Some Problems of deformations on three-step nilpotent Lie groups

Abstract

Let $G$ be an exponential solvable Lie group and $H$ a connected Lie subgroup of $G$. Given any discontinuous group $\mathit{\Gamma}$ for the homogeneous space $\mathscr M = G/H$ and any deformation of $\mathit{\Gamma}$, deformation of discrete subgroups may destroy proper discontinuity of the action on $\mathscr M$ 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 $\mathscr T(\mathit{\Gamma}, G, H)$ 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 $\mathscr T(\mathit{\Gamma}, G, H)$.