## A rigidity Theorem and a Stability Theorem for two-step nilpotent Lie groups

J. Math. Sci. Univ. Tokyo
Vol. 19 (2012), No. 3, Page 281--307.

Baklouti, Ali; ElAloui, Nasreddine;K\'edim, Imed
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 $\mathscr{X}=G/H$ and any deformation of $\Gamma$, the deformed discrete subgroup may fail to be discontinuous for $\mathscr{X}$. 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 $\Gamma$ on $\mathscr{X}$, 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.