### Abstract

A group G of homeomorphisms of the unit circle onto itself is a convergence group if every sequence of elements of G contains a subsequence, say g_{n}, such that either (i) g_{n} → g and g;;1 → g-I uniformly on the circle where g is a homeomorphism, or (ii) g_{n} → Xo and g;; 1 → Y_{0} uniformly on compact infsets of the complements of (Y_{0}) and (x_{o}), respectively, for some points X_{o} and Y_{0} of the circle (possibly X_{o} = Y_{0}). For example, a group of K-quasisymmetric maps, for a fixed K, is a convergence group. We show that if G is an abelian or nondiscrete convergence group, then there is a homeomorphism I such that loGo I-I is a group of Mobius transformation.

Original language | English (US) |
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Pages (from-to) | 87-121 |

Number of pages | 35 |

Journal | Transactions of the American Mathematical Society |

Volume | 318 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1990 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics