Boundary of the braid groups and Markov--Ivanovsky normal form

Abstract

We describe random walk boundaries (in particular, the Poisson--Furstenberg, or PF-boundary) for a vast family of groups in terms of the hyperbolic boundary of a special free subgroup. We prove that almost all trajectories of the random walk (with respect to an arbitrary nondegenerate measure on the group) converge to points of that boundary. This implies the stability (in the sense of \cite{Ver}) of the so-called Markov--Ivanovsky normal form for braids.