Conjugacy p-separability of right-angled Artin groups and applications

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Conjugacy p-separability of right-angled Artin groups and applications

Emmanuel Toinet

To cite this version:

Emmanuel Toinet. Conjugacy p-separability of right-angled Artin groups and applications. 2013.

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Conjugacy p-separability of right-angled Artin groups and applications

Emmanuel Toinet

Abstract

We prove that every subnormal subgroup of p-power index in a right-angled Artin group is conjugacyp-separable. As an application, we prove that every right-angled Artin group is conjugacy separable in the class of torsion-free nilpotent groups. As another application, we prove that the outer automorphism group of a right-angled Artin group is virtually residuallyp-finite. We also prove that the Torelli group of a right-angled Artin group is residually torsion-free nilpotent, hence residuallyp-finite and bi-orderable.

1 Introduction

Let Γ = (V,E) be a finite simplicial graph. Theright-angled Artin group associated to Γ is the groupG_Γ defined by the presentation:

G_Γ =h V |vw =wv,∀ {v,w} ∈E i.

Note that, if Γ is a discrete graph, then GΓ is the free group Fr on r generators (where r = |V|), and if Γ is a complete graph, thenG_Γ is the free abelian group^Zr. Thus, right-angled Artin groups can be seen as in- terpolating between free groups and free abelian groups. Therank of GΓ

is by definition the number of vertices of Γ. A special subgroup of G_Γ is a subgroup generated by a subset W of the set of vertices V of Γ – it is naturally isomorphic to the right-angled Artin group G_Γ(W), where Γ(W) denotes the full subgraph of Γ spanned byW. Letv be a vertex of Γ. The link of v, denoted by link(v), is the subset of V consisting of all vertices that are adjacent tov. Thestar of v, denoted bystar(v), is link(v)∪ {v}.

We refer to [C] for a general survey of right-angled Artin groups.

2010Mathematics Subject Classification. 20F36, 20E26, 20F28.

Key words and phrases. Right-angled Artin group, automorphism group, Torelli group, residual properties, separability properties, pro-ptopology.

Little is known about the automorphim groups of right-angled Artin groups. In 1989, Servatius conjectured a generating set for Aut(G_Γ) (see [Ser]). He proved his conjecture in certain special cases, for example when the graph is a tree. Thereafter Laurence proved the conjecture in the general case (see [L]). Charney and Vogtmann showed that Out(G_Γ) is virtually torsion-free and has finite virtual cohomological dimension (see [CV1]). Day gave a finite presentation for Aut(G_Γ) (see [D1]). More re- cently, Minasyan proved that Out(G_Γ) is residually finite (see [M]). This result was obtained independently by Charney and Vogtmann in [CV2], where they also proved that, for a large class of graphs, Out(G_Γ) satisfies the Tits alternative.

Let K be a class of group. A group G is said to be residually K if for all g ∈ G\{1}, there exists a homomorphism ϕ from G to some group of K such that ϕ(g) 6= 1. Note that if K is the class of all finite groups, this notion reduces to residual finiteness.

For a groupGand forg,h∈G, we use the notationg∼hto mean that g and h are conjugate. A group Gis said to be conjugacy K-separable (or conjugacy separable in the classK) if for allg,h∈G, either g∼h, or there exists a homomorphism ϕ from G to some group of K such that ϕ(g) ≁ ϕ(h). Note that ifK is the class of all finite groups, this notion reduces to conjugacy separability. Clearly, if a group is conjugacy K-separable, then it is residuallyK.

Our focus here is on conjugacy separability in the class of finitep-groups.

Letpbe a prime number. IfKis the class of all finitep-groups, then, instead of saying “Gis residuallyK”, we shall say thatGisresiduallyp-finite. Note that this implies residually finite as well as residually nilpotent. Instead of saying “G is conjugacy K-separable”, we shall say that G is conjugacy p-separable. Following Ivanova (see [I]), we say that a subset S of a group Gisfinitelyp-separable if for everyg∈G\S, there exists a homomorphism ϕfrom G onto a finite p-group P such that ϕ(g) ∈/ ϕ(S). Note that G is conjugacy p-separable if and only if every conjugacy class of G is finitely p-separable.

Examples of groups which are known to be conjugacy p-separable in- clude free groups (see, for example, [LS]) and fundamental groups of ori- ented closed surfaces (see [P]).

There is a connection between these notions and a topology on G, the

“pro-p topology” on G. The pro-p topology on Gis defined by taking the normal subgroups of p-power index in G as a basis of neighbourhoods of

1 (see [RZ]). Equipped with the pro-p topology, Gbecomes a topological group. Observe that G is Hausdorff if and only if it is residually p-finite.

One can show that a subset S of G is closed in the pro-p topology on G if and only if it is finitely p-separable. Thus, G is conjugacy p-separable if and only if every conjugacy class ofGis closed in the pro-ptopology onG.

In [CZ], Chagas and Zalesskii constructed an example of a conjugacy separable group possessing a non conjugacy separable subgroup of finite index. This led them to introduce the notion of “hereditarily conjugacy separable group”. A group Gis said to behereditarily conjugacy separable if every subgroup of finite index in Gis conjugacy separable.

Recall that a subnormal subgroup of a group G is a subgroup H of G such that there exists a finite sequence of subgroups ofG:

H =H₀ < H₁ <... < H_n =G, such thatHi is normal inHi+1 for alli ∈ {0,...,n−1}.

A subgroupH of a groupG is open in the pro-ptopology on G if and only if it is subnormal ofp-power index (see Lemma A.1). This leads us to the following definition, which naturally generalizes that of [CZ]:

Definition 1.1. LetGbe a group. We say thatGis hereditarily conjugacy p-separable if every subnormal subgroup ofp-power index in Gis conjugacy p-separable.

In [M], Minasyan proved that right-angled Artin groups are hereditarily conjugacy separable. Our main theorem is the following:

Theorem 6.15. Every right-angled Artin group is hereditarily conjugacy p-separable.

We will now discuss some applications of Theorem 6.15. The first ap- plication that we mention is an application of Theorem 6.15 to separability properties ofG_Γ:

Corollary 7.1. Every right-angled Artin group is conjugacy separable in the class of torsion-free nilpotent groups.

Let P be a group property. A group G is said to be virtually P if there exists a finite index subgroupH < G such that H has Property P. Combining Theorem 6.15 with a result of Paris (see [P]), we obtain the following:

Corollary 7.4. The outer automorphism group of a right-angled Artin group is virtually residually p-finite.

