Hyperbolic lengths of some filling geodesics on Riemann surfaces with punctures

Abstract

Let $\tilde{S}$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$ and $n\geq 1$. In this paper, we give a quantitative common lower bound for the hyperbolic lengths of all filling geodesics on $\tilde{S}$ generated by two parabolic elements in the fundamental group $\pi_{1}(\tilde{S},a)$.