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Quasiconformal homogeneity of hyperbolic surfaces with fixed-point full automorphisms

Published online by Cambridge University Press:  01 July 2007

PETRA BONFERT–TAYLOR ,
MARTIN BRIDGEMAN ,
RICHARD D. CANARY  and
EDWARD C. TAYLOR
PETRA BONFERT–TAYLOR
Affiliation:
Wesleyan University, Middletown, CT 06459, U.S.A. e-mail: pbonfert@wesleyan.edu
MARTIN BRIDGEMAN
Affiliation:
Boston College, Chestnut Hill, MA 02467, U.S.A. e-mail: bridgem@bc.edu
RICHARD D. CANARY
Affiliation:
University of Michigan, Ann Arbor, MI 48109, U.S.A. e-mail: canary@umich.edu
EDWARD C. TAYLOR
Affiliation:
Wesleyan University, Middletown, CT 06459, U.S.A. e-mail: ectaylor@wesleyan.edu
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Abstract

We show that any closed hyperbolic surface admitting a conformal automorphism with “many” fixed points is uniformly quasiconformally homogeneous, with constant uniformly bounded away from 1. In particular, there is a uniform lower bound on the quasiconformal homogeneity constant for all hyperelliptic surfaces. In addition, we introduce more restrictive notions of quasiconformal homogeneity and bound the associated quasiconformal homogeneity constants uniformly away from 1 for all hyperbolic surfaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 1 , July 2007 , pp. 71 - 84
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Anderson, G., Vamanamurthy, M. and Vuorinen, M.. Conformal Invariants, Inequalities and Quasiconformal Maps (John Wiley & Sons, 1997).Google Scholar
[2]Bonfert-Taylor, P., Canary, R. D., Martin, G. and Taylor, E. C.. Quasiconformal homogeneity of hyperbolic manifolds. Math. Ann. 331 (2005), 281–295.Google Scholar
[3]Bonfert-Taylor, P. and Taylor, E. C.. Hausdorff dimension and limit sets of quasiconformal groups. Mich. Math. J. 49 (2001), 243–257.Google Scholar
[4]Canary, R. D., Epstein, D. B. A. and Green, P.. Notes on Notes of Thurston. In Analytical and Geometric Aspects of Hyperbolic Space. London Mathematical Society Lecture Notes 111 (Cambridge University Press, 1987) pp. 3–92.Google Scholar
[5]Earle, C. J.. Moduli of Surfaces with Symmetries. In Advances in the Theory of Riemann Surfaces. Ann. Math. Stud. (Princeton University Press, 1971).Google Scholar
[6]Gehring, F. W. and Palka, B.. Quasiconformally homogeneous domains. J. Analyse Math. 30 (1976), 172–199.CrossRefGoogle Scholar
[7] T. Jørgensen and Marden, A.. Algebraic and geometric convergence of Kleinian groups. Math. Scand. 66 (1990), 47–72.Google Scholar
[8]MacManus, P., Näkki, R. and Palka, B.. Quasiconformally homogeneous compacta in the complex plane. Michigan Math. J. 45 (1998), 227–241.Google Scholar
[9]MacManus, P., Näkki, R. and Palka, B.. Quasiconformally bi-homogeneous compacta in the complex plane. Proc. London Math. Soc. 78 (1999), 215–240.Google Scholar
[10]Pansu, P.. Quasiisométries des variétés à courbure négative. Thesis (1987).Google Scholar
[11] O. Teichmüller. Ein Verschiebungssatz der quasikonformen Abbildung. Deutsche Math. 7 (1944), 336–343.Google Scholar
[12] J. Väisälä. Lectures on n-Dimensional Quasiconformal Mappings (Springer-Verlag, 1971).Google Scholar
[13]Yamada, A.. On Marden's universal constant of Fuchsian groups. Kodai Math. J. 4 (1981), 266–277.Google Scholar
[14]Yamada, A.. On Marden's universal constant of Fuchsian groups II. J. Analyse Math. 41 (1982), 234–248.Google Scholar