Virtually free groups and integral representations

JID:YJABR

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.1 (1-13) Journal of Algebra ••• (••••) •••–•••

Contents lists available at ScienceDirect

Journal of Algebra www.elsevier.com/locate/jalgebra

Virtually free groups and integral representations Igor Lima a,1 , Pavel Zalesskii b,∗,2 a

Universidade Federal de Goiás, IMTec – Regional Catalão, Av. Dr. Lamartine P. de Avelar, 1120, Setor Universitário, Catalão, GO, 75704-020, Brazil b Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília, DF, Brazil

a r t i c l e

i n f o

Article history: Received 21 May 2016 Available online xxxx Communicated by N. Andruskiewitsch, A. Elduque, E. Khukhro and I. Shestakov

a b s t r a c t Let G = F H be a semidirect product of a free group F and a finite group H. The H-module structure of the abelianization F ab is described in terms of splitting of G as the fundamental group of a graph of finite groups. © 2017 Elsevier Inc. All rights reserved.

Keywords: Virtually free groups Integral representations Bass–Serre theory of groups acting on trees

1. Introduction Let G = F  H be a semidirect product of a free group F and a finite group H. Then the action of H on F induces the action of H on the abelianization F ab of F that turns F ab into an Z[H]-module. What is the structure of this module? Is it decomposable and what are indecomposable components of it? The questions are very natural, nevertheless quite surprisingly this was never studied before. * Corresponding author. 1 2

E-mail addresses: [email protected] (I. Lima), [email protected] (P. Zalesskii). Partially supported by bolsa de pós-doutorado chamada 12/2014 FAPEG/CAPES. Partially supported by PQ n. 301744/2013-0 CNPq.

http://dx.doi.org/10.1016/j.jalgebra.2017.02.015 0021-8693/© 2017 Elsevier Inc. All rights reserved.

JID:YJABR 2

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.2 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

The objective of this paper is to try to answer these questions, but then it has to be explained in what terms. By the result of Karras, Pietrowski, Solitar, Cohen and Scott (see (Chapter IV, Theorem 3.2 [3]), [5,2,8]) G = π1 (G, Γ) is the fundamental group of a graph of finite groups. We shall describe the structure of Z[H]-module F ab in terms of the graph of groups (G, Γ). Note that the finite group H is conjugate to one of the vertex group (see Chapter II Theorem 3.5 in [3]), so we may assume that H = G(v0 ) is a vertex group for some vertex v0 of Γ. Let ϕ : G → H be an epimorphism such that ϕ|H = id with kernel F . Let T be a maximal subtree of Γ. For e ∈ star(v0 ) let Te be a subtree growing from v0 with trunk e. The fundamental group π1 (G, Te ) of the subgraph of groups restricted to Te is a subgroup of G = π1 (G, Γ) and we denote by Fe the kernel of the restriction ϕ|π1 (G,Te ) . Let Me = Feab be the abelianization of Fe regarded as Z[H]-module. Then the main result of the section is the following Theorem 1.1. The abelianization F ab is an extension MT = ⊕e∈star(v0 ) Me by MB = ˜ e , where M ˜ e = Z[H/ϕ(Ge )] for e ∈ / T. ⊕e∈Γ\T M The extension might not split as Example 3.6 shows. Note also that this decomposition is the best possible as the modules Me can be indecomposable (see Example 3.5). Using Theorem 1.1 we can give a criteria for F ab to be a permutation module if H is a p-group. Theorem 1.2. Suppose H is a finite p-group. Then F ab is a permutation Z[H]-module if and only if G is an HNN-extension of H (with finitely many stable letters). Note that Theorem 1.1 also holds for finitely generated virtually free pro-p groups. The proof works mutatis mutandis using the result of Herfort and the second author [4] instead of the result of Karras–Pietrowski–Solitar and the pro-p version of Bass–Serre theory of groups acting on trees from [7]. For the case G free pro-p by cyclic the result was obtained in [6]. A finitely generated pro-p version of Theorem 1.2 is the subject of Theorem 4.9 [4] whose new proof can be obtained from our proof of Theorem 1.2 making all appropriate changes. 2. Preliminaries In this paper we will use the following notation. For A ≤ T ≤ H JT (H/A) = ker(Z[H/A] → Z[H/T ]). 2.1. Definitions and notation for a graph In this subsection we set some definitions and notations for a graph that will be used in the rest of the paper. A graph Γ (oriented) consists of: a non-empty set V = V (Γ)

