Commensurators of thin normal subgroups and abelian quotients

We give an affirmative answer to many cases of a question due to Shalom, which asks if the commensurator of a thin subgroup of a Lie group is discrete.In this paper, let K ă Γ ă G be an infinite normal subgroup of an arithmetic lattice Γ in a rank one simple Lie group G, such that the quotient Q " Γ{K is infinite.We show that the commensurator of K in G is discrete, provided that Q admits a surjective homomorphism to Z.In this case, we also show that the commensurator of K contains the normalizer of K with finite index.We thus vastly generalize a result of the authors 22, which showed that many natural normal subgroups of PSL 2 pZq have discrete commensurator in PSL 2 pRq.