Closed Image Characterizations of Locally Finite Groups via Cellular Automata

Abstract

We prove that a group $G$ is locally finite if and only if, for some (equivalently, every) infinite set $A$, every cellular automaton $A^G\to A^G$ has closed image in the prodiscrete topology. Equivalently, this holds if and only if every linear cellular automaton $V^G\to V^G$ has closed image for some pair $(K,V)$ with $V$ infinite-dimensional over the field $K$ (equivalently, for every such pair). This gives affirmative answers to Open Problems 6 and 7 of Ceccherini-Silberstein and Coornaert. More precisely, if $G$ is not locally finite, then for every infinite set $A$ there is a finite-memory cellular automaton $A^G\to A^G$ with non-closed image, and for every field $K$ and every infinite-dimensional $K$-vector space $V$ there is such a linear cellular automaton $V^G\to V^G$. The common obstruction is constructed on a countable direct-sum alphabet from an infinite ray in a locally finite Cayley graph. A direct-summand argument gives arbitrary vector-space alphabets, while an alphabet-retract principle gives arbitrary infinite set alphabets.