On the Calabi–Markus phenomenon and a rigidity theorem for Euclidean motion groups

Abstract

In this article, we study the rigidity properties of deformation parameters of the natural action of a discontinuous subgroup $\Gamma \subset G$, on a homogeneous space $G/H$, where $H$ stands for a closed subgroup of a Euclidean motion group $G:={\mathrm{O}}_n\left(\mathbb{R}\right)⋉{\mathbb{R}}^n$. That is, we prove the following local (and global) rigidity theorem: the parameter space admits a rigid (equivalently a locally rigid) point if and only if $\Gamma$ is finite. Remarkably, it turns out that $H$ is compact whenever $\Gamma$ is infinite, which makes accessible the study of the corresponding parameter and deformation spaces and their topological and local geometrical features. This shows that the Calabi–Markus phenomenon occurs in this setting. That is, if $H$ is a closed noncompact subgroup of $G$, then $G/H$ does not admit a compact Clifford–Klein form, unless $G/H$ itself is compact. We also answer a question posed by T. Kobayashi. That is, no homogeneous space $G/H$ admits a noncommutative free group as a discontinuous group.