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arXiv · 2411.13858

Zimmer's conjecture for non-split semisimple Lie groups

Jinpeng An·Aaron Brown·Zhiyuan Zhang
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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.

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Jinpeng An, Aaron Brown, Zhiyuan Zhang. 2024-11-21. Zimmer's conjecture for non-split semisimple Lie groups. https://arxiv.org/abs/2411.13858

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