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The Quarterly Journal of Mathematics
, Volume Advance Article – Jan 9, 2018

20 pages

/lp/ou_press/homological-finiteness-properties-of-fibre-products-mpCubnrXkJ

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0033-5606
- eISSN
- 1464-3847
- D.O.I.
- 10.1093/qmath/hax063
- Publisher site
- See Article on Publisher Site

Abstract We study the homological finiteness property FPn of fibre products. A homological version of the n−(n+1)−(n+2) Conjecture is suggested and solved in some cases. Though the Homological 1–2–3 Conjecture is still open, we prove a homological version of the Virtual Surjection Conjecture in the case of virtual surjection on pairs. 1. Introduction In this paper, we study homological finiteness properties FPn of the fibre product P of two epimorphisms of groups f1:G1→Q and f2:G2→Q. In [15], Kuckuck studied the homotopical finiteness property Fn of the fibre product P. The homotopical type Fn was defined by Wall in [21]. We recall that a group G is of type Fn if there is K(G,1)-complex with finite n-skeleton. For n≥2 a group, G has a homotopical type Fn if and only if it is finitely presented and has homological type FPn. The latest means there is a projective resolution of the trivial ZG-module Z with finitely generated projectives in all dimensions ≤n, for more details and properties on the homological property FPn, we refer the reader to the Bieri book [3] and the Brown book [8]. By definition, for epimorphisms of groups f1:G1→Q and f2:G2→Q, the fibre product of f1 and f2 is P={(g,h)∣f1(g)=f2(h)}⊆G1×G2. Alternatively we say that P is the fibre product associated to the short exact sequences Ker(f1)↪G1↠f1Q and Ker(f2)↪G2↠f2Q. In the case when both G1,G2 are finitely presented, Q is of homotopical type F3 and one of Ker(f1) and Ker(f2) is finitely generated Bridson et al. showed in [6] that P is finitely presented. This result is called the 1–2–3 Theorem or sometimes the Asymmetric 1–2–3 Theorem. A symmetric version when f1=f2 was proved earlier by Baumslag et al. in [1]. Some results on finite presentability of twisted fibre products were established by Martínez-Pérez in [17] and involved the use of the Bieri–Strebel–Neumann Σ-invariant. In [14, 15] Kuckuck suggested: The n−(n+1)−(n+2)ConjectureLet N1→G1→Qand N2→G2→Qbe short exact sequences, where Qis of type Fn+2, G1and G2are groups of type Fn+1and N1is of type Fn. Then the fibre product Pis of type Fn+1. In this paper, we discuss a homological version of this conjecture. The Homological n−(n+1)−(n+2)ConjectureLet N1→G1→Qand N2→G2→Qbe short exact sequences, where Qis of type FPn+2, G1and G2are groups of type FPn+1and N1is of type FPn. Then the fibre product Pis of type FPn+1. One of the main results in [15] is the technical [15, Proposition 4.3], and it is proved there by purely topological methods using stacks of complexes and the Borel construction. Our first result, Theorem A, is a homological version of [15, Proposition 4.3]. We prove Theorem A by purely algebraic means (spectral sequences) and observe that the original proof in [15] cannot be translated in homological language that is the fact that the groups are finitely presented was essentially used in [15]. Theorem A. Let n≥1be a natural number, A↪B↠Ca short exact sequence of groups with Aof type FPnand Cof type FPn+1. Assume that there is another short exact sequence of groups A↪B0↠C0with B0of type FPn+1and that there is a group homomorphism