A local rigidity theorem for finite actions on Lie groups and application to compact extensions of Rn

Abstract

Let $G$ be a Lie group, and let $\Gamma$ be a finite group. We show in this article that the space $Hom(\Gamma ,G)/G$ is discrete and—in addition—finite if $G$ has finitely many connected components. This means that in the case in which $\Gamma$ is a discontinuous group for the homogeneous space $G/H$, where $H$ is a closed subgroup of $G$, all the elements of Kobayashi’s parameter space are locally rigid. Equivalently, any Clifford–Klein form of finite fundamental group does not admit nontrivial continuous deformations. As an application, we provide a criterion of local rigidity in the context of compact extensions of ${\mathbb{R}}^n$.