2000/05/23 by Greg McShane, Igor Rivin
Mathematics · #math.GT #math.DG #math.DS #math.NT #msc:57M50 #msc:11J06 #msc:57N05
published as C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 12, 1523--1528 · 9 Pages, 1 figure (the published version does not include the figure for space reasons)
arxiv created 2000/05/23 · arxiv updated 2009/11/30
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..