ON WEAK COMMUTATIVITY IN GROUPS 1

ON WEAK COMMUTATIVITY IN GROUPS DESSISLAVA KOCHLOUKOVA, SAID SIDKI

Abstract. For a group G we study homological and homotopical properties of the group χ(G) = hG, Gψ | [g, g ψ ] = 1 for g ∈ Gi. In particular, we show that the operator χ preserves the soluble of type F P∞ property.

1. Introduction In [30] Sidki associated to an arbitrary group G a new group χ(G) which is defined by two isomorphic copies of G and satisfies some natural commutator relations. It turned out that for G finite, the group χ(G) is always finite and for an arbitrary G, surprisingly χ(G) has a subquotient that is isomorphic to the Schur multiplier H2 (G, Z). By definition χ(G) = hG, Gψ | [g, g ψ ] = 1 for g ∈ Gi, where ψ : G → Gψ is an isomorphism of groups. As shown in [30, Thm. C] if P is one of the following classes of groups : finite π-groups, where π is a set of primes; finite nilpotent groups; solvable groups; perfect groups and G ∈ P then χ(G) ∈ P. Later on Gupta, Rocco and Sidki showed in [19] that if G is finitely generated nilpotent then χ(G) is nilpotent and found bounds on the nilpotency class of χ(G). Recently Lima and Oliveira proved that if G is polycyclic-by-finite then χ(G) is polycyclic-by-finite [22]. Our first result on χ(G) considers the case when G is soluble of homological type F P∞ . Recall that a group G is of type F Pm if there is a projective resolution of the trivial ZG-module Z with finitely generated projectives in dimension ≤ m and G is of type F P∞ if it is of type F Pm for every m. Our first theorem shows that the operator χ preserves the soluble of type F P∞ property. Theorem A Let G be a soluble group of type F P∞ . Then χ(G) is a soluble group of type F P∞ . As shown by Sidki in [30] the group χ(G) has two normal abelian subgroups R(G) and W (G) such that χ(G)/W (G) is isomorphic to a subgroup of G × G × G that contains the commutator subgroup and W (G)/R(G) ' H2 (G, Z). The crucial point of the proof of Lima and Oliveira in [22] is that if G is polycyclic-by-finite then W (G) is finitely generated. We generalize the main idea of this proof by introducing a more complicated homological argument that uses spectral sequences, see Theorem 6.1, which is a homological version of Martinez-Perez’s result [23, Thm. C]. This enables us to prove the following theorem. Theorem B Let G be a group of type F P2 such that G0 /G00 is finitely generated. Then W (G) is finitely generated. 2000 Mathematics Subject Classification. Primary Secondary . 1

2

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

As a corollary we obtain Corollary C. If G is finitely presented and G0 /G00 is finitely generated then χ(G) is finitely presented. Recall that a group G has homotopical type Fm if there is a K(G, 1) complex with finite m-skeleton. In particular a group is of type F2 if and only if it is finitely presented. For m ≥ 2 a group is of type Fm if and only if it is F Pm and finitely presented. Using the link between Σ-theory and results of Bieri and Renz [11], [29] on homological/homotopical properties of subgroups containing the commutator subgroup, we establish the following result. Theorem D Let G be a group of type Fk (respectively type F Pk ) and the commutator subgroup G0 has type Fs (respectively type F Ps ). Suppose that k ≤ 3s + 2. Then χ(G)/W (G) has type Fk (respectively type F Pk ). Furthermore if G0 /G00 is finitely generated and k ≥ 2 we have that χ(G) has type Fk (respectively type F Pk ). As pointed by Bridson, the group χ(G)/W (G) is isomorphic to a subdirect product of G × G × G that maps surjectively on the direct product of any two of the three copies of G. By [4, Thm. A] if G is finitely presented then χ(G)/W (G) is finitely presented. This provides an alternative proof of the homotopical part of Theorem D for k = 2. We apply the results to the group ψ

h3 h3 ν(G) = hG, Gψ | [h1 , hψ = [h1 , hψ = [hh1 3 , (hh2 3 )ψ ]i, 2] 2]

defined by Rocco in [27]. Theorem E Let G be a group. a) If G is finitely presented then ν(G) is finitely presented. b) If G is soluble of type F P∞ then ν(G) is soluble of type F P∞ . In the final section, we consider various examples of soluble groups of type F P∞ . By Theorem A, we know that for G soluble of type F P∞ , the group χ(G) is of type F P∞ , however our examples show that in some cases W (G) cannot be finitely generated. Theorem F a) For the metabelian Baumslag-Solitar group G = ha, b | a−1 ba = bm i for some m ∈ Z \ {0}, we have that W (G) = 1. b) Let H = G × Z, where G is the group from part (a). For m = 2, we have R(H) = 1 and W (H) ' Z ' H2 (H, Z). c) There exists a finitely presented (nilpotent of class 2)-by-abelian group L such that W (L) is infinitely generated. Since H2 (L, Z) is finitely generated, R(L) is infinitely generated too. Remark The example of item c) denoted by L should not be confused with the subgroup of χ(G) denoted by L. The paper is structured as follows. In Section 2 we review the theory of soluble groups of type F P∞ and some properties of the invariant ΣV (Q) and its higher dimensional generalizations. In Section 4 we review general properties of the group χ(G). In Section 3 we prove a result about the Bieri-Strebel Σ-invariant; see Lemma 3.2, that will be the core of the proof of Theorem A in Section 5. Theorem B is

ON WEAK COMMUTATIVITY IN GROUPS

3

proved in Section 6 as a corollary of the homological result Theorem 6.1. Corollary C and Theorem D are proved in Section 7 and as an aplication we show that for the R. Thompson group F the group χ(F ) is of type F P∞ . In Section 8 we prove Theorem E that establishes properties of the group ν(G). Finally in Section 9 we make computations for the groups stated in Theorem F. Acknowledgements The first named author is partially supported by ”bolsa de produtividade em pesquisa”, CNPq, Brazil. The authors thank Martin Bridson for helpful comments on the paper. 2. Preliminaries on Σ-theory and soluble groups of type F Pm 2.1. The Bieri-Strebel invariant. A group G has type F Pm if the trivial ZGmodule Z has projective resolution with all projectives finitely generated in dimension ≤ m. A group has type F P∞ if it is F Pm for every m. For more properties of groups of type F Pm we refer the reader to [6], [16]. Let Q be a finitely generated abelian group. By definition S(Q) = (Hom(Q, R) \ {0})/ ∼, where χ1 ∼ χ2 if the homomorphism χ1 is obtained from the homomorphism χ2 by multiplication with a positive real number. The equivalence class of χ is denoted by [χ]. Observe that Q = Zn ⊕ T for some finite abelian group T , so there is an isomorphism Hom(Q, R) ' Rn and for χ ∈ Hom(Q, R) \ {0} the image in Rn of all non-zero homomorphisms from the equivalence class of χ is an open ray in Rn starting at the origin of Rn . This open ray intersects the unit sphere Sn−1 in exactly one point. This gives a bijection between S(Q) and the unit sphere Sn−1 and this way we can endow S(Q) with a topology. For a non-zero homomorphism χ : Q → R define the monoid Qχ = {q ∈ Q | χ(q) ≥ 0} = χ−1 ([0, ∞)). Thus ZQχ is a subring of the group algebra ZQ. By definition the Bieri-Strebel invariant associated to a finitely generated ZQ-module V is ΣV (Q) = {[χ] ∈ Hom(Q, R) | V is finitely generated as ZQχ -module}. If not stated otherwise the modules considered in this paper are right ones. The complement of ΣV (Q) in S(Q) is denoted by ΣcV (Q). By [12] ΣV (Q) is an open subset of S(Q). Lemma 2.1. [12] Let 0 → V1 → V → V2 → 0 be a short exact sequence of finitely generated ZQ-modules. Then ΣV (Q) = ΣV1 (Q) ∩ ΣV2 (Q); equivalently ΣcV (Q) = ΣcV1 (Q) ∪ ΣcV2 (Q). By [9] there is a link between ΣcV (Q) and valuation theory from commutative algebra. Exploring this link Bieri and Groves proved in [9] that ΣcV (Q) is a rationally defined spherical polyhedron i.e. ΣcV (Q) is a finite union X1 ∪ . . . ∪ Xm , where each Xi is an intersection of finitely many closed rationally defined semispheres in S(Q). We say that V is m-tame if every subset from ΣcV (Q) with at most m points is contained in an open semisphere of S(Q). This is equivalent to whenever χ1 + . . . + χm = 0 for some χ1 , . . . , χm ∈ Hom(Q, R) \ {0} then at least one [χi ] is in ΣV (Q). The F Pm -Conjecture Let V → G → Q be a short exact sequence of groups with V and Q abelian, G finitely generated. Then G is of type F Pm if and only if V is m-tame as ZQ-module.

