On quasiconformal groups (Q1082479)

A group G is a quasiconformal group of \(\bar R{}^ n\) if every \(g\in G\) is a K-quasiconformal homeomorphism of \(\bar R{}^ n\) for some fixed K. It is shown that every quasiconformal group G admits a G-invariant conformal structure, i. e. there is a measurable map \(\mu\) : \(R^ n\to S\) where S is the set of positive definite \(n\times n\)-matrices with determinant 1 such that \(\mu\) satisfies a boundedness condition and such that, given \(g\in G\), \[ \mu (x)=| \det g'(x)|^{-2/n} g'(x)^ T \mu (g(x)) g'(x) \] a.e. in \(R^ n\). This has been proved by \textit{D. Sullivan} [Proc. 1978 Stony Brook Conf. Ann. Stud. 97, 465-496 (1981; Zbl 0567.58015)] for discrete G. A consequence is that if \(n=2\), then every qc group is conjugate by a quasiconformal map to a Möbius group. While this is not true if \(n>2\) [\textit{P. Tukia}, Ann. Acad. Sci. Fenn., Ser. A I 6, 149-160 (1981; Zbl 0443.30026)], some conditions are given for the existence of such a conjugacy. For instance, the conjugacy exists if G satisfies a condition similar to that of a convex co-compact discrete Möbius group of \(\bar R{}^ n\) or if there is a G-invariant conformal structure \(\mu\) which is approximately continuous at a radial limit point of G or continuous at a limit point of G. The paper also contains some auxiliary results on quasiconformal maps, for instance the so-called good approximation theorem for quasiconformal maps, known if \(n=2\), is generalized for every dimension \(n\geq 2\).