Commutators, commensurators, and ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$

Let $H<{\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $\mathrm{\Phi }(H)\mathrm{\ne }H$ 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 ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$ is discrete. We thus obtain a natural family of thin subgroups of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R})$ whose commensurators are discrete, establishing some cases of a conjecture of Shalom.