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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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.

Article information

Source
Kyoto J. Math., Advance publication (2019), 12 pages.

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

Subjects
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

Keywords
Lie group Euclidean motion group proper action discontinuous group rigidity

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., advance publication, 11 June 2019. doi:10.1215/21562261-2019-0018. https://projecteuclid.org/euclid.kjm/1560218486


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