Springer Monographs in Mathematics for Further Volumes: Tullio Ceccherini-Silberstein R Michel Coornaert

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

Cellular Automata and Groups Tullio Ceccherini-Silberstein Michel Coornaert Dipartimento di Ingegneria Institut de Recherche Mathématique Avancée Università del Sannio Université de Strasbourg C.so Garibaldi 107 7 rue René-Descartes 82100 Benevento 67084 Strasbourg Cedex Italy France [email protected] [email protected]

ISSN 1439-7382 ISBN 978-3-642-14033-4 e-ISBN 978-3-642-14034-1 DOI 10.1007/978-3-642-14034-1 Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2010934641

Mathematics Subject Classiﬁcation (2010): 37B15, 68Q80, 20F65, 43A07, 16S34, 20C07

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Springer is part of Springer Science+Business Media (www.springer.com) To Katiuscia, Giacomo, and Tommaso

To Martine and Nathalie Preface

Two seemingly unrelated mathematical notions, namely that of an amenable group and that of a cellular automaton, were both introduced by John von Neumann in the first half of the last century. Amenability, which originated from the study of the Banach-Tarski paradox, is a property of groups gener- alizing both commutativity and finiteness. Nowadays, it plays an important role in many areas of mathematics such as representation theory, harmonic analysis, ergodic theory, geometric group theory, probability theory, and dynamical systems. Von Neumann used cellular automata to serve as theoretical models for self-reproducing machines. About twenty years later, the famous cellular automaton associated with the Game of Life was invented by John Horton Conway and popularized by Martin Gardner. The theory of cellular automata flourished as one of the main branches of computer science. Deep connections with complexity theory and logic emerged from the discovery that some cellular automata are universal Turing machines. AgroupG 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

