Holomorphic Functions and Moduli II

Volume 11 of the series Mathematical Sciences Research Institute Publications pp 113-140

Convergence and Möbius Groups

  • Gaven J. MartinAffiliated withYale University
  • , Pekka TukiaAffiliated withYale University

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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.