J. Aust. Math. Soc.  74 (2003), 43-60
Parametrizing simple closed geodesy on $\Gamma^3\backslash\mathcal{H}$

Thomas A. Schmidt
  Oregon State University
  Corvallis, OR 97331
Mark Sheingorn
  CUNY - Baruch College
  New York, NY 10010

We exhibit a canonical geometric pairing of the simple closed curves of the degree three cover of the modular surface, $\Gamma^3\backslash\mathcal{H}$, with the proper single self-intersecting geodesics of Crisp and Moran. This leads to a pairing of fundamental domains for $\Gamma^3$ with Markoff triples.

The routes of the simple closed geodesics are directly related to the above. We give two parametrizations of these. Combining with work of Cohn, we achieve a listing of all simple closed geodesics of length within any bounded interval. Our method is direct, avoiding the determination of geodesic lengths below the chosen lower bound.
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