Stability of discontinuous groups acting on homogeneous spaces (Q722169)

Let \(G/H\) be a homogeneous space of a connected simply connected nilpotent Lie group \(G\), here \(H \subset G\) is a connected Lie subgroup. Suppose that for some discrete subgroup \(\Gamma \subset G\) the natural action on \(G/H\) is free and proper. In this article the parameter space \(\mathcal R(\Gamma,G,H)\) is investigated. With an additional condition on \(\Gamma, G,H\) (which is called superregularity; it is rather complex and formulated on the level of Lie algebras) a stability theorem is proved in the case when \(G\) is a three-step nilpotent Lie group. Then the notion of strong stability on layers (with respect on some layering of \(\text{Hom}(\Gamma, G)\)) is introduced for general triples \((G,H, \Gamma)\) and it is checked for the case of Heisenberg groups \(G\).