An explicit description of the Teichmüller space as holonomy representations and its applications
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).
scientific article; zbMATH DE number 108661
scientific article