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

TL;DR

This paper introduces a new approach to understanding subgroups of braid groups generated by Birman-Ko-Lee generators, providing criteria for membership and algorithms for element decomposition.

Contribution

It offers an intrinsic description and an algorithm for subgroups generated by Birman-Ko-Lee generators, enhancing the understanding of braid group structure.

Findings

Provides an easy criterion for subgroup membership.

Develops an algorithm to express elements as generator products.

Uses diagrammatic approach to analyze Hurwitz action.

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.