In this paper we only consider finitely generated parafree groups and from now on we assume that to be finitely generated is a part of the definition of parafree groups

WITH CYCLIC EDGE SUBGROUPS

ANDREI JAIKIN-ZAPIRAIN AND ISMAEL MORALES

Abstract. We determine when the fundamental group of a finite graph of groups with cyclic edge subgroups is parafree.

1. Introduction

A group is said to beparafreeif it is residually nilpotent and its quotients by the terms of its lower central series are the same as those of a free group. These groups were introduced by Baumslag [3] who also constructed first examples of non-free parafree groups. In this paper we only consider finitely generated parafree groups and from now on we assume that to be finitely generated is a part of the definition of parafree groups.

Many known examples of parafree groups are isomorphic to amalgamated free products or HNN extensions of free groups [6]. The purpose of this paper is to characterize which amalgamated free products and HNN extensions with cyclic base groups are parafree.

The following statement describes exactly under which circumstances an amal- gamated free product with cyclic amalgam is parafree.

Theorem 1.1. LetU andV be finitely generated groups,16=u∈U and16=v∈V. Consider the amalgamated free productW =U ∗

u=vV. Then W is parafree if and only if the following three conditions hold.

(1) The groups U andV are parafree.

(2) The elementuv⁻¹ is not a proper power in the abelianization ofU∗V. (3) At least one of uorv is not a proper power in U orV, respectively.

A particular case of Theorem 1.1, whereU andV are free, follows from [4, 1]. In the case of HNN extensions with cyclic base groups we obtain the following result.

Theorem 1.2. LetU be a finitely generated group,u∈U\ {1}andα:hui →U a monomorphism. Putv=α(u). Consider the HNN extension W =U∗α. Then W is parafree if and only if the following four conditions hold.

(1) The group U is parafree.

(2) The elementuv⁻¹ is not a proper power in the abelianization ofU. (3) At least one of uorv is not a proper power in U.

Date: December 2021.

2010Mathematics Subject Classification. Primary: 20E06, Secondary: 16K40, 20C07, 20E18, 20E26.

Key words and phrases. Parafree groups, free pro-pgroups, universal division ring of fractions, amalgamated free products, HNN extensions.

1

(4) The image of the elementuis non-trivial in some finite nilpotent quotient of W.

In the case where U is a free group on 2 generators the last condition can be replaced by a simpler one.

Corollary 1.3. LetF2be a free group on two generators,u∈F2\{1}andα:hui → F₂ a monomorphism. Put v = α(u). Consider the HNN extension W = F₂∗α. ThenW is parafree if and only if the following three conditions hold.

(1) The elementuv⁻¹ is not a proper power in the abelianization ofF₂. (2) At least one of uorv is not a proper power in F₂.

(3) The images of u and v in the abelianization of F₂ generate a subgroup isomorphic toZ².

It is tempting to try to formulate a general criterion for the fundamental group of a graph of groups with cyclic edge groups to be parafree. Combining Theorem 1.1 and Theorem 1.2 we obtain the following result.

Corollary 1.4. Let (G,Γ) be a graph of groups over a finite graph Γ and W = π(G,Γ)be its fundamental group. Assume that all vertex subgroupsG(v)(v∈V(Γ)) are finitely generated and all edge subgroups G(e)(e∈E(Γ)) are cyclic. Then W is parafree if and only if the following four conditions hold.

(1) All the vertex subgroups G(v)(v∈V(Γ)) are parafree.

(2) The abelianization ofW is torsion-free of rank rab(W) = X

v∈V(Γ)

rab(G(v))− X

e∈E(Γ)

rab(G(e))−χ(Γ),

whereχ(Γ) =|V(Γ)| − |E(Γ)|-1.

(3) All the centralizers of non-trivial elements inW are cyclic.

(4) For each non-trivial edge subgroup of G(e) (e ∈ E(Γ)) there is a finite nilpotent quotient ofW where the image of this edge subgroup is non-trivial.

It is relatively easy to see that the conditions presented in Theorem 1.1 and Theorem 1.2 are necessary for W to be parafree and they also imply that the quotients of W by the terms of its lower central series are the same as those of a free group. The difficult part in the proofs of both theorems is to show that W is residually nilpotent. We establish this as an application of recent methods developed in [12] which were used for constructing abstract subgroups of free pro-p groups.

The paper is organized as follows. In Section 2 we describe the preliminary results which we use in the paper. Section 3 is devoted to construction of new examples of D_FpG-torsion-free FpG-modules. In Section 4 we prove Theorem 1.1 and Theorem 1.2 and their corollaries.

Acknowledgments

This paper is partially supported by the Spanish Ministry of Science and Inno- vation through the grant MTM2017-82690-P and the “Severo Ochoa Programme for Centres of Excellence in R&D" (CEX2019-000904-S4). The work of the second author is supported by the collaboration grant JAE Intro SOMdM 2020, associated to ICMAT.

2. Preliminaries

2.1. General notation for rings, groups and pro-p groups. All of our rings R will be associative and unitary. All ring homomorphisms will map 1 7→ 1. All R-modules are leftR-modules, unless we say that we consider rightR-modules.

Given a group G, we will denote by G⁰ = [G, G]its commutator subgroup and byG_ab =G/G⁰ its abelianization. The terms of the lower central series of Gare defined as γ₁(G) = G and γ_n+1(G) = [γ_n(G), G] if n ≥ 1 and the terms of the p-lower central series ofG are defined asG1,p =G andGn+1,p = [Gn,p, G]G^p_n,p if n≥1.

We denote byd(G)the minimal number of generators of a groupGand we put rab(G) =d(Gab).

