Boundaries of braid groups and the Markov–Ivanovsky normal form

We describe random walk boundaries (in particular, the Poisson–Furstenberg, or PF-, boundary) for a large family of groups in terms of the hyperbolic boundary of a special normal free subgroup. We prove that almost all the trajectories of a random walk (with respect to an arbitrary non-degenerate measure on the group) converge to points of that boundary. This yields the stability (in the sense of [6]) of the so-called Markov–Ivanovsky normal form [12] for braids.