Narrow Results

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 ℤ^d), 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 describe the satellite groups which correspond to all 2-cocycles of cohomology group associated with the free solvable groups. The homology of the Cayley graph can be used for studying the wide class of groups which include the class of all solvable groups.

Leech’s (co)homology groups of finite cyclic monoids are computed.