## Kyoto Journal of Mathematics

### A local rigidity theorem for finite actions on Lie groups and application to compact extensions of $\mathbb{R}^{n}$

#### Abstract

Let $G$ be a Lie group, and let $\Gamma$ be a finite group. We show in this article that the space $\operatorname{Hom}(\Gamma ,G)/G$ 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 $\mathbb{R}^{n}$.

#### Article information

Source
Kyoto J. Math., Volume 59, Number 3 (2019), 607-618.

Dates
Received: 7 February 2017
Revised: 12 April 2017
Accepted: 25 April 2017
First available in Project Euclid: 11 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1560218486

Digital Object Identifier
doi:10.1215/21562261-2019-0018

Mathematical Reviews number (MathSciNet)
MR3990179

Zentralblatt MATH identifier
07108004

#### Citation

Baklouti, Ali; Bejar, Souhail; Fendri, Ramzi. A local rigidity theorem for finite actions on Lie groups and application to compact extensions of $\mathbb{R}^{n}$. Kyoto J. Math. 59 (2019), no. 3, 607--618. doi:10.1215/21562261-2019-0018. https://projecteuclid.org/euclid.kjm/1560218486

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