Conformal automorphisms of countably connected regions

Short, Ian

doi:10.1090/S1088-4173-2013-00253-1

Conformal automorphisms of countably connected regions
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by Ian Short

Conform. Geom. Dyn. 17 (2013), 1-5

DOI: https://doi.org/10.1090/S1088-4173-2013-00253-1

Published electronically: January 9, 2013

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Abstract:

We prove that the conformal automorphism group of a countably connected circular region of connectivity at least three is either a Fuchsian group or a discrete elementary group of Möbius transformations.

References

Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
Leon Greenberg, Conformal transformations of Riemann surfaces, Amer. J. Math. 82 (1960), 749–760. MR 122988, DOI 10.2307/2372937
Maurice Heins, On the number of 1-1 directly conformal maps which a multiply-connected plane region of finite connectivity $p(>2)$ admits onto itself, Bull. Amer. Math. Soc. 52 (1946), 454–457. MR 16469, DOI 10.1090/S0002-9904-1946-08590-0
Zheng-Xu He and Oded Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. (2) 137 (1993), no. 2, 369–406. MR 1207210, DOI 10.2307/2946541
Steven G. Krantz, Geometric function theory, Cornerstones, Birkhäuser Boston, Inc., Boston, MA, 2006. Explorations in complex analysis. MR 2167675
M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894