Abstract
We show that an Anosov map has a geodesic axis on the curve graph of the torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map. The application of our result is threefold: (a) to determine which word realizes the minimal translation length on the curve graph within a specific class of words, (b) to establish the effective bound on the ratio of translation lengths of an Anosov map on the curve graph to that on Teichmüller space, and (c) to estimate the overall growth of the number of Anosov maps which have a sufficient number of Anosov maps with the same translation length .
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Data availability
Not applicable.
References
Aougab, T., Taylor, S.J.: Pseudo-Anosovs optimizing the ratio of Teichmüller to curve graph translation length. the tradition of Ahlfors-Bers VII. Contemp. Math. 696, 17–28 (2017)
Baik, H., Rafiqi, A., Wu, C.: Is a typical bi-Perron algebraic unit a pseudo-anosov dilatation? Ergod. Theor. Dyn. Syst. 39, 1–6 (2017)
Baik, H., Shin, H.: Minimal asymptotic translation lengths of Torelli groups and pure braid groups on the curve graph. Math. Res. Not. IMRN. 24, 9974–9987 (2018)
Beardon, A.F., Hockman, M., Short, I.: Geodesic continued fractions. Mich. Math. J. 61(1), 133–150 (2012)
Bell, M.C., Webb, R.C.: Polynomial-time algorithms for the curve graph. arXiv preprint arXiv:1609.09392 (2016)
Bers, L.: An extremal problem for quasiconformal mappings and a theorem by Thurston. Acta Mathematica 141(1), 73–98 (1978)
Bowditch, B.H.: Tight geodesics in the curve complex. Inventiones Mathematicae 171(2), 281–300 (2008)
Chowla, S., Cowles, J., Cowles, M.: On the number of conjugacy classes in SL\((2, \mathbb{Z})\). J. Number Theor. 12(3), 372–377 (1980)
Farb, B., Leininger, C.J., Margalit, D.: The lower central series and pseudo-Anosov dilatations. Amer. J. Math. 130(3), 799–827 (2008)
Gadre, V., Tsai, C.Y.: Minimal pseudo-Anosov translation lengths on the complex of curves. Geom. Topol. 15(3), 1297–1312 (2011)
Gadre, V., Hironaka, E., Kent, R.P., Leininger, C.J.: Lipschitz constants to curve complexes. Math. Res. Lett. 20(4), 647–656 (2013)
Harvey, W.J.: Boundary structure of the modular group. Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) pp. 245–251 (1981)
Hatcher, A.: Topology of numbers. Unpublished manuscript, in preparation (2017)
Kin, E., Shin, H.: Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex. Groups Geom. Dyn. 13(3), 883–907 (2019)
Masur, H.A., Minsky, Y.N.: Geometry of the complex of curves I: hyperbolicity. Inventiones Mathematicae 138(1), 103–149 (1999)
Minsky, Y.N.: A geometric approach to the complex of curves on a surface. In: Topology and Teichmüller spaces, pp. 149–158. World Scientific (1996)
Series, C.: Continued fractions and hyperbolic geometry (2015)
Shackleton, K.J.: Tightness and computing distances in the curve complex. Geometriae Dedicata 160(1), 243–259 (2012)
Shin, H., Strenner, B.: Pseudo-Anosov mapping classes not arising from Penner’s construction. Geom. Topol. 19(6), 3645–3656 (2016)
Steinig, J.: A proof of Lagrange’s theorem on periodic continued fractions. Archiv der Mathematik 59(1), 21–23 (1992)
Thurston, W.P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull Am. Math. Soc. 19(2), 417–431 (1988)
Valdivia, A.D.: Asymptotic translation length in the curve complex. New York J. Math. 20, 989–999 (2014)
Webb, R.: Combinatorics of tight geodesics and stable lengths. Trans. Am. Math. Soc. 367(10), 7323–7342 (2015)
Acknowledgements
We are grateful to Mladen Bestvina, John Blackman, Sami Douba, Hongtaek Jung, KyeongRo Kim, Justin Lanier, and Dan Margalit for helpful comments. This work was supported by 2018 Summer-Fall KAIST Undergraduate Research Program. The first author was partially supported by Samsung Science and Technology Foundation grant No. SSTFBA1702-01. The fourth author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2017R1D1A1B03035017).
Funding
This work was supported by 2018 Summer-Fall KAIST Undergraduate Research Program. The first author was partially supported by Samsung Science and Technology Foundation Grant no. SSTFBA1702-01. The fourth author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2017R1D1A1B03035017).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Code availability
The code for calculating translation length is available at the following link: http://www.math.utah.edu/~kwak/codes/TranslationLength.py
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Algorithms for ladders
Algorithms for ladders
In this appendix, we provide two algorithms for ladders: to calibrate a ladder with odd length into one with even length(Algorithm 2), and to generate a ladder bounded by two given edges(Algorithm 3). Here ExtRational is the class representing the set of extended rationals \(\overline{\mathbb {Q}}=\mathbb {Q} \cup \{\frac{1}{0}=\infty \}\).
Rights and permissions
About this article
Cite this article
Baik, H., Kim, C., Kwak, S. et al. On translation lengths of Anosov maps on the curve graph of the torus. Geom Dedicata 214, 399–426 (2021). https://doi.org/10.1007/s10711-021-00622-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-021-00622-1