The level four braid group

TL;DR

This paper investigates the level four congruence subgroup of the braid group, showing it is generated by squares of Dehn twists and pure braids, and explores its symplectic representation.

Contribution

It establishes the equivalence of the level four congruence subgroup with groups generated by squares of specific braids and computes its image under a symplectic representation.

Findings

Level four congruence subgroup equals the group generated by squares of Dehn twists.

Level four congruence subgroup equals the group generated by squares of pure braids.

Computed the image of the point pushing subgroup under the symplectic representation.

Abstract

By evaluating the Burau representation at t=-1, we obtain a symplectic representation of the braid group. We define the congruence subgroups of the braid group to be the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup of the braid group is equal to the group generated by squares of Dehn twists and is also equal to the group generated by squares of pure braids. We also compute the image of the point pushing subgroup under the symplectic representation.