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Hyperbolization of Fibrations

  • Michael Kapovich
Chapter
Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

Let S be a Riemann surface of finite hyperbolic type, \(\Gamma = \pi_1(S)\). We assume that \(\Gamma\) is embedded in \(PSL(2, \mathbb{R})\) as Fuchsian group so that \(S = \mathbb{H}^{2}/\Gamma\). Recall that \(\bar{S} = (\widehat{\mathbb{C}} - {\rm cl}(\mathbb{H}^2))/\Gamma\) has the same marked hyperbolic structure as S and the opposite orientation. These surfaces are identified via projection of the complex conjugation \(Z \mapsto \bar{Z}\) to a map \(S \rightarrow \bar{S}\). There is a continuous length function
$$\ell : \mathcal{T}(S) \times \mathcal{M}\mathcal{L}(S) \rightarrow \mathbb{R}$$
That extends the length function for the closed geodesics in S; see Section 11.16.
Suppose that
$$(\Psi_{n}^{+}, \Psi_{n}^{-} \epsilon \mathcal{T}(S) \times \mathcal{T}(\bar{S})$$
Is a sequence and \((\lambda^+, \lambda^-)\) is a pair of elements in \(\mathcal{M}\mathcal{L}(S) \times \mathcal{M}\mathcal{L}(\bar{S})\) such that there is a number \(L < \infty\) such that for all \(n \geq 0\),
$$\ell(\Psi_{n}^{\pm}, \lambda^{\pm}) \leq L.$$
Consider the sequence \([\rho_n] \epsilon \mathcal{T}(\Gamma)\) that corresponds to \((\Psi_{n}^{+}, \Psi_{n}^{-})\) under that Bers isomorphism.

Keywords

Riemann Surface Conjugacy Class Length Function Closed Geodesic Fuchsian Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, DavisDavisU.S.A.

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