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.