Let F be a pro-pgroup. In this case we denote by d(F) the minimal number of topological generators of F. The completed group algebra of F over Fp is the inverse limit of theFp-group algebrasFp[F/U]

Fp[[F]] = lim

U←−EoF

Fp[F/U],

whereU ranges over open normal subgroup ofF.

We will denote the free product in the category of pro-pgroups by` . 2.2. Parafree groups. IfGis a group we denote by G

bp the pro-pcompletion of G. In this paper we will use the following characterization of parafree groups.

Proposition 2.1. Let G be finitely generated residually nilpotent group. Then G is parafree if and only if G

bp is a free pro-pgroup for every prime p. Moreover, a parafree group is residually-pfor every prime p.

Proof. A group is said to be weaklyp-parafree if its quotients by the terms of its p-lower central series are the same as those of a free group. In these terms, a finitely generatedGwill be parafree if and only if it is residually nilpotent and it is weakly p-parafree for every prime p. We know from [14, Corollary 2.9] that a finitely generatedGis weaklyp-parafree if and only if it has free pro-pcompletion.

This proves the first claim. The second claim of the proposition follows from the fact that finitely generated torsion-free nilpotent groups are residually-pfor every

primep[10, Theorem 2.1].

The following result provides a useful necessary condition for a group to be parafree.

Proposition 2.2. The centralizers of non-trivial elements of a parafree group are cyclic.

Proof. LetA be the centralizer of a non-trivial element uof a parafree group G.

By [5, Theorem 4.2], we know that

(a) a two-generated subgroup of Gis free and (b) an abelian subgroup ofGis cyclic.

Thus, by (a),Ais locally cyclic, and, by (b),Ais cyclic.

2.3. Group algebra and the augmentation ideal. Let Gbe a group andk a commutative ring. The group ring ofGoverkis denoted bykG. We denote byIG

the augmentation ideal ofZGand bykIG=k⊗

Z

IG the augmentation ideal of kG.

Given two groupsH ≤G, we denote bykI_H^G the left ideal of kG generated by kIH. SincekGis a natural free rightkH-module, it follows that the canonical map ofkG-modules

kG ⊗

kH

kIH −→kI_H^G is an isomorphism.

2.4. The fundamental groups of graphs of groups. We refer the reader to [8]

for standard definitions and notions related to graph of groups and their fundamen- tal groups.

By agraph of groups (G,Γ) we mean a connected graph Γ = (V(Γ), E(Γ), ι, τ) together with a function G which assigns to each v ∈ V(Γ) a group G(v), and to each e ∈ E(Γ) a distinguished subgroup G(e) of G(ι(e)) and injective group homomorphism te : G(e) → G(τ(e)). The monomorphisms te are called the edge functions.

In this paper we work mostly with two particular cases: amalgamated free prod- ucts and HNN extensions. The structure of the augmentation ideal of the group algebra of the fundamental group of a graph of groups is described in [7, Lemma 6]. We will describe this result in the cases that interest us.

Proposition 2.3. Let kbe a commutative ring.

(1) LetU andV be two groups,Aa subgroup ofU andα:A→V a monomor- phism. Put W =U∗AV. Then there exists an exact of sequence of kW- modules.

0→kI_A^W −→^γ kI_U^W ⊕kI_V^W −→^p kI_W →0,

whereγ(a) = (a,−a)ifa∈kIAandp(b, c) =b+cifb∈kI_U^W andc∈kI_V^W. (2) Let U be a group, A a subgroup of U and α : A → U a monomorphism.

Put W =U∗α=hU, t:tg=α(g)t ifg∈Ai. Then there exists an exact of sequence ofkW-modules.

0→kI_α(A)^W −→^γ kI_U^W ⊕kW −→^p kI_W →0,

whereγ(a) = (a(1−t), a)ifa∈kI_α(A)^W andp(b, c) =b+c(t−1)ifb∈kI_U^W andc∈kW.

The following proposition will be used several times in the paper.

Proposition 2.4. [2, Corollary 1.14 and Proposition 1.20]Let(H,∆)be a subgraph of groups of a graph of groups (G,Γ), i.e.

(1) ∆ is a connected subgraph of a finite connected graph Γ;

(2) H(v)≤ G(v)ifv∈V(∆),H(e)≤ G(e) ife∈E(∆) and the edge functions of (H,∆)are the restrictions of the edge functions of (G,Γ).

Assume that

H(e) =G(e)∩ H(ι(e))andt_e(H(e)) =t_e(G(e))∩ H(τ(e))if e∈E(∆).

Let ∆0 be a maximal subtree in ∆ and∆0⊆Γ0 a maximal subtree in Γ. Then the canonical map between the fundamental groups of graphs of groups

π(H,∆,∆0)→π(G,Γ,Γ0) is injective.

2.5. The induced map between the augmentation ideals. Given a group homomorphism Ge → G, we obtain an induced map between the augmentation ideals I

Ge → IG. Here we will explain how one can derive information from this induced map back to the initial group homomorphism.

Proposition 2.5. Let φ:Ge →G be a surjective group homomorphism of kernel K. Suppose that the natural homomorphism ofkG-modules

α:kG⊗

kGe

kI

Ge−→kI_G,

defined by the k-linear extension ofa⊗b7→aφ(b), is an isomorphism. Then k⊗_ZKab= 0.

Proof. Observe that by Shapiro’s lemma

k⊗_ZKab∼=H1(K, k)∼=H1(G, kG) = Tore ^k₁^Ge(kG, k).

Applying the right-exact functorkG⊗_k

Ge to the exact sequence of leftkG-modulese 0→kI

Ge→kGe→k→0, we get the exact sequence

0→Tor^k₁^Ge(kG, k)→kG⊗_k

GekI

Ge

−→α kG→k→0.

Sinceαis injective,Tor^k₁^Ge(kG, k) ={0}and we are done.

