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Parametrizing simple closed geodesy on Γ3\ℋ

Part of: Diophantine approximation, transcendental number theory Riemann surfaces

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: toms@math.orst.edu
Mark Sheingorn
Affiliation:
CUNY - Baruch CollegeNew York, NY 10010USA e-mail: marksh@alum.dartmouth.org
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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

References

[1]Cohn, H., ‘Approach to Markoff's minimal forms through modular functions’, Ann. of Math. (2) 61 (1955), 1–12.CrossRefGoogle Scholar
[2]Cohn, H., ‘Markoff geodesics in matrix theory’, in: Number theory with an emphasis on the Markoff spectrum (eds. Pollington, A. D. and Moran, W.), Lecture Notes in Pure and Appl. Math. 147 (Marcel Dekker, New York, 1993), pp. 69–82.Google Scholar
[3]Casson, A. J. and Bleiler, S. A., Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Student Texts 9 (Cambridge University Press, Cambridge, 1988).CrossRefGoogle Scholar
[4]Crisp, D. J. and Moran, W., ‘Single self-intersection geodesics and the Markoff spectrum’, in: Number of theory with an emphasis on the Markoff spectrum (eds. Pollington, A. D. and Moran, W.), Lecture Notes in Pure and Appl. Math. 147 (Marcel Dekker, New York, 1993) pp. 83–93.Google Scholar
[5]Crisp, D. J. and Moran, W., ‘The Markoff spectrum and closed geodesics with one self-intersection on a torus’, preprint, (Flinders University, 1995).Google Scholar
[6]Cusick, T. W. and Flahive, M. E., The Markoff and Lagrange spectra, Mathematical Surveys and Monographs 30 (Amer. Math. Society, Providence R.I., 1989).CrossRefGoogle Scholar
[7]Ford, L. E., ‘A geometrical proof of a theorem of Hurwitz’, Proc. Edinburgh Math. Soc. 35 (1917), 59–65.CrossRefGoogle Scholar
[8]Haas, A., ‘Diophantine approximation on hyperbolic Riemann surfaces’, Acta Math. 156 (1986), 33–82.CrossRefGoogle Scholar
[9]Haas, A., ‘Diophantine approximation on hyperbolic orbifolds’, Duke Math. J. 56 (1988), 531–547.CrossRefGoogle Scholar
[10]Lehner, J. and Sheingorn, M., ‘Simple closed geodesics on H+/Γ(3) arise from the Markov spectrum’, Bull. Amer. Math. Soc.(N.S.) 11 (1984), 359–362.CrossRefGoogle Scholar
[11]Markoff, A. A., ‘Sur les formes quadratiques binaires indéfinie’, Math. Ann. 15 (1879), 381–406.CrossRefGoogle Scholar
[12]Markoff, A. A., ‘Sur les formes quadratiques binaires indéfinies II’, Math. Ann. 17 (1880), 379–400.CrossRefGoogle Scholar
[13]McShane, G., ‘Simple geodesics and a series constant over Teichmüller space’, Inventiones Math. 132 (1998), 607–632.CrossRefGoogle Scholar
[14]McShane, G. and Rivin, I., ‘A norm on homology of surfaces and counting simple geodesics’, Intl. Math. Res. Not. 2 (1995), 61–69.CrossRefGoogle Scholar
[15]Nielsen, J., ‘Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden’, Math. Ann. 78 (1918), 385–397.CrossRefGoogle Scholar
[16]Penner, R. C. and Harer, J. L., Combinatorics of train tracks, Ann. of Math. Studies 125 (Princeton University Press, Princeton, N.J., 1992).CrossRefGoogle Scholar
[17]Series, C., ‘The geometry of Markoff numbers’, Math. Intelligencer 7 (1985), 20–29.CrossRefGoogle Scholar
[18]Sheingorn, M., ‘Characterization of simple closed geodesics of Fricke surfaces’, Duke Math. J. 52 (1985), 535–545.CrossRefGoogle Scholar
[19]Zieschang, H., ‘Minimal geodesics of a torus with a hole’, Izv. Akad. Nauk USSR 50 (1986), 1097–1105; English translation: Math. USSR Izv. 29 (1987), 449–457.Google Scholar