  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.
