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. Let K < Gamma < G be an infinite normal subgroup of an arithmetic lattice Gamma in a rank-one simple Lie group G , such that the quotient Q = Gamma/ 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 2021 result of the authors, which showed that many natural normal subgroups of PSL2(Z) have discrete commensurator in PS2 (R) .