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.
All Science Journal Classification (ASJC) codes
- Geometry and Topology