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Generalized Gottschalk's conjecture for sofic groups and applications

Xuan Kien Phung

Abstract

We establish generalizations of the well-known surjunctivity theorem of Gromov and Weiss as well as the dual-surjunctivity theorem of Capobianco, Kari and Taati for cellular automata (CA) to local perturbations of CA over sofic group universes. We also extend the results to a class of non-uniform cellular automata (NUCA) consisting of global perturbations with uniformly bounded singularity of CA. As an application, we obtain the surjunctivity of algebraic NUCA with uniformly bounded singularity over sofic groups. Moreover, we prove the stable finiteness of twisted group rings over sofic groups to generalize known results on Kaplansky's stable finiteness conjecture for group rings.

Generalized Gottschalk's conjecture for sofic groups and applications

Abstract

We establish generalizations of the well-known surjunctivity theorem of Gromov and Weiss as well as the dual-surjunctivity theorem of Capobianco, Kari and Taati for cellular automata (CA) to local perturbations of CA over sofic group universes. We also extend the results to a class of non-uniform cellular automata (NUCA) consisting of global perturbations with uniformly bounded singularity of CA. As an application, we obtain the surjunctivity of algebraic NUCA with uniformly bounded singularity over sofic groups. Moreover, we prove the stable finiteness of twisted group rings over sofic groups to generalize known results on Kaplansky's stable finiteness conjecture for group rings.
Paper Structure (15 sections, 16 theorems, 51 equations)

This paper contains 15 sections, 16 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main dynamical results
  4. Stable surjunctivity for sofic groups
  5. Stable dual-surjunctivity for sofic groups
  6. Conjectures
  7. Applications
  8. Direct finiteness of algebraic NUCA
  9. Stable finiteness of twisted group rings
  10. Organization of the paper
  11. Sofic groups
  12. Induced local maps of NUCA
  13. Invertibility of stably injective local perturbations of CA
  14. Invertibility of stably post-surjective NUCA over sofic groups with uniformly bounded singularity
  15. Stable surjunctivity of NUCA over sofic groups with uniformly bounded singularity

Key Result

Lemma 1.2

Let $G$ be a countable group and let $M\subset G$ be a finite subset. Let $A$ be a finite alphabet and let $s\in S^G$ where $S= A^{A^M}$. Suppose that $\sigma_s$ is invertible. Then $\sigma_s$ is also stably invertible.

Theorems & Definitions (31)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Theorem A
  • Corollary 1.3
  • Definition 1.4
  • Theorem B
  • Theorem C
  • Theorem D
  • Conjecture 1.5: Stable surjunctivity
  • ...and 21 more