2.6. D-torsion-free modules. LetR ,−→ Dbe an embedding of the ringRinto a division ringD. Let M be anR-module. We say thatM is D-torsion-free if the canonical mapM −→ D ⊗

R

M is injective.

The following provides a more flexible criterion for verifying whether a module is torsion-free.

Lemma 2.6. [12, Lemma 4.1]LetM be aR-module. ThenM isD-torsion-free if and only if there exists aD-moduleN and an injective homomorphism ofR-modules M ,−→N.

Given a homomorphism R → D, where D is a division ring, we define D- dimension of M to be the dimension of theD-vector spaceD ⊗RM. We denote it bydim_DM.

Lemma 2.7. [12, Lemma 4.2]Let 1→M1→M2→M3→0be an exact sequence of R-modules. Assume that

(1) M1 andM3 areD-torsion-free;

(2) dim_DM1 anddim_DM3 are finite; and (3) dim_DM1+ dim_DM3= dim_DM2. ThenM2 is alsoD-torsion-free.

The following lemma will be used in the computation ofdim_DM.

Lemma 2.8. [12, Lemma 4.3] Let M be a D-torsion-free R-module of finite D- dimension. Let L be a non-trivial R-submodule of M. Then dim_D(M/L) <

dim_D(M). Moreover, ifdim_DL= 1, thendim_D(M/L) = dim_DM −1.

2.7. The universal division rings of fraction of the group algebra of a subgroup of a free pro-pgroup. LetFbe a finitely generated free pro-pgroup.

If G is an abstract subgroup of F, then it turns out that FpG has a universal division ring of fractions, denoted by D_FpG. We will not give a formal definition of universal field of fractions which can be found in [13], but we will describe the main properties of the embeddingsFpG⊆ D_FpG, which we will use in this paper.

IfH is a subgroup ofG, then the division closure ofFpH inD_FpGis the universal division ring of fractions ofFpH, and, therefore, we will denote it byD_FpH.

If N is a normal subgroup of Gsuch that G/N ∼=Z, then F^pG is a freeF^pN- module with the basis{tⁱ:i∈Z}, wheret∈GandN tgeneratesG/N. ThusFpG is isomorphic to a skew-polynomial ringFpN[t^±1, σ]with the indeterminatet and coefficients inF^pN. Hereσis an automorphism of the ringF^pN, induced from the automorphism of conjugation-by-t.

Let R be a subring of D_FpG generated byD_FpN and t. It turns out that{tⁱ : i∈Z} is also aD_FpN-basis of R (this is so called theHughes-free property of the embeddingFpG⊆ D_FpG, see [11, 9]). thusRis isomorphic toD_FpN[t^±1, σ].

The following result gives a lower bound for theD_FpG-dimension ofFpIG. Proposition 2.9. [12, Corollary 3.7 and the discussion afterwards] Let G be a finitely generated dense subgroup of a free pro-p group F. Then dim_DF

p GF^pIG ≥ d(F). Moreover, if Gis parafree andF=G

pb, thendimD_{Fp G}FpIG=d(F).

The following result provides an important example ofD_FpG-torsion-free-module.

Proposition 2.10. [12, Proposition 4.8] Let Fbe a free pro-pgroup, H ≤G≤F two subgroups ofF, andAa maximal abelian subgroup ofH. Then theFpG-module FpI_H^G

FpI_A^G isD_FpG-torsion-free.

2.8. Embeddings of abstract groups into free pro-pgroups. In this subsec- tion we detail a method introduced in [12] for ensuring that a map from an abstract group Ge to a free pro-p group is injective. We are particularly interested in the problem of producing families of parafree groups. Let Ge be a candidate to being parafree, meaning thatGe

bpis free for every primep. We want to study whether the canonical map Ge −→Ge

pbis an embedding for some suitable prime p. This would establish the residual nilpotence ofGe and we could conclude thatGe is parafree.

Proposition 2.11. Let Ge be a finitely generated group,Fa finitely generated free pro-p group and φ : Ge → F a group homomorphism. Suppose that we have the following conditions.

(1) The imageG=φ(G)e is dense in F.

(2) The FpG-moduleFpG ⊗

FpGe

FpI

Ge isD_FpG-torsion-free and dim_DF

p GFpG ⊗

FpGe

FpI

Ge=d(F).

(3) The kernel of φis free.

Then the map φis an embedding.

Proof. We will prove that the surjective mapφ:Ge→Gverifies the assumption of Proposition 2.5 to deduce thatkerφhas trivialp-abelianization. Sincekerφis free, the latter would imply thatkerφ= 1and the conclusion would follow.

It is clear that the natural homomorphism ofFpG-modules FpG ⊗

FpGe

I

Ge→FpI_G,

defined bya⊗b7→aφ(b), is surjective. If it was not injective, naming its kernel by Land namingM =FpG ⊗

FpGe

IGe, we would deduce, after applying Lemma 2.8, d(F) = dim_{DFp G}M >dim_{DFp G}(M/L) = dim_{DFp G}(FpIG),

which would contradict Proposition 2.9.

In the setting of our problem, that is, taking someGe with free pro-pcompletion and studying whetherGe −→ Ge

pbis injective, we make a few comments about the three conditions of Proposition 2.11. We takeGto be the image ofGe insideGe

bp. (1) The first condition will be naturally ensured.

(2) The second condition is the hardest part and requires the most technical arguments presented in Section 3.

(3) The third condition is natural from the point of view of the Bass-Serre theory. If we take Ge to be the fundamental group of a graph of groups, and we ensure that kerφ intersects trivially every vertex subgroup, then thekerφwill necessarily be free.

3. Examples of D_FpG-torsion-free modules

In this section we construct two families of examples ofD_FpG-torsion-free mod- ules. The first result is a slight generalization of [12, Proposition 4.10].

