The limit sets of Schottky quasiconformal groups are uniformly perfect

Li, Xiaosheng

doi:10.1090/S0002-9947-05-03870-5

The limit sets of Schottky quasiconformal groups are uniformly perfect
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Trans. Amer. Math. Soc. 357 (2005), 2119-2132 Request permission

Abstract:

In this paper we study Schottky quasiconformal groups. We show that the limit sets of Schottky quasiconformal groups are uniformly perfect, and that the limit set of a given discrete non-elementary quasiconformal group has positive Hausdorff dimension.

References

A. F. Beardon and Ch. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2) 18 (1978), no. 3, 475–483. MR 518232, DOI 10.1112/jlms/s2-18.3.475
Petra Bonfert-Taylor and Edward C. Taylor, Hausdorff dimension and limit sets of quasiconformal groups, Michigan Math. J. 49 (2001), no. 2, 243–257. MR 1852301, DOI 10.1307/mmj/1008719771
Petra Bonfert-Taylor and Edward C. Taylor, The exponent of convergence and a theorem of Astala, Indiana Univ. Math. J. 51 (2002), no. 3, 607–623. MR 1911047, DOI 10.1512/iumj.2002.51.2249
Lester R. Ford, Automorphic functions, Chelsea Publishing Company, New York, 1972.
F. W. Gehring and G. J. Martin, Discrete quasiconformal groups. I, Proc. London Math. Soc. (3) 55 (1987), no. 2, 331–358. MR 896224, DOI 10.1093/plms/s3-55_{2}.331
A. Hinkkanen, Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc. 113 (1993), no. 3, 543–559. MR 1207519, DOI 10.1017/S0305004100076192
P. Järvi and M. Vuorinen, Uniformly perfect sets and quasiregular mappings, J. London Math. Soc. (2) 54 (1996), no. 3, 515–529. MR 1413895, DOI 10.1112/jlms/54.3.515
R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc. 116 (1992), no. 1, 251–257. MR 1106180, DOI 10.1090/S0002-9939-1992-1106180-2
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. (Basel) 32 (1979), no. 2, 192–199. MR 534933, DOI 10.1007/BF01238490
Pekka Tukia, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand J. Math. 23 (1994), no. 2, 157–187. MR 1313451
Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009, DOI 10.1007/BFb0061216
Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174, DOI 10.1007/BFb0077904