On the geometry of stable discontinuous subgroups acting on threadlike homogeneous spaces (Q650496)

Let \(G\) be an (\(n\)-step) threadlike Lie group, i.e., its Lie algebra admits a basis \(\{X,Y_1,\dots,Y_n\}\) with nontrivial Lie brackets \([X,Y_i]=Y_{i+1}\) (for \(1\leq i \leq n-1\)). Let \(G/H\) be a homogeneous space, and let \(\Gamma\) be a non-abelian discontinuous subgroup of \(G\) acting on \(G/H\). The problem of describing deformations was introduced by \textit{T. Kobayashi} [J. Geom. Phys. 12, No. 2, 133--144 (1993; Zbl 0815.57029)]. In the present paper, the authors show that any non-abelian discrete subgroup is stable. One consequence is that any resulting deformation space \({\mathcal I} (\Gamma, G, H)\) is a Hausdorff space, where \(\Gamma\) acts on the threadlike homogeneous space \(G/H\) as a discontinuous subgroup. Whenever the rank of \(\Gamma\) is bigger than three, this space is also shown to be endowed with a smooth manifold structure. But if \(k=3\), then \({\mathcal I} (\Gamma, G, H)\) admits a smooth manifold structure as its open dense subset.