On the quotient of the braid group by commutators of transversal half-twists and its group actions

This paper presents and describes a quotient of the Artin braid group by commutators of transversal half-twists and investigates its group actions. We denote the quotient by B̃n and refer to the groups which admit an action of B̃n as B̃n-groups. The group B̃n is an extension of a solvable group by a symmetric group. We distinguish special elements in B̃n-groups which we call prime elements and we give a criterion for an element to be prime. B̃n-groups appear as fundamental groups of complements of branch curves.