Cellular automata and groups

✍ Scribed by Tullio Ceccherini-Silberstein, Michel Coornaert (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2010
• Tongue: English
• Leaves: 460
• Series: Springer Monographs in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages I-XIX
Cellular Automata....Pages 1-36
Residually Finite Groups....Pages 37-55
Surjunctive Groups....Pages 57-75
Amenable Groups....Pages 77-110
The Garden of Eden Theorem....Pages 111-149
Finitely Generated Amenable Groups....Pages 151-232
Local Embeddability and Sofic Groups....Pages 233-281
Linear Cellular Automata....Pages 283-342
Nets and the Tychonoff Product Theorem....Pages 343-349
Uniform Structures....Pages 351-358
Symmetric Groups....Pages 359-366
Free Groups....Pages 367-377
Inductive Limits and Projective Limits of Groups....Pages 379-381
The Banach-Alaoglu Theorem....Pages 383-385
The Markov-Kakutani Fixed Point Theorem....Pages 387-389
The Hall Harem Theorem....Pages 391-401
Complements of Functional Analysis....Pages 403-408
Ultrafilters....Pages 409-416
Back Matter....Pages 417-439

✦ Subjects

Dynamical Systems and Ergodic Theory