I will discuss a question raised independently by Greenberg and Shalom: can a discrete subgroup of a simple Lie group have dense commensurator and not be a lattice? Both Greenberg and Shalom seem to believe the answer is yes, so we call this the Greenberg-Shalom hypothesis. The hypothesis turns out to imply answers to many long standing questions in group theory, geometry and topology. I'll start off with a simple question about pairs of 2 by 2 matrices first raised in the 50's and 60's by Sanov and Lyndon-Ullman and use that to motivate the idea of commensuration and the Greenberg-Shalom hypothesis.

The talk should be broadly accessible. Parts of the talk represent joint work with (subsets of) Brody, Mj and van Limbeek.