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July 2012 Teichmüller disks and hyperbolic mapping classes
Chaohui Zhang
Hiroshima Math. J. 42(2): 169-187 (July 2012). DOI: 10.32917/hmj/1345467069

Abstract

Let $\tilde{S}$ be an analytically finite Riemann surface of type $(p,n)$ with $3p+n>3$. Let $x\in \tilde{S}$ and $S=\tilde{S}\backslash \{x\}$. Let $\mbox{Mod}_S^x$ denote the $x$-pointed mapping class group of $S$ and $\mbox{Mod}_{\tilde{S}}$ the mapping class group of $\tilde{S}$. Then the natural projection $J:T(S)\rightarrow T(\tilde{S})$ between Teichmüller spaces induces a group epimorphism $I:\mbox{Mod}_S^x\rightarrow \mbox{Mod}_{\tilde{S}}$. It is well known that for a given Teichmüller disk $\tilde{\Delta}$ in $T(\tilde{S})$, there is a family $\mathscr{F}(\tilde{\Delta})$ of Teichmüller disks $\Delta(z)$ in $T(S)$ parametrized by a hyperbolic plane. If $\tilde{\Delta}$ is invariant under a hyperbolic mapping class $\tilde{\theta}$, then all known hyperbolic mapping classes $\theta\in \mbox{Mod}_S^x$ for which $I(\theta)=\tilde{\theta}$ stem from the construction of $\mathscr{F}(\tilde{\Delta})$. We show that if $\tilde{\theta}$ is represented by a product of Dehn twists along two filling simple closed geodesics, then there exist infinitely many hyperbolic mapping classes $\gamma\in \mbox{Mod}_S^x$ with $I(\gamma)=\tilde{\theta}$ so that their invariant Teichmüller disks are not members of $\mathscr{F}(\tilde{\Delta})$. The result contrasts with the original pattern established by I. Kra.

Citation

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Chaohui Zhang. "Teichmüller disks and hyperbolic mapping classes." Hiroshima Math. J. 42 (2) 169 - 187, July 2012. https://doi.org/10.32917/hmj/1345467069

Information

Published: July 2012
First available in Project Euclid: 20 August 2012

zbMATH: 1250.32013
MathSciNet: MR2978301
Digital Object Identifier: 10.32917/hmj/1345467069

Subjects:
Primary: 32G15
Secondary: 30C60, 30F60

Rights: Copyright © 2012 Hiroshima University, Mathematics Program

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Vol.42 • No. 2 • July 2012
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