More precisely, we prove that a subgroup of a finitely generated virtually free group G is a free factor if and only if its closure in the profinite completion of G is a profinite free factor

FREE FACTORS AND PROFINITE COMPLETIONS

ALEJANDRA GARRIDO AND ANDREI JAIKIN-ZAPIRAIN

Abstract. Can one detect free products of groups via their profinite completions? We answer positively among virtually free groups. More precisely, we prove that a subgroup of a finitely generated virtually free group G is a free factor if and only if its closure in the profinite completion of G is a profinite free factor. This generalises results by Parzanchevski and Puder (later also proved by Wilton) for free groups. Our methods are entirely different to theirs, combining homological properties of profinite groups and the decomposition theory of Dicks and Dunwoody.

1. Introduction

One of the most natural and successful ways to study an infinite object is via finite approximations to it. A natural way to approximate an infinite group is to consider larger and larger finite quotients of it. The object that encodes all finite quotients of a group G is its profinite completionG. Indeed, this is the categorical limit of all finite quotients andb the natural maps between them. In order for Gb to be a faithful approximation to G, the latter must be residually finite, in which caseGembeds inG. We restrict ourselves to theb study of these groups.

One then may wonder to what extent a residually finite group is determined by its profi- nite completion. The best possible answer is known asabsolute profinite rigidity: ifGb ∼=Hb then G∼=H. It is straightforward to see that finite groups and finitely generated abelian groups are absolutely profinitely rigid. Beyond that, things get difficult very quickly. There are examples of virtually cyclic groups ([1]) and of finitely generated torsion-free nilpotent groups of class 2 ([8]) that are not absolutely profinitely rigid.

Perhaps the boldest question in this area is that posed by Remeslennikov ([10, Problem 5.48]): Are free non-abelian groups absolutely profinitely rigid?

The answer to this question still seems remote, given current knowledge. Nevertheless, some progress has been made on variants of this question: free groups and surface groups are determined, among limit groups, by their profinite completion ([18, Corollary D], [19, Theorem 2]); certain Kleinian groups and triangle groups are absolutely profinitely rigid ([3, 4]).

Here we study another variant of this question, namely, whether one can detect a free factor of a group from its profinite completion. It is an exercise in the definition of free profinite products ([14, 9.1.1]) to show that ifGis the free product of subgroupsH, K then Gb is the free profinite product of the closuresH, K. One might hope that the converse is also true:

1

Question 1. Let G be a finitely generated residually finite group and H ≤ G. If H is a free profinite factor of G, mustb H be a free factor of G?

This question is surprisingly tricky. The only results so far requireGto be a free group, and then the answer is “yes”. This case was first shown by Parzanchevski and Puder [13]

and later, using different methods, by Wilton [18]. Each of the two proofs rely on beautiful and deep results.

The first uses detailed calculations for the expected number of common fixed points of the image of a finitely generated H under a uniformly random homomorphism G→Sym(n), for each n. In particular, Puder and Parzanchevski obtain that if w∈Gis not a primitive word then, for large enoughn, the average number of fixed points of the image of wunder a uniformly random homomorphismG→Sym(n) is strictly larger than 1. Wilton’s result is a consequence of another result of his that partially answers a question of Gromov: if the fundamental group of a graph of free groups with cyclic edge groups is one-ended and hyperbolic then it contains a surface subgroup.

Here we use entirely different methods to answer Question 1 positively whenGis virtually free.

We need some notation. Let H be a subgroup of a finitely generated group G and γ : H → P a homomorphism to a finite group P. If H is a free factor of G, then the number h(G, H, γ, P) of homomorphisms ˜γ :G → P that extend γ is independent of the choice ofγ; it is simply the number of homomorphisms from the free complement ofH to P. It turns out that the converse is also true when Gis virtually free.

Theorem 1.1. Let G be a finitely generated virtually free group. Let H be a finitely generated subgroup of G and H the closure of H in G. The following are equivalent:b

(a) H is a free factor ofG;

(b) for every finite groupP, the number h(G, H, γ, P) is constant on Hom(H, P);

(c) H is a free profinite factor ofG.b

We have already explained, (a) =⇒ (b). Now notice that Hom(G, P) can be identi- fied with the continuous morphisms Homcont.(G, Pb ), for every finite group P. If Hb = H then Hom(H, P) can also be identified with Homcont.(H, P). In this case condition (b) is equivalent to

(b’) for every finite groupP, the number h(G, H, γ, Pb ) is constant on Homcont.(H, P).

As a first step we show in Proposition 3.1 that (b’) is equivalent to (c). This is a completely general result. The only condition in passing from (b) to (b’) is thatHbe closed in the profinite topology of G. This is ensured, for instance, if G is subgroup separable (every finitely generated subgroup is closed in the profinite topology), also termed LERF.

The meat of the paper is the proof of (c) =⇒ (a). Condition (c) implies that there is a non-inner derivation (or crossed-homomorphism) d:Gb → k[[G]] whose kernel is preciselyb H, where k is a finite commutative ring. This, and the related definitions, are explained in Section 2. The analogous statement in the discrete setting is almost, but not quite, equivalent to (a). In fact, a result of Dicks following Dunwoody [7] states that if there is

a derivation d⁰ : G→ M to a projective k[G]-module M with kernel H, then G splits as a fundamental group of a finite graph of groups, one of whose vertex groups is H, and all of whose edge groups are finite, with conditions on their sizes related to the ring k. We must therefore do two things: first obtain somed⁰ fromd, and second show that the graph of groups decomposition of Gactually makesH a free factor. The second thing is done in Section 6. The first thing is done in Section 5, where it is shown (Corollary 5.6) that, forG virtually free, and k=Z/nZ with square-free nthe left k[G]-module k[[G]] is isomorphicb to a direct union of projective k[G]-modules. This means, as Gis finitely generated, that the image d(G) is contained in a finitely generated projective k[G]-module. In order to obtain Corollary 5.6, we give in Section 4 a criterion for certain rings to have flat profinite completion, so that we may apply some ring-theoretic results from [5]. It might be hoped that our main theorem could be generalised to wider classes of groups if the results of Sections 4 and 5 hold in that setting.

2. Preliminaries

All rings in this paper have a unit, denoted 1, and all ring homomorphisms send the unit to the unit. By a module we will mean a left module if we do not say the opposite.

A nonempty class of finite groups C is a pseudovariety if it is closed under taking subgroups, homomorphic images and finite direct products. The pro-C completion of a groupG is denoted byG_Cband by Gb we denote the profinite completion ofG. The closure of a subgroupH ≤G insideGb is denoted H.

The abstract free product of groupsG₁, G₂is denoted byG₁∗G₂, whileG₁`

CG₁denotes the free pro-Cproduct of two pro-CgroupsG1, G2. Recall thatG1`

CG2is characterized up to isomorphism by the following universal property: there are continuous homomorphisms ϕi : Gi → G1`

CG2 (i = 1,2), and for any pair of continuous homomorphisms ψi : G_i → P (i = 1,2) to a finite group P in C there is a unique continuous homomorphism ψ : G1`

CG2 → P such that ψ◦ϕi = ψi, i = 1,2. The basic properties of free pro-C products can be found in [14, Section 9]. IfC is the pseudovariety of all finite groups, then we write A`

B instead of A`

CB. Let G be a pro-C-group and H a closed subgroup of G. We say that H is a pro-C free factor of G if there exists a closed subgroup K of G such that G=H`

CK.