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

In this book, the universe will always be a group G (in the classical setting the corresponding group was G = Z2) and the alphabet may be finite or infinite. The left multiplication in G induces a natural action of G on the set of configurations which is called the G-shift and all cellular automata will be required to commute with the shift. It was soon realized that the question whether a given cellular automaton is surjective or not needs a special attention. From the dynamical viewpoint, surjectivity means that each configuration may be reached at any time. The first important result in this direction is the celebrated theorem of Moore and Myhill which gives a necessary and sufficient condition for the surjectivity of a cellular automaton with finite alphabet over the group G = Z2.EdwardF. Moore and John R. Myhill proved that such a cellular automaton is surjective if and only if it is pre-injective. As the term suggests it, pre-injectivity is a weaker notion than injectivity. More precisely, a cellular automaton is said to be pre-injective if two configurations are equal whenever they have the same image and coincide outside a finite subset of the group. Moore proved the “surjective ⇒ pre-injective” part and Myhill proved the converse implication shortly after. One often refers to this result as to the Garden of Eden theorem. This biblical terminology is motivated by the fact that, regarding a cellular automaton as a dynamical system with discrete time, a configuration which is not in the image of the cellular automaton may only appear as an initial configuration, that is, at time t =0. The surprising connection between amenability and cellular automata was established in 1997 when Antonio Mach`ı, Fabio Scarabotti and the first author proved the Garden of Eden theorem for cellular automata with finite alphabets over amenable groups. At the same time, and completely inde- pendently, Misha Gromov, using a notion of spacial entropy, presented a more general form of the Garden of Eden theorem where the universe is an amenable graph with a dense holonomy and cellular automata are called maps of bounded propagation. Mach`ı, Scarabotti and the first author also showed that both implications in the Garden of Eden theorem become false if the underlying group contains a subgroup isomorphic to F2. The question whether the Garden of Eden theorem could be extended beyond the class of Preface ix amenable groups remained open until 2008 when Laurent Bartholdi proved that the Moore implication fails to hold for non-amenable groups. As a consequence, the whole Garden of Eden theorem only holds for amenable groups. This gives a new characterization of amenable groups in terms of cellular automata. Let us mention that, up to now, the validity of the Myhill implication for non-amenable groups is still an open problem. Following Walter H. Gottschalk, a group G is said to be surjunctive if every injective cellular automaton with finite alphabet over G is surjective. Wayne Lawton proved that all residually finite groups are surjunctive and that every subgroup of a surjunctive group is surjunctive. Since injectivity implies pre-injectivity, an immediate consequence of the Garden of Eden theorem for amenable groups is that every amenable group is surjunctive. Gromov and Benjamin Weiss introduced a class of groups, called sofic groups, which includes all residually finite groups and all amenable groups, and proved that every sofic group is surjunctive. Sofic groups can be defined in three equivalent ways: in terms of local approximation by finite symmetric groups equipped with their Hamming distance, in terms of local approximation of their Cayley graphs by finite labelled graphs, and, finally, as being the groups that can be embedded into ultraproducts of finite symmetric groups (this last characterization is due to 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 con- taining 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 x Preface 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 cellular automaton with one-dimensional alphabet over a torsion-free group always pre-injective? ∗ ∗∗ The material presented in this book is entirely self-contained. In fact, its reading only requires some acquaintance with undergraduate general topology and abstract algebra. Each chapter begins with a brief overview of its contents and ends with some historical notes and a list of exercises at various difficulty levels. Some additional topics, such as subshifts and cellular automata over subshifts, are treated in these exercises. Hints are provided each time help may be needed. In order to improve accessibility, a few appendices are included to quickly introduce the reader to facts he might be not too familiar with. In the first chapter, we give the definition of a cellular automaton. We present some basic examples and discuss general methods for constructing cellular automata. We equip the set of configurations with its prodiscrete uniform structure and prove the generalized Curtis-Hedlund theorem: a necessary and sufficient condition for a self-mapping of the configuration space to be a cellular automaton is that it is uniformly continuous and commutes with the shift. Chapter 2 is devoted to residually finite groups. We give several equivalent characterizations of residual finiteness and prove that the class of residually finite groups is closed under taking subgroups and projective limits. We establish in particular the theorems, respectively due to Anatoly I. Mal’cev and Gilbert Baumslag, which assert that finitely generated residually finite groups are Hopfian and that their automorphism group is residually finite. Surjunctive groups are introduced in Chap. 3. We show that every subgroup of a surjunctive group is surjunctive and that locally residually finite groups are surjunctive. We also prove a theorem of Gromov which says that limits of surjunctive marked groups are surjunctive. The theory of amenable groups is developed in Chap. 4. The class of amenable groups is closed under taking subgroups, quotients, extensions, and inductive limits. We prove the theorems due to Erling Følner and Alfred Tarski which state the equivalence between amenability, the existence of a Følner net, and the non-existence of a paradoxical decomposition. The Garden of Eden theorem is established in Chap. 5.Itisprovedby showing that both surjectivity and pre-injectivity of the cellular automaton are equivalent to the fact that the image of the configuration space has maxi- Preface xi mal entropy. We give an example of a cellular automaton with finite alphabet over F2 which is pre-injective but not surjective. Following Bartholdi’s construction, we also prove the existence of a surjective but not pre-injective cellular automaton with finite alphabet over any non-amenable group. In Chap. 6 we present the basic elementary notions and results on growth of finitely generated groups. We prove that finitely generated nilpotent groups have polynomial growth. We then introduce the Grigorchuk group and show that it is an infinite finitely generated periodic group of intermediate growth. We show that every finitely generated group of subexponential growth is amenable. We also establish the Kesten-Day characterization of amenability which asserts that a group with a finite (not necessarily symmetric) generat- ing subset is amenable if and only if 0 is in the 2-spectrum of the associated Laplacian. Finally, we consider the notion of quasi-isometry for not necessarily countable groups and we show that amenability is a quasi-isometry invariant. In Chap. 7 we consider the notion of local embeddability of groups into a class of groups. For the class of finite groups, this gives the class of LEF groups introduced by Anatoly M. Vershik and Edward I. Gordon. We discuss several stability properties of local embeddability and show that locally embeddable groups are closed in marked groups spaces. The remaining of the chapter is devoted to the class of sofic groups. We show that the three definitions, namely analytic, geometric, and algebraic, we alluded to before, are equivalent. We then prove the Gromov-Weiss theorem which states that every sofic group is surjunctive. The last chapter is devoted to linear cellular automata. We prove the linear version of the Garden of Eden theorem and show that every sofic group is L-surjunctive. We end the chapter with a discussion on the stable finiteness and the zero-divisors conjectures of Kaplansky and their reformulation in terms of linear cellular automata. Appendix A gives a quick overview of a few fundamental notions and results of topology (nets, compactness, product topology, and the Tychonoff product theorem). Appendix B is devoted to 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 flavour, we consider bipartite graphs and their matchings. We prove Hall’s marriage theorem and its harem version which plays a key role in the proof of Tarski’s theorem on amenability. The Baire theorem, the open mapping theorem, as well as other xii Preface complements of functional analysis including uniform convexity are treated in Appendix I. The last appendix deals with the notions of filters and ultra- filters. We would like to express our deep gratitude to Dr. Catriona Byrne, Dr. Marina Reizakis and Annika Eling from Springer Verlag and to Donatas Akmanaviˇcius for their constant and kindest help at all stages of the editorial process.

