Free subgroups of surface mapping class groups
HTML articles powered by AMS MathViewer
- by James W. Anderson, Javier Aramayona and Kenneth J. Shackleton
- Conform. Geom. Dyn. 11 (2007), 44-55
- DOI: https://doi.org/10.1090/S1088-4173-07-00156-7
- Published electronically: March 15, 2007
- PDF | Request permission
Corrigendum: Conform. Geom. Dyn. 13 (2009), 136-138.
Abstract:
We quantify the generation of free subgroups of surface mapping class groups by pseudo-Anosov mapping classes in terms of their translation distance and the distance between their axes in Teichmüller’s metric. The method makes reference to Teichmüller space only.References
- Joan S. Birman and Caroline Series, Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology 24 (1985), no. 2, 217–225. MR 793185, DOI 10.1016/0040-9383(85)90056-4
- Francis Bonahon, Geodesic laminations on surfaces, Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998) Contemp. Math., vol. 269, Amer. Math. Soc., Providence, RI, 2001, pp. 1–37. MR 1810534, DOI 10.1090/conm/269/04327
- Brian H. Bowditch, Hyperbolic 3-manifolds and the geometry of the curve complex, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 103–115. MR 2185739
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
- Georgios Daskalopoulos, Ludmil Katzarkov, and Richard Wentworth, Harmonic maps to Teichmüller space, Math. Res. Lett. 7 (2000), no. 1, 133–146. MR 1748294, DOI 10.4310/MRL.2000.v7.n1.a12
- Georgios Daskalopoulos and Richard Wentworth, Classification of Weil-Petersson isometries, Amer. J. Math. 125 (2003), no. 4, 941–975. MR 1993745, DOI 10.1353/ajm.2003.0023
- Benson Farb and Lee Mosher, Convex cocompact subgroups of mapping class groups, Geom. Topol. 6 (2002), 91–152. MR 1914566, DOI 10.2140/gt.2002.6.91
- H. Hamidi-Tehrani, On free subgroups of the mapping class groups, Preprint (1997).
- Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
- N. V. Ivanov, Coefficients of expansion of pseudo-Anosov homeomorphisms, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), no. Issled. Topol. 6, 111–116, 191 (Russian, with English summary); English transl., J. Soviet Math. 52 (1990), no. 1, 2819–2822. MR 964259, DOI 10.1007/BF01099245
- Nikolai V. Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, vol. 115, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by E. J. F. Primrose and revised by the author. MR 1195787, DOI 10.1090/mmono/115
- Nikolai V. Ivanov, Mapping class groups, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 523–633. MR 1886678
- R. P. Kent IV, C. J. Leininger, Shadows of mapping class groups: capturing convex co-compactness, arXiv:math.GT/0505114 (2005).
- Howard Masur, On a class of geodesics in Teichmüller space, Ann. of Math. (2) 102 (1975), no. 2, 205–221. MR 385173, DOI 10.2307/1971031
- Howard A. Masur and Michael Wolf, Teichmüller space is not Gromov hyperbolic, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 259–267. MR 1346811
- John McCarthy, A “Tits-alternative” for subgroups of surface mapping class groups, Trans. Amer. Math. Soc. 291 (1985), no. 2, 583–612. MR 800253, DOI 10.1090/S0002-9947-1985-0800253-8
- John McCarthy and Athanase Papadopoulos, Dynamics on Thurston’s sphere of projective measured foliations, Comment. Math. Helv. 64 (1989), no. 1, 133–166. MR 982564, DOI 10.1007/BF02564666
- John D. McCarthy and Athanase Papadopoulos, The mapping class group and a theorem of Masur-Wolf, Topology Appl. 96 (1999), no. 1, 75–84. MR 1701241, DOI 10.1016/S0166-8641(98)00040-6
- Yair N. Minsky, Quasi-projections in Teichmüller space, J. Reine Angew. Math. 473 (1996), 121–136. MR 1390685, DOI 10.1515/crll.1995.473.121
- R. C. Penner, Bounds on least dilatations, Proc. Amer. Math. Soc. 113 (1991), no. 2, 443–450. MR 1068128, DOI 10.1090/S0002-9939-1991-1068128-8
- Igor Rivin, Simple curves on surfaces, Geom. Dedicata 87 (2001), no. 1-3, 345–360. MR 1866856, DOI 10.1023/A:1012010721583
- H. L. Royden, Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 369–383. MR 0288254
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
- Scott Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. of Math. (2) 109 (1979), no. 2, 323–351. MR 528966, DOI 10.2307/1971114
- Scott A. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 357–393. MR 2039996, DOI 10.4310/SDG.2003.v8.n1.a13
Bibliographic Information
- James W. Anderson
- Affiliation: School of Mathematics, University of Southampton, Southampton SO17 1BJ, England
- Email: j.w.anderson@soton.ac.uk
- Javier Aramayona
- Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, England
- MR Author ID: 796736
- Email: jaram@maths.warwick.ac.uk
- Kenneth J. Shackleton
- Affiliation: Institut des Hautes Études Scientifiques, Le Bois-Marie, 35 Route de Chartres, F-91440 Bures-sur-Yvette, France
- Address at time of publication: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan
- Email: kjs2006@alumni.soton.ac.uk; shackleton.k.aa@m.titech.ac.jp
- Received by editor(s): May 15, 2006
- Received by editor(s) in revised form: November 8, 2006
- Published electronically: March 15, 2007
- Additional Notes: The third author was partially supported by a short-term Japan Society for the Promotion of Science post-doctoral fellowship for foreign researchers, number PE05043.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn. 11 (2007), 44-55
- MSC (2000): Primary 20F65; Secondary 57M50
- DOI: https://doi.org/10.1090/S1088-4173-07-00156-7
- MathSciNet review: 2295997