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Parametrizing simple closed geodesy on Γ3\ℋ

Part of: Riemann surfaces Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 April 2009

Thomas A. Schmidt  and
Mark Sheingorn
Thomas A. Schmidt
Affiliation:
Oregon State UniversityCorvallis, OR 97331USA e-mail: [email protected]
Mark Sheingorn
Affiliation:
CUNY - Baruch CollegeNew York, NY 10010USA e-mail: [email protected]
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Abstract

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We exhibit a canonical geometric pairing of the simple closed curves of the degree three cover of the modular surface, Γ3\ℋ, with the proper single self-intersecting geodesics of Crisp and Moran. This leads to a pairing of fundamental domains for Γ3 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.

MSC classification

Secondary: 30F35: Fuchsian groups and automorphic functions 11J70: Continued fractions and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 1 , February 2003 , pp. 43 - 60
Copyright
Copyright © Australian Mathematical Society 2003

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