Geometriae Dedicata

, Volume 63, Issue 2, pp 123–145

Coding of closed geodesics after Gauss and morse

  • Svetlana Katok

DOI: 10.1007/BF00148213

Cite this article as:
Katok, S. Geom Dedicata (1996) 63: 123. doi:10.1007/BF00148213


Closed geodesics associated to conjugacy classes of hyperbolic matrices in SL(2, ℤ) can be coded in two different ways. The geometric code, with respect to a given fundamental region, is obtained by a construction universal for all Fuchsian groups, while the arithmetic code, given by ‘—’ continued fractions, comes from the Gauss reduction theory and is specific for SL(2, ℤ). In this paper we give a complete description of all closed geodesics for which the two codes coincide.

Mathematical Subject Classifications (1991)

Primary: 20H05, 58F17 Secondary: 20H10, 11H55 

Key words

modular group modular surface closed geodesics ‘—’ continued fractions reduction theory 

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Svetlana Katok
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkU.S.A.