On the other hand, combining Theorem 6.15 with a result of Myasnikov (see [My]), we obtain the following:

Corollary 7.6. The outer automorphism group of a right-angled Artin group is residuallyK, whereKis the class of all outer automorphism groups of finitep-groups.

The next application was suggested to the author by Ruth Charney and Luis Paris.

The natural actionAut(G_Γ)→GL_r(^Z) of Aut(G_Γ) onH₁(G_Γ,^Z) gives rise to a homomorphism Out(G_Γ) → GL_r(^Z), whose kernel is called the Torelli group of G_Γ – by analogy with the Torelli group of a mapping class group. In Section 7, we combine well-known results of Bass-Lubotzky (see [BL]), and Duchamp-Krob (see [DK1], [DK2]) with Theorem 6.15 to attain the following:

Theorem 7.14. The Torelli group of a right-angled Artin group is residu- ally torsion-free nilpotent.

Corollary 7.15. The Torelli group of a right-angled Artin group is resid- ually p-finite.

Recall that a group G is said to be bi-orderable if it can be endowed with a total order ≤such that ifg ≤h, thenkg ≤kh and gk ≤hk for all g,h, k∈G.

Corollary 7.16. The Torelli group of a right-angled Artin group is bi- orderable.

Our proof follows closely that of Minasyan (see [M]). Both proofs pro- ceed by induction on the rank ofG_Γ. The key observation is that a right- angled Artin group of rank r can be written as an HNN extension of any of its special subgroups of rankr−1. After passing to an HNN extension of a finite group (which is known to be virtually free), Minasyan applies a theorem of Dyer stating that virtually free groups are conjugacy separable (see [Dy1]).

This paper is organized as follows. In Section 3, we introduce the p- centralizer condition which is the analogue of the centralizer condition in [M], and we prove that a group is hereditarily conjugacy p-separable if and only if it is conjugacyp-separable and satisfies the p-centralizer condi- tion. In Section 4, we prove the following analogue of Dyer’s theorem for conjugacyp-separability:

Theorem 1.2. Every extension of a free group by a finitep-group is con- jugacy p-separable.

Section 5 deals with retractions that are key tools in the proof of our main theorem, which is the object of Section 6.

My gratefulness goes to my Ph.D. thesis advisor, Luis Paris, for his trust, time and advice. I am in debt to Ashot Minasyan for pointing out a mistake in an earlier draft of this work, and for directing me to the paper of Aschenbrenner and Friedl [AF2]. I also wish to thank the referee for his (or her) many useful suggestions.

2 HNN extensions

In this section, we recall the definition and basic properties of HNN ex- tensions (see [LS]).

LetH be a group. Then by the notation:

hH,s,... |r,... i,

we mean the group defined by the presentation whose generators are the generators ofH together withs,... and the relators ofH together withr,...

LetH be a group, and letK be a subgroup ofH. TheHNN extension of H relative to K is the group defined by the presentation:

G =h H,t|t⁻¹kt =k,∀k ∈K i.

Every element of G can be written as a wordx₀t^a1x₁· · ·t^anx_n (n ≥ 0, x₀,...,x_n∈H,a₁,...,a_n∈^Z\{0}). Following Minasyan (see [M]), we will say that the word x0t^a1x1· · ·t^anxn is reduced if x0 ∈ H,x1,...,xn−1 ∈ H\K, and x_n ∈ H. Every element of Gcan be written as a reduced word. Note that our definition of a reduced word is stronger than the definition of a reduced word in [LS].

Lemma 2.1(Britton’s Lemma). If a wordx₀t^a1x₁· · ·t^anx_n is reduced with n≥ 1, then x₀t^a1x₁· · ·t^anx_n 6= 1.

Proof: Proved in [LS] (see Theorem IV.2.1).

Lemma 2.2. If x0t^a1x1· · ·t^anxn and y0t^b1y1· · ·t^bmym are reduced words such that x₀t^a1x₁· · ·t^anx_n =y₀t^b1y₁· · ·t^bmy_m, thenm =n anda_i =b_i for all i ∈ {1,...,n}.

Proof: Proved in [LS] (see Lemma IV.2.3).

Acyclic permutationof the wordt^a1x₁· · ·t^anx_nis a wordt^akx_k· · ·t^anx_n t^a1x1· · ·t^ak−1x_k−1 with k ∈ {1,...,n}. A word t^a1x1· · ·t^anxn is said to be cyclically reduced if any cyclic permutation of t^a1x₁· · ·t^anx_n is reduced.

Note that, if t^a1x₁· · ·t^anx_n is reduced and n ≥ 2, then t^a1x₁· · ·t^anx_n is cyclically reduced if and only ifx_n∈H\K. Every element ofGis conjugate to a cyclically reduced word.

Lemma 2.3 (Collins’ Lemma). If g = t^a1x1· · ·t^anxn (n ≥ 1) and h = t^b1y₁· · ·t^bmy_m (m ≥ 1) are cyclically reduced and conjugate, then there exists a cyclic permutation h^∗ of h and an element α ∈ K such that g = αh^∗α⁻¹.

Proof: Proved in [LS] (see Theorem IV.2.5).

Remark 2.4. There exists a natural homomorphism f : G → H, defined byf(h) =h for all h ∈H, and f(t) = 1.

Remark 2.5. Let P be a group and let ϕ : H → P be a homomorphism.

LetQbe the HNN extension of P relative to ϕ(K):

Q= h P,t |t⁻¹ϕ(k)t= ϕ(k),∀k ∈K i.

Thenϕ induces a homomorphismϕ : G → Q, defined byϕ(h) =ϕ(h) for allh ∈ H, andϕ(t) =t.

Lemma 2.6. With the notations of Remark 2.5, ker(ϕ) is the normal closure of ker(ϕ) in G.

Proof: Proved in [M] (see Lemma 7.5).

The following observation is the key in the proof of our main theorem:

Remark 2.7. Let G be a right-angled Artin group of rank r (r ≥ 1). Let H be a special subgroup of G of rank r−1. In other words, there is a partition ofV, V =W∪{t}, such that H = hWi. Then G can be written as the HNN extension ofH relative to the special subgroup K =hlink(t)i ofH:

G =h H,t|t⁻¹kt= k,∀k ∈K i.

3 Hereditary conjugacy p-separability and p-cen- tralizer condition

We start with an observation that the reader has to keep in mind, be- cause it will be used repeatedly in the rest of the paper: ifHandK are two normal subgroups of p-power index in a group G, then H∩K is a normal subgroup ofp-power index inG.

