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 measurably isomorphic to a relatively measure-preserving action over a standard boundary action.
Citation
Aaron Brown. Federico Hertz. Zhiren Wang. "Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds." Ann. of Math. (2) 196 (3) 941 - 981, November 2022. https://doi.org/10.4007/annals.2022.196.3.2
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