A rigidity theorem and a stability theorem for two-step nilpotent lie groups

Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup & Gamma; for the homogeneous space script:X sign = G/H and any deformation of Γ, the deformed discrete subgroup may fail to be discontinuous for script:X sign. To understand this phenomenon in the case when G is a two-step nilpotent Lie group, we provide a stratification of the deformation space of the action of Γ on script:X sign, which depends upon the dimensions of G-adjoint orbits. As a direct consequence, a rigidity Theorem is given and a certain sufficient condition for the stability property is derived. We also discuss the Hausdorff property of the associated deformation space.