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On extendibility of a map induced by the Bers isomorphism. (English) Zbl 1305.30023
Let \(S\) be a closed Riemann surface of genus \(g\geq 2\). Fix a point \(\hat{z}_0\) of \(S\) and let \(\dot{S}=S-\{\hat{z}_0\}\). Let \(T(S)\) and \(T(\dot{S})\) denote the Teichmüller space of \(S\) and \(\dot{S}\), respectively. Let \((F(S), \pi, T(S))\) be the Bers fiber space over \(T(S)\). Each fiber \(\pi^{-1}(X)\) of \(F(X)\) over a point \(X\) of \(T(S)\) is a domain (a quasidisk) in \(\mathbb{C}\) which is a universal covering space of the marked Riemann surface \(X\) and on which a quasifuchsian group acts as the group of deck transformations. There is a holomorphic isomorphism \(\varphi : F(S)\to T(\dot{S})\) [L. Bers, Acta Math. 130, 89–126 (1973; Zbl 0249.32014)].
A \(d\)-dimensional Teichmüller space is realized as a bounded domain in \(\mathbb{C}^d\) by the Bers embedding. By this fact, \(F(S)\) can be realized as a subset of \(\mathbb{C}^{3g-3}\times\hat{\mathbb{C}}\) which projects to a bounded domain in the first factor. Thus, using the Bers embedding, we can consider the closures of \(T(\dot{S})\) and \(F(S)\).
C. Zhang [Proc. Am. Math. Soc. 123, No. 8, 2451–2458 (1995; Zbl 0826.30030)] proved that \(\varphi\) does not extend continuously to the closures. In the paper under review the authors consider the problem to find a subset of the boundary \(\partial F(S)\) to which \(\varphi\) extends continuously.
Let \(U\) be the upper half plane. We think of it as a universal covering space of \(S\). For each \(X\in T(S)\) let \(h_X: U\to \pi^{-1}(X)\) denote the lift of the Teichmüller mapping \(S\to X\) normalized so that it fixes \(0\), \(1\) and \(\infty\). Note that, since \(\pi^{-1}(X)\) is a Jordan curve and \(h_X\) is quasiconformal, \(h_X\) extends to a homeomorphism between their closures. Define \(r:T(S)\times U\to F(S)\) by \(r(X,z)=(X,h_X(z))\). A point \(x\) in \(\partial U\) is called filling if any hyperbolic geodesic ray in \(U\) ending at \(x\) projects to a geodesic which meets all simple closed geodesics in \(S\). Let \(\mathbb{A}\subset \partial U\) be the set of all filling points. This set \(\mathbb{A}\) is of full Lebesgue measure in the unit circle. The main theorem (Theorem 4.1) of this paper shows that the map \(\varphi\circ r: T(S)\times U\to T(\dot{S})\) has a continuous extension to \(T(S)\times \mathbb{A}\).
To prove this therem, choose any sequence \((p_m,z_m)\) of \(T(S)\times U\) converging to a \((p_{\infty},z_{\infty})\) in \(T(S)\times\mathbb{A}\) and let \(q_m=\varphi\circ r(p_m,z_m)\). The authors prove a lemma (Lemma 4.1) by which we can find a filling lamination \(\lambda\) on \(\dot{S}\) and a sequence of simple closed curves \(\alpha_{m}\) converging to \(\lambda\) in the hyperbolic compactification (in the sense of Gromov) of the curve complex \(\mathcal{C}(\dot{S})\) in such a way that the lengths of \(\tilde{\alpha}_m\) remain bounded, where \(\tilde{\alpha}_m\) is the geodesic representative of \(\alpha_m\) in the quasifuchsian hyperbolic \(3\)-manifold \(N_m=\mathbb{H}^3/\Gamma_m\) determined by \((q_m,\bar{q}_0)\), where \(\bar{q}_0\) is the mirror image of \(q_0=(\dot{S},id)\). This lemma implies that the (extended) length of the filling lamination \(\lambda\) in \(N_{\infty}=\mathbb{H}^3/\Gamma_{\infty}\) is zero, where \(\Gamma_{\infty}\) is an algebraic limit of \(\Gamma_m\). Then \(\Gamma_{\infty}\) is singly degenerate and determined uniquely by \((\lambda,\bar{q}_0)\) by the ending lamination theorem. Finally the authors use the continuous extendability of the universal Cannon-Thurston map by C. J. Leininger et al. [Comment. Math. Helv. 86, No. 4, 769–816 (2011; Zbl 1248.57003)] to show that \(\lambda\) depends only on \(z_{\infty}\). Thus \(q_m=\varphi\circ r(p_m,z_m)\) converges to a point uniquely determined by \((p_{\infty},z_{\infty})\).
MSC:
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
20F67 Hyperbolic groups and nonpositively curved groups
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