Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



On behavior of pairs of Teichmüller geodesic rays

Author: Masanori Amano
Journal: Conform. Geom. Dyn. 18 (2014), 8-30
MSC (2010): Primary 32G15; Secondary 30F60
Published electronically: February 6, 2014
MathSciNet review: 3162997
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we obtain the explicit limit value of the Teichmüller distance between two Teichmüller geodesic rays which are determined by
Jenkins-Strebel differentials having a common end point in the augmented Teichmüller space. Furthermore, we also obtain a condition under which these two rays are asymptotic. This is similar to a result of Farb and Masur.

References [Enhancements On Off] (What's this?)

  • [Abi77] William Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), no. 1, 29–44. MR 0442293
  • [FLP79] Albert Fathi, François Laudenbach, and Valentin Poénaru.
    Travaux de Thurston sur les surfaces, volume 66 of Astérisque.
    Société Mathématique de France, Paris, 1979.
    Séminaire Orsay, With an English summary.
  • [FM10] Benson Farb and Howard Masur, Teichmüller geometry of moduli space, I: distance minimizing rays and the Deligne-Mumford compactification, J. Differential Geom. 85 (2010), no. 2, 187–227. MR 2732976
  • [GM91] Frederick P. Gardiner and Howard Masur, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl. 16 (1991), no. 2-3, 209–237. MR 1099913
  • [Gro81] M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183–213. MR 624814
  • [Gup11] Subhojoy Gupta.
    Asymptoticity of grafting and Teichmüller rays I.
    arXiv:1109.5365v1, 2011.
  • [HM79] John Hubbard and Howard Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, 221–274. MR 523212, 10.1007/BF02395062
  • [HS07] Frank Herrlich and Gabriela Schmithüsen, On the boundary of Teichmüller disks in Teichmüller and in Schottky space, Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys., vol. 11, Eur. Math. Soc., Zürich, 2007, pp. 293–349. MR 2349673, 10.4171/029-1/7
  • [IT92] 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
  • [Iva92] 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
  • [Iva01] Nikolai V. Ivanov, Isometries of Teichmüller spaces from the point of view of Mostow rigidity, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, 2001, pp. 131–149. MR 1819186
  • [Ker80] Steven P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), no. 1, 23–41. MR 559474, 10.1016/0040-9383(80)90029-4
  • [KH95] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374
  • [LM10] Anna Lenzhen and Howard Masur, Criteria for the divergence of pairs of Teichmüller geodesics, Geom. Dedicata 144 (2010), 191–210. MR 2580426, 10.1007/s10711-009-9397-7
  • [LS12] Lixin Liu and Weixu Su.
    The horofunction compactification of Teichmüller metric.
    arXiv:1012.0409v4, 2012.
  • [Mas75] Howard Masur, On a class of geodesics in Teichmüller space, Ann. of Math. (2) 102 (1975), no. 2, 205–221. MR 0385173
  • [Mas80] Howard Masur, Uniquely ergodic quadratic differentials, Comment. Math. Helv. 55 (1980), no. 2, 255–266. MR 576605, 10.1007/BF02566685
  • [Miy08] Hideki Miyachi, Teichmüller rays and the Gardiner-Masur boundary of Teichmüller space, Geom. Dedicata 137 (2008), 113–141. MR 2449148, 10.1007/s10711-008-9289-2
  • [Miy11] Hideki Miyachi.
    Teichmüller space has non-Busemann points.
    arXiv:1105.3070v1, 2011.
  • [Ree81] Mary Rees, An alternative approach to the ergodic theory of measured foliations on surfaces, Ergodic Theory Dynamical Systems 1 (1981), no. 4, 461–488 (1982). MR 662738
  • [Rie02] Marc A. Rieffel, Group 𝐶*-algebras as compact quantum metric spaces, Doc. Math. 7 (2002), 605–651 (electronic). MR 2015055
  • [Str84] Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423
  • [Wal11] Cormac Walsh, The action of a nilpotent group on its horofunction boundary has finite orbits, Groups Geom. Dyn. 5 (2011), no. 1, 189–206. MR 2763785, 10.4171/GGD/122
  • [Wal12] Cormac Walsh,
    The asymptotic geometry of the Teichmüller metric,
    arXiv:1210.5565v1, 2012.

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 32G15, 30F60

Retrieve articles in all journals with MSC (2010): 32G15, 30F60

Additional Information

Masanori Amano
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguroku, Tokyo 152-8551, Japan

Keywords: Teichm\"uller space, Teichm\"uller distance, Teichm\"uller geodesic, augmented Teichm\"uller space, Gardiner-Masur compactification, horofunction
Received by editor(s): April 18, 2013
Received by editor(s) in revised form: August 7, 2013, and September 11, 2013
Published electronically: February 6, 2014
Article copyright: © Copyright 2014 American Mathematical Society