[Paper Review] On Cheeger constants of hyperbolic surfaces

Thomas Budzinski, Nicolas Curien+1 more|arXiv (Cornell University)|Jul 1, 2022

Geometric and Algebraic Topology4 citations

TL;DR

This paper establishes an upper bound of $\frac{2}{\pi} \approx 0.63$ on the Cheeger constant of closed hyperbolic surfaces of large genus, using a random Poisson–Voronoi tessellation with vanishing intensity. The result shows a strict gap between the maximal Cheeger constant of such surfaces and that of the hyperbolic plane ($h=1$), analogous to known gaps in graph theory, and implies that Cheeger's inequality cannot yield optimal spectral bounds for high-genus surfaces.

ABSTRACT

It is a well-known result due to Bollobas that the maximal Cheeger constant of large $d$-regular graphs cannot be close to the Cheeger constant of the $d$-regular tree. We prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by $2/π\approx 0.63...$ which is strictly less than the Cheeger constant of the hyperbolic plane. The proof uses a random construction based on a Poisson--Voronoi tessellation of the surface with a vanishing intensity.

Motivation & Objective

• To resolve a conjecture that the maximal Cheeger constant of large-genus hyperbolic surfaces is strictly less than that of the hyperbolic plane.
• To establish a quantitative upper bound on the Cheeger constant in the limit of large genus, analogous to known results in $d$-regular graph theory.
• To demonstrate that Cheeger's inequality cannot yield optimal spectral gaps for high-genus hyperbolic surfaces.
• To develop a probabilistic method based on Poisson–Voronoi tessellations to analyze isoperimetric properties of hyperbolic surfaces.

Proposed method

• Construct a random Poisson–Voronoi tessellation on a hyperbolic surface with vanishing intensity, modeling the local geometry of the hyperbolic plane.
• Use a limiting construction of the pointless Voronoi tessellation to analyze boundary length densities in the hyperbolic plane.
• Apply a random coloring of Voronoi cells with i.i.d. fair coin flips to generate random subsets $A_\lambda$ of approximately half the area.
• Estimate the expected boundary length $\mathbb{E}[|\partial A_\lambda|]$ using geometric probability and integral estimates over hyperbolic annuli.
• Use concentration and tail bounds to show that with positive probability, $|A_\lambda|$ is close to half the area and $|\partial A_\lambda|$ is small.
• Apply Markov's inequality to derive an upper bound on the Cheeger constant $h(\mathcal{S})$ in terms of $\frac{2}{\pi}(1+\delta)^2/(1-\delta)$, which tends to $\frac{2}{\pi}$ as $\delta \to 0$.

Experimental results

Research questions

• RQ1 Is the maximal Cheeger constant of closed hyperbolic surfaces of large genus strictly less than that of the hyperbolic plane?
• RQ2 Can the Cheeger constant of large-genus hyperbolic surfaces be bounded above by a constant strictly less than 1, as in the graph case?
• RQ3 Does the spectral gap of large-genus hyperbolic surfaces fail to be optimally bounded by Cheeger's inequality due to this gap in the Cheeger constant?
• RQ4 Can a random Poisson–Voronoi tessellation with vanishing intensity be used to derive isoperimetric bounds on hyperbolic surfaces?

Key findings

• The Cheeger constant of any closed hyperbolic surface of genus $g \geq 2$ is bounded above by $\frac{2}{\pi} \approx 0.63$ in the limit as $g \to \infty$, i.e., $\limsup_{g \to \infty} \sup_{\mathcal{S} \in \mathcal{M}_g} h(\mathcal{S}) \leq \frac{2}{\pi}$.
• This upper bound is strictly less than the Cheeger constant of the hyperbolic plane, which is 1, establishing a quantitative gap between the universal cover and large finite covers.
• The result implies that Cheeger's inequality cannot be used to prove optimal spectral gaps for large-genus hyperbolic surfaces, as the Cheeger constant is bounded away from 1.
• The proof relies on a random construction using Poisson–Voronoi tessellations with vanishing intensity, where the expected boundary length of a random half-coloring of cells is shown to be at most $\frac{2}{\pi}(1+\delta)|\mathcal{S}|$ for small $\delta > 0$.
• With positive probability, the random subset $A_\lambda$ has area within $1 \pm \delta$ of half the surface area and boundary length at most $\frac{1}{\pi}(1+\delta)^2|\mathcal{S}|$, leading to the final bound.
• The method avoids complex interplay between different radii by focusing on self-consistent geometric estimates using the intrinsic hyperbolic measure and concentration on large cells.