θ:B0→Bsuch that θ∣A=idA, that is there is a commutative diagram of homomorphisms of groups Then Bis of type FPn+1. The homotopical version of Theorem A [15, Proposition 4.3] was used in [15] to prove several results about the n−(n+1)−(n+2) Conjecture. Here we adopt the same approach and following the recipe suggested in [15], we deduce from Theorem A several results about the Homological n−(n+1)−(n+2) Conjecture. Theorem B. The Homological n−(n+1)−(n+2)Conjecture holds if the second sequence splits. Theorem C. If the Homological n−(n+1)−(n+2)Conjecture holds whenever G2is a finitely generated free group then it holds in general. The proof of the following result uses some properties of the homological Σ-invariants defined by Bieri and Renz in [5]. In Section 2, we will revise the properties of the homological Σ-invariants that will be needed later. Theorem D. Let n≥1be a natural number, N1↪G1↠π1Q, N2↪G2↠π2Qbe short exact sequences of groups, where G1,G2are of type FPn+1, Qis virtually abelian, N1is of type FPkand N2is of type FPlfor some k,l≥0with k+l≥n. Then the fibre product Pof π1and π2is of type FPn+1. Though the general case of the Homological 1–2–3 Conjecture is still open, we solve it in the case when Q is finitely presented. Theorem E. The Homological 1–2–3 Conjecture holds if Qis finitely presented. Our interest in the homological finiteness properties of fibre products stems from our interest in the homological finiteness type of subgroups of direct products of groups. Some results about the homotopical type Fn were conjectured in the case of subdirect products of non-abelian limit groups by Dison in [9, Section 12.5] and in the case of some special subdirect products of groups of type FP∞ by Kochloukova in [12]. Limit groups were defined by Sela and studied by Kharlampovich and Myasnikov under the name fully residually free groups. The class of limit groups played an important role in the solution of the Tarski problem in [11, 20]. The interest in the study of homological and homotopical properties of subdirect products derives from the fact that every finitely generated residually free group embeds as a subgroup of a direct product of finitely many limit groups [2]. The homotopical type of subdirect products of groups was conjectured in [15], where Kuckuck stated the following form of the Virtual Surjection Conjecture. The Virtual Surjection ConjectureLet n≥2be a natural number, G1,…,Gkbe groups of homotopical type Fn, where n≤kand P⊆G1×⋯×Gkbe a subgroup that virtually surjects on every nfactors, that is for every 1≤i1<⋯<in≤kthe image of Punder the canonical projection P→Gi1×Gi2×⋯×Ginhas finite index. Then Pis of type Fn. In [6], Bridson et al. showed that the Virtual Surjection Conjecture holds for n=2 and this was deduced as a corollary of the 1–2–3 Theorem. This was later generalized in [15], where Kuckuck proved that if the (n−1)−n−(n+1) Conjecture holds when Q is virtually nilpotent, then the Virtual Surjection Conjecture holds in general. In [7], Bridson et al. proved that if P is a finitely presented subdirect product of non-abelian limit groups G1,…,Gk such that P∩Gi=1 for every 1≤i≤k, then P virtually surjects on pairs. Later in [12], Kochloukova showed that if furthermore P is of type FPn for some n≤k, then P virtually surjects on every n factors. In this paper, we suggest the following homological version of the Virtual Surjection Conjecture. The Homological Virtual Surjection ConjectureLet n≥2be a