Commensurators of thin normal subgroups and abelian quotients

Abstract

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 $K<\Gamma<G$ 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 $\mathbb{Z}$. 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, which showed that many natural normal subgroups of $\mathrm{PSL}_2(\mathbb{Z})$ have discrete commensurator in $\mathrm{PSL}_2(\mathbb{R})$.