Arboreal structures on groups and the associated boundaries

Abstract

For any countable group with infinite conjugacy classes we construct a family of forests on the group. For each of them there is a random walk on the group with the property that its sample paths almost surely converge to the geometric boundary of the forest in a way that resembles the simple random walks on trees. It allows us to identify the Poisson boundary of the random walk with the boundary of the forest and to show that the group action on the Poisson boundary is free (which, in particular, implies non-triviality of the Poisson boundary). As a consequence we obtain that any countable group carries a random walk such that the stabilizer of almost every point of the Poisson boundary coincides with the hyper-FC-centre of the group, and, more generally, we characterize all normal subgroups which can serve as the pointwise stabilizer of the Poisson boundary of a random walk on a given countable group. Our work is a development of a recent result of Frisch - Hartman - Tamuz - Vahidi Ferdowsi who proved that any group which is not hyper-FC-central admits a measure with a non-trivial Poisson boundary.