$\mathrm{SL}_2$-character varieties of $2$-generated groups and failure of weak integrality
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912135902134272 |
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| author | Church, Benjamin García-Cortés, Francisco |
| author_facet | Church, Benjamin García-Cortés, Francisco |
| contents | Let $\ell$ be a prime number, $k$ a positive integer and consider the group $Γ_{\ell^k} :=\langle a,b\ \vert\ a^{\ell^k(\ell^k-1)}ba^{-\ell^k}b^{-2}\rangle$. We prove that $Γ_{\ell^k}$ is not $\mathrm{SL}_2$-weakly integral with obstruction at exactly the prime $\ell$. We also give a general description of the character varieties of $2$-generated groups with a relation of the form $a^{n_1} b^{m_1} a^{n_2} b^{m_2} = 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathrm{SL}_2$-character varieties of $2$-generated groups and failure of weak integrality Church, Benjamin García-Cortés, Francisco Number Theory Group Theory 20C12, 14M35 Let $\ell$ be a prime number, $k$ a positive integer and consider the group $Γ_{\ell^k} :=\langle a,b\ \vert\ a^{\ell^k(\ell^k-1)}ba^{-\ell^k}b^{-2}\rangle$. We prove that $Γ_{\ell^k}$ is not $\mathrm{SL}_2$-weakly integral with obstruction at exactly the prime $\ell$. We also give a general description of the character varieties of $2$-generated groups with a relation of the form $a^{n_1} b^{m_1} a^{n_2} b^{m_2} = 1$. |
| title | $\mathrm{SL}_2$-character varieties of $2$-generated groups and failure of weak integrality |
| topic | Number Theory Group Theory 20C12, 14M35 |
| url | https://arxiv.org/abs/2410.23233 |