natural number and G1,…,Gkbe groups of homological type FPn, where n≤kand P⊆G1×⋯×Gkbe a subgroup that virtually surjects on every nfactors. Then Pis of type FPn. The first part of Theorem F is a homological version of [15, Theorem 3.10]. The second part of Theorem F follows from the first part, Theorem E and the fact that every virtually nilpotent group is finitely presented. Theorem F. If the Homological (n−1)−n−(n+1)Conjecture holds for Qvirtually nilpotent then the Homological Virtual Surjection Conjecture holds in general. In particular, the Homological Virtual Surjection Conjecture holds for n=2,that is for groups that virtually surject on pairs. Finally, we note that some results on homological finiteness properties of fibre sums of Lie algebras and subdirect sums of Lie algebras were recently established by Kochloukova and Martínez-Pérez in [13]. Though in the Lie algebra case, there are no homotopic methods available, a version of the 1–2–3 Theorem for Lie algebras was proved in [13]. 2. Preliminaries on the homological type FPm and homological Σ-invariants 2.1. Preliminaries on the homological type FPm If not otherwise stated the modules considered in this paper are left ones. Definition. Let S be an associative ring with 1. An S-module M is said to be of type FPn if there is a projective resolution ⋯→Pi→Pi−1→⋯→P0→M→0 with Pi finitely generated for all i≤n. We say that a group G is of type FPn if the trivial ZG-module Z is of type FPn. We will need later the following criterion for modules of type FPn. Lemma 2.1. ([3, Proposition 1.2, Theorem 1.3+remarks]) Let Sbe an associative ring with 1 and n≥1be a natural number. The following are equivalent for S-module M: Mis of type FPn; for a direct product ∏Sof arbitrary many copies of Swe have TorkS(∏S,M)=0for 1≤k≤n−1and Mis finitely presented as S-module; the functor TorkS(−,M)commutes with arbitrary direct product for 0≤k≤n−1. Remark. We will apply the above lemma for S=ZG, where G is a finitely generated group and for M=Z the trivial ZG-module. In this case, M is automatically finitely presented as S-module. The following result is well known and can be deduced after making appropriate modifications to the proof of [3, Proposition 2.7], which uses spectral sequences and Lemma 2.1. A detailed proof can be found in Lima’s PhD thesis [16]. Proposition 2.2. Let A→B→Cbe a short exact sequence of groups. if both Aand Care of type FPn, then Bis of type FPn; if Ais of type FPnand Bis of type FPn+1, then Cis of type FPn+1. 2.2. Preliminaries on the homological Σ-invariants For a finitely generated group G, we define S(G)=Hom(G,R)⧹{0}/∼, where for two characters χ1, χ2∈Hom(G,R)⧹{0} we have χ1∼χ2 if there is a positive real number r such that χ1=rχ2. We write [χ] for the equivalence class of χ with respect to ∼. Thus S(G)≃Sd−1, where d is the torsion-free rank of the abelianization of G. The n-dimensional Bieri–Renz Σ-invariant is defined by Σn(G,Z)={[χ]∣ZisoftypeFPnasZGχ-module}, where Gχ is the monoid {g∈G∣χ(g)≥0}. The following results will be used later in the paper. Theorem 2.3. ([5, Theorem 5.1]). Let n≥1be a natural number, Ga group of type FPnand N⊆Ga normal subgroup such that G/Nis abelian. Then, Nis of type FPnif and only if S(G,N)⊆Σn(G,Z). The following result was published in [10]. As stated in [10], the result was proved (unpublished) by Meinert and generalizes ideas from [18]. Theorem 2.4. ([10, Lemma 9.1]). Let G1,G2be groups of type FPnwith n≥1and χ:G1×G2→Rbe a character