Propriétés combinatoires du bord d’un groupe hyperbolique
Séminaire de théorie spectrale et géométrie, Volume 32 (2014-2015), pp. 73-96.

Le but de ce survol est de présenter les modules combinatoires récemment utilisés pour étudier les propriétés quasi-conformes des bords des groupes hyperboliques. Dans un premier temps, on rappellera quelques résultats et questions de rigidité bien connus qui ont motivés l’introduction de ces outils. Puis on définira les modules combinatoires et la propriété de Loewner combinatoire qui offrent une nouvelle approche pour résoudre des problèmes ouverts depuis longtemps. Enfin, on décrira des applications concrètes de ces outils à travers quelques résultats récents et questions ouvertes.

DOI: 10.5802/tsg.304
Classification: 20F67, 30L10
Keywords: Bord d’un groupe hyperbolique, analyse quasi-conforme, modules combinatoires
Clais, Antoine 1

1 Technion Department of Mathematics 32000 Haifa (Israel)
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Clais, Antoine. Propriétés combinatoires du bord d’un groupe hyperbolique. Séminaire de théorie spectrale et géométrie, Volume 32 (2014-2015), pp. 73-96. doi : 10.5802/tsg.304. http://www.numdam.org/articles/10.5802/tsg.304/

[1] Beeker, Benjamin; Lazarovich, Nir Sphere boundaries of hyperbolic groups (2016) (https://arxiv.org/abs/1512.00866) | MR

[2] Benedetti, Riccardo; Petronio, Carlo Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, 1992, pp. xiv+330 | DOI | MR | Zbl

[3] Bonk, Mario; Kleiner, Bruce Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math., Volume 150 (2002) no. 1, pp. 127-183 | DOI | MR | Zbl

[4] Bonk, Mario; Kleiner, Bruce Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geom. Topol., Volume 9 (2005), pp. 219-246 | DOI | MR | Zbl

[5] Bourdon, Marc Mostow type rigidity theorems (to appear in Handbook of Group Actions)

[6] Bourdon, Marc Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal., Volume 7 (1997) no. 2, pp. 245-268 | DOI | MR | Zbl

[7] Bourdon, Marc; Kleiner, Bruce Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups, Groups Geom. Dyn., Volume 7 (2013) no. 1, pp. 39-107 | DOI | MR

[8] Bourdon, Marc; Pajot, Hervé Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings, Proc. Amer. Math. Soc., Volume 127 (1999) no. 8, pp. 2315-2324 | DOI | MR | Zbl

[9] Bourdon, Marc; Pajot, Hervé Rigidity of quasi-isometries for some hyperbolic buildings, Comment. Math. Helv., Volume 75 (2000) no. 4, pp. 701-736 | DOI | MR | Zbl

[10] Bourdon, Marc; Pajot, Hervé Cohomologie l p et espaces de Besov, J. Reine Angew. Math., Volume 558 (2003), pp. 85-108 | DOI | MR | Zbl

[11] Bowditch, Brian H. Cut points and canonical splittings of hyperbolic groups, Acta Math., Volume 180 (1998) no. 2, pp. 145-186 | DOI | MR | Zbl

[12] Bucher, Michelle; Burger, Marc; Iozzi, Alessandra A dual interpretation of the Gromov-Thurston proof of Mostow rigidity and volume rigidity for representations of hyperbolic lattices, Trends in harmonic analysis (Springer INdAM Ser.), Volume 3, Springer, Milan, 2013, pp. 47-76 | DOI | MR | Zbl

[13] Cannon, James W.; Swenson, Eric L. Recognizing constant curvature discrete groups in dimension 3, Trans. Amer. Math. Soc., Volume 350 (1998) no. 2, pp. 809-849 | DOI | MR | Zbl

[14] Carrasco Piaggio, Matias On the conformal gauge of a compact metric space, Ann. Sci. Éc. Norm. Supér. (4), Volume 46 (2013) no. 3, p. 495-548 (2013) | MR | Zbl

[15] Clais, Antoine Combinatorial Modulus on Boundary of Right-Angled Hyperbolic Buildings, Anal. Geom. Metr. Spaces, Volume 4 (2016), pp. Art. 1 | DOI | MR

[16] Clais, Antoine Conformal dimension on boundary of right-angled hyperbolic buildings (2016) (https://arxiv.org/abs/1602.08611) | MR

[17] Coornaert, Michel; Delzant, Thomas; Papadopoulos, Athanase Géométrie et théorie des groupes, Lecture Notes in Mathematics, 1441, Springer-Verlag, Berlin, 1990, pp. x+165 (Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups], With an English summary) | MR | Zbl

