  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
$\mathrm{PSL}_2(\mathbb{R})$.