JID:YJABR

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.3 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

3

called vertices, a set E = E(Γ) called oriented edges, two maps d0 , d1 : E(Γ) → V , where d0 (e) and d1 (e) are initial and terminal vertices of e, respectively. The star of a vertex v −1 is defined as star(v) = d−1 0 (v) ∪ d1 (v). A maximal subtree T of a graph Γ is a subgraph of Γ which is a tree and contains all vertices of Γ. A group G is said to act on a graph Γ if G acts on Γ, leaves V (Γ) invariant and preserves the incident maps d0 , d1 . This action is called without inversion. We can define then the quotient graph Γ/G of Γ by G in the usual way. 2.2. The fundamental group of a graph of groups A graph of groups (G, Γ) consists of a connected graph Γ equipped with: a group G(v) for each v ∈ V (Γ), and for each e ∈ E(Γ), a group G(e) together with a monomorphisms ∂0,e , ∂1,e : G(e) → G(d0 (e)), G(d1 (e)). If Γ is a tree, then we say that (G, Γ) is a tree of groups. Let T be a maximal subtree of Γ. Let F (G, Γ) = ∗v∈V (Γ) G(v) ∗ F (E \ T ), where F (E \ T ) is a free group on E \ T . The fundamental group π1 (G, Γ, T ) of the graph of groups (G, Γ) at T consists of the quotient of F (G, Γ) by the normal subgroup generated by ∂0,e (ge )−1 ∂1,e (ge ) for e ∈ T and by ∂0,e (ge )−1 e∂1,e (ge )e−1 , for e ∈ E \ T , ge ∈ G(e). This definition is independent of the choice of a maximal subtree T (see Proposition 20, [9]), hence we will use the notation π1 (G, Γ). We mention two particular cases which are related to well known constructions of combinatorial group theory. Let (G, Γ) be a graph of groups, where Γ has one edge and two vertices. Let G1 , G2 be the vertex groups and let H be the edge group. Then π1 (G, Γ) ∼ = G1 ∗H G2 is a free product with amalgamation, see Example 1 page 43 in [9]. Suppose now that (G, Γ) is a graph of groups, and Γ has just one vertex, that is, Γ is a bouquet. Let H be the vertex group. Then π1 (G, Γ) ∼ = HN N (H, G(e), te , e ∈ E), is an HNN-extension with {G(e), e ∈ E} the set of associated subgroups and {te | e ∈ E} the set of stable letters (cf. Example 3 page 43 in [9]). In fact, the fundamental group π1 (G, Γ) is obtained successively by amalgamated free products followed by HNN-extensions. For a group G acting on a tree D there exists a graph of groups (G, Γ) whose fundamental group π1 (G, Γ) is isomorphic to G (where each vertex (or edge) group of (G, Γ) is the stabilizer in G of a certain vertex (or edge) of D). The converse is also true, namely π1 (G, Γ) acts on a standard tree S (with each stabilizer of a vertex (or edge) in G conjugate to the vertex (or edge) group of (G, Γ)) such that S/π1 (G, Γ) = Γ. These results are known as the fundamental theorem of Bass–Serre theory, see Th. 13 in [9]. A fictitious edge of a graph of groups (G, Γ) is an edge e which is not a loop and such that one of the edge maps ∂i,e is an isomorphism. A finite graph of groups (G, Γ) is said to be reduced, if it does not have fictitious edges. Any finite graph of groups can be transformed into a reduced finite graph of groups by the following procedure: If {e} is a geometric edge which is not a loop, we can remove {e}

JID:YJABR

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.4 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

