Some problems of deformations on three-step nilpotent Lie groups

Let G be an exponential solvable Lie group and H a connected Lie subgroup of G. Given any discontinuous group Γ for the homogeneous space M = G/H and any deformation of Γ, deformation of discrete subgroups may destroy proper discontinuity of the action on M as H is not compact (except the case when it is trivial). To interpret this phenomenon in the case when G is a 3-step nilpotent, we provide a layering of Kobayashi’s deformation space T(Γ; G; H) into Hausdorff spaces, which depends upon the dimensions of G-adjoint orbits of the corresponding parameter space. This allows us to establish a Hausdorffness theorem for T(Γ; G; H).