4

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

The F Pm -Conjecture holds for m = 2 [12]. Furthermore if G is of type F Pm and either G = V o Q or V has finite exponent then V is m-tame as ZQ-module [20]. If G is of finite Pr¨ ufer rank the F Pm -Conjecture was proved in [2]. 2.2. Soluble groups of type F P∞ . A soluble group is called constructible if it is in the smallest class G of soluble groups such that 1. the trivial group belongs to G; 2. If H is a normal subgroup of a soluble group G with G/H finite and H ∈ G then G ∈ G; 3. If G is a soluble group and an HNN extension with a base group B and associated subgroups G1 , G2 such that B, G1 ∈ G then G ∈ G. As constructible metabelian groups are of finite Pr¨ ufer rank the F Pm -Conjecture holds for m = ∞. Theorem 2.2. [21] Every soluble group of type F P∞ is constructible. It is easy to see that every constructible group is of type F P∞ . Indeed : 1. [16, Chapter VIII, Prop. 5.1] if H is a subgroup of finite index in G then G has type F Pm if and only if H has type F Pm ; 2. if G is an HNN extension with a base group and associated subgroups all of type F P∞ then G is of type F P∞ . Note that constructible soluble groups are nilpotent-by-abelian-by-finite, hence the study of soluble groups of type F P∞ is reduced to the study of nilpotent-byabelian groups of type F P∞ . In this case complete classification exists in terms of Bieri-Strebel Sigma theory. Theorem 2.3. [13] Let N → G → Q be a short exact sequence of groups with Q abelian, N nilpotent and G finitely generated. Set V = N/N 0 . Then G is constructible if and only if ΣcV (Q) is contained in an open semisphere of S(Q). In particular G is constructible if and only if G/N 0 is constructible. Using Krull dimension we can see that if V is ∞-tame as ZQ-module then its torsion part is finite and dimQ V ⊗ Q < ∞. This implies that for ∞-tame modules V we have that ΣcV (Q) is finite. In particular, by Theorem 2.2 and Theorem 2.3, if N → G → Q is a short exact sequence of groups with Q abelian, N nilpotent and G of type F P∞ then G/N 0 is a constructible metabelian group. Hence V = N/N 0 is ∞-tame as ZQ-module, so ΣcV (Q) is finite. But the converse is not true, if G is finitely generated and ΣcV (Q) is finite this does not imply that G is F P∞ ; actually, this does not imply even that G is F P2 . It is worth noting that the proof of the F Pm -Conjecture for m = 2 in [12] implies that for metabelian groups the property F P2 and finite presentability are equivalent. It is an open problem whether this holds for soluble groups. As shown in [5] this is not true in general. Another corollary of Theorem 2.3 is that every quotient of a soluble group of type F P∞ is again of type F P∞ . Quotients of finitely presented soluble groups are not necessary finitely presented. There is an example of finitely presented nilpotent-byabelian group G with infinitely generated center Z(G), hence the quotient G/Z(G) is not finitely presented, furthermore G/Z(G) is not of type F P2 [1]. 3. Some new lemmas on Sigma theory Lemma 3.1. Let H be a finitely generated abelian group, H0 a subgroup of H and M a finitely generated ZH-module such that H0 acts trivially on M . Let

ON WEAK COMMUTATIVITY IN GROUPS

5

ρ : H → H/H0 be the canonical projection and ρ∗ : S(H/H0 ) → S(H) be the induced map that sends [χ] to [χ ◦ ρ]. Then ρ∗ |ΣcM (H/H0 ) : ΣcM (H/H0 ) → ΣcM (H) is a bijection. Proof. The lemma is equivalent to ΣcM (H) = {[χ] ∈ S(H) | χ(H0 ) = 0, M is not finitely generated as Z(Hχ /H0 ) − module}, which in itself is equivalent to ΣcM (H) ⊆ {[χ] ∈ S(H) | χ(H0 ) = 0}. Thus it remains to show that if χ(H0 ) 6= 0 then [χ] ∈ ΣM (H). Let h ∈ H and choose h0 ∈ H0 such that χ(h0 ) > 0. Then for sufficiently large natural number n we have hhn0 ∈ Hχ = {t ∈ H | χ(t) ≥ 0}. Hence h ∈ Hχ H0 and since h is an arbitrary element of H we have H = Hχ H0 . Then, since H0 acts trivially on M , every generating set of M as ZH-module is a generating set of M as ZHχ -module.  The next lemma is a technical result that will be the core of the proof of Theorem 5.2. In the next lemma we consider the direct sum of induced modules A1 ⊗ZQ1 Z[Q1 × Q2 ] ⊕ A2 ⊗ZQ2 Z[Q1 × Q2 ] that is isomorphic as abelian group to A1 ⊗Z ZQ2 ⊕ A2 ⊗Z ZQ1 via the isomorphism µ : A1 ⊗ZQ1 Z[Q1 × Q2 ] ⊕ A2 ⊗ZQ2 Z[Q1 × Q2 ] → A1 ⊗Z ZQ2 ⊕ A2 ⊗Z ZQ1 given by µ(a1 ⊗ q1 q2 ) = a1 q1 ⊗ q2 and µ(a2 ⊗ q1 q2 ) = a2 q2 ⊗ q1 for a1 ∈ A1 , a2 ∈ A2 , q1 ∈ Q1 , q2 ∈ Q2 . Via this isomorphism we view M = A1 ⊗Z ZQ2 ⊕ A2 ⊗Z ZQ1 as Z[Q1 × Q2 ]-module and m ◦ λ denotes the action of λ ∈ Z[Q1 × Q2 ] on m ∈ M . Lemma 3.2. Let Q1 be a finitely generated abelian group, A1 a finitely generated ZQ1 -module that is m-tame, ψ : Q1 → Q2 an isomorphism of abelian groups, A2 a finitely generated ZQ2 -module such that there is an isomorphism δ : A1 → A2 of abelian groups with δ(a1 ◦ q1 ) = δ(a1 ) ◦ ψ(q1 ) for all q1 ∈ Q1 , a1 ∈ A1 . Consider the Z[Q1 × Q2 ]-module B0 defined by B0 = (A1 ⊗Z ZQ2 ⊕ A2 ⊗Z ZQ1 )/R, where R = h{a2 ◦ (q1 − 1)(x − xψ ) | q1 ∈ Q1 , x ∈ Q1 , a2 ∈ A2 }∪ {a1 ◦ (q2 − 1)(x − xψ ) | q2 ∈ Q2 , x ∈ Q1 , a1 ∈ A1 }∪ ψ {a1 ◦ (q2 − 1)q1 − (aψ 1 ) ◦ (q1 − 1)q2 | a1 ∈ A1 , q1 ∈ Q1 , q2 = q1 }i. Then B0 is m-tame as Z[Q1 × Q2 ]-module.

Proof. Write B1 for the image of A1 ⊗Z ZQ2 in B0 and write B2 for the image of A2 ⊗Z ZQ1 in B0 . Thus, W = B1 ∩B2 , the image of h(hψ )◦(q1 −1) | h ∈ A1 , q1 ∈ Q1 i in B0 , is the same as the image of h(h) ◦ (q2 − 1) | h ∈ A1 , q2 ∈ Q2 i in B0 . Consider the filtration of Z[Q1 × Q2 ]-submodules 0 ⊆ W ⊆ B1 ⊆ B0 . c

By Lemma 2.1, Σ is additive and therefore (3.1)

ΣcB0 (Q1 × Q2 ) = ΣcW (Q1 × Q2 ) ∪ ΣcB1 /W (Q1 × Q2 ) ∪ ΣcB0 /B1 (Q1 × Q2 ).

6

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

Observe that (3.2)

B1 /W ' (A1 ⊗Z ZQ2 ) ⊗ZQ2 Z ' A1

as Z[Q1 × Q2 ]-modules, where A1 is viewed as Z[Q1 × Q2 ]-module with trivial Q2 -action. Similarly B0 /B1 ' B2 /W ' A2

(3.3)

as Z[Q1 × Q2 ]-modules, where A2 is viewed as Z[Q1 × Q2 ]-module with trivial Q1 -action. Given χ ∈ Hom(Q1 × Q2 , R), write χi for χ |Qi . Define ∆1 = {[(χ1 , χ2 )] ∈ S(Q1 × Q2 ) | χ2 = 0, [χ1 ] ∈ ΣcA1 (Q1 )}, ∆2 = {[(χ1 , χ2 )] ∈ S(Q1 × Q2 ) | χ1 = 0, [χ2 ] ∈ ΣcA2 (Q2 )} and ∆3 = {[(χ1 , χ2 )] ∈ S(Q1 × Q2 ) | χ1 = χ2 , [χ1 ] ∈ ΣcA1 (Q1 )}. By Lemma 3.1 and (3.2) (3.4)

ΣcB1 /W (Q1 × Q2 ) = ΣcA1 (Q1 × Q2 ) = ∆1 .

By Lemma 3.1 and (3.3) (3.5)

ΣcB0 /B1 (Q1 × Q2 ) = ΣcA2 (Q1 × Q2 ) = ∆2 .

Let Q3 = {q1−1 q1ψ | q1 ∈ Q1 } ≤ Q1 × Q2 . Then W ' V = (A1 ⊗Z ZQ2 )Aug(ZQ2 )/(A1 ⊗Z ZQ2 )Aug(ZQ2 )Aug(ZQ3 ), where Aug(ZQi ) is the augmentation ideal of ZQi . Note that, since ZQ2 is a Noetherian ring, Aug(ZQ2 ) is finitely generated as ZQ2 -module, hence for some natural number m there is an epimorphism of ZQ2 -modules (ZQ2 )m → Aug(ZQ2 ). Hence there is an epimorphism of Z[Q1 × Q2 ]-modules (A1 ⊗Z ZQ2 )m ' A1 ⊗Z (ZQ2 )m → A1 ⊗Z Aug(ZQ2 ) ' (A1 ⊗Z ZQ2 )Aug(ZQ2 ), that induces an epimorphism of Z[Q1 × Q2 ]-modules ((A1 ⊗Z ZQ2 )/(A1 ⊗Z ZQ2 )Aug(ZQ3 ))m → V ' W. Set D = (A1 ⊗Z ZQ2 )/(A1 ⊗Z ZQ2 )Aug(ZQ3 ). Thus, by the additivity of Σc (see Lemma 2.1) we have (3.6)

ΣcW (Q1 × Q2 ) ⊆ ΣcDm (Q1 × Q2 ) = ΣcD (Q1 × Q2 ).

Observe that there is an isomorphism of Z[Q1 × Q2 ]-modules D ' A1 , where we view A1 as Z[Q1 × Q2 ]-module with Q3 acting trivially; i.e., Q2 acts via the isomorphism ψ : Q1 → Q2 . Then by Lemma 3.1 (3.7)

ΣcD (Q1 × Q2 ) = ∆3 .

Set ∆ = ∆1 ∪ ∆2 ∪ ∆3 . By (3.1), (3.4), (3.5), (3.6) and (3.7) we deduce that (3.8)

ΣcB0 (Q1 × Q2 ) ⊆ ∆.

