# Semi-free subgroups of a profinite surface group

Matan Ginzburg and Mark Shusterman
From the journal Journal of Group Theory

## 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 2^. 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 [7], where, for example, the following is proven.

### Theorem 1.1.

Let F be a nonabelian finitely generated free profinite group, and let HcF be a closed subgroup of infinite index in F. Then there exists a free profinite subgroup HLcF of rank 0.

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 NcΓg be a normal subgroup of infinite index in a profinite surface group of genus g2. Then there exists a semi-free profinite subgroup McΓg 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 Γg/N is not hereditarily just infinite.

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 d(G) the minimal size of a generating set of G.

### Definition 2.1.

An infinite profinite group G is called just infinite if for every {1}NcG the quotient G/N 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.

### 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 HKcG is open.

### Definition 2.3.

Given groups G,A,B and surjective homomorphisms

α:AB,β:GB,

we define the embedding problem(G,A,B,α,β) as the problem of finding a homomorphism φ:GA such that β=αφ. 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 (G,A,B,α,β) is called finite if A is finite. Note that since α is surjective, B is also finite.

### Definition 2.4.

An embedding problem (G,A,B,α,β) is called split if there is a homomorphism γ:BA such that αγ=idB. In such a case we have AKer(α)B.

### Definition 2.5.

A profinite group G of rank 0 is called semi-free if every finite split embedding problem (G,A,B,α,β) has a proper solution.

### Definition 2.6.

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

Γg=x1,,xg,y1,,yg|i=1g[xi,yi]=1.

The set x1,,xg,y1,,yg from such a presentation will be called a surface basis.

### Fact 2.7.

An open subgroup H of a genus g profinite surface group Γ is a profinite surface group of genus [Γ:H](g-1)+1.

### Claim 2.8.

Let E(G,A,B,α,β) be an embedding problem and let φ:GA be a solution of E. If Ker(α)Im(φ), then φ is a proper solution.

### Proof.

Let aA. Denote b=α(a) and let gG be such that β(g)=b. Then

α(a)=β(g)=α(φ(g)),

therefore

aφ(g)-1Ker(α)Im(φ).

As Im(φ) is a subgroup, we conclude that aIm(φ) so Im(φ)=A. ∎

### 3 Semi-free subgroup

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

### Lemma 3.1.

Let

Γ=xi,yi|i=1g[xi,yi]=1

be a profinite surface group, and let NcΓ. Consider the diagram

where A,B are finite groups, α is a surjection, and β¯|N=β is a surjection as well. Denote by

=(N,A,B,α,β),¯=(Γ,A,B,α,β¯)

the two finite embedding problems in the above diagram. Let φ:ΓA be a solution to E¯, and set

K:=Ker(α),n:=|Kφ(Γ)|,s:=d(K).

Suppose that gsn+s and that for all 1i,jsn+s,

φ(xi)=φ(xj),φ(yi)=φ(yj),xix1-1N.

Then E admits a proper solution.

### Proof.

Choose a set of generators B:={k1,,ks} of K. Define η(x1)=φ(x1) and

η(xin+j+1)=φ(xsn+i+1)ki+1=φ(x1)ki+1,0is-1, 1jn.

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

η(xi)=φ(xi),η(yj)=φ(yj),sn+1<ig, 1jg.

Since

[φ(xsn+i+1)ki+1,φ(ysn+i+1)]Kφ(Γ),0is-1,

we get

[η(xin+2),η(yin+2)]n=1=[φ(xin+2),φ(yin+2)]n,0is-1.

Therefore

i=1g[η(xi),η(yi)]=[η(x1),η(y1)]i=0s-1[η(xin+2),η(yin+2)]ni=sn+2g[η(xi),η(yi)]=[φ(x1),φ(y1)]i=0s-1[φ(xin+2),φ(yin+2)]ni=sn+2g[φ(xi),φ(yi)]=i=1g[φ(xi),φ(yi)]=1.

Thus η extends to a homomorphism. Since x1-1xiN for 1isn+s, we conclude that kj+1=η(x1-1xjn+2)η(N) for 0js-1, hence η(N) contains 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 Γg be a profinite surface group of genus g, let

=(Γg,A,B,α,β)

be a finite split embedding problem, and suppose that g2|A|2|B|. Then E has a proper solution, and Γg has a surface basis x1,,xg,y1,,yg such that

xix1-1,yiy1-1Ker(β),1im,m=2|A|2|B|.

