Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and $\Gamma$ a discontinuous group for the homogeneous space $\mathscr{X}=G/H$, which means that $\Gamma$ is a discrete subgroup of $G$ acting properly discontinuously and fixed point freely on $\mathscr{X}$. The subject of the talk is to to deal with some questions related to the geometry of the parameter and the deformation spaces of the action of $\Gamma$ on $\mathscr{X}$, when the group $G$ is solvable. The local rigidity conjecture in the nilpotent case and the analogue of the Selberg-Weil-Kobayashi rigidity Theorem in such non-Riemannian setting is also discussed.
Fanny Kassel