Compact quotients of reductive homogeneous spaces.

Let $G$ be be a semisimple Lie group, $H$ a reductive subgroup of $G$ and $\mathrm{\Gamma }$ a discrete subgroup acting properly and cocompactly on $G/H$. I will present a joint work with Fanny Kassel, where we prove that $\mathrm{\Gamma }$ satisfies a sharp form of the Benoist—Kobayashi properness criterion. When $H$ has co-rank $1$ in $G$, it follows that $\mathrm{\Gamma }$ is an Anosov subgroup of $G$ and that small deformations of $\mathrm{\Gamma }$ keep acting properly.