4

from the edge set of Γ, and identify d0 (e) and d1 (e) in a new vertex y. Let Γ be the finite graph given by V (Γ ) = {y} V (Γ) \{d0 (e), d1 (e)} and E(Γ ) = E(Γ) \{e}, and let (G  , Γ ) denote the finite graph of groups based on Γ given by G (y) = G(d1 (e)) if ∂0,e is an isomorphism, and G (y) = G(d0 (e)) if ∂0,e is not an isomorphism; so we have the natural embeddings ηi : G(di (e)) −→ G (y) and the natural epimorphism α : Γ −→ Γ and for any edge f  = α(f ) of Γ such that di (f  ) = y we define ∂i,f  = ηi ∂i,f . This procedure can be continued until ∂0,e , ∂1,e are not surjective for all edges not defining loops. The resulting finite graph of groups (Grd , Γrd ) is reduced and has the same fundamental group as before the reduction process. Lemma 2.1. A reduced graph of finite groups (G, Γ) has only one vertex if and only if G = π1 (G, Γ) is an HNN-extension of the vertex group. Proof. ‘If’ is obvious, we prove the converse. Suppose π1 (G, Γ) is an HNN-extension of the vertex group HN N (G(v), G(e), te ). Suppose on the contrary that Γ has another vertex w. Since G(w) is finite, G(w) is conjugate to a subgroup of G(v). Consider a standard graph S associated to G = π1 (G, Γ) on which G acts with S/G = Γ. Then G(w) stabilizes a certain vertex w0 and also stabilizes a certain vertex v0 , where w = Gw0 and v = Gv0 . It follows that G(w) stabilizes the path [w0 , v0 ]. Let e0 be the first edge of this path with Gd0 (e0 ) = Gd1 (e0 ) (it exists since v = w). Then the stabilizer of the edge e0 is G(w) and so putting e = Ge0 ∈ Γ we have G(e) = G(w). This contradicts the reducibility of (G, Γ). 2 3. The general case According to the Bass–Serre theory of groups acting on trees G acts on a tree S with S/G = Γ with vertex and edge stabilizers conjugate to corresponding vertex and edge groups G(v) and G(e). Put E = E(Γ), V = V (Γ). By the result of Chiswell (see Theorem 5.4 in Chapter I of [3], [1]) S gives rise to a short exact sequence of permutation modules 0

⊕e∈E Z[G/G(e)]

⊕v∈V Z[G/G(v)]

Z

0.

Applying ⊗Z[F ] Z to it we obtain the following long exact sequence of Z[G]-modules ...0

H1 (F, Z)

ι1

⊕e∈E Z[G/G(e)F ]

π1

⊕v∈V Z[G/G(v)F ]

Z

0.

(1) Lemma 3.1. Suppose Γ has just one vertex, i.e. suppose that G = HN N (H, G(e), E(Γ)) is an HNN-extension with the set of stable letters E(Γ). Then F ab is isomorphic, as Z[H]-module, to a permutation module ⊕e∈E(Γ) Z[H/G(e)].

JID:YJABR

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.5 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

5

Proof. Since Γ has just one vertex, the exact sequence above looks as follows: 0

H1 (F, Z)

ι1

⊕e∈E(Γ) Z[G/G(e)F ]

π1

Z[G/HF ]

Z

0.

Recalling that G = F  H and so G/F G(e) = H/G(e) the exact sequence can be rewritten as H1 (F, Z)

0

⊕e∈E(Γ) Z[H/G(e)]

ι1

Since F ab = H1 (F, Z), the result follows.

π1

0.

2

Lemma 3.2. Suppose Γ consists of one edge e and two vertices v0 , v1 . i.e. that G = H ∗G(e) G(v1 ). Then F ab is isomorphic, as H-module, to Me = JG(v1 ) (H/G(e)). Proof. In this case the exact sequence above looks as follows: 0

H1 (F, Z)

ι1

Z[G/G(e)F ]

π1

Z[G/G(v1 )F ] ⊕ Z[G/HF ]

ϕ1

Z

0.

