictive, up to taking a larger .r0 , again if necessary, to suppose that B (r ) .Ω ⊂ BS (r0 ). Thus, .Y |BS (r0 ) C V S 0 . Let .ε > 0 be such that ε
1 −

1 d|B(2r0 )|+1

(1 − ε)−1 < 1 +

.

which yields

1 . d|B(2r0 )|

(8.30)

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 ∈ Q(ir 0 ) and .q ∈ Q((i + 1)r ), V 0

.

) ( 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 Q(ir0 ) → V Q((i+2)r0 ) is the identity on .V Q((i+2)r0 ) . More precisely, .νi+1 ◦ μi : V 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 .

dim(Z) ≥ d|Q(3r0 )|.

(8.32)

Let .Q' ⊂ Q(3r0 ) be as in (7.59) and set .Q' := uq ' ∈Q' B(q ' , r0 ). Note that .Q' ⊆ Q(2r0 ) so that |Q(2r0 )| = |Q' | · |BS (r0 )| + |Q(2r0 ) \ Q' |.

.

(8.33)

8.15 Stable Finiteness of Group Algebras

415

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 ) C 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 ) ) ( ( dim(Z) ≤ dim Z|Q' + dim Z|Q(2r0 )\Q' ≤ |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(3r0 )| ≤ |Q(2r0 )| −

.

|Q' | . 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. u n

8.15 Stable Finiteness of Group Algebras Let R be a ring and denote by .1R its unity element.

416

8 Linear Cellular Automata

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. leftinvertible) 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 leftinvertible, 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 . 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. u n 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, u n 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 .m := 0R (resp. .mn := mn+1 for 0 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.

8.15 Stable Finiteness of Group Algebras

417

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; (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. u n 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 finite-dimensional. 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 : K Z → K Z defined by .τ (x)(n) := x(n + 1) − x(n) for all .x ∈ K Z and .n ∈ Z. Observe that Z Z .τ is surjective. Indeed, given .y ∈ K , the configuration .x ∈ K defined by

x(n) :=

.

⎧ ⎪ ⎪0 ⎨

y(0) + y(1) + · · · + y(n − 1) ⎪ ⎪ ⎩y(n) + y(n + 1) + · · · + y(−1)

if n = 0, if n > 0, if n < 0,

418

8 Linear Cellular Automata

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; (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 Proposiu n tion 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. O

.

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. .O

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 zerodivisor) 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.

8.16

Zero-Divisors in Group Algebras

419

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

α(g) :=

.

⎧ ⎪ ⎪ ⎨1

−1 ⎪ ⎪ ⎩0

if g = 1G , if g = g0 ,

and

β(g) :=

{ 1

if g ∈ / {1G , g0 }.

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 .x 2 = x. An idempotent .x ∈ R is said to be proper if .x /= 0 and .x /= 1. 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) = x 2 − x = 0. If R has no zerodivisors, 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. u n Remarks 8.16.3 (a) A ring without proper idempotents may admit zero-divisors. 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 ) .Mat(2 (R)) is directly finite by Propo( ring 10 sition 8.15.2. However, the matrices . 0 0 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.

420

8 Linear Cellular Automata

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. 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 left-invariant total orderings on .G1 and .G2 respectively. Then the lexicographic ordering .≤ on .G1 × G2 defined by

(g1 , g2 ) ≤ (h1 , h2 ) ⇐⇒

.

⎧ ⎪ ⎪ ⎨g1 ≤1 g2 or ⎪ ⎪ ⎩g = g and h ≤ h 1 2 1 2 2

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 torsionfree 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.

8.16

Zero-Divisors in Group Algebras

421

(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

g ≤ h ⇐⇒

.

⎧ ⎪ ⎪ρ(g) ≤2 ρ(h) ⎨ or ⎪ ⎪ ⎩ρ(g) = ρ(h) and 1 ≤ g −1 h G 1

is a left-invariant total ordering on G. (g) Let R be any ring. Consider the Heisenberg group ⎧ ⎨

⎛

1y .HR := M(x, y, z) = ⎝0 1 ⎩ 00

⎞ z x⎠ 1

:

⎫ ⎬

x, y, z ∈ R . ⎭

(cf. Example 4.6.5). The kernel of the group homomorphism of .HR onto .R 2 given by .M(x, y, z) |→ (x, y) is the normal subgroup ⎧⎛ ⎨ 10 ⎝0 1 .N := ⎩ 00

⎞ z 0⎠ 1

:

⎫ ⎬

z∈R . ⎭

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

422

8 Linear Cellular Automata

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). u n Corollary 8.16.7 The group .Homeo+ (R) of orientation-preserving homeomorphisms of .R is orderable. .O 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. u n 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 (αβ)(g) =

Σ

.

α(h1 )β(h2 ) = α(a)β(b).

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. u n 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. .O 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) :=

Σ

.

α(h)x(gh)

h∈G

for all .α ∈ K[G], .x ∈ K G , 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) : K G → K G is pre-injective.

8.17 A Characterization of Amenability in Terms of Linear Cellular Automata

423

Proof By Proposition 8.2.5, the linear cellular automaton .Ψ (α) is not pre-injective if and only if there exists a nonzero element .β ∈ K[G] such that .Ψ (α)(β) = 0. As Ψ (α)(β)(g) =

Σ

.

h∈G

α(h)β(gh) =

Σ

α(h)β ∗ (h−1 g −1 ) = (αβ ∗ )(g −1 ),

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. u n 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 : K G → K G is preinjective. O

.

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 G → K G is surjective. .τ : K .O 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 : K G → K G is surjective for any field K.

8.17 A Characterization of Amenability in Terms of Linear Cellular Automata In Corollary 8.10.2, it was shown that if G is a group containing a free subgroup of rank 2 then there exists a linear cellular automaton over G with finite-dimensional alphabet space which is pre-injective but not surjective. In fact, the following theorem shows that the existence of such a linear cellular automaton extends to any non-amenable group. Theorem 8.17.1 Let G be a non-amenable group and let K be a field. Then there exist a finite-dimensional vector space V over K and a linear cellular automaton G → V G which is pre-injective but not surjective. .τ : V

424

8 Linear Cellular Automata

For the proof of Theorem 8.17.1, we first need to establish some auxiliary results. The first ones are of a purely combinatorial nature. Lemma 8.17.2 Let n and p be integers such that .1 ≤ p ≤ n − 1. Then one has ( ) Σ ) p ( n n−1−j = . p p−j

(8.35)

.

j =0

Proof By successive applications of Pascal’s formula, we get ( ) ( ) ( ) n n−1 n−1 . = + p p p−1 ) ( ) ( ) ( n−2 n−2 n−1 + + = p−1 p−2 p .. . =

) ) ( ) ( ) ( ( n−1−p n−1 n−2 n−3 + ··· + . + + 0 p p−1 p−2 u n

This gives us (8.35).

Lemma 8.17.3 Let S be a finite set with cardinality .|S| ≥ 2. Then there exist a finite set D and a family .(Es )s∈S of subsets of D such that, for all .T ⊂ S and .s ∈ T , one has |Es \

U

.

t∈T \{s}

Et | >

|D| . (1 + log |S|)|T |

(8.36)

Proof Let .Sym(S) denote the symmetric group of S and consider the subset .D ⊂ Sym(S) × Sym(S) consisting of all pairs .(σ, c), where .σ ∈ Sym(S) and c is one of the cycles appearing in the factorization of .σ as a product of cycles with disjoint supports (cf. Proposition C.2.3). For .s ∈ S, let .Es ⊂ D denote the subset consisting of all .(σ, c) ∈ D such that s belongs to the support of c. Let us first estimate the cardinality of D. Given a subset .Z ⊂ S with cardinality .|Z| = i, .1 ≤ i ≤ |S|, there are .i!/i = (i − 1)! cycles whose support is Z. Such a cycle can be extended to a permutation of S in .(|S| − i)! ways. We deduce that |D| =

.

) |S| ( |S| Σ Σ |S| 1 (i − 1)!(|S| − i)! = |S|! . i i i=1

i=1

8.17 A Characterization of Amenability in Terms of Linear Cellular Automata

425

As |S| Σ 1 .

i=1

1 1 = 1 + + ··· +

.

|D| , (1 + log |S|)|T |

which is precisely (8.36).

U

u n

Note that the sets constructed in the proof of U Lemma 8.17.3 satisfy . s∈S Es = D. However, one can add the further condition . s∈S Es C D, thus obtaining the following sharpening of Lemma 8.17.3. Lemma 8.17.4 Let S be a finite set with cardinality .|S| U ≥ 2. Then there exist a finite set D and a family .(Es )s∈S of subsets of D satisfying . s∈S Es C D such that, for all .T ⊂ S and .s ∈ T , one has |Es \

U

.

