Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert

Mickaël Crampon; Ludovic Marquis

doi:10.5802/ambp.330

Mickaël Crampon ¹ ; Ludovic Marquis ²

¹ Universidad de Santiago de Chile, Departamento de Matemática y Ciencia de la Computación, Av. Las Sophoras 173 - Estación Central, Santiago de Chile Chile
² IRMAR 263 Av. du Général Leclerc CS 74205 35042 Rennes Cedex France

Annales mathématiques Blaise Pascal, Tome 20 (2013) no. 2, pp. 363-376.

Résumé
Abstract

On montre un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert. Plus précisément, en toute dimension $n$ , il existe une constante $ε_{n} > 0$ telle que, pour tout ouvert proprement convexe $Ω$ , pour tout point $x \in Ω$ , tout groupe discret engendré par un nombre fini d’automorphismes de $Ω$ qui déplacent le point $x$ de moins de $ε_{n}$ est virtuellement nilpotent.

We prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant $ε_{n} > 0$ such that, for any properly convex open set $Ω$ and any point $x \in Ω$ , any discrete group generated by a finite number of automorphisms of $Ω$ , which displace $x$ at a distance less than $ε_{n}$ , is virtually nilpotent.

MR Zbl

DOI : 10.5802/ambp.330

Classification : 22E40, 22F50, 57M99
Mot clés : Géométrie de Hilbert, lemme de Margulis, action géométriquement finie
Keywords: Hilbert’s geometry, lemma of Margulis, action geometrically finite

Affiliations des auteurs :

Mickaël Crampon ¹ ; Ludovic Marquis ²

¹ Universidad de Santiago de Chile, Departamento de Matemática y Ciencia de la Computación, Av. Las Sophoras 173 - Estación Central, Santiago de Chile Chile
² IRMAR 263 Av. du Général Leclerc CS 74205 35042 Rennes Cedex France

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     title = {Un lemme de {Kazhdan-Margulis-Zassenhaus} pour les g\'eom\'etries de {Hilbert}},
     journal = {Annales math\'ematiques Blaise Pascal},
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     publisher = {Annales math\'ematiques Blaise Pascal},
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Mickaël Crampon; Ludovic Marquis. Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert. Annales mathématiques Blaise Pascal, Tome 20 (2013) no. 2, pp. 363-376. doi : 10.5802/ambp.330. https://ambp.centre-mersenne.org/articles/10.5802/ambp.330/

Bibliographie
Cité par

[1] Werner Ballmann; Mikhael Gromov; Viktor Schroeder Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkhäuser Boston Inc., Boston, MA, 1985 | MR | Zbl

[2] Yves Benoist Automorphismes des cônes convexes, Invent. Math., Volume 141 (2000) no. 1, pp. 149-193 | DOI | MR | Zbl

[3] Yves Benoist Convexes divisibles. II, Duke Math. J., Volume 120 (2003) no. 1, pp. 97-120 | DOI | MR | Zbl

[4] Yves Benoist Convexes divisibles. I, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, 2004, pp. 339-374 | MR | Zbl

[5] Yves Benoist Convexes divisibles. III, Ann. Sci. École Norm. Sup. (4), Volume 38 (2005) no. 5, pp. 793-832 | Numdam | MR | Zbl

[6] Yves Benoist Convexes divisibles. IV. Structure du bord en dimension 3, Invent. Math., Volume 164 (2006) no. 2, pp. 249-278 | DOI | MR | Zbl

[7] Yves Benoist Convexes hyperboliques et quasiisométries, Geom. Dedicata, Volume 122 (2006), pp. 109-134 | DOI | MR | Zbl

[8] Jean-Paul Benzécri Sur les variétés localement affines et localement projectives, Bull. Soc. Math. France, Volume 88 (1960), pp. 229-332 | Numdam | MR | Zbl

[9] A. Bosché Symmetric cones, the Hilbert and Thompson metrics, ArXiv e-prints (2012)

[10] E. Breuillard; B. Green; T. Tao The structure of approximate groups, ArXiv e-prints (2011) | MR

[11] Herbert Busemann The geometry of geodesics, Academic Press Inc., New York, N. Y., 1955 | MR | Zbl

[12] Suhyoung Choi Convex decompositions of real projective surfaces. I. $π$ -annuli and convexity, J. Differential Geom., Volume 40 (1994) no. 1, pp. 165-208 http://projecteuclid.org/getRecord?id=euclid.jdg/1214455291 | MR | Zbl

[13] Suhyoung Choi Convex decompositions of real projective surfaces. II. Admissible decompositions, J. Differential Geom., Volume 40 (1994) no. 2, pp. 239-283 http://projecteuclid.org/getRecord?id=euclid.jdg/1214455537 | MR | Zbl

[14] Suhyoung Choi The Margulis lemma and the thick and thin decomposition for convex real projective surfaces, Adv. Math., Volume 122 (1996) no. 1, pp. 150-191 | DOI | MR | Zbl

