A classification of $\mathbb R$-Fuchsian subgroups of Picard modular groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2017
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| _version_ | 1866912674286141440 |
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| author | Parkkonen, Jouni Paulin, Frédéric |
| author_facet | Parkkonen, Jouni Paulin, Frédéric |
| contents | Given an imaginary quadratic extension $K$ of $\mathbb Q$, we classify the maximal nonelementary subgroups of the Picard modular group $\operatorname{PU}(1,2;\mathcal O_K)$ preserving a totally real totally geodesic plane in the complex hyperbolic plane $\mathbb H^2_\mathbb C$. We prove that these maximal $\mathbb R$-Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius $Δ$ of the corresponding $\mathbb R$-circle lies in $\mathbb N-\{0\}$, then the stabilizer arises from the quaternion algebra $\Big(\!\begin{array}{c} Δ\,,\, |D_K|\\\hline\mathbb Q\end{array} \!\Big)$. We thus prove the existence of infinitely many orbits of $K$-arithmetic $\mathbb R$-circles in the hypersphere of $\mathbb P_2(\mathbb C)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1707_00154 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | A classification of $\mathbb R$-Fuchsian subgroups of Picard modular groups Parkkonen, Jouni Paulin, Frédéric Number Theory Complex Variables 11F06, 11R52, 20H10, 20G20, 53C17, 53C55 Given an imaginary quadratic extension $K$ of $\mathbb Q$, we classify the maximal nonelementary subgroups of the Picard modular group $\operatorname{PU}(1,2;\mathcal O_K)$ preserving a totally real totally geodesic plane in the complex hyperbolic plane $\mathbb H^2_\mathbb C$. We prove that these maximal $\mathbb R$-Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius $Δ$ of the corresponding $\mathbb R$-circle lies in $\mathbb N-\{0\}$, then the stabilizer arises from the quaternion algebra $\Big(\!\begin{array}{c} Δ\,,\, |D_K|\\\hline\mathbb Q\end{array} \!\Big)$. We thus prove the existence of infinitely many orbits of $K$-arithmetic $\mathbb R$-circles in the hypersphere of $\mathbb P_2(\mathbb C)$. |
| title | A classification of $\mathbb R$-Fuchsian subgroups of Picard modular groups |
| topic | Number Theory Complex Variables 11F06, 11R52, 20H10, 20G20, 53C17, 53C55 |
| url | https://arxiv.org/abs/1707.00154 |