ON WEAK COMMUTATIVITY IN GROUPS

7

Let Y be any subset of at most m-elements in ∆. Define Y1 = ρ∗1 (Y \ ∆2 ) ⊆ S(Q1 ), Y2 = ρ∗2 (Y \ ∆1 ) ⊆ S(Q2 ), where ρ∗i : {[(χ1 , χ2 )] ∈ S(Q1 × Q2 ) | χi 6= 0} → S(Qi ) is induced by the inclusion map ρi : Qi → Q1 × Q2 . Then ψ ∗ (Y2 ) ⊆ S(Q1 ), where ψ ∗ : S(Q2 ) → S(Q1 ) is induced by the isomorphism ψ : Q1 → Q2 and the cardinality of Y1 ∪ ψ ∗ (Y2 ) is at most m. Note that by construction Y1 ∪ ψ ∗ (Y2 ) ⊆ ΣcA1 (Q1 ). The fact that A1 is m-tame implies the existence of q1 ∈ Q1 such that for every [χ1 ] ∈ Y1 ∪ ψ ∗ (Y2 ) we have χ1 (q1 ) > 0. Similarly there is q2 ∈ Q2 such that for every [χ2 ] ∈ Y2 ∪ ψ ∗ (Y1 ) we have χ2 (q2 ) > 0. Finally for q˜ = (q1 , q2 ) we have [χ] ∈ Y implies χ(˜ q ) > 0. In particular, Y lies in an open semisphere in S(Q1 ×Q2 ) and by (3.8) any subset of ΣcB0 (Q1 × Q2 ) with at most m elements, lies in an open semisphere in S(Q1 × Q2 ). This completes the proof.  4. Preliminaries on the group χ(G) In this section, we follow the notations from [30] and review some basic properties of the group χ(G). Let Gψ be an isomorphic copy of G. By definition χ(G) = hG, Gψ | [g, g ψ ] = 1 for g ∈ Gi, D = D(G) = [G, Gψ ], L = L(G) = [G, ψ], W = W (G) = D ∩ L, R = R(G) = [G, L, Gψ ], where the triple commutator is left normed and by [30, Lemma 4.1.3] the above groups are normal and ψ-invariant in χ(G). By [30, Prop. 4.1.4, Lemma 4.1.6 (ii)] χ(G)/DL ' G/G0 and W is central in DL. Let ρ : χ(G) → G × G × G be the homomorphism defined by ρ(g) = (g, g, 1), ρ(g ψ ) = (1, g, g) for g ∈ G. Then by [30, Prop. 4.1.4] Ker(ρ) = W and by [28, Lemma 2.2] W/R ' H2 (G, Z). It is worth noting that the last equality follows from the description of the Schur multiplier in [26]. Observe that ρ(D) = 1 × [G, G] × 1 and ρ(L) = h(g, 1, g −1 ) | g ∈ Gi. Hence (1, 1, [g1 , g2 ]) = ((g1 , 1, g1−1 )(g2 , 1, g2−1 )(g2−1 g1−1 , 1, g1 g2 )) ∈ ρ(L), so 1 × 1 × [G, G] ⊆ ρ(L) and similarly [G, G] × 1 × 1 ⊆ ρ(L). The last two containments imply (4.1)

{(g1 , g2 , g3 ) | g2 ∈ [G, G], g1 g3 ∈ [G, G]} = ρ(DL) ⊆ Im(ρ).

8

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

Then [G, G] × [G, G] × [G, G] ⊆ ρ(DL) ⊆ Im(ρ) = {(g1 , g2 , g3 ) | g1 g2−1 g3 ∈ [G, G]}. 5. The case when G is nilpotent-by-abelian-by-finite In this section we assume that the group G has a chain of normal subgroups N ≤ M ≤ G such that N is nilpotent, M/N abelian and G/M finite. Then, the chain M 0 ≤ N ≤ M ≤ G has the property M 0 is nilpotent, M/M 0 is abelian and G/M is finite. Therefore, we can assume that N ≤ G0 and denote G/N by Q. Lemma 5.1. If G is nilpotent-by-abelian-by-finite then χ(G) is nilpotent-by-abelianby-finite. Proof. Set E = E(G) = ρ−1 (N × N × N ) ⊆ ρ−1 ([G, G] × [G, G] × [G, G]) ⊆ D.L. Note that DL/E ⊆ χ(G)/E ' Im(ρ)/ρ(E) ⊆ (G × G × G)/ρ(E) = (G × G × G)/(N × N × N ) ' Q × Q × Q is abelian-by-finite. Since W = Ker(ρ) is central in DL and E/W ' ρ(E) is a subgroup of the nilpotent group N × N × N , we deduce that E is nilpotent of class at most the nilpotency class of N plus one. Since χ(G)/E is abelian-by-finite we are done.  Theorem 5.2. If G is nilpotent-by-abelian-by-finite of type F P∞ then χ(G) is of type F P∞ . Proof. We use the notation from the proof of Lemma 5.1. By the definition of E in Lemma 5.1 we see that E contains the normal closure of N, N ψ in χ(G). Thus, χ(G/N ) maps onto χ(G)/E. Also as the commutator subgroup E 0 = [E, E] contains the normal closure of N 0 = [N, N ] and (N 0 )ψ in χ(G), it follows that χ(G/N 0 ) maps onto χ(G)/E 0 . Consider the natural projections χ(G) → χ(G/N 0 ) → χ(G)/E 0 . By Theorem 2.3 χ(G) is of type F P∞ if and only if χ(G)/E 0 is of type F P∞ . Furthermore every quotient of a soluble group of type F P∞ is of type F P∞ , so to prove the theorem is equivalent to show that χ(G/N 0 ) is of type F P∞ . Again as G/N 0 is of type F P∞ it suffices to prove the theorem for G/N 0 . Thus we can assume from now on that N 0 = 1, so N is abelian. Set A1 = N ⊆ G ⊆ χ(G), A2 = N ψ ⊆ Gψ ⊆ χ(G) and note that since Q is abelian-by-finite by [22] χ(G) is polycyclic-by-finite, hence χ(G)/hA1 , A2 iχ(G) ' χ(Q) is polycyclic − by − f inite. Define T = χ(G)/E 0 , A0 = hA1 , A2 iT ⊆ T, where we have identified A1 and A2 with their images in T . Since the commutator E 0 is ψ-invariant we have an involution of T induced by ψ. By abuse of notation we call ψ this automorphism of T . Note that there is a natural projection δ : χ(Q) = χ(G)/hA1 , A2 iχ(G) → T /A0 ,

ON WEAK COMMUTATIVITY IN GROUPS

9

hence T /A0 is polycyclic-by-finite. Note that A0 ⊆ E0 = E/[E, E] ⊆ T and since E0 /A0 is a subgroup of the polycyclic-by-finite group T /A0 , the group E0 /A0 is a finitely generated abelian group. Note that T /E0 ' χ(G)/E ' Im(ρ)/(N × N × N ) ⊆ Q × Q × Q. Let Q0 be an abelian normal subgroup of finite index in Q and T0 /E0 be the preimage of Q0 × Q0 × Q0 in T /E0 . Thus T0 is a subgroup of finite index in T containing E0 and T0 /E0 is abelian. By the additivity of Σc (see Lemma 2.1) and the fact that ΣcE0 /A0 (Q) = ∅ (since E0 /A0 is finitely generated abelian) we deduce that ΣcE0 (T0 /E0 ) = ΣcA0 (T0 /E0 ) ∪ ΣcE0 /A0 (T0 /E0 ) = ΣcA0 (T0 /E0 ) ∪ ∅ = ΣcA0 (T0 /E0 ). Thus E0 is ∞-tame as Z(T0 /E0 )-module if and only if A0 is ∞-tame as Z(T0 /E0 )module. Note that T0 /E0 ' Q1 × Q2 , where Q1 ' {(q, q, 1) | q ∈ Q0 } ' Q0 and Q2 = Qψ 1 = {(1, q, q) | q ∈ Q0 }. Recall the commutator calculus [ab, c] = ([a, c]b )[b, c] and [a, bc] = [a, c]([a, b]c ), where [a, b] = a−1 b−1 ab. Let h ∈ A1 , g ∈ T0 . Then since [g, g ψ ] = 1 = [h, hψ ] we have in T 1 = [hg, hψ g ψ ] = ([h, hψ g ψ ]g ).[g, hψ g ψ ] =

(5.1)

([h, g ψ ]g ).([h, hψ ]g

ψ

g

ψ

ψ

).[g, g ψ ].[g, hψ ]g = ([h, g ψ ]g ).[g, hψ ]g .

Let g ∈ T0 belongs to the image of G in T . We write q1 for the image of g in Q1 and q2 for the image of g ψ in Q2 , thus q2 = q1ψ . For a ∈ A0 , t ∈ T0 , q(t) the image of t in T0 /E0 we write a ◦ q(t) for at = t−1 at. Then by (5.1) we have h ◦ (q2 − 1)q1 = (hψ ) ◦ (q1 − 1)q2 .

(5.2)

Note that for x ∈ Q1 we have (h ◦ x)ψ = hψ ◦ (xψ ). Then by (5.2) applied for h ◦ x instead of h we get h ◦ (x(q2 − 1)q1 ) = (hψ ) ◦ (xψ (q1 − 1)q2 ).

(5.3)

On other hand acting on (5.2) on the right by x we get h ◦ ((q2 − 1)q1 x) = (hψ ) ◦ ((q1 − 1)q2 x).

(5.4)

Then by (5.3) and (5.4) and since T0 /E0 ' Q1 × Q2 is commutative we deduce that (hψ ) ◦ ((q1 − 1)(x − xψ )q2 ) = 0, hence A2 ◦ Aug(ZQ1 )(x − xψ ) = 0 for every x ∈ Q1 .

(5.5) Similarly

A1 ◦ Aug(ZQ2 )(x − xψ ) = 0 for every x ∈ Q1 .

