Rigidity theorems show cocompact lattices are the unique infinite finitely generated commensurated subgroups up to commensurability in their commensurators, including for automorphism groups of trees.
Reid, Distal actions on coset spaces in totally disconnected locally compact groups, J
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Lattices determined by their commensurator
Rigidity theorems show cocompact lattices are the unique infinite finitely generated commensurated subgroups up to commensurability in their commensurators, including for automorphism groups of trees.