Proposition 3.1. Let H₁ andH₂ be two finitely generated subgroups of a finitely generated free pro-p group F. Consider A = H₁ ∩H₂ and suppose that A is a maximal abelian subgroup of H₁. LetG=hH₁, H₂i and let

J={(x,−x) :x∈FpI_A^G} ≤FpI_H^G

1⊕FpI_H^G

2. Then the FpG-module

M = FpI_H^G

1⊕FpI_H^G

2

J isD_FpG-torsion-free and

dim_D

Fp GM = dim_D

Fp H1FpI_H1+ dim_D

Fp H2FpI_H2−1.

Before giving the proof, we shall make an observation. If A is an abelian sub- group ofF, as occurs in the previous proposition, the universal division F^pA-ring of fractions D_FpA is the field of fractions Qore(FpA) of the commutative domain FpA. Given a nontrivial ideal I of FpA, we see that the Qore(FpA)-vector space Qore(FpA) ⊗

FpA

Iis one-dimensional. This implies that dimD_{Fp A}I= dimQ_ore(FpA)Qore(FpA) ⊗

FpA

I= 1. (1)

Proof of Proposition 3.1. We consider theFpG-submodule of M defined by L= (FpI_A^G⊕FpI_H^G

2)/J.

We want to apply Lemma 2.7 to the short exact sequence 0→L→M →M/L→0

ofFpG-modules. SinceL∼=FpI_H^G₂≤FpG, then it isD_FpG-torsion-free and dim_{DFp G}L= dim_{DFp H2}FpIH₂.

By the observation (1),

1 = dim_{DFp A}FpIA= dim_{DFp G}FpI_A^G= dim_{DFp G}J.

So it is clear, by Lemma 2.8, that

dimD_{Fp G}M = dimD_{Fp G}FpI_H^G₁+ dimD_{Fp G}FpI_H^G₂−1 =

dim_{DFp H1}FpIH₁+ dim_{DFp H2}FpIH₂−1.

On the other side, the quotient M/L is isomorphic to FpI_H^G

1/FpI_A^G, which is D_FpG-torsion-free by Proposition 2.10. Again, by Lemma 2.8,

dim_D

Fp GM/L= dim_D

Fp GFpI_H^G

1−dim_D

Fp GFpI_A^G= dim_D

Fp H1FpI_H1−1.

Therefore, the short exact sequence0→L→M →M/L→0 of FpG-modules satisfies the requirements of Lemma 2.7; and the conclusion follows.

We now turn to the study of torsion-free modules that have the form of an augmentation ideal of a cyclic HNN extension.

Our point is to prove that a certain module, which may have the form R^m/R(u1, . . . , um)

for some group ringR, isDR-torsion-free. When extending thisR-module with co- efficients in a bigger ring, someu_imay become invertible. The following elementary lemma simply studies this scenario, which we shall encounter many times.

Lemma 3.2. Let R be a unital ring andM be aR-module. Letm0∈M and letu be a unit ofR. Then there is an isomorphism ofR-modules

γ: M ⊕R (m0, u) −→M given by

γ(m, r) =m−ru⁻¹m₀, with inverse

γ⁻¹(m) = (m,0).

We can now state the second result of this section.

Proposition 3.3. Let H ≤ G be subgroups of F. Suppose that we can write G = N o(t) = hN, ti for some H ≤ N ≤ G and some t ∈ G. Let u ∈ H be an element which generates a maximal abelian subgrouphuiinH and suppose that v=tut⁻¹∈H. Then theFpG-moduleM defined by

M = FpI_H^G⊕FpG

FpG(v−1−t(u−1), v−1) isD_FpG-torsion-free.

Proof. The proof is relatively long and, for convenience, it is divided in several intermediate claims.

LetR be the subring of D_FpG generated by D_FpN andt. The structure of this ring is explained in Subsection 2.7.

Claim 3.4. The natural map R ⊗

FpG

M → D_FpG ⊗

FpG

M

is injective.

Proof. Applying Lemma 3.2, we obtain that R ⊗

FpG

M ∼=R ⊗

FpGFpI_H^G and D_FpG ⊗

FpG

M ∼=D_FpG ⊗

FpGFpI_H^G. Observe that

R ⊗

FpGF^pI_H^G ∼=R ⊗

FpHF^pIH and D_FpG ⊗

FpGF^pI_H^G∼=D_FpG ⊗

FpHF^pIH. SinceRis a directD_FpH-summand ofD_FpG we obtain that the map

R ⊗

FpHFpI_H→ D_FpG ⊗

FpHFpI_H

is injective. This proves the claim.

Claim 3.5. We have the following equality of subsets ofD_FpN ⊗

FpNFpI_H^N, 1 ⊗

FpNFpI_H^N \

D_FpN ⊗

FpN

(u−1) = 1 ⊗

FpNFpN(u−1). (2) Proof. By assumption, hui is a maximal abelian subgroup of H. By Proposition 2.10, theFpN-module

M0= FpI_H^N FpN(u−1) isD_FpN-torsion-free. This implies that the canonical map

M₀−→ D_FpN ⊗

FpN

M₀

is injective. This gives the claim.

We denoteH^tn=tⁿHt⁻ⁿ, which are also subgroups ofN havinghu^tnias max- imal abelian subgroup.