Et | >

t∈T \{s}

|D| . (1 + log |S|)|T |

(8.38)

Proof By Lemma 8.17.3, there exist a finite set .D '' and a family .(Es'' )s∈S of subsets of .D '' satisfying |Es'' \

U

.

Et'' | >

t∈T \{s}

|D '' | (1 + log |S|)|T |

(8.39)

for all .s ∈ T ⊂ S. Let k be a positive integer and consider the sets .D ' := D '' × {1, 2, . . . , k} and .Es' := Es'' × {1, 2, . . . , k} ⊂ D ' , .s ∈ S. As .|D ' | = k|D '' | and ' '' .|Es | = k|Es | for all .s ∈ S, we deduce from (8.39) that, for k large enough, |Es' \

U

.

t∈T \{s}

Et' | >

|D ' | + 1 (1 + log |S|)|T |

for all .s ∈ T ⊂ S. Then the sets .D := D ' u {d0 }, where .d0 ∈ / D ' is a new element, ' and .Es := Es ⊂ D, .s ∈ S, have the required properties. u n We shall also need an auxiliary result which gives a sufficient condition for the non-emptyness of the complement of a finite union of algebraic hypersurfaces in n when the ground field K is sufficiently large. We first treat the case of the .K complement of a single hypersurface. Lemma 8.17.5 Let n be a positive integer and let K be a field. Let .f ∈ K[X1 , . . . , Xn ] be a non-zero polynomial. Suppose that K is either infinite or finite with cardinality .|K| ≥ 1 + deg(f ). Then there exist .a1 , . . . , an ∈ K such that .f (a1 , . . . , an ) /= 0. Proof We proceed by induction on the number n of indeterminates. For .n = 1, this follows from the fact that a non-zero polynomial .f ∈ K[X1 ] has at most .deg(f )

8.17 A Characterization of Amenability in Terms of Linear Cellular Automata

427

zeroes in K. Suppose now .n ≥ 2. Since .f /= 0, there exists .i ∈ {1, . . . , n} such that one of the non-zero terms of f contains a positive power of .Xi . Without loss of generality, we can assume .i = n. We then have f = gm Xnm + gm−1 Xnm−1 + · · · + g0 ,

.

for some .m ≥ 1 and .gm , gm−1 , . . . , g0 ∈ K[X1 , . . . , Xn−1 ] with .gm /= 0. Observe that .deg(gm ) ≤ deg(f ). By induction, there exist .a1 , . . . , an−1 ∈ K such that .gm (a1 , . . . , an−1 ) /= 0. As the polynomial h := gm (a1 , . . . , an−1 )Xnm + gm−1 (a1 , . . . , an−1 )Xnm−1

.

+ · · · + g0 (a1 , . . . , an−1 ) ∈ K[Xn ] is non-zero with degree at most .deg(f ), there exists .an ∈ K such that .h(an ) /= 0. We then have .f (a1 , . . . , an−1 , an ) = h(an ) /= 0. u n Lemma 8.17.6 Let .n, r be positive integers and let K be a field. Let .f1 , . . . , fr ∈ K[X1 , . . . , Xn ] be non-zero polynomials. Suppose that K is either infinite or finite Σ with cardinality .|K| ≥ 1 + ri=1 deg(fi ). Then there exist .a1 , . . . , an ∈ K such that .fi (a1 , . . . , an ) /= 0 for all .1 ≤ i ≤ r. Proof It suffices to apply Lemma 8.17.5 with .f := f1 · · · fr .

u n

Proof of Theorem 8.17.1 Since G is not amenable, it follows from Theorem 4.9.2 that there exists a finite subset .S0 ⊂ G such that .|F S0 | ≥ 2|F | for all finite .F ⊂ G. Note that .|S0 | ≥ 2 and that |F S0k | = |F S0k−1 S0 | ≥ 2|F S0k−1 | = 2|F S0k−2 S0 | ≥ 22 |F S0k−2 | ≥ · · · ≥ 2k |F |

.

for all .k ∈ N and .F ⊂ G finite. Choose .k ∈ N large enough so that .2k ≥ 1 + k log |S0 | and set .S := S0k . Since .|S| = |S0k | ≤ |S0 |k , we then have |F S| = |F S0k | ≥ 2k |F | ≥ (1 + k log |S0 |)|F | ≥ (1 + log |S|)|F |

.

(8.40)

for all finite .F ⊂ G. By Lemma 8.17.4, there exist a finite set D and a family .(Es )s∈S of subsets of D such that the set U .Z := Es s∈S

satisfies ZCD

.

(8.41)

428

8 Linear Cellular Automata

and such that, for all .s ∈ T ⊂ S, the set U

Es,T := Es \

.

Et

t∈T \{s}

satisfies |Es,T | ≥

.

|D| . (1 + log |S|)|T |

(8.42)

Let K be a field. Then .V := K D is a finite-dimensional vector space over K. For every subset .E ⊂ D, we shall regard .K E as a vector subspace of .V by using the injective linear map .K E → K D = V given by the identifications .K E = K E ×{0} ⊂ K E × K D\E = K D . Consider a family .(αs )s∈S of linear maps .αs : V → V and the linear cellular automaton .τ : V G → V G defined by τ (x)(g) :=

Σ

.

αs (x(gs))

s∈S

for all .x ∈ V G and .g ∈ G. Note that each linear map .αs , .s ∈ S, is described by a matrix .Ms : D × D → K indexed by .D × D and with entries in K. Let us show that, for K large enough, the family of linear maps .(αs )s∈S can be chosen in such a way that the linear cellular automaton .τ has the required properties, i.e., is pre-injective but not surjective. Let us first impose that (C1) .αs (V ) ⊂ K Es for each .s ∈ S. This amounts to requiring that, for each .s ∈ S, all entries of the matrices .Ms whose column index is not in .Es are equal to 0. Condition (C1) implies that the associated linear cellular automaton .τ is not surjective. Indeed, setting .W := K Z , we have .W C V by (8.41). As τ (x)(g) =

Σ

.

αs (x(gs)) ∈ W

s∈S

for all .x ∈ V G and .g ∈ G, we deduce that .τ (V G ) ⊂ W G C V G , so that .τ is not surjective. For .s ∈ S and .T ⊂ S with .s ∈ T , denote by .πs,T : V → K Es,T the projection onto the vector subspace .K Es,T ⊂ V and consider the linear map .αs,T : V → K Es,T defined by .αs,T := πs,T ◦ αs .

8.17 A Characterization of Amenability in Terms of Linear Cellular Automata

429

We want in addition that the family .(αs )s∈S satisfies the following condition: (C2) for every family .T = (Ts )s∈S of subsets of D such that .s ∈ Ts for all .s ∈ S and such that Σ . |Es,Ts | ≥ |D|, s∈S

one has n .

ker(αs,Ts ) = {0}.

(8.43)

s∈S

Note that Condition (8.43) is equivalent to the injectivity of the linear map ΨT : V → V S given by .v |→ (αs,Ts (v))s∈S . To see that it is possible, for K sufficiently large, to find a family .(αs )s∈S also satisfying (C2), first observe that the matrix .Ms,Ts representing .αs,Ts is the matrix indexed by .Es,Ts × D obtained from .Ms by suppressing all the rows of .Ms whose index is not in .Es,Ts . It follows that the injectivity of .ΨT is equivalent to the condition that the matrix .MT obtained by concatenating “vertically" all the matrices .Ms,Ts , .s ∈ S, has rank .|D|. On the other hand, .MT has rank .|D| if and only if at least one of the minors of order .|D| of .MT is non-zero. As each of these minors is a non-zero polynomial in the entries of .MT , we deduce from Lemma 8.17.6 that there exists a positive integer .N = N(|S|) such that if the field K is either infinite or finite with cardinality .|K| ≥ N , the family .(αs )s∈S can be chosen in such a way that it also satisfies (C2). Let us show that .τ is then pre-injective. Suppose that .x ∈ V [G] ⊂ V G is a configuration with non-empty finite support .F ⊂ G and let .y := τ (x). Define .ρ : F S → {1, 1/2, 1/3, . . . , 1/|S|} by .

ρ(g) :=

.

1 1 = . |{s ∈ S : g ∈ F s}| |{f ∈ F : g ∈ f S}|

for all .g ∈ F S. We then have .

Σ Σ Σ Σ ( ρ(f s)) = ( f ∈F s∈S

ρ(f s)) =

g∈F S s∈S:g∈F s

Σ

1 = |F S|.

g∈F S

Therefore, there exists .f0 ∈ F such that Σ .

s∈S

ρ(f0 s) ≥

|F S| . |F |

(8.44)

430

8 Linear Cellular Automata

By combining (8.44) and (8.40), we get Σ .

ρ(f0 s) ≥ 1 + log |S|.

