On free profinite subgroups of free profinite monoids

TL;DR

This paper proves that the maximal subgroup of the minimal ideal in finitely generated free profinite monoids is a free profinite group, extending to certain varieties of finite groups.

Contribution

It establishes that these maximal subgroups are free profinite groups, answering a question from 1997 and generalizing to varieties of finite groups with specific properties.

Findings

Maximal subgroups of minimal ideals are free profinite groups.

Results apply to free pro-H monoids for certain varieties H.

Extends previous understanding of structure in free profinite monoids.

Abstract

We answer a question of Margolis from 1997 by establishing that the maximal subgroup of the minimal ideal of a finitely generated free profinite monoid is a free profinite group. More generally if $\mathbf H$ is variety of finite groups closed under extension and containing $\mathbb Z/p\mathbb Z$ for infinitely may primes $p$, the corresponding result holds for free pro-$\bar{\mathbf H}$ monoids.