Arboreal structures on groups and the associated boundaries

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