Claim 3.6. For alln∈Z, we have the following equality of subsets ofD_FpN ⊗

FpNFpI_G, 1 ⊗

FpNFptⁿI_H^N \

D_FpN ⊗

FpN

tⁿ(u−1) = 1 ⊗

FpNFptⁿN(u−1). (3) Proof. The same way we had the equality (2), we can derive, for the same reasons,

1 ⊗

FpNFpI^N

H^t−n

\ D_FpN ⊗

FpN

(u^t−n−1) = 1 ⊗

FpNFpN(u^t−n−1). (4) Notice that FpI_G has a righthti-module structure by multiplication. This induces a righthti-module structure onD_FpN ⊗

FpNFpI_G. SinceNEG, thentnormalisesN. We have the equations of subsets ofD_FpN ⊗

FpNFpI_G: 1 ⊗

FpNFpN(u−1)

!

t^m= 1⊗Fpt^mN(u^t−m−1),

and

1 ⊗

FpNFpI_H^N

!

t^m= 1⊗Fpt^mI_Ht−m.

As a consequence, applying the multiplication-by-tⁿautomorphism ofD_FpN ⊗

FpNFpIG

to the equation (4), we get (3).

Notice the following decomposition ofFpN-modules FpI_H^G=M

n∈Z

FptⁿI_H^N,

which yields to the following decomposition ofFp-vector spaces D_FpN ⊗

FpNFpI_H^G∼=M

n∈Z

D_FpN ⊗

FpNFptⁿI_H^N. Claim 3.7. We have the following equation of subsets ofD_FpN ⊗

FpNFpI_H^G, 1 ⊗

FpNFpI_H^G \

D_FpN ⊗

FpNFpG(v−1−t(u−1)) = 1 ⊗

FpNFpG(v−1−t(u−1)). (5) Proof. Let us take an elementw that belongs to the left-hand side. This element will have the form

w=

n₂

X

k=n₁

c_k⊗t^k(v−1−t(u−1)), for somec_k ∈ D_FpN, and will also belong to1 ⊗

FpNFpI_H^G. We rewrite w=c_n1⊗tⁿ¹(v−1)+

n2

X

k=n₁+1

c_k⊗t^k(v−1)−c_k−1⊗t^k(u−1)

+c_n2⊗tⁿ²⁺¹(u−1).

Sincew∈1 ⊗

FpNFpI_H^G, we can look at the highest powertⁿ²⁺¹ to deduce that c_n2⊗tⁿ²⁺¹(u−1)∈1 ⊗

FpNFptⁿ²⁺¹I_H^N \

D_FpN ⊗

FpN

tⁿ²⁺¹(u−1).

By (3), this implies that

cn₂⊗tⁿ²⁺¹(u−1)∈1 ⊗

FpNFptⁿ²⁺¹N(u−1),

so c_n2 ∈FpN. Let n₁+ 1≤k ≤n₂. Inspecting again the expression of wat the component with powert^k, we have that

ck⊗t^k(v−1)−c_k−1⊗t^k(u−1)∈1 ⊗

FpNFpt^kI_H^N. If we knew thatc_k ∈FpN, then it would follow that

c_k−1⊗t^k(u−1)∈1 ⊗

FpNFpt^kI_H^N \

D_FpN ⊗

FpN

t^k(u−1).

By (3), this means that

c_k−1⊗t^k(u−1)∈1 ⊗

FpNF^ptⁿN(u−1), and this implies thatc_k−1∈FpN.

We have proven that ifck ∈FpN, forn1 < k≤n2, then ck−1∈FpN. Since we also know thatcn₂ ∈FpN, an inductive argument gives thatck∈FpN for everyk, meaning that

w∈1 ⊗

FpNFpN[t^±, σ] (v−1−t(u−1)) = 1 ⊗

FpNFpG(v−1−t(u−1)).

This proves that 1 ⊗

FpNFpI_H^G \

D_FpN ⊗

FpNFpG(v−1−t(u−1))⊆1 ⊗

FpNFpG(v−1−t(u−1)), one inclusion of (5). The reverse inclusion is trivial, so equation (5) is proven.

Claim 3.8. The natural map

α:M →R ⊗

FpG

M

is injective.

Proof. Observe that there is a canonical isomorphism ofFpN-modules R∼=D_FpN[t^±1, σ]∼=D_FpN ⊗

FpNFpN[t^±, σ] =D_FpN ⊗

FpNFpG.

This extends to a canonical isomorphism ofFpN-modules

ψ:

R ⊗

FpGFpI_H^G R ⊗

FpGFpG(v−1−t(u−1)) −→

D_FpN ⊗

FpNFpI_H^G D_FpN ⊗

FpNFpG(v−1−t(u−1)). Therefore , there is a commutative triangle of canonicalFpN-homomorphisms

FpI_H^G FpG(v−1−t(u−1))

R⊗

Fp GFpI_H^G R⊗

Fp G

FpG(v−1−t(u−1))

D_{Fp N} ⊗

Fp NFpI_H^G D_{Fp N} ⊗

Fp N

FpG(v−1−t(u−1)),

α⁰ η

ψ

The canonical mapηis injective due to (5). Sinceψis an isomorphism, this implies thatα⁰ is injective.

Let(x, y)∈FpI_H^G⊕FpGbe such that(x, y)+FpG(v−1−t(u−1), v−1)belongs to the kernel ofα. Then1⊗(x, y) =c⊗(v−1−t(u−1), v−1)for somec∈ D_FpG. Notice that

1⊗x=c⊗(v−1−t(u−1))∈1 ⊗

FpGFpI_H^G \ R ⊗

FpG

(v−1−t(u−1)).

From the injectivity ofα⁰, this implies that c⊗(v−1−t(u−1))∈1 ⊗

FpGFpG(v−1−t(u−1)),

so c∈FpGand then(x, y)∈FpG(v−1−t(u−1), v−1).Thusαis injective and the claim is demonstrated.

From Claims 3.8 and 3.4, we conclude thatM isD_FpG-torsion-free.

