×

zbMATH — the first resource for mathematics

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 \(m\geq 2\).

MSC:
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.