If k is a ring and G is a group, we denote by k[G] the group ring of G over k. The augmentation ideal of k[G] is the kernel of the canonical map k[G] → k, and it is denoted by I_k[G].

If G is a profinite group and k is a finite ring the completed group ring k[[G]] of G over k is the inverse limit of k[G/U], where U is a normal open subgroup of G. The augmentation ideal of k[[G]] is the kernel of the canonical map k[[G]] → k, and it is denoted by I_k[[G]]. It is a free k-module on the profinite space {g−1 | g ∈ G} and, if T is a profinite space generating Gand containing the identity, then I_k[[G]] is generated as a profinite k[[G]]−module by{t−1|t∈T} (see [14, 6.3.2]). If A is a closed subgroup ofG we denote by I_k[[A]]^G the left closed ideal ofI_k[[G]] generated byI_k[[A]].

Let G be a group, k a ring and M a k[G]-module. A derivation or crossed homo- morphism is a map d :G → M such that d(xy) = xd(y) +d(x) for all x, y ∈ G. The derivation is innerif there is somem∈M such that d= ad_m:x→(1−x)m.

Let Γ be a profinite group, ka profinite ring andAa discrete k[[Γ]]-module. Acontin- uous derivationd: Γ→Ais a continuous map that satisfiesd(gh) =gd(h) +d(g) for all g, h∈Γ. Denote by Der(Γ, A) the set of continuous derivations from Γ toA.

The following Lemma is well-known to experts, but we could not find a reference with the exact statement. It uses similar ideas to [14, Proposition 9.3.8].

Lemma 2.1. Let kbe a finite ring andC an extension-closed pseudovariety of finite groups containing the group (k,+). LetΓbe a pro-C group and assume that Γ = ∆₁`

C∆₂. Then I_k[[Γ]] =I_k[[∆^Γ

1]]⊕I_k[[∆^Γ

2]]. Proof. To simplify notation, write I := I_k[[Γ]] and Ji := I_k[[∆^Γ

i]], for i = 1,2. Since Γ is generated by ∆₁ and ∆₂, the augmentation idealI is generated as a profinitek[[Γ]]-module by {δ−1|δ ∈∆1∪∆2}. So evaluating inI gives a canonical surjective k[[Γ]]-morphism π :J₁⊕J₂ →I. To prove the claim, we will find an inverseρ:I →J₁⊕J₂ of π.

For i = 1,2 and each open normal subgroup U Eo Γ, denote by J_i,U := I_k[[∆^Γ/U

iU/U]]

the image of J_i in k[Γ/U]. Note that J₁ ⊕J₂ = lim←−UEoΓJ_1,U ⊕J_2,U. Since Γ acts on J_1,U ⊕J_2,U ∈ C, the group H_U := (J_1,U ⊕J_2,U)oΓ is a pro-C group. For i= 1,2, define φi,U : ∆i →HU by δ 7→(δU −U, δ) whereδU −U ∈Ji,U. By definition ofJi,U, these are continuous homomorphisms. The universal property of free pro-C products yields a unique continuous homomorphism ψ_U : Γ → H_U extending φ_1,U and φ_2,U. Denote, respectively, by pJ and pΓ the projections of HU to J1,U ⊕J2,U and Γ. Since pΓ◦φi,U = id∆i, the universal property of free pro-C products applied to id_Γ ensures thatp_Γ◦ψ_U = id_Γ. Now define ρU :I →J1,U⊕J2,U by γ−17→ pJ(ψU(γ)) forγ ∈Γ and extending by k-linearity.

This is indeed a profinite k[[Γ]]-module morphism because, asψ_U is a homomorphism, for everyγ, ∈Γ we have

ρU((γ−1)) =ρU(γ−1−(−1)) =pJ(ψU(γ))−pJ(ψU())

=p_Γ(ψ_U())p_J(ψ_U(γ)) +p_J(ψ_U())−p_J(ψ_U())

=p_J(ψ_U(γ)) =ρ_U(γ).

Let ρ : I → J₁ ⊕J₂ be the map given by the universal property of inverse limits for J,{J_1,U ⊕J2,U |U EoΓ} and {ρ_U |U EoΓ}. Observe that, for i= 1,2,and every UEoΓ and δ ∈∆_i, we have ρ_U(π(δ−1)) =ρ(δ−1) =δU −U, so ρ_U ◦π is the projection map J1⊕J2 →J1,U ⊕J2,U and thusρ◦π = idΓ, as required.

Recall that a supernatural number is a formal product n = Q

pp^n(p) where p runs through all prime numbers and n(p) ∈ N∪ {∞}. By convention, n < ∞ and ∞+n = n+∞ = ∞+∞ = ∞ for all n ∈ N. A supernatural number m = Q

pp^m(p) is said to divide another onen=Q

pp^n(p) ifm(p)≤n(p) for every primep. Thelowest common multiple of a collection of supernatural numbers {n_i = Q

pp^n(i,p) | i ∈ I} is defined as

lcm{n_i:i∈I}:=Q

pp^n(p) wheren(p) = max{n(i, p)|i∈I}. IfH is a closed subgroup of a profinite group Γ, we put |Γ :H|= lcm{|Γ :HN|:N EoΓ}.

Let Γ be a profinite group with closed subgroupsA, B ≤Γ andp a prime number. The subgroup A acts continuously on Fp[[Γ/B]] = lim←−NEoΓFp[Γ/BN] by left coset multiplica- tion: am=a(mN)_NEoΓ= (amN)_NEoΓ where

am_N =a



 X

gBN∈Γ/BN

f_N,ggBN



= X

gBN∈Γ/BN

f_N,gagBN, f_N,g ∈Fp.

The above then means that, for each NEoΓ,Fp[Γ/BN] decomposes as an A-module into Fp[Γ/BN]∼= Y

AgBN∈A\Γ/B

Fp[AgBN]∼= Y

AgBN∈A\Γ/B

Fp[A/A∩B^gN]

(since we are using left cosets, conjugation is done on the left: B^g = gBg⁻¹). Taking the inverse limit, we obtain that the A-module (and even the Fp[[A]]-module) Fp[[Γ/B]]

decomposes as

Fp[[Γ/B]]∼= Y

AgB∈A\Γ/B

Fp[[A/A∩B^g]]. (1)

Denote byFp[[Γ/B]]^Athe set{m∈Fp[[Γ/B]] : am=m for everya∈A}of fixed points of A. By (1), this set is reduced to 0 if and only ifFp[[A/A∩B^g]]^A={0} for everyg∈Γ.

This observation will help in proving the following.

