Congruence subgroups from representations of the three-strand braid group

Ng and Schauenburg proved that the kernel of a (2+1)-dimensional topological quantum field theory representation of SL(2,Z) is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group B3 with finite image enjoys a congruence subgroup property. In particular, we show that in dimensions two and three, when the projective order of the image of the braid generator σ1 is between 2 and 5 the kernel projects onto a congruence subgroup of PSL(2,Z) and compute its level. However, we prove that for three dimensional representations, the projective order is not enough to decide the congruence property. For each integer of the form 2ℓ≥6 with ℓ odd, we construct a pair of non-congruence subgroups associated with three-dimensional representations having finite image and σ1 mapping to a matrix with projective order 2ℓ. Our technique uses classification results of low dimensional braid group representations, and the Fricke-Wohlfahrt theorem in number theory.