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Quasiconformally homogeneous planar domains. (English) Zbl 1191.30008

The authors explore the ambient quasiconformal homogeneity of planar domains and their boundaries. They show that the quasiconformal homogeneity of a domain \( D\) and its boundary \( E\) implies that the pair \( (D,E)\) is in fact quasiconformally bi-homogeneous. They also give a geometric and topological characterization of the quasiconformal homogeneity of \( D\) or \( E\) under the assumption that \( E\) is a Cantor set captured by a quasicircle. A collection of examples is provided to demonstrate that certain assumptions are the weakest possible. The notion of uniform perfectness in the sense of Pommerenke has an important role in this paper.

MSC:

30C62 Quasiconformal mappings in the complex plane
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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[1] Petra Bonfert-Taylor, Martin Bridgeman, Richard D. Canary, and Edward C. Taylor, Quasiconformal homogeneity of hyperbolic surfaces with fixed-point full automorphisms, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 1, 71 – 84. · Zbl 1133.30012
[2] Petra Bonfert-Taylor, Richard D. Canary, Gaven Martin, and Edward Taylor, Quasiconformal homogeneity of hyperbolic manifolds, Math. Ann. 331 (2005), no. 2, 281 – 295. · Zbl 1063.30020
[3] P. Bonfert-Taylor, R.D. Canary, G. Martin, E.C. Taylor and M. Wolf, Ambient quasiconformal homogeneity of planar domains, preprint. · Zbl 1198.30017
[4] P. Bonfert-Taylor, G. Martin, A. Reid and E. Taylor, Teichmüller mappings, quasiconformal homogeneity, and non-amenable covers of Riemann surfaces, to appear in Pure and Applied Mathematics Quarterly. · Zbl 1239.30007
[5] P. Bonfert-Taylor and E. Taylor, Quasiconformal homogeneity of limit sets, preprint. · Zbl 0999.30029
[6] F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172 – 199. · Zbl 0349.30019
[7] Peter W. Jones and Stanislav K. Smirnov, Removability theorems for Sobolev functions and quasiconformal maps, Ark. Mat. 38 (2000), no. 2, 263 – 279. · Zbl 1034.30014
[8] Robert Kaufman and Jang-Mei Wu, On removable sets for quasiconformal mappings, Ark. Mat. 34 (1996), no. 1, 141 – 158. · Zbl 0862.30020
[9] Pekka Koskela, Removable sets for Sobolev spaces, Ark. Mat. 37 (1999), no. 2, 291 – 304. · Zbl 1070.46502
[10] Pekka Koskela and Tomi Nieminen, Quasiconformal removability and the quasihyperbolic metric, Indiana Univ. Math. J. 54 (2005), no. 1, 143 – 151. · Zbl 1078.30016
[11] Paul MacManus, Catching sets with quasicircles, Rev. Mat. Iberoamericana 15 (1999), no. 2, 267 – 277. · Zbl 0944.30013
[12] Paul MacManus, Raimo Näkki, and Bruce Palka, Quasiconformally homogeneous compacta in the complex plane, Michigan Math. J. 45 (1998), no. 2, 227 – 241. · Zbl 0960.30017
[13] Paul MacManus, Raimo Näkki, and Bruce Palka, Quasiconformally bi-homogeneous compacta in the complex plane, Proc. London Math. Soc. (3) 78 (1999), no. 1, 215 – 240. · Zbl 0974.30012
[14] Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. · Zbl 0627.30039
[15] Curtis T. McMullen, Kleinian groups and John domains, Topology 37 (1998), no. 3, 485 – 496. · Zbl 0897.30021
[16] Raimo Näkki and Jussi Väisälä, John disks, Exposition. Math. 9 (1991), no. 1, 3 – 43. · Zbl 0757.30028
[17] Ch. Pommerenke, On uniformly perfect sets and Fuchsian groups, Analysis 4 (1984), no. 3-4, 299 – 321. · Zbl 0501.30036
[18] Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. (Basel) 32 (1979), no. 2, 192 – 199. · Zbl 0393.30005
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