(5.6) Note that

A0 = A1 ◦ Z[Q1 × Q2 ] + A2 ◦ Z[Q1 × Q2 ] = A1 ◦ Z[Q2 ] + A2 ◦ Z[Q1 ].

10

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

By (5.2), (5.5) and (5.6) A0 is a surjective image of B0 = (A1 ⊗Z ZQ2 ⊕ A2 ⊗Z ZQ1 )/R, where R = h{a2 ◦ (q1 − 1)(x − xψ ) | q1 ∈ Q1 , x ∈ Q1 , a2 ∈ A2 }∪ {a1 ◦ (q2 − 1)(x − xψ ) | q2 ∈ Q2 , x ∈ Q1 , a1 ∈ A1 }∪ ψ {a1 ◦ (q2 − 1)q1 − (aψ 1 ) ◦ (q1 − 1)q2 | a1 ∈ A1 , q1 ∈ Q1 , q2 = q1 }i. By Lemma 3.2 B0 is ∞-tame as Z[Q1 ×Q2 ]-module, hence its quotient A0 is ∞-tame as Z[Q1 × Q2 ]-module. 

Remark The proof of Theorem 5.2 together with Lemma 3.2 show that if N/N 0 is m-tame as ZQ0 -module then E/E 0 is m-tame as Z[Q1 × Q2 ]-module. Proof of Theorem A Since a soluble group G of type F P∞ is nilpotent-byabelian-by-finite then by Lemma 5.1 and Theorem 5.2 χ(G) is a soluble group of type F P∞ . 6. A generalization of the Lima-Oliveira approach In this section we generalise the approach used by Lima and Oliveira in [22] to prove that if G is polycyclic-by-finite then χ(G) is polycyclic-by-finite. The new ingredient is the following result. Theorem 6.1. Let G be a group of type F P2 and S = h(h, h−1 ) | h ∈ Gi ≤ G × G. Suppose that G0 /G00 is finitely generated. Then H2 (S, Z) is finitely generated. Proof. Let P : . . . → P2 → P1 → P0 → Z → 0 be a free resolution of the trivial right ZG-module Z with Pi finitely generated for i ≤ 2. Consider the complex P ⊗ZG0 Z : . . . → P2 ⊗ZG0 Z → P1 ⊗ZG0 Z → P0 ⊗ZG0 Z → Z → 0, where Pi ⊗ZG0 Z is a free Z[G/G0 ]-module for all i and is finitely generated for i ≤ 2. Since G/G0 is a finitely generated abelian group we have that Z[G/G0 ] is a Noetherian ring, hence (6.1) Hi (G0 , Z) ' Hi (P⊗ZG0 Z) is finitely generated as Z[G/G0 ]-module for i ≤ 2. Note that G0 × G0 ⊆ S. Consider the Lyndon-Hoschild-Serre spectral sequence = Hp (S/(G0 × G0 ), Hq (G0 × G0 , Z)) converging to Hp+q (S, Z). Note that by K¨ unneth formula for any q ≥ 0 M Hq (G0 × G0 , Z) ' Hi (G0 , Z) ⊗Z Hq−i (G0 , Z). 2 Ep,q

0≤i≤q 2 Ep,q

2 We will analyse for p + q = 2 and will show that Ep,q is a finitely generated 0 0 abelian group for p + q = 2 . First using that S/(G × G ) ' G/G0 we get 2 E2,0 = H2 (S/(G0 × G0 ), H0 (G0 × G0 , Z)) = H2 (G/G0 , Z)

is a finitely generated abelian group since G/G0 is finitely presented. Note that 2 E1,1 = H1 (S/(G0 × G0 ), H1 (G0 × G0 , Z)) ' H1 (G/G0 , V1 × V2 ),

where for q ∈ G/G0 the action of q on V1 = G0 /G00 is via the conjugation action and on V2 = G0 /G00 is via the conjugation action of q −1 . Thus V1 × V2 is

ON WEAK COMMUTATIVITY IN GROUPS

11

finitely generated as Z[G/G0 ]-module with the described above action. By Noetherianess of Z[G/G0 ] it follows that V1 × V2 has type F P∞ as Z[G/G0 ]-module, hence Hi (G/G0 , V1 × V2 ) is a finitely generated abelian group for all i ≥ 0, in particular 2 E1,1 = H1 (G/G0 , V1 × V2 ) is finitely generated. 2 Consider E0,2 . Note that by K¨ unneth formula M 2 E0,2 = H0 (S/(G0 ×G0 ), H2 (G0 ×G0 , Z)) = H0 (G/G0 , Hi (G0 , Z)⊗Z Hq−i (G0 , Z)) 0≤i≤2

M

'

H0 (G/G0 , Hi (G0 , Z) ⊗Z Hq−i (G0 , Z)).

0≤i≤2

Observe that by (6.1) H0 (G/G0 , H2 (G0 , Z) ⊗Z H0 (G0 , Z)) ' H0 (G/G0 , H2 (G0 , Z)) is a finitely generated abelian group. Similarly H0 (G/G0 , H0 (G0 , Z) ⊗Z H2 (G0 , Z)) is a finitely generated abelian group. Finally consider H0 (G/G0 , H1 (G0 , Z) ⊗Z H1 (G0 , Z)) ' H0 (G/G0 , V1 ⊗Z V2 ). Since V1 ' V2 is finitely generated as an abelian group we see that V1 ⊗ V2 is a finitely generated an abelian group, so H0 (G/G0 , V1 ⊗ V2 ) is finitely generated as an abelian group. Actually this is the only place we use essentially that G0 /G00 2 is finitely generated as an abelian is finitely generated. Thus we proved that Ep,q ∞ group for all p + q = 2, hence Ep,q is finitely generated as an abelian group for all p + q = 2. By the convergence of the spectral sequence we deduce that H2 (S, Z) is finitely generated.  Remark 1. In the proof of the above result it is important to study when H0 (Q, V1 ⊗Z V2 ) is finitely generated, where Q is finitely generated abelian group and V2 is equipped with Q-action antipodal to the Q-action on V1 , but as abelian groups V1 and V2 are the same. In [10] a criterion for finite generation as ZQmodule (via the diagonal Q-action) of tensor powers of modules is established. The method introduced there implies that for any two finitely generated ZQ-modules M1 and M2 we have that M1 ⊗Z M2 is finitely generated via the diagonal Q-action if and only if for every real valuation v : Z → R ∪ ∞ we have for the valuation ∆-invariants defined in [10] that ∆vV1 (Q) ∩ −∆vV2 (Q) = {0}. By the link between the Σ and ∆ theory [9] if [χ] ∈ / ΣV1 (Q) then there is a positive real number r such that rχ ∈ ∆vV1 (Q) for some valuation v. Since [rχ] = [χ] without loss of generality r = 1. Note that since the Q-action on V2 is the antipodal compared to the Qaction on V1 we see that ∆vV1 (Q) = −∆vV2 (Q), hence χ ∈ ∆vV1 (Q) ∩ −∆vV2 (Q) 6= {0}, a contradiction. Thus S(Q) = ΣV1 (Q) and by [12] this implies that V1 is a finitely generated abelian group. This shows that for the modules from the proof of the above theorem V1 ⊗Z V2 is finitely generated as ZQ-module via the diagonal Q-action if and only if V1 (and hence V2 ) is finitely generated. Remark 2. In the above theorem the condition that G is of type F P2 could be substituted with G is finitely generated and H0 (G/G0 , H2 (G0 , Z)) is a finitely generated as abelian group. Remark 3. If we assume the stronger condition that G0 is finitely generated and G is finitely presented the above theorem follows immediately from of [23, Thm. C]

12

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

that shows that S is finitely presented. Note that in the metabelian case this was established earlier in [3]. Proof of Theorem B Write W for W (G) and L for L(G), recall that W ⊆ L. Observe that by [22, Prop. 1] L/L0 is finitely generated. Since W/(L0 ∩ W ) is a subgroup of the finitely generated abelian group L/L0 we get that W/(L0 ∩ W ) is finitely generated abelian, hence of type F P∞ . Recall that χ(G)/W is a subgroup of G × G × G. Note that S = L/W ' ρ(L) = h{(g, 1, g −1 ) | g ∈ G}i ' h{(g, g −1 ) | g ∈ G}i, so by Theorem 6.1 S has the property that H2 (S, Z) is finitely generated. Since L0 ∩ W → L → L/(L0 ∩ W ) is a stem extension (i.e. a central extension with central part inside the commutator) it follows that L0 ∩ W is a quotient of H2 (L/W, Z). Indeed let µ : F → L be an epimorphism of groups with F free and let R = µ−1 (W ). Then µ−1 (L0 ∩ W ) = µ−1 (L0 ) ∩ µ−1 (W ) = (F 0 Ker(µ)) ∩ R = (F 0 ∩ R)Ker(µ), so µ(F 0 ∩ R) = µ((F 0 ∩ R)Ker(µ)) = µ(µ−1 (L0 ∩ W )) = L0 ∩ W . On other hand µ([R, F ]) = [µ(R), µ(F )] = [W, L] = 1, so µ induces an epimorphism θ : (F 0 ∩ R)/[R, F ] → L0 ∩ W. The Hopf formula H2 (L/W, Z) ' (F 0 ∩ R)/[R, F ] completes the proof of the fact that L0 ∩ W is a quotient of H2 (L/W, Z), hence L0 ∩ W is finitely generated. As already shown W/(L0 ∩ W ) is finitely generated, so W is finitely generated. This completes the proof of Theorem B. Remark 4. By Remark 1 and Remark 2 the proof of the above theorem works under the weaker (but technical) conditions that G is finitely generated, H0 (G/G0 , H2 (G0 , Z)) is finitely generated and H0 (G/G0 , V1 ⊗ V2 ) is finitely generated, where V1 and V2 are defined in the proof of Theorem 6.1. 7. More on the homological and homotopical type of χ(G) 7.1. Preliminaries on the higher dimensional Σ-invariants. In this subsection we consider some results on higher dimensional Σ-invariants that will be used only in section 7. The Bieri-Strebel invariant from subsection 2.1 was generalised first for finitely generated groups in [14] and later in [11], [29] higher dimensional homological and homotopical invariants were defined. By definition for a finitely generated group G the homological invariant in dimension n is Σn (G, Z) = {[µ] ∈ S(G) | Z has type F Pn as ZGµ − module}, where as before S(G) = Hom(G, R) \ {0}/ ∼ but G is not assumed abelian. The homotopical invariant Σn (G) is more complicated to explain, it is defined only for groups G of homotopical type Fn i.e. there is a K(G, 1) complex with finite nskeleton. Note that the property F2 is equivalent with finite presentability and for n ≥ 3 a group is of type Fn if and only if G is F Pn and is finitely presented. As well though the definition of Σn (G, Z) requires only that G is finitely generated, if Σn (G, Z) 6= ∅ we have that G is F Pn . For us it is important to know that Σ1 (G) = Σ1 (G, Z) and Σn (G) = Σn (G, Z) ∩ Σ2 (G). We will need later on the following results.

