Computing self-intersections of closed geodesics on finite-sheeted covers of the modular surface

Authors:
J. Lehner and M. Sheingorn

Journal:
Math. Comp. **44** (1985), 233-240

MSC:
Primary 11F06; Secondary 11J06, 20H10, 30F35

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771045-7

MathSciNet review:
771045

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Abstract: An algorithm is given for deciding whether a closed geodesic on a finite-sheeted cover of the modular surface has self-intersections; if it does, the algorithm gives them in the order they occur in traversing the geodesic. The following general result on geodesics is proved: any closed geodesic on a Riemann surface *R* can be lifted to a simple closed geodesic on some finite-sheeted cover of *R*. In the last two sections the connection with the stabilizer (under the modular group) of a Markov quadratic irrationality is discussed.

**[1]**A. F. Beardon,*The structure of words in discrete subgroups of 𝑆𝐿(2,𝐶)*, J. London Math. Soc. (2)**10**(1975), 201–211. MR**0382633**, https://doi.org/10.1112/jlms/s2-10.2.201**[2]**A. F. Beardon, J. Lehner & M. Sheingorn, "Closed simple geodesics on Riemann surfaces and the Markov spectrum." (To be published.)**[3]**Joan S. Birman and Caroline Series,*An algorithm for simple curves on surfaces*, J. London Math. Soc. (2)**29**(1984), no. 2, 331–342. MR**744104**, https://doi.org/10.1112/jlms/s2-29.2.331**[4]**J. Birman & C. Series, "Simple curves have Hausdorff dimension one." (Preprint.)**[5]**J. W. S. Cassels,*An introduction to Diophantine approximation*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR**0087708****[6]**J. F. Koksma,*Diophantische Approximationen*, Springer-Verlag, Berlin-New York, 1974 (German). Reprint. MR**0344200****[7]**Morris Newman,*A note on Fuchsian groups*, Illinois J. Math.**29**(1985), no. 4, 682–686. MR**806474****[8]**Robert A. Rankin,*Modular forms and functions*, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR**0498390****[9]**John C. Stillwell,*Classical topology and combinatorial group theory*, Graduate Texts in Mathematics, vol. 72, Springer-Verlag, New York-Berlin, 1980. MR**602149****[10]**Don Zagier,*On the number of Markoff numbers below a given bound*, Math. Comp.**39**(1982), no. 160, 709–723. MR**669663**, https://doi.org/10.1090/S0025-5718-1982-0669663-7

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771045-7

Article copyright:
© Copyright 1985
American Mathematical Society