Commensurated subgroups in tree almost automorphism groups

Abstract

We prove that the almost automorphism groups AAut$({\mathcal{T}}_{d,k})$ admit exactly three commensurability classes of closed commensurated subgroups. Our proof utilizes an independently interesting characterization of subgroups of AAut$({\mathcal{T}}_{d,k})$ 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 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.