ON WEAK COMMUTATIVITY IN GROUPS

13

Theorem 7.1. [29], [11] Let G be a group of type Fm (respectively F Pm ) and N be a subgroup of G that contains the commutator of G. Then N is of type Fm (respectively F Pm ) if and only if S(G, N ) = {[µ] ∈ S(G) | µ(N ) = 0} ⊆ Σm (G) ( respectively Σm (G, Z)). Theorem 7.2. [24], [25] Let m1 , m2 be non-negative integers and G1 and G2 be groups of type Fm1 +m2 +1 (respectively F Pm1 +m2 +1 ). Let [µ1 ] ∈ Σm1 (G1 ) (respectively Σm1 (G1 , Z) ) and [µ2 ] ∈ Σm2 (G2 ) (respectively Σm2 (G2 , Z)). Then [(µ1 , µ2 )] ∈ Σm1 +m2 +1 (G1 × G2 ) (respectively Σm1 +m2 +1 (G1 × G2 , Z)). In [7] Bieri and Geoghegan proved that a direct product formula holds for the Sigma invariants Σn (G, K), where K is a field, and in [8] Bieri, Geoghegan and Kochloukova showed that for the Thompson group x

F = hx0 , x1 , x2 , . . . | xi j = xi+1 for 0 ≤ j < ii we have that Σn (F, Q) = Σn (F, Z) for every n ≥ 1. As observed and used in [8] this implies that the direct product formula for Sigma invariants with coefficients in Z holds for the Thompson group F. The precise result is stated in the following theorem. Theorem 7.3. Let F1 , . . . , Fk be isomorphic copies of the Thompson group F . Then for any n ≥ 1 we have Σn (F1 × . . . × Fk , Z)c = ∪i1 +i2 +...+ik =n Σi1 (F1 , Z)c ∗ Σi2 (F2 , Z)c ∗ . . . ∗ Σik (Fk , Z)c , where ∗ denotes the spherical join. In particular if µj ∈ Hom(Fj , R) \ {0} for 1 ≤ j ≤ k then [(µ1 , . . . , µk )] ∈ Σn (F1 × . . . × Fk , Z)c if and only if there are i1 + i2 + . . . + ik = n such that [µj ] ∈ Σij (Fj , Z)c for all 1 ≤ j ≤ k. In [8] Bieri, Geoghegan and Kochloukova proved that Σn (F, Z)c = conv≤2 Σ1 (F, Z)c = Σ2 (F, Z)c for every n ≥ 2, hence Σn (F, Z)c lies in an open semisphere of S(F ). Combining this result with Theorem 7.3 we obtain Corollary 7.4. For every m ≥ 1 we have {[(µ1 , −µ1 , µ1 )] ∈ S(F × F × F, Z) | [µ1 ] ∈ S(F )} ⊆ Σm (F × F × F, Z).

7.2. Proof of Corollary C, Theorem D and some examples. Proposition 7.5. If G is finitely presented then χ(G)/R(G) is finitely presented. Proof. Write W for W (G) and R for R(G). Since H2 (G, Z) ' W/R is a finitely generated abelian group it is sufficient to show that χ(G)/W is finitely presented. Let H = χ(G)/W ' {(g1 , g2 , g3 ) | g1 , g2 , g3 ∈ G, g1 g2−1 g3 ∈ G0 } ⊆ G × G × G =: D. Note that H contains the commutator subgroup of D and for a homomorphism δ = (δ1 , δ2 , δ3 ) ∈ Hom(D, R) \ {0} we have that δ(H) = 0 if and only if δ1 + δ2 = 0 = δ2 + δ3 , δ1 6= 0. By Theorem 7.2 for every µ = (µ1 , µ2 ) ∈ Hom(G × G, R) such that µ1 6= 0, µ2 6= 0 we have that [µ] ∈ Σ1 (G × G). Then using again Theorem 7.2

14

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

for every µ3 ∈ Hom(G, R) \ {0} we have that [(µ, µ3 )] ∈ Σ2 (D). In particular, for µi = δi for 1 ≤ i ≤ 3, we obtain S(D, H) = {[δ] ∈ S(D) | δ(H) = 0} ⊆ Σ2 (D). Hence by Theorem 7.1 H is finitely presented.



Proof of Corollary C By Theorem B W = W (G) is finitely generated. Hence R = R(G) is finitely generated abelian and so is finitely presented. By Proposition 7.5 χ(G)/R(G) is finitely presented. Hence χ(G) is finitely presented. Proof of Theorem D Recall that H = χ(G)/W (G) = {(g1 , g2 , g3 ) ∈ G×G×G | g1 g2−1 g3 ∈ G0 }. Hence S(G × G × G, H) = {[µ] ∈ S(G × G × G) | µ(H) = 0} = (7.1)

{[µ] ∈ S(G × G × G) | µ = (µ1 , −µ1 , µ1 ), µ1 ∈ Hom(G, R) \ {0}}.

Since the commutator G0 has type Fs (respectively F Ps ) by Theorem 7.1 Σs (G) = S(G) (respectively Σs (G, Z) = S(G)). Then for µ = (µ1 , −µ1 , µ1 ) 6= 0 we have that [µ1 ] ∈ Σs (G) (respectively Σs (G, Z)) and [−µ1 ] ∈ Σs (G) (respectively Σs (G, Z)) . Thus by Theorem 7.2 we deduce that [(µ1 , −µ1 )] ∈ Σt (G × G) ( respectively Σt (G × G, Z)), where t = min{k, 2s + 1}. Then applying Theorem 7.2 again we get that [(µ1 , −µ1 , µ1 )] ∈ Σd (G × G × G) (respectively Σd (G × G × G, Z)), where d = min{k, s + t + 1} = k. Thus S(G × G × G, H) ⊆ Σk (G × G × G) (respectively Σk (G × G × G, Z)). Then by Theorem 7.1 H has homotopical type Fk (respectively F Pk ). Finally if G0 /G00 is finitely generated (this holds for example if s ≥ 1) and k ≥ 2 by Theorem B W (G) is finitely generated abelian, so is of type F∞ . Hence χ(G) has the same homotopical (respectively homological) type as H = χ(G)/W (G). This completes the proof of Theorem D. Example Let F be the Thompson group with infinite presentation x

hx0 , x1 , x2 , . . . | xi j = xi+1 for 0 ≤ j < ii. By [18] F is of type F∞ i.e. is finitely presented and has type F P∞ . The invariant Σ1 (F )c has exactly two non-antipodal points. Since Σ1 (F, Z) = Σ1 (F ) 6= S(F ) we see by Theorem 7.1 that F 0 is not finitely generated. But F 0 is simple [17], so F 0 = F 00 . Thus by Theorem B W (F ) is finitely generated and χ(F ) is finitely presented. We claim that χ(F ) is of type F P∞ . It suffices to show that H = χ(F )/W (F ) is of type F P∞ . By (7.1) and by Corollary 7.4 we deduce that S(F × F × F, H) ⊆ Σk (F × F × F, Z) for every k ≥ 1. Then by Theorem 7.1 H has type F Pk for every k.

ON WEAK COMMUTATIVITY IN GROUPS

15

8. The group ν(G) Let ν(G) be the group defined in [27] as ψ

h3 h3 = [hh1 3 , (hh2 3 )ψ ]i. ν(G) = hG, Gψ | [h1 , hψ = [h1 , hψ 2] 2]

By [27] there is a central subgroup ∆(G) of ν(G) such that ν(G)/∆(G) ' χ(G)/R(G)

(8.1)

and ∆(G) is a quotient of H2 (χ(G)/R(G), Z). Proof of Theorem E a) By Proposition 7.5 χ(G)/R(G) is finitely presented, hence H2 (χ(G)/R(G), Z) is finitely generated and hence ∆(G) is finitely generated abelian. Thus ν(G) is finitely presented if and only if ν(G)/∆(G) ' χ(G)/R(G) is finitely presented. b) Soluble groups of type F P∞ are finitely presented and nilpotent-by-abelianby-finite. By the proof of part a) ∆(G) is finitely generated abelian, hence of type F P∞ . Then ν(G) is a soluble group of type F P∞ if and only if ν(G)/∆(G) ' χ(G)/R(G) is a soluble group of type F P∞ . Finally apply Lemma 5.1 and Theorem 5.2. 9. Some examples In this section we consider various examples included in Theorem F. First we state some properties of the group χ(G) for a general group G from [30]. By [30, Lemma 4.1.6] (9.1)

[g1ψ , g2 ] = [g1 , g2ψ ] and [g1ψ g1−1 , [g2 , g3ψ ]] = 1 for g1 , g2 , g3 ∈ G.

Note that this implies ψ

[g1 , g2ψ ][g3 ,g4 ] = [g1 , g2ψ ][g3 ,g4 ] .

