The level $d$ principal congruence subgroup of $\textrm{SL}(n;\mathbb{Z})$

Nao Imoto; Ryoma Kobayashi

The level $d$ principal congruence subgroup of $\textrm{SL}(n;\mathbb{Z})$

Geometric Topology 2023-06-27 v2 Group Theory

Authors: Nao Imoto , Ryoma Kobayashi

Abstract

The abelianization of the level $d$ principal congruence subgroup $\Gamma_d(n)$ of $\textrm{SL}(n;\mathbb{Z})$ was determined by Lee-Szczarba. By this result and a result of Tits, we can obtain a minimal generating set for $\Gamma_d(n)$ . In this paper, we give a minimal generating set for $\Gamma_d(n)$ and determine the abelianization of $\Gamma_d(n)$ , without using the results of Tits and Lee-Szczarba. Moreover, we give three theorems about $\Gamma_d(n)$ .

Keywords

principal congruence subgroup special linear group abelianization minimal generating set integer matrices

Cite

@article{arxiv.2212.13181,
  title  = {The level $d$ principal congruence subgroup of $\textrm{SL}(n;\mathbb{Z})$},
  author = {Nao Imoto and Ryoma Kobayashi},
  journal= {arXiv preprint arXiv:2212.13181},
  year   = {2023}
}

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15 pages

The level $d$ principal congruence subgroup of $\textrm{SL}(n;\mathbb{Z})$

Abstract

Keywords

Cite

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