Convergence and Möbius Groups
 Gaven J. Martin,
 Pekka Tukia
 … show all 2 hide
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.
 Ahlfors, LV, Sario, L (1960) Riemann surfaces.
 Beardon, AF, Maskit, B (1974) Limit points of Kleinian groups and finite sided fundamental polyhedra. Acta. Math. pp. 132
 Freedman, M, Skora, R (1987) Strange actions of groups on spheres. J. Diff. Geom. 26: pp. 7598
 Gehring, FW (1962) Rings and quasiconformal mappings in space. Trans. A.M.S. pp. 103
 Gehring, FW, Martin, GJ (1987) Discrete quasiconformal groups I, II. Proc. London Math. Soc. 55: pp. 331358 CrossRef
 Hempel, J (1976) 3manifolds, Ann. Math. Stud.
 Morgan, JW, Bass, H (1984) The Smith conjecture.
 Martin, GJ, Gehring, FW (1986) Generalizations of Kleinian.
 G. J. Martin and R. Skora. Group actions on To appear.
 Maskit, B (1967) A characterization of Schottky groups. J. D’Analyse Math. pp. 19
 Scott, GP (1984) Strong annulus and torus theorems and the enclosing property of characteristic submanifolds of three manifolds. Quart. J. Math. Oxford 2: pp. 35
 Tukia, P (1985) On isomorphisms of geometrically finite Möbius groups.
 Tukia, P (1972) On discrete groups of the unit disk and their isomorphisms. Ann. Acad. Sci. Fenn. Ser. A.I. Math. pp. 504
 Tukia, P (1973) Extension of boundary homeomorphisms of discrete groups of the unit disk. Ann. Acad. Sci. Fenn. Ser. A. I. pp. 548
 P. Tukia. On quasiconformal groups ,To appear, J. D’Analyse Math.
 P. Tukia. Homeomorphic conjugates of Fuchsian groups ,To appear.
 Tukia, P, Väisälä, J (1981) Lipschitz and quasiconformal approximation and extension. Ann. Acad. Sci. Fenn. Ser. A. I. pp. 6
 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
 Additional Links
 Topics
 Industry Sectors
 eBook Packages
 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
Continue reading...
To view the rest of this content please follow the download PDF link above.