(9.2) Indeed ψ

−1 −ψ g4 g3 g4ψ

[g1 , g2ψ ][g3 ,g4 ] = [g1 , g2ψ ]g3

−ψ −ψ g4 g3 g4ψ

= [g1 , g2ψ ]g3

−ψ

= [g1 , g2ψ ](g4 g3 )

ψ ψ −1 −1 −1 −1 −1 ψ ψ [g1 , g2ψ ](g4 g3 ) g3 g4 = [g1 , g2ψ ]g3 g4 g3 g4 = [g1 , g2ψ ](g3 g4 g3 ) g4 −1 −1 −1 −1 ψ [g1 , g2ψ ](g3 g4 g3 g4 ) = [g1 , g2ψ ]g3 g4 g3 g4 = [g1 , g2ψ ][g3 ,g4 ] .

g3 g4ψ

=

=

Furthermore by [30] (9.3) ψ [g1 , g2ψ ] . . . [g2k−1 , g2k ] ∈ W (G) if and only if [g1 , g2 ] . . . [g2k−1 , g2k ] = 1 in G. 1. First we study the case when G is a metabelian Baumslag-Solitar group i.e. G = ha, b | a−1 ba = bm i for some m ∈ Z \ {0}. If m 6= 1 it is easy to check that H2 (G, Z) = 0 either by a direct calculation using a free resolution of the trivial ZG-module Z coming from Fox derivations [16, exer. 3, p.45] or directly by considering stem extensions. Set A = hai and B = hbiG . Lemma 9.1. The group χ(G) has the following properties : 1. [B, B ψ ] = 1 = [A, Aψ ]. ψ m b2 a and [a1 , bψ = [a1 , bψ 2. [a1 , bψ 1] 1 ] for a1 ∈ A, b1 , b2 ∈ B. Hence B 1 ] = [a1 , b1 ] ψ ψ and B centralize the abelian group [B , A]. 3. R(G) = [[G, [G, ψ]], Gψ ] = 1. Furthermore W (G) = 1 if m 6= 1 and W (G) ' Z if m = 1.

16

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

Proof. For any element b0 ∈ B we have [b0 , bψ 0 ] = 1, hence for s ∈ Z we have ψ s ψ ) , b ], hence [B, B ] = 1. Similarly [A, Aψ ] = 1. ] = [(b 1 = [bs0 , bψ 0 0 0 −1 1−mz Let a1 = az ∈ A = hai, b1 , b2 ∈ B, hence for b3 = [a1 , b2 ] = a−1 1 b2 a1 b2 = b2 we get ψ ψ b3 ψ ψ b3 b2 b2 = [ab12 , (bψ [a1 , bψ 1 ) ] = [a1 b3 , b1 ] = [a1 , b1 ] [b3 , b1 ] = [a1 , b1 ] . 1] z

−m Then b3 b−1 centralizes [a1 , bψ 1 ]. Since b2 is an arbitrary element of B we 2 = b2 ψ ψ ψ ψ deduce that B centralizes [a1 , b1 ] and B ψ centralizes [a1 , bψ 1 ] = [a1 , b1 ] = [a1 , b1 ]. On other hand ψ ψ a ψ m ψ a a m [a1 , bψ for a1 ∈ A = hai, b1 ∈ B. 1 ] = [a1 , b1 ] = [a1 , b1 ] = [a1 , b1 ] = [a1 , b1 ]

Note that we have proved parts 1 and 2 from Lemma 9.1. Using (9.2) and the fact that [a2 , b2 ] ∈ B for a2 ∈ A, b2 ∈ B we obtain that ψ

[a2 ,b2 ] [a2 ,b2 ] [a1 , bψ = [a1 , bψ = [a1 , bψ 1] 1] 1 ] for a1 , a2 ∈ A, b1 , b2 ∈ B.

Thus [A, B ψ ] = [Aψ , B] is abelian. Recall that [B, B ψ ] = 1 = [A, Aψ ]. Then i

D(G) = [G, Gψ ] = [BA, B ψ Aψ ] = [A, B ψ ] = {[a, bψ ]m | i ∈ Z}. The following claim completes the proof of part 3 of Lemma 9.1. ψ −1 ψ Claim a) The commutators [a−1 1 a1 , b1 ] and [b1 b1 , a1 ] are both centralized by ψ ψ −1 ψ −1 ψ a m a m B, B ψ and Aψ , [a−1 and [b−1 for 1 a1 , b1 ] = [a1 a1 , b1 ] 1 b1 , a1 ] = [b1 b1 , a1 ] a1 ∈ A, b1 ∈ B; b) R(G) = 1; c) W (G) = 1 for m 6= 1 and W (G) ' Z for m = 1.

Proof of the Claim. a) Note that for b4 = [a−1 1 , b1 ] we have by Lemma 9.1, part 2, that (9.4) ψ ψ ψ ψ −1 aψ ψ 1 [a−1 1 a1 , b1 ] = [a1 , b1 ] [a1 , b1 ] = b4 [b4 , a1 ][a1 , b1 ] commutes with B and B . ψ ψ To show that [a−1 1 a1 , b1 ] commutes with A consider ψ

ψ

ψ

ψ

ψ ψ a a a (b4 [b4 , aψ = ba4 [b4 , aψ = 1 ][a1 , b1 ]) 1 ] [a1 , b1 ]

(9.5)

ψ

ψ m ψ a ψ a ψ m ba4 [b4 , aψ 1 ] [a1 , b1 ] = b4 [b4 , a ][b4 , a1 ] [a1 , b1 ] .

Then making calculation in the abelian group [A, B ψ ] we have by (9.4), (9.5) that ψ ψ ψ ψ ψ [[a−1 1 a1 , b1 ], a ] = [b4 [b4 , a1 ][a1 , b1 ], a ] =

(9.6)

ψ

ψ ψ −1 a m−1 ψ (b4 [b4 , aψ (b4 [b4 , aψ = [b4 , aψ ][b4 , aψ [a1 , b1 ]m−1 . 1 ][a1 , b1 ]) 1 ][a1 , b1 ]) 1] (ms −1)/ms

Suppose a1 = as , so we have b4 = b1

s

(m −1)/m

m−1 ψ [b4 , aψ ][b4 , aψ [a1 , b1 ]m−1 = [b1 1]

[b1 , aψ ](m

s

−1)/ms

and by Lemma 9.1 (2) we have (ms −1)/ms

, aψ ][b1 s

(m−1)(m [b1 , aψ 1]

[b1 , aψ ](m

(9.7)

s

s

−1)/ms

−1)/ms

m−1 ψ , aψ [a1 , b1 ]m−1 = 1]

m−1 [aψ = 1 , b1 ] s

(1−m)/m [b1 , aψ . 1]

Now by Lemma 9.1 (2) (9.8)

ψ ψ s ψ 1+m+...+m [b1 , aψ 1 ] = [b1 , a1 ] = [b1 , a ] = [b1 , a]

s−1

= [b1 , aψ ](m

s

−1)/(m−1)

,

ON WEAK COMMUTATIVITY IN GROUPS

17

hence by (9.6), (9.7), (9.8) ψ ψ ψ (m [[a−1 1 a1 , b1 ], a ] = [b1 , a ]

s

−1)/ms −(ms −1)/ms

=1

Note that by (9.4) ψ −1 ψ a a −1 ψ m −1 ψ a m [a−1 1 a1 , b1 ] = [(a1 a1 ) , b1 ] = [a1 a1 , b1 ] = [a1 a1 , b1 ] . ψ −1 ψ ψ ψ To show that [b−1 1 b1 , a1 ] commutes with G = A B set b5 = [b1 , a1 ] ∈ B. First note that

(9.9)

bψ

ψ

ψ ψ ψ −1 b1 ψ ψ ψ 1 [b−1 1 b1 , a1 ] = [b1 , a1 ] [b1 , a1 ] = b5 [b1 , a1 ] = b5 [b1 , a1 ] ∈ BB [B , A].

By Lemma 9.1 (2) BB ψ [B ψ , A] is an abelian group, hence B and B ψ centralize ψ [b−1 1 b1 , a1 ]. Thus by (9.1), (9.9) and Lemma 9.1 (2) ψ

ψ

ψ a a a [b−1 = b5 [b5 , aψ ][bψ = b5 [b5 , aψ ][bψ 1 b1 , a1 ] 1 , a1 ] 1 , a1 ] = ψ m ψ ψ ψ b5 [bψ 5 , a][b1 , a1 ] ∈ B[B , A] ⊆ BB [B , A].

(9.10)

Observe that by Lemma 9.1 (2) the conjugation with a on B and [B ψ , A] is the same as taking mth power, hence ψ −1 ψ a m [b−1 1 b1 , a1 ] = [b1 b1 , a1 ] . s

Then by (9.9) and (9.10) and assuming a1 = as , so b5 = b1−m , we have 1 ψ

ψ −1 ψ ψ −1 −1 ψ [[b−1 [b1 b1 , a1 ]a = 1 b1 , a1 ], a ] = [b1 b1 , a1 ] s

ψ −1 m ψ m−1 (b5 [bψ b5 [b5 , aψ ][bψ = [b1−m , aψ ][b1 , (aψ )s ]m−1 = 1 , a1 ]) 1 , a1 ] = [b5 , a ][b1 , a1 ] 1 s

s−1

[b1 , aψ ]1−m [b1 , aψ ](1+m+...+m

(9.11)

)(m−1)

= 1.

b) Note that (9.12)

ψ b1 −1 ψ −1 ψ −1 ψ −1 ψ (a1 b1 )−1 (a1 b1 )ψ = (a−1 1 a1 ) (b1 b1 ) = (a1 a1 )[a1 a1 , b1 ](b1 b1 ).

