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Almost all simple lengths are unique up to isometry

Primarily AI-generated textHuman understanding: some partsmath.GT — Geometric Topologymath.NT — Number Theory

Contributed by Nhat Minh Doan ↗

Version 1 / Oct 17, 2026 / CC BY-SA 4.0

Abstract

For every fixed hyperbolic torus with one cusp or one geodesic boundary, we prove that, among simple curves of length at most LL, the number of isometry orbits and the number of distinct lengths differ by o(L2)o(L^2). Thus almost every complete length level is a single isometry orbit of at most six curves, and multiplicity converges in mean to the generic orbit size. The modular specialization is exactly Chen's density theorem for Markoff uniqueness. We also prove the full Schmutz--Schaller bound for every punctured torus having a simple trace that is not an algebraic integer, including any transcendental trace, and for dense families with integral traces. More strongly, every rational length class on these surfaces contains at most two curves. Transcendental twists give multiplicity one, while symmetric families with geodesic boundary attain the sharp bounds four and six. The density proof extends Chen's method using the stable-norm estimates of Doan-Li-Nguyen.

Provenance statement

Substantial parts of the arguments and text were generated with OpenAI's ChatGPT through Codex. The starting points were the papers of Chen \cite{Chen} and Doan--Li--Nguyen \cite{DLN}, together with the author's earlier drafts motivated by Hugo Parlier's question about once-punctured tori with bounded simple-length multiplicity. Assistance included literature searches, proof development, drafting, revision, source checks, LaTeX compilation, and internal cross-checks by parallel model agents. These checks do not constitute independent peer review or verification by a proof assistant. The author does not claim to have independently verified every generated argument.

