Teichmüller spaces for pointed Fuchsian groups
type: Item
instance of: scholarly article
title: Teichmüller spaces for pointed Fuchsian groups
zbMATH Open document ID: 1213.30077
author: Yuliang Shen
author: Ming Feng Sun
published in: Science China. Mathematics
publication date: 2011-02-25T00:00:00Z
review text: Let \(G\) be a finitely generated Fuchsian group of the first kind, which acts on the hyperbolic plane \(\mathbb{H}\). Then \(\mathbb{H}/G\) is an analytically finite orbifold of type \((g,n)\), where \(g\) is the genus of \(\mathbb{H}/G\) and \(n\) is number of punctures and cone points of \(\mathbb{H}/G\). Let \(L^{\infty}(G)\) be the set of all Beltrami differentials \(\mu\) satisfying \[ \frac{(\mu\circ g)\overline g'}{g'}=\mu\;\;\text{for all}\;\;g\in G. \] Let \(M(G)\) denote the open unit ball in \(L^{\infty}(G)\). For any \(\mu\in M(G)\), the fundamental theorem of \textit{L. V. Ahlfors} and \textit{L. Bers} [Ann. Math. (2) 72, 385--404 (1960; Zbl 0104.29902)] states that there is a unique quasiconformal mapping \(w^{\mu}\) of \(\widehat{\mathbb{C}}\) that fixes \(0,1,\infty\), is conformal on \(\widehat{\mathbb{C}}\backslash \mathbb{H}\), and satisfies the Beltrami equation \[ \partial _{\bar{z}}w=\mu \partial _zw \] in \(\mathbb{H}\). Two elements \(\mu,\nu\in M(G)\) are said to be equivalent if \(w^{\mu}|_{\mathbb{R}}=w^{\mu}|_{\mathbb{R}}\). The Teichmüller space \(T(G)\) is the quotient space \(M(G)/\! \sim\) under this equivalence relation. It is well-known that \(T(G)\) admits a unique complex manifold structure so that the natural projection \(\Phi_G:M(G)\rightarrow T(G)\) is holomorphic with local holomorphic sections. Following \textit{C. J. Earle} [Duke Math. J. 36, 409--415 (1969; Zbl 0176.38401)], \(\Phi_G:M(G)\rightarrow T(G)\) has a global holomorphic section if and only if dim\(_{\mathbb{C}}T(G)\leq 1\). In [Isr. J. Math. 16, 237--257 (1973; Zbl 0279.32009)], \textit{I. Kra} introduced the so-called pointed Teichmüller space \(T(\mathcal{G})\) for a pointed Fuchsian group \(\mathcal{G}\) corresponding to \(G\). More precisely, the \((n+1)\)-tuple \[ \mathcal{G}=\{G;z_1,\ldots, z_n\} \] is said to be an \(n\)-pointed Fuchsian group if \(G\) is as mentioned and \(z_1,\ldots, z_n\) are \(n\) inequivalent points in \[ \mathbb{H}_G=\mathbb{H}\back…
reviewed by: C. Zhang
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 5857651
zbMATH Keywords: Fuchsian group
zbMATH Keywords: pointed Fuchsian group
zbMATH Keywords: Teichmüller space
zbMATH Keywords: holomorphic section
MaRDI profile type: Publication
full work available at URL: s11425-010-3126-4
OpenAlex ID: W2117269981
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DOI: 10.1007/S11425-010-3126-4
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description: scientific article; zbMATH DE number 5857651