such that χ∣(G1×1)≠0andχ∣(1×G2)≠0. If [χ∣(G1×1)]∈Σk(G1,Z)and [χ∣(1×G2)]∈Σl(G2,Z)for some k,l≥0with k+l<n, then [χ]∈Σk+l+1(G1×G2,Z). The above result was one of the reasons to believe in special formula for calculating Σ-invariants of direct product of groups, known as the direct product conjecture for sigma invariants. As shown by Bieri and Geoghegan in [4], this conjecture holds for the invariants Σn(G,R), where R is a field and Σn(G,R)={[χ]∣RisoftypeFPnasRGχ-module}. It turned out that the conjecture is wrong in general, Schütz proved in [19] that the conjecture does not hold for Σn(G,Z) when n≥4. 3. Homological version of a result of Kuckuck In this section, we prove a homological version of [15, Proposition 4.3] with algebraic methods. The starting point is the following result that is based on the classical Lyndon–Hoschild–Serre spectral sequence. Theorem 3.1. Let Ibe an index set, n≥1a natural number and A↪B↠Ca short exact sequence of groups. Furthermore, we assume that Ais of type FPnand Bis of type FPn+1, Mis a free ZB-module and consider the LHS spectral sequence Ep,q2=Hp(C,Hq(A,∏α∈IMα))converging to Hp+q(B,∏α∈IMα), where Mα=Mfor α∈I. Then En+1,0n+1=En+1,02=Hn+1(C,H0(A,∏α∈IMα)),E0,nn+1=E0,n2=H0(C,Hn(A,∏α∈IMα))and the differential dn+1,0n+1:Hn+1(C,H0(A,∏α∈IMα))⟶H0(C,Hn(A,∏α∈IMα))is surjective. Proof First we assume n=1. In this case, d2,02:E2,02→E0,12. Since B is FP2 by Lemma 2.1 we have H1(B,∏α∈IMα)=∏α∈IH1(B,Mα)=0, hence by the convergence of the LHS spectral sequence E0,1∞=0. Note that all differentials that enter and leave E0,1j are zero if j≥3, so 0=E0,1∞=E0,13=E0,12/Im(d2,02), hence d2,02 is surjective. From now on we assume that n≥2. We split the proof in several steps. Step 1. Note that by Proposition 2.2 (b) C is of type FPn+1. Since A is of type FPn, by Lemma 2.1, we have that Hq(A,∏α∈IMα)=∏α∈IHq(A,Mα)for0≤q≤n−1. Now since Mα=M is a free ZB-module for α∈I, we have Hq(A,Mα)=0 for q≥1 and M≅⨁β∈J(ZB)βforanindexsetJ, where (ZB)β=ZB for β∈J. Since direct sum commutes with tensor product for all α∈I, we have H0(A,Mα)≅Z⊗ZA(⨁β∈J(ZB)β)≅⨁β∈J(Z(B/A))β≅⨁β∈J(ZC)β≕M˜, where (ZC)β=ZC for β∈J. Thus Ep,q2={0if1≤q≤n−1Hp(C,∏α∈IM˜α)ifq=0, (3.1) where M˜α=M˜ for all α∈I and M˜ is a free ZC-module. Since C is of type FPn+1, by Lemma 2.1 Hp(C,∏α∈IM˜α)≅∏α∈IHp(C,M˜α)for0≤p≤n (3.2) and since M˜α is a free ZC-module, we have that Hp(C,M˜α)=0,forallα∈Iandp≥1. (3.3) It follows by (3.2) and (3.3) that Hp(C,∏α∈IM˜α)=0,for1≤p≤n. (3.4) Thus we obtain by (3.1) that Ep,q2=0,if1≤q≤n−1,orq=0and1≤p≤n. (3.5) Step 2. Consider the differentials where E−i,n+i−1i=0 since −i<0. By definition E0,ni+1=ker(d0,ni)im(di,n+1−ii)=E0,niim(di,n+1−ii). (3.6) By (3.5) Ei,n+1−i2=0,if2≤i≤n. (3.7) On other hand, for i≥n+2, we have n+1−i<0 and this implies that Ei,n+1−i2=0 for i≥n+2. Using this and (3.7), we conclude that Ei,n+1−i2=0 if i≠n+1 and i≥2. Hence Ei,n+1−ii=0ifi≠n+1andi≥2, and this implies that im(di,n+1−ii)=0ifi≠n+1andi≥2. (3.8) By (3.6) and (3.8), we obtain that E0,ni+1=E0,ni,ifi≠n+1andi≥2, hence E0,n2=E0,n3=⋯=E0,nn=E0,nn+1andE0,nn+2=E0,nn+3=⋯=E0,n∞. (3.9) This implies that dn+1,0n+1:En+1,0n+1→E0,nn+1 has as codomain E0,n2=H0(C,Hn(A,∏α∈IMα)). Step 3. We will show that En+1,0n+1=En+1,02. Consider the differentials For i≥2, we have En+1+i,1−ii=0, since 1−i<0. Then, im(dn+1+i,1−ii)=0 and so En+1,0i+1=ker(dn+1,0i)im(dn+1+i,1−ii)=ker(dn+1,0i). (3.10) Now, by (3.5), En+1−i,i−1i=0 if 1≤i−1≤n−1. Then ker(dn+1,0i)=En+1,0i if 2≤i≤n and this implies En+1,0i+1=En+1,0i