Convergence and Möbius Groups

  • Gaven J. Martin
  • Pekka Tukia
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 11)

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 Rn+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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AS]
    L. V. Ahlfors and L. Sario. “Riemann surfaces”, Princeton Univ. Press, (1960).MATHGoogle Scholar
  2. [BM]
    A. F. Beardon and B. Maskit. Limit points of Kleinian groups and finite sided fundamental polyhedra ,Acta. Math. 132 (1974).Google Scholar
  3. [FS]
    M. Freedman and R. Skora. Strange actions of groups on spheres ,J. Diff. Geom. 26 (1987) 75–98.MathSciNetGoogle Scholar
  4. [Ge]
    F. W. Gehring. Rings and quasiconformal mappings in space ,Trans. A.M.S. 103 (1962).Google Scholar
  5. [GM]
    F. W. Gehring and G.J. Martin. Discrete quasiconformal groups I, II ,Proc. London Math. Soc. (3) 55 (1987) 331–358.MathSciNetMATHCrossRefGoogle Scholar
  6. [He]
    J. Hempel. “3-manifolds”, Ann. Math. Stud. 86, Princeton Univ. Press, (1976).Google Scholar
  7. [MB]
    J. W. Morgan and H. Bass. “The Smith conjecture”, Academic Press, (1984).MATHGoogle Scholar
  8. [MG]
    G. J. Martin and F. W. Gehring. Generalizations of Kleinian groups M.S.R.I. Preprint. (1986).Google Scholar
  9. [MS]
    G. J. Martin and R. Skora. Group actions on To appear.Google Scholar
  10. [Ma]
    B. Maskit. A characterization of Schottky groups ,J. D’Analyse Math 19, (1967).Google Scholar
  11. [Sc]
    G. P. Scott. Strong annulus and torus theorems and the enclosing property of characteristic submanifolds of three manifolds ,Quart. J. Math. Oxford (2) 35 (1984).Google Scholar
  12. [Tul]
    P. Tukia. On isomorphisms of geometrically finite Möbius groups ,I.H.E.S. Publ. No. 61 (1985).Google Scholar
  13. [Tu2]
    P. Tukia. On discrete groups of the unit disk and their isomorphisms ,Ann. Acad. Sci. Fenn. Ser. A.I. Math 504 (1972).Google Scholar
  14. [Tu3]
    P. Tukia. Extension of boundary homeomorphisms of discrete groups of the unit disk ,Ann. Acad. Sci. Fenn. Ser. A.I. 548 (1973)Google Scholar
  15. [Tu4]
    P. Tukia. On quasiconformal groups ,To appear, J. D’Analyse Math.Google Scholar
  16. [Tu5]
    P. Tukia. Homeomorphic conjugates of Fuchsian groups ,To appear.Google Scholar
  17. [TV]
    P. Tukia and J. Väisälä. Lipschitz and quasiconformal approximation and extension ,Ann. Acad. Sci. Fenn. Ser. A.I. 6 (1981).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Gaven J. Martin
    • 1
  • Pekka Tukia
    • 1
  1. 1.Yale UniversityNew HavenUSA

Personalised recommendations