Lemma 2.2. Let Γ be a profinite group with closed subgroups A, B ≤ Γ and p a prime number. Then Fp[[Γ/B]]^A={0} if and only if p^∞ divides|A:A∩B^g|for everyg∈Γ.

In order to prove the lemma we need the following result on finite groups.

Lemma 2.3. Let G be a finite group, H ≤ G and N E G. The image of the map Fp[G/H]^G →Fp[G/N H]^G is trivial if and only if p divides |N H :H|.

Proof. Observe that every element of Fp[G/H]^G is a constant function f X

gH∈G/H

gH for somef ∈Fp and that its image inFp[G/N H]^G is X

gN H∈G/N H

(|N H:H|f)gN H.

Proof of Lemma 2.2. By the observation made before the statement, we can suppose with- out loss of generality that A= Γ. For ease of exposition, assume also that Γ is countably based, so that Fp[[Γ/B]] = lim←−^i∈NFp[Γ/BN_i] where Γ = N₀ ≥ N₁ ≥ . . . is a countable chain of open normal subgroups such thatT

i∈NN_i={1}. The general case is not difficult, just more tedious to notate.

Assume first that p^∞ divides |Γ : B|. Let m = (m_i)i∈N ∈ Fp[[Γ/B]]^Γ, so m_i ∈ Fp[[Γ/BN_i]]^Γ for each i ∈ N. Since p^∞ divides |Γ : B|, p divides |BN_i : BN_i+1| for infinitely many values of i. By Lemma 2.3, for these i,mi = 0. Thus,m= 0.

Conversely, if the p-part of |Γ :B| is finite, there must be somek such that p does not divide |BN_k:BNi|for alli≥k. Put

mi= 1

|BN_k :BNi|

X

gBNi∈Γ/BNi

gBNi ∈Fp[Γ/BNi] and m= (mi)∈Fp[[Γ/B]].

It is clear that 06=m∈Fp[[Γ/B]]^Γ.

We finish the section with another auxiliary result that we will need later.

Lemma 2.4. Let C be a pseudovariety of finite groups and G a finitely generated pro-C group. Then there is a chainG > N₁ > N₂ > . . . of open characteristic subgroups ofGwith trivial intersection such that for every i and φ_i∈Epi(G, G/N_i) we have that kerφ_i =N_i. Proof. For each i ∈ N, let N_i be the intersection of all open normal subgroups of G of index at most i. Since G is finitely generated, there are finitely many such subgroups, so N_i is a closed subgroup of finite index and therefore open. By construction, each N_i is characteristic and they form a descending chain whose intersection is trivial. Now, for any i∈N, ifφ∈Epi(G, G/Ni), thenNi ≤kerφbecauseφis a homomorphism, and kerφ≤Ni

as it is onto G/N_i.

3. Free profinite factors

Throughout this section, C denotes an extension-closed pseudovariety of finite groups.

Subgroups of pro-C groups are closed and homomorphisms are continuous, unless stated otherwise.

The main result of this section is a general criterion for a given closed subgroup H of a pro-C groupGto be a free pro-Cfactor in terms of extending homomorphismsH →P ∈ C toG. It is the key to showing that (b) and (c) in Theorem 1.1 are equivalent, as explained in the introduction.

Recall that given H ≤ G and a homomorphism γ : H → P to a finite group P ∈ C, we denote by h(G, H, γ, P) the number of homomorphisms ˜γ :G→P such that ˜γ|_H =γ.

The number of epimorphisms GP that extendγ is denoted bye(G, H, γ, P).

Proposition 3.1. Let G be a finitely generated pro-C group and H a closed finitely gen- erated subgroup of G. Then H is a free pro-C factor of Gif and only if for every group P from C, the number h(G, H, γ, P) is independent of the choice of γ ∈Hom(H, P).

In the language of [13], the result says that H is a free pro-C factor of Gif and only if H is measure-preserving, meaning that for every P ∈ C, if ϕ ∈ Hom(G, H) is a random homomorphism chosen uniformly, thenϕ_|H is uniformly distributed in Hom(H, P).

In order to prove Proposition 3.1 we will study whether an isomorphism between closed subgroups of a profinite group can be extended to an automorphism of the ambient group.

Proposition 3.2. LetGbe a finitely generated pro-Cgroup andα:H1 →H2 a continuous isomorphism between two closed subgroups of G. Then the following are equivalent.

(a) there existsα˜∈Aut(G) such that α˜_|H1 =α;

(b) for every group P ∈ C and every γ ∈Hom(H₂, P), h(G, H1, γ◦α, P) =h(G, H2, γ, P);

(c) for every group P ∈ C and every γ ∈Hom(H₂, P), e(G, H1, γ◦α, P) =e(G, H2, γ, P);

(d) for every group P ∈ C and every γ ∈Hom(H₂, P),

if e(G, H₂, γ, P)6= 0, then e(G, H₁, γ◦α, P)6= 0.

Remark. Since G is finitely generated, by a well-known result of Nikolov and Segal [12], any homomorphism in Hom(G, P) is continuous. Thus, for non-continuous γ the condi- tions (b), (c) and (d) hold trivially as all numbers involved are 0.

Proof. The implications (a) =⇒(b) and (c) =⇒(d) are immediate.

For a homomorphism β ∈ Hom(H, P) and a subgroup Imβ ≤ Q ≤ P, we denote by β^Q the corestriction of β to Q; that is, the homomorphism in Hom(H, Q) such that β^Q(h) =β(h) for everyh∈H.

The implication (b) =⇒ (c) is proved by induction on the order of P and taking into account that

h(G, H₁, γ◦α, P) = X

Q≤P

e(G, H₁, γ^Q◦α, Q) and h(G, H₂, γ, P) = X

Q≤P

e(G, H₂, γ^Q, Q).

Let us prove the implication (d) =⇒ (a). By Lemma 2.4, there is a chain G > N₁ >

N2 > . . . of characteristic open subgroups of G with trivial intersection such that for each i, if φ_i ∈ Epi(G, G/N_i) then kerφ_i = N_i. For each i ∈ N, put P_i = G/N_i, denote by δi : G → Pi the canonical map and by γi = (δi)|H₂ the natural map H2 → Pi. By assumption, there exists φi ∈ Epi(G, Pi) such that φ|H₁ = γi ◦α. Since kerφi = Ni we obtain an automorphism of P_i, denotedφ_i, which satisfies that φ_i◦(δ_i)_|H1 =γ_i◦α. This shows that the set S_i ={τ ∈Aut(P_i) :τ◦(δ_i)_|H1 =γ_i◦α} is not empty. Now, for j > i, the maps δ_ji :P_j → P_i induce maps S_j → S_i that turn (S_i)i∈N into a directed system of non-empty sets. Their inverse limit S = lim←−iSi is therefore not empty and it is easy to check that any ˜α∈S is an automorphism ofGsuch that ˜α_|H1 =α, as required.

Proof of Proposition 3.1. The “only if ”part is clear. Let us show the “if ” part.

