On discontinuous groups acting on (ℍ2n+1r × ℍ2n+1r)/Δ

Let ℍ2n+1r be the reduced Heisenberg Lie group, G = ℍ2n+1r × ℍ2n+1r be the (4n + 2)-dimensional Lie group and ΔG = {(x,x) ∈ G : x ∈ ℍ2n+1r} be the diagonal subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/ΔG, we provide a layering of the parameter space ℛ(Γ, G, ΔG), which is shown to be endowed with a smooth manifold structure, we also show that the stability property holds. On the other hand, a local (and hence a global) rigidity theorem is obtained. That is, the parameter space ℛ(Γ, G, ΔG) admits a rigid point if and only if Γ is finite and this is also equivalent to the fact that the deformation space is Hausdorff.