(8.45)

s∈S

For each .s ∈ S, let us set .Ts := {t ∈ S : f0 s ∈ F t}. Observe that .s ∈ Ts and .|Ts | = 1/ρ(f0 s). From (8.42), we deduce that, |Es,Ts | ≥

.

|D|ρ(f0 s) |D| = (1 + log |S|)|Ts | (1 + log |S|)

for all .s ∈ S. By using (8.45), this gives us Σ .

|Es,Ts | ≥ |D|.

(8.46)

s∈S

Thus, by applying Condition (C2) to the family .(Ts )s∈S , we deduce that the linear map .V → V S , given by .v |→ (αs,Ts (v))s∈S , is injective. Since .x(f0 ) /= 0, this implies that there exists .s0 ∈ S such that αs0 ,Ts0 (x(f0 )) /= 0.

.

As y(f0 s0−1 ) = τ (x)(f0 s0−1 ) Σ = αs (x(f0 s0−1 s))

.

s∈S

= αs0 (x(f0 )) +

Σ

αs (x(f0 s0−1 s))

s∈S\{s0 }

= αs0 (x(f0 )) +

Σ

s∈Ts0 \{s0 }

αs (x(f0 s0−1 s)) +

Σ

αs (x(f0 s0−1 s)),

s∈S\Ts0

this implies that πs0 ,Ts0 (y(f0 s0−1 )) = πs0 ,Ts0 (αs0 (x(f0 ))) = αs0 ,Ts0 (x(f0 )) /= 0.

.

Therefore .y(f0 s0−1 ) /= 0 and hence .y /= 0. This shows that the restriction of .τ to .V [G] is injective, i.e., .τ is pre-injective. Thus, the linear cellular automaton .τ has the required properties. This proves Theorem 8.17.1 in the case when K is sufficiently large, i.e., either infinite or finite with cardinality .|K| ≥ N. Suppose now that K is any finite field. By the classification theorem of finite fields, there exist a prime p and an integer .n ≥ 1 such that .|K| = pn . Moreover, for any integer .m ≥ n, there exists a field extension L of K such that .|L| = pm . Choose

Notes

431

m large enough so that .pm ≥ N. By the first part of the proof, there exist a finitedimensional vector space V over L and a linear cellular automaton .τ : V G → V G which is pre-injective but not surjective. Note that V is a vector space over K with finite dimension .dimK (V ) = pm−n dimL (V ) < ∞ and that .τ is K-linear. Thus, G → V G is also a linear cellular automaton when regarding V as a vector .τ : V space over K. The fact that .τ is pre-injective but not surjective does not depend of the ground field. u n Corollary 8.17.7 Let G be a group. Then the following conditions are equivalent: (a) G is amenable; (b) for any field K and any finite-dimensional vector space V over K, every preinjective linear cellular automaton .τ : V G → V G is surjective; (c) there exists a field K such that, for any finite-dimensional vector space V over K, every pre-injective linear cellular automaton .τ : V G → V G is surjective; (d) for any finite field K and any finite-dimensional vector space V over K, every pre-injective linear cellular automaton .τ : V G → V G is surjective; (e) there exists a finite field K such that, for any finite-dimensional vector space V over K, every pre-injective linear cellular automaton .τ : V G → V G is surjective; (f) for any finite set A, every pre-injective cellular automaton .τ : AG → AG is surjective. Proof The implication (a) . =⇒ (b) follows from the linear version of the Garden of Eden theorem (Theorem 8.9.6), the implications (b) . =⇒ (c) and (d) . =⇒ (e) are obvious, and the implications (c) . =⇒ (a) and (e) . =⇒ (a) follow from Theorem 8.17.1. We deduce that the five conditions (a), (b), (c), (d), and (e) are equivalent. On the other hand, (f) . =⇒ (d) is obvious, while (a) . =⇒ (f) follows from the Garden of Eden theorem (Theorem 5.3.1). Consequently, all six conditions are equivalent. u n

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 d-dimensional 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

432

8 Linear Cellular Automata

elements in .EndK (V )[G] and as elements in .EndK[G] (V [G]) were given in Section 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 Kalgebras .K[Z] ∼ = K(t, t −1 ] and .LCA(Z; K d ) ∼ = Matd (K[t, t −1 ]). In [LiM, d Sect. 1.6] 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. In the same way as for entropy, it can be shown, as an application of the Ornstein-Weiss convergence theorem (see [OrW], [LiW], [Gro6], [Ele], [Kri], [CCK]) that one can replace the .lim sup in (8.18) by .lim provided one assumes that X is G-invariant. 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 infinitedimensional vector space V over a field K there exist a bijective cellular automaton G → V G which is not invertible (cf. Theorem 8.12.1 and Remark 8.12.3) and .τ : V 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]. See also [CCD]. 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

Notes

433

asked whether this property remains true for fields of characteristic .p > 0. This question remains open up to now and is commonly known as Kaplansky’s stable finiteness conjecture. 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 Szabó [ES1], using the notion of von Neumann dimension for continuous regular rings. The proof of Elek and Szabó’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 sofic group. This yields an extension of the Elek and Szabó 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 torsion-free 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 indicable. The group .SL2 (R) is orderable since it has a natural faithful action by orientationpreserving 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 [Mor, 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 biorderable. 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

434

8 Linear Cellular Automata

and right invariant and a field K, the vector subspace of .K G consisting of all maps .x : G → K whose support is a well-ordered 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’cevNeumann 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 zerodivisors 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 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-divisor conjecture. This implies in particular that all torsion-free virtually solvable groups satisfy the Kaplansky zero-divisor 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 : K G → K G is pre-injective. This reformulation of the Kaplansky zero-divisor conjecture in terms of linear cellular automata was given in Sect. 6 of [CeC1]. Relations between linear cellular automata and Kaplansky’s stable finiteness conjecture have been further explored by X.K. Phung in [Phu1], [Phu8], [Phu9], and [Phu11]. In particular, in [Phu8, Theorem B] it shown that Kaplansky’s stable finiteness conjecture is a consequence of Gottschalk’s surjunctivity conjecture. In other words, every surjunctive group is L-surjunctive and therefore satisfies Kaplansky’s stable finiteness conjecture. All the results presented in Sect. 8.17 are due to L. Bartholdi (see [Bar2] and [Bar3]). Theorem 8.17.1 answers positively Open Problem (OP-13) in the first edition of this book. As a generalization of the classical cellular automata (i.e., over finite alphabets) and linear cellular automata, a notion of algebraic cellular automaton was introduced in [CeC12] and further investigated in collaboration with Phung in [CCP1], [CCP2], and [CCP4], as well as in [Phu1], [Phu2], [Phu3], [Phu4], [Phu7], [Phu8], [Phu9], and [Phu5]). In this framework, the alphabet set is an algebraic variety or, more generally, the set of rational points of a scheme, and the local defining maps are required to be algebraic morphisms. In the above mentioned papers, several surjunctivity results and Garden of Eden type theorems have been established for certain classes of algebraic cellular automata.

Exercises

435

Exercises 8.1 Let G be a group and let R be a ring. Suppose that G and R have both cardinality 2. Is the ring R[G] isomorphic to the product ring Z/2Z × Z/2Z? 8.2 Let G be a group and let R be a nonzero ring. (a) Show that the ring R[G] is finite if and only if both G and R are finite. (b) Show that the ring R[G] is commutative if and only if both G and R are commutative. 8.3 (The Augmentation Morphism) Σ Let G be a group and let R be a ring. Define the map ε : R[G] → R by ε(α) := g∈G α(g) for all α ∈ R[G]. (a) (b) (c) (d)

Show that ε is surjective. Show that ε is a ring morphism. Show that ε is a left R-module morphism. Show that the kernel of ε is a free left R-module admitting as a basis the set of elements of the form g − 1G with g ∈ G and g /= 1G .

8.4 (The Center of a Group Ring) The center of a ring A is the commutative subring B of A consisting of all the elements b ∈ A that satisfy ab = ba for all a ∈ A. Let G be a group and let R be a ring. Let β ∈ R[G]. Show that β is in the center of R[G] if and only if β is constant on each conjugacy class of G and takes all of its values in the center of R. 8.5 (Group Rings of ICC-Groups) Let G be a group and let R be a non-zero ring. Show that G is an ICC-group if and only if the center of R[G] is equal to the center of R. 8.6 Let G be a group, let K be a field, and let d be a positive integer. Show that the K-algebras Matd (K[G]) and Matd (K)[G] are isomorphic. 8.7 Let G be a group and let S ⊂ G be a non-empty finite subset. Denote by the standard scalar product on R[G] ⊂ l2 (G) and let || · || denote the associated norm. (a) Show that for all x ∈ R[G] and λ ∈ R, one has =

.

