The automorphism group of a strongly irreducible subshift on a group

TL;DR

This paper investigates the automorphism groups of strongly irreducible subshifts over infinite groups, extending classical theorems and establishing new embedding results under various conditions.

Contribution

It generalizes Ryan's and Kim-Roush's theorems to broader classes of subshifts and groups, introducing a new marker lemma applicable to all strongly irreducible subshifts.

Findings

The center of the automorphism group is generated by shifts from the group's center.

Automorphism groups of full F_k-shifts embed into automorphism groups of certain subshifts.

Embedding results depend on properties like nonamenability and the strong topological Markov property.

Abstract

We study the automorphism group $\operatorname{Aut}(X)$ of a non-trivial strongly irreducible subshift $X$ on an arbitrary infinite group $G$ and generalize classical results of Ryan, Kim and Roush. We generalize Ryan's theorem by showing that the center of $\operatorname{Aut}(X)$ is generated by shifts by elements of the center of $G$ modded out by the kernel of the shift action. We generalize Kim and Roush's theorem by showing that if the free group $F_k$ of rank $k\geq 1$ embeds into $G$, then the automorphism group of any full $F_k$-shift embeds into $\operatorname{Aut}(X)$. If $X$ is an SFT, or more generally, if $X$ satisfies the strong topological Markov property, then we can weaken the conditions on $G$. In this case we show that the automorphism group of any full $\mathbb{Z}$-shift embeds into $\operatorname{Aut}(X)$ provided $G$ is not locally finite, and that the automorphism group of any full $F_k$-shift embeds into $\operatorname{Aut}(X)$ whenever $G$ is nonamenable. Our results rely on a new marker lemma which is valid for any nonempty strongly irreducible subshift on an infinite group. We remark that our results are new even for $G=\mathbb{Z}$ as they do not require the subshift to be an SFT.