Linear cellular automata, duality and sofic groups

TL;DR

This paper constructs non-pre-injective, surjective linear cellular automata on non-amenable groups, and establishes duality relations linking injectivity and surjectivity properties in cellular automata over sofic groups.

Contribution

It provides a positive solution to an open problem for non-amenable groups and introduces a duality framework connecting injectivity and surjectivity in linear cellular automata.

Findings

Constructed non-pre-injective, surjective linear cellular automata on non-amenable groups.

Reproved that cellular automata over sofic groups are injective iff post-surjective.

Established duality between kernel and image of matrices over group rings.

Abstract

We produce for arbitrary non-amenable group $G$ and field $K$ a non-pre-injective, surjective linear cellular automaton. This answers positively Open Problem (OP-14) in Ceccherini-Silberstein and Coornaert's monograph "Cellular Automata and Groups". We also reprove in a direct manner, for linear cellular automata, the result by Capobianco, Kari and Taati that cellular automata over sofic groups are injective if and only if they are post-surjective. These results come from considerations related to matrices over group rings: we prove that a matrix's kernel and the image of its adjoint are mutual orthogonals of each other. This gives rise to a notion of "dual cellular automaton", which is pre-injective if and only if the original cellular automaton is surjective, and is injective if and only if the original cellular automaton is post-surjective.