Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Conformal automorphisms of countably connected regions

Author: Ian Short
Journal: Conform. Geom. Dyn. 17 (2013), 1-5
MSC (2010): Primary 30C20, 30C35; Secondary 30F35, 30F45
Published electronically: January 9, 2013
MathSciNet review: 3005739
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777 (85d:22026)
  • [2] Leon Greenberg, Conformal transformations of Riemann surfaces, Amer. J. Math. 82 (1960), 749-760. MR 0122988 (23 #A319)
  • [3] 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 0016469 (8,21d)
  • [4] 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 (96b:30015),
  • [5] Steven G. Krantz, Geometric function theory, Cornerstones, Birkhäuser Boston Inc., Boston, MA, 2006. Explorations in complex analysis. MR 2167675 (2006e:30001)
  • [6] M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894 (22 #5712)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 30C20, 30C35, 30F35, 30F45

Retrieve articles in all journals with MSC (2010): 30C20, 30C35, 30F35, 30F45

Additional Information

Ian Short
Affiliation: Department of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA, United Kingdom

Keywords: Circular region, conformal automorphism, discrete, Fuchsian group, Möbius transformation
Received by editor(s): July 25, 2012
Published electronically: January 9, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society