Finite generation of iterated wreath products

TL;DR

This paper characterizes when infinitely iterated wreath products of finite transitive groups are topologically finitely generated, linking it to the finite generation of associated profinite abelian groups, and explores implications for wreath powers and branch groups.

Contribution

It provides a necessary and sufficient condition for the finite generation of infinite iterated wreath products based on profinite abelian groups, and constructs a finitely generated branch group with maximal subgroups of infinite index.

Findings

Infinite iterated wreath product is finitely generated iff associated profinite abelian group is finitely generated.

Wreath power of a finite transitive group has bounded generators if perfect, grows linearly if non-perfect.

Constructed a finitely generated branch group with maximal subgroups of infinite index.

Abstract

Let $(G_n,X_n)$ be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product $...\wr G_2\wr G_1$ is topologically finitely generated if and only if the profinite abelian group $\prod_{n\geq 1} G_n/G'_n$ is topologically finitely generated. As a corollary, for a finite transitive group $G$ the minimal number of generators of the wreath power $G\wr...\wr G\wr G$ ($n$ times) is bounded if $G$ is perfect, and grows linearly if $G$ is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index, answering [2,Question 14].