Abelian and nondiscrete convergence groups on the circle

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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 gn, such that either (i) gn → g and g;;1 → g-I uniformly on the circle where g is a homeomorphism, or (ii) gn → Xo and g;; 1 → Y0 uniformly on compact infsets of the complements of (Y0) and (xo), respectively, for some points Xo and Y0 of the circle (possibly Xo = Y0). 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 languageEnglish (US)
Pages (from-to)87-121
Number of pages35
JournalTransactions of the American Mathematical Society
Issue number1
StatePublished - Mar 1990
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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