Zimmer's conjecture for non-split semisimple Lie groups

Abstract

We prove many new cases of Zimmer's conjecture for actions by lattices in non-$\mathbb{R}$-split semisimple Lie groups $G$. By prior arguments, Zimmer's conjecture reduces to studying certain probability measures invariant under a minimal parabolic subgroup for the induced $G$-action. Two techniques are introduced to give lower bounds on the dimension of a manifold $M$ admitting a non-isometric action. First, when the Levi component of the stabilizer of the measure has higher-rank simple factors, cocycle superrigidity provides a lower bound on the dimension of $M$. Second, when certain fiberwise coarse Lyapunov distributions are one-dimensional, a measure rigidity argument provides additional invariance of the measure if the associated root spaces are higher-dimensional.