Observing that HF = G, G/F G(e) = H/G(e) and G/G(v1 )F = H/ϕ(G(v1 )) the exact sequence can be rewritten as 0

H1 (F, Z)

ι1

Z[H/G(e)]

π1

im(π1 )

0

with im(π1 ) ∼ = Z[H/ϕ(G(v1 ))], since ϕ1 is surjective and is identity restricted to the direct summand Z = Z[G/HF ]. Hence the result. 2 The next proposition gives a complete description of the Z[H]-module F ab in the case Γ is a tree and all vertex groups G(v) are isomorphic to H. Proposition 3.3. 3 Suppose Γ is a tree and every vertex group is isomorphic to H. Fix the orientation of Γ such that the initial vertex d0 (e) is closer to v0 than the terminal  vertex d1 (e) for every e ∈ E(Γ). Then the Z[H]-module F ab ∼ = e∈E(Γ) Me , where ∼ Me = Jϕ(G(d1 (e))) (H/ϕ(G(e))). Proof. Assume first that Γ is finite. We shall use induction on number n of edges of Γ. Note that Lemma 3.2 is the base of induction. Suppose that the result holds for any tree of groups with number of edges less than n. 3 For some time the work was carried out jointly with Ismael Lins, in particular this proposition was obtained jointly with him.

JID:YJABR 6

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.6 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

˜ be ˜ Γ) Let v be a pending vertex of Γ distinct from v0 with an incident edge ev . Let (G, ˜ ˜ = a subtree of Γ obtained by removing v and ev , that is, V (Γ) = V (Γ) − {v} and E(Γ) E(Γ) − {ev }. Now subdivide ev into two edges e1 , ev (abusing notation) introducing a new vertex v1 incident to ev and e1 (see the picture above). Put G(e1 ) = G(v1 ) ∼ = H and w.l.o.g we assume that G(v1 ) = H = G(e1 ) (indeed, since every vertex group G(v) ∼ =H by hypothesis, G = F  G(v) for any v). Then G = π(G, Γ) can be decomposed as an amalgamated free product over H = G(e1 ) ˜ ∗H (H ∗G(e ) G(v)). ˜ Γ) G = π(G, v By the Kurosh Subgroup Theorem (Theorem 14 in [9]), we have ˜ x ) ∗ ∗y (F ∩ (H ∗G(e ) G(v))y ) ∗ F˜ ˜ Γ) F = ∗x (F ∩ π(G, v ˜ y runs through a ˜ Γ), where x runs through a system of coset representatives F \ G/π(G, ˜ system of coset representatives F \G/(H ∗A0 H0 ) and F is a free subgroup of F . Note that ˜ = 1 and |F \ G/(H ∗G(e ) G(v))| = 1 because G = F H and H ∗G(e ) G(v) ˜ Γ)| |F \ G/π(G, v v contains H. This implies that F˜ = {1} (by Exercise 2 in [9] page 57). Hence ˜ ∗ (F ∩ (H ∗G(e ) G(v))) ˜ Γ)) F = (F ∩ π1 (G, v and its abelianization is ˜ ab ⊕ (F ∩ (H ∗G(e ) G(v)))ab . ˜ Γ)) F ab = (F ∩ π1 (G, v ˜ is the kernel of the restriction of ϕ to the subgroup π1 (G, ˜ ˜ Γ) ˜ Γ) Note that F ∩ π1 (G, and by the induction hypothesis its abelianization is isomorphic, as Z[H]-module, to  ˜ Me . Furthermore, F ∩ (H ∗G(ev ) G(v)) is the kernel of the restriction of ϕ to e∈E(Γ)

JID:YJABR

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.7 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

7

the subgroup H ∗G(e0 ) G(v) and by the base of induction (Lemma 3.2) its abelianization F ab is isomorphic, as H-module, to Me0 ∼ = JG(v) (H/G(e0 )). The result is proved in this case.  Suppose now that Γ is infinite. Then Γ = i∈I Ti is a union of finite subtrees Ti containing v0 . Then G = π1 (G, Γ) = lim π1 (G, Ti ) −→

is a direct limit of the fundamental groups π1 (G, Ti ) of subgraphs of groups restricted to Ti . Hence F = lim Fi , where Fi = F ∩ π1 (G, Ti ) and so F ab = lim Fiab . By the −→ −→ ∼ previous case Fiab = Me . Since a direct sum commutes with direct limits we e∈E(T ) i  get F ab ∼ Me and the proposition is proved. 2 = e∈E(Γ)

