Title: Character Rigidity for Non-Uniform Lattices
Abstract:
I will discuss a joint work with Alon Dogon, Michael Glasner, Liam Hanany and Arie Levit.
It has been a long-standing open problem to extend the Nevo-Stuck-Zimmer theorem and character rigidity for all higher-rank lattices, without any property (T) assumption. The Nevo-Stuck-Zimmer result ensures that every ergodic pmp action of a higher-rank lattice with property (T) is either essentially transitive or essentially free. Character rigidity is the claim that every character of such a lattice is either supported on the center, or has a finite dimensional GNS representation. Up until today, these results were generally proved only for lattices in higher-rank semisimple groups that have some factor with property (T).
Our result establishes these theorems for all higher-rank non-uniform lattices. The proof extends ideas behind a remarkable result of Peterson and Thom, which proves character rigidity for some cases of irreducible lattices in products of rank-1 groups.