The adelic closure of triangle groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916946429083648 |
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| author | Calegari, Frank |
| author_facet | Calegari, Frank |
| contents | Motivated by questions arising from billiard trajectories in the regular $n$-gon, McMullen defined a pair of functions $κ$ and $δ$ on the cusps $c$ of the corresponding triangle group $Δ_n$ inside $\mathrm{SL}_2({\mathcal{O}})$, where ${\mathcal{O}} = \mathbf{Z}[ζ_n+ ζ^{-1}_n]$. McMullen asks for which $n$ these functions are congruence, that is, when they only depend on the image of the cusp $c \in \mathbf{P}^1(\mathcal{O})$ in $\mathbf{P}^1(\mathcal{O}/d)$ for some integer $d$. In this note, we answer McMullen's questions. We obtain our results by computing the exact closure of $Δ_n \subset \mathrm{SL}_2({\mathcal{O}})$ inside $\mathrm{SL}_2(\widehat{\mathcal{O}})$, where $\widehat{\mathcal{O}}$ is the profinite completion of ${\mathcal{O}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20374 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The adelic closure of triangle groups Calegari, Frank Number Theory Geometric Topology Motivated by questions arising from billiard trajectories in the regular $n$-gon, McMullen defined a pair of functions $κ$ and $δ$ on the cusps $c$ of the corresponding triangle group $Δ_n$ inside $\mathrm{SL}_2({\mathcal{O}})$, where ${\mathcal{O}} = \mathbf{Z}[ζ_n+ ζ^{-1}_n]$. McMullen asks for which $n$ these functions are congruence, that is, when they only depend on the image of the cusp $c \in \mathbf{P}^1(\mathcal{O})$ in $\mathbf{P}^1(\mathcal{O}/d)$ for some integer $d$. In this note, we answer McMullen's questions. We obtain our results by computing the exact closure of $Δ_n \subset \mathrm{SL}_2({\mathcal{O}})$ inside $\mathrm{SL}_2(\widehat{\mathcal{O}})$, where $\widehat{\mathcal{O}}$ is the profinite completion of ${\mathcal{O}}$. |
| title | The adelic closure of triangle groups |
| topic | Number Theory Geometric Topology |
| url | https://arxiv.org/abs/2407.20374 |