Commutators, commensurators, and ${\mathrm{PSL}}_2(\mathbb{Z})$

Abstract

Let $H<{\mathrm{PSL}}_2(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $\Phi (H)\ne H$ denote a term of the lower central series or the derived series of $H$. In this paper, we prove that the commensurator of $\Phi (H)$ 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.

20 pages, appendix removed. To appear in the Journal of Topology

References in corpus (1)

• Notes on Thin Matrix Groups

Cited by in corpus (1)

• Commensurators of thin normal subgroups and abelian quotients