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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Abelian and nondiscrete convergence groups on the circle


Author: A. Hinkkanen
Journal: Trans. Amer. Math. Soc. 318 (1990), 87-121
MSC: Primary 30C62; Secondary 20H10, 22A99, 30F35
MathSciNet review: 1000145
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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 $ {{\text{g}}_n}$, such that either (i) $ {{\text{g}}_n} \to {\text{g}}$ and $ {\text{g}}_n^{ - 1} \to {{\text{g}}^{ - 1}}$ uniformly on the circle where $ {\text{g}}$ is a homeomorphism, or (ii) $ {{\text{g}}_n} \to {{\text{x}}_0}$ and $ {\text{g}}_n^{ - 1} \to {{\text{y}}_0}$ uniformly on compact subsets of the complements of $ \{ {{\text{y}}_0}\} $ and $ \{ {{\text{x}}_0}\} $, respectively, for some points $ {{\text{x}}_0}$ and $ {{\text{y}}_0}$ of the circle (possibly $ {{\text{x}}_0}{\text{ = }}{{\text{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 $ f$ such that $ f \circ G \circ {f^{ - 1}}$ is a group of Màbius transformations.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1000145-X
PII: S 0002-9947(1990)1000145-X
Article copyright: © Copyright 1990 American Mathematical Society