Finitude géométrique en géométrie de Hilbert
[Geometrical finiteness in Hilbert geometry]
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2299-2377.

We study the notion of geometrical finiteness for those Hilbert geometries defined by strictly convex sets with 𝒞 1 boundary.

In Gromov-hyperbolic spaces, geometrical finiteness is defined by a property of the group action on the boundary of the space. We show by constructing an explicit counter-example that this definition has to be strenghtened in order to get equivalent characterizations in terms of the geometry of the quotient orbifold, similar to those obtained by Brian Bowditch in hyperbolic geometry.

On étudie la notion de finitude géométrique pour certaines géométries de Hilbert définies par un ouvert strictement convexe à bord de classe 𝒞 1 .

La définition dans le cadre des espaces Gromov-hyperboliques fait intervenir l’action du groupe discret considéré sur le bord de l’espace. On montre, en construisant explicitement un contre-exemple, que cette définition doit être renforcée pour obtenir des définitions équivalentes en termes de la géométrie de l’orbifold quotient, similaires à celles obtenues par Brian Bowditch en géométrie hyperbolique.

DOI: 10.5802/aif.2914
Classification: 22E40,  20F67,  20F65,  53C60
Keywords: Hilbert geometry, geometrical finiteness, Gromov-hyperbolic space, discrete sub-group of Lie groups, convex projective manifold
Crampon, Mickaël 1; marquis, Ludovic 2

1 Universidad de Santiago de Chile Departamento de Matemática Y Ciencia de la Computación Avenida Las Sophoras 173 Estación Central, Santiago de Chile (Chile)
2 Université de Rennes 1 Institut de Recherche Mathématique de Rennes IRMAR - UMR 6625 du CNRS 263, avenue du Général Leclerc, CS 74205 35042 Rennes Cédex (France)
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Crampon, Mickaël; marquis, Ludovic. Finitude géométrique en géométrie de Hilbert. Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2299-2377. doi : 10.5802/aif.2914. http://www.numdam.org/articles/10.5802/aif.2914/

[1] Ahlfors, Lars V. Fundamental polyhedrons and limit point sets of Kleinian groups, Proc. Nat. Acad. Sci. U.S.A., Volume 55 (1966), pp. 251-254 | DOI | MR | Zbl

[2] Beardon, Alan F.; Maskit, Bernard Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math., Volume 132 (1974), pp. 1-12 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[11] Birkes, David Orbits of linear algebraic groups, Ann. of Math. (2), Volume 93 (1971), pp. 459-475 | DOI | MR | Zbl

[12] Bowditch, B. H. Geometrical finiteness for hyperbolic groups, J. Funct. Anal., Volume 113 (1993) no. 2, pp. 245-317 | DOI | MR | Zbl

[13] Bowditch, B. H. Geometrical finiteness with variable negative curvature, Duke Math. J., Volume 77 (1995) no. 1, pp. 229-274 | DOI | MR | Zbl

[14] Busemann, Herbert The geometry of geodesics, Academic Press Inc., New York, N. Y., 1955, pp. x+422 | MR | Zbl

[15] Busemann, Herbert; Kelly, Paul J. Projective geometry and projective metrics, Academic Press Inc., New York, N. Y., 1953, pp. viii+332 | MR | Zbl

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

[17] Choi, Suhyoung The convex real projective manifolds and orbifolds with radial ends : the openness of deformations, 2010 (Preprint)

[18] Colbois, B.; Vernicos, C.; Verovic, P. L’aire des triangles idéaux en géométrie de Hilbert, Enseign. Math. (2), Volume 50 (2004) no. 3-4, pp. 203-237 | MR | Zbl

[19] Colbois, Bruno; Vernicos, Constantin 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] Conze, J.-P.; Guivarc’h, Yves Limit sets of groups of linear transformations, Sankhyā Ser. A, Volume 62 (2000) no. 3, pp. 367-385 Ergodic theory and harmonic analysis (Mumbai, 1999) | MR | Zbl

[21] Cooper, Daryl; Long, Darren; Tillmann, Stephan On convex projective manifolds and cusps, 2011 (Preprint)