The centralizer conditon was first introduced by Chagas and Zalesskii as a sufficient condition for a conjugacy separable group to be hereditar- ily conjugacy separable (see [CZ]). Thereafter Minasyan showed that this condition is also necessary; that is, a group is hereditarily conjugacy sep- arable if and only if it is conjugacy separable and satisfies the centralizer condition (see [M]). We make the following definition, which naturally generalizes that of [M]:

Definition 3.1. We say that Gsatisfies the p-centralizer condition (pCC) if, for every normal subgroup H of p-power index in G, and for all g ∈G, there exists a normal subgroupK of p-power index inG such thatK < H, and:

C_G/K(ϕ(g)) ⊂ ϕ(C_G(g)H), where ϕ : G → G/K denotes the canonical projection.

We shall show that a group G is hereditarily conjugacy p-separable if and only if it is conjugacy p-separable and satisfies the p-centralizer con- dition (see Proposition 3.6). If H is a subgroup of G, andg ∈ G, we set C_H(g) = {h ∈ H |gh= hg}. For technical reasons, we have to introduce the following definitions:

Definition 3.2. Let G be a group, H be a subgroup of G, andg ∈ G. We say that the pair(H, g) satisfies thep-centralizer condition inG(pCCG) if, for every normal subgroup K of p-power index inG, there exists a normal subgroup L of p-power index in G such thatL < K, and:

C_ϕ(H)(ϕ(g)) ⊂ ϕ(C_H(g)K),

where ϕ : G → G/L denotes the canonical projection. We say that H satisfies the p-centralizer condition in G(pCC_G) if the pair (H, g) satisfies the p-centralizer condition in G for allg ∈G.

If G is a group, H is a subgroup of G, and g ∈ G, we set: g^H = {αgα⁻¹ |α∈H}. In order to prove Proposition 3.6, we need the following

statements, which are the analogues of some statements obtained in [M]

(Lemma 3.4, Corollary 3.5, and Lemma 3.7 respectively):

Lemma 3.3. Let Gbe a group,H be a subgroup ofG, andg ∈G. Suppose that the pair(G, g) satisfies pCC_G, and thatg^Gis finitely p-separable inG.

If C_G(g)H is finitely p-separable in G, then g^H is also finitely p-separable in G.

Proof: Leth∈Gsuch thath /∈g^H. Ifh /∈g^G, then, sinceg^G is finitely p-separable in G, there exists a homomorphismϕ from G onto a finite p- group P such that ϕ(h) ∈/ ϕ(g^G). In particular, ϕ(h) ∈/ ϕ(g^H). Thus we can assume thath∈g^G. Letα∈Gbe such thath=αgα⁻¹. Suppose that C_G(g)∩α⁻¹H 6=∅. Let k ∈C_G(g)∩α⁻¹H. Then αk ∈ H, and h =αgα⁻¹

= αkg(αk)⁻¹ ∈ g^H – which is a contradiction. Thus C_G(g)∩α⁻¹H = ∅, i.e. α⁻¹ ∈/ C_G(g)H. As C_G(g)H is finitely p-separable in G, there exists a normal subgroup K of p-power index in G such thatα⁻¹ ∈/ C_G(g)HK.

Now the conditionpCC_G implies that there exists a normal subgroupLof p-power index inG such thatL < K, and:

C_G/L(ϕ(g)) ⊂ϕ(C_G(g)K),

whereϕ: G→G/Ldenotes the canonical projection. We claim thatϕ(h)

∈/ ϕ(g^H). Indeed, if there is β ∈ H such that ϕ(h) = ϕ(βgβ⁻¹), then ϕ(α⁻¹β)ϕ(g) = ϕ(α⁻¹β)ϕ(β⁻¹hβ) = ϕ(α⁻¹hα)ϕ(α⁻¹β) = ϕ(g)ϕ(α⁻¹β), i.e. ϕ(α⁻¹β)∈C_G/L(ϕ(g)). But thenϕ(α⁻¹)∈C_G/L(ϕ(g))ϕ(H)⊂ϕ(C_G(g) KH). Henceα⁻¹ ∈C_G(g)HKL=C_G(g)HK (because L < K) – which is

a contradiction.

Corollary 3.4. Let G be a conjugacy p-separable group satisfying pCC, and H be a subgroup of G such that C_G(h)H is finitely p-separable in G for all h ∈ H. ThenH is conjugacyp-separable. Moreover, for all h ∈H, h^H is finitely p-separable in G.

Proof: Let h ∈ H. Since G satisfies pCC, the pair (G, h) satisfies pCCG. Since G is conjugacy p-separable, h^G is finitely p-separable in G.

Lemma 3.3 now implies thath^H is finitelyp-separable inG. Thereforeh^H

is finitelyp-separable in H.

Lemma 3.5. Let Gbe a group, H be a subgroup of G, andg ∈ G. Let K be a normal subgroup ofp-power index in G. Ifg^H∩K is finitelyp-separable in G, then there exists a normal subgroup L of p-power index in G such that L < K, and:

C_ϕ(H)(ϕ(g)) ⊂ ϕ(C_H(g)K),

where ϕ : G → G/L denotes the canonical projection.

Proof: Note thatH∩K is of finite index nin H. Actually, H∩K is of p-power index inH(because _H∩K^H ≃ ^KH_K < _K^G), but this is not needed here.

There exist α₁,...,α_n ∈ H such that H = ⊔ⁿ_i=1α_i(H∩K). Up to renum- bering, we can assume that there exists l ∈ {0,...,n} such that α⁻¹_i gα_i ∈ g^H∩K for alli∈ {1,...,l}and α⁻¹_i gαi ∈/ g^H∩K for alli∈ {l+1,...,n}. By the assumptions, there exists a normal subgroupLof p-power index in Gsuch that α⁻¹_i gα_i ∈/ g^H∩KL for all i∈ {l+1,...,n}. Up to replacing L by L∩K, we can assume that L < K. Let ϕ : G → G/L be the canonical projec- tion. Leth∈C_ϕ(H)(ϕ(g)). There exists h∈H such that h =ϕ(h). There existi ∈ {1,...,n}and k ∈H∩K such thath =α_ik. We have ϕ(h⁻¹gh) = ϕ(h)⁻¹ϕ(g)ϕ(h) =ϕ(g). Thush⁻¹gh∈gL. But thenα⁻¹_i gαi=kh⁻¹ghk⁻¹

∈ kgLk⁻¹ = kgk⁻¹L ⊂ g^H∩KL. Therefore i ≤ l. Then there exists β ∈ H∩K such that: α⁻¹_i gα_i = βgβ⁻¹. This is to say that α_iβ ∈C_H(g), and thenh =α_ik= (α_iβ)(β⁻¹k)∈C_H(g)(H∩K)⊂C_H(g)K. We have shown

thatC_ϕ(H)(ϕ(g))⊂ϕ(C_H(g)K).

