The topological decomposition of inverse limits of iterated wreath products of finite Abelian groups

Let (A_i)_i ≤1 be an arbitrary infinite sequence of nontrivial finite Abelian transitive groups such that the topological rank p of the infinite Cartesian product of these groups is finite. We consider the corresponding inverse limit W D:::{A₂} A₁ of iterated permutational wreath products. By using the geometric language of automorphisms of a certain rooted tree we show that the group W is topologically generated by the union S∪Š of two sets each containing ρ elements and such that both the group generated by S and the group generated by Š are free Abelian groups of rank ρ. Moreover, the group G generated by the above union decomposes as a semidirect product of the group generated by one of these sets and the normal closure of the other set, and the semigroup generated by this union is a free product of semigroups generated by S and Š, respectively. We derive other algebraic properties of the group G. In particular, the condition for the groups W to be finitely generated as profinite groups and some nontrivial results concerning topological ranks of their subgroups are given in this way. For example, for every n > 1 our construction gives an explicit and naturally defined finitely generated subgroup of an inverse limit of iterated wreath products of finite Abelian groups such that the rank of this subgroup and the topological rank of its topological closure differ by n.