## Abelian and nondiscrete convergence groups on the circle

HTML articles powered by AMS MathViewer

- by A. Hinkkanen
- Trans. Amer. Math. Soc.
**318**(1990), 87-121 - DOI: https://doi.org/10.1090/S0002-9947-1990-1000145-X
- PDF | 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.

## References

- F. W. Gehring and G. J. Martin,
*Discrete quasiconformal groups. I*, Proc. London Math. Soc. (3)**55**(1987), no. 2, 331–358. MR**896224**, DOI 10.1093/plms/s3-55_{2}.331 - A. Hinkkanen,
*Uniformly quasisymmetric groups*, Proc. London Math. Soc. (3)**51**(1985), no. 2, 318–338. MR**794115**, DOI 10.1112/plms/s3-51.2.318 - A. Hinkkanen,
*The structure of certain quasisymmetric groups*, Mem. Amer. Math. Soc.**83**(1990), no. 422, iv+87. MR**948926**, DOI 10.1090/memo/0422 - Gaven J. Martin and Pekka Tukia,
*Convergence and Möbius groups*, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 113–140. MR**955836**, DOI 10.1007/978-1-4613-9611-6_{9} - Pekka Tukia,
*Homeomorphic conjugates of Fuchsian groups*, J. Reine Angew. Math.**391**(1988), 1–54. MR**961162**, DOI 10.1515/crll.1988.391.1

## Bibliographic 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