Finitude géométrique en géométrie de Hilbert  [ Geometrical finiteness in Hilbert geometry ]
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, p. 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 : https://doi.org/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
@article{AIF_2014__64_6_2299_0,
     author = {Crampon, Micka\"el and marquis, Ludovic},
     title = {Finitude g\'eom\'etrique en g\'eom\'etrie de Hilbert},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {6},
     year = {2014},
     pages = {2299-2377},
     doi = {10.5802/aif.2914},
     mrnumber = {3331168},
     zbl = {06387341},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2014__64_6_2299_0}
}
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/item/AIF_2014__64_6_2299_0/

[1] Ahlfors, Lars V. Fundamental polyhedrons and limit point sets of Kleinian groups, Proc. Nat. Acad. Sci. U.S.A., Tome 55 (1966), pp. 251-254 | Article | MR 194970 | Zbl 0132.30801

[2] Beardon, Alan F.; Maskit, Bernard Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math., Tome 132 (1974), pp. 1-12 | Article | MR 333164 | Zbl 0277.30017

[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., Tome 141 (2000) no. 1, pp. 149-193 | Article | MR 1767272 | Zbl 0957.22008

[5] Benoist, Yves Convexes divisibles. II, Duke Math. J., Tome 120 (2003) no. 1, pp. 97-120 | Article | MR 2010735 | Zbl 1037.22022

[6] Benoist, Yves Convexes divisibles. I, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai (2004), pp. 339-374 | MR 2094116 | Zbl 1084.37026

[7] Benoist, Yves Convexes divisibles. III, Ann. Sci. École Norm. Sup. (4), Tome 38 (2005) no. 5, pp. 793-832 | Article | Numdam | MR 2195260 | Zbl 1085.22006

[8] Benoist, Yves Convexes divisibles. IV. Structure du bord en dimension 3, Invent. Math., Tome 164 (2006) no. 2, pp. 249-278 | Article | MR 2218481 | Zbl 1107.22006

[9] Benoist, Yves Convexes hyperboliques et quasiisométries, Geom. Dedicata, Tome 122 (2006), pp. 109-134 | Article | MR 2295544 | Zbl 1122.20020

[10] Benzécri, Jean-Paul Sur les variétés localement affines et localement projectives, Bull. Soc. Math. France, Tome 88 (1960), pp. 229-332 | Numdam | MR 124005 | Zbl 0098.35204

[11] Birkes, David Orbits of linear algebraic groups, Ann. of Math. (2), Tome 93 (1971), pp. 459-475 | Article | MR 296077 | Zbl 0198.35001

[12] Bowditch, B. H. Geometrical finiteness for hyperbolic groups, J. Funct. Anal., Tome 113 (1993) no. 2, pp. 245-317 | Article | MR 1218098 | Zbl 0789.57007

[13] Bowditch, B. H. Geometrical finiteness with variable negative curvature, Duke Math. J., Tome 77 (1995) no. 1, pp. 229-274 | Article | MR 1317633 | Zbl 0877.57018

[14] Busemann, Herbert The geometry of geodesics, Academic Press Inc., New York, N. Y. (1955), pp. x+422 | MR 75623 | Zbl 0112.37002

[15] Busemann, Herbert; Kelly, Paul J. Projective geometry and projective metrics, Academic Press Inc., New York, N. Y. (1953), pp. viii+332 | MR 54980 | Zbl 0052.37305

[16] Choi, Suhyoung Convex decompositions of real projective surfaces. II. Admissible decompositions, J. Differential Geom., Tome 40 (1994) no. 2, pp. 239-283 http://projecteuclid.org/euclid.jdg/1214455537 | MR 1293655 | Zbl 0822.53009

[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), Tome 50 (2004) no. 3-4, pp. 203-237 | MR 2116715 | Zbl 1079.53110

[19] Colbois, Bruno; Vernicos, Constantin Bas du spectre et delta-hyperbolicité en géométrie de Hilbert plane, Bull. Soc. Math. France, Tome 134 (2006) no. 3, pp. 357-381 | Numdam | MR 2245997 | Zbl 1117.53034

[20] Conze, J.-P.; Guivarc’H, Yves Limit sets of groups of linear transformations, Sankhyā Ser. A, Tome 62 (2000) no. 3, pp. 367-385 (Ergodic theory and harmonic analysis (Mumbai, 1999)) | MR 1803464 | Zbl 1115.37305

[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, Tome 20 (2013) no. 2, pp. 363-376 http://ambp.cedram.org/item?id=AMBP_2013__20_2_363_0 | Article | Numdam | MR 3138033 | Zbl 1282.22007

[23] Crampon, Mickaël; Marquis, Ludovic Le flot géodésique des quotients géométriquement finis des géométries de Hilbert, Pacific J. Math., Tome 268 (2014) no. 2, pp. 313-369 | Article | MR 3227438

[24] Goldman, William M. Convex real projective structures on compact surfaces, J. Differential Geom., Tome 31 (1990) no. 3, pp. 791-845 http://projecteuclid.org/euclid.jdg/1214444635 | MR 1053346 | Zbl 0711.53033

