[Paper Review] Random Construction of Riemann Surfaces

Robert Brooks, Eran Makover|ArXiv.org|Jun 28, 2001

Geometric and Algebraic Topology8 references4 citations

TL;DR

This paper introduces a random model for constructing finite-area Riemann surfaces using 3-regular graphs with orientations, enabling control over global geometric invariants. It proves that as genus increases, typical surfaces from this model almost surely have large first eigenvalue, large Cheeger constant, large systole, and logarithmic diameter—key results established via spectral graph theory and probabilistic analysis of graph cycles.

ABSTRACT

In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi surfaces. And in this construction we can control the geometry of the compact Riemann surface by the geometry of the graph. We show that almost all such surfaces have large first eigenvalue and large Cheeger constant.

Motivation & Objective

• To understand the typical geometric structure of compact Riemann surfaces of large genus.
• To develop a tractable random model for Riemann surfaces that allows control over global geometric invariants such as the first eigenvalue, Cheeger constant, systole, and diameter.
• To establish that these geometric properties are tightly controlled in the random model by leveraging spectral properties of 3-regular graphs.
• To prove that the random construction yields surfaces satisfying a 'large cusps' condition with probability approaching one as genus increases.

Proposed method

• Associate each 3-regular graph with an orientation to construct a finite-area hyperbolic surface $S^O(\Gamma,\mathcal{O})$ via gluing ideal hyperbolic triangles.
• Form the conformal compactification $S^C(\Gamma,\mathcal{O})$ of $S^O(\Gamma,\mathcal{O})$, which yields compact Riemann surfaces dense in the moduli space.
• Use the spectral gap of the graph $\Gamma$ to control the first eigenvalue $\lambda_1(S^C(\Gamma,\mathcal{O}))$ via known graph-surface spectral correspondence.
• Apply probabilistic graph theory—specifically, the asymptotic Poisson distribution of short cycles—to analyze cusp geometry and injectivity radius.
• Use Lemma 3.1 on asymptotic independence of short cycles to show that obstructions to large cusps occur with vanishing probability as $n \to \infty$.
• Establish diameter bounds using the Cheeger constant and volume growth, leveraging the relation $\text{diam}(M) \leq 2[r_0 + \frac{1}{h(M)} \log(\frac{\text{vol}(M)}{2B(r_0)})]$.

Experimental results

Research questions

• RQ1 What is the typical geometric structure of a random compact Riemann surface of large genus?
• RQ2 Can the spectral and geometric invariants of random Riemann surfaces be controlled via a combinatorial model based on 3-regular graphs?
• RQ3 How likely is it that a random surface from this model has large first eigenvalue, large Cheeger constant, large systole, and logarithmic diameter?
• RQ4 What is the role of cusp geometry in the spectral properties of the resulting surfaces?

Key findings

• As $n \to \infty$, the probability that $S^O(\Gamma,\mathcal{O})$ satisfies the 'large cusps' condition tends to 1, ensuring robustness of geometric invariants.
• The first eigenvalue $\lambda_1(S^C(\Gamma,\mathcal{O}))$ is bounded below by a positive constant $C_1$ with probability tending to 1.
• The Cheeger constant $h(S^C(\Gamma,\mathcal{O}))$ is bounded below by a positive constant $C_2$ with probability tending to 1.
• The systole $\text{syst}(S^C(\Gamma,\mathcal{O}))$ is bounded below by a positive constant $C_3$ with probability tending to 1.
• The diameter of $S^C(\Gamma,\mathcal{O})$ is bounded above by $C_4 \log(\text{genus})$ with probability tending to 1.
• Short geodesics and small cusps occur with positive asymptotic probability, but are typically far apart due to the asymptotic independence of cycle structures in random 3-regular graphs.