Double coset problem for parabolic subgroups of braid groups

TL;DR

This paper presents the first comprehensive solution to the double coset problem for a broad class of parabolic subgroups in braid groups, significantly advancing understanding of their algebraic structure.

Contribution

It provides the first solution to the DCP for parabolic subgroups with connected Coxeter graphs, and extends this to all parabolic subgroups of braid groups.

Findings

Solved the DCP for all parabolic subgroups of braid groups

Established methods for connected Coxeter graph cases

Extended solutions to all parabolic subgroups

Abstract

We provide the first solution to the double coset problem (DCP) for a large class of natural subgroups of braid groups, namely for all parabolic subgroups which have a connected associated Coxeter graph. Update: We succeeded to solve the DCP for all parabolic subgroups of braid groups.