We are now ready to prove:

Proposition 3.6. A group is hereditarily conjugacyp-separable if and only if it is conjugacy p-separable and satisfies pCC.

Proof: Suppose thatGis conjugacyp-separable and satisfiespCC. Let H be a subnormal subgroup of p-power index in G. Thus H is closed in the pro-p topology on G (because G\H = ∪{gH | g /∈ H}). Let h ∈ H. The set C_G(h)H is a finite union of left cosets modulo H and thus is closed in the pro-p topology on G. Corollary 3.4 now implies that H is conjugacy p-separable. Therefore G is hereditarily conjugacy p-separable.

Suppose now that G is hereditarily conjugacy p-separable. In particular, Gis conjugacy p-separable. We shall show that Gsatisfies pCC. Let g ∈ G. Let K be a normal subgroup of p-power index in G. Let H = Khgi.

Since K < H, [G : H] is a power of p. As _K^G is a finite p-group, every subgroup of it is subnormal. Thus H is subnormal in G. Therefore H is conjugacyp-separable. Note that g^G∩K =g^K = g^H ⊂ H. Asg^H is closed in the pro-ptopology onH, it is closed in the pro-ptopology onG, because the topology induced onH by the pro-p topology onGcoincides with the pro-p topology on H (see, for example, [RZ2], Corollary 5.8). The result

now follows from lemma 3.5.

4 Extensions of free groups by finite p-groups are conjugacy p-separable

We start with an observation that the reader has to keep in mind be- cause it will be used repeatedly in the proof of Theorem 4.2: if ϕ : G

→ H is a homomorphism from a group G to a group H, whose kernel is torsion-free, then the restriction ofϕto any finite subgroup ofGis injective.

We need the following lemma:

Lemma 4.1. Let G = G₁∗ · · · ∗G_n be a free product of n conjugacy p- separable groups G1,...,Gn. Let g, h ∈G\ {1} be two non-trivial elements of finite order inG such thatg ≁h. There exists a homomorphism ϕfrom Gonto a finite p-group P such thatϕ(g) ≁ ϕ(h).

Proof: Sincegis of finite order inG, there existsi∈ {1,...,n}such that gis conjugate to an element of finite order inG_i. Thus we may assume that g belongs to Gi. Similarly, we may assume that there exists j in {1,...,n}

such that h belongs to G_j. Suppose that i 6= j. Let ϕ : G_i → P be a homomorphism fromG_i onto a finite p-groupP such that ϕ(g) 6= 1. Let ϕe : G → P be the natural homomorphism extending ϕ. Then ϕ(g)e ≁ ϕ(h).e Suppose thati=j. Thengandhare not conjugate inG_i – otherwise they would be conjugate in G. Since G_i is conjugacy p-separable, there exists a homomorphismϕ : G_i → P from G_i onto a finite p-group P such that ϕ(g)≁ϕ(h). Letϕe: G→P be defined as above. We haveϕ(g)e ≁ϕ(h).e In Section 4, by a graph, we mean a unoriented graph, possibly with loops or multiple edges.

Recall that agraph of groups is a connected graph Γ = (V, E), together with a functionG which assigns:

• to each vertexv ∈V, a groupGv,

• and to each edge e = {v,w} ∈ E, a group G_e together with two injective homomorphismsα_e : G_e → G_v and β_e : G_e → G_w – we are not assuming thatv 6=w –,

(see [Se], see also [Dy1]). The groups G_v (v ∈ V) are called the vertex groups of Γ, the groups G_e (e ∈ E) are called the edge groups of Γ. The monomorphisms αe and βe (e ∈ E) are called the edge monomorphisms.

The images of the edge groups under the edge monomorphisms are called theedge subgroups.

Choose disjoint presentations Gv = h Xv | Rv i for the vertex groups of Γ. Choose a maximal tree T in Γ. Assign a direction to each edge of Γ. Let {te |e ∈ E} be a set in one-to-one correspondence with the set of edges of Γ, and disjoint from theXv (v ∈ V). The fundamental group of the above graph of groups Γ is the group G_Γ defined by the presentation whose generators are:

X_v (v ∈V), t_e (e∈ E)

(called vertex and edge generators respectively) and whose relations are:

R_v (v ∈V), t_e = 1 (e ∈T),

t_eα_e(g_e)t⁻¹_e =β_e(g_e),∀g_e ∈ G_e (e∈E).

(called vertex, tree, and edge relations respectively). One can prove that this is well-defined – that is, independent of our choice ofT, etc. Note that it suffices to write the edge relations for g_e in a set of generators for G_e.

Convention: The groups G_v (v ∈ V) and G_e (e ∈ E) will be regarded as subgroups ofG_Γ.

Let{Γi}i∈I be a collection of connected and pairwise disjoint subgraphs of Γ. We may define a graph of groups Γ^∗ from Γ by contracting Γ_i to a point for all i ∈ I, as follows. The graph Γ^∗ is obtained from Γ by con- tracting Γ_i to a pointp_i for all i∈ I. The function G^∗ is obtained from G by using the fundamental group of Γ_i for the vertex group at p_i, and by composing the edge monomorphisms of Γ by the natural inclusions of the vertex groups of Γ_i into the fundamental group of Γ_i, if necessary. Clearly, GΓ is isomorphic to the fundamental groupGΓ^∗ of Γ^∗.

