On quotients of congruence subgroups of braid groups

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.