Tools used

OpenAI
ChatGPTVersion Astra 6.0, Sol 6.1

References

  1. M. Aigner, Markov's Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings , Springer, Cham, 2013.
  2. G. E. Andrews, A lower bound for the volume of strictly convex bodies with many boundary lattice points , Trans. Amer. Math. Soc. 106 (1963), 270--279.
  3. A. Avila, Almost reducibility and absolute continuity I , arXiv:1006.0704 (https://arxiv.org/abs/1006.0704) (2010).arXiv
  4. A. Avila, Global theory of one-frequency Schrödinger operators , Acta Math. 215 (2015), 1--54.
  5. E. Banaian, Orderings on kk-Markov numbers , Ramanujan J. 71 (2026), Article 12. doi:10.1007/s11139-026-01468-5 (https://doi.org/10.1007/s11139-026-01468-5).DOI
  6. V. Bangert, Mather sets for twist maps and geodesics on tori , Dynamics Reported, Vol. 1, 1988, 1--56.
  7. V. Bangert, Minimal geodesics , Ergodic Theory Dynam. Systems 10 (1990), 263--286.
  8. V. Bangert, Geodesic rays, Busemann functions and monotone twist maps , Calc. Var. Partial Differential Equations 2 (1994), 49--63.
  9. A. Baragar, On the unicity conjecture for Markoff numbers , Canad. Math. Bull. 39 (1996), 3--9.
  10. J. P. Bell, A generalised Skolem--Mahler--Lech theorem for affine varieties , J. Lond. Math. Soc. (2) 73 (2006), 367--379.
Show all 103 referencesShow fewer references
  1. J. P. Bell, Corrigendum: A generalised Skolem--Mahler--Lech theorem for affine varieties , J. Lond. Math. Soc. (2) 78 (2008), 267--272.
  2. J. Bourgain, A. Gamburd and P. Sarnak, Markoff triples and strong approximation , C. R. Math. Acad. Sci. Paris 354 (2016), 131--135.
  3. J. Bourgain, A. Gamburd and P. Sarnak, Strong approximation and Diophantine properties of Markoff triples , J. Amer. Math. Soc. 39 (2026), 177--204.
  4. J. P. Bell, D. Ghioca and T. J. Tucker, The dynamical Mordell--Lang problem for étale maps , Amer. J. Math. 132 (2010), 1655--1675.
  5. C. Burrin, A. Nevo, R. Rühr and B. Weiss, Effective counting for discrete lattice orbits in the plane via Eisenstein series , Enseign. Math. 66 (2020), 259--304. arXiv:1905.01493v1 (https://arxiv.org/abs/1905.01493v1). Numbered results are cited from this preprint version.arXiv
  6. F. Bonahon, The geometry of Teichmüller space via geodesic currents , Invent. Math. 92 (1988), 139--162.
  7. I. Borosh, More numerical evidence on the uniqueness of Markov numbers , BIT 15 (1975), 351--357.
  8. B. H. Bowditch, A proof of McShane's identity via Markoff triples , Bull. Lond. Math. Soc. 28 (1996), 73--78.
  9. B. H. Bowditch, Markoff triples and quasifuchsian groups , Proc. Lond. Math. Soc. (3) 77 (1998), 697--736.
  10. C. A. Brown, An almost linear time algorithm testing whether the Markoff graph modulo pp is connected , arXiv:2401.00630v3 (https://arxiv.org/abs/2401.00630v3) (2024).arXiv
  11. E. Banaian and A. Sen, A generalization of Markov numbers , Ramanujan J. 63 (2024), 1021--1055.
  12. J. O. Button, Markoff numbers, principal ideals and continued fraction expansions , J. Number Theory 87 (2001), 77--95.
  13. J. O. Button, The uniqueness of the prime Markoff numbers , J. Lond. Math. Soc. (2) 58 (1998), 9--17.
  14. J. W. S. Cassels, An Introduction to Diophantine Approximation , Cambridge Tracts in Mathematics and Mathematical Physics 45, Cambridge Univ. Press, 1957.
  15. J. W. S. Cassels, Local Fields , London Math. Soc. Student Texts 3, Cambridge Univ. Press, 1986.
  16. F.-J. Chen and Y.-G. Chen, On the Frobenius conjecture for Markoff numbers , J. Number Theory 133 (2013), 2363--2373.
  17. W. Y. Chen, Nonabelian level structures, Nielsen equivalence, and Markoff triples , Ann. of Math. (2) 199 (2024), 301--443.
