Lattices determined by their commensurator

TL;DR

This paper investigates the structure of finitely generated commensurated subgroups within certain dense subgroups of totally disconnected groups, establishing rigidity results that identify these subgroups with the original lattice in various geometric contexts.

Contribution

It proves that under broad conditions, finitely generated commensurated subgroups are virtually contained in the original lattice, often uniquely determined up to commensurability.

Findings

Every finitely generated commensurated subgroup of C is virtually contained in Γ.

In specific cases, Γ is the only infinite finitely generated commensurated subgroup of C up to commensurability.

The results apply to automorphism groups of trees and other geometric structures, settling related classification problems.

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.