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.
Willis, Simple groups of automorphisms of trees determined by their actions on finite subtrees, 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.