Maximal Fuchsian subgroups of the $d=2$ Bianchi group
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912840622800896 |
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| author | Lee, Anthony |
| author_facet | Lee, Anthony |
| contents | Let $Γ$ denote the $d = 2$ Bianchi group $\operatorname{PSL}(2,\mathbb{Z}[\sqrt{-2}])$. We give an explicit description of all conjugacy classes of maximal nonelementary Fuchsian subgroups of $Γ$ as integral orders of certain indefinite quaternion algebras over $\mathbb{Q}$. Using this description, we also provide the covolumes corresponding to each conjugacy class. As an application, we compute the limit $\lim_{x\to\infty} \frac{Π(x)}{x}$ where $Π(x)$ counts the number of primitive totally geodesic immersed surfaces in the manifold $Γ\backslash\mathbb{H}^3$ with area less than $x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15700 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Maximal Fuchsian subgroups of the $d=2$ Bianchi group Lee, Anthony Number Theory Geometric Topology Let $Γ$ denote the $d = 2$ Bianchi group $\operatorname{PSL}(2,\mathbb{Z}[\sqrt{-2}])$. We give an explicit description of all conjugacy classes of maximal nonelementary Fuchsian subgroups of $Γ$ as integral orders of certain indefinite quaternion algebras over $\mathbb{Q}$. Using this description, we also provide the covolumes corresponding to each conjugacy class. As an application, we compute the limit $\lim_{x\to\infty} \frac{Π(x)}{x}$ where $Π(x)$ counts the number of primitive totally geodesic immersed surfaces in the manifold $Γ\backslash\mathbb{H}^3$ with area less than $x$. |
| title | Maximal Fuchsian subgroups of the $d=2$ Bianchi group |
| topic | Number Theory Geometric Topology |
| url | https://arxiv.org/abs/2601.15700 |