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. In this paper we describe a quotient of the Artin braid group by commutators of transversal half-twists and investigate its group actions. These groups turned out to be extremely important in describing fundamental groups of complements of branch curves (see, e.g., [Te]). The description here is completely independent from the algebraic-geometrical background and provides an algebraic study of the groups involved, using a topological approach to the braid group. We denote the quotient by B̃n and refer to the groups which admit an action of B̃n as B̃n-groups. In particular, we study P̃n, the image of the pure braid group in B̃n, and prove that B̃n is an extension of a solvable group by a symmetric group. The main results on the structure of B̃n are Theorem 6.4 and Corollary 6.5. We distinguish special elements in B̃n-groups which we call prime elements, compute the action of halftwists on prime elements (§2-§4), and finally we give a criterion for an element to be prime (see Proposition 7.1). This criterion will be applied to the study of fundamental groups of complements of branch curves. Fundamental groups related to algebraic varieties are very important in classification problems and in topological studies in algebraic geometry. These groups are very difficult to compute. The group B̃n and the B̃n-groups appeared when we were computing such groups. It turns out that all new examples of fundamental 1991 Mathematics Subject Classification. 20F36.