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On quasiconformal groups. (English) Zbl 0603.30026
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 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 $$n>2$$ [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 $$n\geq 2$$.

##### MSC:
 30C62 Quasiconformal mappings in the complex plane
##### Keywords:
K-quasiconformal homeomorphism; conformal structure
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##### References:
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