4. Proofs of the main results

4.1. Amalgamated products. Now we are ready to prove Theorem 1.1.

Theorem 4.1. Let Hf1 and Hf2 be finitely generated groups. Let 16=u1∈Hf1 and let16=u2∈Hf2. Consider the following amalgamated product of cyclic amalgam

Ge=Hf1 ∗

u₁=u₂Hf2∼= Hf1∗Hf2

hhu1u⁻¹₂ ii.

ThenGe is parafree if and only if the three following conditions hold.

(1) Hf1 andHf2 are parafree.

(2) The elementu₁u⁻¹₂ of Hf₁∗Hf₂ is not a proper power in the abelianization of Hf1∗Hf2.

(3) There is at least onei∈ {1,2}such that ui is not a proper power in Hfi. Remark. The condition (2) can be substituted by the condition

(2’) rab(G) =e rab(Hf1) +rab(Hf2)−1.

The proof of the theorem shows that the condition (3) can be substituted by the condition

(3’) All the centralizers of non-trivial elements inW are cyclic.

Proof. We first prove that the conditions are necessary. Let us assume thatGe is parafree.

Both Hf1 and Hf2 are subgroups of Ge and hence they are residually nilpotent.

We want to show that, for all primesp,Hf1

pbandHf2

pbare free. By Proposition 2.1, this would imply thatHf1 andHf2are parafree.

Fix a primep. Observe that d(Ge

bp)≥d(Hf₁

pb) +d(Hf₂

pb)−1. (6)

Consider the closureH_iof the image ofHf_iunder the canonical mapGe→Ge

bp. Since Ge is parafree,Ge

pbis free pro-p, and so bothH1 andH2 are free pro-p, because any closed subgroup of a free pro-pgroup is again free. Note thatHf1 andHf2generate G. Hence the canonical mape f :H1`

H2→Ge

pbis onto.

Since Ge is parafree, the images of u1 and u2 in Gf

bp are non-trivial. Therefore, kerf 6={1}. We also recall that the groupsH1`H2 andGe

bpare free pro-p, so d(Ge

bp)≤d(H1) +d(H2)−1. (7) Since H_i is a quotient ofHf_i

bp, thend(H_i) ≤ d(fH_i

bp). The later observation in combination with (6) and (7) yieldsd(Hi) =d(fHi

pb). The pro-pgroup Hi is free, so the canonical map Hf_i

bp → H_i is an isomorphism. HenceHf_i

bp is free pro-pand the first condition is proved.

The previous argument also shows that, for all primesp, d(Ge

bp) =d(Hf₁

pb) +d(Hf₂

pb)−1, and this implies the second condition.

Finally, suppose that u1 =v₁ⁿ¹ in Hf1, and that u2 =v₂ⁿ² in Hf2 with n1, n2 ≥ 2. Since u_i 6= 1 and Hf_i is torsion-free, then hvii are infinite cyclic. Consider

the subgroup A generated by v1 and v2. By Proposition 2.4, A is isomorphic to hv1i ∗_vⁿ1

1 =vⁿ₂² hv2i. This group is non-abelian and belongs to the centralizer of u1=u2inG. This contradicts Proposition 2.2. This shows that the third conditione holds.

We now verify that the three given conditions are sufficient. There is a canonical isomorphism

Geab∼= Hf_1ab⊕Hf_2ab u₁−u₂ .

From the fact thatu1−u2is not a proper power inGeab, we see thatGeabis torsion- free of rankr_ab(Hf₁) +r_ab(Hf₂)−1. Therefore for any primep,

d(Ge

bp) =d(Hf1

pb) +d(Hf2 pb)−1.

Let us fix an arbitrary primepfrom this point on. Consider the canonical map φ:Ge−→Ge

pb.

Claim 4.2. The pro-pgroupGe

pbis free.

Proof. SinceHf1andHf2are parafree, the pro-pgroupsHf1pbandHf2pbare free. thus Ge

bpis a quotient of the free pro-pgroup Hf1 bp`

Hf2

bp by the closed normal subgroup generated byu1u⁻¹₂ . Observe that u1u⁻¹₂ is primitive inHf1pb

`

Hf2pb. Hence,Ge

pbis

free pro-p.

Claim 4.3. The restrictions ofφto each Hf_i are injective.

Proof. To check this claim, we consider each restriction φi=φ|

Hf_i:Hfi−→Hi,

whereH_i=φ(fH_i). The subgroupsH₁andH₂of the free pro-pgroupGe

pbare closed.

Hence they both are free pro-pgroups.

Since the inducedφi

pb:Hfipb−→Hi are surjective maps of free pro-pgroups, d(fHi

pb)≥d(Hi), for alli∈ {1,2}. (8) Furthermore, by the universal property of the coproduct, there is a continuous homomorphism

f :H₁a

H₂−→Ge

pb

which sendsHi to each corresponding copy in Ge

bp. Notice that Imf contains both H1 andH2, so G≤Imf, implying thatf is surjective.

We will show thatφi

pb(i= 1,2) are isomorphisms. This will imply thatφi are injective, because, sinceHfi are parafree,Hi is a subgroup of Hfi

pb. Since

d(H1) +d(H2)≥d(Ge

pb) =d(Hf1

bp) +d(fH2 bp)−1, d(H₁) = d(fH₁

pb) or d(H₂) = d(Hf₂

pb). Without loss of generality, we assume that d(H1) =d(Hf1

pb). The fact thatH1 and Hf1

pbare free pro-pimplies thatφ1 pbis an isomorphism. Furthermore, sinceφ(u1)6= 1inHj andf(φ(u1)φ(u2)⁻¹) = 1inGe

pb; thenf has non-trivial kernel. Recall, in addition, thatGe

bp is free pro-p, so d(H₁) +d(H₂)−1≥d(Ge

pb) =d(Hf₁

bp) +d(Hf₂

pb)−1,

and henced(H2) =d(Hf2

bp)andφ2

pbis also an isomorphism.

