Kahn and Markovic solved the surface subgroup problem and Ehrenpreis conjecture making use of exponential mixing of the geodesic flow on compact hyperbolic manifolds. The geodesic flow may be thought of as homogeneous dynamics on a diagonal subgroup $H< G$ acting on $G/\Gamma$, although the proof by Cal Moore only refers to the unit tangent bundle (which is where the geodesic flow lives). At least in the case of $PSL_2(\mathbb{R})$ (for hyperbolic surfaces), this is the same thing.

In turn, the surface subgroup problem tells us many more interesting things about $\Gamma$ when $\Gamma$ is a closed hyperbolic 3-manifold group.