Appears in collection : 2024 - T2 - WS3 - Actions of large groups, geometric structures, and the Zimmer program

I will discuss a question raised independently by Greenberg and Shalom: Can an infinite discrete subgroup of a simple Lie group have dense commensurator and not be a lattice? I will explain the surprising connections between this question and other long-standing open problems, and discuss recent progress on special cases of the question. This is joint work with (subsets of) Brody, Fisher, and Mj.