The equivariant fundamental group, uniformization of real algebraic curves, and global complex analytic coordinates on Teichmüller spaces. (Q1872806)

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.