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June 2016 On the Calabi–Markus phenomenon and a rigidity theorem for Euclidean motion groups
Ali Baklouti, Souhail Bejar
Kyoto J. Math. 56(2): 325-346 (June 2016). DOI: 10.1215/21562261-3478898


In this article, we study the rigidity properties of deformation parameters of the natural action of a discontinuous subgroup ΓG, on a homogeneous space G/H, where H stands for a closed subgroup of a Euclidean motion group G:=On(R)Rn. That is, we prove the following local (and global) rigidity theorem: the parameter space admits a rigid (equivalently a locally rigid) point if and only if Γ is finite. Remarkably, it turns out that H is compact whenever Γ is infinite, which makes accessible the study of the corresponding parameter and deformation spaces and their topological and local geometrical features. This shows that the Calabi–Markus phenomenon occurs in this setting. That is, if H is a closed noncompact subgroup of G, then G/H does not admit a compact Clifford–Klein form, unless G/H itself is compact. We also answer a question posed by T. Kobayashi. That is, no homogeneous space G/H admits a noncommutative free group as a discontinuous group.


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Ali Baklouti. Souhail Bejar. "On the Calabi–Markus phenomenon and a rigidity theorem for Euclidean motion groups." Kyoto J. Math. 56 (2) 325 - 346, June 2016.


Received: 30 September 2014; Revised: 24 December 2014; Accepted: 11 March 2015; Published: June 2016
First available in Project Euclid: 10 May 2016

zbMATH: 06591222
MathSciNet: MR3500844
Digital Object Identifier: 10.1215/21562261-3478898

Primary: 22E27
Secondary: 22E40, 57S30

Rights: Copyright © 2016 Kyoto University


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Vol.56 • No. 2 • June 2016
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