Powers of half-twists and congruence subgroups of braid groups
Paolo Bellingeri, Celeste Damiani, Oscar Ocampo, Charalampos Stylianakis
TL;DR
This work clarifies how level-$m$ congruence subgroups $B_n[m]$ of braid groups interact with Coxeter-type subgroups $ _n(\sigma_1^m)$. A precise finiteness criterion is established: the quotient $B_n[m]/ _n(\sigma_1^m)$ is finite if and only if $(m-2)(n-2)<4$, with explicit descriptions in the seven finite-index cases, and it is infinite otherwise in which case the quotient contains a free subgroup. The paper also investigates related finite quotients obtained by combining commutator subgroups, proving finiteness in several parametric regimes and deriving new cyclic quotients; it recovers known results for $m=2$ and provides new case-by-case finite quotients. Finally, it computes the Abelianisation of $ _n(\sigma_1^m)$ in the finite-index cases using Coxeter–Todd and Reidemeister–Schreier methods (aided by GAP), yielding explicit rank data for several $(n,m)$ and illuminating the abelian structure of these Coxeter-type subgroups. Together, these results deepen the understanding of how congruence and Coxeter-type subgroups cohabit braid groups and supply explicit finite quotients and abelian invariants for further study.
Abstract
In this work, we study the relationship between congruence subgroups $B_n[m]$ and $\mathcal{N}_n(σ_1^m)$ the normal closure of $σ_1^m$, where $σ_1$ is the classical generator of $B_n$. We characterize the conditions under which $\mathcal{N}_n(σ_1^m)$ 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 $B_n[m]/\mathcal{N}_n(σ_1^m)$ 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.