Fields of definition for triangle groups as Fuchsian groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916550110347264 |
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| author | Calegari, Frank Chen, Qiankang |
| author_facet | Calegari, Frank Chen, Qiankang |
| contents | The compact hyperbolic triangle group $Δ(p,q,r)$ admits a canonical representation to $\mathrm{PSL}_2(\mathbf{R})$ with discrete image which is unique up to conjugation. The trace field of this representation is \[K = \mathbf{Q}(\cos(π/p), \cos(π/q), \cos(π/r)).\] We prove that there are exactly eleven such groups which are conjugate to subgroups of $\mathrm{PSL}_2(K)$. Moreover, we prove that there are no additional compact hyperbolic triangle groups which are conjugate to subgroups of $\mathrm{PSL}_2(L)$ for any totally real field $L$. This answers a question first raised by Waterman and Machlachlan, and also resolves (in the positive) five (interrelated) recent conjectures of McMullen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fields of definition for triangle groups as Fuchsian groups Calegari, Frank Chen, Qiankang Geometric Topology Group Theory Number Theory The compact hyperbolic triangle group $Δ(p,q,r)$ admits a canonical representation to $\mathrm{PSL}_2(\mathbf{R})$ with discrete image which is unique up to conjugation. The trace field of this representation is \[K = \mathbf{Q}(\cos(π/p), \cos(π/q), \cos(π/r)).\] We prove that there are exactly eleven such groups which are conjugate to subgroups of $\mathrm{PSL}_2(K)$. Moreover, we prove that there are no additional compact hyperbolic triangle groups which are conjugate to subgroups of $\mathrm{PSL}_2(L)$ for any totally real field $L$. This answers a question first raised by Waterman and Machlachlan, and also resolves (in the positive) five (interrelated) recent conjectures of McMullen. |
| title | Fields of definition for triangle groups as Fuchsian groups |
| topic | Geometric Topology Group Theory Number Theory |
| url | https://arxiv.org/abs/2501.01780 |