Characterization of finitely generated infinitely iterated wreath products

Abstract

Given a sequence of $(G_i{)}_{i\in \mathbb{N}}$ of finite transitive groups of degree $n_i$, let $W_{\infty }$ be the inverse limit of the iterated permutational wreath products of the first m groups. We prove that $W_{\infty }$ is (topologically) finitely generated if and only if ${\prod }_{i=1}^{\infty }(G_i/G_i^{\prime })$ is finitely generated and the growth of the minimal number of generators of $G_i$ is bounded by $d...n_1...n_{i-1}$ for a constant $d$. Moreover we give a criterion to decide whether $W_{\infty }$ is positively finitely generated.