1 ΣΣ |x(g) − x(gs)|2 + λ||x||2 . 2 g∈G s∈S

(b) Show that if λ > 0 then the linear cellular automaton τ : R G → R G defined by τ := ΔS + λ IdRG is both pre-injective and surjective. 8.8 Recall that a monoid M is said to be left-cancellative (resp. right-cancellative) if given r, s, t ∈ M such that tr = ts (resp. rt = st) then r = s. A monoid which is both left and right-cancellative is said to be cancellative.

436

8 Linear Cellular Automata

(a) Show that every finite left-cancellative (resp. finite right-cancellative, resp. finite cancellative) monoid is a group. (b) Let G be a group and let S ⊂ G be a subset. Show that the following conditions are equivalent: (i) the subgroup H ⊂ G generated by S is infinite; (ii) the submonoid M ⊂ G generated by S is infinite. G 8.9 Let G be a group and let S be a non-empty finite subset of G. Let ΔC S: C → G C denote the associated complex discrete Laplacian. Show that the following conditions are equivalent:

(i) (ii) (iii) (iv)

ΔC S is surjective; the subgroup of G generated by S is infinite; the submonoid of G generated by S is infinite; ΔC S is pre-injective.

8.10 || Let (Ri )i∈I be a family of directly finite rings. Show that the product ring P := i∈I Ri is directly finite. 8.11 Show that every subring of a directly finite (resp. stably finite) ring is itself directly finite (resp. stably finite). 8.12 (Hopfian Modules) 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. One says that the module M is co-Hopfian if every injective endomorphism of M is surjective. Show that if the module M is either Hopfian or co-Hopfian, then the ring EndR (M) is directly finite. 8.13 (Noetherian and Artinian Modules) Let R be a ring and let M be a left Rmodule. One says that the module M is Noetherian if its submodules satisfy the ascending chain condition, i.e., every increasing sequence N1 ⊂ N2 ⊂ . . .

.

of submodules of M stabilizes (i.e. there is an integer i0 ≥ 1 such that Ni = Ni0 for all i ≥ i0 ). One says that the module M is Artinian if its submodules satisfy the descending chain condition, i.e., every decreasing sequence N1 ⊃ N2 ⊃ . . .

.

of submodules of M stabilizes (i.e. there is an integer i0 ≥ 1 such that Ni = Ni0 for all i ≥ i0 ). Show that if the module M is Noetherian (resp. Artinian), then it is Hopfian (resp. co-Hopfian). 8.14 (Projective Modules) 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 of left R-modules - → M, f : P → M and any surjective homomorphism of left R-modules g : M there exists a homomorphism of left R-modules h : P → M such that f = g ◦ h.

Exercises

437

(a) Show that every free left R-module is projective. (b) Suppose that the module P is projective. Show that P is Hopfian if and only if the ring EndR (P ) is directly finite. 8.15 Let R be a ring and let d ≥ 1 be an integer. Equip R d with its natural structure of left R-module. Show that R d is Hopfian if and only if the ring Matd (R) is directly finite. 8.16 (Injective Modules) Let R be a ring and let Q be a left R-module. One says that the module Q is injective if for every homomorphism f : M → Q and any injective homomorphism g : M → N of left R-modules, there exists a homomorphism h : N → Q such that f = h ◦ g. Suppose that the module Q is injective. Show that Q is co-Hopfian if and only if the ring EndR (Q) is directly finite. 8.17 Let M be a left module over a ring R and let N be a submodule of M. Show that M is Noetherian (resp. Artinian) if and only if the modules N and M/N are both Noetherian (resp. Artinian). 8.18 (Noetherian Rings) 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 that R d is Noetherian as a left R-module for each integer d ≥ 1. (b) Show that every left Noetherian ring is stably finite. 8.19 Show that every division ring is stably finite. 8.20 (UPR-Rings) One says that a ring R has the unique rank property (URP for short), or that R is a UPR-ring, if it satisfies the following condition: if m and n are positive integers such that R m and R n are isomorphic as left R-modules, then one has m = n. (a) Show that every stably finite nonzero ring is a URP-ring. (b) Show that all finite nonzero rings, all commutative nonzero rings, and all division rings are URP-rings. (c) Show that a ring R is a URP-ring if and only if it satisfies the following condition: if m and n are positive integers such that there exist matrices A ∈ Matm,n (R) and B ∈ Matn,m (R) such that AB = Im and BA = In , then one has m = n. (d) Show that a ring R is a URP-ring if and only if it satisfies the following condition: if m and n are positive integers such that R m and R n are isomorphic as right R-modules, then one has m = n. (e) Suppose that R and S are rings such that there exists a ring morphism ϕ : R → S. Show that if S is a URP-ring then R is also a URP-ring. 8.21 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.

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8 Linear Cellular Automata

8.22 (Monoid Rings) The notion of a group ring extends in an obvious way to monoids. More precisely, given a ring R and a monoid M, one defines the monoid ring R[M] as being the set of all maps with finite support α : M → R equipped with pointwise addition and the convolution product defined by (αβ)(m) :=

Σ

.

α(m1 )β(m2 )

m1 ,m2 ∈M m1 m2 =m

for all α, β ∈ R[M] and m ∈ M. One verifies that this defines a ring structure on R[M] as in the case of group rings. (a) Let R be a ring and let M be a monoid. Show that if R has URP then R[M] has URP. (b) Consider the symmetric monoid M := Map(N) consisting of all maps f : N → N with the composition of maps as the monoid operation. Show that if R is any URP-ring , then the ring R[M] has URP but is not directly finite. 8.23 (von Neumann Regular Rings) Let R be a ring. An element a ∈ R is called von Neumann regular if there exists an element x ∈ R such that a = axa. The ring R is said to be von Neumann regular if every element of R is regular. (a) (b) (c) (d) (e) (f) (g)

Show that every idempotent element in R is von Neumann regular. Show that 0R and 1R are von Neumann regular. Show that every invertible element in R is von Neumann regular. Show that every division ring is von Neumann regular. Prove that the ring Z is not von Neumann regular. Is every finite commutative ring von Neumann regular? Let R be a ring and let a ∈ R. Show that the following conditions are equivalent: (1) a is von Neumann regular; (2) there exists an idempotent element e ∈ R such that Ra = Re; (3) there exists an idempotent element e ∈ R such that aR = eR. (h) Let V be a vector space over a field K and let R := EndK (V ) denote the endomorphism ring of V . Show that the ring R is von Neumann regular. (i) Is every von Neumann regular ring directly finite? 8.24 (Unit-Regular Rings) Let R be a ring. An element a ∈ R is called unitregular if there exists an invertible element u ∈ R such that a = aua. The ring R is said to be unit-regular if every element of R is unit-regular. (a) Let R be a ring. Show that every unit-regular element of R is von Neumann regular (cf. Exercise 8.23). (b) Show that every unit-regular ring is von Neumann regular. (c) Let R be a ring. Show that the following elements are all unit-regular: 0R , any invertible element in R, any idempotent element in R. (d) Show that every division ring is unit-regular. (e) Let R be a ring and let a be a nonzero unit-regular element in R. Suppose that a is not invertible in R. Show that a is a zero-divisor.

Exercises

439

(f) Let R be a nonzero unit-regular ring without zero-divisors. Show that R is a division ring. (g) Let R be a ring and a ∈ R. Show that the following conditions are equivalent: (1) a is unit-regular; (2) there exists an invertible element u ∈ R and an idempotent element e ∈ R such that a = ue; (3) there exists an invertible element u ∈ R and an idempotent element e ∈ R such that a = eu. (h) Let V be a finite-dimensional vector space over a field K. Let R := EndK (V ) denote the ring of endomorphisms of V . Show that the ring R is unit-regular. (i) Let K be a field and let d ≥ 1 be an integer. Show that the ring Matd (K) of d × d matrices with entries in K is unit-regular. (j) Show that every unit-regular ring is directly finite. (k) Is every directly finite ring unit-regular? (l) Is every von Neumann regular ring unit-regular? 8.25 Let K be a field. Show that every finite-dimensional K-algebra is stably finite. 8.26 Let G be an amenable group and let F be a right Følner net for G. Let V be a finite-dimensional vector space over some field K. Suppose that X and Y are vector subspaces of V G . Show that one has .

mdimF (X + Y ) ≤ mdimF (X) + mdimF (Y ).

(8.47)

8.27 Prove that every finite group is L-surjunctive by using the characterization of L-surjunctivity in terms of stable finiteness of group rings. 8.28 Prove that every abelian group is L-surjunctive by using the characterization of L-surjunctivity in terms of stable finiteness of group rings. 8.29 Let G be a group and let V be a vector space over a field K. Let τ : V G → V G be an linear cellular automaton. (a) Let H be a subgroup of G. Show that the set of configurations that are fixed by H , i.e., the set .