Claim 4.4. The mapφ:Ge−→Ge

pbis injective.

Proof. We denote G = φ(G).e We want to apply Proposition 2.11 to the map φ:Ge−→G⊆Ge

pb.

Notice first that, by Claim 4.3, the kernel ofφintersects trivially the subgroups Hf1 andHf2. Therefore, by the Bass-Serre theory, the kernel acts freely on a tree, and so it is free.

We consider the correspondingFpG-module M =FpG ⊗

FpGe

FpI

Ge. ConsiderA=hφ(u1)iand

J={(x,−x) :x∈FpI_A^G} ≤FpI_H^G₁⊕FpI_H^G₂. Then, by Proposition 2.3(1)

M ∼= FpI_H^G

1⊕FpI_H^G

2

J .

Without loss of generality, we suppose thatu₁is not a proper power in Hf₁. By Claim 4.3, Hfi ∼=Hi, soHi (i= 1,2) are parafree and φ(u1)is not a proper power inH1. HenceA=hu1iis a maximal abelian subgroup ofH1.

By Proposition 3.1,M isD_FpG-torsion-free with dimension dim_{DFp G}M = dim_{DFp H1}FpIH₁+ dim_{DFp H2}FpIH₂−1.

In addition, by Proposition 2.9,

dim_{DFp H1}FpIH₁+ dim_{DFp H2}FpIH₂−1 =d(H1

bp) +d(H2

pb)−1 =d(Ge

pb).

It follows that dim_DF

p GM = d(Ge

pb). We can apply Proposition 2.11 to conclude

thatφ:Ge−→Gis injective.

The last claim implies thatGeis residually nilpotent. Moreover, we already know that eachGe

bp is free, soGe is parafree by Proposition 2.1.

4.2. HNN extensions. Now we prove Theorem 1.2.

Theorem 4.5. Let He be a finitely generated group. Let u, v ∈He\ {1}. Consider the following cyclic HNN extension ofHe

Ge= He∗ hti hhtut⁻¹v⁻¹ii.

ThenGe is parafree if and only if the four following conditions hold.

(1) The group He is parafree.

(2) The elementuv⁻¹ is not a proper power inHe_ab. (3) At least one of uorv is not a proper power in He.

(4) The image of the elementuis non-trivial in some finite nilpotent quotient of G.e

Remark. The condition (2) can be substituted by the condition

(2’) rab(G) =e rab(He).

The proof of the theorem shows that the condition (3) can be substituted by the condition

(3’) All the centralizers of non-trivial elements inW are cyclic.

Proof. First let us show that he given conditions are necessary. Assume thatGe is parafree.

The groupHe is a subgroup ofGe and hence it is residually nilpotent. We want to show that, for all primesp,He

pbis free. By Proposition 2.1, this would imply that He is parafree.

Fix a primep. Observe that d(Ge

pb)≥d(He

bp).

Consider the closure H of the image of He under the canonical map Ge → Ge

pb. Since Ge is parafree, Ge

bp is free pro-p, and so H is also free pro-p. Note that He and t generate G. Hence, the canonical mape f : H`

Zp → Ge

bp, which sends the generator 1 ofZp tot, is onto.

Since Ge is parafree, the images of u and v in Ge

pb are non-trivial. Therefore, kerf 6={1}. Thus, since the groupsH`

Zp andGe

bpare free pro-p, d(Ge

bp)≤d(H).

Since H is a quotient of He

pb, d(H) = d(He

bp), and since, H is free pro-p, the canonical map He

pb→H is an isomorphism. Hence,He

bp is free pro-p, and the first condition is proved.

The previous argument also shows that for all primep, d(Ge

pb) =d(He

bp).

This implies the second condition.

Now suppose thatu=wⁿ¹ andv =wⁿ₂² in He with n1, n2 ≥2. Since u, v 6= 1 and He is torsion-free, then hwii are infinite cyclic. Consider the subgroup A of Ge generated by w1, w2 and t. By Proposition 2.4, A is isomorphic to the HNN extensionA⁰ =hw1, w2, t|twⁿ₁¹t⁻¹=w₂ⁿ²i. The centralizer ofwⁿ₁¹ = (t⁻¹w2t)ⁿ² in A⁰ containsw1 and t⁻¹w2t, so it is not abelian. Thus the centralizer of uin Ge is not abelian. This contradicts Proposition 2.2. This shows that the third condition holds. The forth condition holds becauseGeis parafree.

Now we are going to verify that these four conditions are sufficient forGe to be parafree. First of all, it is clear that

Geab∼= Heab

hu−vi⊕ hti.

From the fact thatu−vis not a proper power inHeab, we see thatGeabis torsion-free of the same rank asHeab. Therefore,d(Ge

pb) =d(He

bp). In addition, we also see that t is primitive inGeab. Let us fix a primepsuch that the image of the elementuis non-trivial in some finitep-quotient oG. Consider the canonical mape φ:Ge−→Ge

pb. Claim 4.6. The pro-pgroupGe

pbis free and the element φ(t)is primitive inGe

pb.

Proof. The pro-p group Ge

bp is the quotient of He

pb`

Zp by the closed subgroup generated bytut⁻¹v⁻¹. SinceHe is parafree,He

pb, is free pro-p. ThusHe

pb`

Zpis free pro-p. Observe also that the elementtut⁻¹v⁻¹ is primitive inHe

pb`

Zp. Hence,Ge

pb

is free pro-p.

We nameH=φ(H).e

Claim 4.7. The restriction ofφtoHe is injective.