For a ∈ star(v0 ) let Ta be the maximal subtree growing from v0 with trunk a (i.e. a ∈ E(Ta ) and E(Ta )∩star(v0 ) = {a}). The fundamental group π1 (G, Ta ) of the subgraph of groups restricted to Ta is a subgroup of G = π1 (G, Γ) and we denote by Fa the kernel of the restriction ϕ|π1 (G,Ta ) . Let Ma = Faab be the abelianization of Fa regarded as Z[H]-module.  Theorem 3.4. Suppose Γ is a tree. Then the Z[H]-module F ab ∼ = a∈star(v0 ) Ma decomposes as a direct sum of Z[H]-modules Ma . Proof.

We can rewrite the long exact sequence (1) as in the first line of the following commutative diagram and for every Ta we have the exact sequence as in the second line of the diagram. ...0

H1 (F, Z)

...0

H1 (Fa , Z)

⊕a∈star(v0 ) ⊕e∈E(Ta ) Z[G/G(e)F ]

ι1

π1

⊕a∈star(v0 ) ⊕v∈V (Ta )\{v0 } Z[G/G(v)F ] ⊕ Z

π2

⊕v∈V (Ta )\{v0 } Z[G/G(v)F ] ⊕ Z

γ1

α1

ι2

⊕e∈E(Ta ) Z[G/G(e)F ]

ϕ

Z

0

Z

0

β1 ϕa

where vertical maps are natural inclusions. In the second line of the diagram above the image of π2 is a submodule of ⊕v∈V (Tv ) Z[G/G(v)F ] ⊕ Z that coincides with ker(ϕa ), because of the exactness of the sequence. As ϕ is surjective and is the identity restricted to the direct summand Z and

JID:YJABR

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.8 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

8

so is ϕa for each a, it follows that ⊕v∈V (Ta )\{v0 } Z[G/G(v)F ] ⊕ Z = ker(ϕa ) ⊕ Z and ker(ϕa ) ∼ = ⊕v∈V (Ta )\{v0 } Z[G/G(v)F ]. Therefore, the second row of the diagram can be rewritten as ...0

H1 (F, Z)

⊕a∈star(v0 ) ⊕e∈E(Ta ) Z[G/G(e)F ]

ι1

...0

π1

⊕a∈star(v0 ) ⊕v∈V (Ta )\{v0 } Z[G/G(v)F ] ⊕ Z

γ1

α1

H1 (Fa , Z)

⊕e∈E(Ta ) Z[G/G(e)F ]

ι2

Z

0

0

0

β1

ker(ϕa )

π2

ϕa

and so the first row of the diagram can be rewritten as ...0

H1 (F, Z)

ι3

⊕a∈star(v0 ) ⊕e∈E(Ta ) Z[H/G(e)]

π3

⊕a∈star(v0 ) ker(ϕa )

0

By the exactness of the sequence and injectivity of ι3 we have H1 (F, Z) = ker(π3 ) and ker(⊕a∈star(v0 ) ⊕e∈E(Ta ) Z[H/G(e)] → ⊕a∈star(v0 ) ker(ϕa )) = ⊕a∈star(v0 ) ker(⊕e∈E(Ta ) Z[H/G(e)] → ker(ϕa )), as ker commutes with direct sums. Then Ma = H1 (Fa , Z) = ker(⊕e∈E(Ta ) Z[H/G(e)] → ker(ϕa )), and so H1 (F, Z) = ⊕a∈star(v0 ) Ma . 2 The next example shows that the Z[H]-summands Ma of F ab in Theorem 3.4 can be indecomposable. We will denote by d(G) the minimal number of generators of a group G. Example 3.5. Consider the following graph of groups C2 × C2

