Typical hyperbolic surfaces have an optimal spectral gap

TL;DR

The paper proves that typical compact hyperbolic surfaces of large genus have a near-optimal spectral gap: for any $\epsilon>0$, the Weil–Petersson measure on $\mathcal{M}_g$ satisfies $\mathbb{P}_g^{\mathrm{WP}}(\lambda_1 \ge \tfrac{1}{4}-\epsilon) \to 1$ as $g\to\infty$. It develops a geometry-flavored trace method inspired by Friedman's work on random regular graphs, recasting spectral controls in terms of averaged geometric counts via Ramanujan-type functions and stability under (pseudo)convolution. A central innovation is the construction and analysis of Weil–Petersson volume functions $V_g^{\mathrm{all}}(\ell)$ and $V_g^{\mathbf{T}}(\ell)$, decomposing by local topological types and writing them as integrals over bordered Teichmüller spaces; these admit asymptotic expansions in powers of $1/g$ with coefficients that can be described explicitly. The work introduces new coordinates on Teichmüller space tailored to specific local types (e.g., generalized eights) and leverages Wolpert’s formula to obtain precise Jacobians and length-function expressions, enabling a robust FR-type analysis for hyperbolic surfaces. Overall, the results establish that near-optimal spectral gaps are not only possible but prevalent among high-genus random hyperbolic surfaces, mirroring Friedman's probabilistic proof in graph theory and offering tools potentially relevant to the Selberg $1/4$ conjecture landscape.

Abstract

The first non-zero Laplace eigenvalue of a hyperbolic surface, or its spectral gap, measures how well-connected the surface is: surfaces with a large spectral gap are hard to cut in pieces, have a small diameter and fast mixing times. For large hyperbolic surfaces (of large area or large genus $g$, equivalently), we know that the spectral gap is asymptotically bounded above by $\frac 14$. The aim of these talks is to present joint work with Nalini Anantharaman, where we prove that most hyperbolic surfaces have a near-optimal spectral gap. That is to say, we prove that, for any $ε> 0$, the Weil--Petersson probability for a hyperbolic surface of genus $g$ to have a spectral gap greater than $\frac 14- ε$ goes to one as $g$ goes to infinity. This statement is analogous to Alon's 1986 conjecture for regular graphs, proven by Friedman in 2003. I will present our approach, which shares many similarities with Friedman's work, and introduce new tools and ideas that we have developed in order to tackle this problem.

Figures (16)

• Figure 1: The dumbbell example.
• Figure 2: Relationship between the spectral gap and other properties of the surface.
• Figure 3: The construction of the Bolza surface.
• Figure 4: Breaking the figure-eight in simpler loops, for graphs and hyperbolic surfaces.
• Figure 5: Examples of simple loops on a surface of genus $3$: non-separating on the left, and separating into a surfaces of genus $1$ and $2$ on the right.

Theorems & Definitions (36)

• Theorem 1.1
• Remark 1.2: The Selberg $\frac{1}{4}$ conjecture
• Definition 2.1: friedman2003
• Proposition 2.2: anantharaman2023
• Remark 2.3
• Definition 2.4
• Theorem 2.5
• Proposition 2.6
• Lemma 2.7
• proof