Classification of Smooth Actions by Higher Rank Lattices in Critical Dimensions

The Zimmer program asks how lattices in higher rank semisimple Lie groups may act smoothly on compact manifolds. Below a certain critical dimension, the recent proof of the Zimmer conjecture by Brown-Fisher-Hurtado asserts that, for SL(n,R) with n\geq 3 or other higher rank R-split semisimple Lie groups, the action is trivial up to a finite group action. In this talk, we will explain what happens in the critical dimension for higher rank R-split semisimple Lie groups. For example, non-trivial actions by lattices in SL(n,R), n\geq 3, on (n-1)-dimensional manifolds are isomorphic to the standard action on RP^{n-1} up to a finite quotient group and a finite covering. This is a joint work with Aaron Brown and Federico Rodriguez Hertz.