Abstract
Following the notion of stability introduced by T. Kobayashi and S. Nasrin in [14], we show in the context of a threadlike Lie group G that any non-Abelian discrete subgroup is stable. One consequence is that any resulting deformation space a"(Gamma,G,H) is a Hausdorff space, where I" acts on the threadlike homogeneous space G/H as a discontinuous subgroup. Whenever k = rank(Gamma) > 3, this space is also shown to be endowed with a smooth manifold structure. But if k = 3, then a"(Gamma,G,H) admits a smooth manifold structure as its open dense subset. These phenomena are strongly linked to the features of adjoint orbits of the basis group G on the parameter space a">(Gamma,G,H) (which is semi-algebraic in this case) and specifically to their dimensions, as it will be seen throughout the paper. This also allows to provide a proof of the Local Rigidity Conjecture in this setup.