Conformal dimension and Gromov hyperbolic groups with 2–sphere boundary

Bonk, Mario¹ ; Kleiner, Bruce¹

¹ Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109, USA

Geometry & topology, Tome 9 (2005) no. 1, pp. 219-246.

Résumé

Suppose $G$ is a Gromov hyperbolic group, and ${\partial }_{\infty }G$ is quasisymmetrically homeomorphic to an Ahlfors $Q$–regular metric 2–sphere $Z$ with Ahlfors regular conformal dimension $Q$. Then $G$ acts discretely, cocompactly, and isometrically on ${ℍ}^3$.

Détail

DOI : 10.2140/gt.2005.9.219

Keywords: Gromov hyperbolic groups, Cannon's conjecture, quasisymmetric maps

Bibliographie

[1] C J Bishop, J T Tyson, Conformal dimension of the antenna set, Proc. Amer. Math. Soc. 129 (2001) 3631

[2] M Bonk, B Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002) 127

[3] M Bonk, B Kleiner, Rigidity for quasi–Möbius group actions, J. Differential Geom. 61 (2002) 81

[4] M Bonk, P Koskela, S Rohde, Conformal metrics on the unit ball in Euclidean space, Proc. London Math. Soc. $(3)$ 77 (1998) 635

[5] M Bourdon, H Pajot, Rigidity of quasi-isometries for some hyperbolic buildings, Comment. Math. Helv. 75 (2000) 701

[6] M Bourdon, H Pajot, Cohomologie $l_p$ et espaces de Besov, J. Reine Angew. Math. 558 (2003) 85

[7] S Buyalo, Volume entropy of hyperbolic graph surfaces, Ergodic Theory Dynam. Systems 25 (2005) 403

[8] J Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999) 428

[9] M Coornaert, Mesures de Patterson–Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993) 241

[10] P Hajłasz, P Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000)

[11] J Heinonen, Lectures on analysis on metric spaces, Universitext, Springer (2001)

[12] J Heinonen, P Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998) 1

[13] S Keith, T Laakso, Conformal Assouad dimension and modulus, Geom. Funct. Anal. 14 (2004) 1278

[14] J Kinnunen, N Shanmugalingam, Regularity of quasi-minimizers on metric spaces, Manuscripta Math. 105 (2001) 401

[15] T J Laakso, Ahlfors $Q$–regular spaces with arbitrary $Q>1$ admitting weak Poincaré inequality, Geom. Funct. Anal. 10 (2000) 111

[16] F Paulin, Un groupe hyperbolique est déterminé par son bord, J. London Math. Soc. $(2)$ 54 (1996) 50

[17] S Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Selecta Math. $($N.S.$)$ 2 (1996) 155

[18] S Semmes, Some novel types of fractal geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press (2001)

[19] D Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. (1979) 171

[20] D Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, from: "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. 97, Princeton Univ. Press (1981) 465

[21] P Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986) 318

[22] J Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Math. 23 (1998) 525

[23] J T Tyson, Metric and geometric quasiconformality in Ahlfors regular Loewner spaces, Conform. Geom. Dyn. 5 (2001) 21