Quotients of braid groups by their congruence subgroups

TL;DR

This paper investigates the structure of quotients of braid groups formed by their congruence subgroups, connecting these quotients to symplectic congruence subgroups and addressing an open problem in the field.

Contribution

It determines the image of congruence subgroups in the integral Burau representation and characterizes the resulting braid group quotients in terms of symplectic congruence subgroups.

Findings

Identified the image of congruence subgroups in GL(n, Z)

Characterized braid group quotients via symplectic congruence subgroups

Addressed an open problem posed by Dan Margalit

Abstract

The congruence subgroups of braid groups arise from a congruence condition on the integral Burau representation $B_n \to \operatorname{GL}_{n}(\mathbb Z)$. We find the image of such congruence subgroups in $\operatorname{GL}_{n}(\mathbb Z)$-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.