Subgroups of braid groups generated by Birman-Ko-Lee generators

Abstract

We define a Young subgroup of the braid group as a subgroup generated by an arbitrary subset of the Birman-Ko-Lee generators. We give an intrinsic description of such subgroups which yields, in particular, an easy criterion to decide membership. We also give an algorithm to write an element of a Young subgroup as a product of the generators. Our methods are based on analyzing the Hurwitz action on tuples over free groups via a diagrammatic approach.