Congruence subgroups of braid groups

TL;DR

This paper studies the structure of prime level congruence subgroups of braid groups, providing generators, new presentations, and finite generating sets for specific cases, advancing understanding of their algebraic properties.

Contribution

It offers a new description of generators for prime level congruence subgroups and presents a novel presentation of the symplectic group over finite fields.

Findings

Generators for prime level congruence subgroups identified

New presentation of the symplectic group over finite fields

Finite generating set for the level-3 congruence subgroup of B_3

Abstract

In this paper we give a description of the generators of the prime level congruence subgroups of braid groups. Also, we give a new presentation of the symplectic group over a finite field, and we calculate symmetric quotients of the prime level congruence subgroups of braid groups. Finally, we find a finite generating set for the level-3 congruence subgroup of the braid group on 3 strands.