On quotients of congruence subgroups of braid groups

TL;DR

This paper investigates the structure of congruence subgroups of braid groups via the integral Burau representation, exploring their quotients and generalizations beyond symmetric groups.

Contribution

It extends previous work by analyzing quotients of congruence subgroups without relying on generating sets, introducing new families of non-symmetric quotients.

Findings

Constructed explicit elements in preimages of transpositions.

Identified new quotient families not isomorphic to symmetric groups.

Generalized previous results on level four congruence subgroups.

Abstract

The integral Burau representation provides a map from the braid group into a group of integral matrices. This allows for a definition of congruence subgroups of the braid group as the preimage of the usual principal congruence subgroups of integral matrices. We explore the structure these congruence subgroups by examining some of the quotients that may arise in the series induced by divisibility of levels. We build on the work of Stylianakis on symmetric quotients of congruence subgroups, which itself generalizes the quotient of the braid group by the pure braid group. We accomplish this by utilizing results of Newman on integral matrices and explicitly finding elements in the preimage of any transposition. Our generalization is made possible by avoiding the use of a generating set for congruence subgroups. We find further generalizations based on results of Brendle and Margalit as well as Kordek and Margalit on the level four congruence subgroup. This gives families of quotients which are not isomorphic to symmetric groups.