Asymptotics of shortest filling closed multi-geodesics

TL;DR

The paper analyzes the asymptotics of the shortest filling closed multi-geodesic on closed hyperbolic surfaces, establishing a sharp uniform relation between this length and the genus plus a logarithmic sum over short geodesics, via the bound $\pi(g-1)+\mathrm{R}(X)\le \mathcal{L}_{sys}^{fill}(X)\le 300g+12\mathrm{R}(X)$ with $\mathrm{R}(X)=\sum_{\gamma: \ell(\gamma)<1} \log(1/\ell(\gamma))$. The core method connects minimal-length filling graphs to filling geodesics through dual 4-valent graphs, and employs trigon decompositions, immersion theory, and Gauss-Bonnet arguments to derive explicit length controls, yielding an explicit upper bound $\ell(G)\le 150g+6\mathrm{R}(X)$. In the random regime, the work shows linear-in-$g$ behavior for typical Weil-Petersson random surfaces and analogous results for Brooks-Makover surfaces, with moment and concentration bounds obtained via Mirzakhani’s integration formula. Collectively, these results bridge filling-geodesic geometry and random hyperbolic geometry, revealing that the filling-geodesic length is governed by topological complexity and the distribution of short geodesics, and it scales linearly with genus in typical random models.

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

Figures (19)

• Figure 1: Modification near a common edge of two disks
• Figure 2: Annulus shaped trigon
• Figure 3: A monogon and a bigon
• Figure 4: An immersed bigon
• Figure 5: A triangle move

Theorems & Definitions (65)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 3.14
• Proposition 2.1
• Theorem 2.2: martelli2015spines
• Lemma 2.3
• proof
• proof : Proof of Proposition \ref{['prop:min_prop']}