On bireversible automata and commensurators of groups in automorphisms of their Cayley graphs

TL;DR

This paper explores the structure of automorphism groups of Cayley graphs, showing they can be built from bireversible automata, and investigates implications for the types of groups these automata can generate.

Contribution

It demonstrates that automorphism groups fixing a vertex are unions of groups generated by bireversible automata and analyzes their properties and limitations.

Findings

Cyclic subgroups of these automorphism groups are undistorted.

Certain groups cannot be generated by bireversible automata.

The class of groups generated by bireversible automata is strictly smaller than those generated by invertible and reversible automata.

Abstract

If $G$ is a finitely generated group and $X$ is a Cayley graph of $G$, denote by $\mathcal{C}_1^X(G)$ the subgroup of all automorphisms of $X$ commensurating $G$ and fixing the vertex corresponding to the identity. Building on the work of Macedo\'{n}ska, Nekrashevych and Sushchansky, we observe that $\mathcal{C}_1^X(G)$ can be expressed as a directed union of groups generated by bireversible automata. We use this to to show that every cyclic subgroup of $\mathcal{C}_1^X(G)$ is undistorted and to obtain a necessary condition on $G$ for $\mathcal{C}_1^X(G)$ not to be locally finite. As a consequence, we prove that several families of groups cannot be generated by bireversible automata and show that the set of groups generated by bireversible automata is strictly contained in the set of groups generated by invertible and reversible automata.