Lattices determined by their commensurator

Abstract

Let $\Gamma$ be a finitely generated cocompact lattice of a totally disconnected locally compact group $G$, and $C$ a dense subgroup of $G$ that contains and commensurates $\Gamma$. We study the problem of describing all finitely generated commensurated subgroups of $C$. We establish general rigidity results ensuring every finitely generated commensurated subgroup of $C$ is virtually contained in $\Gamma$. In more concrete situations, in fact we conclude that up to commensurability, $\Gamma$ is the only infinite finitely generated commensurated subgroup of $C$. For instance this last conclusion holds when $G$ is the automorphism group of a tree. This settles in particular the problem whether two non-commensurable cocompact tree lattices may have the same commensurator. Further applications include commensurators of cocompact lattices in other groups of automorphisms of trees, as well as commensurators of graph product of finite groups in automorphism groups of right-angled building.