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.


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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