C4 C2

C4 × C4 = H C2

with the fundamental group G = (C4 ∗C2 (C2 × C2 )) ∗C2 (C4 × C4 ), where amalgamated subgroups C2 are identified with the left and the right factors of C2 ×C2 respectively. The identification of the edge group with a subgroup of C4 ×C4 of order 2 can be arbitrary, for example we can identify C2 with the square of generator of the left factor of C4 × C4 . Let H = C4 × C4 be the right factor. Clearly, an isomorphism of the left factor C4 to a direct factor of H that intersects the amalgamated subgroups trivially induces an epimorphism G −→ H with the free kernel F , i.e. G = F  H. Since d(G) ≥ d(Gab ), H is abelian, d(G) ≥ d(F ab /[F ab , H]) + d(H), d(G) = 3 and d(H) = 2, we have d(F ab /[F ab , H]) = 1. Therefore F ab is a cyclic indecomposable Z[H]-module. Proof of Theorem 1.1. Let T be a maximal subtree of Γ and GT = π1 (G, T ) be the fundamental group of a graph of groups over T . Let FT be the kernel of the restriction

JID:YJABR

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.9 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

9

of ϕ to GT . Then FT is a subgroup of F normalized by H and GT = FT  H. So by Theorem 3.4, the Z[H]-module (FT )ab ∼ = ⊕a∈star(v0 ) Ma . The group G is an HNN-extension of GT with associated subgroups G(e) and stable letters te where e runs through edges not belonging to T . This HNN-extension acts on an associated tree S that gives rise to a short exact sequence of permutation modules 0

⊕e∈E(Γ)\T Z[G/G(e)]

Z[G/GT ]

Z

0.

Applying ⊗Z[F ] Z to it we get the following long exact sequence H1 (FT , Z)

0

H1 (F, Z)

⊕e∈E(Γ)\T Z[G/G(e)F ]

Z[G/GT F ]

Z

0

and since G = GT F and G/G(e)F ∼ = H/ϕ(G(e)) we can rewrite this sequence as 0

FTab = H1 (FT , Z)

H1 (F, Z)

⊕e∈Γ\T Z[H/ϕ(G(e))]

0.

2

The next example shows that the extension in Theorem 1.1 might not split in general. Example 3.6. Consider the following graph of groups C2 C2 × C2

C2 × C2 = H C2

The fundamental group G = F  H, where H = C2 × C2 and the lower edge group C2 identified with one of the factors of C2 × C2 and of H. The upper edge group C2 identified with the same factor C2 of C2 ×C2 and conjugate to the same factor C2 of H by the stable letter. Hence using d(G) ≥ d(Gab ) and commutativity of H it follows d(G) ≥ d(F ab /[F ab , H]) +d(H), d(H) = 2 and 3 = d(G) = d(H) +1 so that d(F ab /[F ab , H]) = 1. Therefore F ab is cyclic indecomposable Z[H]-module. 4. The case of F ab is a permutation module We shall need a general lemma on splitting of an exact sequence of permutation modules for the proof of Theorem 1.2. The idea of the proof of the lemma is due to Peter H Symonds. We shall use the notation M ↑H L = Z[H] ⊗Z[L] M and M ↓L for the induced from L to H and restricted from H to L modules, respectively.

JID:YJABR 10

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.10 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

Lemma 4.1. Let H be a finite group and let 0

A

B

C

0

be a short exact sequence of Z[H]-modules. Suppose that A and C are permutation Z[H]-modules. Then B = A ⊕ C. Proof. It is sufficient to show that Ext1Z[H] (C, A) = 0. There are two sets I and J of subgroups of H such that A = ⊕K∈I Z[H/K] and C = ⊕L∈J Z[H/L], because A and C are permutation Z[H]-modules. It follows that Ext1Z[H] (C, A) = Ext1Z[H] (⊕L∈J Z[H/L], ⊕K∈I Z[H/K]) = ⊕L∈J ⊕K∈I Ext1Z[H] (Z[H/L], Z[H/K]), where in the second equality we used Ext properties (commutativity with direct sum). We have that Ext1Z[H] (Z[H/L], Z[H/K]) = Ext1Z[L] (Z, Z[H/K] ↓H L) , by Frobenius reciprocity (see Lemma 1.7.32 and Proposition 1.8.32 in [10]). Using Mackey Decomposition Theorem: g Z[H/K] ↓H L = ⊕KgL∈K\H/L Z[L/(K ∩ L)],

and that Ext commutes with ⊕ on the second variable we have 1 g Ext1Z[L] (Z, Z[H/K] ↓H L ) = ⊕KgL∈K\H/L ExtZ[L] (Z, Z[L/(K ∩ L)]