  18. W. Y. Chen, Almost all Markoff numbers are unique , Hexagon preprint 2610.00281v2 (2026). https://hexagonmath.org/pdf/2610.00281v2.link
  19. S. Cantat and S. U. Jang, Orbits of automorphism groups of affine surfaces over pp-adic fields , arXiv:2410.08579v1 (https://arxiv.org/abs/2410.08579v1) (2024).arXiv
  20. H. Cohn, Approach to Markoff's minimal forms through modular functions , Ann. of Math. (2) 61 (1955), 1--12.
  21. H. Cohn, Representation of Markoff's binary quadratic forms by geodesics on a perforated torus , Acta Arith. 18 (1971), 125--136.
  22. H. Cohn, Markoff forms and primitive words , Math. Ann. 196 (1972), 8--22.
  23. S. Cantat and J. Xie, Algebraic actions of discrete groups: the pp-adic method , Acta Math. 220 (2018), 239--295.
  24. S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation , J. Reine Angew. Math. 359 (1985), 55--89; correction, 360 (1985), 214.
  25. P. T. Dang, T. T. Dao and N. M. Doan, The torus with asymptotically the fewest simple closed geodesics , Hexagon preprint 2610.00159v1 (2026). https://hexagonmath.org/pdf/2610.00159v1.link
  26. N. M. Doan, X. Li and V. Nguyen, Stable-norm boundary geometry on once-punctured tori , arXiv:2605.14574v3 (https://arxiv.org/abs/2605.14574v3) (2026).arXiv
  27. J. Eddy, E. Fuchs, M. Litman, D. E. Martin and N. Tripeny, Connectivity of Markoff mod-pp graphs and maximal divisors , Proc. Lond. Math. Soc. (3) 130 (2025), no. 2, Paper No. e70027.
  28. A. Eskin, M. Mirzakhani and A. Mohammadi, Effective counting of simple closed geodesics on hyperbolic surfaces , J. Eur. Math. Soc. 24 (2022), 3059--3108.
  29. V. Erlandsson and J. Souto, Counting curves in hyperbolic surfaces , Geom. Funct. Anal. 26 (2016), 729--777.
  30. V. Erlandsson and J. Souto, Mirzakhani's Curve Counting and Geodesic Currents , Progress in Mathematics 345, Birkhäuser, Cham, 2022.
  31. B. Farb, Some problems on mapping class groups and moduli space , Problems on Mapping Class Groups and Related Topics, Proc. Sympos. Pure Math. 74, Amer. Math. Soc., 2006, 11--55.
  32. H. Federer, Real flat chains, cochains and variational problems , Indiana Univ. Math. J. 24 (1974/75), 351--407.
  33. E. Fuchs, J. Gaster, D. Martínez-Granado and M. Rabideau, Perspectives on Markov numbers , BIRS workshop report 25w5411, 26--31 January 2025. https://www.birs.ca/workshops/2025/25w5411/report25w5411.pdf.link
  34. D. Fisac, Markov's conjecture on integral necklaces , Bull. Lond. Math. Soc. 57 (2025), 4122--4131.
  35. A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces , Astérisque 66--67, Soc. Math. France, Paris, 1979.
  36. V. V. Fock, Dual Teichmüller spaces , arXiv:dg-ga/9702018v3 (https://arxiv.org/abs/dg-ga/9702018v3) (1997, revised 1998).arXiv
  37. G. Frobenius, Über die Markoffschen Zahlen , Sitzungsber. Königl. Preuss. Akad. Wiss. Berlin (1913), 458--487.
  38. J. Gaster, Boundary slopes for the Markov ordering on relatively prime pairs , Adv. Math. 403 (2022), Paper No. 108377. arXiv:2107.13499 (https://arxiv.org/abs/2107.13499).arXiv
  39. A. O. Gelfond, Sur le septième problème de Hilbert , Bull. Acad. Sci. URSS, VII Sér., Cl. Sci. Math. Nat. (1934), no. 4, 623--634. https://www.mathnet.ru/eng/im4924.link
  40. Y. Gyoda and K. Matsushita, Generalization of Markov Diophantine equation via generalized cluster algebra , Electron. J. Combin. 30 (2023), no. 4, Paper No. 4.10.
  41. A. Gamburd, M. Magee and R. Ronan, An asymptotic formula for integer points on Markoff--Hurwitz varieties , Ann. of Math. (2) 190 (2019), 751--809.
  42. W. M. Goldman, Trace coordinates on Fricke spaces of some simple hyperbolic surfaces , Handbook of Teichmüller Theory, Vol. II, IRMA Lect. Math. Theor. Phys. 13, EMS, 2009, 611--684.
