Simple curves on hyperbolic tori

TL;DR

This paper introduces a new valuation-based approach to analyze simple geodesics on hyperbolic punctured tori, providing insights into their structure and asymptotic counts.

Contribution

It develops a valuation and norm on homology classes, extending to real coefficients, and analyzes their variation over moduli space.

Findings

Established a valuation associating homology classes with geodesic lengths

Extended the valuation to a norm on homology with real coefficients

Derived sharp asymptotic estimates for the number of simple geodesics of bounded length

Abstract

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..