## Abelian and nondiscrete convergence groups on the circle

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- by A. Hinkkanen PDF
- Trans. Amer. Math. Soc.
**318**(1990), 87-121 Request permission

## 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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## Additional Information

- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**318**(1990), 87-121 - MSC: Primary 30C62; Secondary 20H10, 22A99, 30F35
- DOI: https://doi.org/10.1090/S0002-9947-1990-1000145-X
- MathSciNet review: 1000145