[18] Ghys, Étienne; de la Harpe, Pierre Espaces métriques hyperboliques, Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988) (Progr. Math.), Volume 83, Birkhäuser Boston, Boston, MA, 1990, pp. 27-45 | DOI | Zbl

[19] Gromov, Mikhael Hyperbolic groups, Essays in group theory (Math. Sci. Res. Inst. Publ.), Volume 8, Springer, New York, 1987, pp. 75-263 | DOI | MR | Zbl

[20] Haïssinsky, Peter Empilements de cercles et modules combinatoires, Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 6, pp. 2175-2222 | Numdam | MR | Zbl

[21] Haïssinsky, Peter Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités [d’après Mostow, Pansu, Bourdon, Pajot, Bonk, Kleiner...], Astérisque (2009) no. 326, p. Exp. No. 993, ix, 321-362 (2010) (Séminaire Bourbaki. Vol. 2007/2008) | Numdam | MR | Zbl

[22] Heinonen, Juha Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001, pp. x+140 | DOI | MR | Zbl

[23] Heinonen, Juha; Koskela, Pekka Quasiconformal maps in metric spaces with controlled geometry, Acta Math., Volume 181 (1998) no. 1, pp. 1-61 | DOI | MR | Zbl

[24] Keith, Stephen; Laakso, Tomi J. Conformal Assouad dimension and modulus, Geom. Funct. Anal., Volume 14 (2004) no. 6, pp. 1278-1321 | DOI | MR | Zbl

[25] Kleiner, Bruce The asymptotic geometry of negatively curved spaces : uniformization, geometrization and rigidity, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 743-768 | MR | Zbl

[26] Lafont, Jean-François Rigidity of hyperbolic P-manifolds : a survey, Geom. Dedicata, Volume 124 (2007), pp. 143-152 | DOI | MR | Zbl

[27] Loewner, Charles On the conformal capacity in space, J. Math. Mech., Volume 8 (1959), pp. 411-414 | MR | Zbl

[28] Mackay, John M. Spaces and groups with conformal dimension greater than one, Duke Math. J., Volume 153 (2010) no. 2, pp. 211-227 | DOI | MR | Zbl

[29] Mackay, John M. Conformal dimension via subcomplexes for small cancellation and random groups (2014) (to appear in Math. Annalen., https://arxiv.org/abs/1409.0802) | MR

[30] Mackay, John M.; Tyson, Jeremy T. Conformal dimension, University Lecture Series, 54, American Mathematical Society, Providence, RI, 2010, pp. xiv+143 (Theory and application) | MR | Zbl

[31] Mackay, John M.; Tyson, Jeremy T.; Wildrick, Kevin Modulus and Poincaré inequalities on non-self-similar Sierpiński carpets, Geom. Funct. Anal., Volume 23 (2013) no. 3, pp. 985-1034 | DOI | MR | Zbl

[32] Markovic, Vladimir Criterion for Cannon’s conjecture, Geom. Funct. Anal., Volume 23 (2013) no. 3, pp. 1035-1061 | DOI | MR | Zbl

[33] Mostow, George D. Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. (1968) no. 34, pp. 53-104 | Numdam | MR | Zbl

[34] Pansu, Pierre Dimension conforme et sphère à l’infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math., Volume 14 (1989) no. 2, pp. 177-212 | MR | Zbl

[35] Pansu, Pierre Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2), Volume 129 (1989) no. 1, pp. 1-60 | DOI | MR | Zbl

[36] Paulin, Frédéric Un groupe hyperbolique est déterminé par son bord, J. London Math. Soc. (2), Volume 54 (1996) no. 1, pp. 50-74 | DOI | MR | Zbl

[37] Sullivan, Dennis On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics : Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) (Ann. of Math. Stud.), Volume 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465-496 | MR | Zbl

[38] Thurston, William P. The Geometry and Topology of Three-Manifolds (1980) (Notes of Princeton University, http://library.msri.org/books/gt3m/)

[39] Tyson, Jeremy T. Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Math., Volume 23 (1998) no. 2, pp. 525-548 | MR | Zbl

[40] Väisälä, Jussi Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971, pp. xiv+144 | MR

[41] Väisälä, Jussi Quasi-Möbius maps, J. Analyse Math., Volume 44 (1984/85), pp. 218-234 | DOI | MR

[42] Xie, Xiangdong Quasi-isometric rigidity of Fuchsian buildings, Topology, Volume 45 (2006) no. 1, pp. 101-169 | DOI | MR | Zbl

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