Commensurated subgroups in tree almost automorphism groups

TL;DR

This paper classifies the types of commensurated subgroups in tree almost automorphism groups, revealing a limited set of classes and implications for subgroup embeddings, with applications to Thompson groups.

Contribution

It provides a complete classification of commensurated subgroups in tree almost automorphism groups and characterizes subgroups with only periodic elements based on boundary dynamics.

Findings

Exactly three commensurability classes of closed commensurated subgroups exist.

Thompson's group T has no nontrivial commensurated subgroups besides finite groups and itself.

Results imply rigidity in embedding these groups into locally compact groups.

Abstract

We prove that the tree almost automorphism groups admit exactly three commensurability classes of closed commensurated subgroups. Our proof utilizes an independently interesting characterization of subgroups of the tree almost automorphism groups which contain only periodic elements in terms of the dynamics of the action on the boundary of the tree. Our results further cover several interesting finitely generated subgroups of the tree almost automorphism groups, including the Thompson groups $F$, $T$, and $V$. We show in particular that Thompson's group $T$ has no commensurated subgroups other than the finite subgroups and the entire group. As a consequence, we derive several rigidity results for the possible embeddings of these groups into locally compact groups.