Complex hyperbolic Fenchel-Nielsen coordinates
Given a compact surface \(\Sigma_g\), Fenchel and Nielsen described how to put coordinates on the Teichmuller space of marked hyperbolic structures \(\Sigma_g\). Fix a \textit{pants decomposition}: that is, a system of curves \(\{\gamma_1, \dots, \gamma_{3g-3}\}\) whose complement is a collection of \(2g-2\) \(3\)-holed spheres. To each curve \(\gamma_i\), we associate a \textit{length} \(l_i \in \mathbb R^+\) and a \textit{twist} \(k_i \in \mathbb R\). The \(6g-6\) parameters \((l_i, k_i)\), \(1 \leq i \leq 3g-3\) form the Fenchel-Nielsen coordinates. Viewing Teichmuller spaces as the variety of (conjugacy classes of) discrete, faithful, purely loxodromic (hyperbolic) representations from \(\pi_1(\Sigma_g)\) into \(SL(2,\mathbb R)\), the length and twist parameters can be read off as traces of matrices and cross-ratios of fixed points. If one considers \textit{quasi-Fuchsian} representations, the length and twist parameters become complex, and it is no longer easy to determine which coordinates yield discrete representations. In this paper, Fenchel-Nielsen coordinates are given for the complex hyperbolic quasi-Fuchsian representation space, that is, a part of the \(SU(2, 1)\)-representation variety. The authors consider two cases: 1. The representation is irreducible, in which case there are \(16g-16\) (real) parameters. 2. The representation preserves a complex line, in which case there are \(8g-6\) parameters. The authors define 3 types of parameters: complex length, twist-bend, and cross-ratio.
scientific article; zbMATH DE number 5544208