# Abstract

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

## 1 Introduction

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

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 [7], where, for example, the following is proven.

## Theorem 1.1.

*Let F be a nonabelian finitely generated free profinite group, and let *

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 [1], where it is shown that a group is free profinite if and only if it is projective and semi-free. As shown in [8], 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.

## Theorem 1.2.

*Let *

*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

Our method also gives the following analog of the aforementioned results of Shusterman.

## Theorem 1.3.

*Weakly maximal subgroups of profinite surface groups are semi-free profinite.*

Weakly maximal subgroups (see Definition 2.2) were also studied in the context of branch groups, for instance in [4].

## 2 Preliminaries

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 *G*.

## Definition 2.1.

An infinite profinite group *G* is called *just infinite* if for every *G* is open.
We say that *G* is *hereditarily just infinite* if every open normal subgroup of it is just infinite.

## Definition 2.2.

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

## Definition 2.3.

Given groups

we define the *embedding problem**solution* to the problem. If moreover φ is surjective, then it is called a *proper solution*.
An embedding problem *finite* if *A* is finite. Note that since α is surjective, *B* is also finite.

## Definition 2.4.

An embedding problem *split* if there is a homomorphism

## Definition 2.5.

A profinite group *G* of rank *semi-free* if every finite split embedding problem

## Definition 2.6.

The *profinite surface group of genus g* is the group given by the profinite presentation

The set *surface basis*.

## Fact 2.7.

*An open subgroup H of a genus g profinite surface group Γ is a profinite surface group of genus *

## Claim 2.8.

*Let *

## Proof.

Let

therefore

As

## 3 Semi-free subgroup

We need the following variant of [5, Lemma 6.1].

## Lemma 3.1.

*Let*

*be a profinite surface group, and let *

*where *

*the two finite embedding problems in the above diagram.
Let *

*Suppose that *

*Then *

## Proof.

Choose a set of generators *K*. Define

Let η coincide with φ for all other generators of Γ, that is,

Since

we get

Therefore

Thus η extends to a homomorphism. Since *K*, so the result follows by invoking Claim 2.8.
∎

We also need the following generalization of [2, Lemma 2.2].

## Lemma 3.2.

*Let *

*be a finite split embedding problem, and suppose that *

*Furthermore, if *

*then the embedding problem *

## Proof.

Let

and thus

Each of the pairs

such that

Suppose

and so we can replace

By repeating this process with

Since

## Corollary 3.3.

*The finite split embedding problem*

*has a proper solution once *

*g*, and

*G*is any profinite group.

## Proof.

Write

has a proper solution

for every

such that

By the universal property of free products, φ is a solution to the original embedding problem.
Since *K* is contained in

## Theorem 3.4.

*Let *

*N*is semi-free.

## Proof.

Let

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

such that

where *F* is a profinite surface group of genus

Note that

Applying Lemma 3.2 to *F*, the extended embedding problem and *m*, we obtain a proper solution φ, and a surface basis *F* such that

If *i* with

The homomorphism β factors modulo

Note that

and that

so β even factors through

where

Let

be the quotient map, and set *N* or is open in *F*. The former is impossible as there exists

Hence *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

and so by Corollary 3.3 the desired proper solution exists. ∎

Repeating the above proof verbatim, one obtains the following.

## Theorem 3.5.

*Let *

*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.

# Acknowledgements

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.

### References

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P. A. Zalesskii,
Profinite surface groups and the congruence kernel of arithmetic lattices in

**Received:**2018-04-15

**Revised:**2018-05-07

**Published Online:**2018-06-08

**Published in Print:**2018-09-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston