Greenberg-Shalom's Commensurator Hypothesis and Applications

TL;DR

This paper explores the Greenberg-Shalom hypothesis, which posits that infinite discrete subgroups with dense commensurators in semisimple Lie groups are lattices, and discusses its implications and extensions.

Contribution

It analyzes the implications of the Greenberg-Shalom hypothesis and extends its scope to semisimple algebraic groups over various fields.

Findings

Implications of the Greenberg-Shalom hypothesis analyzed

Extension of the hypothesis to algebraic groups over different fields

Discussion of conditions under which subgroups are lattices

Abstract

We discuss many surprising implications of a positive answer to a question raised in some cases by Greenberg in the $`70$s and more generally by Shalom in the early $2000$s. We refer to this positive answer as the Greenberg-Shalom hypothesis. This hypothesis then says that any infinite discrete subgroup of a semisimple Lie group with dense commensurator is a lattice in a product of some factors. For some applications it is natural to extend the hypothesis to cover semisimple algebraic groups over other fields as well.