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April 2019 A classification problem on mapping classes on fiber spaces over Teichmüller spaces
Yingqing Xiao, Chaohui Zhang
Osaka J. Math. 56(2): 213-227 (April 2019).

Abstract

Let $\tilde{S}$ be an analytically finite Riemann surface which is equipped with a hyperbolic metric. Let $S=\tilde{S}\backslash \{\mbox{one point}\ x\}$. There exists a natural projection $\Pi$ of the $x$-pointed mapping class group Mod$_S^x$ onto the mapping class group Mod$(\tilde{S})$. In this paper, we classify elements in the fiber $\Pi^{-1}(\chi)$ for an elliptic element $\chi\in \mbox{Mod}(\tilde{S})$, and give a geometric interpretation for each element in $\Pi^{-1}(\chi)$. We also prove that $\Pi^{-1}(t_a^n\circ \chi)$ or $\Pi^{-1}(t_a^n\circ \chi^{-1})$ consists of hyperbolic mapping classes provided that $t_a^n\circ \chi$ and $t_a^n\circ \chi^{-1}$ are hyperbolic, where $a$ is a simple closed geodesic on $\tilde{S}$ and $t_a$ is the positive Dehn twist along $a$.

Citation

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Yingqing Xiao. Chaohui Zhang. "A classification problem on mapping classes on fiber spaces over Teichmüller spaces." Osaka J. Math. 56 (2) 213 - 227, April 2019.

Information

Published: April 2019
First available in Project Euclid: 3 April 2019

zbMATH: 07080080
MathSciNet: MR3934971

Subjects:
Primary: 30F40
Secondary: 32G05

Rights: Copyright © 2019 Osaka University and Osaka City University, Departments of Mathematics

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Vol.56 • No. 2 • April 2019
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