Character varieties on a four-holed sphere
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910280401813504 |
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| author | Shin, Eunju |
| author_facet | Shin, Eunju |
| contents | For each $\mathbf{k}\in\mathbb{C}^4$, let $V_\mathbf{k}$ be the character variety on a four-holed sphere and $Γ_\mathbf{k}$ the group generated by the Vieta involution maps. First, under a certain condition, we find a fundamental domain for $Γ_\mathbf{k}$-action on $V_\mathbf{k}$, expressed via inequalities. Second, we show that it is decidable whether or not two integral solutions for $V_\mathbf{k}$ are in the same $Γ_\mathbf{k}$-orbit and in the same mapping class group orbit. To achieve our goals, we introduce graphs corresponding to the $Γ_\mathbf{k}$-orbits and the mapping class group orbits, and classify their restricted global shapes by analyzing the limited local edge configurations at each vertex, using a descent argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16614 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Character varieties on a four-holed sphere Shin, Eunju Number Theory 11D79, 11D25 For each $\mathbf{k}\in\mathbb{C}^4$, let $V_\mathbf{k}$ be the character variety on a four-holed sphere and $Γ_\mathbf{k}$ the group generated by the Vieta involution maps. First, under a certain condition, we find a fundamental domain for $Γ_\mathbf{k}$-action on $V_\mathbf{k}$, expressed via inequalities. Second, we show that it is decidable whether or not two integral solutions for $V_\mathbf{k}$ are in the same $Γ_\mathbf{k}$-orbit and in the same mapping class group orbit. To achieve our goals, we introduce graphs corresponding to the $Γ_\mathbf{k}$-orbits and the mapping class group orbits, and classify their restricted global shapes by analyzing the limited local edge configurations at each vertex, using a descent argument. |
| title | Character varieties on a four-holed sphere |
| topic | Number Theory 11D79, 11D25 |
| url | https://arxiv.org/abs/2308.16614 |