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

TL;DR

This paper characterizes the boundary behavior of random walks on braid groups using hyperbolic geometry, demonstrating convergence to a boundary and stability of a specific normal form.

Contribution

It introduces a new description of random walk boundaries for braid groups via hyperbolic boundaries, establishing convergence and stability of the Markov--Ivanovsky normal form.

Findings

Almost all random walk trajectories converge to the boundary.

The Markov--Ivanovsky normal form is stable under random walks.

Boundary description applies to a broad family of groups.

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.