Nearly optimal spectral gaps for random Belyi surfaces

TL;DR

The paper proves that a random hyperbolic surface from the Brooks-Makover model has a spectral gap exceeding $\tfrac{1}{4}-\varepsilon$ for any $\varepsilon>0$, resolving the nearly optimal spectral gap conjecture in this model. It introduces a new geometric description of Brooks-Makover surfaces via degree-$6n$ covers and leverages a polynomial-method-like strong convergence for PSL$(2,\mathbb{Z})$ actions to transfer spectral-gap information from the modular surface to random surfaces. The authors develop a detailed combinatorial framework using $X$-labeled graphs to count fixed points of representations, obtaining precise asymptotics and norm bounds that feed into the probabilistic convergence result. The work completes the picture of nearly optimal spectral gaps across the major random models of closed hyperbolic surfaces and provides robust tools for analyzing spectral quantities in random hyperbolic geometry.

Abstract

In this paper, we show that for any $ε>0$, a random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than $\frac{1}{4}-ε$, confirming the nearly optimal spectral gap conjecture in this model.

Figures (9)

• Figure 1: The standard hyperbolic ideal triangle $T$
• Figure 2: Orientation-preserving
• Figure 3: Tilings
• Figure 4: Two examples for $S^O(\sigma,\tau)$
• Figure 5: Two $X-$labeled graphs

Theorems & Definitions (47)

• Theorem 1.1
• Proposition 1.2: =Proposition \ref{['p-count']}
• Proposition 1.3: =Proposition \ref{['p-bound']}
• Definition
• Definition
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Proposition 2.4