  43. D. S. Gorshkov, Geometry of Lobachevskii in connection with certain questions of arithmetic , J. Soviet Math. 16 (1981), 788--820; English translation of Zap. Nauchn. Sem. LOMI 67 (1977), 39--85.
  44. M. Gromov, Structures métriques pour les variétés riemanniennes , Textes Mathématiques 1, CEDIC/Fernand Nathan, Paris, 1981.
  45. N. Hanada, Number of systoles of once-punctured torus and four-punctured sphere , arXiv:1612.08267 (https://arxiv.org/abs/1612.08267) (2016).arXiv
  46. R. Hines, An infinite product on the Teichmüller space of the once-punctured torus , arXiv:2001.05557v3 (https://arxiv.org/abs/2001.05557v3) (2020).arXiv
  47. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers , 6th ed., Oxford Univ. Press, 2008.
  48. S. U. Jang, Residual transitivity implies minimality for Markoff surfaces over pp-adic integers, by means of pp-adic flows , arXiv:2502.18976v1 (https://arxiv.org/abs/2502.18976v1) (2025).arXiv
  49. V. Jarník, Über die Gitterpunkte auf konvexen Kurven , Math. Z. 24 (1926), 500--518.
  50. A. Ya. Khinchin, Continued Fractions , Dover, Mineola, NY, 1997; reprint of the 1964 Univ. of Chicago Press edition.
  51. D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds , Ann. of Math. (2) 148 (1998), 339--360.
  52. K. Lee, L. Li, M. Rabideau and R. Schiffler, On the ordering of the Markov numbers , Adv. in Appl. Math. 143 (2023), Paper No. 102453.
  53. C. Lagisquet, E. Pelantová, S. Tavenas and L. Vuillon, On the Markov numbers: fixed numerator, denominator, and sum conjectures , Adv. in Appl. Math. 130 (2021), Paper No. 102227.
  54. M. L. Lang and S. P. Tan, A simple proof of the Markoff conjecture for prime powers , Geom. Dedicata 129 (2007), 15--22. arXiv:math/0508443 (https://arxiv.org/abs/math/0508443).arXiv
  55. A. Markoff, Sur les formes quadratiques binaires indéfinies , Math. Ann. 15 (1879), 381--406.
  56. A. Markoff, Sur les formes quadratiques binaires indéfinies. II , Math. Ann. 17 (1880), 379--399.
  57. D. Massart, Stable norms of surfaces: local structure of the unit ball at rational directions , Geom. Funct. Anal. 7 (1997), 996--1010.
  58. J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems , Math. Z. 207 (1991), 169--207.
  59. D. McGinn, Generalized Markoff equations and Chebyshev polynomials , J. Number Theory 152 (2015), 1--20. doi:10.1016/j.jnt.2014.12.006 (https://doi.org/10.1016/j.jnt.2014.12.006).DOI
  60. G. McShane, Convexity and Aigner's conjectures , arXiv:2101.03316v1 (https://arxiv.org/abs/2101.03316v1) (2021).arXiv
  61. G. McShane, Simple geodesics and a series constant over Teichmüller space , Invent. Math. 132 (1998), 607--632.
  62. J. S. Milne, Algebraic Number Theory , version 3.08 (2020), course notes, available at https://www.jmilne.org/math/CourseNotes/ANT.pdf.link
  63. M. Mirzakhani, Simple Geodesics on Hyperbolic Surfaces and the Volume of the Moduli Space of Curves , Ph.D. thesis, Harvard University, 2004.
  64. M. Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces , Ann. of Math. (2) 168 (2008), 97--125.
  65. W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations , 2nd revised ed., Dover, New York, 1976.
  66. G. McShane and H. Parlier, Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space , Geom. Topol. 12 (2008), 1883--1919. arXiv:math/0701835 (https://arxiv.org/abs/math/0701835).arXiv
  67. G. McShane and H. Parlier, Simple closed geodesics of equal length on a torus , Geometry of Riemann Surfaces, London Math. Soc. Lecture Note Ser. 368, Cambridge Univ. Press, 2010, 268--282. doi:10.1017/CBO9781139194266.014 (https://doi.org/10.1017/CBO9781139194266.014).DOI
