Graph and wreath products of cellular automata

Abstract

We prove that the set of subgroups of the automorphism group of a two-sided full shift is closed under countable graph products. We introduce the notion of a group action without $A$-cancellation (for an abelian group $A$), and show that when $A$ is a finite abelian group and $G$ is a group of cellular automata whose action does not have $A$-cancellation, the wreath product $A \wr G$ embeds in the automorphism group of a full shift. We show that all free abelian groups and free groups admit such cellular automata actions. In the one-sided case, we prove variants of these results with reasonable alphabet blow-ups.