= ⊕KgL∈K\H/L H 1 (K g ∩ L, Z) by Shapiro’s lemma. But H 1 (K g ∩ L, Z) = Hom(K g ∩ L, Z) = 0 since K g ∩ L is finite. Therefore Ext1Z[H] (C, A) = 0. Hence the result. 2 Proof of Theorem 1.2. One direction is the subject of Lemma 3.1. For the converse, suppose that F ab = H1 (F, Z) is a permutation Z[H]-module (i.e. there is a set I of some subgroups of H such that F ab = ⊕K∈I Z[H/K]). W.l.o.g we may assume that (G, Γ) is reduced (see the end of Subsection 2.2). Then by Lemma 1.2 we just need to prove that Γ has one vertex only. Suppose on the contrary Γ has more than one vertex. Let v0 be a vertex such that G(v0 ) = H. Then identifying G/G(e)F with H/G(e) and G/G(v)F with H/G(v) in the long exact sequence (1) of permutation Z[H]-modules we have ...0

H1 (F, Z)

ι

⊕e∈E Z[H/G(e)]

π

⊕v∈V \{v0 } Z[H/G(v)] ⊕ Z

ϕ

Z

0

The image of π is a submodule of ⊕v∈V \{v0 } Z[H/G(v)] ⊕ Z that coincides with ker(ϕ), because of the exactness of the sequence. As ϕ is surjective and is identity restricted to the direct summand Z, we have ⊕v∈V \{v0 } Z[H/G(v)] ⊕ Z = ker(ϕ) ⊕ Z and so

JID:YJABR

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.11 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

11

im(π) = ker(ϕ) ∼ = ⊕v∈V \{v0 } Z[H/G(v)]. Therefore, the long exact sequence above can be rewritten as a short exact sequence of permutation Z[H]-modules 0

H1 (F, Z)

ι

⊕e∈E Z[H/G(e)]

π

im(π)

0

 which splits by Lemma 4.1. It follows that ⊕e∈E Z[H/G(e)] ∼ = H1 (F, Z) ⊕v∈V \{v0 } Z[H/ G(v)]. Recall that G = π1 (G, Γ, T ) is the fundamental group of a graph of finite groups (G, Γ) with respect to a choice of some maximal subtree T of Γ (see Subsection 2.2). Let Γ be a quotient graph of Γ obtained by collapsing T . Define a graph of groups (G, Γ), putting G(¯ v ) = H to be the unique vertex group, G(¯ e) = G(e) for edges e ∈ / T to be the edge groups, where e¯ is the image of e in Γ, together with monomorphisms ∂0,¯e , ∂1,¯e : G(¯ e) = G(e) → H defined as the compositions ∂ i,¯e = ∂i,e ◦ ϕ (i = 0, 1), where ϕ : G −→ H is the natural projection. It follows that the fundamental group G = π1 (G, Γ) is just an HNN-extension HN N (H, G(¯ e), E \ E(T )) with the set of stable letters E \ E(T ). Then the restrictions ϕ|G(v) define the natural morphism of graph of groups (G, Γ) −→ (G, Γ) (cf. the presentation of G = πi (G, Γ, T ) in Subsection 2.2) that induces a natural homomorphism η : G −→ G. Let (G, T ) be the subgraph of groups of (G, Γ) restricted to T and GT = π1 (G, T ) be its fundamental group. Put FT = GT ∩ F . Then G = G/(FTG ). To see this one can think of G = πi (G, Γ, T ) as an HNN-extension HN N (GT , G(e), E \ E(T )) and the map η as induced by the natural epimorphism GT −→ GT /FT = H. Put F = F/(FTG ). Then we have the following commutative diagram with exact rows and columns. 0

0

0

H1 (F , Z)

¯ ι

⊕e∈E(Γ) Z[H/G(e)]

1

0

π ¯

0

0

im(π)

