Tokyo Journal of Mathematics

Pseudo-Anosov Maps and Pairs of Filling Simple Closed Geodesics on Riemann Surfaces

Chaohui ZHANG

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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 $m\geq 2$.

Article information

Tokyo J. Math., Volume 35, Number 2 (2012), 469-482.

First available in Project Euclid: 23 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 30C60 30F60: Teichmüller theory [See also 32G15]


ZHANG, Chaohui. Pseudo-Anosov Maps and Pairs of Filling Simple Closed Geodesics on Riemann Surfaces. Tokyo J. Math. 35 (2012), no. 2, 469--482. doi:10.3836/tjm/1358951331.

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  • Bers, L., Fiber spaces over Teichmüller spaces, Acta Math. 130 (1973), 89–126.
  • Bers, L., An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), 73–98.
  • A. Fathi, Dehn twists and pseudo-Anosov diffeomorphisms, Invent. Math. 87 (1987), 129–151.
  • Fathi, A., Laudenbach, F. and Poenaru, V., Travaux de Thurston sur les surfaces, Seminaire Orsay, Asterisque, 66–67, Soc. Math. de France, 1979.
  • Farb, B., Leininger, C. and Margalit, D., The lower central series and pseudo-Anosov dilatations, Amer. J. Math. 130 (2008), 799–827.
  • Kra, I., On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981), 231–270.
  • Masur, H. and Minsky, Y., Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138 (1999), 103–149.
  • Nag, S., Non-geodesic discs embedded in Teichmüller spaces, Amer. J. Math. 104 (1982), 339–408.
  • Thurston, W. P., On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N. S.) 19 (1988), 417–431.
  • Wang, S., Wu, Y. and Zhou, Q., Pseudo-Anosov maps and simple closed curves on surfaces, Math. Proc. Camb. Phil. Soc. 128 (2000), 321–326.
  • Zhang, C., Singularities of quadratic differentials and extremal Teichmüller mappings defined by Dehn twists, Aust. J. Math. 3 (2009), 275–288.
  • $\underline{\qquad\qquad}$, Commuting mapping clases and their actions on the circle at infinity, Acta Math. Sinica 52 (2009), 471–482.
  • $\underline{\qquad\qquad}$, Pseudo-Anosov maps and fixed points of boundary homeomorphisms compatible with a Fuchsian group, Osaka J. Math. 46 (2009), 783–798.
  • $\underline{\qquad\qquad}$, On products of Pseudo-Anosov maps and Dehn twists of Riemann surfaces with punctures, J. Aust. Math. Soc. 88 (2010), 413–428.
  • $\underline{\qquad\qquad}$, Pseudo-Anosov maps and pairs of filling simple closed geodesics on Riemann surfaces, II, Preprint, 2011.
  • $\underline{\qquad\qquad}$, On pseudo-Anosov maps with small dilatations on punctured Riemann spheres, JP Journal of Geometry and Topology 11 (2011), 117–145.
  • $\underline{\qquad\qquad}$, Length estimates for combined simple and filling closed geodesics on hyperbolic Riemann surfaces, Preprint, (2012).
  • $\underline{\qquad\qquad}$, Dehn twists combined with pseudo-Anosov maps, Kodai Math. J. 34 (2011), 367–382.
  • P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge. Philos. Soc. 53(1957), 568-575.