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

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

MathSciNet review:
833700

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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]**H. Cohn,*Approach to Markoff's minimal forms through modular functions*, Ann. of Math. (2)**61**(1955), 1-12. MR**0067935 (16:801e)****[2]**J. F. Koksma,*Diophantische Approximationen*, Chelsea, New York, n.d.**[3]**A. Haas,*Diophantine approximation on hyperbolic Riemann surfaces*, Acta Math. (Uppsala) (to appear). MR**822330 (87h:11063)****[4]**J. Lehner and M. Sheingorn,*Simple closed geodesics on**arise from the Markov spectrum*, Bull. Amer. Math. Soc. (N.S.)**11**(1984), 359-362. MR**752798 (86b:11033)****[5]**O. Perron,*Die Lehre von den Kettenbrucken*, Teubner, Stuttgart, 1954. MR**0064172 (16:239e)****[6]**A. Schmidt,*Minimum of quadratic forms with respect to Fuchsian groups*. I, J. Reine Angew. Math.**286/7**(1976), 341-368. MR**0457358 (56:15566)****[7]**C. Series,*The geometry of Markoff numbers*, Math. Intelligencer**7**(1985), 20-30. MR**795536 (86j:11069)****[8]**M. Sheingorn,*Characterization of simple closed geodesics on Fricke surfaces*, Duke Math. J.**52**(1985), 535-545. MR**792188 (87b:11034)****[9]**D. Zagier,*On the number of Markov numbers below a given bound*, Math. Comp.**39**(1982), 709-723. MR**669663 (83k:10062)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
11F99,
11J06,
30F25

Retrieve articles in all journals with MSC: 11F99, 11J06, 30F25

Additional Information

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

Article copyright:
© Copyright 1986
American Mathematical Society