The second integral homology of ${\rm SL}_2(\mathbb{Z}[1/n])$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912669355737088 |
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| author | Mirzaii, Behrooz Ramos, Bruno Reis Verissimo, Thiago |
| author_facet | Mirzaii, Behrooz Ramos, Bruno Reis Verissimo, Thiago |
| contents | In this article, we explore the second integral homology, or Schur multiplier, of the special linear group ${\rm SL}_2(\mathbb{Z}[1/n])$ for a positive integer $n$. We definitively calculate the group structure of $H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ when $n$ is divisible by one of the primes $2$, $3$, $5$, $7$ or $13$. For a general $n > 1$, we offer a partial description by placing the homology group within an exact sequence, and we investigate its rank. Finally, we propose a conjectural structure for $H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ when $n$ is not divisible by any of those specific primes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The second integral homology of ${\rm SL}_2(\mathbb{Z}[1/n])$ Mirzaii, Behrooz Ramos, Bruno Reis Verissimo, Thiago K-Theory and Homology 19G99 19G99, 20G10, In this article, we explore the second integral homology, or Schur multiplier, of the special linear group ${\rm SL}_2(\mathbb{Z}[1/n])$ for a positive integer $n$. We definitively calculate the group structure of $H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ when $n$ is divisible by one of the primes $2$, $3$, $5$, $7$ or $13$. For a general $n > 1$, we offer a partial description by placing the homology group within an exact sequence, and we investigate its rank. Finally, we propose a conjectural structure for $H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ when $n$ is not divisible by any of those specific primes. |
| title | The second integral homology of ${\rm SL}_2(\mathbb{Z}[1/n])$ |
| topic | K-Theory and Homology 19G99 19G99, 20G10, |
| url | https://arxiv.org/abs/2503.12190 |