Thomas Koberda, Mahan Mj

Let $H<{\mathrm{PSL}}_{2}(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $\mathrm{\Phi}(H)\ne H$ denote a term of the lower central series or the derived series of $H$. In this paper, we prove that the commensurator of $\mathrm{\Phi}(H)$ in ${\mathrm{PSL}}_{2}(\mathbb{R})$ is discrete. We thus obtain a natural family of thin subgroups of ${\mathrm{PSL}}_{2}(\mathbb{R})$ whose commensurators are discrete, establishing some cases of a conjecture of Shalom.