Graph and wreath products of cellular automata
Ville Salo
TL;DR
This work investigates which groups embed into automorphism groups of full shifts and proves three main advances. First, the class ${\mathcal G}$ of isomorphism classes of subgroups of ${\mathrm Aut}(\Sigma^{\mathbb{Z}})$ is closed under countable graph products. Second, it introduces the concepts of ${A}$-cancellation and unique visits, showing that for finite abelian ${A}$ and ${G\in \mathcal G}$ acting without ${A}$-cancellation, the wreath product ${A}\wr G$ embeds in ${\mathrm Aut}(\Sigma^{\mathbb{Z}})$; in particular, all free abelian groups and free groups admit such cellular automata actions. Third, the paper develops robust machinery (including conveyor-belt constructions, property P, and closure under direct products) to extend these results to one-sided shifts, yielding that right-angled Coxeter and Artin groups appear in the one-sided setting, and providing concrete embeddings for $A\wr F_n$ and $A\wr \mathbb{Z}^n$ as well as for various graph products. Together, these contributions illuminate how symbolic-dynamic automorphism groups encode rich algebraic structures with broad implications for the embedding problem and the construction of large algebraic subgroups in dynamical systems.
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.