A local rigidity theorem for finite actions on Lie groups and application to compact extensions of \(\mathbb{R}^n\). (English) Zbl 1427.22009
Summary: 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 \(\mathbb{R}^n\).
MSC:
| 22E27 | Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) |
| 22E40 | Discrete subgroups of Lie groups |
| 57S30 | Discontinuous groups of transformations |
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