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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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