  68. E. Makover, H. Parlier and C. J. Sutton, Constructing metrics on a 2-torus with a partially prescribed stable norm , Manuscripta Math. 139 (2012), 515--534.
  69. G. McShane and I. Rivin, A norm on homology of surfaces and counting simple geodesics , Internat. Math. Res. Notices 1995, no. 2, 61--69. arXiv:math/0005222 (https://arxiv.org/abs/math/0005222).arXiv
  70. G. McShane and I. Rivin, Simple curves on hyperbolic tori , C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1523--1528.
  71. C. O'Sullivan, Patterns in the Markov numbers and their generalizations , preprint (2026), arXiv:2609.14149v1 (https://arxiv.org/abs/2609.14149v1).arXiv
  72. H. Parlier, Simple closed geodesics and the study of Teichmüller spaces , Handbook of Teichmüller Theory, Vol. IV, EMS, 2014, 113--134.
  73. M. Perling, Unfocused notes on the Markoff equation and T-singularities , arXiv:2210.12982v1 (https://arxiv.org/abs/2210.12982v1) (2022).arXiv
  74. B. Poonen, pp-adic interpolation of iterates , Bull. Lond. Math. Soc. 46 (2014), 525--527. https://math.mit.edu/~poonen/papers/p-iteration.pdf.link
  75. I. Rivin, Simple curves on surfaces , Geom. Dedicata 87 (2001), 345--360.
  76. R. T. Rockafellar, Convex Analysis , Princeton Mathematical Series 28, Princeton Univ. Press, 1970.
  77. M. Rabideau and R. Schiffler, Continued fractions and orderings on the Markov numbers , Adv. Math. 370 (2020), Paper No. 107231.
  78. R. Schneider, Convex Bodies: The Brunn--Minkowski Theory , 2nd expanded ed., Encyclopedia of Mathematics and its Applications 151, Cambridge Univ. Press, 2014.
  79. P. Schmutz Schaller, The modular torus has maximal length spectrum , Geom. Funct. Anal. 6 (1996), 1057--1073. doi:10.1007/BF02246996 (https://doi.org/10.1007/BF02246996).DOI
  80. P. Schmutz, Systoles of arithmetic surfaces and the Markoff spectrum , Math. Ann. 305 (1996), 191--203. doi:10.1007/BF01444218 (https://doi.org/10.1007/BF01444218).DOI
  81. P. Schmutz Schaller, Geometry of Riemann surfaces based on closed geodesics , Bull. Amer. Math. Soc. (N.S.) 35 (1998), 193--214.
  82. C. Series, The geometry of Markoff numbers , Math. Intelligencer 7 (1985), no. 3, 20--29.
  83. C. Series, The modular surface and continued fractions , J. Lond. Math. Soc. (2) 31 (1985), 69--80.
  84. A. Srinivasan, Markoff numbers and ambiguous classes , J. Théor. Nombres Bordeaux 21 (2009), 757--770.
  85. The Stacks Project Authors, The Stacks Project , Lemma 10.50.2, Tag 00IA, https://stacks.math.columbia.edu/tag/00IA.link
  86. R. Strassmann, Über den Wertevorrat von Potenzreihen im Gebiet der pp-adischen Zahlen , J. Reine Angew. Math. 159 (1928), 13--28.
  87. D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics , Acta Math. 149 (1982), 215--237.
  88. K. Spalding and A. P. Veselov, Lyapunov spectrum of Markov and Euclid trees , Nonlinearity 30 (2017), 4428--4453.
  89. A. Sorrentino and A. P. Veselov, Markov numbers, Mather's β\beta function and stable norm , Nonlinearity 32 (2019), 2147--2156. arXiv:1707.03901 (https://arxiv.org/abs/1707.03901).arXiv
  90. W. P. Thurston, Minimal stretch maps between hyperbolic surfaces , 1986 preprint; arXiv:math/9801039 (https://arxiv.org/abs/math/9801039) (1998 version).arXiv
  91. T. Tao and B. Yang, Markov's equation is not partition regular , arXiv:2510.10151v1 (https://arxiv.org/abs/2510.10151v1) (2025).arXiv
  92. D. Zagier, On the number of Markoff numbers below a given bound , Math. Comp. 39 (1982), 709--723.
  93. Y. Zhang, Congruence and uniqueness of certain Markoff numbers , Acta Arith. 128 (2007), 295--301.

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  1. v1Submitted by Nhat Minh DoanInitial depositCurrentOct 17, 2026