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The geometry of Markoff numbers. (English) Zbl 0566.10024

This article offers a very fine exposition of the result of Andrew Haas to the effect that the Markoff quadratic irrationalities and simple closed geodesics on the punctured torus H/\(\Gamma\) ’ are identified. In addition, the article contains a discussion of the analogue of the necessary dynamics for Haas’ theorem in the case of a torus, uniformized by the plane. Some interesting historical data are provided for this case, some of it concerning ancient astronomy. The article also discusses the author’s own work on the description of the exact connection between continued fractions and the tracing of geodesics on the modular surface, H/\(\Gamma\) (1). The paper ends with the interesting speculation that perhaps there is also a nice structure for geodesics on a punctured torus that self-intersect precisely once.
One typo: The citations for references 6 (Haas’ paper) and 7 are reversed.
Reviewer: M.Sheingorn

MSC:

11J04 Homogeneous approximation to one number
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
11F06 Structure of modular groups and generalizations; arithmetic groups
30F99 Riemann surfaces
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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References:

[1] E. B. Christoffel,Observatio Arithmetica, Annali di Mathe- matica, 2nd series, 6(1875), 148–152.
[2] H. Cohn, Approach to Markoff’s minimal forms through modular functions.Ann. Math. 61(1955), 1–12. · Zbl 0064.04303
[3] H. Cohn, Representation of Markoff’s binary quadratic forms by geodesics on a perforated torus.Ada Arithmetica XVIII(1971), 125–136. · Zbl 0218.10041
[4] L. E. Dickson, Studies in the theory of numbers. Chicago: 1930. · JFM 56.0154.01
[5] D. Fowler, Anthyphairetic ratio and Eudoxan proportion.Archive for History of Exact Sciences 24(1981), 69–72.
[6] A. Haas, Diophantine approximation on hyperbolic Rie-mann surfaces,Bull. A.M.S. 12(1984), 359–362.
[7] J. Lehner, M. Scheingorn, Simple closed geodesics on H+/r(3) arise from the Markoff spectrum, preprint.
[8] A. A. Markoff, Sur les formes binaires indefinies, I,Math. Ann. 15(1879), 281–309; II, 17(1880), 379–400. · JFM 11.0147.01
[9] C. Series, The modular surface and continued fractions. J. London Math. Soc. (1984). · Zbl 0545.30001
[10] A. L. Schmidt, Minimum of quadratic forms with respect to Fuchsian groups I.J. Reine Angew. Math. 286/7 (1976), 341–368. · Zbl 0332.10015
[11] H. J. S. Smith, Note on continued fractions.Messenger of Mathematics, 2nd series, 6(1876), 1–14.
[12] E. C. Zeeman, An algorithm for Eudoxan and anthi- phairetic ratios, preprint.
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