Quotients of braid groups by their congruence subgroups

Abstract

The congruence subgroups of braid groups arise from a congruence condition on the integral Burau representation $B_n\to {\mathrm{GL}}_n(\mathbb{Z})$. We find the image of such congruence subgroups in ${\mathrm{GL}}_n(\mathbb{Z})$-an open problem posed by Dan Margalit. Additionally, we characterize the quotients of braid groups by their congruence subgroups in terms of symplectic congruence subgroups.

Important correction: The previous version incorrectly used the reduced Burau representation