By Claim (a) and (9.12) ψ ψ χ(G) χ(G) [G, [G, ψ]] ⊆ h[a−1 , [b−1 | a1 ∈ A, b1 ∈ Bi = 1 a1 , b1 ] 1 b1 , a1 ] i

i

ψ ψ m m h[a−1 , [b−1 | a1 ∈ A, b1 ∈ B, i ∈ Zi =: S. 1 a1 , b1 ] 1 b1 , a1 ]

Then using again Claim (a) R(G) = [[G, [G, ψ]], Gψ ] ⊆ [S, Gψ ] = 1. c) Finally R(G) is the kernel of the map W (G) → H2 (G, Z), so W (G) ' H2 (G, Z). Note that H2 (G, Z) = 0 if m 6= 1 and H2 (G, Z) ' Z if m = 1.  2. Let G be the group from the previous example where m = 2 and K = hzi ' Z, H = G × K. We keep the notations from part 1 i.e. G = BA, where B is the normal closure of b and A is the cyclic group generated by a. We first establish some properties of χ(H).

18

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

Lemma 9.2. The following commutator identities hold in the group χ(H), where H =G×K : 1. [[B, z ψ ], [B, aψ ]] = 1 and [[Aψ , z], [Aψ , B]] = 1 , hence [[B ψ , z], [B ψ , a]] = 1 and [[A, z ψ ], [A, B ψ ]] = 1; ψ ψ ψ 2. [bψ 1 , z, a1 ] = [b1 , a1 , z] = 1 and [a1 , z, b1 ] = [a1 , b1 , z] = 1 for a1 ∈ A, b1 ∈ B; ψ 3. [B , z] = 1; 4. [B ψ , a, z]ψ = [B ψ , a, z ψ ] = [B ψ , z, aψ ] = [B ψ , z, a]ψ = 1. Proof. 1. By (9.1) and (9.2) for every b1 , b2 ∈ B and [b2 , a] = b0 ∈ B we have [b1 , z ψ ][b2 ,a

ψ

]

ψ b0 b0 b0 ψ = [(bψ = [b1 , z ψ ][b2 ,a] = [b1 , z ψ ]b0 = [bψ 1 ) , z ] = [b1 , z] = [b1 , z ]. 1 , z]

By (9.1) and Lemma 9.1 (2) for a1 , a2 ∈ A, b1 ∈ B we have ψ

a2 z a2 z 2z [aψ = [aψ = [aψ 1 , b1 ] 1 , b1 ] 1 , b1 ]

and ψ

za2 z = [aψ [aψ 1 , b1 ] 1 , b1 ]

ψ

aψ 2

ψ ψ

ψ

ψ

a2 z a2 z 2z 2z = [aψ = [aψ = [aψ = [aψ 1 , b1 ] 1 , b1 ] 1 , b1 ] 1 , b1 ] .

ψ

ψ

ψ

a2 z za2 [a2 ,z] Hence [aψ = [aψ , so [aψ = [aψ 1 , b1 ] 1 , b1 ] 1 , b1 ] 1 , b1 ]. 2. Note that using part 1 we have ψ

ψ ψ ψ ψ a1 a1 a1 [b1 ,a1 ] ψ [bψ = [(bψ [b1 , a1 , z] = [bψ 1 , z] 1 ) , z ] = [b1 [b1 , a1 ], z] = [b1 , z] 1 , z][b1 , a1 , z]. ψ a1 On other hand [bψ = [bψ 1 , z] 1 , z][b1 , z, a1 ], hence ψ [bψ 1 , z, a1 ] = [b1 , a1 , z].

(9.13)

Observe that for a1 ∈ A and b1 ∈ B using part 1 we obtain ψ ψ ψ ψ b1 b1 b1 [aψ = [(aψ 1 , z][a1 , z, b1 ] = [a1 , z] 1 ) , z ] = [a1 [a1 , b1 ], z] = ψ

ψ [a1 ,b1 ] ψ [aψ [a1 , b1 , z] = [aψ 1 , z] 1 , z][a1 , b1 , z],

hence ψ [aψ 1 , z, b1 ] = [a1 , b1 , z].

(9.14)

By Lemma 9.2 (1) and (9.2) we deduce for b2 ∈ B, a2 ∈ A that ψ

ψ [a2 ,b2 ] [a2 ,b2 ] [aψ = [aψ hence [aψ 1 , z] = [a1 , z] 2 , z] 2 , z, [a2 , b2 ]] = 1.

Observe that [A, B] = B, hence for an arbitrary element b1 ∈ B we have [aψ 1 , z, b1 ] = 1.

(9.15) By the Witt identity we have (9.16)

ψ

[bψ , a−1 , z]a [z, b−ψ , a]b [a, z −1 , bψ ]z = 1.

Recall that in section 4 the map ρ:H →H ×H ×H was defined by ρ(h) = (h, h, 1), ρ(hψ ) = (1, h, h) and Ker(ρ) = W (H). Observe that [z, b−1 ] = 1 in H, hence by (9.3)[z, b−ψ ] ∈ W (H) and so [z, b−ψ , a] ∈ W (H). Since [z, b−ψ , a] ∈ W (H) is centralized by the group D(H)L(H) defined in section 4 and ρ(D(H)L(H)) contains H 0 × H 0 × H 0 = B × B × B = ρ(hB, B ψ , [B ψ , a]i) we ψ deduce that [z, b−ψ , a]b = [z, b−ψ , a]. Using this, [a, z −1 ] = 1 and (9.16) we deduce (9.17)

[bψ , a−1 , z]a [z, b−ψ , a] = 1.

ON WEAK COMMUTATIVITY IN GROUPS

19

Note that (9.18)

[bψ , a−1 , z]a = [[bψ , a−1 ]a , z a ] = [[a, bψ ], z].

Then by (9.14), (9.17) and (9.18) (9.19)

[aψ , z, b] = [aψ , b, z] = [a, bψ , z] = [bψ , a−1 , z]a = [z, b−ψ , a]−1 .

Note that by the Witt identity and since [bψ , b] = 1 = [b, z] we have [bψ , z, b] = 1, −1 hence [bψ , z]b = [bψ , z] and (9.20)

[z, b−ψ , a] = [[bψ , z]b

−ψ

−1

, a] = [[bψ , z]b , a] = [bψ , z, a].

Thus by (9.15), (9.19), (9.20) we have 1 = [aψ , z, b] = [z, b−ψ , a]−1 = [bψ , z, a]−1 . Similarly for a1 ∈ A, b1 ∈ B (9.21)

−ψ −1 −1 1 = [aψ = [bψ . 1 , z, b1 ] = [z, b1 , a1 ] 1 , z, a1 ]

Finally (9.13), (9.14) and (9.21) complete the proof of Lemma 9.2 (2). 3. Note that z ψ z −1 ∈ L(H) centralizes W (H) and [b, z −ψ , a], [b, z −ψ ] ∈ W (H) by (9.3). Hence by Lemma 9.2 (2) ψ

[b, z −ψ , a]z = [b, z −ψ , a]z = [[b, (z −ψ )]z , az ] = ψ

(9.22)

[[b, z −ψ ]z , a] = [z ψ , b, a] = [z, bψ , a] = [[bψ , z]−1 , a] = 1,

(9.23)

[z ψ , a−1 , b]a = [[z ψ , a−1 ]a , ba ] = [a, z ψ , b2 ] = [aψ , z, b2 ] = 1.

Then since [a, b−1 ] = b and by (9.22), (9.23) and the Witt identity ψ

[b, z −ψ , a]z [z ψ , a−1 , b]a [a, b−1 , z ψ ]b = 1 we have [b, z ψ ]b = 1 hence [b, z ψ ] = 1. 4. [B ψ , a, z]ψ = [B, aψ , z ψ ] = [B ψ , a, z ψ ] and [B ψ , z, a]ψ = [B, z ψ , aψ ] = [B ψ , z, aψ ]. Finally part 2 completes the proof.  Lemma 9.3. M = hB, B ψ iχ(H) = hB, B ψ , [B ψ , a]i is an abelian group, χ(H) = M oχ(Q) is abelian-by-(nilpotent of class 2) and R(H) = 1, where Q = ha, zi ' Z2 . Proof. Note that by Lemma 9.1 and Lemma 9.2 [hB, B ψ i, χ(H)] = hB, B ψ , [B ψ , a]i and [hB, B ψ i, χ(H), χ(H)] = hB, B ψ , [B ψ , a], [B ψ , a, z], [B ψ , a, z ψ ]i = hB, B ψ , [B ψ , a]i. Hence M = hB, B ψ iχ(H) = hB, B ψ , [B ψ , a]i and by Lemma 9.1 M is abelian. Thus χ(H) = M o χ(Q), where Q = ha, zi ' Z2 , hence χ(Q) is nilpotent of class 2 and R(Q) = 1 (this is a particular case of Lemma 9.1). It is easy to check that ρ(M ) = B × B × B and since M ' B × B ψ × [B ψ , a] ' B × B × B we deduce that Ker(ρ) ∩ M = 1. On other hand Ker(ρ) ∩ χ(Q) = W (Q) ' H2 (Q, Z) ' Z, so Ker(ρ) ∩ χ(Q) = h[aψ , z]i ' Z.

20

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

Finally since ρ(M ) ∩ ρ(χ(Q)) = 1 we deduce that W (H) = Ker(ρ) = (Ker(ρ) ∩ M )(Ker(ρ) ∩ χ(Q)) = h[aψ , z]i ' Z. Since W (H)/R(H) ' H2 (H, Z) ' Z, where the last isomorphism follows from the K¨ unneth formula, we deduce that R(H) = 1 as required. This completes the proof.  3. Our final example is a group L with infinitely generated W (L). Let p be an odd prime number. By [28, Table 1, ix] there is a finite p-group H(p) = B(p)×hb3,p i such that R(H(p)) 6= 1. Furtheremore b3,p has order p and B(p) = hb1,p , b2,p i is non-abelian, [b1,p , b2,p ] is central in B(p), both b1,p and b2,p have order p. By [28, Table 1, ix] R(H(p)) has order p and W (H(p)) has order p5 . Consider a finite abelian group Q(p) ⊆ Aut(H(p)) generated by the automorphisms a1,p , a2,p and a3,p defined by a

1+δi,j

bi,pj,p = bi,p

(9.24)

.

