Deforming discontinuous subgroups of reduced Heisenberg groups

Abstract

Let $G={\mathbb{H}}_{2n+1}^r$ be the $(2n+1)$-dimensional reduced Heisenberg group, and let $H$ be an arbitrary connected Lie subgroup of $G$. Given any discontinuous subgroup $\Gamma \subset G$ for $G/H$, we show that resulting deformation space $\mathcal{T}(\Gamma ,G,H)$ of the natural action of $\Gamma$ on $G/H$ is endowed with a smooth manifold structure and is a disjoint union of open smooth manifolds. Unlike the setting of simply connected Heisenberg groups, we show that the stability property holds and that any discrete subgroup of $G$ is stable, following the notion of stability. On the other hand, a local (and hence global) rigidity theorem is obtained. That is, the related parameter space $\mathcal{R}(\Gamma ,G,H)$ admits a rigid point if and only if $\Gamma$ is finite.