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 Γ be a finite group. We show in this article that the space Hom(Γ,G)/G is discrete and—in addition—finite if G has finitely many connected components. This means that in the case in which Γ 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 Rn.