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December 2012 Pseudo-Anosov Maps and Pairs of Filling Simple Closed Geodesics on Riemann Surfaces
Chaohui ZHANG
Tokyo J. Math. 35(2): 469-482 (December 2012). DOI: 10.3836/tjm/1358951331

Abstract

Let $S$ be a Riemann surface of finite area with at least one puncture $x$. Let $a\subset S$ be a simple closed geodesic. In this paper, we show that for any pseudo-Anosov map $f$ of $S$ that is isotopic to the identity on $S\cup \{x\}$, the pair $(a, f^m(a))$ of geodesics fills $S$ for $m\geq 3$. We also study the cases of $0<m\leq 2$ and show that if $(a,f^2(a))$ does not fill $S$, then there is only one geodesic $b$ such that $b$ is disjoint from both $a$ and $f^2(a)$. In fact, $b=f(a)$ and $\{a,f(a)\}$ forms the boundary of an $x$-punctured cylinder on $S$. As a consequence, we show that if $a$ and $f(a)$ are not disjoint, then $(a,f^m(a))$ fills $S$ for any $m\geq 2$.

Citation

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Chaohui ZHANG. "Pseudo-Anosov Maps and Pairs of Filling Simple Closed Geodesics on Riemann Surfaces." Tokyo J. Math. 35 (2) 469 - 482, December 2012. https://doi.org/10.3836/tjm/1358951331

Information

Published: December 2012
First available in Project Euclid: 23 January 2013

zbMATH: 1266.32018
MathSciNet: MR3058719
Digital Object Identifier: 10.3836/tjm/1358951331

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

Rights: Copyright © 2012 Publication Committee for the Tokyo Journal of Mathematics

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Vol.35 • No. 2 • December 2012
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