An explicit description of the Teichmüller space as holonomy representations and its applications
type: Item
instance of: scholarly article
title: An explicit description of the Teichmüller space as holonomy representations and its applications
zbMATH Open document ID: 0805.32011
author: Takayuki Okai
published in: Hiroshima Mathematical Journal
publication date: 1993-04-19T00:00:00Z
review text: Let \(\Sigma_ g\) be the closed oriented surface of genus \(g\geq 2\). We will treat with the space of representations \(\Hom(\pi_ 1 (\Sigma_ g), \text{PSU} (1,1))^{e= 2-2g}/ \text{conj}\), where \(\text{PSU}(1,1)= \{ {{a\;b} \brack {\bar b\;\bar a}}\in \text{PSL} (2,\mathbb{C})\}\) = the group of orientation preserving isometries on the Poincaré disk and ``\(e= 2- 2g\)'' stands for the connected component consisting of all representations with the Euler number of the associated oriented \(\mathbb{R} P^ 1\text{-bundles} = 2-2g\). The Teichmüller space \({\mathcal T}_ g\) of \(\Sigma_ g\) is naturally identified with this representation space via holonomy representations after uniformizations. We fix a marking of \(\Sigma_ g\) and a pants decomposition \(P_ 1\cup \cdots\cup P_{2g- 2}= \Sigma_ g\) associated to it. Hence we have a Fenchel-Nielsen coordinate \(\{(l,\tau)\}\) as a parametrization of \({\mathcal F}_ g\). Depending on this marking, we describe a global section of the trivial principal \(\text{PSU}(1,1)\)-bundle \[ \Hom (\pi_ 1 (\Sigma_ g), \text{PSU} (1,1))^{e=2- 2g}\to {\mathcal T}_ g= \Hom(\pi_ 1 (\Sigma_ g), \text{PSU} (1,1))^{e= 2-2g}/ \text{conj}. \] More explicitly we write down the image of a generator system of \(\pi_ 1(\Sigma_ g)\) in \(\text{PSU}(1,1)\) under a holonomy representation \(\rho(l,\tau)\) corresponding to each \((l,\tau)\) (Theorem 1). As an application, the length of the closed geodesic in each free homotopy class of (based) closed curves \(\omega\) in a Riemann surface with Fenchel-Nielsen coordinate \((l,\tau)\) is expressed by the trace of \(\rho(l, \tau)(\omega)\) (Theorem 3.1). On the other hand the parameters \(\tau\) of Nielsen twists are described in terms of lengths of closed geodesics (Theorem 3.2). Combining these two, we get a formula of describing the transformation of the mapping class group on the coordinate \(\{(l,\tau)\}\) (Theorem 3.3).
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 30F60
zbMATH DE Number: 108661
zbMATH Keywords: Teichmüller space
zbMATH Keywords: holonomy representations
zbMATH Keywords: Fenchel-Nielsen coordinate
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206392902
OpenAlex ID: W1539926754
DOI: 10.32917/HMJ/1206392902
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description: scientific article; zbMATH DE number 108661