Powers of half-twists and congruence subgroups of braid groups

In this work, we study the relationship between congruence subgroups B n [ m ] B_n[m] and N n ( σ 1 m ) \mathcal {N}_n(\sigma _1^m) the normal closure of σ 1 m \sigma _1^m , where σ 1 \sigma _1 is the classical generator of B n B_n . We characterize the conditions under which N n ( σ 1 m ) \mathcal {N}_n(\sigma _1^m) has finite index in B n [ m ] 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 ] / N n ( σ 1 m ) 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.