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

Authors:
A. F. Beardon, J. Lehner and M. Sheingorn

Journal:
Trans. Amer. Math. Soc. **295** (1986), 635-647

MSC:
Primary 11F99; Secondary 11J06, 30F25

MathSciNet review:
833700

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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 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.

**[1]**Harvey Cohn,*Approach to Markoff’s minimal forms through modular functions*, Ann. of Math. (2)**61**(1955), 1–12. MR**0067935****[2]**J. F. Koksma,*Diophantische Approximationen*, Chelsea, New York, n.d.**[3]**Andrew Haas,*Diophantine approximation on hyperbolic Riemann surfaces*, Acta Math.**156**(1986), no. 1-2, 33–82. MR**822330**, 10.1007/BF02399200**[4]**J. Lehner and M. Sheingorn,*Simple closed geodesics on 𝐻⁺/Γ(3) arise from the Markov spectrum*, Bull. Amer. Math. Soc. (N.S.)**11**(1984), no. 2, 359–362. MR**752798**, 10.1090/S0273-0979-1984-15307-2**[5]**Oskar Perron,*Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche*, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1954 (German). 3te Aufl. MR**0064172****[6]**Asmus L. Schmidt,*Minimum of quadratic forms with respect to Fuchsian groups. I*, J. Reine Angew. Math.**286/287**(1976), 341–368. MR**0457358****[7]**Caroline Series,*The geometry of Markoff numbers*, Math. Intelligencer**7**(1985), no. 3, 20–29. MR**795536**, 10.1007/BF03025802**[8]**Mark Sheingorn,*Characterization of simple closed geodesics on Fricke surfaces*, Duke Math. J.**52**(1985), no. 2, 535–545. MR**792188**, 10.1215/S0012-7094-85-05228-7**[9]**Don Zagier,*On the number of Markoff numbers below a given bound*, Math. Comp.**39**(1982), no. 160, 709–723. MR**669663**, 10.1090/S0025-5718-1982-0669663-7

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1986-0833700-7

Article copyright:
© Copyright 1986
American Mathematical Society