Quasisymmetric groups

Markovic, Vladimir¹

¹ University of Warwick, Institute of Mathematics, Coventry CV4 7AL, United Kingdom

Journal of the American Mathematical Society, Tome 19 (2006) no. 3, pp. 673-715

Résumé

One of the first problems in the theory of quasisymmetric and convergence groups was to investigate whether every quasisymmetric group that acts on the sphere $\textbf {S}^{n}$, $n>0$, is a quasisymmetric conjugate of a Möbius group that acts on $\textbf {S}^{n}$. This was shown to be true for $n=2$ by Sullivan and Tukia, and it was shown to be wrong for $n>2$ by Tukia. It also follows from the work of Martin and of Freedman and Skora. In this paper we settle the case of $n=1$ by showing that any $K$-quasisymmetric group is $K_1$-quasisymmetrically conjugated to a Möbius group. The constant $K_1$ is a function $K$.

Détail

DOI : 10.1090/S0894-0347-06-00518-2

Bibliographie

[1] Abikoff, William, Earle, Clifford J., Mitra, Sudeb Barycentric extensions of monotone maps of the circle 2004 1 20

[2] Bowditch, Brian H. A topological characterisation of hyperbolic groups J. Amer. Math. Soc. 1998 643 667

[3] Casson, Andrew, Jungreis, Douglas Convergence groups and Seifert fibered 3-manifolds Invent. Math. 1994 441 456

[4] Douady, Adrien, Earle, Clifford J. Conformally natural extension of homeomorphisms of the circle Acta Math. 1986 23 48

[5] Epstein, D. B. A., Marden, A., Markovic, V. Quasiconformal homeomorphisms and the convex hull boundary Ann. of Math. (2) 2004 305 336

[6] Freedman, Michael H., Skora, Richard Strange actions of groups on spheres J. Differential Geom. 1987 75 98

[7] Gabai, David Convergence groups are Fuchsian groups Bull. Amer. Math. Soc. (N.S.) 1991 395 402

[8] Gehring, F. W., Martin, G. J. Discrete quasiconformal groups. I Proc. London Math. Soc. (3) 1987 331 358

[9] Gehring, F. W., Martin, G. J. Discrete convergence groups 1987 158 167

[10] Gehring, F. W., Palka, B. P. Quasiconformally homogeneous domains J. Analyse Math. 1976 172 199

[11] He, Zheng-Xu, Schramm, Oded Fixed points, Koebe uniformization and circle packings Ann. of Math. (2) 1993 369 406

[12] Heinonen, Juha, Koskela, Pekka Definitions of quasiconformality Invent. Math. 1995 61 79

[13] Hinkkanen, A. Uniformly quasisymmetric groups Proc. London Math. Soc. (3) 1985 318 338

[14] Hinkkanen, A. Abelian and nondiscrete convergence groups on the circle Trans. Amer. Math. Soc. 1990 87 121

[15] Hinkkanen, A. The structure of certain quasisymmetric groups Mem. Amer. Math. Soc. 1990

[16] Katok, Svetlana Fuchsian groups 1992

[17] Martin, Gaven J. Discrete quasiconformal groups that are not the quasiconformal conjugates of Möbius groups Ann. Acad. Sci. Fenn. Ser. A I Math. 1986 179 202

[18] Martin, Gaven J., Tukia, Pekka Convergence and Möbius groups 1988 113 140

[19] Pommerenke, Ch. Boundary behaviour of conformal maps 1992

[20] Sullivan, Dennis On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions 1981 465 496

[21] Thurston, William P. Three-dimensional geometry and topology. Vol. 1 1997

[22] Tukia, Pekka On two-dimensional quasiconformal groups Ann. Acad. Sci. Fenn. Ser. A I Math. 1980 73 78

[23] Tukia, Pekka A quasiconformal group not isomorphic to a Möbius group Ann. Acad. Sci. Fenn. Ser. A I Math. 1981 149 160

[24] Tukia, Pekka Homeomorphic conjugates of Fuchsian groups J. Reine Angew. Math. 1988 1 54