## Algebraic & Geometric Topology

### Asymptotics of a class of Weil–Petersson geodesics and divergence of Weil–Petersson geodesics

Babak Modami

#### Abstract

We show that the strong asymptotic class of Weil–Petersson geodesic rays with narrow end invariant and bounded annular coefficients is determined by the forward ending laminations of the geodesic rays. This generalizes the recurrent ending lamination theorem of Brock, Masur and Minsky. As an application we provide a symbolic condition for divergence of Weil–Petersson geodesic rays in the moduli space.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 1 (2016), 267-323.

Dates
Revised: 5 April 2015
Accepted: 5 May 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841109

Digital Object Identifier
doi:10.2140/agt.2016.16.267

Mathematical Reviews number (MathSciNet)
MR3470702

Zentralblatt MATH identifier
1334.30018

#### Citation

Modami, Babak. Asymptotics of a class of Weil–Petersson geodesics and divergence of Weil–Petersson geodesics. Algebr. Geom. Topol. 16 (2016), no. 1, 267--323. doi:10.2140/agt.2016.16.267. https://projecteuclid.org/euclid.agt/1510841109

#### References

• R,L Bishop, B O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969) 1–49
• M,R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999)
• J,F Brock, The Weil–Petersson metric and volumes of $3$–dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16 (2003) 495–535
• J Brock, H Masur, Y Minsky, Asymptotics of Weil–Petersson geodesic, I: Ending laminations, recurrence, and flows, Geom. Funct. Anal. 19 (2010) 1229–1257
• J Brock, H Masur, Y Minsky, Asymptotics of Weil–Petersson geodesics, II: Bounded geometry and unbounded entropy, Geom. Funct. Anal. 21 (2011) 820–850
• P Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics 106, Birkhäuser, Boston (1992)
• I Chavel, Riemannian geometry, 2nd edition, Cambridge Studies in Advanced Mathematics 98, Cambridge Univ. Press (2006)
• J Cheeger, D,G Ebin, Comparison theorems in Riemannian geometry, revised edition, AMS Chelsea Publishing, Providence (2008)
• P Eberlein, Geodesic flows on negatively curved manifolds, I, Ann. of Math. 95 (1972) 492–510
• R,L Foote, Regularity of the distance function, Proc. Amer. Math. Soc. 92 (1984) 153–155
• E Klarreich, The boundary at infinity of the curve complex, preprint (1999) Available at \setbox0\makeatletter\@url http://www.msri.org/people/members/klarreic/curvecomplex.ps {\unhbox0
• H,A Masur, Y,N Minsky, Geometry of the complex of curves, I: Hyperbolicity, Invent. Math. 138 (1999) 103–149
• H,A Masur, Y,N Minsky, Geometry of the complex of curves, II: Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902–974
• Y,N Minsky, Quasi-projections in Teichmüller space, J. Reine Angew. Math. 473 (1996) 121–136
• B Modami, Prescribing the behavior of Weil–Petersson geodesics in the moduli space of Riemann surfaces, J. Topol. Anal. 7 (2015) 543–676
• R,C Penner, J,L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton Univ. Press (1992)
• S,A Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space, Surv. Differ. Geom. 8, International Press, Somerville, MA (2003) 357–393
• S,A Wolpert, Behavior of geodesic-length functions on Teichmüller space, J. Differential Geom. 79 (2008) 277–334
• S,A Wolpert, Extension of the Weil–Petersson connection, Duke Math. J. 146 (2009) 281–303
• S,A Wolpert, Families of Riemann surfaces and Weil–Petersson geometry, CBMS Regional Conference Series in Mathematics 113, Amer. Math. Soc. (2010)
• S,A Wolpert, Understanding Weil–Petersson curvature, from: “Geometry and analysis, 1”, (L Ji, editor), Adv. Lect. Math. 17, Int. Press, Somerville, MA (2011) 495–515
• S,A Wolpert, Geodesic-length functions and the Weil–Petersson curvature tensor, J. Differential Geom. 91 (2012) 321–359