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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.17566 |
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| _version_ | 1866915751679492096 |
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| author | Gao, Yue Wang, Zhongzi Wu, Yunhui |
| author_facet | Gao, Yue Wang, Zhongzi Wu, Yunhui |
| contents | In this paper, we investigate the asymptotics of shortest filling closed multi-geodesics of closed hyperbolic surfaces as systole $\to 0$ or as genus $\to \infty$. We first show that for a closed hyperbolic surface $X_g$ of genus $g$, the length of a shortest filling closed multi-geodesic of $X_g$ is uniformly comparable to $$\left(g+\sum\limits_{\textit{closed geodesic }γ\subset X_g, \ \ell(γ)<1}\log \left(\frac{1}{\ell(γ)}\right)\right).$$ As an application, we show that as $g\to \infty$, a Weil-Petersson random hyperbolic surface has a shortest closed multi-geodesic of length uniformly comparable to $g$. We also show that this is true for a random hyperbolic surface in the Brooks-Makover model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics of shortest filling closed multi-geodesics Gao, Yue Wang, Zhongzi Wu, Yunhui Geometric Topology Differential Geometry In this paper, we investigate the asymptotics of shortest filling closed multi-geodesics of closed hyperbolic surfaces as systole $\to 0$ or as genus $\to \infty$. We first show that for a closed hyperbolic surface $X_g$ of genus $g$, the length of a shortest filling closed multi-geodesic of $X_g$ is uniformly comparable to $$\left(g+\sum\limits_{\textit{closed geodesic }γ\subset X_g, \ \ell(γ)<1}\log \left(\frac{1}{\ell(γ)}\right)\right).$$ As an application, we show that as $g\to \infty$, a Weil-Petersson random hyperbolic surface has a shortest closed multi-geodesic of length uniformly comparable to $g$. We also show that this is true for a random hyperbolic surface in the Brooks-Makover model. |
| title | Asymptotics of shortest filling closed multi-geodesics |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2508.17566 |