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Journal of the American Mathematical Society

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Quasisymmetric groups

Author: Vladimir Markovic
Journal: J. Amer. Math. Soc. 19 (2006), 673-715
MSC (2000): Primary 20H10, 37F30
Published electronically: January 25, 2006
MathSciNet review: 2220103
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Abstract: One of the first problems in the theory of quasisymmetric and convergence groups was to investigate whether every quasisymmetric group that acts on the sphere $ {\bf S}^{n}$, $ n>0$, is a quasisymmetric conjugate of a Möbius group that acts on $ {\bf S}^{n}$. This was shown to be true for $ n=2$ by Sullivan and Tukia, and it was shown to be wrong for $ n>2$ by Tukia. It also follows from the work of Martin and of Freedman and Skora. In this paper we settle the case of $ n=1$ by showing that any $ K$-quasisymmetric group is $ K_1$-quasisymmetrically conjugated to a Möbius group. The constant $ K_1$ is a function $ K$.

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Vladimir Markovic
Affiliation: University of Warwick, Institute of Mathematics, Coventry CV4 7AL, United Kingdom

Received by editor(s): December 15, 2004
Published electronically: January 25, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.