Quasisymmetric groups

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 S^n, n > 0, is a quasisymmetric conjugate of a Möbius group that acts on 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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