Commensurators of normal subgroups of lattices

TL;DR

This paper investigates the arithmeticity of normal subgroups within lattices in semisimple Lie groups, providing a positive answer to a question about dense commensurators and extending previous results.

Contribution

It generalizes prior work by proving that normal subgroups of lattices have dense commensurators are arithmetic, broadening understanding of subgroup structures in Lie groups.

Findings

Normal subgroups of lattices have dense commensurators and are arithmetic.

Extends previous results from specific cases to more general settings.

Provides a positive answer to Greenberg-Shalom's question in this context.

Abstract

We study a question of Greenberg-Shalom concerning arithmeticity of discrete subgroups of semisimple Lie groups with dense commensurators. We answer this question positively for normal subgroups of lattices. This generalizes a result of the second author and T. Koberda for certain normal subgroups of arithmetic lattices in SO(n,1) and SU(n,1).