A tree bijection for the moduli space of genus-0 hyperbolic surfaces with boundaries

TL;DR

The paper develops a genus-0 tree-bijection for the moduli space of hyperbolic surfaces with boundaries, extending the spine construction to include geodesic boundaries and a distinguished cusp. It proves that WP volumes decompose into polytopes associated to decorated trees and uses an inclusion-exclusion approach over anti-Delaunay constraints to obtain explicit volume formulas; these lead to generating-function relations that encode the string equation and connect to topological recursion. A novel distance statistic, the distance-difference D, is computed exactly via a distance-dependent three-point function, revealing precise metric information for Weil-Petersson random surfaces and linking modular geometry to continuum limits such as the Brownian sphere. The results establish a concrete bridge between hyperbolic geometry, combinatorial tree encodings, and spectral-curve techniques, with potential implications for scaling limits, recursion relations, and metric properties of WP random surfaces with many boundaries or cusps.

Abstract

The Weil-Petersson volume of genus-g hyperbolic surfaces with geodesic boundaries is known since work of Mirzakhani to be polynomial in the boundary lengths. We provide a bijective proof of this fact in the genus-0 case in the presence of a distinguished cusp. It is based on a generalization of a recent tree bijection, by the first author and Curien, to the setting with geodesic boundaries, requiring an extension of the Bowditch-Epstein-Penner spine construction. As an application of our tree bijection we establish an explicit formula for the distance-dependent three-point function, which records an exact metric statistic measuring the difference of two geodesic distances among a triple of distinguished cusps in a Weil-Petersson random surface. We conclude with a discussion of the relevance of this function to the topological recursion of Weil-Petersson volumes and metric properties of Weil-Petersson random surfaces with many boundaries or cusps.

Figures (25)

• Figure 1: Illustration of a genus-$0$ hyperbolic surface $\mathsf{X}$ with cusps and geodesic boundaries (left). The bijection in this work considers the spine construction in the extended hyperbolic surface $\check{\mathsf{X}}$ (right), obtained by attaching funnels to the geodesic boundaries of $\mathsf{X}$. The tree in question corresponds to the cut locus of points that have multiple length-minimizing geodesics to a horocycle around the distinguished cusp, called the origin.
• Figure 2: Example of the tree $\mathfrak{t}$ associated to a surface $\mathsf{X} \in \mathcal{M}_{0,1+n}(\mathbf{L})$, with in this case $n=10$ boundaries (of which $6$ are cusps and $4$ are geodesic) besides the origin cusp. The spine of the extended hyperbolic surface $\check{\mathsf{X}}$ has the structure of a bicolor plane tree $\mathfrak{t} \in \mathfrak{T}_n$.
• Figure 3: Left: Example of an anti-Delaunay tree $\tilde{\mathfrak{t}}\in \widetilde{\mathfrak{T}}_{12}(\mathbf{L})$. Middle: The contribution to the Weil-Petersson volume for several low-degree vertices. Right: all five trees in $\widetilde{\mathfrak{T}}_4(\mathbf{L})$.
• Figure 4: A hyperbolic sphere $\mathsf{X}$ with three distinguished cusps equipped with their unit-length horocycles $h_1,h_2,h_3$. The distance difference $D(\mathsf{X})$ measures the length difference between the two length-minimizing geodesics.
• Figure 5: The Bouttier-Di Francesco-Guitter bijection relates bipartite planar maps with a distinguished vertex (shown here in blue), $n$ labeled faces, which are properly bicolored plane trees with $n$ labeled white vertices and red vertices that are decorated by the graph distances.

Theorems & Definitions (52)

• Theorem 1
• Corollary 2
• proof
• Theorem 3
• Corollary 4: String equation
• Theorem 5: distance-dependent three-point function
• Corollary 6
• proof : Proof of Corollary \ref{['cor:distvar']}
• Theorem 7
• Lemma 8