Fix(H ) := {x ∈ V G : gx = x for all g ∈ H }

is a vector subspace of V G . (b) Suppose that H is a finite index subgroup of G and that V is finite-dimensional. Show that the vector space Fix(H ) is finite-dimensional. (c) Deduce from (b) that all residually finite groups are L-surjunctive. 8.30 Show that every virtually L-surjunctive group is L-surjunctive. 8.31 Let G be a group and let K be a field. Let H be a subgroup of G. Let T ⊂ G be a complete set of representatives for the left cosets of H in G. (a) Show that K[G] is a free right K[H ]-module with base T . (b) Suppose that H has finite index n := [G : H ] in G. Show that there is an injective ring homomorphism K[G] → Matn (K[H ]).

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(c) Use (b) to recover the fact that every virtually L-surjunctive group is Lsurjunctive (cf. Exercise 8.30). 8.32 Let R be a ring and let x ∈ R. Show that x is an idempotent if and only if 1R − x is an idempotent. 8.33 (Boolean Rings) A ring is caled Boolean if all of its elements are idempotents. Let R be a Boolean ring. Let x ∈ R. Show that 2x = 0R and x = −x. Show that R is commutative and unit-regular. Show that every subring of R is Boolean. Show that every quotient ring of R is Boolean. Show that the following conditions are all equivalent: (1) R has no zero-divisors; (2) R has no non-trivial idempotents; (3) R has at most two elements. (f) Show that if R is finite then |R| = 2n for some integer n ≥ 0. (g) Let I be a prime ideal of R. Show that I is a maximal ideal of R and that the quotient ring R/I is a field with two elements. (a) (b) (c) (d) (e)

8.34 Let R be a nonzero ring and let G be a group. Show that the ring R[G] is Boolean if and only if the ring R is Boolean and the group G is trivial. 8.35 Let R be a ring and let G be a group. ΣLet H be a finite subgroup of G. Consider the element α ∈ R[G] defined by α := h∈H h. (a) Show that α 2 = |H |2 α. (b) Suppose that H is normal in G. Show that α is in the center of R[G]. (c) Suppose that R is a nonzero ring and α is in the center of R[G]. Show that H is normal in G. (d) Suppose that R is a nonzero ring, |H | is invertible in R[G], and H /= {1G }. Show that the element μ ∈ R[G] defined by μ := |H |−1 α is a non-trivial idempotent of R[G]. 8.36 (Reduced Rings) A ring R is called reduced if a 2 = 0 implies a = 0 for all a ∈ R. An element a ∈ R is called nilpotent if there exists an integer n ≥ 1 such that a n = 0R . (a) (b) (c) (d)

Show that if a ring R has no zero-divisors then R is reduced. Give an example of a reduced ring that has zero-divisors. Show that a ring is reduced if and only if it has no nonzero nilpotent elements. Let G be a finite group and let K be a field such that the characteristic of K divides |G|. Show that the ring K[G] is not reduced.

8.37 (Strongly Regular Rings) A ring R is called strongly regular if for every a ∈ R, there exists x ∈ R such that a = a 2 x. (a) Show that a commutative ring is strongly regular if and only if it is von Neumann regular. (b) Show that a ring is strongly regular if and only if it is von Neumann regular and reduced.

Exercises

441

8.38 (Reversible Rings) A ring R is called reversible if ab = 0R implies ba = 0R for all a, b ∈ R. (a) Show that every reversible ring is directly finite. (b) Let V be a vector space over a field K such that dimK (V ) ≥ 2. Show that the ring EndK (V ) is not reversible. (c) Let K be a field and let n ≥ 2 be an integer. Show that the ring of n × n matrices with entries in K is directly finite but not reversible. (d) Let K be a field. Show that the group ring K[Sym3 ] is directly finite but not reversible. (e) Show that every reduced ring is reversible. (f) Show that if a ring R has no zero-divisors then R is reversible. (g) Give examples of rings that are reversible but not reduced. 8.39 Let K be a field and let V be a vector space over K. (a) Show that the ring EndK (V ) has no zero-divisors (resp. is reduced, resp. is reversible) if and only if dimK (V ) ≤ 1. (b) Show that the ring EndK (V ) is directly finite if and only if dimK (V ) < ∞. 8.40 Let G be a group. Show that G is a unique-product group if and only if it satisfies the following condition: for every non-empty finite subset A of G, there exists g ∈ G such that g can be uniquely written in the form g = a1 a2 with a1 , a2 ∈ A. 8.41 Let G be a group. (a) Let A be a finite subgroup of G. Show that for every g ∈ A, there are exactly |A| distinct ordered pairs (a, b) with a, b ∈ A such that g = ab. (b) Use (a) to show that every unique-product group is torsion-free. 8.42 Show that every subgroup of a unique-product group is a unique-product group. 8.43 Let G be a group. Suppose that G is locally unique-product, i.e., every finitely generated subgroup of G is a unique-product group. Show that G is a uniqueproduct group. 8.44 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. 8.45 Let G be a group. Suppose that G is residually unique-product, i.e., for every g ∈ G \ {1G }, there exist a unique-product group H and a group homomorphism φ : G → H such that φ(g) /= 1H . (a) Let Ω be a finite subset of G. Show that there exist a unique-product group H and a group homomorphism φ : G → H such that the restriction of φ to Ω is injective. (b) Deduce from (a) that G is a unique-product group.

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8 Linear Cellular Automata

8.46 Show that every group which is locally embeddable into the class of uniqueproduct groups is a unique-product group. 8.47 Let G be a group. Suppose that every non-trivial finitely generated subgroup of G admits a non-trivial unique-product quotient group. Show that G is a uniqueproduct group. 8.48 (Indicable Groups) A group is said to be indicable if it is either trivial or admits an infinite cyclic quotient. A group is said to be locally indicable if all of its finitely generated subgroups are indicable. (a) (b) (c) (d) (e)

Show that every free group is indicable and locally indicable. Show that every torsion-free abelian group is locally indicable. Give an example of a torsion-free abelian group that is not indicable. Give an example of an indicable abelian group that is not locally indicable. Show that every locally indicable group is a unique-product group.

8.49 Let G be an orderable group and let ≤ be a left-invariant total ordering on G. The subset P ⊂ G defined by P := {g ∈ G : 1G < g} is called the positive cone of (G, ≤). (a) (b) (c) (d)

Let g, h ∈ G. Show that one has g < h if and only if g −1 h ∈ P . Show that P is a subsemigroup of G, i.e., P P ⊂ P . Show that P −1 = {g ∈ G : g < 1G }. Show that G is the disjoint union of the sets P , P −1 , and {1G }.

8.50 Let G be a group. Show that G is orderable if and only if there exists a subsemigroup P of G such that G is the disjoint union of the sets P , P −1 , and {1G }. 8.51 Let G be a bi-orderable group and let ≤ be a total ordering on G that is both left and right invariant. Consider the subset P ⊂ G defined by P := {g ∈ G : 1G < g}. Show that hP h−1 = P for all h ∈ G. 8.52 Let G be a group. Show that G is bi-orderable if and only if there exists a subsemigroup P of G such that hP h−1 = P for all h ∈ G and G is the disjoint union of the sets P , P −1 , and {1G }. 8.53 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 .

.

(8.48)

Show that g = 1G . 8.54 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.

Exercises

443

8.55 Let G be a group and let Z(G) denote the center of G. Suppose that G contains a subgroup N ⊂ Z(G) such that both N and G/N are bi-orderable. Show that G is also bi-orderable. 8.56 Show that every torsion-free nilpotent group is bi-orderable. 8.57 || Let (Gi )i∈I be a family of bi-orderable groups. Show that the product group G := i∈I Gi is bi-orderable. 8.58 Show that the limit of a projective system of orderable (resp. bi-orderable) groups is orderable (resp. bi-orderable). 8.59 (The Klein Bottle Group) Consider the group G given by the presentation G := . Thus, G is the quotient group G = F /N, where F is the free group based on two generators x and y, and N is the normal closure of yxy −1 x in F . Let ρ : F → G denote the quotient homomorphism. (a) Consider the elements a, b ∈ G defined by a := ρ(x) and b := ρ(y). Show that every element g ∈ G can be written in the form g = a n bm for some n, m ∈ Z. (b) Show that the group G is isomorphic to the semidirect product Z xα Z, where α : Z → Aut(Z) is the group homomorphism given by α(m)(n) := (−1)m n for all m, n ∈ Z. More precisely, show that the map ψ : Z xα Z → G, defined by ψ(n, m) := a n bm for all (n, m) ∈ Z xα Z, is a group isomorphism. (c) Show that G is torsion-free. (d) Show that G is metabelian and hence solvable. (e) Show that G is orderable. (f) Show that G is not bi-orderable. (g) Show that G is not nilpotent. (h) Show that G is not commutative-transitive. 8.60 (The Passman-Promislow-Hantzsche-Wendt Group) G given by the presentation

Consider the group

G := .

