Hyperbolization of Fibrations

Part of the Modern Birkhäuser Classics book series (MBC)


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.


Riemann Surface Conjugacy Class Length Function Closed Geodesic Fuchsian Group 


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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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