We give a topological interpretation of the free metabelian group, following the plan
described in [11, 12]. Namely, we represent the free metabelian group with d-generators as
an extension of the group of the first homology of the d-dimensional lattice (as Cayley graph
of the group ℤd), with a canonical 2-cocycle. This construction opens a possibility to study
metabelian groups from new points of view; in particular to give useful normal forms of the
elements of the group, leading to applications to the random walks, and so on. We also …