Random walks on three-strand braids and on related hyperbolic groups

We investigate the statistical properties of random walks on the simplest nontrivial braid group B₃, and on related hyperbolic groups. We provide a method using Cayley graphs of groups allowing us to compute explicitly the probability distribution of the basic statistical characteristics of random trajectories - the drift and the return probability. The action of the groups under consideration in the hyperbolic plane is investigated, and the distribution of a geometric invariant - the hyperbolic distance - is analysed. It is shown that a random walk on B₃ can be viewed as a 'magnetic random walk' on the group PSL(2,Z).