Set G(p) = H(p) o Q(p). Consider the group L ≤ SL5 (Z[ 21 ]) generated by the matrices      1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0  1 1 0 0 0          B1 =  0 0 1 0 0 , B2 = 0 1 1 0 0 , B3 = 0 0 0 0 0 1 0  0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1

0 1 0 0 0

0 0 1 0 0

0 0 0 1 1

 0 0  0  0 1

0 1 0 0 0

0 0 1 0 0

0 0 0 2 0

 0 0  0  0 1

and  2 0  A1 =  0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

  1 0 0 0 1 0    0  , A2 = 0 0 0 0  0 1 0 0

0 0 1 2

0 0

  0 0 1 0 0 0    0 0  , A3 = 0 0  1 0 0 0 1

Thus L = B o Q, where B is the normal closure of hB1 , B2 , B3 i in L, so B is nilpotent group of class 2 and Q is the abelian group generated by the matrices A1 , A2 and A3 . Then (9.25)

δ

i,j [Bi , Aj ] = Bi−1 A−1 . j B i Aj = B i

By (9.24) and (9.25) there is a homomorphism of groups θp : L → G(p) defined by θp (Ai ) = ai,p and θp (Bi ) = bi,p for 1 ≤ i ≤ 3 and p an odd prime number. Note that θp (B) = B(p). Furthermore the conjugation action of Q on B induces an action of Q on χ(B) by (bψ )q = (bq )ψ whenever b ∈ B, q ∈ Q. Consider the homomorphism ρp : χ(L) → χ(B) o Q → χ(B(p)) o Q(p) where the first homomorphism j : χ(L) → χ(B) o Q identifies q with q ψ for q ∈ Q and the second homomorphismχ(B) o Q → χ(B(p)) o Q(p) is induced by θp and sends bψ to θp (b)ψ . As observed before R(H(p)) 6= 1 and R(H(p)) is generated by [[b1,p , b2,p ]ψ , b3,p ], so [[b1,p , b2,p ]ψ , b3,p ] has order p. Note that for every odd prime p ρp ([[B1 , B2 ]ψ , B3 ]) = [[b1,p , b2,p ]ψ , b3,p ],

ON WEAK COMMUTATIVITY IN GROUPS

21

hence [[B1 , B2 ]ψ , B3 ] ∈ χ(L) has infinite order. On other hand by (9.3) [[B1 , B2 ]ψ , B3 ] ∈ W (L) since [B1 , B2 , B3 ] = 1 in L. Note that [[B1 , B2 ]ψ , B3 ]B3 = [[B1 , B2 ], B3ψ ]B3 = [[B1 , B2 ]B3 , (B3ψ )B3 ] = [[B1 , B2 ], B3ψ ], hence j([[B1 , B2 ]ψ , B3 ]A3 ) = [([B1 , B2 ]A3 )ψ , B3A3 ] = [[B1 , B2 ]ψ , B32 ] = [[B1 , B2 ]ψ , B3 ]2 and

k 1 Z[ ] ' h[[B1 , B2 ]ψ , B3 ]A3 | k ∈ Zi ⊆ j(W (L)). 2 Thus we have proved that W (L) is not finitely generated. Observe that 1 1 1 V = B/B 0 ' Z[ ] ⊕ Z[ ] ⊕ Z[ ] 2 2 2 and Q ' Z3 = hA1 , A2 , A3 i and the conjugation action of Ai on V is multiplication on the i-th factor with 2. Thus V is ∞-tame as ZQ-module and L is of type F P∞ . By Theorem 5.2 and Lemma 5.1 χ(L) is of type F P∞ and is (nilpotent of class 3)by-abelian. Since L is soluble of type F P∞ we deduce that L is finitely presented, so W (L)/R(L) ' H2 (L, Z) is finitely generated. Hence R(L) is infinitely generated.

Remark We note that in the above example W (L) has Krull dimension 1 as ZQ-module and actually this is an upper bound for the Krull dimension of W (G) when G is nilpotent-by-abelian of type F P∞ . Indeed by the proof of Theorem A there is a short exact sequence N → χ(G) → Q of groups, where N is nilpotent, Q is abelian and N/[N, N ] is ∞-tame as ZQ-module. Since N is nilpotent, any central subgroup A of N that is normal in G is ∞-tame as ZQ-module, hence ⊗i A is finitely generated as ZQ-module via the diagonal Q-action for every i ≥ 1. Denote the Krull dimension of A as ZQ-module by s. Then by [15, Lemma 3] i(s − 1) ≤ h(Q) for every i ≥ 1, where h(Q) is the torsion-free rank of Q. Applying this for A = W (G) we obtain s ≤ 1 as claimed. References [1] Abels, H. An example of a finitely presented solvable group. Homological group theory (Proc. Sympos., Durham, 1977), pp. 205 - 211, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press, Cambridge-New York, 1979 [2] Aberg, H. Bieri-Strebel valuations (of finite rank). Proc. London Math. Soc. (3) 52 (1986), no. 2, 269 - 304 [3] Baumslag, G.; Bridson, M. R.; Holt, D. F.; Miller III, Ch. F. Finite presentation of fibre products of metabelian groups, J. Pure Appl. Algebra 181 (1) (2003) 15 - 22 [4] Bridson, M. R.; Howie, J.; Miller III, C. F.; Short, H.; On the finite presentation of subdirect products and the nature of residually free groups, American Journal of Mathematics 135 (4) (2013), 891 - 933 [5] Bestvina, M.; Brady, N. Morse theory and finiteness properties of groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 445 - 470 [6] Bieri, R. Homological dimension of discrete groups. Second edition. Queen Mary College Mathematical Notes. Queen Mary College, Department of Pure Mathematics, London, 1981 [7] Bieri, R.; Geoghegan, R. Sigma Invariants of Direct Products of Groups, Groups, Geometry and Dynamics, 4 (2010), 251 - 261 [8] Bieri, R.; Geoghegan, R.; Kochloukova, D. The Sigma Invariants of Thompson’s Group F, Groups, Geometry and Dynamics, 4 (2010), 263 - 273 [9] Bieri, R.; Groves, J. R. J. The geometry of the set of characters induced by valuations. J. Reine Angew. Math. 347 (1984), 168 - 195 [10] Bieri, R.; Groves, J. R. J. Tensor powers of modules over finitely generated abelian groups. J. Algebra 97 (1985), no. 1, 68 - 78

22

DESSISLAVA KOCHLOUKOVA, SAID SIDKI

[11] Bieri, R.; Renz, B. Valuations on free resolutions and higher geometric invariants of groups. Comment. Math. Helv. 63 (1988), no. 3, 464 - 497 [12] Bieri, R.; Strebel, R. Valuations and finitely presented metabelian groups. Proc. London Math. Soc. (3) 41 (1980), no. 3, 439 - 464 [13] Bieri, R.; Strebel, R. A geometric invariant for nilpotent-by-abelian-by-finite groups. J. Pure Appl. Algebra 25 (1982), no. 1, 1 - 20 [14] Bieri, R.; Neumann, W. D.; Strebel, R.A geometric invariant for discrete groups. Invent. Math. 90, (1987), 451 - 477 [15] Brookes, C. J. B. Finitely presented groups and the finite generation of exterior powers. Combinatorial and geometric group theory (Edinburgh, 1993), 16 - 28, London Math. Soc. Lecture Note Ser., 204, Cambridge Univ. Press, Cambridge, 1995 [16] Brown, K. S. Cohomology of groups. Graduate Texts in Mathematics, 87. Springer-Verlag, New York, 1994 [17] Brown, K. S. Finiteness properties of groups, J. Pure Appl. Algebra 44 (1987), no. 1-3, 45 75 [18] Brown, K. S.; Geoghegan, R. An infinite-dimensional torsion-free FP∞ group, Invent. Math. 77 (1984), no. 2, 367 - 381 [19] Gupta, N.; Rocco, N.; Sidki, S. Diagonal embeddings of nilpotent groups. Illinois J. Math. 30 (1986), no. 2, 274 - 283 [20] Kochloukova, D. H. The F Pm -conjecture for a class of metabelian groups. J. Algebra 184 (1996), no. 3, 1175 - 1204 [21] Kropholler, P. H. On groups of type F P∞ . J. Pure Appl. Algebra 90 (1993), no. 1, 55 - 67 [22] Lima, B. C. R.; Oliveira, R. N. Weak commutativity between two isomorphic polycyclic groups, to appear in J. Group Theory, arXiv 1409.5511 [23] Mart´ınez-P´ erez, C. Finite presentability of normal fibre products. J. Pure Appl. Algebra 218 (2014), no. 8, 1373 - 1384 ¨ [24] Meinert, H. Uber die h¨ oheren geometnschen Invananten f¨ ur endhehe direkte Produkte von Gruppen, Diplomarbeit, Universitat Frankfurt a M (1990) [25] Meinert, H. The geometric invariants of direct products of virtually free groups. Comment. Math. Helv. 69 (1994), no. 1, 39 - 48 [26] Miller, C. The second homology group of a group; relations among commutators, Proc. Amer. Math. Soc. 3 (1952), 588 - 595 [27] Rocco, N. R. On a construction related to the nonabelian tensor square of a group. Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), no. 1, 63 - 79 [28] Rocco, N. R. On weak commutativity between finite p-groups, p odd. J. Algebra 76 (1982), no. 2, 471 - 488 [29] Renz, B. Geometrische Invarianten und Endlichkeitseigenschaften von Gruppen. Dissertation, Universitt Frankfurt a.M. (1988). [30] Sidki, S. On weak permutability between groups. J. Algebra 63 (1980), no. 1, 186 - 225 State University of Campinas (UNICAMP), Campinas, Brazil; University of Bras´ılia (UnB), Bras´ılia, Brazil