On extendibility of a map induced by the Bers isomorphism.

*(English)*Zbl 1305.30023Let \(S\) 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 \((p_{\infty},z_{\infty})\).

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 \((p_{\infty},z_{\infty})\).

Reviewer: Toshihiro Nakanishi (Matsue)

##### 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 |

##### References:

[1] | Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. · Zbl 0272.30012 |

[2] | Lipman Bers, Fiber spaces over Teichmüller spaces, Acta. Math. 130 (1973), 89 – 126. · Zbl 0249.32014 · doi:10.1007/BF02392263 · doi.org |

[3] | Lipman Bers, Finite-dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 131 – 172. · Zbl 0485.30002 |

[4] | Lipman Bers, An inequality for Riemann surfaces, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 87 – 93. · Zbl 0575.30039 |

[5] | J. F. Brock, Continuity of Thurston’s length function, Geom. Funct. Anal. 10 (2000), no. 4, 741 – 797. · Zbl 0968.57011 · doi:10.1007/PL00001637 · doi.org |

[6] | Jeffrey F. Brock, Richard D. Canary, and Yair N. Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. (2) 176 (2012), no. 1, 1 – 149. · Zbl 1253.57009 · doi:10.4007/annals.2012.176.1.1 · doi.org |

[7] | Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. · Zbl 0770.53001 |

[8] | Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. · Zbl 0754.30001 |

[9] | E. Klarreich, The boundary at infinity of the curve complex and relative Teichmüller spaces, preprint. · Zbl 1003.53053 |

[10] | Christopher J. Leininger, Mahan Mj, and Saul Schleimer, The universal Cannon-Thurston map and the boundary of the curve complex, Comment. Math. Helv. 86 (2011), no. 4, 769 – 816. · Zbl 1248.57003 · doi:10.4171/CMH/240 · doi.org |

[11] | Bernard Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381 – 386. · Zbl 0587.30043 · doi:10.5186/aasfm.1985.1042 · doi.org |

[12] | Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103 – 149. · Zbl 0941.32012 · doi:10.1007/s002220050343 · doi.org |

[13] | Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. · Zbl 0892.30035 |

[14] | Yair N. Minsky, Teichmüller geodesics and ends of hyperbolic 3-manifolds, Topology 32 (1993), no. 3, 625 – 647. · Zbl 0793.58010 · doi:10.1016/0040-9383(93)90013-L · doi.org |

[15] | Yair N. Minsky, On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds, J. Amer. Math. Soc. 7 (1994), no. 3, 539 – 588. · Zbl 0808.30027 |

[16] | Ken’ichi Ohshika, Limits of geometrically tame Kleinian groups, Invent. Math. 99 (1990), no. 1, 185 – 203. · Zbl 0747.53033 · doi:10.1007/BF01234417 · doi.org |

[17] | Chaohui Zhang, Nonextendability of the Bers isomorphism, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2451 – 2458. · Zbl 0826.30030 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.