  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.