if 2≤i≤n. Hence En+1,02=En+1,03=⋯=En+1,0n=En+1,0n+1 and so En+1,0n+1=En+1,02=Hn+1(C,H0(A,∏α∈IMα)). Step 4. By the convergence of the LHS spectral sequence, there is a filtration 0=Φ−1Hn⊆Φ0Hn⊆⋯⊆Φn−1Hn⊆ΦnHn=Hn, (3.11) such that Ep,q∞≅ΦpHn/Φp−1Hnforp+q=n, (3.12) where Hn denotes Hn(B,∏α∈IMα). By (3.5), we deduce that Ep,q2=0 if p+q=nandp≠0. Hence Ep,q∞=0,ifp+q=nandp≠0. (3.13) Using the filtration (3.11), by (3.12) and (3.13), we get 0=Φ−1Hn⊆Φ0Hn=⋯=Φn−1Hn=ΦnHn=Hn. Then E0,n∞≅Φ0Hn/Φ−1Hn=Hn=Hn(B,∏α∈IMα). (3.14) Note that by now we have used only the fact that A is FPn and C is FPn+1 but we have not used that B is of type FPn+1. Since B is of type FPn+1, by Lemma 2.1 and since for all α∈I,Mα=M is a free ZB-module, we obtain that Hn(B,∏α∈IMα)≅∏α∈IHn(B,Mα)=∏α∈I0=0 and so E0,n∞=0. (3.15) By (3.9) and (3.15), we have E0,nn+2=0. (3.16) Consider the differentials and note that E0,nn+2=ker(d0,nn+1)im(dn+1,0n+1)=E0,nn+1im(dn+1,0n+1). Then by (3.16), we obtain that dn+1,0n+1 is surjective.□ Proposition 3.2. Let n≥1be a natural number, Ba group with a normal subgroup Asuch that Ais of type FPnand C=B/Ais of type FPn+1. Then Bis of type FPn+1if and only if for any direct product of copies of ZBthe map dn+1,0n+1:Hn+1(C,H0(A,∏ZB))⟶H0(C,Hn(A,∏ZB))is surjective, where dn+1,0n+1is the differential from the LHS spectral sequence Ep,q2=Hp(C,Hq(A,∏ZB))converging to Hp+q(B,∏ZB). Proof Note that one of the directions is contained in Theorem 3.1. Assume from now on that the differential dn+1,0n+1 is surjective. We apply the notations of the proof of Theorem 3.1 with M=ZB. Note that in the proof of (3.14), we have not used that B is of type FPn+1. Then we can apply (3.14) for M=ZB and deduce by Lemma 2.1 that BisoftypeFPn+1ifandonlyifE0,n∞=0. Note that in (3.9) of the proof of Theorem 3.1 we have not used that B is FPn+1. By (3.9), we have E0,n2=E0,nn+1,E0,nn+2=E0,n∞. Thus, BisoftypeFPn+1ifandonlyif0=E0,nn+2. Since d0,nn+1:E0,nn+1→E−n−1,2nn+1=0 is the zero map, E0,nn+2=ker(d0,nn+1)/im(dn+1,0n+1)=E0,nn+1/im(dn+1,0n+1) is the cokernel of dn+1,0n+1. Thus, 0=E0,nn+2ifandonlyifdn+1,0n+1issurjective. □ Corollary 3.3. Let n≥1be a natural number, A→B→Cbe a short exact sequence of groups such that Ais of type FPnand Cis of type FPn+1. Then if H0(C,Hn(A,∏ZB))=0for any direct product ∏ZBthen Bis of type FPn+1; if Cis of type FPn+2then BisoftypeFPn+1ifandonlyifH0(C,Hn(A,∏ZB))=0for any direct product ∏ZB. Proof Part (a) follows directly from Proposition 3.2, since in this case the co-domain of dn+1,0n+1 is 0, so dn+1,0n+1 is surjective. To prove part (b) note that if C is of type FPn+2 then Hn+1(C,∏ZC)=∏Hn+1(C,ZC)=∏0=0. Since A is finitely generated H0(A,−) commutes with direct products, hence Hn+1(C,H0(A,∏ZB))≃Hn+1(C,∏H0(A,ZB))=Hn+1(C,∏ZC)=0. Thus the domain of the map dn+1,0n+1 is 0, so dn+1,0n+1 is the zero map and dn+1,0n+1 is surjective if and only if H0(C,Hn(A,∏ZB))=0. Finally by Proposition 3.2 B is FPn+1 if and only if dn+1,0n+1 is surjective.□ Below we restate and prove Theorem A. Theorem 3.4. Let n≥1be a natural number, A↪B↠Ca short exact sequence of groups with Aof type FPnand Cof type FPn+1. Assume that there is another short exact sequence of groups A↪B0↠C0with B0of type FPn+1, and that there is a group homomorphism θ:B0→Bsuch that θ∣A=idA, that is there is a commutative diagram of homomorphisms of groups Then Bis of type FPn+1. Proof We break the proof in several steps. Step 1. Consider the LHS spectral sequence Ep,q2=Hp(C0,Hq(A,ZB))⇒pHp+q(B0,ZB), that