Some remarks on Teichmüller spaces and modular groups
type: Item
instance of: scholarly article
title: Some remarks on Teichmüller spaces and modular groups
zbMATH Open document ID: 1173.30029
author: Yun Hu
publication date: 2009-09-03T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 5601623
zbMATH Keywords: Teichmüller space
zbMATH Keywords: modular group
zbMATH Keywords: Beltrami coefficient
zbMATH Keywords: quasiconformal mapping
MaRDI profile type: Publication
full work available at URL: s10114-009-7443-x
OpenAlex ID: W2151854627
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DOI: 10.1007/S10114-009-7443-X
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5601623