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

HTML articles powered by AMS MathViewer

- by J. Lehner and M. Sheingorn PDF
- Math. Comp.
**44**(1985), 233-240 Request permission

## 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.

## References

- A. F. Beardon,
*The structure of words in discrete subgroups of $\textrm {SL}(2,\,C)$*, J. London Math. Soc. (2)**10**(1975), 201–211. MR**382633**, DOI 10.1112/jlms/s2-10.2.201
A. F. Beardon, J. Lehner & M. Sheingorn, "Closed simple geodesics on Riemann surfaces and the Markov spectrum." (To be published.)
- 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**, DOI 10.1112/jlms/s2-29.2.331
J. Birman & C. Series, "Simple curves have Hausdorff dimension one." (Preprint.)
- 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** - J. F. Koksma,
*Diophantische Approximationen*, Springer-Verlag, Berlin-New York, 1974 (German). Reprint. MR**0344200** - Morris Newman,
*A note on Fuchsian groups*, Illinois J. Math.**29**(1985), no. 4, 682–686. MR**806474** - Robert A. Rankin,
*Modular forms and functions*, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR**0498390** - John C. Stillwell,
*Classical topology and combinatorial group theory*, Graduate Texts in Mathematics, vol. 72, Springer-Verlag, New York-Berlin, 1980. MR**602149** - 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

## Additional Information

- © Copyright 1985 American Mathematical Society
- 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