is this is the standard Lyndon–Hoschild–Serre spectral sequence applied for the short exact sequence A→B0→C0 and the ZB0-module ZB, where we view ZB as a ZB0-module via θ. Note that Hq(A,ZB)=0 for q≥1 since ZB is a free ZA-module and H0(A,ZB)≅Z⊗ZAZB≅Z(B/A)≅ZC. (3.17) It follows that Ep,q2={0ifq≥1Hp(C0,ZC)ifq=0, (3.18) hence the spectral sequence collapses and En,0∞=En,02=Hn(C0,ZC) for every n≥0. By the convergence of the LHS spectral sequence, there is a filtration 0=Φ−1Hn⊆Φ0Hn⊆⋯⊆Φn−1Hn⊆ΦnHn=Hn, such that Ep,q∞≅ΦpHn/Φp−1Hnforp+q=n, where Hn denotes Hn(B0,ZB). Thus 0=Φ−1Hn=Φ0Hn=⋯=Φn−1Hn⊆ΦnHn=Hn (3.19) and so the homomorphism φ:Hn=Hn(B0,ZB)→En,0∞≃Hn(C0,ZC)isanisomorphismforn≥0, (3.20) where φ is induced by π0:B0↠C0 and by the homomorphism π#:ZB↠ZC that itself is induced by π. Step 2. Let I be an index set. Consider the LHS spectral sequence Ep,q2=Hp(C0,Hq(A,∏α∈I(ZB)α))⇒pHp+q(B0,∏α∈I(ZB)α), associated to the short exact sequence of groups A→B0→C0, where (ZB)α=ZB for α∈I and ZB is ZB0-module via θ. Since A is of type FPn by Lemma 2.1, it follows that Hq(A,∏α∈I(ZB)α)=∏α∈IHq(A,(ZB)α)for0≤q≤n−1. (3.21) Furthermore, Hq(A,(ZB)α)=Hq(A,ZB)=0 for q≥1, since ZB is a free ZA-module. Then Ep,q2=0if1≤q≤n−1. (3.22) Then since Ep,q∞ is a subquotient of Ep,q2, we obtain that Ep,q∞=0if1≤q≤n−1. (3.23) Observe that C0 is of type FPn+1 by Proposition 2.2, hence by Lemma 2.1 the functor Hn(C0,−) commutes with direct products. Since A is finitely generated the functor H0(A,−) commutes with direct products. This together with (3.17) implies the isomorphisms En,02=Hn(C0,H0(A,∏α∈I(ZB)α))≅Hn(C0,∏α∈IH0(A,(ZB)α)) ≅∏α∈IHn(C0,H0(A,(ZB)α))≅∏α∈IHn(C0,(ZC)α), (3.24) where (ZC)α=ZC and (ZB)α=ZB, for every α∈I. Step 3. Consider the differentials Then En+i,1−ii=0, since 1−i<0, and furthermore En−i,i−1i=0 for 2≤i≤n by (3.22). Note that En−i,i−1i=0 for i>n, since in this case n−i<0. Thus En,0i+1=ker(δn,0i)im(δn+i,1−ii)=En,0iforeveryi≥2. This together with (3.24) implies En,0∞=En,02≃Hn(C0,∏α∈I(ZC)α). (3.25) By the convergence of the spectral sequence, there is a filtration 0=Λ−1Hn⊆Λ0Hn⊆⋯⊆Λn−1Hn⊆ΛnHn=Hn, (3.26) such that Ep,q∞≅ΛpHn/Λp−1Hnforp+q=n, (3.27) where to simplify the notation we denote Hn(B0,∏α∈I(ZB)α) by Hn. By (3.23) and (3.27), we have 0=Λ−1Hn⊆Λ0Hn=⋯=Λn−1Hn⊆ΛnHn=Hn=Hn(B0,∏α∈I(ZB)α). Thus, on the one hand E0,n∞≅Λ0Hn/Λ−1Hn≅Λ0Hn=Λn−1Hn. And on the other hand En,0∞≅ΛnHn/Λn−1Hn=Hn/Λn−1Hn≅Hn/E0,n∞. Then we have a short exact sequence of groups E0,n∞↪Hn↠θ^En,0∞, (3.28) where the epimorphism θ^ is induced by the epimorphism π0:B0↠C0 e π#:ZB↠ZC, where π# is a ring epimorphism induced by the epimorphism of groups π. Step 4. We claim that θ^:Hn(B0,∏α∈I(ZB)α)→Hn(C0,∏α∈I(ZC)α) is an isomorphism. In fact, by (3.20), there is a group isomorphism φ that induces an isomorphism Πφ:∏α∈IHn(B0,(ZB)α)→∼∏α∈IHn(C0,(ZC)α). Since B0 and C0 are FPn+1, we have that both functors Hn(B0,−) and Hn(C0,−) commute with direct products. Then, Πφ induces the homomorphism of groups θ^. Then by (3.28) E0,n∞=Ker(θ^)=0. (3.29) Step 5. Consider the differentials for i≥2 Then by (3.22), Ei,n+1−ii=0 for 2≤i≤n and Ei,n+1−ii=0 for i≥n+2 since n+1−i<0. Furthermore, E−i,n+i−1i=0, since −i<0 . Thus, we have E0,ni+1=ker(δ0,ni)im(δi,n+1−ii)≅E0,niforeveryi≥2andi≠n+1. Then E0,n2≅⋯≅E0,nn+1andE0,nn+2≅⋯≅E0,n∞. (3.30) By (3.29) and (3.30), we have that E0,nn+2≅E0,n∞=0. (3.31) Step 6. Consider the differentials Note that E−n−1,2nn+1=0, since −n−1<0, then ker(δ0,nn+1)=E0,nn+1. This together with (3.31) implies 0=E0,nn+2≔ker(δ0,nn+1)im(δn+1,0n+1)=E0,nn+1im(δn+1,0n+1). Then im(δn+1,0n+1)=E0,nn+1 and we conclude that δn+1,0n+1:En+1,0n+1⟶E0,nn+1issurjective. (3.32) As in the proof of Theorem 3.1, δn+1,0n+1 has the following domain and co-domain: Step 7. By the naturality of the LHS spectral sequence, we have the following commutative diagram of group homomorphisms: (3.33) where μ1 and μ2 are induced by ν:C0→C and ψn+1,0n+1 is the differential of the LHS spectral sequence Hp(C,Hq(A,∏α∈I(ZB)α))⇒pHp+q(B,∏α∈I(ZB)α) associated to the short exact sequence A→B→C. Set V=Hn(A,∏α∈I(ZB)α). Recall that ν:C0→C is induced by θ and V is a left ZC0-module via the homomorphism ν. Thus, we have H0(C0,V)≅VAug(ZC0)V=VAug(Z(im(ν)))VandH0(C,V)≅VAug(ZC)V, where Aug denotes the augmentation ideal of the appropriate group algebra. Then μ2:VAug(Z(im(ν)))V⟶VAug(ZC)V is the enlargement homomorphism, hence is surjective. By the commutative diagram (3.33), since δn+1,0n+1 and μ2 are both surjective, we deduce that ψn+1,0n+1 is surjective too, and hence by Proposition 3.2, we have that B is of type FPn+1.□ Note that several results in [15] are deduced as corollaries of the technical [15, Proposition 4.3]. As we proved the homological version Theorem 3.4 of [15, Proposition 4.3], we will deduce in the following propositions that the homological versions of several results of [15] hold too. The proofs of the following results will use significantly Theorem 3.4 plus ideas from [15]. The following proposition implies Theorem B. As the statement of Proposition 3.5 shows, in the case when the second short exact sequence splits, there is no need to assume that Q is of type FPn+2 as in the Homological n−(n+1)−(n+2) Conjecture. Proposition 3.5. Let n≥1be a natural number, N1→G1→Qand N2→G2→Qbe short exact sequences with G1and G2groups of type FPn+1such that N1is of type FPnand the second sequence splits. Then the fibre product P, associated to the above short exact sequences, is of homological type FPn+1. In particular, the Homological n−(n+1)−(n+2)Conjecture holds if the second sequence splits. Proof As in the proof of [15, Corollary 4.6], there is a homomorphism ϕ:G1→P whose restriction to N1 is the identity map. Then we obtain the following exact diagram: Finally by Theorem 3.4, P is of type FPn+1.□ The following corollary proves Theorem C. The second part of Corollary 3.6 will be used in the proof of Theorem 4.1. Corollary 3.6. If the Homological n−(n+1)−(n+2)Conjecture holds whenever G2is a finitely generated free group, then it holds in general. If the Homological n−(n+1)−(n+2)Conjecture holds whenever G2is a finitely generated free group and Qis finitely presented, then it holds if Qis finitely presented without restrictions on G2. Proof We prove the first statement, the proof of the second statement is the same. Let N1↪G1↠π1Q and N2↪G2↠π2Q be short exact sequences of groups such that N1 is of type FPn, G1 and G2 are of type FPn+1 and Q is of type FPn+2. Let P be the fibre product P={(g1,g2)∈G1×G2:π1(g1)=π2(g2)}, and let p:F↠G2 be an epimorphism, where F is a finitely generated free group. Consider the short exact sequences of groups N1↪G1↠π1Q and ker(π2◦p)↪F↠π2◦pQ, and denote by P′ the fibre product P′={(g1,f)∈G1×F:π1(g1)=π2◦p(f)}. By assumption, the homological n−(n+1)−(n+2) Conjecture holds whenever the middle group in the second exact sequence is finitely generated and free, so P′ is of type FPn+1. Consider the following commutative diagram, whose rows are short exact sequences where p2′:P′↠F is given by p2′(g1,f)=f, p2:P↠G2 is defined by p2(g1,g2)=g2 and idG1×p:P′→P is restriction of idG1×p:G1×F→G1×G2. Then by Theorem 3.4 P is of type FPn+1.