# 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.

## Authors

Aaron Brown

Department of Mathematics, Northwestern University, Evanston, IL 60208, USA

Federico Rodriguez Hertz

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Zhiren Wang

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA