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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.10329 |
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| _version_ | 1866909784076189696 |
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| author | Liu, Jinsong Shan, Xu Wang, Lang Yang, Yaosong |
| author_facet | Liu, Jinsong Shan, Xu Wang, Lang Yang, Yaosong |
| contents | Let $\mathcal{M}_g$ be the moduli space of hyperbolic surfaces of genus g endowed with the Weil-Petersson metric. In this paper, we introduce a function $L(g)$ of genus $g$ and call the geodesics whose length less than $L(g)$ short geodesics. We compute the growth rate on the volume of the subset of hyperbolic surfaces with short geodesics. In particular, when $g$ approaches infinity, if $L(g)$ also approaches infinity, then the volume of surfaces characterized by short geodesics is equal to $V_g$ almost surely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10329 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The growth rate on the volume of $\mathcal{M}_g^{<L(g)}$ Liu, Jinsong Shan, Xu Wang, Lang Yang, Yaosong Geometric Topology 32G15, 57M50 Let $\mathcal{M}_g$ be the moduli space of hyperbolic surfaces of genus g endowed with the Weil-Petersson metric. In this paper, we introduce a function $L(g)$ of genus $g$ and call the geodesics whose length less than $L(g)$ short geodesics. We compute the growth rate on the volume of the subset of hyperbolic surfaces with short geodesics. In particular, when $g$ approaches infinity, if $L(g)$ also approaches infinity, then the volume of surfaces characterized by short geodesics is equal to $V_g$ almost surely. |
| title | The growth rate on the volume of $\mathcal{M}_g^{<L(g)}$ |
| topic | Geometric Topology 32G15, 57M50 |
| url | https://arxiv.org/abs/2509.10329 |