Ifπ : G_Γ→H is a homomorphism fromG_Γ to a groupH, such that the restriction ofπ to each edge subgroup of Γ is injective, then we may define a graph of groups Γ^′ from Γ by composing with π, as follows. The vertex set of Γ^′ isV, and the edge set of Γ^′ isE. The vertex groups of Γ^′ are the groups G^′_v =π(G_v) (v ∈ V), and the edge groups of Γ^′ are the groups G^′_e

= G_e (e ∈ E). The edge monomorphisms are the monomorphisms α^′_e = π◦α_eandβ_e^′ =π◦β_e (e∈E). PresentG_Γ andG_Γ′ using the same symbols for edge generators and with the same choice of maximal tree. There exist two homomorphisms, π_V : G_Γ → G_Γ′ and π_E : G_Γ′ → H, such that the

diagram:

G_Γ

πV

π //H

G_Γ′

πE

==

||

||

||

||

commutes, and that the restriction of π_E to each vertex group of G_Γ′ is injective. The homomorphismπV is given by:

(π_V)_|Gv =π_|Gv,∀ v ∈V, πV(te) =te,∀e∈ E.

And the homomorphismπ_E is given by:

(π_E)_|G′

v = (id_H)_|G′

v,∀ v ∈V, π_E(t_e) = π(t_e), ∀e∈ E.

In [Dy1], Dyer proved that every extension of a free group by a finite group is conjugacy separable. The following theorem is the analogue of Dyer’s theorem for conjugacy p-separability.

Theorem 4.2. Every extension of a free group by a finitep-group is con- jugacy p-separable.

Proof: Our proof is inspired by that of Dyer (see [Dy1]). Let Gbe an extension of a free group by a finitep-group. In other words, there exists a short exact sequence:

1 ^//F ^//G ^{π /}/P ^//1,

whereF is a free group, and P is a finitep-group. Letg ∈ G. Leth ∈G such thatg ≁h.

Step 1: We show that we may assume that G satisfies a short exact sequence:

1 ^//F ^//G ^{π //}C_p^{n //}1,

where F is a free group, n ≥1, C_pⁿ denotes the cyclic group of order pⁿ, and π(g) =π(h).

SinceGis an extension of a free group by a finitep-group,Gis residually p-finite by [G], Lemma 1.5. Therefore, if g = 1, then g^G = {1} is finitely p-separable inG. On the other hand, if g is of infinite order inG, theng^G is finitelyp-separable inGby [I], Proposition 5. Therefore we may assume

thatg6= 1 and thatgis of finite order inG. Similarly, we may assume that h6= 1 and thath is of finite order inG. Ifπ(g) andπ(h) are not conjugate inP, we are done. Thus, up to replacingh by a conjugate of itself, we may assume that π(g) = π(h). Since ker(π) = F is torsion-free, g and h have the same orderpⁿ (n ∈^N∗). LetH =Fhgi. Note that H is a subgroup of p-power index in G, and that g and h belong to H. As ^G_F = P is a finite p-group, every subgroup of it is subnormal. Thus H is subnormal in G.

Then we may replaceG byH, by [I], Proposition 4¹, so as to assume that Gsatisfies the short exact sequence:

1 ^//F ^//G ^{π /}/C_p^{n /}/1.

Now, Gis the fundamental group of a graph of groups Γ, whose vertex groups are all finite groups, by [S], Theorem. As π_|Gv is injective for all v

∈ V,G_v is isomorphic to a subgroup ofC_pⁿ for allv ∈ V. From now on, the groupsG_v (v ∈V) will be regarded as subgroups of C_pⁿ.

Step 2: We show that we may assume that all edge groups are non- trivial, that if two different vertices are connected by an edge, then the corresponding edge group is a proper subgroup of C_pⁿ, and that g and h belong to two different vertex groups.

First, we show that we may assume that all edge groups are non-trivial.

Indeed, Let Γ₀be the subgraph of Γ whose vertices are all the vertices of Γ, and whose edges are the edges of Γ for which the edge group is non-trivial.

Let Γ1,...,Γr be the connected components of Γ0. Let Γ^∗ be the graph of groups obtained from Γ by contracting Γ_i to a point for all i ∈ {1,...,r}.

LetT be a maximal tree of Γ^∗. ThenG is isomorphic to the fundamental group G^∗ of Γ^∗. Observe that G^∗ is the free product of the free group on {te|e∈E\T}and the fundamental groups of the Γ_i (i∈ {1,...,r}). Thus, it suffices to consider the case where Γ = Γ_i (i∈ {1,...,r}), by Lemma 4.1.

Since each Γ_i (i∈ {1,...,r}) is a graph of groups whose edge groups are all non-trivial, the first part of the assertion is proved.

Now, we show that we may assume that if two different vertices are connected by an edge, then the corresponding edge group is a proper sub- group of C_pⁿ. Indeed, let Γ₀ be the subgraph of Γ whose vertices are all

1Strictly speaking, it follows from the proof of [I], Proposition 4, that, if there exists a homomorphismϕ : H →P fromH onto a finitep-group P such that ϕ(g) ≁ ϕ(h), then there exists a homomorphismψ: G→QfromGonto a finitep-groupQsuch that ψ(g)≁ψ(h). The exact statement of [I], Proposition 4, is slightly different.

the vertices of Γ, and whose edges are the edges of Γ for which the edge group is isomorphic toC_pⁿ. Let Γ₁,...,Γ_r be the connected components of Γ₀. Choose a maximal tree T_i in Γ_i, for all i ∈ {1,...,r}. Let Γ^∗ be the graph of groups obtained from Γ by contracting Ti to a point for all i ∈ {1,...,r}. Then G is isomorphic to the fundamental groupG^∗ of Γ^∗. Note that a vertex group of Γ^∗ is either a vertex group of Γ, or the fundamental group of T_i, for some i ∈ {1,...,r}, in which case it is isomorphic to C_pⁿ (because each T_i (i ∈ {1,...,r}) is a tree of groups whose vertex and edge groups are all equal to C_pⁿ). Thus, we may replace Γ by Γ^∗, so that the second part of the assertion is proved.

Since g is of finite order in G, there exists a vertex v of Γ, an element g₀ of finite order in the vertex group G_v ofv, and an element α ofG such that g = αg₀α⁻¹. Similarly, there exists a vertex w of Γ, an element h₀ of finite order in the vertex group G_w of w, and an element β of G such thath =βh₀β⁻¹. As C_pⁿ is abelian, we have: π(g₀) = π(h₀). Thus, up to replacingg byg₀ andh byh₀, we may assume thatg belongs toG_v, andh belongs toG_w. Since π_|Gv is injective, and π(g) = π(h), we havev 6= w.

Step 3: We show that we may assume that Γ has exactly two vertices, and that all edges join these two vertices.

