On the number and location of short geodesics in moduli space

A closed Teichmüller 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 R>ε>0, independent of g, so that the L-short geodesics in ℳ_g all lie in the intersection of the ε-thick part and the R-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.