Quasisymmetric groups

Markovic, Vladimir

doi:10.1090/S0894-0347-06-00518-2

Quasisymmetric groups
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by Vladimir Markovic

J. Amer. Math. Soc. 19 (2006), 673-715

DOI: https://doi.org/10.1090/S0894-0347-06-00518-2

Published electronically: January 25, 2006

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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 $\textbf {S}^{n}$, $n>0$, is a quasisymmetric conjugate of a Möbius group that acts on $\textbf {S}^{n}$. 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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