Liam Hanany

Character rigidity of higher-rank lattices

n the 1970s, Grigory Margulis established his normal subgroup theorem, which asserts that for any irreducible lattice in a higher-rank semi-simple Lie group, every normal subgroup is either finite-index or central. Since then, this result has been extended and generalized in a variety of directions. One example is a dynamical result of Garrett Stuck and Robert J. Zimmer, which shows that under the additional assumption of Kazhdan's property (T), every ergodic probability-measure-preserving action of such lattices is either essentially transitive or essentially free. It has since been conjectured that the same result holds without the property (T) assumption.

I will give a brief overview of these results, and will then talk about a joint work with Alon Dogon, Michael Glasner, Yuval Gorfine and Arie Levit. We prove that non-uniform irreducible lattices in higher-rank semi-simple Lie groups are character rigid and deduce the conclusion of the Stuck-Zimmer theorem in this case.