Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds

Abstract

We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is smaller than $r(G)$, then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most $m(G)$, we show there is a quasi-invariant measure on the manifold such that the action is measurable isomorphic to a relatively measure preserving action over a standard boundary action.

arXiv commentMinor corrections and improvements in exposition

References in corpus (3)

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• Growth of maps, distortion in groups and symplectic geometry
• Groups acting on manifolds: around the Zimmer program

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• Conformal actions of higher rank lattices on compact pseudo-Riemannian manifolds
• Smooth Factors of Projective Actions of Higher Rank Lattices and Rigidity
• Superrigidity, arithmeticity, normal subgroups: results, ramifications and directions
• On classification of higher rank Anosov actions on compact manifold
• Zimmer's conjecture for lattice actions: the $\mathrm{SL}(n,\mathbb{C})$-case