This GAP package provides methods for computing with finite-index subgroups of the modular groups SL_2(ℤ) and PSL_2(ℤ). This includes, but is not limited to, computation of the generalized level, index or cusp widths. It also implements algorithms described in [Hsu96] and [HL14] for testing if a given group is a congruence subgroup. Hence it differs from the Congruence package [DJKV18], which can be used - among other things - to construct canonical congruence subgroups of SL_2(ℤ).
A convenient way to represent finite-index subgroups of SL_2(ℤ) is by specifying the action of generator matrices of SL_2(ℤ) on the right cosets by right multiplication. For example, one could choose the generators
[ 0 -1 ] [ 1 1 ]
S = [ 1 0 ], T = [ 0 1 ]
and represent a subgroup as a tuple of transitive permutations (σ_S,σ_T) describing the action of S and T. This is exactly the way this package internally treats such subgroups. We use the convention that 1 corresponds to the coset of the identity matrix. Note that such a representation as a tuple of permutations is only unique up to relabelling of the cosets, i.e. up to simultaneous conjugation (fixing the 1 coset by our convention).
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