Double coset problem for parabolic subgroups of braid groups
  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. 
Arkadius Kalka, Mina Teicher and Boaz Tsaban
math.GR