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 measurably isomorphic to a relatively measure-preserving action over a standard boundary action.

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Authors

Aaron Brown

Northwestern University, Evanston, IL

Federico Rodriguez Hertz

Pennsylvania State University, State College, PA

Zhiren Wang

Pennsylvania State University, State College, PA