Powers of half-twists and congruence subgroups of braid groups

TL;DR

This paper explores the structure of congruence subgroups in braid groups, characterizing when certain normal closures have finite index, and constructing new finite quotients with explicit generators and subgroup properties.

Contribution

It provides a detailed analysis of the relationship between congruence subgroups and normal closures in braid groups, including conditions for finite index and explicit generators.

Findings

Normal closures have finite index under specific conditions.

Finite quotients are explicitly generated and characterized.

Infinite index cases contain free subgroups.

Abstract

In this work, we study the relationship between congruence subgroups $B_n[m]$ and $\mathcal{N}_n(\sigma_1^m)$ the normal closure of $\sigma_1^m$, where $\sigma_1$ is the classical generator of $B_n$. We characterize the conditions under which $\mathcal{N}_n(\sigma_1^m)$ has finite index in $B_n[m]$ and provide explicit generators for these finite quotients. For the cases where the index is infinite, we show that $B_n[m]/\mathcal{N}_n(\sigma_1^m)$ contains a free subgroup. Additionally, we compute the Abelianisation of Coxeter braid subgroups in the finite index cases and construct new finite quotients using commutators of congruence subgroups.