A criterion of proper action on some compact extensions of \(\mathbb{R}^n\) and applications. (English) Zbl 07668717
Let \(G\) be the semi-direct product group \(K \rtimes\mathbb R^n\) where \(K\) designates a compact subgroup \(K\) of \(GL(n, \mathbb R)\), let \(H\) be a closed subgroup of \(G\) and \(\Gamma\) be a discontinuous group for the homogeneous space \(\Xi= G/H\). In this work, the authors establish a geometrical criterion of the proper action of \(\Gamma\) on \(\Xi\), which requires an accurate description of the structure of closed connected subgroups of Euclidean motion groups. To this end, many elementary linear algebra results appear and they record some facts about discrete subgroups of Euclidean motion groups. As a consequence, they establish a criterion for the existence of a compact Clifford-Klein form \(\Gamma\backslash \Xi\) and study the Calabi-Markus phenomenon.
Reviewer: Béchir Dali (Bizerte)
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 |
References:
| [1] | Arhangel’skii, A. and Tkachenko, M., Topological Groups and Related Structures: An Introduction to Topological Algebra, , Vol. 1 (Springer Science and Business Media, 2008). · Zbl 1323.22001 |
| [2] | Baklouti, A. and Bejar, S., On Calabi-Markus phenomenon and a rigidity theorem for Euclidean motion groups, Kyoto. J. Math.56(2) (2016) 325-346. · Zbl 1404.22019 |
| [3] | Baklouti, A. and Bejar, S., Variants of stability concepts for Euclidean motion groups, Int. J. Math.28(6) (2017) 1750046. · Zbl 1370.57016 |
| [4] | Baklouti, A., Bejar, S. and Fendri, R., A rigidity theorem for finite actions on Lie groups and application to compact extensions of \(\mathbb{R}^n,\) Kyoto J. Math.59(3) (2019) 607-618. · Zbl 1427.22009 |
| [5] | Baklouti, A., Bejar, S. and Dhahri, K., Deforming discontinuous groups for Heisenberg motion groups, Int. J. Math.30(9) (2019) 1950045. · Zbl 1471.22006 |
| [6] | Baklouti, A., Deformation Theory of Discontinuous Groups, , Vol. 72 (De Gruyter, 2022). · Zbl 1542.22001 |
| [7] | Bourbaki, N., Elements of Mathematics, Lie Groups and Lie Algebras: Chapters 1-3 (Springer Science & Business Media, 1989). · Zbl 0672.22001 |
| [8] | Hilgert, J. and Neeb, K.-H., Structure and Geometry of Lie Groups, (Springer Science, Business Media, LLC, 2012). https://doi.org/10.1007/978-0-387-84794-8. · Zbl 1229.22008 |
| [9] | Kobayashi, T., Proper action on homogeneous space of reductive type, Math. Ann.285 (1989) 249-263. · Zbl 0662.22008 |
| [10] | Kobayashi, T., Discontinuous groups acting on homogeneous spaces of reductive type, in Proc. Conf. Representation Theorie of Lie Groups and Lie Algebras () (World Scientific, 1992), pp. 59-75. · Zbl 1193.22010 |
| [11] | Kobayashi, T., On discontinuous groups on homogeneous space with non-compact isotropy subgroups, J. Geom. Phys.12 (1993) 133-144. · Zbl 0815.57029 |
| [12] | Kobayashi, T., Criterion for proper actions on homogeneous spaces of reductive groups, J. Lie Theory6 (1996) 147-163. · Zbl 0863.22010 |
| [13] | Kobayashi, T., Discontinuous groups for non-Riemannian homogeneous space, Mathematics Unlimited-2001 and Beyond, eds. Engquist, B. and Schmid, W. (Springer, 2001), pp. 723-747. · Zbl 1023.53031 |
| [14] | Kobayashi, T. and Yoshino, T., Compact Clifford-Klein forms of symmetric spaces — Revisited, Pure Appl. Math. Quart.1(13) (2005) 591-663. · Zbl 1145.22011 |
| [15] | \(Wü\) stner, M., A connected Lie group equals the square of the exponential image, J. Lie Theory13 (2003) 307-309. · Zbl 1022.22005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
