Cellular Automata and Groups

This unique book provides a self-contained exposition of the theory of cellular automata on groups and explores its deep connections with recent developments in geometric and combinatorial group theory, amenability, symbolic dynamics, the algebraic theory of group rings, and other branches of mathematics and theoretical computer science. The topics treated include the Garden of Eden theorem for amenable groups, the Gromov–Weiss surjunctivity theorem, and the solution of the Kaplansky conjecture on the stable finiteness of group rings for sofic groups.

Entirely self-contained and now in its second edition, the volume includes 10 appendices and more than 600 exercises, the solutions of which are presented in the companion book Exercises in Cellular Automata and Groups (2023) by the same authors.  It will appeal to a large audience, including specialists and newcomers to the field.

1 Cellular Automata.- 2 Residually Finite Groups.- 3 Surjunctive Groups.- 4 Amenable Groups.- 5 The Garden of Eden Theorem.- 6 Finitely Generated Groups.- 7 Local Embeddability and So?c Groups.- 8 Linear Cellular Automata.- Appendix A: Nets and the Tychonoff Product Theorem.- Appendix B: Uniform Structures.- Appendix C: Symmetric Groups.- Appendix D: Free Groups.- Appendix E: Inductive Limits and Projective Limits of Groups.- Appendix G: The Markov-Kakutani Fixed Point Theorem.- Appendix I: Complements of Functional Analysis.

Tullio Ceccherini-Silberstein graduated from the University of Rome La Sapienza in 1990 and obtained his PhD in mathematics from the University of California at Los Angeles in 1994. Since 1997 he has taught at the University of Sannio, Benevento (Italy). His main interests include harmonic and functional analysis, geometric and combinatorial group theory, ergodic theory and dynamical systems, and theoretical computer science. He is an editor of the journal Groups, Geometry, and Dynamics, published by the European Mathematical Society. He has published more than 90 research papers, 9 monographs, and 4 conference proceedings.

Professor Michel Coornaert taught mathematics at the University of Strasbourg from 1992 until 2021. His research interests are in geometry, topology, group theory and dynamical systems. He is the author of many Springer volumes, including Topological Dimension and Dynamical Systems (2015), Cellular Automata and Groups (2010), Symbolic Dynamics and Hyperbolic Groups (1993) and Géométrie et théorie des groupes (1990).