Holomorphic Functions and Moduli II
Volume 11 of the series Mathematical Sciences Research Institute Publications pp 113140
Convergence and Möbius Groups
 Gaven J. MartinAffiliated withYale University
 , Pekka TukiaAffiliated withYale University
Abstract
In this paper we collect together a few results on the topological conjugacy of convergence groups to conformal or Möbius groups. Convergence groups were first introduced in [G.M. I,II] and their basic properties were established as well as the classification of the structure of elementary convergence groups. Convergence groups have been found to be a natural generalization of Kleinian or Möbius groups to the topological category. In particular if one wishes to topologically characterize Möbius groups amongst groups of homeomorphisms of Sn, the unit sphere of R ^{n}+1, then one is led to the necessary condition that such a group is a convergence group. This is not in general a sufficient condition when n ≥ 2 as there are many nonstandard convergence groups, see [G.M. I,II], [F.S.] and [M.G.] for a variety of examples. We will see however that under certain reasonable restrictions the condition of being a convergence group will suffice in dimension two and three.
 Title
 Convergence and Möbius Groups
 Book Title
 Holomorphic Functions and Moduli II
 Book Subtitle
 Proceedings of a Workshop held March 13–19, 1986
 Pages
 pp 113140
 Copyright
 1988
 DOI
 10.1007/9781461396116_9
 Print ISBN
 9781461396130
 Online ISBN
 9781461396116
 Series Title
 Mathematical Sciences Research Institute Publications
 Series Volume
 11
 Series ISSN
 09404740
 Publisher
 Springer US
 Copyright Holder
 SpringerVerlag New York Inc.
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 Editors

 D. Drasin ^{(1)} ^{(6)}
 C. J. Earle ^{(2)} ^{(6)}
 F. W. Gehring ^{(3)} ^{(6)}
 I. Kra ^{(4)} ^{(6)}
 A. Marden ^{(5)} ^{(6)}
 Editor Affiliations

 1. Department of Mathematics, Purdue University
 6. Mathematical Sciences Research Institute
 2. Department of Mathematics, Cornell University
 3. Department of Mathematics, University of Michigan
 4. Department of Mathematics, State University of New York at Stony Brook
 5. Department of Mathematics, University of Minnesota
 Authors

 Gaven J. Martin ^{(7)}
 Pekka Tukia ^{(7)}
 Author Affiliations

 7. Yale University, New Haven, CT, 06520, USA
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