[15] Suhyoung Choi The deformation spaces of projective structures on 3-dimensional Coxeter orbifolds, Geom. Dedicata, Volume 119 (2006), pp. 69-90 | DOI | MR | Zbl

[16] Suhyoung Choi The convex real projective manifolds and orbifolds with radial ends : the openness of deformations, ArXiv e-prints (2010)

[17] Suhyoung Choi; William Goldman Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc., Volume 118 (1993) no. 2, pp. 657-661 | DOI | MR | Zbl

[18] Suhyoung Choi; William Goldman The classification of real projective structures on compact surfaces, Bull. Amer. Math. Soc. (N.S.), Volume 34 (1997) no. 2, pp. 161-171 | DOI | MR | Zbl

[19] Bruno Colbois; Constantin Vernicos Bas du spectre et delta-hyperbolicité en géométrie de Hilbert plane, Bull. Soc. Math. France, Volume 134 (2006) no. 3, pp. 357-381 | Numdam | MR | Zbl

[20] D. Cooper; D. Long; S. Tillmann On Convex Projective Manifolds and Cusps, ArXiv e-prints (2011)

[21] M. Crampon; L. Marquis Finitude géométrique en géométrie de Hilbert, ArXiv e-prints (2012)

[22] William Goldman Convex real projective structures on compact surfaces, J. Differential Geom., Volume 31 (1990) no. 3, pp. 791-845 http://projecteuclid.org/getRecord?id=euclid.jdg/1214444635 | MR | Zbl

[23] William Goldman Projective geometry on manifolds (2010) (Notes from a course given in 1988)

[24] M. Gromov Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) (London Math. Soc. Lecture Note Ser.), Volume 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1-295 | MR

[25] Pierre de la Harpe On Hilbert’s metric for simplices, Geometric group theory, Vol. 1 (Sussex, 1991) (London Math. Soc. Lecture Note Ser.), Volume 181, Cambridge Univ. Press, Cambridge, 1993, pp. 97-119 | DOI | MR | Zbl

[26] Dennis Johnson; John J. Millson Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven, Conn., 1984) (Progr. Math.), Volume 67, Birkhäuser Boston, Boston, MA, 1987, pp. 48-106 | MR | Zbl

[27] Victor Kac; Èrnest Vinberg Quasi-homogeneous cones, Mat. Zametki, Volume 1 (1967), pp. 347-354 | MR | Zbl

[28] Michael Kapovich Convex projective structures on Gromov-Thurston manifolds, Geom. Topol., Volume 11 (2007), pp. 1777-1830 | DOI | MR | Zbl

[29] D. A. Každan; G. A. Margulis A proof of Selberg’s hypothesis, Mat. Sb. (N.S.), Volume 75 (117) (1968), pp. 163-168 | MR | Zbl

[30] Bas Lemmens; Cormac Walsh Isometries of polyhedral Hilbert geometries, J. Topol. Anal., Volume 3 (2011) no. 2, pp. 213-241 | DOI | MR | Zbl

[31] G. A. Margulis Discrete groups of motions of manifolds of nonpositive curvature, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 2 (1975), pp. 21-34 | MR | Zbl

[32] Ludovic Marquis Espace des modules marqués des surfaces projectives convexes de volume fini, Geom. Topol., Volume 14 (2010) no. 4, pp. 2103-2149 | DOI | MR | Zbl

[33] Ludovic Marquis Exemples de variétés projectives strictement convexes de volume fini en dimension quelconque, Enseign. Math. (2), Volume 58 (2012), pp. 3-47 | DOI | MR

[34] Ludovic Marquis Finite volume convex projective surface. (Surface projective convexe de volume fini.), Ann. Inst. Fourier, Volume 62 (2012) no. 1, pp. 325-392 | DOI | Numdam | MR | Zbl

[35] V. D. Milman; A. Pajor Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$ -dimensional space, Geometric aspects of functional analysis (1987–88) (Lecture Notes in Math.), Volume 1376, Springer, Berlin, 1989, pp. 64-104 | DOI | MR | Zbl

[36] M. S. Raghunathan Discrete subgroups of Lie groups, Springer-Verlag, New York, 1972 (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68) | MR | Zbl

[37] Constantin Vernicos Introduction aux géométries de Hilbert, Actes de Séminaire de Théorie Spectrale et Géométrie. Vol. 23. Année 2004–2005 (Sémin. Théor. Spectr. Géom.), Volume 23, Univ. Grenoble I, Saint, 2005, pp. 145-168 | Numdam | MR | Zbl

[38] Jacques Vey Sur les automorphismes affines des ouverts convexes saillants, Ann. Scuola Norm. Sup. Pisa (3), Volume 24 (1970), pp. 641-665 | Numdam | MR | Zbl

[39] Hans Zassenhaus Beweis eines Satzes über diskrete Gruppen., Abh. math. Sem. Hansische Univ., Volume 12 (1938), pp. 289-312 | DOI | Zbl

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