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, )