## Rigidity of Discontinuous Actions on Diamond Homogeneous Spaces

J. Math. Sci. Univ. Tokyo
Vol. 23 (2016), No. 2, Page 381--403.

K\'edim, Imed
Rigidity of Discontinuous Actions on Diamond Homogeneous Spaces
Let $G=\mathbb R^n \ltimes H_{2n+1}$ be the diamond group, $H$ a closed Lie subgroup of $G$ and $\Gamma$ a discontinuous subgroup for the homogeneous space $G/H$. We study in this paper some rigidity properties of the discontinuous action of $\Gamma$ on $G/H$. Namely, we show that the strong local rigidity fails to hold on the corresponding parameter space. In addition, when particularly $H$ is dilation-invariant, the local rigidity also fails to hold.