Mark Shusterman:
Free profinite groups and their subgroups

Abstract:
We derive sufficient conditions for a subgroup of a free profinite group
to be free profinite itself. In the abstract case, the classical theorem
of Nielsen and Schreier tells us that every subgroup of a free group is
free. In the profinite case, counterexamples exist and we develop a rather
general and elementary technique for proving freeness of subgroups that
gives much of the results previously derived from the diamond theorem and
the wreath product approach, which are the most advanced results in this
area.

Our setup fixes a normal subgroup N of a free (finitely generated) pro-
finite group F and examines the quotient. Almost all of the theorems in
this direction say that if F/N is not "general enough" in a sense, N
(and intermediate subgroups) will be free. We measure this generality
using a combinatorial invariant called Schreier's index formula, and show
that if it is not satisfied, freeness is guaranteed for an abundance of
subgroups.

We use our technique to reprove some known results like the generalization
of a group-theoretic analogue of Weissauer's theorem, and the case of pro-
nilpotent (the pro-p case is new) quotient. We also prove some new results
that the known methods do not seem to reach, like the case of solvable,
prosupersolvable or finite-length (in the sense of http://arxiv.org/abs/1203.4217)
quotients. Finally, we carry over our results to the countably generated
case allowing applications to the theory of Hilbertian fields.