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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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##### References:
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