Kyoto Journal of Mathematics

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

Ali Baklouti, Souhail Bejar, and Ramzi Fendri

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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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 57S30: Discontinuous groups of transformations

Lie group Euclidean motion group proper action discontinuous group rigidity


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.

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