Vershik, A. M.; Malyutin, A. V. Boundaries of braid groups and the Markov-Ivanovsky normal form. (English. Russian original) Zbl 1155.60033 Izv. Math. 72, No. 6, 1161-1186 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 6, 105-132 (2008). Summary: 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 [Russ. Math. Surv. 55, No. 4, 667–733 (2000; Zbl 0991.37005)] for braids. Cited in 1 Document MSC: 60J50 Boundary theory for Markov processes 20F36 Braid groups; Artin groups 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 37A50 Dynamical systems and their relations with probability theory and stochastic processes Citations:Zbl 0991.37005 PDFBibTeX XMLCite \textit{A. M. Vershik} and \textit{A. V. Malyutin}, Izv. Math. 72, No. 6, 1161--1186 (2008; Zbl 1155.60033); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 6, 105--132 (2008) Full Text: DOI arXiv