Stationary actions of higher rank lattices on von Neumann algebras

Abstract: 

I will talk about a recent joint work with Remi Boutonnet in which we show that for higher rank lattices (e.g. SL(3, Z)), the left regular representation is weakly contained in any weakly mixing unitary representation. This strengthens Margulis’ normal subgroup theorem (1978), Stuck-Zimmer’s stabilizer rigidity result (1992) as well as Peterson’s character rigidity result (2014). We also prove that Uniformly Recurrent Subgroups (URS) of higher rank lattices are finite, answering a question of Glasner-Weiss (2014). The main novelty of our work is a structure theorem for stationary actions of higher rank lattices on von Neumann algebras.