Rome and Strasbourg Tullio Ceccherini-Silberstein Michel Coornaert Contents

1 Cellular Automata ...... 1 1.1 TheConﬁgurationSetandtheShiftAction...... 1 1.2 The Prodiscrete Topology ...... 3 1.3 PeriodicConﬁgurations ...... 3 1.4 CellularAutomata...... 6 1.5 MinimalMemory...... 14 1.6 CellularAutomataoverQuotientGroups...... 15 1.7 Induction and Restriction of Cellular Automata ...... 16 1.8 CellularAutomatawithFiniteAlphabets...... 20 1.9 TheProdiscreteUniformStructure...... 22 1.10 Invertible Cellular Automata ...... 24 Notes...... 27 Exercises ...... 29

2 Residually Finite Groups ...... 37 2.1 DeﬁnitionandFirstExamples ...... 37 2.2 Stability Properties of Residually Finite Groups ...... 40 2.3 ResidualFinitenessofFreeGroups ...... 42 2.4 HopﬁanGroups...... 44 2.5 Automorphism Groups of Residually Finite Groups ...... 45 2.6 Examples of Finitely Generated Groups Which Are Not Residually Finite ...... 47 2.7 DynamicalCharacterizationofResidualFiniteness...... 50 Notes...... 51 Exercises ...... 52

3 Surjunctive Groups ...... 57 3.1 Deﬁnition ...... 57 3.2 Stability Properties of Surjunctive Groups ...... 58 3.3 Surjunctivity of Locally Residually Finite Groups ...... 59

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3.4 MarkedGroups ...... 61 3.5 ExpansiveActionsonUniformSpaces...... 64 3.6 Gromov’sInjectivityLemma...... 65 3.7 Closedness of Marked Surjunctive Groups ...... 67 Notes...... 68 Exercises ...... 68

4 Amenable Groups ...... 77 4.1 MeasuresandMeans...... 77 4.2 PropertiesoftheSetofMeans...... 82 4.3 MeasuresandMeansonGroups...... 83 4.4 Deﬁnition of Amenability ...... 85 4.5 Stability Properties of Amenable Groups ...... 88 4.6 SolvableGroups...... 92 4.7 TheFølnerConditions...... 94 4.8 ParadoxicalDecompositions ...... 98 4.9 TheTheoremsofTarskiandFølner...... 99 4.10 The Fixed Point Property ...... 103 Notes...... 105 Exercises ...... 106

5 The Garden of Eden Theorem ...... 111 5.1 Garden of Eden Conﬁgurations and Garden of Eden Patterns 111 5.2 Pre-injectiveMaps...... 112 5.3 StatementoftheGardenofEdenTheorem ...... 114 5.4 Interiors, Closures, and Boundaries ...... 115 5.5 Mutually Erasable Patterns ...... 121 5.6 Tilings ...... 122 5.7 Entropy...... 125 5.8 ProofoftheGardenofEdenTheorem ...... 128 5.9 Surjunctivity of Locally Residually Amenable Groups ...... 131 5.10 A Surjective but Not Pre-injective Cellular Automaton over F2 ...... 133 5.11 A Pre-injective but Not Surjective Cellular Automaton over F2 ...... 133 5.12 A Characterization of Amenability in Terms of Cellular Automata...... 135 5.13 Garden of Eden Patterns for Life ...... 136 Notes...... 138 Exercises ...... 139

6 Finitely Generated Amenable Groups ...... 151 6.1 TheWordMetric...... 151 6.2 LabeledGraphs...... 153 6.3 CayleyGraphs...... 156 6.4 GrowthFunctionsandGrowthTypes...... 160 Contents xv

