Torsion-freeness of the abelianization of congruence subgroups B_n[m]

Establish whether the abelianization of the level m congruence subgroup of the braid group, B_n[m]/[B_n[m], B_n[m]], is torsion-free for arbitrary integers n and m ≥ 2.

Background

The level m congruence subgroup B_n[m] is defined as the kernel of the mod m reduction of the integral Burau representation. The quotient B_n[m]/[B_n[m], B_n[m]] is the abelianization of B_n[m].

This torsion-freeness question is directly connected to crystallographic properties established in the paper: under the hypothesis that B_n[m]/[B_n[m], B_n[m]] is torsion-free (for n odd and m prime), the group B_n/[B_n[m], B_n[m]] is shown to be crystallographic (Theorem 3.3). Known positive cases include B_n[2]/[B_n[2], B_n[2]] being free abelian of rank \binom{n}{2}, and for n = 3, m = 3 or 4, the groups are torsion-free with ranks 4 and 6, respectively.