Free subgroups of finitely generated free profinite groups

TL;DR

This paper advances understanding of free subgroups in finitely generated free profinite groups, establishing new conditions for freeness and addressing conjectures related to field theory and algebraic structures.

Contribution

It provides new criteria for subgroup freeness in free profinite groups and proves a conjecture connecting subgroup structure to field Hilbertianity.

Findings

Subgroups containing the normal closure of a finite word are free.

Infinite index subgroups are contained in infinitely generated free profinite subgroups.

A general sufficient condition for subgroups to be free profinite is established.

Abstract

We give new and improved results on the freeness of subgroups of free profinite groups: A subgroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subgroup of a finitely generated nonabelian free profinite group is contained in an infinitely generated free profinite subgroup. These results are combined with the twisted wreath product approach of Haran, an observation on the action of compact groups, and a rank counting argument to prove a conjecture of Bary-Soroker, Fehm, and Wiese, thus providing a quite general sufficient condition for subgroups to be free profinite. As a result of our work, we are able to address a conjecture of Jarden on the Hilbertianity of fields generated by torsion points of abelian varieties.