Boundaries and random walks on finitely generated infinite groups

We prove that almost every path of a random walk on a finitely generated non-amenable group converges in the compactification of the group introduced by W.J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of 1-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on work of V. Kaimanovich together with visibility properties of Floyd boundaries. We also prove a related statement about convergence of certain sequences of points, for example quasi-geodesic rays or orbits of 1-Lipschitz maps.