Closed geodesics on a Riemann surface with application to the Markov spectrum

Beardon, A. F.; Lehner, J.; Sheingorn, M.

doi:10.1090/S0002-9947-1986-0833700-7

Closed geodesics on a Riemann surface with application to the Markov spectrum
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by A. F. Beardon, J. Lehner and M. Sheingorn PDF

Trans. Amer. Math. Soc. 295 (1986), 635-647 Request permission

Abstract:

This paper determines those Riemann surfaces on which each nonsimple closed geodesic has a parabolic intersection—that is, an intersection in the form of a loop enclosing a puncture or a deleted disk. An application is made characterizing the simple closed geodesic on $H/\Gamma (3)$ in terms of the Markov spectrum. The thrust of the situation is this: If we call loops about punctures or deleted disks boundary curves, then if the surface has "little" topology, each nonsimple closed geodesic must contain a boundary curve. But if there is "enough" topology, there are nonsimple closed geodesics not containing boundary curves.

References

Harvey Cohn, Approach to Markoff’s minimal forms through modular functions, Ann. of Math. (2) 61 (1955), 1–12. MR 67935, DOI 10.2307/1969618

Diophantische Approximationen

Andrew Haas, Diophantine approximation on hyperbolic Riemann surfaces, Acta Math. 156 (1986), no. 1-2, 33–82. MR 822330, DOI 10.1007/BF02399200
J. Lehner and M. Sheingorn, Simple closed geodesics on $H^{+}/\Gamma (3)$ arise from the Markov spectrum, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 359–362. MR 752798, DOI 10.1090/S0273-0979-1984-15307-2
Oskar Perron, Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1954 (German). 3te Aufl. MR 0064172
Asmus L. Schmidt, Minimum of quadratic forms with respect to Fuchsian groups. I, J. Reine Angew. Math. 286(287) (1976), 341–368. MR 457358, DOI 10.1515/crll.1976.286-287.341
Caroline Series, The geometry of Markoff numbers, Math. Intelligencer 7 (1985), no. 3, 20–29. MR 795536, DOI 10.1007/BF03025802
Mark Sheingorn, Characterization of simple closed geodesics on Fricke surfaces, Duke Math. J. 52 (1985), no. 2, 535–545. MR 792188, DOI 10.1215/S0012-7094-85-05228-7
Don Zagier, On the number of Markoff numbers below a given bound, Math. Comp. 39 (1982), no. 160, 709–723. MR 669663, DOI 10.1090/S0025-5718-1982-0669663-7