Asymptotics of shortest filling closed multi-geodesics

Abstract

In this paper, we investigate the asymptotics of shortest filling closed multi-geodesics of closed hyperbolic surfaces as systole $\to 0$ or as genus $\to \infty$. We first show that for a closed hyperbolic surface $X_g$ of genus $g$, the length of a shortest filling closed multi-geodesic of $X_g$ is uniformly comparable to $$\left(g+\sum _{\text{closed geodesic }\gamma \subset X_g,\ell (\gamma )<1}\log \left(\frac{1}{\ell (\gamma )}\right)\right).$$ As an application, we show that as $g\to \infty$, a Weil-Petersson random hyperbolic surface has a shortest closed multi-geodesic of length uniformly comparable to $g$. We also show that this is true for a random hyperbolic surface in the Brooks-Makover model.

30 Pages, 19 Figures, Comments Welcome