## On behavior of pairs of Teichmüller geodesic rays

HTML articles powered by AMS MathViewer

- by Masanori Amano
- Conform. Geom. Dyn.
**18**(2014), 8-30 - DOI: https://doi.org/10.1090/S1088-4173-2014-00261-6
- Published electronically: February 6, 2014
- PDF | Request permission

## 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

- William Abikoff,
*Degenerating families of Riemann surfaces*, Ann. of Math. (2)**105**(1977), no. 1, 29–44. MR**442293**, DOI 10.2307/1971024 - 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. - 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** - 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**, DOI 10.1080/17476939108814480 - 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** - Subhojoy Gupta. Asymptoticity of grafting and Teichmüller rays I.
*arXiv:1109.5365v1*, 2011. - John Hubbard and Howard Masur,
*Quadratic differentials and foliations*, Acta Math.**142**(1979), no. 3-4, 221–274. MR**523212**, DOI 10.1007/BF02395062 - 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**, DOI 10.4171/029-1/7 - 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 - 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,
*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**, DOI 10.1090/trans2/202/11 - Steven P. Kerckhoff,
*The asymptotic geometry of Teichmüller space*, Topology**19**(1980), no. 1, 23–41. MR**559474**, DOI 10.1016/0040-9383(80)90029-4 - 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**, DOI 10.1017/CBO9780511809187 - Anna Lenzhen and Howard Masur,
*Criteria for the divergence of pairs of Teichmüller geodesics*, Geom. Dedicata**144**(2010), 191–210. MR**2580426**, DOI 10.1007/s10711-009-9397-7 - Lixin Liu and Weixu Su. The horofunction compactification of Teichmüller metric.
*arXiv:1012.0409v4*, 2012. - 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 Masur,
*Uniquely ergodic quadratic differentials*, Comment. Math. Helv.**55**(1980), no. 2, 255–266. MR**576605**, DOI 10.1007/BF02566685 - Hideki Miyachi,
*Teichmüller rays and the Gardiner-Masur boundary of Teichmüller space*, Geom. Dedicata**137**(2008), 113–141. MR**2449148**, DOI 10.1007/s10711-008-9289-2 - Hideki Miyachi. Teichmüller space has non-Busemann points.
*arXiv:1105.3070v1*, 2011. - Mary Rees,
*An alternative approach to the ergodic theory of measured foliations on surfaces*, Ergodic Theory Dynam. Systems**1**(1981), no. 4, 461–488 (1982). MR**662738**, DOI 10.1017/s0143385700001383 - Marc A. Rieffel,
*Group $C^*$-algebras as compact quantum metric spaces*, Doc. Math.**7**(2002), 605–651. MR**2015055** - 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**, DOI 10.1007/978-3-662-02414-0 - 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**, DOI 10.4171/GGD/122 - Cormac Walsh,
*The asymptotic geometry of the Teichmüller metric*,*arXiv:1210.5565v1*, 2012.

## Bibliographic Information

**Masanori Amano**- Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguroku, Tokyo 152-8551, Japan
- Email: amano.m.ab@m.titech.ac.jp
- 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
- © Copyright 2014 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**18**(2014), 8-30 - MSC (2010): Primary ~32G15; Secondary ~30F60
- DOI: https://doi.org/10.1090/S1088-4173-2014-00261-6
- MathSciNet review: 3162997