Title: Smooth ergodic theory and rigidity of lattice actions
Abstract: In joint work with David Fisher and Sebastian Hurtado, we recently established Zimmer’s conjecture on the finiteness of lattice actions: for n>=3, given a lattice subgroup of SL(n, R), and action on a manifold of dimension at most (n-2) factors through the action of a finite group.
In this course, I will outline the proof of the above theorem in the case of cocompact lattices. Using Strong Property (T) and the superrigidity theorem of Margulis, we will reduce the above theorem to our main theorem, a statement on the slow growth of the derivative cocycle.
Our main theorem is a theorem in smooth dynamics whose proof employs many tools from classical smooth ergodic theory adapted to actions of higher-rank abelian groups including Lyapunov exponents, invariant manifolds, and metric entropy. After explaining the statement of the main theory and the reductions mentioned above, I will present a complete proof of our main theorem using the above tools from smooth ergodic theory.