[22] Crampon, Mickaël; Marquis, Ludovic Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert, Ann. Math. Blaise Pascal, Volume 20 (2013) no. 2, pp. 363-376 | DOI | Numdam | MR | Zbl

[23] Crampon, Mickaël; Marquis, Ludovic Le flot géodésique des quotients géométriquement finis des géométries de Hilbert, Pacific J. Math., Volume 268 (2014) no. 2, pp. 313-369 | DOI | MR

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

[25] Goldman, William M. Projective geometry on manifolds, 2010 (Note)

[26] Greenberg, L. Fundamental polyhedra for kleinian groups, Ann. of Math. (2), Volume 84 (1966), pp. 433-441 | DOI | MR | Zbl

[27] Guivarc’h, Yves Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire, Ergodic Theory Dynam. Systems, Volume 10 (1990) no. 3, pp. 483-512 | DOI | MR | Zbl

[28] Hamilton, Emily Geometrical finiteness for hyperbolic orbifolds, Topology, Volume 37 (1998) no. 3, pp. 635-657 | DOI | MR | Zbl

[29] de la Harpe, Pierre 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

[30] Humphreys, James E. Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975, pp. xiv+247 (Graduate Texts in Mathematics, No. 21) | MR | Zbl

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

[32] Koszul, J.-L. Déformations de connexions localement plates, Ann. Inst. Fourier (Grenoble), Volume 18 (1968) no. fasc. 1, pp. 103-114 | DOI | Numdam | MR | Zbl

[33] Lee, Jaejeong Fundamental domains of convex projective structures, ProQuest LLC, Ann Arbor, MI, 2008, pp. 118 http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3329634 Thesis (Ph.D.)–University of California, Davis | MR

[34] Marden, Albert On finitely generated Fuchsian groups, Comment. Math. Helv., Volume 42 (1967), pp. 81-85 | DOI | MR | Zbl

[35] Marden, Albert The geometry of finitely generated kleinian groups, Ann. of Math. (2), Volume 99 (1974), pp. 383-462 | DOI | MR | Zbl

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

[37] Marquis, Ludovic Exemples de variétés projectives strictement convexes de volume fini en dimension quelconque, Enseign. Math. (2), Volume 58 (2012) no. 1-2, pp. 3-47 | DOI | MR | Zbl

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

[39] McMullen, Curtis T. Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math. Inst. Hautes Études Sci. (2002) no. 95, pp. 151-183 | DOI | Numdam | MR | Zbl

[40] Raghunathan, M. S. Discrete subgroups of Lie groups, Springer-Verlag, New York-Heidelberg, 1972, pp. ix+227 (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68) | MR | Zbl

[41] Ratcliffe, John G. Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, 149, Springer, New York, 2006, pp. xii+779 | MR | Zbl

[42] Rosenlicht, Maxwell On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. Amer. Math. Soc., Volume 101 (1961), pp. 211-223 | DOI | MR | Zbl

[43] Serre, Jean-Pierre Cohomologie des groupes discrets, Séminaire Bourbaki, 23ème année (1970/1971), Exp. No. 399, Springer, Berlin, 1971, p. 337-350. Lecture Notes in Math., Vol. 244 | Numdam | MR | Zbl

[44] Socié-Méthou, Edith Caractérisation des ellipsoï des par leurs groupes d’automorphismes, Ann. Sci. École Norm. Sup. (4), Volume 35 (2002) no. 4, pp. 537-548 | DOI | Numdam | MR | Zbl

[45] Thurston, William P. The geometry and topology of three-manifold (Lecture notes)

[46] Thurston, William P. Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997, pp. x+311 (Edited by Silvio Levy) | MR | Zbl

[47] Vernicos, Constantin 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-Martin-d’Hères, 2005, pp. 145-168 | Numdam | Zbl

[48] Vinberg, È. B. Invariant convex cones and orderings in Lie groups, Funktsional. Anal. i Prilozhen., Volume 14 (1980) no. 1, p. 1-13, 96 | DOI | MR | Zbl

[49] Vinberg, È. B.; Kac, V. G. Quasi-homogeneous cones, Mat. Zametki, Volume 1 (1967), pp. 347-354 | MR | Zbl

[50] Yaman, Asli A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math., Volume 566 (2004), pp. 41-89 | DOI | MR | Zbl

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