The geometry of Markoff numbers. (English) Zbl 0566.10024

Series, Caroline

Math. Intell. 7, No. 3, 20-29 (1985).

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)

Keywords:

Markoff spectrum; self-intersections; Markoff quadratic irrationalities; simple closed geodesics; punctured torus; continued fractions; modular surface

References:

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[3]	Cohn, H., Representation of Markoff’s binary quadratic forms by geodesics on a perforated torus, Ada Arithmetica, XVIII, 125-136 (1971) · Zbl 0218.10041
[4]	L. E. Dickson, Studies in the theory of numbers. Chicago: 1930. · JFM 56.0154.01
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