Zimmer's conjecture for non-split semisimple Lie groups

TL;DR

This paper advances Zimmer's conjecture for non-split semisimple Lie groups by establishing new cases using measure rigidity and cocycle superrigidity techniques to determine minimal manifold dimensions for group actions.

Contribution

It introduces novel methods to analyze invariant measures and provides new lower bounds on manifold dimensions for non-split semisimple Lie group actions, extending Zimmer's conjecture.

Findings

New cases of Zimmer's conjecture proved for non-$ ext{ extbackslash}mathbb{R}$-split groups

Lower bounds on manifold dimensions established using measure rigidity

Techniques involving cocycle superrigidity and Lyapunov distributions developed

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.