On the number and location of short geodesics in moduli space

TL;DR

This paper investigates the distribution and quantity of short geodesics in the moduli space of Riemann surfaces, establishing bounds on their location and number that are independent of genus g.

Contribution

It proves that L-short geodesics are confined to a specific intersection of thick and thin parts of moduli space, and provides polynomial bounds on their count.

Findings

L-short geodesics lie in a bounded intersection of thick and thin regions.

Number of L-short geodesics is bounded by polynomials in g with degrees depending on L.

Bounds on geodesic count grow with L, tending to infinity as L increases.

Abstract

A closed Teichmuller geodesic in the moduli space M_g of Riemann surfaces of genus g is called L-short if it has length at most L/g. We show that, for any L > 0, there exist e_2 > e_1 > 0, independent of g, so that the L-short geodesics in M_g all lie in the intersection of the e_1-thick part and the e_2-thin part. We also estimate the number of L-short geodesics in M_g, bounding this from above and below by polynomials in g whose degrees depend on L and tend to infinity as L does.