.

Thus, G is the quotient group G = F /N, where F is the free group based on two generators x, y and N is the normal closure of the set {xy 2 x −1 y 2 , yx 2 y −1 x 2 } in F . Let ρ : F → G denote the quotient group homomorphism. Let a, b, c ∈ G be the elements defined by a := ρ(x), b := ρ(y), and c := ρ(yx) = ba. (a) Show that the subgroup H ⊂ G generated by the set {a 2 , b2 , c2 } is abelian. (b) Show that every g ∈ G can be written in the form g = a i bj ck a ε1 bε2 ,

.

(8.49)

where i, j, k ∈ 2Z and ε1 , ε2 ∈ {0, 1}. (c) Show that the subgroup H is normal in G and that the group G/H is isomorphic to Z/2Z × Z/2Z.

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8 Linear Cellular Automata

(d) For n ≥ 1, denote by Isom(Rn ) (resp. Isom+ (Rn )) the group of affine isometries (resp. orientation-preserving affine isometries) of the affine Euclidean space Rn . Let σ ∈ Isom(R) (resp. η ∈ Isom(R)) denote the symmetry of the real line around 0 (resp. 1/2), so that τ := η ◦ σ ∈ Isom+ (R) is the translation by 1. Consider the elements f1 , f2 ∈ Isom+ (R3 ) respectively defined by f1 := τ × σ ×σ and f2 := σ ×τ ×η. Show that there exists a unique group homomorphism ϕ : G → Isom+ (R3 ) such that ϕ(a) = f1 and ϕ(b) = f2 . (e) Show that ϕ is injective. (f) Show that the map ψ : Z3 → H , defined by ψ(p, q, r) := a 2p b2q c2r for all (p, q, r) ∈ Z3 , is a group isomorphism. (g) Let g ∈ G. Show that there exist unique i, j, k ∈ 2Z and ε1 , ε2 ∈ {0, 1} satisfying (8.49). (h) Show that the group G is torsion-free. (i) Show that the group G is not orderable. 8.61 (Fibonacci Groups) Let n ≥ 2 be an integer and consider the group Gn given by the presentation Gn := .

.

Thus, Gn is the quotient group Gn = Fn /Nn , where Fn is the free group based on the generators x1 , x2 , . . . , xn and Nn is the normal closure in Fn of the set {x1 x2 x3−1 , x2 x3 x4−1 , . . . , xn−1 xn x1−1 , xn x1 x2−1 }.

.

Let ρ : Fn → Gn denote the quotient group homomorphism. Let a1 , a2 , . . . , an ∈ Gn be the elements defined by ai := ρ(xi ), i = 1, 2, . . . , n. (a) (b) (c) (d) (e) (f)

Show that Gn is generated by a1 and a2 . Show that G2 is a trivial group. Show that G3 is isomorphic to the quaternion group of order 8. Show that G4 is cyclic of order 5. Show that G5 is cyclic of order 11. Show that G6 is isomorphic to the Passman-Promislow-Hantzsche-Wendt group from Exercise 8.60. (g) Let An := Gn /[Gn , Gn ] denote the abelianization of Gn . Show that An is finite of cardinality |An | = fn − 1 − (−1)n , where (fn )n≥1 is the Fibonacci sequence (cf. Exercise 5.37) satisfying the initial conditions f1 := 1 and f2 := 3. Deduce that Gn is non-trivial for n ≥ 3. (h) Show that the group Gn is not trivial for n ≥ 3. (i) Show that the group Gn is not orderable for n ≥ 3. 8.62 (Diffuse Groups) Let G be a group and let A be a subset of G. An element a ∈ A is called an extremal element of A if, for every element g ∈ G \ {1G }, one / A. A group G is said to be diffuse if every non-empty has either ga ∈ / A or g −1 a ∈ finite subset of G admits an extremal element.

Exercises

445

(a) Show that every orderable group is diffuse. (b) Show that every diffuse group is a unique-product group. 8.63 Show that every locally orderable (resp. locally bi-orderable) group is orderable (resp. bi-orderable). 8.64 Show that every torsion-free locally nilpotent group is bi-orderable. 8.65 Let G be a group. Show that G is orderable (resp. bi-orderable) if and only if the following holds: for every finite symmetric subset F ⊂ G such that 1G ∈ F there exists a subset PF ⊂ F satisfying conditions (i)–(iii) (resp. (i)–(iv)), where (i) (ii) (iii) (iv)

if g, h ∈ PF and gh ∈ F , then gh ∈ PF ; PF ∩ PF−1 = ∅; PF ∪ PF−1 ∪ {1G } = F ; if h ∈ F , g ∈ PF and hgh−1 ∈ F , then hgh−1 ∈ PF .

8.66 Show that every group which is locally embeddable into the class of orderable (resp. bi-orderable) groups is orderable (resp. bi-orderable). 8.67 Show that every group which is residually orderable (resp. residually biorderable) is orderable (resp. bi-orderable). 8.68 Let G1 and G2 be two groups and let R be a ring. Let Φ : R[G1 × G2 ] → (R[G1 ])[G2 ] be the map defined by setting .

(Φ(α)(g2 )) (g1 ) := α(g1 , g2 )

for all α ∈ R[G1 × G2 ] and (g1 , g2 ) ∈ G1 × G2 . Show that Φ is a ring isomorphism from R[G1 × G2 ] onto (R[G1 ])[G2 ]. 8.69 (NZD-Groups) Recall that a group G is called an NZD-group if the ring R[G] has no zero-divisors whenever R is a ring with no zero-divisors. (a) Show that the direct product of two NZD-groups is an NZD-group. (b) Show that every group which is locally embeddable into the class of NZDgroups is an NZD-group. (c) Show that every group which is locally NZD is NZD. (d) Show that every group which is residually NZD is NZD. 8.70 Let G be an amenable group, let F = (Fj )j ∈J be a right Følner net for G, and let V be a finite-dimensional 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). 8.71 (The Ore Condition) Let R be a ring. Suppose that R is a domain, i.e., a nonzero ring without zero-divisors. Let S := R \ {0R } denote the set of nonzero elements in R. One says that R satisfies the right-Ore condition or that R is a rightOre domain if for all s, t ∈ S, there exist s ' , t ' ∈ S such that ss ' = tt ' .

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8 Linear Cellular Automata

(a) A commutative domain is called an integral domain. Show that every integral domain satisfies the right-Ore condition. (b) Show that if R is a right-Ore domain then there exists a division ring D containing a subring isomorphic to R. (c) Show that every right-Ore domain is stably finite. 8.72 Let G be a free group and let R be a (possibly non-commutative) domain. (a) Show that G is a unique-product group. (b) Show that the ring R[G] is a domain. (c) Suppose that the free group G is not abelian. Show that the ring R[G] satisfies neither the right nor the left Ore condition. 8.73 Let G be an amenable group and let K be a field. Suppose that the ring K[G] has no zero-divisors. (a) Show that K[G] satisfies the right-Ore condition. (b) Show that there exists a division ring D such that K[G] is isomorphic to a subring of D. 8.74 (The Ring of Non-Commutative Formal Power Series) Let X be a set and let X∗ denote the free monoid generated by X (cf. Sect. D.1). Let R be a ring. We equip the set ∗

R[[X]] := R X = {f : X∗ → R}

.

with pointwise addition and the convolution product respectively defined by (f + g)(w) := f (w) + g(w) and (f g)(w) :=

Σ

.

f (u)g(v)

(8.50)

u,v∈X∗ uv=w

for all f, g ∈ R[[X]] and w ∈ X∗ . Given a word w ∈ X∗ , we write |w| for the length of w and we denote by δw ∈ R[[X]] the Dirac function at w defined by setting δw (w ' ) = 1 (resp. δw (w ' ) = 0) if w ' = w (resp. w ' /= w) for all w ' ∈ X∗ . (a) Show that R[[X]] is a ring. (b) For each integer n ≥ 1, set In := {f ∈ R[[X]] : f (w) = 0 for all w ∈ X∗ such that |w| ≤ n − 1}.

.

Show that In is a two-sided ideal of R[[X]] and that one has In+1 ⊂ In for all n ≥ 1. (c) Show that Im In ⊂ Im+n for all integers m, n ≥ 1.

Exercises

447

(d) Let f ∈ I1 and set g := 1 + f ∈ R[[X]]. Show that the series h := Σ+∞ n n n=0 (−1) f is convergent for the prodiscrete topology on R[[X]] and that one has gh = hg = 1. (e) For each integer n ≥ 1 set Gn := 1 + In = {1 + f : f ∈ In } ⊂ R[[X]].

.

