Geodesic
Boundaries of braid groups and the Markov--Ivanovsky normal form
Izvestiya. Mathematics , Tome 72 (2008) no. 6, pp. 1161-1186.

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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. 
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A. M. Vershik; A. V. Malyutin. Boundaries of braid groups and the Markov--Ivanovsky normal form. Izvestiya. Mathematics , Tome 72 (2008) no. 6, pp. 1161-1186. http://geodesic.mathdoc.fr/item/IM2_2008_72_6_a4/

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