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

TL;DR

This paper investigates the structure of certain subgroups of PSL(2,Z), proving that their commensurators in PSL(2,R) are discrete, which contributes to understanding the nature of thin subgroups and supports a conjecture of Shalom.

Contribution

It demonstrates that for specific subgroups derived from principal congruence subgroups, their commensurators are discrete, providing new examples of thin subgroups with this property.

Findings

The commensurator of er(H) in PSL(2,R) is discrete.

Identifies a family of thin subgroups with discrete commensurators.

Supports cases of Shalom's conjecture.

Abstract

Let $H<\mathrm{PSL}_2(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $\Phi(H)\neq H$ 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.