(f) (g) (h) (i)

Show that Gn is a normal subgroup of U (R[[X]]) and that Gn+1 ⊂ Gn for all n ≥ 1. Show that [Gm , Gn ] ⊂ Gm+n for all m, n ≥ 1. Show that Gn /Gn+1 is contained in the center of G1 /Gn+1 for all n ≥ 1. Show that C i (G1 ) ⊂ Gi+1 for all i ∈ N. Show that the group G1 is residually nilpotent.

8.75 (Residual Nilpotency of Free Groups) Let X be a set and let F = F (X) denote the free group based on X. Let R be a ring. We keep the notation introduced in Exercise 8.74. Moreover, for x ∈ X and m ∈ N, we denote by Im (x) the set consisting of all f ∈ R[[X]] such that f (w) = 0R for all w ∈ X∗ \ {x n : n ≥ m}. (a) Let x ∈ X. Show that I0 (x) is a subring of R[[X]] and that (Im (x))m∈N is a decreasing sequence of two-sided ideals of I0 (x). (b) Let x ∈ X and let k ∈ Z. Show that there exists hx,k ∈ I2 (x) such that (1 + δx )k = 1 + kδx + hx,k .

.

(c) Show that there is a unique group homomorphism ϕ : F → G1 such that ϕ(x) = 1 + δx for all x ∈ X. (d) Suppose that the ring R has characteristic 0 (i.e., k := k1R /= 0R for all k ∈ Z \ {0}). Show that ϕ is injective. (e) Show that every free group is residually nilpotent. 8.76 (Bi-orderability of Free Groups) (a) Let (A, ≤A ) and (Y, ≤Y ) be two ordered sets. Suppose that ≤Y is a wellordering. Define a relation ≤ on AY as follows. Let f, g ∈ AY and set Y0 := {y ∈ Y : f (y) /= g(y)} ⊂ Y . If f and g are distinct, then Y0 is non-empty and, since ≤Y is a well-ordering on Y , there exists a least element y0 in Y0 . We then write f < g if f (y0 ) 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 The Banach-Alaoglu Theorem

495

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 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 have .B(X∗ ) ⊂ x∈X Ix by definition of the operator norm. Let X := x∈X R with the product topology. Then it is clear that the topology us equip .R || 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∗ ). Consequently, .B(X∗ ) is closed in .RX . Since . x∈X Ix is compact by Tychonoff theorem (Theorem A.5.2), we deduce that .B(X∗ ) is compact. u n

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.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3

497

498

G The Markov-Kakutani Fixed Point Theorem

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 |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 U .K ⊂ Ωx . (G.2) x∈F

If we take .α := minx∈F αx , then .α > 0 and |λ| < α ⇒ λK ⊂ V

.

u n

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. u n 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 1Σ k (f (x) − f k+1 (x)). n n−1

xn :=

.

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 xn =

.

1 1 x − f n (x) n n

G

The Markov-Kakutani Fixed Point Theorem

499

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. u n 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 n .

Fix(f ) /= ∅.

f ∈F

This shows that there is a point in K which is fixed by all elements of .F .

u n

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). 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 leftneighborhood of y) is the subset .NR (x) ⊂ Y (resp. .NL (y) ⊂ X) defined by (see Figs. H.3 and H.4): 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) :=

U

.

a∈A

NR (a)

and

NL (B) = NLG (B) :=

U

NL (b).

b∈B

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3

501

502

H

The Hall Harem Theorem

Fig. H.1 The bipartite graph = (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 )} .G

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

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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503

Fig. H.4 The left-neighborhood .NL (y1 ) ⊂ X of the vertex .y1 ∈ Y in the bipartite graph .G of Fig. H.1

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 |

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. 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 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}.

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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 ' ), 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 G' G' ' ' ' ' ' .|N R (A )| ≥ |A | since .NR (A ) = NR (A ) \ {y0 }. As .|X | = n − 1, it follows

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505

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 ' , and .E '' := E ∩ (X '' × Y '' ). We claim that the left Hall condition .Y 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 left-perfect 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). 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

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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 n has the 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 . u n 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. Note that an equivalence class is reduced to a single edge .(x, y) if and only if (x, y) ∈ MX ∩ MY .

.

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507

Fig. H.6 The bipartite graph = (X, Y, E), where .X = Y := N and .E := {(0, n + 1) : n ∈ N} ∪ {(n + 1, n) : n ∈ N} .G

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 :=

U

.

M(C ),

C

where .C runs over all equivalence classes in .M, is the required perfect matching for G. u n

.

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The Hall Harem Theorem

509

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 .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)). u n 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. u n

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 Fig. H.8 The bipartite graph = (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 .G

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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 |NL (B)| ≥ k1 |B| for all finite subsets A ⊂ X and B ⊂ Y.

(H.5)

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. 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 . Consider the new bipartite graph .G ' = (X' , Y ' , E ' ), where .X' := uki=1 Xi , .Y ' := Y , and E ' := {(φi (x), y) : (x, y) ∈ E, i = 1, 2, . . . , k} ⊂ X' × Y ' .

.

Let .A' be a finite subset of .X' and denote by .A' the set of .x ∈ X such that 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 '

|NLG (B ' )| = k|NLG (B ' )| ≥ k

.

(

) 1 ' |B | = |B ' | k

(H.7)

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

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511

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 . n u

Notes The Hall marriage theorem was first established for finite bipartite graphs by P. Hall [HallP1] and then extended to infinite locally finite bipartite graphs by M. Hall [HallM]. 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 = ∅

(I.1)

for all .n ≥ 1. Then ⎛ .

Int ⎝

U

⎞ Xn ⎠ = ∅.

(I.2)

n≥1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3

513

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I Complements of Functional Analysis

Proof Letn 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

BX (x0 , r0 )

n

.

⎛ ⎝

n

⎞ An ⎠ /= ∅.

(I.3)

n≥1

As .A1 is dense and open we can find .x1 ∈ BX (x0 , r0 ) { .

BX (x1 , r1 ) ⊂ OB X (x0 , r0 ) 0 < r1
0 such that

A1

r0 2.

Continuing this way, we produce by induction a sequence .(xn )n≥1 in X and a sequence .(rn )n≥1 in .R such that { .

BX (xn+1 , rn+1 ) ⊂ OB X (xn , rn ) 0 < rn+1
0 such that T (OB X (0, 1)) ⊃ OB Y (0, r).

.

(I.6)

For all U.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 ⊂ T (OB X (0, 1/2n+1 )). .OB Y 0, 2n

(I.9)

Let .y0 ∈ OB Y (0, r) and let us show that there exists .x ' ∈ OB X (0, 1) such that 1 ' .y0 = T (x ). We deduce from (I.9) (with .n = 0) that there exists .x0 ∈ OB X (0, ) 2 r r such that .||y0 −T (x0 )|| < 2 . Set .y1 := y0 −T (x0 ) and observe that .y1 ∈ OB Y (0, 2 ).

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I Complements of Functional Analysis

Continuing this way, we find a sequence .(yn )n≥1 in Y and a sequence .(xn )n≥1 in X such that ) ( r , (I.10) .yn+1 = yn − T (xn ) ∈ OB Y 0, 2n+1 ( ) 1 .xn ∈ OB X 0, (I.11) 2n+1 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 u n 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.

I Complements of Functional Analysis

517

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)

admits a unique solution .x ∈ X. Indeed (I.15) is equivalent to x=

.

1 (T (x) − y). λ

Moreover, .|| λ1 (T (x1 ) − y) − λ1 (T (x2 ) − y)|| ≤ 1 . |λ| ||T ||

1 |λ| ||T || · ||x1 − x2 ||

(I.16) for all .x1 , x2 ∈ X

and < 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 −λ 1Id )−1 || . For .y ∈ X we have that the linear Eq. (I.15) can be written as 0 X .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. u n

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 || || || .||x − y|| > ε implies || 2 || < 1 − δ for all x, y ∈ X with ||x||, ||y|| ≤ 1.

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Let Z be a nonempty set not reduced to a single point. Then the Banach space l1 (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 || || || x + y ||2 1 1 2 || || || 2 || + 4 ||x − y|| = 4 ( + ) 1 ( + + 2 + + − 2) 4 ) 1( 2||x||2 + 2||y||2 = 4

=

.

≤ 1. Therefore, .

|| || || x + y ||2 1 2 || || || 2 || ≤ 1 − 4 ||x − y|| .

Let now ε > 0 and set δ := 1 − 2

1 2

(I.18)

√ 4 − ε2 . Suppose that ||x − y|| > ε. This

implies 1 − 14 ||x − y||2 < 1 − ε4 = (1 − δ)2 . From (I.18) we then deduce that || x+y u n 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 .A'1 , A'2 ∈ ' ' ' ' 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 Fréchet filter on X. If we consider the directed set .(N, ≤), a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3

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subset .A ⊂ N is residual if and only if its complement .N \ A is finite. Therefore, the residual filter on .N equals the Fréchet filter on .N. (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 cannot both n belong to .F ; (F-5) if .A1 , A2 , . . . , An ∈ F , .n ≥ 1, then . ni=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 . u n .

