Deforming discontinuous subgroups for threadlike homogeneous spaces

Abstract

Let G be an exponential solvable Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for the homogeneous space \({\fancyscript{M}=G/H}\) and any deformation of Γ, the deformed discrete subgroup may utterly destroy its proper discontinuous action on \({\fancyscript{M}}\) as H is not compact (except the case when it is trivial). To understand this specific issue, we provide an explicit description of the parameter and the deformation spaces of any abelian discrete Γ acting properly discontinuously and fixed point freely on G/H for an arbitrary H of a threadlike nilpotent Lie group G. The topological features of deformations, such as the local rigidity and the stability are also discussed. Whenever the Clifford-Klein form Γ\G/H in question is assumed to be compact, these spaces are cutely determined and unlike the case of Heisenberg groups, the deformation space fails in general to be a Hausdorff space. We show further that this space admits a smooth manifold as its open dense subset.