Surface Projective Convexe de volume fini
[Convex projective surface of finite volume]
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 325-392.

A convex projective surface is the quotient of a properly convex open Ω of the projective real space 2 () by a discrete subgroup Γ of SL 3 (). We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann’s measure. We deduce that, if Ω is not a triangle, then Ω is strictly convex, with 𝒞 1 boundary and that a convex projective surface S is of finite volume if and only if the dual surface is of finite volume.

Une surface projective convexe est le quotient d’un ouvert proprement convexe Ω de l’espace projectif réel 2 () par un sous-groupe discret Γ de SL 3 (). Nous donnons plusieurs caractérisations du fait qu’une surface projective convexe est de volume fini pour la mesure de Busemann. On en déduit que si Ω n’est pas un triangle alors Ω est strictement convexe, à bord 𝒞 1 et qu’une surface projective convexe S est de volume fini si et seulement si la surface duale est de volume fini.

DOI: 10.5802/aif.2707
Classification: 57M50,  52C20,  22E40
Keywords: Surface, Hilbert’s geometry, Hyperbolic geometry, Lattice, Discrete subgroup of Lie group
Marquis, Ludovic  1

1 tabacckludge ’Ecole Normale Supérieure de Lyon Unité de Mathématiques Pures et Appliquées 46, allée d’Italie 69364 Lyon Cedex 07 (France)
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Marquis, Ludovic . Surface Projective Convexe de volume fini. Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 325-392. doi : 10.5802/aif.2707. https://aif.centre-mersenne.org/articles/10.5802/aif.2707/

[1] Benoist, Yves Automorphismes des cônes convexes, Invent. Math., Volume 141 (2000), pp. 149-193 | DOI | MR

[2] Benoist, Yves Convexes hyperboliques et fonctions quasisymétriques, Publ. Math. IHES, Volume 97 (2003), pp. 181-237 | DOI | Numdam | MR

[3] Benoist, Yves Convexes hyperpoliques et quasiisométries, Geometriae Dedicata, Volume 122 (2006), pp. 109-134 | DOI | MR

[4] Benoist, Yves Sous-groupes discrets des groupes de Lie, European Summer School in Group Theory Luminy (7-18 July 1997)

[5] Benzécri, Jean-Paul Sur les variétés localement affines et localement projectives, Bulletin de la Société Mathématique de France, Volume 88 (1960), pp. 229-332 | Numdam | MR | Zbl

[6] Borel, Armand Compact Clifford-Klein forms of symmetric spaces, Topology, Volume 2 (1963), pp. 111-122 | DOI | MR | Zbl

[7] Burago, Dmitri; Burago, Yuri; Ivanov, Sergei A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001 | MR

[8] Choi, Suhyoung Convex decompositions of real projective surfaces. II : Admissible decompositions, J. Differential Geom, Volume 40 (1994), pp. 239-283 | MR | Zbl

[9] Colbois, Bruno; Vernicos, Constantin; Verovic, Patrick L’aire des triangles idéaux en géométrie de Hilbert, L’enseignement mathématique, Volume 50 (2004), pp. 203-237 | MR

[10] Colbois, Bruno; Vernicos, Constantin; Verovic, Patrick Area of ideal triangles and gromov hyperbolicity in hilbert geometry, A paraître

[11] Goldman, William Convex real projective structures on compact surfaces, J. Differential Geom., Volume 31 (1990), pp. 791-845 | MR | Zbl

[12] Johnson, D.; Millson, John Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in Geometry and Analysis Progr. Math, Volume 67 (1984), pp. 48-106 | MR | Zbl

[13] Kac, Victor; Vinberg, Ernest Borisovich Quasi-homogeneous cones, Math. Notes, Volume 1 (1967), pp. 231-235 | DOI | Zbl

[14] Kapovich, Misha Convex projective structures on Gromov-Thurston manifolds, Geometry and Topology, Volume 11 (2007), pp. 1777-1830 | DOI | MR

[15] Lee, Jaejeong Convex fundamental domains for properly convex real projective structures (preprint)

[16] Marquis, Ludovic Espace de Modules Marqués des Surfaces Projectives Convexes de Volume Fini (preprint arxiv.org/abs/0910.5839)

[17] Richards, Ian On the classification of noncompact surfaces, Trans. Amer. Math. Soc., Volume 106 (1963), pp. 259-269 | DOI | MR | Zbl

[18] Vey, Jacques Sur les automorphismes affines des ouverts convexes saillants, Anna Scuola Normale Superiore di Pisa, Volume 24 (1970), pp. 641-665 | Numdam | MR | Zbl

[19] Vinberg, Èrnest Borisovich The theory of convex homogeneous cones, Trudy Moskov. Mat. Obšč., Volume 12 (1963), pp. 303-358 | MR | Zbl

[20] Vinberg, Èrnest Borisovich The structure group of automorphisms of a homogeneous convex cone, Trudy Moskov. Mat. Obšč., Volume 13 (1965), pp. 56-81 | MR | Zbl

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