Simple curves on hyperbolic tori

Details Simple curves on hyperbolic tori Simple curves on hyperbolic tori

2000-05-23

arxiv.org/abs/math/0005220v1

C. R. Acad. Sci. Paris S\'er. I Math. 320 (1995), no. 12, 1523--1528

Mathematics

Geometric Topology

9 Pages, 1 figure (the published version does not include the figure for space reasons)

Scientific paper

We describe a new approach to the study of the set of all simple geodesics on a hyperbolic punctured torus. We introduce a valuation on the first integral homology group of the torus. This valuation associates to each homology class the length of the unique simple geodesic in it. We show that this valuation extends to a norm on the homology with real coefficients. We analyze the structure of this norm, and its variation over the moduli space of punctured tori. These results are applied to obtain sharp asymptotic estimates on the number of simple geodesics of bounded length..

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