On symbolic group varieties and dual surjunctivity

TL;DR

This paper investigates algebraic group cellular automata over sofic and amenable groups, establishing conditions under which surjectivity and pre-injectivity are equivalent, thus extending classical Garden of Eden theorems to algebraic settings.

Contribution

It extends the dual version of Gottschalk's Conjecture and Gromov's Garden of Eden theorem to algebraic group cellular automata over sofic and amenable groups.

Findings

Post-surjective automata are weakly pre-injective over sofic groups with uncountable fields.

Surjective automata are weakly pre-injective over amenable groups.

Pre-injective automata are surjective over amenable groups.

Abstract

Let $G$ be a group. Let $X$ be an algebraic group over an algebraically closed field $K$. Denote by $A=X(K)$ the set of rational points of $X$. We study algebraic group cellular automata $\tau \colon A^G \to A^G$ whose local defining map is induced by a homomorphism of algebraic groups $X^M \to X$ where $M$ is a finite memory. When $G$ is sofic and $K$ is uncountable, we show that if $\tau$ is post-surjective then it is weakly pre-injective. Our result extends the dual version of Gottschalk's Conjecture for finite alphabets proposed by Capobianco, Kari, and Taati. When $G$ is amenable, we prove that if $\tau$ is surjective then it is weakly pre-injective, and conversely, if $\tau$ is pre-injective then it is surjective. Hence, we obtain a complete answer to a question of Gromov on the Garden of Eden theorem in the case of algebraic group cellular automata.