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Pseudo-Anosov maps and pairs of filling simple closed geodesics on Riemann surfaces. (English) Zbl 1266.32018
Summary: 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 \(m\geq 2\).

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces
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[1] Bers, L., Fiber spaces over Teichmüller spaces, Acta Math. 130 (1973), 89-126. · Zbl 0249.32014
[2] Bers, L., An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), 73-98. · Zbl 0389.30018
[3] A. Fathi, Dehn twists and pseudo-Anosov diffeomorphisms, Invent. Math. 87 (1987), 129-151. · Zbl 0618.58027
[4] Fathi, A., Laudenbach, F. and Poenaru, V., Travaux de Thurston sur les surfaces, Seminaire Orsay , Asterisque, 66-67, Soc. Math. de France, 1979.
[5] Farb, B., Leininger, C. and Margalit, D., The lower central series and pseudo-Anosov dilatations, Amer. J. Math. 130 (2008), 799-827. · Zbl 1187.37060
[6] Kra, I., On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981), 231-270. · Zbl 0477.32024
[7] Masur, H. and Minsky, Y., Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138 (1999), 103-149. · Zbl 0941.32012
[8] Nag, S., Non-geodesic discs embedded in Teichmüller spaces, Amer. J. Math. 104 (1982), 339-408. · Zbl 0497.32018
[9] Thurston, W. P., On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N. S.) 19 (1988), 417-431. · Zbl 0674.57008
[10] 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. · Zbl 0959.57015
[11] Zhang, C., Singularities of quadratic differentials and extremal Teichmüller mappings defined by Dehn twists, Aust. J. Math. 3 (2009), 275-288. · Zbl 1182.30078
[12] \(\underline{\qquad\qquad}\), Commuting mapping clases and their actions on the circle at infinity, Acta Math. Sinica 52 (2009), 471-482. · Zbl 1199.30262
[13] \(\underline{\qquad\qquad}\), Pseudo-Anosov maps and fixed points of boundary homeomorphisms compatible with a Fuchsian group, Osaka J. Math. 46 (2009), 783-798. · Zbl 1179.30048
[14] \(\underline{\qquad\qquad}\), On products of Pseudo-Anosov maps and Dehn twists of Riemann surfaces with punctures, J. Aust. Math. Soc. 88 (2010), 413-428. · Zbl 1194.30049
[15] \(\underline{\qquad\qquad}\), Pseudo-Anosov maps and pairs of filling simple closed geodesics on Riemann surfaces, II, Preprint, 2011.
[16] \(\underline{\qquad\qquad}\), On pseudo-Anosov maps with small dilatations on punctured Riemann spheres, JP Journal of Geometry and Topology 11 (2011), 117-145. · Zbl 1229.32015
[17] \(\underline{\qquad\qquad}\), Length estimates for combined simple and filling closed geodesics on hyperbolic Riemann surfaces, Preprint, (2012).
[18] \(\underline{\qquad\qquad}\), Dehn twists combined with pseudo-Anosov maps, Kodai Math. J. 34 (2011), 367-382. · Zbl 1236.32008
[19] P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge. Philos. Soc. 53 (1957), 568-575. · Zbl 0080.02903
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