Abstract
A Riemann surface M is said to be K-quasiconformally homogeneous if, for every two points p, q ∈ M, there exists a K-quasiconformal homeomorphism f: M→M such that f(p) = q. In this paper, we show there exists a universal constant K > 1 such that if M is a K-quasiconformally homogeneous hyperbolic genus zero surface other than ⅅ2, then K ≥ K. This answers a question by Gehring and Palka [10]. Further, we show that a non-maximal hyperbolic surface of genus g ≥ 1 is not K-quasiconformally homogeneous for any finite K ≥ 1.
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References
L. Ahlfors, Lectures on Quasiconformal Mappings, 2nd edition, Amer. Math. Soc., Providence, RI, 2006.
S. Bochner, Fortsetzung Riemannscher Flächen, Math. Ann. 98 (1927), 406–421.
P. Bonfert-Taylor, R. Canary, G. Martin and E. Taylor, Quasiconformal homogeneity of hyperbolic manifolds, Math. Ann. 331 (2005), 281–295.
P. Bonfert-Taylor, M. Bridgeman, R. Canary and E. Taylor, Quasiconformal homogeneity of hyperbolic surfaces with fixed-point full automorphisms, Math. Proc. Camb. Phil. Soc. 143 (2007), 71–74.
P. Bonfert-Taylor and E. Taylor, Quasiconformally homogeneous planar domains, Conform. Geom. Dyn. 12 (2008), 188–198.
P. Bonfert-Taylor, R. Canary, G. Martin, E. Taylor and M. Wolf, Ambient quasiconformal homogeneity of planar domains, Ann. Acad. Sci. Fenn. 35 (2010), 275–283.
P. Buser and H. Parlier, The distribution of simple closed geodesics on a Riemann surface, in Complex Analysis and its Applications, Osaka Munic. Univ. Press, Osaka 2007, pp. 3–10.
B. Farb and D. Margalit, A Primer on Mapping Class Groups, 2008.
A. Fletcher and V. Markovic, Quasiconformal Maps and Teichmüller Theory, Oxford Univ. Press, 2007.
F. Gehring and B. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199.
P. MacManus, R. Näkki and B. Palka, Quasiconformally bi-homogeneous compacta in the complex plane, Proc. London Math. Soc. (3) 78 (1999), 215–240.
H. Parlier, Hyperbolic polygons and simple closed geodesics, Enseign.Math. (2) 52 (2006), 295–317.
J. Ratcliffe, Foundations of Hyperbolic Manifolds, 2nd edition, Springer-Verlag, Berlin, 2006.
W. Thurston, Three-Dimensional Geometry and Topology, Volume 1, Princeton Univ. Press, 1997.
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The first author was supported by Marie Curie grant MRTN-CT-2006-035651 (CODY).
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Kwakkel, F., Markovic, V. Quasiconformal homogeneity of genus zero surfaces. JAMA 113, 173–195 (2011). https://doi.org/10.1007/s11854-011-0003-1
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DOI: https://doi.org/10.1007/s11854-011-0003-1