Indeed, choose a maximal tree T in Γ. There is a path P inT joining v to w. Choose an edgee on this path. ThenT \ {e} is the disjoint union of two trees, T_v and T_w – with v ∈ T_v and w ∈ T_w. Let Γ_v be the full subgraph of Γ generated by the vertices ofT_v, and Γ_w be the full subgraph of Γ generated by the vertices ofTw. Let Γ^∗be the graph of groups obtained from Γ by contracting Γ_v to a pointv^∗ and Γ_w to a pointw^∗. Observe that Γ^∗ has exactly two vertices and that all edges join these two vertices. The vertex groups of Γ^∗ are the fundamental groups of Γ_v and Γ_w, respectively.

The edge groups of Γ^∗ are non-trivial proper subgroups of C_pⁿ. And G is isomorphic to the fundamental groupG^∗ of Γ^∗. Now, since the restriction of π to each edge subgroup of Γ^∗ is injective, we may define a graph of groups Γ^′ from Γ^∗ by composing withπ, as described above. Denote byG^′ the fundamental group of Γ^′. There exist two homomorphisms, π_V : G → G^′ and πE : G^′ → Cpⁿ, such that the diagram:

G

πV

π //C_pⁿ

G^′

πE

==

{{ {{ {{ {

commutes, and that the restriction of π_E to each vertex group of Γ^′ is

injective. Consequently, ker(πE) is free by [Se], II, 2.6., Lemma 8.

Setg^′ =π_V(g), andh^′ =π_V(h). Asg^′ andh^′ have orderpⁿ, the vertex groups of Γ^′ are equal toC_pⁿ. The edge groups of Γ^′ are non-trivial proper subgroups of Cpⁿ. Observe that g^′ and h^′ belong to two different vertex groups, and thatg^′ (resp. h^′) is not conjugate to an element of one of the edge groups. Let e be an edge of Γ^′. Theng^′ and h^′ are not conjugate in G^′_v∗_G^′_eG^′_w, by [MKS], Theorem 4.6. Observe thatG^′ is an HNN extension (in the general sense) ofG^′_v∗_G^′_eG^′_w with stable letterst_a(a∈E\ {e}), and associated subgroupsα^′_a(G^′_a) andβ_a^′(G^′_a) (a∈ E\ {e}). Thereforeg^′ and h^′ are not conjugate in G^′ (see, for example, [Dy2], Theorem 3). Thus, we may replace Γ by Γ^′,Gby G^′,gby g^′, and hby h^′, so as to assume that Γ has two vertices and that all edges join these two vertices.

Step 4: We show that we may assume that Γ has at most two edges.

Suppose that Γ has more than two edges. Choose a maximal treeT in Γ – that is, an edge of Γ. PresentG_v =hg|g^pn = 1i,G_w =hh |h^pn = 1i, and Gas described above. Choose an edgee∈E\T.

•

v •w

T

e

The edge relations corresponding toecan be reduced to the following:

t_eα_e(g_e)t⁻¹_e =β_e(g_e),

whereg_e is a generator ofG_e. Letp^sbe the order of G_e(s∈ {1,...,n−1}).

Then α_e(g_e) generates a subgroup of order p^s of G_v. But there exists a unique subgroup of order p^s in G_v; it is cyclic, generated by g^pr, wherer

= n−s. Thus, up to replacing g_e by the preimage of g^pr under α_e, we may assume thatαe(ge) =g^pr. There exists k∈ ^N, such that pand k are coprime, and thatβ_e(g_e) =h^kpr. Therefore the edge relation corresponding to ecan be written:

teg^prt⁻¹_e =h^kpr,

where r ∈ {1,...,n−1}, k ∈ ^N, and p and k are coprime. Now, since π : G→ C_pⁿ satisfies π(g) = π(h), we have: π(g)^pr =π(h)^kpr =π(g)^kpr, and then π(g)^(k−1)pr = 1 (in Cpⁿ). As π(g) has order pⁿ in Cpⁿ, we deduce thatp^n−r dividesk−1. There existsa ∈^Z such that k= ap^n−r + 1. We conclude that the edge relation corresponding toecan be written:

teg^prt⁻¹_e =h^pr, wherer ∈ {1,...,n−1}.

LetH be the normal subgroup ofG generated by the elements:

g,h,t_a (a ∈E\ {e}), t^pe.

ThenHhas index p inG, andgandhbelong toH. Thus we may replaceG byHby [I], Proposition 4. LetG₀be the fundamental group of the graph of groups Γ\{e}. SetG₀=hX0|R₀i, where the presentation is as fundamental group of the graph of groups Γ\ {e}. SetG_i =tⁱ_eG₀t⁻ⁱ_e =hXi |R_ii, for all i∈ {1,...,p−1}. Clearly{1,t_e,...,t^p−1e }is a Schreier transversal forH inG.

The Reidemeister-Schreier method yields the presentation:

H =h X₀,X₁,...,X_p−1,u |R₀,R₁,...,R_p−1,g^p₁^r =h^p₀^r,g^p₂^r = h^p₁^r,..., g_p−1^pr

=h^p_p−2^r ,ug₀^pru⁻¹ =h^p_p−1^r i,

whereu=t^p_e,g_i=tⁱ_egt⁻ⁱ_e (i∈ {0,...,p−1}), andh_j =t^j_eht^−j_e (j∈ {0,...,p−1}).

Replace g by g0, and h by h1. Observe that H is the fundamental group of a graph of groups Γ, as follows. The graphe Γ has 2pe vertices, say v₀, w₀, v₁, w₁,...,v_p−1, w_p−1, and p|E| edges. Let Γe_i be the full subgraph of eΓ generated by{vi, w_i} for all i∈ {0,...,p−1}. Then Γe_i is isomorphic to Γ\ {e}. There is one edge joining w₀ to v₁, one edge joining w₁ to v₂,..., one edge joining w_p−2 to v_p−1, and one edge joining v₀ to w_p−1, and the edge groups associated to these egdes are isomorphic to G_e. Note that g belongs to the vertex group ofv₀ andh belongs to the vertex group of w₁.

v•₀ w•₀

v•₁ w•₁

v•₂ w•₂

v•₃ w•₃

v•₄ w•₄

T T T T T

Let Γ^∗be the graph of groups obtained fromeΓ by contractingΓei to a point for alli∈ {1,...,p−1}. ThenGis isomorphic to the fundamental group of Γ^∗. The graph Γ^∗ hasp vertices, sayv₀,...,v_p−1. There is one edge joining v₀ to v₁, one edge joining v₁ to v₂,..., one edge joining v_p−2 to v_p−1, and one edge joiningv₀ to v_p−1, and the edge groups associated to these edges are all isomorphic toG_e. Note that gbelongs to the vertex group ofv₀ and

h belongs to the vertex group ofv1. v•0

v•1

v•2

v•3

v•4

Let T be the maximal tree T = v₀v₁· · ·v_p−2v_p−1. Then T \ {v0v₁} is the disjoint union of two trees : v0 and v1v2· · ·vp−2vp−1. Set Γ^∗₁ = v0 and Γ^∗₂

= v₁v₂· · ·v_p−2v_p−1. Let Λ be the graph of groups obtained from Γ^∗ by contracting Γ^∗_i to a point for all i ∈ {1,2}. Let Λ^′ be the graph of groups obtained from Λ by composing withπ. As in Step 3, we may replace Γ by Λ^′, so as to assume that Γ has two vertices and two edges joining these two vertices.