Let us first assume thatH is finite. LetP be a group fromCandγ ∈Hom(H, P). Then, sinceh(G, H, γ, P) is constant on Hom(H, P),

h(G, H, γ, P)· |Hom(H, P)|=|Hom(G, P)|. (2) Let α :K →H be an isomorphism, γ ∈ Hom(H, P) and consider the free pro-C product Γ =G`

CK. By the universal property, every homomorphism Γ→P is determined by two homomorphisms G→ P and K →P. This observation gives the first and last equalities in the following:

h(Γ, H, γ, P) =h(G, H, γ, P)· |Hom(K, P)|

=h(G, H, γ, P)· |Hom(H, P)|⁽²⁾= |Hom(G, P)|=h(Γ, K, γ◦α, P).

By Proposition 3.2, there exists ˜α ∈ Aut(Γ) that extendsα. In particular, there exists a subgroup G⁰ in Γ such that Γ =G⁰`

CH. By [15, Corollary 7.1.3 (a)], there exists g ∈Γ such that K^g ≤G⁰ orK^g ≤H. The second possibility cannot occur because the normal subgroup of Γ generated by K has trivial intersection with H. Thus K^g ≤ G⁰, and so, G∼= Γ/hK^Γi ∼= (G⁰/hK^G0i)`

CH.Hence H is a free pro-C factor ofG.

Now assume thatH is arbitrary. LetN be an open normal subgroup ofG.

We put

H_N⁰ =H/(N∩H) and G⁰_N =G/h(N ∩H)^Gi.

We see H_N⁰ as a subgroup of G⁰_N. Let P ∈ C and γ ∈ Hom(H_N⁰ , P). We denote by

˜

γ ∈Hom(H, P) the canonical lift ofγ. Then we have thath(G⁰_N, H_N⁰ , γ, P) =h(G, H,γ, P˜ ).

Hence h(G⁰_N, H_N⁰ , γ, P) is constant on Hom(H_N⁰ , P). Using what we have proved before, H_N⁰ is a free pro-C factor of G⁰_N. Let dbe the number of generators of G/H^G. Then we have just proved that the set

S_N ={(h₁, . . . , h_d)∈(G⁰_N)^d:G⁰_N =hh₁, . . . , h_dia

C

H_N⁰ } is not empty.

Observe that{S_N}form an inverse system of compact sets. Thus, there exists (g₁, . . . , g_d)∈ lim←−NS_N ⊂G^d. It is clear thatG=hg₁, . . . , g_di`

CH.

4. Profinitely flat modules

In this section we establish a connection between coherence of rings and flatness of profi- nite completions. In the context of commutative rings the relation between completions, flatness and coherence was studied in [9].

Recall thatR is calledleft coherent if every finitely generated submodule of a free left R-module is finitely presented. If M is anR-module, its profinite completion Mcis the R-module lim←−N≤_fMM/N where N ranges over all submodules of M of finite index (as a subgroup). Observe that Mc is a R-module: givenb r = (r_I), where r_I ∈R/I and I is an ideal of R of finite index, and m= (m_N) ∈Mc, where m_N ∈M/N and N is a submodule of M of finite index, definer·m= (r_Ann(M/N)·m_N).

The goal of this section is to prove the following:

Theorem 4.1. Let R be a finitely generated residually finite ring such that all finitely presented R-modules are residually finite. The following are equivalent:

(1) The profinite completion Rb of R is right flat.

(2) R is left coherent.

Before proving the theorem, we need some auxiliary results.

Lemma 4.2 (See Proposition 3.2.5 of [14]). Let A → B → C → 0 be an exact sequence of left R-modules. Then the induced sequence of left R-modulesb Ab→ Bb → Cb → 0 is also exact.

Lemma 4.3. Let M be a leftR-module andα_M :Rb⊗_RM →M , rc ⊗m7→r·m the natural map of left R-modules.b

(1) If M is finitely generated, then α_M is onto.

(2) If M is finitely presented, then α_M is an isomorphism.

(3) An R-module morphism γ :M1→M2 induces the following commutative diagram Rb⊗_RM₁ −−−→^Id⊗γ Rb⊗_RM₂

↓_αM

1 ↓_αM

2

Mc₁ −→^bγ Mc₂ .

Proof. Suppose thatM is finitely generated, sayM =Rm₁+· · ·+Rm_d. Thenα_M(1⊗M) = M and α_M(Rb⊗_RM) = Rmb ₁+· · ·+Rmb _d. As each Rmb _i is closed in Mc, their sum is a closed subset ofMcthat contains M, so must be all ofMc.

Let M be finitely presented, with free presentation R^e →R^d→M → 0 and e, d finite.

Tensoring withRband using Lemma 4.2 we obtain the following commutative diagram with exact top and bottom rows:

Rb⊗_RR^e → Rb⊗_RR^d → Rb⊗_RM →0

↓ ↓ ↓_αM

Rb^e → Rb^d → Mc →0 .

The two left downward arrows are isomorphisms, because tensor products and completions commute with finite direct sums. A diagram chase shows thatαM is also an isomorphism.

The last item is equivalent tobγ being a R-homomorphism, which is clear.b The next lemma shows that the condition that all finitely presented left R-modules are residually finite plays an analogous role to the Artin-Rees Lemma forI-adic completions of finitely generated modules over commutative Noetherian rings. In particular, this lemma is classically used to show that if 0→A→B→C →0 is a short exact sequence of finitely generated R-modules whereR is commutative and Noetherian and I is some proper ideal ofR, then theI-adic completion ofAis the same as the closure ofAin theI-adic topology ofB. This guarantees that the induced sequence 0→Ab→Bb →Cb→0 is exact. Moreover, it is also used to show that theI-adic completion ofR is flat as aR-module.

Lemma 4.4. Let R be a finitely generated ring such that all finitely presented left R- modules are residually finite. Then, for every d >0and every finitely generated submodule L≤R^d, the profinite topology of L coincides with the subspace topology from the profinite topology of R^d.

In particular, given an exact sequence 0→L→R^d→M →0 of left R-modules with L finitely generated, the induced sequence

0→Lb→Rb^d→Mc→0 of left R-modules, is exact.b

Proof. The second statement follows directly from the first one and Lemma 4.2, because the kernel of the map to Mcis the closure ofLin R^d.

To prove the first statement, let L ≤ R^d be a finitely generated submodule. Then R^d/L is finitely presented, so, by assumption, residually finite. In other words, L is the intersection of all finite index submodules ofR^dthat containL. We need to show that for every K ≤L of finite index, there is some U ≤R^d of finite index such that L∩U ≤K.

Since R is a finitely generated ring,K is a finitely generated submodule. Therefore R^d/K is residually finite and, asL/K is finite, there is someU ≥K of finite index inR^davoiding any set of non-trivial coset representatives of K in L. This means that U ∩L = K, as

required.

Proof of Theorem 4.1. (1) =⇒ (2). Let L be a finitely generated submodule of a free module R^k. We want to show that L is also finitely presented. Consider the following exact sequence of left R-modules

0→L→R^k →M →0.

SinceRbis right flat and taking into account Lemma 4.2 we obtain the commutative diagram 0 → Rb⊗_RL → Rb⊗_RR^k → Rb⊗_RM →0

↓^αL ↓_α

Rk ↓_αM

Lb −→^β Rb^k → Mc →0.

