Characterization of finitely generated infinitely iterated wreath products

TL;DR

This paper characterizes when infinitely iterated wreath products of finite transitive groups are finitely generated, linking their generation properties to the structure of their abelian quotients and growth conditions.

Contribution

It provides necessary and sufficient conditions for the finite generation of infinitely iterated wreath products, including criteria for positive finite generation.

Findings

W_ _infty is finitely generated iff the product of abelian quotients is finitely generated.

Bounded growth of minimal generators of G_i is crucial.

A criterion for positive finite generation of W_ _infty is established.

Abstract

Given a sequence of $(G_i)_{i \in \N}$ 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 $\prod_{i=1}^{\infty} (G_i/G_i')$ 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.