We show that every closed normal subgroup of infinite index in a profinite surface group Γ is contained in a semi-free profinite normal subgroup of Γ. This answers a question of Bary-Soroker, Stevenson, and Zalesskii
The classical theorem of Nielsen and Schreier states that every subgroup of a free group is free. Trying to extend this result to profinite groups fails, for example . This naturally gives rise to the question of finding conditions upon which a subgroup of a free profinite group is free.
Some results in this direction are known, for instance, Melnikov’s characterization of normal subgroups of free profinite groups, and Haran’s diamond theorem. Results of a slightly different flavor have been obtained by Shusterman in , where, for example, the following is proven.
Let F be a nonabelian finitely generated free profinite group, and let be a closed subgroup of infinite index in F. Then there exists a free profinite subgroup of rank .
In particular, weakly maximal subgroups (see Definition 2.2) are free.
In this work we consider an analog of the above for profinite surface groups. These groups show up as étale fundamental groups of curves over an algebraically closed field of characteristic 0.
We will be interested in semi-free profinite subgroups (of profinite surface groups), a notion introduced in , where it is shown that a group is free profinite if and only if it is projective and semi-free. As shown in , projectivity of a subgroup of a profinite surface group is equivalent to a simple condition on its index (as a supernatural number). Henceforth, we will focus on semi-freeness.
Our main result is the following.
Let be a normal subgroup of infinite index in a profinite surface group of genus . Then there exists a semi-free profinite subgroup that contains N.
This answers a question raised by Bary-Soroker, Stevenson, and Zalesskii (in [2, Remark 4.1]), who used their diamond theorem to establish the special case where is not hereditarily just infinite.
Our method also gives the following analog of the aforementioned results of Shusterman.
Weakly maximal subgroups of profinite surface groups are semi-free profinite.
In this section we give the basic definitions and claims that will be used in the rest of this paper. We will work in the category of profinite groups, namely we assume that every subgroup is closed, every homomorphism is continuous, and so on. For a finitely generated profinite group G we denote by the minimal size of a generating set of G.
An infinite profinite group G is called just infinite if for every the quotient is finite. Equivalently, every nontrivial normal subgroup of G is open. We say that G is hereditarily just infinite if every open normal subgroup of it is just infinite.
Let H be a closed subgroup of infinite index in a profinite group G. We say that H is weakly maximal in G if every is open.
Given groups and surjective homomorphisms
we define the embedding problem as the problem of finding a homomorphism such that . Such a homomorphism φ is called a solution to the problem. If moreover φ is surjective, then it is called a proper solution. An embedding problem is called finite if A is finite. Note that since α is surjective, B is also finite.
An embedding problem is called split if there is a homomorphism such that . In such a case we have .
A profinite group G of rank is called semi-free if every finite split embedding problem has a proper solution.
The profinite surface group of genus g is the group given by the profinite presentation
The set from such a presentation will be called a surface basis.
An open subgroup H of a genus g profinite surface group Γ is a profinite surface group of genus .
Let be an embedding problem and let be a solution of . If , then φ is a proper solution.
Let . Denote and let be such that . Then
As is a subgroup, we conclude that so . ∎
3 Semi-free subgroup
We need the following variant of [5, Lemma 6.1].
be a profinite surface group, and let . Consider the diagram
where are finite groups, α is a surjection, and is a surjection as well. Denote by
the two finite embedding problems in the above diagram. Let be a solution to , and set
Suppose that and that for all ,
Then admits a proper solution.
Choose a set of generators of K. Define and
Let η coincide with φ for all other generators of Γ, that is,
Thus η extends to a homomorphism. Since for , we conclude that for , hence contains K, so the result follows by invoking Claim 2.8. ∎
We also need the following generalization of [2, Lemma 2.2].
Let be a profinite surface group of genus g, let
be a finite split embedding problem, and suppose that . Then has a proper solution, and has a surface basis such that
Furthermore, if is such that and
then the embedding problem is properly solvable.
Let be a surface basis of . Let γ be a section of α and set . As before, put , , and . Note that
Each of the pairs can attain at most values, whence by the pigeonhole principle (since ) there are
Suppose . Then
and so we can replace with a new surface basis
By repeating this process with , we obtain a surface basis , of such that for , and so
Since , we can apply Lemma 3.1 (with if necessary) and the result follows. ∎
The finite split embedding problem
has a proper solution once , is a profinite surface group of genus g, and G is any profinite group.
Write for . According to Lemma 3.2, the finite split embedding problem
has a proper solution . Let be the map defined by
for every . There exists a unique homomorphism
By the universal property of free products, φ is a solution to the original embedding problem. Since K is contained in , it is also contained in so by Claim 2.8, φ is a proper solution. ∎
Let be a normal subgroup of the profinite surface group of genus such that is hereditarily just infinite. Then N is semi-free.
be a finite split embedding problem for N. We shall prove it has a proper solution.
Using [3, Lemma 1.2.5 (c)], we can extend our embedding problem to a subgroup
where . By Fact 2.7, F is a profinite surface group of genus
Applying Lemma 3.2 to F, the extended embedding problem and m, we obtain a proper solution φ, and a surface basis of F such that
If for all , then by Lemma 3.2, our original embedding problem has a proper solution. Assume henceforth that for some i with .
The homomorphism β factors modulo
so β even factors through
be the quotient map, and set . Since is hereditarily just infinite, is just infinite, so either equals N or is open in F. The former is impossible as there exists such that
Hence is an open subgroup of F, so M can be seen as an open subgroup of .
We can now write the original embedding problem as
so it is sufficient to properly solve it for M. Applying the Kurosh theorem (see [6, Theorem D.3.1]), we find that , where
and so by Corollary 3.3 the desired proper solution exists. ∎
Repeating the above proof verbatim, one obtains the following.
Let be a surface group and weakly maximal. Then N is semi-free profinite.
Funding statement: The authors were partially supported by a grant of the Israel Science Foundation with cooperation of UGC no. 40/14.
The authors would like to thank Lior Bary-Soroker and Pavel Zalesskii for useful remarks and discussions. Mark Shusterman is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship.
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