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

[26] Greenberg, L. Fundamental polyhedra for kleinian groups, Ann. of Math. (2), Tome 84 (1966), pp. 433-441 | Article | MR 200446 | Zbl 0161.27405

[27] Guivarc’H, Yves Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire, Ergodic Theory Dynam. Systems, Tome 10 (1990) no. 3, pp. 483-512 | Article | MR 1074315 | Zbl 0715.60008

[28] Hamilton, Emily Geometrical finiteness for hyperbolic orbifolds, Topology, Tome 37 (1998) no. 3, pp. 635-657 | Article | MR 1604903 | Zbl 0915.32006

[29] De La Harpe, Pierre On Hilbert’s metric for simplices, Geometric group theory, Vol. 1 (Sussex, 1991), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 181 (1993), pp. 97-119 | Article | MR 1238518 | Zbl 0832.52002

[30] Humphreys, James E. Linear algebraic groups, Springer-Verlag, New York-Heidelberg (1975), pp. xiv+247 (Graduate Texts in Mathematics, No. 21) | MR 396773 | Zbl 0471.20029

[31] Kapovich, Michael Convex projective structures on Gromov-Thurston manifolds, Geom. Topol., Tome 11 (2007), pp. 1777-1830 | Article | MR 2350468 | Zbl 1130.53024

[32] Koszul, J.-L. Déformations de connexions localement plates, Ann. Inst. Fourier (Grenoble), Tome 18 (1968) no. fasc. 1, pp. 103-114 | Article | Numdam | MR 239529 | Zbl 0167.50103

[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 2712298

[34] Marden, Albert On finitely generated Fuchsian groups, Comment. Math. Helv., Tome 42 (1967), pp. 81-85 | Article | MR 217287 | Zbl 0156.30703

[35] Marden, Albert The geometry of finitely generated kleinian groups, Ann. of Math. (2), Tome 99 (1974), pp. 383-462 | Article | MR 349992 | Zbl 0282.30014

[36] Marquis, Ludovic Espace des modules marqués des surfaces projectives convexes de volume fini, Geom. Topol., Tome 14 (2010) no. 4, pp. 2103-2149 | Article | MR 2740643 | Zbl 1225.32022

[37] Marquis, Ludovic Exemples de variétés projectives strictement convexes de volume fini en dimension quelconque, Enseign. Math. (2), Tome 58 (2012) no. 1-2, pp. 3-47 | Article | MR 2985008 | Zbl 1284.57021

[38] Marquis, Ludovic Surface projective convexe de volume fini, Ann. Inst. Fourier (Grenoble), Tome 62 (2012) no. 1, pp. 325-392 http://aif.cedram.org/item?id=AIF_2012__62_1_325_0 | Article | Numdam | MR 2986273 | Zbl 1254.57015

[39] Mcmullen, Curtis T. Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math. Inst. Hautes Études Sci. (2002) no. 95, pp. 151-183 | Article | Numdam | MR 1953192 | Zbl 1148.20305

[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 507234 | Zbl 0254.22005

[41] Ratcliffe, John G. Foundations of hyperbolic manifolds, Springer, New York, Graduate Texts in Mathematics, Tome 149 (2006), pp. xii+779 | MR 2249478 | Zbl 1106.51009

[42] Rosenlicht, Maxwell On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. Amer. Math. Soc., Tome 101 (1961), pp. 211-223 | Article | MR 130878 | Zbl 0111.17902

[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 422504 | Zbl 0273.57022

[44] Socié-Méthou, Edith Caractérisation des ellipsoï des par leurs groupes d’automorphismes, Ann. Sci. École Norm. Sup. (4), Tome 35 (2002) no. 4, pp. 537-548 | Article | Numdam | MR 1981171 | Zbl 1040.32021

[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 University Press, Princeton, NJ, Princeton Mathematical Series, Tome 35 (1997), pp. x+311 (Edited by Silvio Levy) | MR 1435975 | Zbl 0873.57001

[47] Vernicos, Constantin Introduction aux géométries de Hilbert, Actes de Séminaire de Théorie Spectrale et Géométrie. Vol. 23. Année 2004–2005, Univ. Grenoble I, Saint-Martin-d’Hères (Sémin. Théor. Spectr. Géom.) Tome 23 (2005), pp. 145-168 | Numdam | Zbl 1100.53031

[48] Vinberg, È. B. Invariant convex cones and orderings in Lie groups, Funktsional. Anal. i Prilozhen., Tome 14 (1980) no. 1, p. 1-13, 96 | Article | MR 565090 | Zbl 0452.22014

[49] Vinberg, È. B.; Kac, V. G. Quasi-homogeneous cones, Mat. Zametki, Tome 1 (1967), pp. 347-354 | MR 208470 | Zbl 0163.16902

[50] Yaman, Asli A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math., Tome 566 (2004), pp. 41-89 | Article | MR 2039323 | Zbl 1043.20020