Teichmüller spaces for pointed Fuchsian groups
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 \[ \mathbb{H}_G=\mathbb{H}\backslash \{\text{all elliptic fixed points of}\;G\}. \] Two elements \(\mu,\nu\) in \(M(G)\) are said to be \(\mathcal{G}\)-equivalent if they are equivalent in \(M(G)\) and if, for each \(z_i\), \(i=1,2,\ldots, n\), \(w^{\mu}(z_i)=w^{\nu}(z_i)\). The \(n\)-pointed Teichmüller space \(T(\mathcal{G})\) is defined as quotient space of \(M(G)\) under this equivalence relation. We thus obtain a natural projection \(\Phi_{\mathcal{G}}:M(G)\rightarrow T(\mathcal{G})\), which together with the projection \(\Phi_G:M(G)\rightarrow T(G)\), also defines a natural projection \(\Phi:T(\mathcal{G})\rightarrow T(G)\) via \[ \Phi_G=\Phi\circ \Phi_{\mathcal{G}}. \] While the holomorphic projection \(\Phi_G:M(G)\rightarrow T(G)\) is well-understood [cf. \textit{C. J. Earle} and \textit{I. Kra}, Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers, 107--124 (1974; Zbl 0307.32016); Acta Math. 137, 49--79 (1976; Zbl 0331.32018)], in the paper under review, the authors study the projections \(\Phi_{\mathcal{G}}\) and \(\Phi\), and prove the following result: Theorem. \((1)\) The projection \(\Phi_{\mathcal{G}}:M(G)\rightarrow T(\mathcal{G})\) admits a global holomorphic section if and only if \(n=1\) and \(G\) is torsion free and of type \((0,3)\); and \((2)\) the projection \(\Phi:T(\mathcal{G})\rightarrow T(G)\) has no holomorphic section unless \(G\) is exceptional in the sense that \(2g+n\leq 4\). The authors also generalize some results of \textit{L. Bers} [Acta Math. 130, 89--126 (1973; Zbl 0249.32014)] to the case of the \(n\)-pointed Teichmüller space \(T(\mathcal{G})\).
scientific article; zbMATH DE number 5857651
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