By Lemma 4.3,αLis onto. Sinceα_R^k is bijective, the commutativity of the diagram implies that α_L should be injective, and so, bijective.

Assume that Lis generated by delements. Therefore we have the following short exact sequence

0→I →R^d→L→0.

Tensoring with Rb and noting thatRb is flat gives the short exact sequence 0→Rb⊗_RI →Rb⊗_RR^d→Rb⊗_RL→0.

Let I be the closure of I in Rb^d and denote by αI :Rb⊗_RI → I the composition of αI

and the map Ib→ I. Then we also have, upon taking completions and applying Lemmas 4.2 and 4.3 the commutative diagram

0 → Rb⊗_RI → Rb⊗_RR^d → Rb⊗_RL →0

↓_αI ↓_α

Rd ↓_αL

0 → I → Rb^d → Lb →0

where α_Rd and α_L are isomorphisms and γ, φ are continuous. The commutativity of the diagram implies that αI is also an isomorphism.

As R is countable, I is also countable. Enumerate the elements of I = {i_k}_k∈N and define, for each k∈Nthe sets

Ik=

k

X

j=1

Rij and Tk=

k

X

j=1

Rb⊗ij.

Thus,I =∪_k∈Nα_I(T_k) is a countable union of compact submodules. SinceI is compact, by the Baire category theorem, there exists n∈Nsuch that αI(Tn) contains an open subset ofI. Hence for somem∈N,αI(Tm) =I. This implies that the closure ofIm=αI(1⊗Im) in R^k contains I = α_I(1⊗I). As R^k/I_m is finitely presented, it is residually finite, by assumption on R. In other words, I_m is closed in R^k, which yieldsI = I_m and thus L is finitely presented.

(2) =⇒(1). We wish to show that Tor^R₁(R, Mb ) = 0 for every left R-module M. Since every module is a direct limit of finitely presented modules, and Tor^R₁ commutes with direct limits (in both variables), we may assume without loss of generality that M is finitely presented. Suppose thatM is defined by the short exact sequence

0→L→R^d→M →0

where L is finitely generated. As R is coherent, L is in fact finitely presented. Tensoring this exact sequence withRb we obtain the long exact sequence

0→Tor^R₁(R, Mb )→Rb⊗_RL−→ⁱ Rb⊗_RR^{d p}−→Rb⊗_RM →0.

Then by Lemmas 4.3 and 4.4, we obtain the following commutative diagram with exact top and bottom rows

0 →Tor^R₁(R, M)b −→^β Rb⊗_RL −→ⁱ Rb⊗_RR^d −→^p Rb⊗_RM →0

↓_αL ↓_α

Rd ↓_αM

0 → Lb −→^η Rb^d −→^π Mc →0,

whereα_L,α_M andα_Rd are isomorphisms, because theR-modulesL,M andR^dare finitely presented. A diagram chase shows that Tor^R₁(R, M) = 0, as required.b The condition on R “each finitely presented module is residually finite” can also be encoded in a condition of flatness of some module.

Proposition 4.5. Let R be a finitely generated residually finite ring. The following are equivalent:

(1) the quotient module R/Rb is right flat;

(2) every finitely presented R-module is residually finite andRb is right flat;

(3) every finitely presented R-module is residually finite andR is left coherent.

Proof. The last two items are equivalent by Theorem 4.1, so we show that the first two are equivalent. Assume that R/Rb is right flat. SinceR is also flat, Rb is flat as well. Let M

be a finitely presented left R-module. Tensoring the short exact sequence 0 →R→ Rb→ R/Rb →0 with M yields the following commutative diagram with exact upper sequence:

Tor^R₁(R, M)b → Tor^R₁(R/R, Mb ) → R⊗_RM −→^γ Rb⊗_RM → (R/R)b ⊗_RM →0

↓^δ ↓^αM M −→^σ Mc

where γ, δ and σ are the canonical maps. By Lemma 4.3, αM is an isomorphism.

It is obvious that the diagram commutes. Since Rb and R/Rb are flat, Tor^R₁(R, Mb ) = Tor^R₁(R/R, Mb ) = 0. Thus,σ is an embedding, and soM is residually finite.

Now assume that every finitely presented leftR-module is residually finite andRbis right flat. This means that in the previous diagram Tor^R₁(R, Mb ) = 0 and, asσ is an embedding,

Tor^R₁(R/R, Mb ) = 0 as well.

Remark 4.6. LetG be a topologically finitely generated profinite group and R =Fp[[G]].

By the result of Nikolov and Segal [12], the natural map R → Rb is an isomorphism. So R/Rb = 0 is flat, but obviously R is not always coherent. Thus, the condition that R is finitely generated is not superficial.

5. Completed group algebras of completions of virtually free groups We now apply the results of the previous section, together with results from [5], to show that the completed group algebra Fp[[G]] of a finitely generated virtually free group is ab direct union of free Fp[G]-modules. This will be a key ingredient in the next section, to show that ifGb splits as a profinite free product, then so doesG.

Definition 5.1. A free left ideal ring, or left fir for short, is a ring in which all left ideals are free left R-modules of unique rank. Right firsare defined correspondingly and a fir is a left and right fir.

Note that if R is a fir then every finitely generated left (or right) ideal is a finitely presented left (or right) R-module.

Proposition 5.2([5] Theorem A.6 and Corollary A.7, p. 554). For a ring R, the following are equivalent:

(1) R is left coherent;

(2) every finitely generated left ideal of R is a finitely presented left R-module.

Furthermore, over a left coherent ring, the intersection of any pair of finitely generated submodules of a free left module is finitely generated.

Proposition 5.3. Let F be a finitely generated free group and p a prime. ThenFp[[F]]b is a right flat Fp[F]-module.

Proof. First observe that F\p[F] ∼= Fp[[Fb]]. The ring Fp[F] is a fir ([2]). Thus, it is also coherent. We claim that every finitely presented Fp[F]-module is residually finite. If this holds, then, observing thatF\p[F]∼=Fp[[Fb]], Theorem 4.1 implies that Fp[[Fb]] is right flat.

To show the claim, let M ∼= Rⁿ/I be a finitely presented module, with n ∈ N and I finitely generated. We wish to show that given any z∈Rⁿ\I there is a submoduleL⊇I such that z /∈ L and Rⁿ/L is finite. We proceed by induction on n. If n = 1, then [16, Theorem 4.3] implies that for any z∈R\I there is a finitely generated idealL⊇I such that z /∈L andR/L is finite.

