I have been trying to learn about congruence groups. 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 (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.

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.


  • 4
    $\begingroup$ "the generators"? you probably mean "some generating family". It's a finite index subgroup in $SL_2(Z[i])$, so using standard generators from the latter (this does not use any algebraic group theory), you can get generators of the congruence subgroup as well with standard methods. (It's the kernel of a homomorphism onto $SL_2(Z/5Z)$ which has order 120, so I expect it will yield some generating subset much larger than necessary.) $\endgroup$ – YCor May 26 '18 at 15:56
  • $\begingroup$ @YCor I don't even know what the "standard" generators are. Sage will rather mechanically produce generating sets for invertible $2 \times 2$ matrices over $\mathbb{Z}$. Not sure what they'll say about the Bianch group specified here. $\endgroup$ – john mangual May 26 '18 at 16:09
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    $\begingroup$ Over an arbitrary Euclidean ring, $SL_2$ is generated by elementary matrices $e_{12}(r)$, $e_{21}(r)$ where $r$ ranges over the ring. So $\{e_{12}(1),e_{12}(i),e_{21}(1),e_{21}(i)\}$ generates $SL_2(Z[i])$. Hence $\{e_{12}(1),e_{12}(i),s\}$ also generates, where $s=[[0,-1][1,0]]$. $\endgroup$ – YCor May 26 '18 at 17:08
  • $\begingroup$ Generating subsets of such congruence subgroups have possibly been addressed in generating algebraic K-theory by the way. $\endgroup$ – YCor May 26 '18 at 17:09
  • $\begingroup$ @YCor I'm guessing elementary operations are like $(x,y) \mapsto (x+r \times y,y)$ and $(x,y) \mapsto (x, x+r \times y)$. And it's sufficient to have $r \in \{ \pm 1, \pm i \}$. it's amazing how quickly that turns into K-theory. $\endgroup$ – john mangual May 26 '18 at 21:46

$PSL_2(\mathbb{O}_d)$ 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, Ann. Fac. Sci. Toulouse, Math. (6) 23, No. 5, 1063-1092 (2014). ZBL1322.57015.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, Trans. Am. Math. Soc. 353, No. 8, 3229-3246 (2001). ZBL0986.20049.

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.

Görner, Matthias, Regular tessellation link complements, Exp. Math. 24, No. 2, 225-246 (2015). ZBL1319.57002.

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 $PSL_2(\mathbb{O}_d)$.


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