## Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group.(English)Zbl 0562.30018

Let G be a Fuchsian group of the upper half-plane U and let $$f: R\to R$$ be a k-quasisymmetric map which is G-compatible (i.e. $$fgf^{-1}$$ is a Möbius transformation if $$g\in G)$$. The main theorem of the paper says that then f can be extended to a K-quasiconformal and G-compatible homeomorphism of U with K depending only on k. The central idea is to prove this first for such G for which hyperbolic disks of radius r for some $$r=r(k)$$ do not contain pairs of G-equivalent points. This is extended for all finitely generated G by means of Teichmüller’s extremal mapping. A normal family argument finally gives it for all G.
This is valid only for $$n=1$$ but for $$n\geq 1$$ the following parallel theorem is given. Let $$f: R^ n\to R^ n$$ be a G-compatible homeomorphism for some group G of Möbius transformations of $$\bar R^ n$$. Then f can be extended to a G-compatible map of $$R^ n\times [0,\infty)$$. The extension is not in general a homeomorphism but if f is quasiconformal, then it satisfies a pseudo-isometric condition in $$R^ n\times (0,\infty)$$ (in the hyperbolic metric).
Some applications are also given. The most important of these concerns the Bers embedding of a Teichmüller space into the space of holomorphic quadratic differentials and says that if q is a quadratic differential for G which is in the universal Teichmüller space then q is in the Teichmüller space of G.
Recently, Douady and Earle [Conformally natural extension of homeomorphisms of the circle (to appear)] have obtained a considerably simpler proof of the main theorem of the paper.

### MSC:

 30C62 Quasiconformal mappings in the complex plane 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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