Congruence subgroups from representations of the three-strand braid group

摘要

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 B-3 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 sigma(1) 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 sigma(1) 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. (C) 2017 Elsevier Inc. All rights reserved.