Stationary measures and invariant subsets of homogeneous spaces (III)

Abstract

Let $G$ be a real Lie group, $\Lambda$ be a lattice in $G$ and $\Gamma$ be a compactly generated closed subgroup of $G$. If the Zariski closure of the group $\mathrm{Ad} (\Gamma)$ is semisimple with no compact factor, we prove that every $\Gamma$-orbit closure in $G/\Lambda$ is a finite volume homogeneous space. We also establish related equidistribution properties.