We study the notion of geometrical finiteness for those Hilbert geometries defined by strictly convex sets with 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 .
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.
Mot clés : géométrie de Hilbert, finitude géométrique, espace Gromov-hyperbolique, sous-groupes discrets des groupes de Lie, variété projective convexe
Keywords: Hilbert geometry, geometrical finiteness, Gromov-hyperbolic space, discrete sub-group of Lie groups, convex projective manifold
Crampon, Mickaël 1; marquis, Ludovic 2
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TY - JOUR AU - Crampon, Mickaël AU - marquis, Ludovic TI - Finitude géométrique en géométrie de Hilbert JO - Annales de l'Institut Fourier PY - 2014 SP - 2299 EP - 2377 VL - 64 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2914/ DO - 10.5802/aif.2914 LA - fr ID - AIF_2014__64_6_2299_0 ER -
%0 Journal Article %A Crampon, Mickaël %A marquis, Ludovic %T Finitude géométrique en géométrie de Hilbert %J Annales de l'Institut Fourier %D 2014 %P 2299-2377 %V 64 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2914/ %R 10.5802/aif.2914 %G fr %F AIF_2014__64_6_2299_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2914/
[1] 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] Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math., Volume 132 (1974), pp. 1-12 | DOI | MR | Zbl
[3] Sous-groupes discrets des groupes de Lie, European Summer School in Group Theory, 1997 (Luminy July 7-18)
[4] Automorphismes des cônes convexes, Invent. Math., Volume 141 (2000) no. 1, pp. 149-193 | DOI | MR | Zbl
[5] Convexes divisibles. II, Duke Math. J., Volume 120 (2003) no. 1, pp. 97-120 | DOI | MR | Zbl
[6] Convexes divisibles. I, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, 2004, pp. 339-374 | MR | Zbl
[7] Convexes divisibles. III, Ann. Sci. École Norm. Sup. (4), Volume 38 (2005) no. 5, pp. 793-832 | DOI | Numdam | MR | Zbl
[8] Convexes divisibles. IV. Structure du bord en dimension 3, Invent. Math., Volume 164 (2006) no. 2, pp. 249-278 | DOI | MR | Zbl
[9] Convexes hyperboliques et quasiisométries, Geom. Dedicata, Volume 122 (2006), pp. 109-134 | DOI | MR | Zbl
[10] Sur les variétés localement affines et localement projectives, Bull. Soc. Math. France, Volume 88 (1960), pp. 229-332 | Numdam | MR | Zbl
[11] Orbits of linear algebraic groups, Ann. of Math. (2), Volume 93 (1971), pp. 459-475 | DOI | MR | Zbl
[12] Geometrical finiteness for hyperbolic groups, J. Funct. Anal., Volume 113 (1993) no. 2, pp. 245-317 | DOI | MR | Zbl
[13] Geometrical finiteness with variable negative curvature, Duke Math. J., Volume 77 (1995) no. 1, pp. 229-274 | DOI | MR | Zbl
[14] The geometry of geodesics, Academic Press Inc., New York, N. Y., 1955, pp. x+422 | MR | Zbl
[15] Projective geometry and projective metrics, Academic Press Inc., New York, N. Y., 1953, pp. viii+332 | MR | Zbl
[16] 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] The convex real projective manifolds and orbifolds with radial ends : the openness of deformations, 2010 (Preprint)
[18] L’aire des triangles idéaux en géométrie de Hilbert, Enseign. Math. (2), Volume 50 (2004) no. 3-4, pp. 203-237 | MR | Zbl
[19] 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] 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] On convex projective manifolds and cusps, 2011 (Preprint)
[22] 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] 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] 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] Projective geometry on manifolds, 2010 (Note)
[26] Fundamental polyhedra for kleinian groups, Ann. of Math. (2), Volume 84 (1966), pp. 433-441 | DOI | MR | Zbl
[27] 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] Geometrical finiteness for hyperbolic orbifolds, Topology, Volume 37 (1998) no. 3, pp. 635-657 | DOI | MR | Zbl
[29] 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] Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975, pp. xiv+247 (Graduate Texts in Mathematics, No. 21) | MR | Zbl
[31] Convex projective structures on Gromov-Thurston manifolds, Geom. Topol., Volume 11 (2007), pp. 1777-1830 | DOI | MR | Zbl
[32] Déformations de connexions localement plates, Ann. Inst. Fourier (Grenoble), Volume 18 (1968) no. fasc. 1, pp. 103-114 | DOI | Numdam | MR | Zbl
[33] 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] On finitely generated Fuchsian groups, Comment. Math. Helv., Volume 42 (1967), pp. 81-85 | DOI | MR | Zbl
[35] The geometry of finitely generated kleinian groups, Ann. of Math. (2), Volume 99 (1974), pp. 383-462 | DOI | MR | Zbl
[36] 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] 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] Surface projective convexe de volume fini, Ann. Inst. Fourier (Grenoble), Volume 62 (2012) no. 1, pp. 325-392 | DOI | Numdam | MR | Zbl
[39] 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] 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] Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, 149, Springer, New York, 2006, pp. xii+779 | MR | Zbl
[42] 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] 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] 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] The geometry and topology of three-manifold (Lecture notes)
[46] 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] 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] 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] Quasi-homogeneous cones, Mat. Zametki, Volume 1 (1967), pp. 347-354 | MR | Zbl
[50] A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math., Volume 566 (2004), pp. 41-89 | DOI | MR | Zbl
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