0

im(π)

0



H1 (F, Z)

ι

⊕e∈E(T ) Z[H/G(e)]



⊕e∈E\E(T ) Z[H/G(e)]

π

α

0

H1 (FT , Z)

0

ι

⊕e∈E(T ) Z[H/G(e)]

0

π

0

Note that  restricted to the summand ⊕e∈E\E(T ) Z[H/G(e)] in the middle entry is an isomorphism. On the other hand α is an isomorphism onto the left factor of the middle term. It follows that identifying the left column with its image in the middle column we have the following exact sequence

JID:YJABR

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.12 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

12

0 −→ H1 (FT , Z) −→ H1 (F, Z) = H1 (FT , Z)



⊕e∈E\E(T ) Z[H/G(e)]

−→ ⊕e∈E(Γ) Z[H/G(e)] ∼ = H1 (F , Z) −→ 0 with the third arrow restricted to ⊕e∈E\E(T ) Z[H/G(e)] being the identity map. So the sequence splits and therefore so does the left column of the diagram. Thus H1(F, Z) ∼ = H1 (FT , Z) ⊕ H1 (F , Z). Now taking H-coinvariants mod p of the diagram we have 0

0

0

H1 (F , Fp )H

¯ ιH

π ¯H

0

0

im(πH )

0

im(πH )

0

H

(1 )H

0

⊕e∈E(Γ) Fp

H1 (F, Fp )H

ιH

⊕e∈E(T ) Fp



⊕e∈E\E(T ) Fp

πH

αH

0

H1 (FT , Fp )H

0

ιH

⊕e∈E(T ) Fp

πH

0

0

Note that πH restricted on the left summand of the middle term is an isomorphism. Indeed since T is a tree, the short sequence 0 −→ ⊕e∈E(T ) Fp −→ ⊕v∈V Fp −→ Fp −→ 0 associated to it is exact and our πH restricted to this left summand is the middle arrow of it. It follows that ιH (H1 (F, Fp )H ) intersects trivially this left summand. This implies that H restricted to ιH (H1 (F, Fp )H ) is an isomorphism and therefore so is (1 )H . Hence H1 (FT , Fp )H = 0 and since H is a finite p-group, H1 (FT , Fp ) = 0. It follows that FT is trivial and so T must consists of one vertex only. 2 References [1] I.M. Chiswell, Exact sequences associated with a graph of groups, J. Pure Appl. Algebra 8 (1976) 385–400. [2] D.E. Cohen, Groups with free subgroups of finite index, in: Conference on Group Theory, University of Wisconsin–Parkside, 1973, in: Lecture Notes in Mathematics, vol. 319, Springer, 1972, pp. 26–44. [3] W. Dicks, Groups, Trees and Projective Modules, Springer-Verlag, Berlin, 1980. [4] W. Herfort, P.A. Zalesskii, Virtually free pro-p groups, Publ. Math. IHES 118 (2013) 193–211. [5] A. Karras, A. Pietrowski, D. Solitar, Finite and infinite cyclic extensions of free groups, J. Aust. Math. Soc. 16 (1973) 458–466.

JID:YJABR

AID:16126 /FLA

[m1L; v1.205; Prn:6/03/2017; 11:30] P.13 (1-13)

I. Lima, P. Zalesskii / Journal of Algebra ••• (••••) •••–•••

13

[6] A.L.P. Porto, P. Zalesskii, Free-by-finite pro-p groups and integral p-adic representations, Arch. Math. 97 (2011) 225–235. [7] L. Ribes, P. Zalesskii, Pro-p trees, in: M. du Sautoy, D. Segal, A. Shalev (Eds.), New Horizons in Pro-p Groups, in: Progress in Mathematics, vol. 184, Birkhäuser, Boston, 2000. [8] G.P. Scott, An embedding theorem for groups with a free subgroup of finite index, Bull. Lond. Math. Soc. 6 (1974) 304–306. [9] J-P. Serre, Trees, Springer-Verlag, Berlin, 1980. [10] A. Zimmermann, Representation Theory: A Homological Algebra Point of View, Algebra and Applications, Springer, 2014.