On the irreducible action of $ \text{PSL}(2,\mathbb{R})$ on the 3-dimensional Einstein universe

Masoud Hassani

doi:10.1016/j.crma.2017.10.003

Group theory/Differential geometry

On the irreducible action of

PSL (2, R)

on the 3-dimensional Einstein universe
[Sur l'action irréductible de

PSL (2, R)

sur l'univers d'Einstein de dimension 3]

Présenté par : the Editorial Board

Masoud Hassani ^1,²

¹ Université d'Avignon, Campus Jean-Henri-Fabre, 301, rue Baruch-de-Spinoza, BP 21239, 84916 Avignon cedex 9, France
² University of Zanjan, Faculty of Science, Department of Mathematics, University blvd, Zanjan, Iran

Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1133-1137.

Résumé
Abstract

Dans cet article, nous étudions l'action irréductible de $PSL (2, R)$ dans $PSL (5, R)$ . Cette action préserve une forme quadratique de signature $(2, 3)$ . Elle sur agit donc conformément sur l'univers d'Einstein $E {in}^{1, 2}$ de dimension 3, ainsi que sur son complément dans $R P^{4}$ . Ce travail complète l'étude préliminaire dans [2], et est un élément de la classification des actions sur $E {in}^{1, 2}$ de cohomogenéité un [5].

In this paper, we study the irreducible representation of $PSL (2, R)$ in $PSL (5, R)$ . This action preserves a quadratic form with signature $(2, 3)$ . Thus, it acts conformally on the 3-dimensional Einstein universe $E {in}^{1, 2}$ . We describe the orbits induced in $E {in}^{1, 2}$ and its complement in $R P^{4}$ . This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on $E {in}^{1, 2}$ [5].

Reçu le : 2017-06-26
Accepté le : 2017-10-03
Publié le : 2017-10-19

DOI : 10.1016/j.crma.2017.10.003

Affiliations des auteurs :

Masoud Hassani ^{1, 2}

¹ Université d'Avignon, Campus Jean-Henri-Fabre, 301, rue Baruch-de-Spinoza, BP 21239, 84916 Avignon cedex 9, France
² University of Zanjan, Faculty of Science, Department of Mathematics, University blvd, Zanjan, Iran

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     author = {Masoud Hassani},
     title = {On the irreducible action of $ \text{PSL}(2,\mathbb{R})$ on the 3-dimensional {Einstein} universe},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1133--1137},
     publisher = {Elsevier},
     volume = {355},
     number = {11},
     year = {2017},
     doi = {10.1016/j.crma.2017.10.003},
     language = {en},
}

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Masoud Hassani. On the irreducible action of $ \text{PSL}(2,\mathbb{R})$ on the 3-dimensional Einstein universe. Comptes Rendus. Mathématique, Volume 355 (2017) no. 11, pp. 1133-1137. doi : 10.1016/j.crma.2017.10.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.10.003/

Bibliographie
Cité par

[1] T. Barbot; V. Charette; T. Drumm; W.M. Goldman; K. Melnick A primer on the ( $2 + 1$ ) Einstein universe (D.V. Alekseevsky; H. Baum, eds.), Recent Developments in Pseudo-Riemannian Geometry, ESI Lectures in Mathematics and Physics, 2008, pp. 179-229

[2] B. Collier; N. Tholozan; J. Toulisse The geometry of maximal representations of surface groups into $S O (2, n)$ | arXiv

[3] A.J. Di Scala; T. Leistner Connected subgroups of $S O (2, n)$ acting irreducibly on $R^{2, n}$ , Isr. J. Math., Volume 182 (2011), pp. 103-122

[4] C. Frances Géometrie et dynamique lorentziennes conformes, ENS Lyon, France, 2002 (PhD thesis)

[5] M. Hassani, Cohomogeneity one actions on the three-dimensional Einstein universe, in preparation.

Cité par Sources :

^☆ This work is funded by the French Ministry of Foreign Affairs, through Campus France. Grant number 878286E. It is also part of project MATH AMSUD 2017, Project No. 38888QB – GDAR.

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