Lattice subgroups acting on manifolds

Abstract

We discuss recent progress in understanding rigidity properties of smooth actions of higher-rank lattices. We primarily discuss questions of existence in low dimensions (Zimmer’s conjecture), classification in the smallest possible dimension, and further classification assuming dynamical properties of the action. Two common themes arise in the proofs: (1) dynamical properties of the lattice action are mimicked by certain measures on an induced $G$-space; (2) such measures often exhibit additional rigidity properties. Throughout, we state some open problems and possible directions for future research.