End of the proof: Present G_v =hg|g^pn= 1i, G_w =hh|h^pn = 1i, and Gas described above. There are two cases:

Case 1: Γ has one edge.

In this case,Gis an amalgamated product of two finite abelianp-groups.

SinceGis residuallyp-finite, Gis conjugacyp-separable by [I], Theorem 2.

Thus, there exists a homomorphismϕfromGonto a finitep-groupP such thatϕ(g) ≁ ϕ(h).

Case 2: Γ has two edges.

We have:

G=h g,h,t|g^pn = 1,h^pn = 1,g^pr = h^pr,tg^pst⁻¹ =h^ps i, wherer ∈ {1,...,n−1},s∈ {1,...,n−1}. Let:

A=C_pⁿ×C_p^s × · · · ×C_p^s

| {z } p^r−1

×C_p^r

Set m = p^r+ 1. Present each factor of this product in the natural way, using generators x1,...,xm respectively. Let α be the automorphism of A defined by:

α(x₁) =x₁x₂x_m

α(xi) = xi+1,∀i∈ {2,...,m−2}

α(x_m−1) = (x₂· · ·x_m−1)⁻¹ α(x_m) =x_m

It is easily seen thatα has orderm−1 = p^r. We have:

α⁰(x₁) =x₁, α¹(x1) =x1x2xm, α²(x₁) =x₁x₂x₃x²_m,

...

α^m−2(x1) =x1x2x3· · ·xm−1x^m−2_m .

LetB =A⋊hαi be the semidirect product of Aby hαi. Note that B is a finite p-group. Letϕ: G →B be the homomorphism defined by:

ϕ(g) = x₁, ϕ(h) = x₁x_m,

ϕ(t) = α.

Observe that the conjugacy class of ϕ(g) in B is ϕ(g)^B = {α^k(x₁) | k ∈ {0,...,m−2}}. Thus,ϕ(g) and ϕ(h) are not conjugate inB. Corollary 4.3. Let P be a finite p-group. Let A be a subgroup of P. Let Qbe the HNN extension of P relative to A:

Q = h P, t| t⁻¹at=a, ∀ a∈ A i.

ThenQ is hereditarily conjugacy p-separable.

Proof: Let R be an arbitrary subgroup of Q. Let f : Q → P be the natural homomorphism. We have ker(f)∩P ={1}. Therefore ker(f) is free by [KS], Theorem 6. That is, Q is an extension of a free group by a finite p-group. Thus R is itself an extension of a free group by a finite p-group. ThereforeR is conjugacy p-separable by Theorem 4.2.

Remark 4.4. It is known that a fundamental group of a graph of groups, whose vertex groups are all finitep-groups is residually p-finite if and only if it is an extension of a free group by a finite p-group (see, for example, [AF1], Lemma 3.3). Thus, as an immediate consequence of Theorem 4.2, we have that a fundamental group of a graph of groups whose vertex groups are all finitep-groups is conjugacyp-separable if and only if it is residually p-finite.

5 Retractions

In this section, we shall prove several results on retractions that will allow us to control the growth of the intersection of Lemma 6.5 in finite p-group quotients ofG_Γ.

Definition 5.1. Let G be a group, and H be a subgroup of G. We say that H is a retract of G if there exists a homomorphism ρ_H : G→ G such that ρ_H(G) =H andρ_H(h) =h for allh ∈ H. The homomorphism ρ_H is called a retraction ofG onto H.

Remark 5.2. IfGis a right-angled Artin group, and H =hWi is a special subgroup ofG, then H is a retract ofG. A retraction of GontoH is given by:

ρH(v) =

v if v∈W 1 ifv∈V \W

Lemma 5.3. Let Gbe a group, andH be a subgroup ofG. Suppose thatH is a retract ofG. Let ρH be a retraction of G onto H. Let N be a normal subgroup of G such that ρ_H(N) ⊂ N. Then ρ_H induces a retraction ρ_H : G/N → G/N of G/N onto the canonical image H of H in G/N, defined by: ρ_H(gN) = ρ_H(g)N for all gN ∈ G/N.

Proof : Proved in [M] (see Lemma 4.1).

Remark 5.4. LetGbe a group, and letH,H^′ be two subgroups ofG. Sup- pose that H and H^′ are retracts of G and that the corresponding retrac- tions,ρH and ρ_H^′, commute. ThenρH(H^′) =ρ_H^′(H) =H∩H^′. Moreover H∩H^′ is a retract of G. A retraction of G onto H∩H^′ is given by ρ_H∩H^′

=ρ_H ◦ρ_H′ =ρ_H′◦ρ_H.

Proposition 5.5. Let G be a group and H₁,...,H_n be n subgroups of G.

Suppose that H₁,...,H_n are retracts ofG and that the corresponding retrac- tions pairwise commute. Then, for every normal subgroup K of p-power index in G, there exists a normal subgroup N of p-power index in G such that N < K andρ_Hi(N)⊂N for all i∈ {1,...,n}. Consequently, for every i ∈ {1,...,n}, the retraction ρ_Hi induces a retraction ρ_Hi of G/N onto the canonical imageH_i of H_i in G/N.

Proof: Proved in [M] (see Proposition 4.3 and Remark 4.4).

Lemma 5.6. LetGbe a group, and letH,H^′ be two subgroups ofG. Sup- pose thatHandH^′are retracts ofGand that the corresponding retractions, ρ_H and ρ_H′, commute. Let N be a normal subgroup of G such that ρ_H(N)

⊂N and ρ_H′(N) ⊂ N. Then ϕ(H∩H^′) = ϕ(H)∩ϕ(H^′), where ϕ : G → G/N denotes the canonical projection.

