Abelianization of $\text{SL}_2$ over Dedekind domains of arithmetic type
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914082099036160 |
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| author | Mirzaii, Behrooz Ramos, Bruno R. Verissimo, Thiago |
| author_facet | Mirzaii, Behrooz Ramos, Bruno R. Verissimo, Thiago |
| contents | We determine the exact group structure of the abelianization of $\text{SL}_2(A)$, where $A$ is a Dedekind domain of arithmetic type with infinitely many units. In particular, our results show that $\text{SL}_2(A)^\text{ab}$ is finite, with exponent dividing $12$ when $\text{char}(A)=0$, and dividing $6$ when $\text{char}(A)>0$. As illustrative cases, we compute $\text{SL}_2(A)^\text{ab}$ explicitly for instances where $A$ is the ring of integers of a real quadratic field or a cyclotomic extension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_12638 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Abelianization of $\text{SL}_2$ over Dedekind domains of arithmetic type Mirzaii, Behrooz Ramos, Bruno R. Verissimo, Thiago Number Theory Group Theory 20J06, 11F75 We determine the exact group structure of the abelianization of $\text{SL}_2(A)$, where $A$ is a Dedekind domain of arithmetic type with infinitely many units. In particular, our results show that $\text{SL}_2(A)^\text{ab}$ is finite, with exponent dividing $12$ when $\text{char}(A)=0$, and dividing $6$ when $\text{char}(A)>0$. As illustrative cases, we compute $\text{SL}_2(A)^\text{ab}$ explicitly for instances where $A$ is the ring of integers of a real quadratic field or a cyclotomic extension. |
| title | Abelianization of $\text{SL}_2$ over Dedekind domains of arithmetic type |
| topic | Number Theory Group Theory 20J06, 11F75 |
| url | https://arxiv.org/abs/2506.12638 |