ModularGroup

Algorithms for computing with finite-index subgroups of $\mathrm{(P)SL}_2(\mathbb{Z})$

This package implements finite-index subgroups of $\mathrm{(P)SL}_2(\mathbb{Z})$ and various algorithms for working with them. These subgroups are stored as tuples of permutations s and t which describe the action of the generator matrices

$$ S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \quad \mathrm{and} \quad T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} $$

on the right cosets.

Usage: Copy this folder into your GAP package directory (usually something like /opt/gap/gap4r8/pkg) and load it via

LoadPackage("ModularGroup");

Usage example: The two main objects of this package are called ModularSubgroup and ProjectiveModularSubgroup. A ModularSubgroup can be constructed in two different ways:

• either by specifying the action as permutations s and t and calling the constructor as follows:

G := ModularSubgroup(s,t);

• or by providing a list of generator matrices:

G := ModularSubgroup([ .. ]);

The former option is recommended since the latter implicitly computes the coset graph from the generators which might be time-consuming. Moreover, no check is performed if the generated group actually has finite index!

Projective modular subgroups can be constructed either by specifying permutations like above and executing:

G := ProjectiveModularSubgroup(s,t);

Or they can be derived from subgroups in $\mathrm{SL}_2(\mathbb{Z})$ by projecting them to $\mathrm{PSL}_2(\mathbb{Z})$ via

H := Projectivization(G);

Having constructed a (projective) modular subgroup, you can apply the various operations this package implements (such as testing if the given group is a congruence subgroup via IsCongruence(G)). For more details and a full list of the provided operations, please refer to the documentation. One explicit example is given below:

# it is assumed that the package has been loaded as described above
gap> G := ModularSubgroup(
> (1,2,5,3)(4,8,10,9)(6,11,7,12),
> (1,4)(2,6)(3,7)(5,10)(8,12,9,11)
> );
<modular subgroup of index 12>
gap> IsCongruence(G);
true
gap> GeneratorsOfGroup(G);
[ [ [ 1, -2 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 2, 1 ] ] ]
gap> GeneralizedLevel(G);
2
gap> Cusps(G);
[ infinity, 0, 1 ]