Cellular Automata and Groups

Cellular automata were introduced in the first half of the last century by John von Neumann who used them as theoretical models for self-reproducing machines. The authors present a self-contained exposition of the theory of cellular automata on groups and explore its deep connections with recent developments in geometric group theory, symbolic dynamics, and other branches of mathematics and theoretical computer science. The topics treated include in particular the Garden of Eden theorem for amenable groups, and the Gromov-Weiss surjunctivity theorem as well as the solution of the Kaplansky conjecture on the stable finiteness of group rings for sofic groups. The volume is entirely self-contained, with 10 appendices and more than 300 exercises, and appeals to a large audience including specialists as well as newcomers in the field. It provides a comprehensive account of recent progress in the theory of cellular automata based on the interplay between amenability, geometric and combinatorial group theory, symbolic dynamics and the algebraic theory of group rings which are treated here for the first time in book form.

first and unique book dedicated to cellular automata and groups 90% of the material presented appears for a first time entirely self-contained with more than 300 exercises appeals to a large audience including specialists as well as newcomers in the field Includes supplementary material: sn.pub/extras

Autorentext

Tullio Ceccherini-Silberstein obtained his BS in Mathematics (1990) from the University of Rome "La Sapienza" and his PhD in Mathematics (1994) from UCLA. Currently, he is professor of Mathematical Analysis at the University of Sannio (Benevento). He is an Editor of the EMS journal "Groups, Geometry, and Dynamics". He has authored more than 90 research articles in Functional and Harmonic Analysis, Group Theory, Ergodic Theory and Dynamical Systems, and Theoretical Computer Science and has co-authored 5 monographs on Harmonic Analysis and Representation Theory. Fabio Scarabotti obtained his BS in Mathematics (1989) and his PhD in Mathematics (1994) from the University of Rome "La Sapienza". Currently, he is professor of Mathematical Analysis at the University of Rome "La Sapienza". He has authored more than 40 research articles in Harmonic Analysis, Group Theory, Combinatorics, Ergodic Theory and Dynamical Systems, and TheoreticalComputer Science and has co-authored 4 monographs on Harmonic Analysis and Representation Theory. Filippo Tolli obtained his BS in Mathematics (1991) from the University of Rome "La Sapienza" and his PhD in Mathematics (1996) from UCLA. Currently, he is professor of Mathematical Analysis at the University of Roma Tre. He has authored more than 30 research articles in Harmonic Analysis, Group Theory, Combinatorics, Lie Groups and Partial Differential Equations and has co-authored 4 monographs on Harmonic Analysis and Representation Theory.

Inhalt
Cellular Automata.- Residually Finite Groups.- Surjunctive Groups.- Amenable Groups.- The Garden of Eden Theorem.- Finitely Generated Amenable Groups.- Local Embeddability and Sofic Groups.- Linear Cellular Automata.- Nets and the Tychonoff Product Theorem.- Uniform Structures.- Symmetric Groups.- Free Groups.- Inductive Limits and Projective Limits of Groups.- The Banach-Alaoglu Theorem.- The Markov-Kakutani Fixed Point Theorem.- The Hall Harem Theorem.- Complements of Functional Analysis.- Ultrafilters.