Character Rigidity and Ergodic Actions of Non-uniform Higher Rank Lattices

The theory of rigidity for lattices in higher rank semisimple Lie groups is a powerful and exciting subject, combining methods from algebra, number theory, geometry and dynamics. One of the most celebrated results is Margulis' normal subgroup theorem, which states that any normal subgroup of a higher rank lattice is either central or of finite index. Conjectures due to Connes, as well as Stuck-Zimmer, suggest significant strengthenings of this statement in the context of ergodic theory and representation theory of higher rank lattices. Connes' conjecture states that any extremal, positive definite, normalised central function (also known as a character) defined on a lattice is either the trace of a finite dimensional representation, or vanishes off the center.

I will present a recent joint work with Glasner, Gorfine, Hanany and Levit which proves this conjecture for non-uniform lattices in products of rank-1 groups over arbitrary local fields of uneven characteristic, as well as Lie groups with infinite center. Together with previous results of other authors, this fully establishes Connes and Stuck-Zimmer's conjectures for all non-uniform lattices.