Deforming discontinuous subgroups of reduced Heisenberg groups

Abstract

Let G=H2n+1r be the (2n+1)-dimensional reduced Heisenberg group, and let H be an arbitrary connected Lie subgroup of G. Given any discontinuous subgroup Γ⊂G for G/H, we show that resulting deformation space T(Γ,G,H) of the natural action of Γ 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 R(Γ,G,H) admits a rigid point if and only if Γ is finite.