Proof. To verify this, consider the closed subgroupH ≤Ge

pb. SinceGe

pbis free, the pro-pgroupH must be free. We notice that the epimorphismφ:He −→H induces a continuous epimorphismφ

pb:He

pb−→H. In particular, d(He

pb)≥d(H). (9)

Furthermore, by the universal property of the coproduct, there is a continuous homomorphism

f :Ha

Zp −→Ge

bp, which sendsH toGe

pb, by inclusion; andZ^p to the cyclic pro-pgroup generated by φ(t). Since the image off contains bothH andφ(t), it follows thatImf contains φ(G), soe f must be a surjective. In addition, it has a nontrivial kernel; since φ(t)φ(u)φ(t)⁻¹φ(v)⁻¹= 1 and φ(u)6= 1, by assumption. Here we have used that the canonical mapι:H∗Zp−→H`

Zp is injective.

This verifies thatf is a surjective and non-injective continuous homomorphism between free pro-pgroups. Hence,

d(H) + 1 =d(H) +d(Zp) =d Ha

Zp

> d(Ge

bp).

This, in addition to (9), implies thatd(He

pb) =d(H). Soφ

bp:He

pb→His a continuous epimorphism between free pro-pgroups of the same rank, which implies that it is an isomorphism. SinceHe is parafree, it is residually-p. henceφis also injective.

Claim 4.8. The mapφ:Ge−→Ge

pbis injective.

Proof. We denote G = φ(G).e We want to apply Proposition 2.11 to the map φ:Ge−→G⊆Ge

bp. We already know, from Claim 4.6, thatGe

bp is free. Notice also that, by Claim 4.7, the kernel of φintersects trivially the subgroup He of G. Thee Bass-Serre theory implies that the kernel is free.

We define a continuous homomorphismq:Ge

pb−→Zpsuch thatq(φ(t)) = 1, and q(φ(h)) = 0ifh∈H. The restrictione q|G verifies that its its kernelkerq|Gcontains H =φ(He).

We now consider theFpG-module M =FpG ⊗

FpGe

FpI

Ge. By Proposition 2.3(2), thisFpG-module is isomorphic to

M ∼= FpI_H^G⊕FpG

FpG(φ(v)−1−φ(t)(φ(u)−1), φ(v)−1).

Without loss of generality, we suppose thatuis not a proper power inH. Sincee φ:He −→H is an isomorphism,His parafree andφ(u)is not a proper power inH.

From Proposition 2.2, we deduce that that hφ(u)iis a maximal abelian subgroup ofH. By Proposition 3.3, this implies thatM isD_FpG-torsion-free.

Using Lemma 3.2, we have the following isomorphisms ofD_FpG-modules D_FpG ⊗

FpG

M ∼=D_FpG ⊗

FpGFpI_H^G∼=D_FpG ⊗

FpHFpIH ∼=D_FpG ⊗

D_{Fp H} D_FpH ⊗

FpHFpIH

! .

The combination of these isomorphisms with Proposition 2.9 yields to dimD_{Fp G}M = dimD_{Fp H}FpIH =d(He

pb) =d(Ge

pb).

We can apply Proposition 2.11 to conclude thatφ:Ge−→Gis injective.

The last claim implies thatGeis residually nilpotent. Moreover, we already know that every pro-pcompletionGe

pbis free, soGe is parafree by Proposition 2.1.

Proof of Corollary 1.3. It is clear that the three conditions of the corollary imply the four conditions of Theorem 1.2. Therefore, they are sufficient.

Let us show that they are necessary. Assume thatW is parafree. Again, in view of Theorem 1.2 we have to show only the third condition. Assume that the images ofuandvin the abelianization ofF₂ generate a subgroup not isomorphic toZ².

Let p be a prime. Since W is parafree we consider W inside W

pb. Let N be the normal closed subgroup of (F₂)

pb generated by uv⁻¹. Then in view of (1), (F2)

pb/N ∼=Zp, and so u∈ N. Observe that uv⁻¹ =utu⁻¹t⁻¹ in W

pb. Thus, uis contained in the closed normal subgroup ofW

pbgenerated byutu⁻¹t⁻¹. Since,W

pb

is a pro-pgroup,u= 1. This is a contradiction. Therefore, the images ofuandv in the abelianization ofF2generate a subgroup isomorphic toZ². 4.3. The fundamental group of graph of groups. In this subsection we prove Corollary 1.4.

Proof of Corollary 1.4. First assume thatW is parafree. We want to show that the four condition of the corollary hold. The conditions (3) and (4) hold becauseW is parafree. Let us show the conditions (1) and (2). We will argue by induction on the number of edges|E(Γ)|. If|E(Γ)|= 0, the claim is obvious.

Now we assume that we have proved that (1) and (2) hold if|E(Γ)| ≤nand we consider the case|E(Γ)|=n+ 1. Lete∈E(Γ).

IfΓ\ {e}= ∆1∪∆2 is disconnected then, by Proposition 2.4, W ∼=U∗_G(e)V, whereU ∼=π(G,∆₁)andV ∼=π(G,∆₂).

If G(e) = {1}, then W ∼= U ∗ V, and so, since W is parafree, U and V are also parafree. If G(e) ∼= Z, then U and V are parafree by Theorem 4.1. Since

|E(∆1)|,|E(∆2)| ≤ n, we can apply the induction. Thus, the four conditions of Corollary 1.4 hold for U and V. This implies immediately that conditions (1) and (2) also hold forW.

IfΓ\ {e}= ∆ is connected, then, by Proposition 2.4, W ∼=U∗t_e whereU ∼=π(G,∆).

If G(e) = {1}, then W ∼=U ∗Z, and so, since W is parafree, U is also parafree.

IfG(e)∼=Z, then U is parafree by Theorem 4.5. Since |E(∆)| ≤n, we can apply the induction. Thus, the four conditions of Corollary 1.4 holds forU. This implies immediately that the conditions (1) and (2) holds also forW.