Counting surface subgroups in cusped hyperbolic 3-manifolds
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908867846209536 |
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| author | Han, Xiaolong Hans Rao, Zhenghao Wan, Jia |
| author_facet | Han, Xiaolong Hans Rao, Zhenghao Wan, Jia |
| contents | Let $M =\mathbb{H}^3/Γ$ be a finite-volume, noncompact hyperbolic 3-manifold. We show that the number of quasi-Fuchsian surface subgroups of $Γ$ (up to conjugacy and commensurability) of genus at most $g$ is bounded both above and below by functions of the form $(cg)^{2g}$. As a corollary, for all $h\geq 4$, the number of purely pseudo-Anosov closed surface subgroups of genus at most $g$ of the mapping class group $\mathrm{Mod}(S_{h,0})$ is bounded below by $(Cg)^{2g}$ for a universal constant $C$. In contrast, for some $g \geq 2$, we construct infinitely many conjugacy classes of genus-$g$ surface subgroups of $Γ$ with accidental parabolics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20098 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting surface subgroups in cusped hyperbolic 3-manifolds Han, Xiaolong Hans Rao, Zhenghao Wan, Jia Geometric Topology Differential Geometry Group Theory 57M50, 57K32, 20H10, 32G15 Let $M =\mathbb{H}^3/Γ$ be a finite-volume, noncompact hyperbolic 3-manifold. We show that the number of quasi-Fuchsian surface subgroups of $Γ$ (up to conjugacy and commensurability) of genus at most $g$ is bounded both above and below by functions of the form $(cg)^{2g}$. As a corollary, for all $h\geq 4$, the number of purely pseudo-Anosov closed surface subgroups of genus at most $g$ of the mapping class group $\mathrm{Mod}(S_{h,0})$ is bounded below by $(Cg)^{2g}$ for a universal constant $C$. In contrast, for some $g \geq 2$, we construct infinitely many conjugacy classes of genus-$g$ surface subgroups of $Γ$ with accidental parabolics. |
| title | Counting surface subgroups in cusped hyperbolic 3-manifolds |
| topic | Geometric Topology Differential Geometry Group Theory 57M50, 57K32, 20H10, 32G15 |
| url | https://arxiv.org/abs/2602.20098 |