Proof: Proved in [M] (see Lemma 4.5).

The next statement is the analogue of Lemma 4.6 in [M]:

Corollary 5.7. Let Gbe a group and H1,...,Hnbe nsubgroups ofG. Sup- pose thatH₁,...,H_n are retracts ofGand that the corresponding retractions ρ_H1,...,ρ_Hn pairwise commute. Then, for every normal subgroup K of p- power index in G, there exists a normal subgroup N of p-power index inG such thatN < K andρ_Hi(N) ⊂N, for alli∈ {1,...,n}. Moreover, ifϕ: G

→ G/N denotes the canonical projection, then ϕ(T_n

i=1H_i) =T_n

i=1ϕ(H_i).

Proof: By Proposition 5.5, there exists a normal subgroupN ofp-power index in G such that N < K and ρ_Hi(N) ⊂ N for all i ∈ {1,...,n}. We denote byϕ: G → G/N the canonical projection. We argue by induction on k ∈ {1,...,n} to prove that ϕ(Tk

i=1Hi) = Tk

i=1ϕ(Hi). If k = 1, then the result is trivial. Thus we can assume that k ≥ 2 and that the result has been proved fork−1. We setH^′ =T_k−1

i=1 H_i. By Remark 5.4, H^′ is a retract ofG. A retraction ofGonto H^′ is given by ρ_H′ =ρ_H1◦...◦ρ_Hk−1. We have:

ρ_H′(N) =ρ_H1(...(ρ_Hk−2(ρ_Hk−1(N))))

⊂ρ_H1(...(ρ_Hk−2(N))) ...

⊂ρ_H1(N)

⊂N.

The retractionsρ_H^′ andρ_Hk commute, so we can apply Lemma 5.6 to con- clude that ϕ(H^′ ∩H_k) = ϕ(H^′)∩ϕ(H_k). By the induction hypothesis, ϕ(H^′) =Tk−1

i=1 ϕ(H_i). Finally ϕ(Tk

i=1H_i) =Tk

i=1ϕ(H_i).

In the following lemmas, Gis a group, and A, B are two subgroups of G. We assume thatA and B are retracts of Gand that the corresponding retractions,ρ_Aand ρ_B, commute.

Lemma 5.8. Let x,y ∈G. We set α =ρ_A(ρ_B(x)x⁻¹)xρ_B(x⁻¹) (∈ AxB) and β =ρ_A(ρ_B(y)y⁻¹)yρ_B(y⁻¹) (∈ AyB). Then the following are equiva- lent:

1. y ∈ AxB, 2. β ∈α^A∩B.

Proof: Proved in [M] (see Lemma 5.1).

Lemma 5.9. Let x ∈ G. We set α = ρ_A(ρ_B(x)x⁻¹)xρ_B(x⁻¹) (∈ AxB) andγ =ρ_A(ρ_B(x)x⁻¹) (∈ A). Then we have:

A∩xBx⁻¹ =γ⁻¹C_A∩B(α)γ.

Proof: Proved in [M] (see Lemma 5.2).

The next five statements are the analogues of some statements in [M]

(Lemma 5.3, Corollary 5.4, Lemma 5.5, Lemma 5.6, and Lemma 5.7 re- spectively):

Lemma 5.10. Let x ∈G. We set: α =ρ_A(ρ_B(x)x⁻¹)xρ_B(x⁻¹) (∈ AxB).

If α^A∩B is finitely p-separable in G, then AxB is also finitely p-separable in G.

Proof: Lety∈Gsuch thaty /∈AxB. We setβ=ρ_A(ρ_B(y)y⁻¹)yρ_B(y⁻¹).

By Lemma 5.8, we have β /∈ α^A∩B. Since α^A∩B is finitely p-separable in G, there exists a normal subgroupK of p-power index in Gsuch that, ifψ : G → G/K denotes the canonical projection, we have: ψ(β) ∈/ ψ(α^A∩B)

= ψ(α)^ψ(A∩B). By Corollary 5.7, there exists a normal subgroup N of p-power index in G such that N < K, ρ_A(N) ⊂ N, ρ_B(N) ⊂ N and, if ϕ : G → G/N denotes the canonical projection, then: ϕ(A ∩B) = ϕ(A)∩ϕ(B). Assume that ϕ(β) ∈ ϕ(α)^ϕ(A∩B). Let g ∈ A∩B be such that ϕ(β) = ϕ(g)ϕ(α)ϕ(g)⁻¹. Then β ∈ gαg⁻¹N. SinceN < K, we ob- tain β ∈ gαg⁻¹K. But this contradicts the fact that ψ(β) ∈/ ψ(α)^ψ(A∩B). Therefore we have: ϕ(β) ∈/ ϕ(α)^ϕ(A∩B) i.e. ϕ(β) ∈/ ϕ(α)^{ϕ(A)∩ϕ(B)}. We set A = ϕ(A) and B = ϕ(B). By Lemma 5.3, ρ_A induces a retraction ρ_A of G/N onto A and ρ_B induces a retraction ρ_B of G/N onto B. We set: x

=ϕ(x) and y =ϕ(y). We have: ϕ(α) = ρ_A(ρ_B(x)x⁻¹)xρ_B(x⁻¹) and ϕ(β)

= ρ_A(ρ_B(y)y⁻¹)yρ_B(y⁻¹). By Lemma 5.8, we have y /∈ AxB i.e. ϕ(y) ∈/

ϕ(AxB).

Corollary 5.11. Let G be a group, and A, B be two subgroups of G.

Suppose that G is residually p-finite. If A and B are retracts of G, such that the corresponding retractions commute, thenAB is finitelyp-separable in G.

Proof: We apply Lemma 5.10 to x= 1.

Lemma 5.12. Let G be a group, and A be a subgroup of G. Suppose that Gis residually p-finite and that A is a retract of G. Then if a subset S of Ais closed in the pro-ptopology on A, it is also closed in the pro-ptopology on G.

Proof: We denote byS the closure ofS inG– equipped with the pro-p topology. We shall show that S ⊂ S. By Corollary 5.11, Ais closed in G.

ThereforeS ⊂A. Let a∈ G\S. We can assume that a∈ A. There exists a homomorphismψfromAonto a finitep-group P such thatψ(a)∈/ ψ(S).

We set: ϕ=ψ◦ρ_A. We have: ϕ(a) =ψ(a) ∈/ ψ(S) =ϕ(S). Thena /∈S.