The rank and generating set for inverse limits of wreath products of Abelian groups

Abstract

We derive an exact formula for the topological rank d(W) of the inverse limit \({W = \ldots \wr A_2 \wr A_1}\) of iterated wreath products of arbitrary nontrivial finite Abelian groups. By using the language of automorphisms of a spherically homogeneous rooted tree, we construct and study a topological generating set for W with cardinality \({d(A_1) + \rho'}\) , where \({\rho'}\) is the topological rank of the profinite Abelian group \({A_2 \times A_3 \times \cdots}\) . In particular, if the group A 1 is cyclic, this approach gives a minimal generating set for W.