Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] A. F. Beardon, "The structure of words in discrete subgroups of $ {\text{SL}}(2,C)$," J. London Math. Soc. (2), v. 10, 1975, pp. 201-211. MR 0382633 (52:3515)
  • [2] A. F. Beardon, J. Lehner & M. Sheingorn, "Closed simple geodesics on Riemann surfaces and the Markov spectrum." (To be published.)
  • [3] J. Birman & C. Series, "An algorithm for simple curves on surfaces," J. London Math. Soc. (2), v. 29, 1984, pp. 331-342. MR 744104 (85m:57002)
  • [4] J. Birman & C. Series, "Simple curves have Hausdorff dimension one." (Preprint.)
  • [5] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Univ. Press, Cambridge, 1957. MR 0087708 (19:396h)
  • [6] J. F. Koksma, Diophantische Approximationen, Springer-Verlag, Berlin, 1936; reprinted, Chelsea, New York. MR 0344200 (49:8940)
  • [7] M. Newman, "A note on fuchsian groups," Illinois J. Math. (To appear). MR 806474 (87d:20063)
  • [8] R. A. Rankin, Modular Forms, Cambridge Univ. Press, Cambridge, 1977. MR 0498390 (58:16518)
  • [9] J. Stillwell, Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, No. 72, Springer-Verlag, Berlin and New York, 1980. MR 602149 (82h:57001)
  • [10] D. Zagier, "On the number of Markoff numbers below a given bound," Math. Comp., v. 39, 1982, pp. 709-723. MR 669663 (83k:10062)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11F06, 11J06, 20H10, 30F35

Retrieve articles in all journals with MSC: 11F06, 11J06, 20H10, 30F35


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1985-0771045-7
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society