Fuchsian groups and compact hyperbolic surfaces

Abstract

We present a topological proof of the following theorem of Benoist-Quint: for a finitely generated non-elementary discrete subgroup $\Gamma\_1$ of PSL$(2, \mathbb R)$ with no parabolics, and for a cocompact lattice $\Gamma\_2$ of PSL$(2, \mathbb R)$, any $\Gamma\_1$ orbit on $\Gamma\_2$ \ PSL$(2, \mathbb R)$ is either finite or dense.