I have been trying to learn about [congruence groups](https://en.wikipedia.org/wiki/Congruence_subgroup). Here is an example:


\begin{eqnarray\*} \Gamma\big(1+2i\big) &=& \text{SL}\_2\big(\mathbb{Z}[i]\big)(1+2i) \\ \\
&=& \left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) 
: ad-bc = 1 \text{ and }
\left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \equiv 
\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \pmod{1+2i} 
 \right\} \\ \\
&\subseteq & \text{SL}\_2\big(\mathbb{Z}[i]\big) \end{eqnarray\*}


While the proof is uses the theory of algebraic groups, we can prove that is finitely generated. This particular case is look elementary, these are $2 \times 2$ invertible matrices, $\mathbb{Z}[i]$ is a Eucliean domain, and we can solve $ad-bc = 1$ by finding two [primes](https://en.wikipedia.org/wiki/Table_of_Gaussian_integer_factorizations) (e.g. $6+i $ or $ 2+3i$) and looking for their greatest common divisor. 


Since this group is finitely generated, how can I find a generating set? What are the generators? Even computer code would be helpful.




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In order to specify what I am looking for here is the result for $\text{SL}\_2(\mathbb{Z})$: 


$$ \text{SL}\_2(\mathbb{Z}) \simeq \big\{ S,T : S^2 = 1,\; (ST)^3 = 1 \big\} \quad\text{with}\quad 
S = \left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right)
\text{ and } 
T = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right)$$


At least over $\mathbb{Z}$ there an exact sequence relating the congruence groups to the special linear group over finite fields:


$$ 1 \to \Gamma(N) \to \Gamma \to \text{PSL}\_2\big(\mathbb{Z}/N\mathbb{Z}\big) \to 1 $$


And so it's likely that congruence groups of $\text{SL}\_2(\mathbb{Z})$ should have finite presentations. Even if I do something slightly more inefficient and use the exact sequence.




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Also 


* [Are congruence subgroups of the modular group finitely presented?](https://mathoverflow.net/questions/105236/are-congruence-subgroups-of-the-modular-group-finitely-presented)



What are the $2 \times 2$ matrix generators of $\text{SL}_2\big(\mathbb{Z}[i]\big)(2+i)$?
$PSL\_2(\mathbb{O}\_d)$ acts on the upper half-space model of hyperbolic 3-space in a nice way, namely the quotient can be viewed as a finite volume 3-orbifold. Since all principal congruence subgroups are finite index in $PSL\_2(\mathbb{O}\_d)$, all principal congruence subgroups are finitely generated and correspond to some (finite-sided/geometrically finite) finite volume orbifold cover of $H^3/PSL\_2(\mathbb{O}\_d)$. (There are probably more direct ways of showing this, but might help frame this discussion.) 


As the OP notes in the question, after specifying a $d$, one can draw a direct analogy between group, and $PSL\_2(\mathbb{Z})$ acting on $H^2$. There is natural question, "Which principal congruence subgroups of $PSL\_2(\mathbb{Z})$ are genus zero (aka fill to a sphere)?" One 3d analog of this question is "Which principal congruence subgroups of $PSL\_2(\mathbb{O}\_d)$ are homeomorphic to $S^3\setminus L$ for some link $L$ in $S^3$?" 


Happily, over a series of papers the subsets of collection of Baker, Goerner and Reid provide a complete solution to this question. More relevantly, $PSL\_2(\mathbb{Z}[i])(2+i)$ is one of the links they study.


Proposition 3.1 and Lemma 3.2 of


*Baker, Mark D.; Reid, Alan W.*, [**Principal congruence link complements**](http://dx.doi.org/10.5802/afst.1436), Ann. Fac. Sci. Toulouse, Math. (6) 23, No. 5, 1063-1092 (2014). [ZBL1322.57015](https://zbmath.org/?q=an:1322.57015).<http://www.numdam.org/item/AFST_2014_6_23_5_1063_0>


discusses this question directly. It gives a six generator presentation of $PSL\_2(\mathbb{Z}[i])(2+i)$. The index of this group is confirmed using a magma computation. Also the generating set is minimal as the abelianization of this group has rank 6. Selecting the correct coset representative from each of these elements in $PSL\_2(\mathbb{Z}[i])$ should give you group you are looking for. 


Some of the other papers in there line of attack are here if you are interested:


*Baker, Mark D.*, [**Link complements and the Bianchi modular groups**](http://dx.doi.org/10.1090/S0002-9947-01-02555-7), Trans. Am. Math. Soc. 353, No. 8, 3229-3246 (2001). [ZBL0986.20049](https://zbmath.org/?q=an:0986.20049).


*Baker, Mark D.*, Link complements and integer rings of class number greater than one, Topology ’90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 55-59 (1992). [ZBL0768.57005](https://zbmath.org/?q=an:0768.57005).


*Görner, Matthias*, [**Regular tessellation link complements**](http://dx.doi.org/10.1080/10586458.2014.986310), Exp. Math. 24, No. 2, 225-246 (2015). [ZBL1319.57002](https://zbmath.org/?q=an:1319.57002).


 Baker, Mark D., Goerner, Matthias, and Reid, Alan W., **All principal congruence link groups** arXiv preprint arXiv:1802.01275 (2018).


Note that as the title suggests, the last paper gives a complete classification of principal congruence link complements. Of course, the methods of these papers which make a good guess for what the generators of the principal congruence subgroups are can be adjusted, especially if the pair $(d,I)$ is not on their lists. In this case, one would definitely need more than just a trace 2 generator corresponding to each cusp. Nevertheless, the magma code can be adapted to more general searches for generating sets of principal congruence groups of $PSL\_2(\mathbb{O}\_d)$. 


