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.
Zimmer, Linear representations and arithmeticity of lattices in products of trees, Essays in geometric group theory, Ramanujan Math
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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.