Suppose the claim true for n−1 ≥ 1 and let z ∈ Rⁿ \I. Denote by R₁ the first summand in Rⁿ. If z /∈R1+I, we consider the quotient Rⁿ/(R1+I) ∼=Rⁿ⁻¹/I1 where Rⁿ⁻¹ =Rⁿ/R₁ and I₁ =R₁+I/R₁. Since R₁ and I are finitely generated R-modules,I₁ is a finitely generated submodule ofRⁿ⁻¹ that does not contain the imagez1 ofzinRⁿ⁻¹. By inductive hypothesis, there is a finitely generated submodule L₁ ⊇ I₁ of Rⁿ⁻¹ such thatz₁ ∈/ L₁ and Rⁿ⁻¹/L₁ is finite. The preimage ofL₁ inRⁿ is the desired submodule of Rⁿ. Ifz∈R1+I, consider its image z1 under the isomorphismR1+I/I∼=R1/R1∩I, so z₁ ∈/ R₁ ∩I. Since Fp[F] is coherent, Proposition 5.2 guarantees that R₁∩I is a finitely generated ideal ofR1. Theorem 4.3 of [16] then gives a finitely generated idealL1 ⊇R1∩I ofR₁ not containingz₁ and such thatR₁/L₁ is finite. The preimage of L₁ inR₁+I is our

desired L.

Proposition 5.4 (See Proposition 1.4.5 in [5]). Let R be a fir and M a left R-module.

Then M is flat if and only if every finitely generated submodule of M is free.

Putting together the last two results yields the following:

Corollary 5.5. Let F be a finitely generated free group. Then every finitely generated submodule of the Fp[F]-module Fp[[Fb]] is free.

Corollary 5.6. Let G be a finitely generated virtually free group. Then the Fp[G]-module Fp[[G]]b is isomorphic to a direct union of free Fp[G]-modules.

Proof. LetF be a free subgroup ofG of finite index and denote by F the closure of F in G. Thenb F ∼= Fb because F is of finite index in G. Now, Fp[[G]]b ∼= Fp[G]⊗_Fp[F]Fp[[F]].

By Corollary 5.5, Fp[[Fb]] =Fp[[ ¯F]] is isomorphic to a direct union of free Fp[F]-modules, therefore, as tensor products commute with direct unions,Fp[[G]]b ∼=Fp[G]⊗_Fp[F]Fp[[F]] is

isomorphic to a direct union of free Fp[G]-modules.

Corollary 5.7. Let G be a finitely generated virtually free group. Let nbe a product of a finite set of different primes and k=Z/nZ. Then the k[G]-modulek[[G]]b is isomorphic to a direct union of projective k[G]-modules.

6. Free factors of virtually free groups

This section is devoted to the proof of the equivalence of items a) and c) in Theorem 1.1. Recall that this equivalence is the following:

Theorem 6.1. Let G be a finitely generated group that is virtually free and let H ≤ G be finitely generated. Denote by H the closure of H in the profinite completion Gb of G.

Then Gb splits as a free profinite product Gb=H`

K if and only ifG=H∗Lsplits as an abstract free product.

The “if” part follows from the facts that “profinite completion commutes with free product”; i.e., Hb`

Lb = H\∗L and that the profinite topology of H∗L induces the full profinite topologies onH and L(see [14, Corollary 3.1.6]).

The rest of this section is devoted to the “only if” implication. It relies on a decomposi- tion theorem due to Dicks and Dunwoody ([7, 6]). To set notation, we recall the relevant Bass–Serre theory, mostly following [6].

6.1. Prerequisites from Bass–Serre Theory. A graph Y =V(Y)tE(Y) is the dis- joint union of a setV =V(Y) of verticesand a setE =E(Y) of edges, given with two maps i, t:E →V that associate to each edge its initiali(e), respectively,terminal t(e) vertex. Loops (i(e) =t(e)) are allowed. In other words, we consider directed multigraphs.

Y is connected if it is connected as an undirected graph. A tree is a connected graph which has no undirected cycles. By Zorn’s Lemma, every connected graph has a maximal subtree. For our purposes, Y will always be finite.

A graph of groups (G, Y) consists of a graph Y, vertex groups {G(v) : v ∈ V(Y)}

and edge groups {G(e) :e∈E(Y)} and, for each e∈ E(Y), two group morphisms i_e, t_e : G(e) → G(i(e)),G(t(e)). If all the morphisms ie,te, e ∈ E(Y) are injective, the graph of groups is called faithful.

Given a connected graph of groups (G, Y) and a maximal subtree T of Y, thefunda- mental group G=π(G, Y, T) of (G, Y) with respect to T is the group with presentation:

• generators{G(v), q_e:v∈V(Y), e∈E(Y)};

• relations of eachG(v), v∈V(Y);

• relations (i_e(g))^qe =t_e(g) for each g∈ G(e), e∈E(Y);

• relationsqe= 1 if e∈E(T)

One of the first results of Bass–Serre theory (see [6, II.2.4], [17, I.5.20]) is that any other maximal subtree yields an isomorphic fundamental group. Another result ([6, II.2.7]) states that if (G, Y) is connected and faithful, then the canonical mapsG(v) →G, v∈V(Y) are injective and we may identify each G(v) with its image.

Remark 6.2. Let (G, Y) be a connected faithful graph of groups with maximal subtree T. Suppose there is some edge e in T with i(e) = v, t(e) = w and such that one of ie or t_e is an isomorphism. Without loss of generality, assume that i_e is an isomorphism, so that we may identify G_e = G_v. Then, in the fundamental group π(G, Y, T) we have that (G_v)^qe =G_v =te(G_v)≤ G_w.

Consider the graph Y⁰ and maximal tree T⁰ obtained from Y by identifying v and w along e (leaving the other possible edges between v and w as loops). Then π(G, Y, T) = π(G, Y⁰, T⁰) becauseG_v ≤ G_w inπ(G, Y, T).

Doing this procedure for all edgeseofT as above, we may assume that ifG=π(G, Y, T), then for every edge eof T, neitheri_e nort_e are isomorphisms.

Since choosing a different maximal subtree does not change the isomorphism type of the group, we may further assume that G is the fundamental group of a connected faithful graph of groups such that for every edge e joining different vertices neither i_e nor t_e are isomorphisms.

Let G act on a tree X and let S ⊂ X be a connected transversal for the G-action.

This is a subset of X (in general, not a subgraph) that is in bijection with G\X, any two of whose vertices are connected by a path in S and such that i(e) ∈ S for every e ∈ S.

For each e∈ E(S), there is a unique vertex of S in the same orbit as t(e), so we choose qe ∈G such that qe·t(e) ∈S, taking qe = 1 ift(e)∈ S. The collection (qe :e∈E(S)) is a connecting family for S. For each s∈S, denote by G_s its stabiliser inG. Note that G_qe·t(e) = qeG_t(e)q⁻¹_e and that for every e ∈ E(S), Ge = G_i(e)∩G_t(e) = G_i(e)∩q_e⁻¹Guqe

whereu=qe·t(e), so thatG^qe^e ≤G_qe·t(e). These are exactly the conjugation relations that appear in the definition of the fundamental group of a graph of groups.

Let (G, Y) be a connected graph of groups with maximal subtreeT. Writeπ =π(G, Y, T).

Thestandard treeof (G, Y, T) is the graph ˜X = ˜X(G, Y, T) with, respectively, vertex and edge sets

V( ˜X) = G

v∈V(Y)

π/G(v), E( ˜X) = G

e∈E(Y)

π/Ge

and incidence maps given by

i(gG(e)) =gG(i(e)), t(gG(e)) =gqeG(t(e)).

