A tree bijection for the moduli space of genus-0 hyperbolic surfaces with boundaries
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912756914978816 |
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| author | Budd, Timothy Meeusen, Thomas Zonneveld, Bart |
| author_facet | Budd, Timothy Meeusen, Thomas Zonneveld, Bart |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A tree bijection for the moduli space of genus-0 hyperbolic surfaces with boundaries Budd, Timothy Meeusen, Thomas Zonneveld, Bart Geometric Topology Mathematical Physics Combinatorics Probability 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. |
| title | A tree bijection for the moduli space of genus-0 hyperbolic surfaces with boundaries |
| topic | Geometric Topology Mathematical Physics Combinatorics Probability |
| url | https://arxiv.org/abs/2512.09722 |