Powers of half-twists and congruence subgroups of braid groups

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.

arXiv comment10 pages. Comments are welcome. To be published in AMS Contemporary Mathematics