Generators for the level $m$ congruence subgroups of braid groups

TL;DR

This paper proves that for sufficiently large n, the level m congruence subgroup of the braid group is generated by specific elements, extending previous results and solving a notable problem in the field.

Contribution

It establishes a generating set for the level m congruence subgroups of braid groups, generalizing prior work and addressing a problem posed by Margalit.

Findings

Generated by mth powers of half-twists and the braid Torelli group

Solves a problem of Margalit

Generalizes previous results by Assion, Brendle--Margalit, Nakamura, Stylianakis, and Wajnryb

Abstract

We prove for $m\geq1$ and $n\geq5$ that the level $m$ congruence subgroup $B_n[m]$ of the braid group $B_n$ associated to the integral Burau representation $B_n\to\mathrm{GL}_n(\mathbb{Z})$ is generated by $m$th powers of half-twists and the braid Torelli group. This solves a problem of Margalit, generalizing work of Assion, Brendle--Margalit, Nakamura, Stylianakis and Wajnryb.