The asymptotic behavior of Jenkins-Strebel rays

Amano, Masanori

doi:10.1090/S1088-4173-2014-00268-9

The asymptotic behavior of Jenkins-Strebel rays
HTML articles powered by AMS MathViewer

by Masanori Amano PDF

Conform. Geom. Dyn. 18 (2014), 157-170 Request permission

Abstract:

In this paper, we consider the asymptotic behavior of two Teichmüller geodesic rays determined by Jenkins-Strebel differentials, and we obtain a generalization of a theorem by the author in On behavior of pairs of Teichmüller geodesic rays, 2014 . We also consider the infimum of the asymptotic distance up to choice of base points of the rays along the geodesics. We show that the infimum is represented by two quantities. One is the detour metric between the end points of the rays on the Gardiner-Masur boundary of the Teichmüller space, and the other is the Teichmüller distance between the end points of the rays on the augmented Teichmüller space.

References

William Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), no. 1, 29–44. MR 442293, DOI 10.2307/1971024
William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, vol. 820, Springer, Berlin, 1980. MR 590044
Masanori Amano, On behavior of pairs of Teichmüller geodesic rays, Conform. Geom. Dyn. 18 (2014), 8–30. MR 3162997, DOI 10.1090/S1088-4173-2014-00261-6
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
Subhojoy Gupta. Asymptoticity of grafting and Teichmüller rays II. Geom. Dedicata, to appear.
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, 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
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
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
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 asymptotic geometry of the Teichmüller metric. arXiv:1210.5565v1, 2012.