The equivariant fundamental group, uniformization of real algebraic curves, and global complex analytic coordinates on Teichmüller spaces. (English) Zbl 1054.30042
The author constructs explicitly an Earle’s slice by using a co-compact torsion free Fuchsian group \(\Gamma\), acting on the upper-half plane \(\mathbb H\), so that there is an anticonformal automorphism \(\tau:{\mathbb H} \to {\mathbb H}\) satisfying \(\tau^{2} \in \Gamma\) and \(\tau \Gamma \tau^{-1} = \Gamma\). In this way, he obtains a global complex parameter for the Teichmüller space of \(\Gamma\) which restricts to a global real paramater for \(\langle \Gamma, \tau \rangle\), that is, for the (real) Teichmüller space of real algebraic curves.
Reviewer: Ruben A. Hidalgo (Valparaiso)
MSC:
| 30F60 | Teichmüller theory for Riemann surfaces |
| 30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |
| 30F50 | Klein surfaces |
| 14H30 | Coverings of curves, fundamental group |
| 14H15 | Families, moduli of curves (analytic) |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
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