Furthermore, if NcΓg is such that β(N)=B and

xix1-1N,1im,

then the embedding problem E¯=E(N,A,B,α,β|N) is properly solvable.

### Proof.

Let z1,,zg,w1,,wg be a surface basis of Γg. Let γ be a section of α and set φ=γβ. As before, put K=Ker(α), s=d(K), and n=|Kφ(Γg)|=|A|. Note that

s|K|=|A||B|

and thus

sn+s|A|2+|A||B|2|A|2|B|=m.

Each of the pairs (φ(zi),φ(wi)) can attain at most |B|2 values, whence by the pigeonhole principle (since gm|B|2) there are

1j1<j2<<jmg

such that

φ(zj1)==φ(zjm),φ(wj1)==φ(wjm).

Suppose j11. Then

1=i=1g[zi,wi]=[zj1,wj1][i=1j1-1[zi,wi]][zj1,wj1]i=j1+1g[zi,wi]=[zj1,wj1]i=1j1-1[zi[zj1,wj1],wi[zj1,wj1]]i=j1+1g[zi,wi]

and so we can replace {zi,wi}i=1g with a new surface basis

zj1,z1[zj1,wj1],,zj1-1[zj1,wj1],zj1+1,,zg,
wj1,w1[zj1,wj1],,wj1-1[zj1,wj1],wj1+1,,wg.

By repeating this process with j2,,jm, we obtain a surface basis x1,,xg, y1,,yg of Γg such that xi=zji,yi=wji for 1im, and so

φ(x1)==φ(xm),φ(y1)==φ(ym).

Since sn+sm, we can apply Lemma 3.1 (with N=Γg if necessary) and the result follows. ∎

### Corollary 3.3.

The finite split embedding problem

has a proper solution once g2|K|2|H|3, Γg is a profinite surface group of genus g, and G is any profinite group.

### Proof.

Write H0 for β(Γg). According to Lemma 3.2, the finite split embedding problem

has a proper solution φ1. Let φ2:GKH be the map defined by

φ2(f)=(1,β|G(f))

for every fG. There exists a unique homomorphism

φ:ΓgGKH

such that

φ(γ)=φ1(γ),φ(f)=φ2(f)for all γΓg and all fG.

By the universal property of free products, φ is a solution to the original embedding problem. Since K is contained in Im(φ1), it is also contained in Im(φ) so by Claim 2.8, φ is a proper solution. ∎

### Theorem 3.4.

Let NcΓg be a normal subgroup of the profinite surface group of genus g2 such that Γg/N is hereditarily just infinite. Then 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

NFoΓg

such that

[Γg:F]2|K|2|H|3+m-1g-1,

where m=2|K|2|H|. By Fact 2.7, F is a profinite surface group of genus

h=[Γg:F](g-1)+1.

Note that

h-m2|K|2|H|3.

Applying Lemma 3.2 to F, the extended embedding problem and m, we obtain a proper solution φ, and a surface basis x1,,xh,y1,,yh of F such that

xjx1-1,yjy1-1Ker(β),1jm.

If xix1-1N for all 1im, then by Lemma 3.2, our original embedding problem has a proper solution. Assume henceforth that xix1-1N for some i with 1im.

The homomorphism β factors modulo

L=x2x1-1,,xmx1-1,y2y1-1,,ymy1-1F.

Note that

F/Lx1,xm+1,,xh,y1,ym+1,,yh|[x1,y1]mi=m+1h[xi,yi]=1

and that

β([x1,y1]m)=β([x1,y1])m=1,

so β even factors through

F/L,[x1,y1]mFxi,yi|i=m+1h[xi,yi]=1x1,y1[x1,y1]m=1Γh-mG,

where G=x1,y1[x1,y1]m=1.

Let

ψ:FF/L,[x1,y1]mF

be the quotient map, and set M=ψ(N). Since Γg/N is hereditarily just infinite, F/N is just infinite, so ψ-1(M) either equals N or is open in F. The former is impossible as there exists 1im such that

xix1-1Ker(ψ)ψ-1(M),xix1-1N.

Hence ψ-1(M) is an open subgroup of F, so M can be seen as an open subgroup of Γh-mG.

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 MΓtG¯, where

th-m2|K|2|H|3

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

Repeating the above proof verbatim, one obtains the following.

### Theorem 3.5.

Let Γg be a surface group and NcΓg weakly maximal. Then N is semi-free profinite.

Communicated by Pavel A. Zalesskii

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.

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