Generalized Gottschalk's conjecture for sofic groups and applications

TL;DR

This paper generalizes key surjunctivity theorems for cellular automata over sofic groups, extends results to non-uniform automata, and proves stable finiteness of twisted group rings, broadening understanding of algebraic and dynamical properties of sofic groups.

Contribution

It extends surjunctivity and dual-surjunctivity theorems to local perturbations and non-uniform automata over sofic groups, and proves stable finiteness of twisted group rings.

Findings

Surjunctivity of algebraic NUCA over sofic groups.

Stable finiteness of twisted group rings over sofic groups.

Generalization of Gromov and Weiss's surjunctivity results.

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.