That ˜X is indeed a tree is a key result of Bass–Serre theory ([6, I.5.3], [17, I.5.12]).

Theorem 6.3 (Structure Theorem of Bass–Serre theory). Let Gact on a connected graph X with connected transversal S ⊂ X, maximal subtree T ⊂ S and connecting family (q_e ∈G:e∈E(S)). Then there is a faithful connected graph of groups (G, G\X) defined by G(s) =Gs for s∈S and ie, te:Ge →G_i(e), G_(qe·t(e)) given, respectively, byg 7→g, g^qe, for each e∈E(S).

There is a surjective homomorphism ϕ :π = π(G, G\X, T) → G, uniquely determined by the inclusion maps G(v)→Gand qe →qe, v∈V(Y), e∈E(S).

There is an equivariant universal covering map from the standard tree X(G, G\X, T˜ ) to X, given by gG(s)7→ϕ(g)·s, g∈π, s∈S.

The map X˜ →X is bijective if and only ifϕ is an isomorphism.

Let G =π(G, Y, T) be the fundamental group of a connected faithful graph of groups, with respect to a maximal subtree T. Since we can identify each G(v), v ∈V(Y) with its image in G, we writeG_y :=G(y) for each y∈Y. For every e∈E(Y), write i(G_e) ≤G_i(e) and t(G_e) ≤G_t(e) for the respective images of G_e under i_e, t_e. Conjugating byq_e induces an isomorphismi(G_e)→t(G_e), which is the identity ife∈T, in which case we identifyG_e withi(Ge) =t(Ge).

Proposition 6.4(Proposition II.3.1 of [6]). With the above notation,G_i(e)∩G_t(e)^qe =ie(G(e)) for any e∈E(Y). Given any v, w∈V(Y), let P be the set of edges of T in the geodesic joining v, w. Then Gv∩Gw =T

e∈PGe.

Remark 6.5. Suppose that G = π(G, Y, T) is the fundamental group of some connected faithful graph of groups and that H ≤ G is one of the vertex groups. H is a free factor of G if all edge groups incident with H have trivial image in H. By Proposition 6.4, this

holds if H∩G^gv is trivial for every vertex group distinct from H and every g ∈ G and if H∩H^g is trivial for everyg∈G\H.

The last result that we need is the crucial link between derivations and decompositions into fundamental groups of graphs of groups. A derivation d:G→ P is innerif there is somep∈P such thatd(g) = (1−g)p for allg∈G.

Theorem 6.6 (Theorem III.4.6 of [6]). Let G be a group and R a ring. Suppose thatP is a projective R[G]-module, and d:G→P is a derivation such that Gis generated by kerd together with finitely many elements. Then G decomposes as the fundamental group of a faithful connected finite graph of groups with finite edge groups, kerd as one of the vertex groups, and such that the derivation dis inner when restricted to each vertex group.

Note that ifdis inner when restricted to vertex groups, thendis also inner on conjugates of vertex groups: for everyf ∈Gandg∈G_vwe haved(f gf⁻¹) = (1−f gf⁻¹)d(f)+f d(g) = (1−f gf⁻¹)(d(f) +f p).

6.2. Proof of Theorem 6.1, continued. Suppose thatGis finitely generated, virtually free andH ≤G satisfies thatGb=H`

K for some closed subgroup K of G.b

Let U be a free, normal subgroup of finite index in Gand let k=Z/nZ wheren is the product of all prime factors of|G:U|. Lemma 2.1 yields a continuous homomorphism

φ:I_k[[G]]b →I_k[[K]]^Gb

of k[[G]]-modules, which is simply projection to the second component. Note that theseb are alsok[G]-modules. SinceG is finitely generated, the image of {1−g:g∈G} ⊆I_k[[

G]]b

underφis a finitely generatedk[G]-module. By Corollary 5.7, this finitely generatedk[G]- module lies in a direct union of projective k[G]-modules (which we can take to be finitely generated, since every module is the direct union of its finitely generated submodules).

Let P be a projective finitely generated k[G]-module containing φ(I_k[G]) and denote by ϕ:I_k[G]→P the restriction ofφtoI_k[G], with image in P.

Then ϕinduces a derivation

d:G→P, g7→ϕ(1−g).

Notice that ϕ(1−g) = 0 exactly when g ∈ H∩G. Since G is virtually free, all finitely generated subgroups are closed in the profinite topology (that is,Gis LERF), soH∩G=H.

We have thus obtained a derivation d:G→ P to a projective k[G]-module whose kernel is exactly H. Theorem 6.6 then implies that Gdecomposes as the fundamental group of a faithful connected finite graph of groupsY = (V, E), with finite edge groups, withH =G_u for some u∈V and such that the derivationdis when restricted to any conjugate of any vertex group.

Note that by Remark 6.2, we can assume that for every e∈E with distinct endpoints v, w, neither of the injections Ge → Gv, Gw is surjective. Following Remark 6.5, we will show thatH∩G^g_v = 1 for everyg∈Gand H∩H^g = 1 for everyg∈G\H.

Case 1: v6=u andG_v is infinite. First we reduce to the case whereHis finite. Recall thatU was taken to be a free, normal subgroup of finite index inG. ThenH⁰ :=H/(H∩U) is finite and, since H ∩U is torsion-free, it intersects trivially all edge groups, so that G⁰ := G/hH ∩Ui^G has the same presentation as G, with the only addition that H is replaced by the finite group H⁰. In particular, each vertex groupGv distinct fromH maps isomorphically to a subgroup ofG⁰, which, abusing notation, we shall also call G_v.

We then have cG⁰ =H⁰`

K =H⁰`

K and so I_k[[

Gc⁰]] = I_k[[H^Gc0 0]]⊕I_k[[K]]^Gc0 . This gives the following commutative diagram:

G ,→ Gb ,→ I_k[[

G]]b

→φ I_k[[K]]^Gb

↓ ↓ ↓ ↓

G⁰ ,→ cG⁰ ,→ I_k[[cG0]]

φ⁰

→ I_k[[K]]^Gb

where the downward arrows are the quotient maps by the appropriate image ofhH∩Ui^G. Note that dis the composition of all maps on the top row of the diagram. Denote byd⁰ the composition of the maps on the bottom row. This is in particular a derivation on G⁰, whose kernel isH⁰, and that is inner on every conjugate of G_v ≤G⁰ for everyv∈V.

Fix now some v 6= u such that Gv is infinite and denote by L any conjugate of Gv. Because d⁰ is inner on L, there is some m ∈ I_k[[

Gc⁰]] such that d⁰(g) = (1−g)m for every g∈L. Recalling that 1−g=d⁰(g) +

I_k[H^Gc0 _]−part of 1−g

, we have that 1−g≡(1−g)m modulo I_k[H]^Gc0 for every g ∈ L. Equivalently, (1−g)(1−m) ≡ 0 modulo I_k[H^Gc0 0] for every g∈L, so the image of 1−m ink[[cG⁰/H⁰]] is a fixed point of L.

