Embedding properties of hereditarily just infinite profinite wreath products

TL;DR

This paper explores the embedding properties and structural characteristics of infinitely iterated wreath products of finite simple groups, revealing their universality and conditions for being co-Hopfian.

Contribution

It constructs finitely generated hereditarily just infinite profinite groups with specified composition factors and analyzes their embedding universality and co-Hopfian properties.

Findings

Existence of universal finitely generated hereditarily just infinite profinite groups.

Characterization of when these wreath products are co-Hopfian.

Identification of conditions for embedding various profinite groups.

Abstract

We study infinitely iterated wreath products of finite permutation groups with respect to product actions. In particular, we prove that, for every non-empty class of finite simple groups $\mathcal{X}$, there exists a finitely generated hereditarily just infinite profinite group $W$ with composition factors in $\mathcal{X}$ such that any countably based profinite group with composition factors in $\mathcal{X}$ can be embedded into $W$. Additionally we investigate when infinitely iterated wreath products of finite simple groups with respect to product actions are co-Hopfian or non-co-Hopfian.