Some remarks on Teichmüller spaces and modular groups
In this paper, the author studies the fixed point sets of subgroups of modular groups acting on Teichmüller spaces. Let \(\Gamma\) be a Fuchsian group acting on the upper half plane \(\mathbb U\) and let \(M(\Gamma)\) be the set of all Beltrami coefficients \(\mu\) for \(\Gamma\) on \(\mathbb U\). Then the Teichmüller space \(T(\Gamma)\) is the set of all the equivalence classes \([\mu]\) of the Beltrami coefficients. Let \(\text{Mod}(\Gamma)\) denote the modular group of \(\Gamma\) and let \(H\) be a subgroup of \(\text{Mod}(\Gamma)\). We denote by \(T(\Gamma)^H\) the set of points fixed by the action of \(H\). For any \(\mu\in M(\Gamma)\) let \(w_\mu\) denote the unique quasiconformal mapping of \(\mathbb U\) onto itself with the fixed points \(0\), \(1\) and \(\infty\), and which satisfies the Beltrami equation \(\partial_{\bar z}w= \mu\partial_zw\). For any \(\mu\in M(G)\), we denote \(\Gamma_\mu=\{\gamma_\mu=w_\mu\circ\gamma\circ w_\mu^{-1}:\gamma\in\Gamma\}\) (\(\Gamma_\mu\) is again a Fuchsian group). Set \(G_\mu=\{\alpha_{\mu,w}:\langle w\rangle\in H\}\), where \(\alpha_{\mu,w}\) is such that \(\alpha_{\mu,w}\circ w_\mu\circ w^{-1}\) is equivalent to \(w_\mu\). Let \(\tilde\Gamma_\mu\) be the group generated by \(\Gamma\) and \(G_\mu\). We assume that \(\Gamma\) is not an abelian group and state the main result of the paper: \(T(\Gamma)^H\) is biholomorphic to \(T(\tilde\Gamma_\mu)\) under the isomorphism \(\chi([w_\mu]): T(\Gamma)\longrightarrow T(\Gamma_\mu)\). The proof is based on a result proved by Tukia and reproved by Douady--Earle.
scientific article; zbMATH DE number 5601623
scientific article