Asymptotics of shortest filling closed multi-geodesics

TL;DR

This paper studies the asymptotic behavior of shortest filling multi-geodesics on hyperbolic surfaces, revealing their length scales with genus and random surface models, and providing uniform comparisons.

Contribution

It establishes uniform asymptotic estimates for shortest filling multi-geodesics as genus grows and in random surface models, extending understanding of geometric properties.

Findings

Shortest filling multi-geodesic length scales with genus g

As g→∞, shortest multi-geodesic length is comparable to g

Results hold for Weil-Petersson and Brooks-Makover random surfaces

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 $$\left(g+\sum\limits_{\textit{closed geodesic }\gamma\subset X_g, \ \ell(\gamma)<1}\log \left(\frac{1}{\ell(\gamma)}\right)\right).$$ 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.