Quasisymmetric groups

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 $K$.