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

.

(1) .F ⊂ F ' ; (2) .A0 ∈ F ' .

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521

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 .A ∈ F0 two sets such that .Ai ∩ A ⊂ Bi . We have i i n n ' .B1 ∩ B2 ⊃ (A1 ∩ A1 ) (A2 ∩ A'2 ) = (A1 ∩ A2 ) (A'1 ∩ A'2 ). But .A1 ∩ A2 belongs to the filter .F , and .A'1 ∩ A'2 ∈ F0 as both .A'1 and .A'2 contain ' ' .A0 . It follows that .B1 ∩ B2 ∈ F . This shows (F-3). It follows that .F is a filter. u n Proof of Theorem J.1.6 Consider the set .Φ0 consisting of all filters on X containing .F0 . This is a nonempty set, partially ordered by inclusion. Let .Φ be a totally U - := ordered subset of .Φ0 . We claim that .F F ∈Φ F is an upper bound for .Φ. We - belongs to .Φ0 . As .∅ ∈ only have to show that .F / F for all .F ∈ Φ we also have - and .B ⊂ X - satisfies condition (F-1). Let now .A ∈ F .∅ ∈ / F . This shows that .F be such that .A ⊂ B. Then there exists .F ∈ Φ such that .A ∈ F . As .F is a filter, -. This shows that .F - satisfies we have .B ∈ F , by (F-2). It then follows that .B ∈ F condition (F-2). Finally, suppose that .A, B ∈ F . Then there exist .FA , FB ∈ Φ such that .A ∈ FA and .B ∈ FB . As .Φ is totally ordered, up to exchanging A and B we can suppose that .FA ⊂ FB . We then have .A, B ∈ FB and therefore -. Therefore .F - satisfies condition .A ∩ B ∈ FB , by (F-3). It follows that .A ∩ B ∈ F (F-3) as well. This shows that .Φ0 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. u n 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 '

522

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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). u n

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 , . . . , n An , n ≥ 1, n be elements in ω and denote by A , . . . , A their closures. As A , 2 n 1 i=1 Ai ⊃ nn A = / ∅ (by (F-1) and (F-5)), we have that the family (A) in X has the A∈ω i i=1 finite intersection property and therefore, by compactness of X, it has a nonempty n 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 (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 n .

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,

J

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523

there exists x ∈ X such that ω converges to x, equivalently, Fx ⊂ ω. Let i ∈ I . by (F-1) and (F-3), we n have Ci ∩ V /= ∅ for all V ∈ Fx . As Ci is closed, we have x ∈ Ci . Thus x ∈ i∈I Ci , and (J.3) follows. u n 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 Fréchet 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 the present example.

i→F

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 f −1 (V ) belongs to ω. Suppose the contrary. As ω is an ultrafilter, by (UF) we have X \ f −1 (V ) = {x ∈ X : f (x) ∈ / V } ∈ ω. Setting U := f (X \ f −1 (V )), 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. u n 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) :=

524

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(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 to x→F

F and as F −1 (W ) ⊃ F −1 (U × V ) = f −1 (U ) ∩ g −1 (V ), we deduce from (F-2) u n 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 ). u n 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 := (y1 − Proof Suppose, by contradiction, that y1 > y2 . Set r := y1 −y 3 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 ) = ∅ u n since f1 ≤ f2 . This contradicts (F-2). It follows that y1 ≤ y2 .

Recall that given a set X, we denote by l∞ (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 : l∞ (X) → R satisfying m(1) = 1 and m(x) ≥ 0 for all x ∈ l∞ (E) such that x ≥ 0, is called a mean on X (cf. Definition 4.1.4). Corollary J.2.10 Let X be a set and let ω be an ultrafilter on X. For every f ∈ l∞ (X) there exists a unique y0 ∈ R such that f (x) −→ y0 . Moreover, the map x→ω

mω : l∞ (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 ∈ l∞ (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

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525

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 ∈ l∞ (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 Propositions J.2.7 and 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 u n 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 ˇ Hausdorff space, called the Stone-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 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 one-to-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 .M P(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 : M P(X) → M (X). Then one has .Φ −1 (μω ) = mω for all ultrafilters .ω on X.

Open Problems

The problems listed below appeared as open problems in the first edition of this book. Some of them have been since solved (and we mark them by (SP)), while the others, to the best of our knowledge, remain still open (and we mark them by (OP)). (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 G G with respect to the prodiscrete topology? .τ (A ) is not closed in .A (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 G with respect to the prodiscrete topology? .V (OP-8) Let G and H be two quasi-isometric groups. Suppose that G is surjunctive. Is it true that H is surjunctive? (SP-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? © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3

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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? (SP-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 G → V G which is pre-injective but not surjective? .τ : V (SP-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 G → V G which is surjective but not pre-injective? .τ : V (OP-15) (Kaplansky’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 automaton .τ : V G → V G is surjective? (OP-16) (Kaplansky’s zero-divisor 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 : K G → K G is pre-injective? (SP-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 3.24), 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 Grigorchuk 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 3.22 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,

Open Problems

(OP-7)

(SP-9)

(SP-13)

(SP-14)

(OP-15)

(OP-16) (SP-17)

529

when G is locally finite, then, for any set A, the image of every cellular automaton τ : AG → AG is closed in AG (cf. Exercise 3.23 or [CeC11, Proposition 4.1]). 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]). The answer is affirmative as proved by L. Bartholdi in [Bar3, Theorem 1.1] (cf. Theorem 8.17.1). Note that when G contains a nonabelian free subgroup this was already shown in Proposition 5.11.1. The answer is affirmative, as proved by L. Bartholdi in [Bar3, Theorem 1.1] and [Bar2, Theorem 1.1] (cf. Theorem 8.17.1). Note that when G contains a nonabelian free subgroup this was already shown in Corollary 8.10.2. The answer is affirmative, as proved by L. Bartholdi in [Bar3, Corollary 1.5] (cf. Exercise 8.98). The arguments combine Bartholdi’s solution to (OP-13) together with his notion of duality for linear cellular automata (cf. Exercises 8.97). Note that when G contains a nonabelian free subgroup this was already shown in Corollary 8.11.2. In [Phu8, Theorem B] (see also [CCP5, Theorem 1.1]) it shown that Kaplansky’s stable finiteness conjecture is a consequence of Gottschalk’s surjunctivity conjecture (OP-5). See the discussion in the notes at the end of Chap. 8. The answer is negative. A counterexample, actually a non-orderable diffuse group (cf. Exercise 8.62), can be found in the appendix of [KioR], written by N. Dunfield.

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List of Symbols

Symbol ε < ∼ ||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− π+ σ (T ) ψq,r Ω −E

Definition the empty word the dominance relation in the set of growth functions γ : N → [0, +∞) the equivalence relation in the set of growth functions γ : N → [0, +∞) the lp -norm of a linear map T : lp (E) → lp (E) the zero element of the ring R the identity element of the group G the unity element of the ring R the growth function of the group G relative to the finite symmetric generating subset S ⊂ G the discrete laplacian on the group G associated with the subset S ⊂ G the restriction of ΔS to the Hilbert space l2 (G) the E-boundary of the subset Ω ⊂ G the isoperimetric constant of the group G with respect to the finite symmetric generating subset S ⊂ G the growth rate of the group G with respect to the finite symmetric generating subset S ⊂ G the label of the edge e ∈ E in a labeled graph G = (Q, E) the label of the path π in a labeled graph G = (Q, E) the initial vertex of the path π in an S-labeled graph 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 Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-43328-3

Page 477 208 208 241 377 3 377 206 10 248 155 238 215 200 200, 293 200 200 517 345 155

541

542 Symbol 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 dSG , dS dQ D(G) (D i (G))i≥0 entF (X) F (X) Fn Fix(α) Fix(H ) G = gx G = (X, Y, E) HR ICA(G; A) IdX l(w) lp (E) l∞ (E) lG S (g), lS (g) LCA(G; V ) LCA(G, H ; V )

List of Symbols Definition 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 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 → A G 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 : V G → V G 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

Page 155 477 2 345 199 199 14 17 128 202 329 199 201 127 127 165 482 482 329 4 485 2 501 128 25 3 477 240 112 198 373 376

List of Symbols Symbol L(X) Ln (X) L (X) (p)

MS Matd (R) mdimF (X) M (E) N (Γ ) NL (B) ⊂ X NL (y) ⊂ X NR (A) ⊂ Y NR (x) ⊂ Y P (E) PM (E)

per(B) per(G ) per(X) pern (X) Per(X) Q(r)

Q = (Q, E)

R[G] R op Sym(X) Sym0 (X) Sym+ 0 (X) Symn Sym+ n Sym(X,