Skip to main content
Log in

Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space

Geometriae Dedicata Aims and scope Submit manuscript

Cite this article


We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmüller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston’s stretch maps and prove the following: (1) On the Teichmüller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and define a geodesic metric on this space. (2) On the same space, the arc and the curve metrics coincide when the length of the boundary component is \(\le 4{\text {arcsinh}}(1)\), but differ when the boundary length is large. We further apply our stretch map generalization to construct novel Thurston geodesics on the Teichmüller spaces of closed hyperbolic surfaces, and use these geodesics to show that the sum-symmetrization of the Thurston metric fails to exhibit Gromov hyperbolicity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  1. There are, in fact, uncountably many leaves in \(\lambda \), and only finitely many lie on the boundary.

  2. For the Thurston boundary of Teichmüller spaces of surfaces with boundary, see [2].

  3. This measured lamination is unique when \(S=S_{1,1}\).


  1. Alessandrini, D., Disarlo, V.: Generalized stretch lines for surfaces with boundary. arXiv:1911.10431

  2. Alessandrini, D., Liu, L., Papadopoulos, A., Su, W.: The horofunction compactification of Teichmüller spaces of surfaces with boundary. Topol. Appl. 208, 160–191 (2016)

    Article  Google Scholar 

  3. Brock, J., Farb, B.: Curvature and rank of Teichmüller space. Am. J. Math. 128(1), 1–22 (2006)

    Article  Google Scholar 

  4. Buser, P.: Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, vol. 106. Birkhäuser Boston Inc., Boston (1992)

    MATH  Google Scholar 

  5. Coornaert, M., Delzant, T., Papadopoulos, A.: Géométrie et théorie des groupes. In: Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics, vol. 1441. Springer (1990)

  6. Danciger, J., Guéritaud, F., Kassel, F.: Margulis spacetimes via the arc complex. Invent. Math. 204(1), 133–193 (2016)

    Article  MathSciNet  Google Scholar 

  7. Dumas, D., Lenzhen, A., Rafi, K., Tao, J.: Coarse and fine geometry of the Thurston metric. Forum Math. Sigma 8. Paper No. e28, 58 pp (2020)

  8. Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, New York (1987)

    Google Scholar 

  9. Guéritaud, F., Kassel, F.: Maximally stretched laminations on geometrically finite hyperbolic manifolds. Geom. Topol. 21(2), 693–840 (2017)

    Article  MathSciNet  Google Scholar 

  10. Huang, Y., Papadopoulos, A.: Optimal Lipschitz maps on bordered hyperbolic surfaces and the Thurston metric theory of Teichmüller space (in preparation)

  11. Huang, Y., Sun, Z.: McShane identities for higher Teichmüller theory and the Goncharov–Shen potential. Memoirs of the American Mathematical Society (to appear)

  12. Kerckhoff, S.P.: Earthquakes are analytic. Commun. Math. Helv. 60, 17–30 (1985)

    Article  MathSciNet  Google Scholar 

  13. Lenzhen, A., Rafi, K., Tao, J.: The shadow of a Thurston geodesic to the curve graph. J. Topol. 8, 1085–1118 (2015)

    Article  MathSciNet  Google Scholar 

  14. Liu, L., Papadopoulos, A., Su, W., Théret, G.: On length spectrum metrics and weak metrics on Teichmüller spaces of surfaces with boundary. Ann. Acad. Sci. Fenn. Math. 35(1), 255–274 (2010)

    Article  MathSciNet  Google Scholar 

  15. Papadopoulos, A.: Sur le bord de Thurston de l’espace de Teichmüller d’une surface non compacte. Math. Ann. 282, 353–359 (1988)

    Article  MathSciNet  Google Scholar 

  16. Papadopoulos, A., Su, W.: Thurston’s metric on Teichmüller space and the translation distances of mapping classes. Ann. Acad. Sci. Fenn. Math. 41(2), 867–879 (2016)

    Article  MathSciNet  Google Scholar 

  17. Papadopoulos, A., Théret, G.: Shortening all the simple closed geodesics on surfaces with boundary. Proc. Am. Math. Soc. 138, 1775–1784 (2010)

    Article  MathSciNet  Google Scholar 

  18. Papadopoulos, A., Théret, G.: Some Lipschitz maps between hyperbolic surfaces with applications to Teichmüller theory. Geom. Dedicata 150(1), 233–247 (2011)

    Article  MathSciNet  Google Scholar 

  19. Papadopoulos, A., Yamada, S.: Deforming hexagons and the arc and the Thurston metric on Teichmüller space. Monatsh. Math. 172(1), 97–120 (2017)

    Article  Google Scholar 

  20. Parlier, H.: Lengths of geodesics on Riemann surfaces with boundary. Ann. Acad. Sci. Fenn. Math. 30(2), 227–236 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Rafi, K.: Hyperbolicity in Teichmüller space. Geom. Topol. 18(5), 3025–3053 (2014)

    Article  MathSciNet  Google Scholar 

  22. Thurston, W.P.: The geometry and topology of three-manifolds. Princeton, Lecture notes (1979). Available at

  23. Thurston, W.: Minimal stretch maps between hyperbolic surfaces. arXiv:9801039 [math.GT] (1986)

  24. Thurston, W.P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. 19, 417–431 (1988)

    Article  MathSciNet  Google Scholar 

  25. Walsh, C.: The horoboundary and isometry group of Thurston’s Lipschitz metric. In: Handbook of Teichmüller theory, vol. IV, IRMA Lectures in Mathematics and Theoretical Physics, vol. 19. European Mathematical Society, Zurich (2014)

Download references


We thank Kasra Rafi and David Dumas for correspondence and for his interest in our work, François Guéritaud for his very thoughtful and enlightening responses to our many questions and Vincent Alberge who read a preliminary version of this paper. We also thank the referee whose corrections and suggestions substantially improved this paper.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Yi Huang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, Y., Papadopoulos, A. Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space. Geom Dedicata 214, 465–488 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification