Uniform two-generator presentations for $SL_n(\mathbb{Z})$ with polynomial complexity bounds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913063832125440 |
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| author | Biswas, Arindam |
| author_facet | Biswas, Arindam |
| contents | We give a uniform explicit construction of finite two-generator presentations for the special linear groups over the integers in all ranks at least three. The construction builds on the generating-pair work of Conder--Liversidge--Vsemirnov and on a standard Tietze-elimination observation pointed out by Button. It recovers Trott's odd-rank generating pair and extends the same monomial/transvection form uniformly to even rank by a sign correction. After rebalancing, the construction has quadratic transvection words, quartically many relators, and sextic total relator length. We also derive several consequences, including infinite--infinite and finite--finite variants, consequences for congruence quotients, a presentation for the projective quotient, and an exact relator count, valid for both the unbalanced and balanced presentations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_17777 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform two-generator presentations for $SL_n(\mathbb{Z})$ with polynomial complexity bounds Biswas, Arindam Group Theory 20F05, 20H05 We give a uniform explicit construction of finite two-generator presentations for the special linear groups over the integers in all ranks at least three. The construction builds on the generating-pair work of Conder--Liversidge--Vsemirnov and on a standard Tietze-elimination observation pointed out by Button. It recovers Trott's odd-rank generating pair and extends the same monomial/transvection form uniformly to even rank by a sign correction. After rebalancing, the construction has quadratic transvection words, quartically many relators, and sextic total relator length. We also derive several consequences, including infinite--infinite and finite--finite variants, consequences for congruence quotients, a presentation for the projective quotient, and an exact relator count, valid for both the unbalanced and balanced presentations. |
| title | Uniform two-generator presentations for $SL_n(\mathbb{Z})$ with polynomial complexity bounds |
| topic | Group Theory 20F05, 20H05 |
| url | https://arxiv.org/abs/2604.17777 |