Stability of Discontinuous Groups Acting on Homogeneous Spaces

Suppose given a nilpotent connected simply connected Lie group $G$, a connected Lie subgroup $H$ of $G$, and a discontinuous group $\Gamma$ for the homogeneous space $\mathscr{M}=G/H$. In this work we study the topological stability of the parameter space $\mathscr{R}(\Gamma,G,H)$ in the case where $G$ is three-step. We prove a stability theorem for certain particular pairs $(\Gamma,H)$. We also introduce the notion of strong stability on layers making use of an explicit layering of $\mathrm{Hom}(\Gamma,G)$ and study the case of Heisenberg groups.