Any fixed point ofLink[[cG⁰/H⁰]] is also a fixed point of the closureL incG⁰, becauseL is dense inL (asG⁰ is residually finite) and the action ofGc⁰ on k[[cG⁰/H⁰]] is continuous.

Now, H⁰ =H⁰ is finite, so for every NEoGc⁰ and g∈Gc⁰ we have

|L:L∩N|=|L:L∩(H⁰)^gN| · |L∩(H⁰)^gN :N| ≤ |L:L∩(H⁰)^gN| · |H⁰|.

Therefore p^∞ divides lcm

NEoGc⁰|L : L∩(H⁰)^gN| if and only if it divides lcm

NEocG⁰|L : L∩N|=|L|.

Because G⁰ is LERF, the profinite completion of L coincides with its closure L in Gc⁰. As L is virtually torsion-free, L contains a copy of bZ; in particular, p^∞ divides|L|. Now Lemma 2.2 implies that L has no non-trivial fixed points when it acts on k[[cG⁰/H⁰]]. In particular, (1−m) is congruent to 0 ink[[cG⁰/H⁰]], which is equivalent to 1−m∈I_k[H^cG0 0].

Recall that the kernel of d⁰ is H⁰, so d⁰(h) = (1−h)m = 0, that is, hm = m, for all h∈L∩H⁰.

Let G⁰/N be a finite quotient of G⁰ in which H⁰ injects. The image me ∈ k[G⁰/N] of m is a finite sum me = P

γ∈G⁰/Nr_γγ. Since m ≡ 1 in k[[Gc⁰/H⁰]], the coefficients of γ ∈ H⁰N/N ∼= H⁰ in me must sum to 1: P

γ∈H⁰N/Nrγ = 1. Moreover, because me is invariant under H⁰∩L, we must have thatrγ =r_hγ for all h∈H⁰∩L and γ ∈G⁰/N. In

particular,

1 = X

γ∈H⁰N/N

rγ = X

δ∈H⁰N/(H⁰∩L)N

rδ|H⁰∩L|

,

so|H⁰∩L|is invertible in k.

By the choice of k, the invertible elements are those that are coprime with all primes that divide |G : U|. But |H⁰ ∩L| divides |H⁰| = |HU : U|, which divides |G : U|. So H⁰∩L= 1.

This means that the image ofH∩L≤GinG⁰is trivial, which implies thatH∩L≤H∩U. But U is torsion-free, so in fact H∩L= 1.

Case 2: v 6=u and G_v is finite. Like in the previous case, denote by L any conjugate of Gv in G. By [14, Theorem 9.5.1], since L is finite, it is conjugate in Gb to a subgroup of H or K. If the latter, we are done, as any conjugate of K intersects H trivially , by [11, Corollary 3.1.2]. The same result guarantees that a conjugate of H distinct from H has trivial intersection withH (see also [14, Theorem 9.1.12]). So we are left with the case L≤H, which means thatL≤H becauseH is closed in the profinite topology ofG.

Suppose that L=G^gv, so that Lis the stabiliser of gv in the standard tree. Let gw be the next vertex in the path between gv and u in the standard tree. Since G^gv ≤H =G_u, we also have thatG^g_v ≤G^g_wand therefore G_v≤G_w, where inclusion is induced by the edge homomorphisms of the edge e∈E betweenv, w∈V. This contradicts the assumption on G that we made following Remark 6.2.

Case 3: v = u. Here we must show that H^g ∩H = 1 for every g /∈ H. No such g is contained in H, as H =G∩H. By [14, Theorem 9.1.12], this implies that H^g∩H = 1,

which finishes the argument.

References

[1] G. Baumslag, Residually finite groups with the same finite images, Compos. Math., Volume 29 (1974), pp. 249-252.

[2] G. M. Bergman, Coproducts and some universal ring constructions. Trans. Amer. Math. Soc. 200 (1974), 33–88.

[3] M. R. Bridson, D. B. McReynolds, A. W. Reid, R. Spitler. Absolute profinite rigidity and hyperbolic geometry. Ann. Math., Volume 192 (2020) no. 3, pp. 679-719.

[4] M. R. Bridson, D. B. McReynolds, A. W. Reid, R. Spitler, On the profinite rigidity of triangle groups.

arXiv:2004.07137.

[5] P. M. Cohn, Free rings and their relations. Second edition. London Mathematical Society Monographs, 19. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985.

[6] W. Dicks, Groups, Trees and Projective Modules. Lecture Notes in Mathematics. Vol. 790. Springer- Verlag Berlin Heidelberg New York, 1980.

[7] , On splitting augmentation ideals. Proc. Amer. Math. Soc. 83 (1981), no. 2, 221?227.

[8] F. J. Grunewald, R. Scharlau, A note on finitely generated torsion-free nilpotent groups of class 2, Journal of Algebra, Volume 58, Issue 1, 1979, 162–175.

[9] B. Hassan Coherent rings and completion. Master’s thesis, University of Glasgow (2011,http://theses.

gla.ac.uk/2514/1/2011HassanBMScR.pdf.

[10] E.I. Khukhro and V.D. Mazurov (eds.). The Kourovka notebook. Unsolved problems in group theory, 18th ed., Ross. Akad. Sci. Sib. Div., Inst. Math., Novosibirsk 2014, 227 pp.

[11] Zalesski˘i, P. A., Mel’nikov, O. V., Subgroups of profinite groups acting on trees. (Russian) Mat. Sb.

(N.S.) 135(177) (1988), no. 4, 419439, 559; translation in Math. USSR-Sb. 63 (1989), no. 2, 405424.

[12] N. Nikolov, D. Segal, On finitely generated profinite groups. I. Strong completeness and uniform bounds. Ann. of Math. (2) 165 (2007), 171–238.

[13] D. Puder, O. Parzanchevski, Measure preserving words are primitive. J. Amer. Math. Soc. 28 (2015), 63–97.

[14] L. Ribes, P. Zalesskii.Profinite groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.

A Series of Modern Surveys in Mathematics, Vol. 40, 2nd edition, Springer-Verlag, Berlin, 2010.

[15] L. Ribes, Profinite graphs and groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.

A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 66. Springer, Cham, 2017.

[16] A. Rosenmann, S. Rosset, Ideals of finite codimension in free algebras and the fc-localization. Pacific J. Math. 162 (1994), 351–371.

[17] J.-P. Serre, Trees. Springer, Berlin, Heidelberg. 1980.

[18] H. Wilton, Essential surfaces in graph pairs. J. Amer. Math. Soc. 31 (2018), no. 4, 893–919.

[19] H. Wilton, On the profinite rigidity of surface groups and surface words. Comptes Rendus.

Math´ematique. 359 (2021), no.2, 119–122.

Departamento de Matem´aticas, Universidad Aut´onoma de Madrid and Instituto de Ciencias Matem´aticas, CSIC-UAM-UC3M-UCM

E-mail address: [email protected] E-mail address: [email protected]