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Transactions of the American Mathematical Society

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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 $ 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.

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Article copyright: © Copyright 1986 American Mathematical Society

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