6.5 TheGrowthRate ...... 168 6.6 Growth of Subgroups and Quotients ...... 170 6.7 A Finitely Generated Metabelian Group with Exponential Growth ...... 173 6.8 GrowthofFinitelyGeneratedNilpotentGroups...... 175 6.9 The Grigorchuk Group and Its Growth ...... 178 6.10 The Følner Condition for Finitely Generated Groups ...... 191 6.11 Amenability of Groups of Subexponential Growth ...... 192 6.12 The Theorems of Kesten and Day ...... 193 6.13 Quasi-Isometries ...... 204 Notes...... 214 Exercises ...... 217 7 Local Embeddability and Sofic Groups ...... 233 7.1 Local Embeddability ...... 234 7.2 Local Embeddability and Ultraproducts ...... 243 7.3 LEF-GroupsandLEA-Groups ...... 246 7.4 TheHammingMetric...... 251 7.5 SoficGroups...... 254 7.6 Sofic Groups and Metric Ultraproducts of Finite Symmetric Groups...... 260 7.7 A Characterization of Finitely Generated Sofic Groups ...... 265 7.8 SurjunctivityofSoficGroups ...... 272 Notes...... 275 Exercises ...... 278 8 Linear Cellular Automata ...... 283 8.1 TheAlgebraofLinearCellularAutomata ...... 284 8.2 Configurations with Finite Support ...... 288 8.3 Restriction and Induction of Linear Cellular Automata ...... 289 8.4 GroupRingsandGroupAlgebras ...... 291 8.5 Group Ring Representation of Linear Cellular Automata . . . . 294 8.6 ModulesoveraGroupRing...... 299 8.7 MatrixRepresentationofLinearCellularAutomata...... 301 8.8 The Closed Image Property ...... 305 8.9 The Garden of Eden Theorem for Linear Cellular Automata . 308 8.10 Pre-injective but not Surjective Linear Cellular Automata . . . 314 8.11 Surjective but not Pre-injective Linear Cellular Automata . . . 315 8.12 Invertible Linear Cellular Automata ...... 317 8.13 Pre-injectivity and Surjectivity of the Discrete Laplacian . . . . 321 8.14 Linear Surjunctivity ...... 324 8.15 Stable Finiteness of Group Algebras ...... 327 8.16 Zero-Divisors in Group Algebras and Pre-injectivity ofOne-DimensionalLinearCellularAutomata ...... 330 Notes...... 335 Exercises ...... 338 xvi Contents

A Nets and the Tychonoﬀ Product Theorem ...... 343 A.1 DirectedSets ...... 343 A.2 Nets in Topological Spaces ...... 343 A.3 Initial Topology ...... 346 A.4 Product Topology ...... 346 A.5 TheTychonoﬀProductTheorem...... 347 Notes...... 349

B Uniform Structures ...... 351 B.1 UniformSpaces ...... 351 B.2 UniformlyContinuousMaps...... 353 B.3 ProductofUniformSpaces...... 355 B.4 The Hausdorﬀ-Bourbaki Uniform Structure on Subsets ...... 356 Notes...... 358

C Symmetric Groups ...... 359 C.1 TheSymmetricGroup...... 359 C.2 Permutations with Finite Support ...... 360 C.3 Conjugacy Classes in Sym0(X)...... 362 C.4 TheAlternatingGroup...... 363

D Free Groups ...... 367 D.1 ConcatenationofWords...... 367 D.2 DeﬁnitionandConstructionofFreeGroups...... 367 D.3 ReducedForms ...... 373 D.4 PresentationsofGroups...... 375 D.5 TheKleinPing-PongTheorem...... 376

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

F The Banach-Alaoglu Theorem ...... 383 F.1 Topological Vector Spaces ...... 383 F.2 The Weak-∗ Topology ...... 384 F.3 The Banach-Alaoglu Theorem ...... 384

G The Markov-Kakutani Fixed Point Theorem ...... 387 G.1 StatementoftheTheorem...... 387 G.2 ProofoftheTheorem...... 387 Notes...... 389

H The Hall Harem Theorem ...... 391 H.1 BipartiteGraphs...... 391 H.2 Matchings...... 393 Contents xvii

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

I Complements of Functional Analysis ...... 403 I.1 TheBaireTheorem...... 403 I.2 TheOpenMappingTheorem ...... 404 I.3 SpectraofLinearMaps ...... 406 I.4 UniformConvexity...... 407

J Ultraﬁlters ...... 409 J.1 FiltersandUltraﬁlters...... 409 J.2 LimitsAlongFilters...... 412 Notes...... 415

Open Problems ...... 417 Comments...... 418

References ...... 421

List of Symbols ...... 429

Index ...... 433 Notation

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

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