Finite generation of iterated wreath products in product action

TL;DR

This paper proves that certain infinitely iterated wreath products of finite perfect transitive groups are topologically finitely generated under specific conditions, with explicit generators provided for particular cases.

Contribution

It establishes finite generation of iterated wreath products in product action for non-regular groups and provides explicit generators for special sequences.

Findings

Infinitely iterated wreath products are topologically finitely generated.

The bounds on generators have correct asymptotic behavior.

Explicit generators are constructed for specific sequences.

Abstract

Let $\mathcal{S}$ be a sequence of finite perfect transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated wreath product in product action of the groups in $\mathcal{S}$ is topologically finitely generated, provided that the actions of the groups in $\mathcal{S}$ are not regular. We prove that our bound has the right asymptotic behaviour. We also deduce that other infinitely iterated mixed wreath products of groups in $\mathcal{S}$ are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterated wreath products in product action of special sequences $\mathcal{S}$.