A norm on homology of surfaces and counting simple geodesics

TL;DR

This paper introduces a new norm on the homology of punctured tori with hyperbolic metrics and uses it to analyze the asymptotic growth of simple geodesics of bounded length.

Contribution

It defines a novel homology norm for punctured tori and applies it to derive asymptotic formulas for counting simple geodesics.

Findings

Established a norm on homology of punctured tori

Derived asymptotic growth rates for simple geodesics

Connected homology norms with geometric geodesic counting

Abstract

We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.