□ The following result is Theorem D from the introduction. Theorem 3.7. Let n≥1, N1↪G1↠π1Qand N2↪G2↠π2Qbe short exact sequences of groups, where G1,G2are of type FPn+1, Qis virtually abelian, N1is of type FPkand N2is of type FPlfor some k,l≥0with k+l≥n. Then the fibre product Pof π1and π2is of type FPn+1. Proof We assume first that Q is abelian. Then P◁(G1×G2) and (G1×G2)/P is abelian. By Theorem 2.3, to complete the proof we have to show that for every character χ:G1×G2→R with χ(P)=0, we have [χ]∈Σn+1(G1×G2,Z). Let χ be such a character. Observe that G1×G2=(G1×1)P and G1×G2=P(1×G2). Thus, χ∣(G1×1)≠0, otherwise χ:(G1×1)P→R would be the zero character. Similarly, χ∣(1×G2)≠0. Since Q is abelian, G1/N1,G2/N2 are abelian. Note that N1∪N2⊆P and χ(P)=0 imply that χ∣(G1×1)(N1)=0 and χ∣(1×G2)(N2)=0. Let k′,l′≥0 be such that k′≤k, l′≤l and k′+l′=n. Since N1 is of type FPk and N2 is of type FPl, then N1 is of type FPk′ and N2 is of type FPl′. Thus, by Theorem 2.3, [χ∣(G1×1)]∈Σk′(G1,Z) and [χ∣(1×G2)]∈Σl′(G2,Z). By Theorem 2.4 [χ]∈Σk′+l′+1(G1×G2,Z)=Σn+1(G1×G2,Z). Now we consider the general case, that is there is a normal abelian subgroup A of finite index in Q. Consider the short exact sequences of groups N1→π1−1(A)↠π˜1A and N2→π2−1(A)↠π˜2A, where π˜i is the restriction of πi to πi−1(A) for 1≤i≤2. Since [G1:π1−1(A)]=[Q:A]<∞ and G1 is of type FPn+1, it follows that π1−1(A) is of type FPn+1. Similarly π2−1(A) is of type FPn+1. By the abelian case discussed above, the fibre product P˜ of π˜1 and π˜2 is of type FPn+1. Note that P˜=P∩(π1−1(A)×π2−1(A)), hence [P:P˜]<∞. Since going up or down with a finite index does not change the homological type, we deduce that P is of type FPn+1.□ We finish the section with the proof of the first part of Theorem F. Theorem 3.8. If n≥2and the Homological (n−1)−n−(n+1)Conjecture holds whenever Qis virtually nilpotent then the Homological Virtual Surjection Theorem holds. Proof The proof is similar to the proof of [15, Theorem 3.10], where instead of [15, Proposition 4.3], we apply Theorem 3.4 and we swap the numbers n and k in the proof of [15, Claim]. We sketch the proof. Let P⊆G1×⋯×Gk be as in the statement of the Homological Virtual Surjection Theorem. Then by substituting each Gi with a subgroup of finite index if necessary, we can assume that P⊆G1×⋯×Gk is a subdirect product that is pi(P)=Gi for every 1≤i≤k. By [6, Proposition 3.2] or [15, Lemma 3.2] we obtain that Gi/(P∩Gi) is virtually nilpotent for every i. Let T=p1,…,k−1(P), where p1,…,k−1:G1×⋯×Gk→G1×⋯×Gk−1 is the canonical projection and N1,…,k−1=P∩T. As in the proof of [15, Theorem 3.10] the fact that P virtually surjects on n factors implies that N1,…,k−1⊆G1×⋯×Gk−1 virtually surjects on n−1 factors. If the Homological Virtual Surjection Theorem holds for smaller values of k, that is we use induction on k, then N1,…,k−1 is of type FPn−1. Furthermore, T⊆G1×⋯×Gk−1 virtually surjects on n-tupples if n≤k−1, so by induction on k the group T is of type FPn. If n≥k we get that n=k and in this case, the Homological Virtual Surjection Theorem obviously holds. As in the proof of [15, Theorem 3.10] and using [15, Lemma 2.3] for the subdirect product P⊆T×Gk, we deduce that P is the fibre product associated to the short exact sequences N1,…,k−1→T→Q and Nk→Gk→Q, where Nk=P∩Gk. Thus, Q≃Gk/Nk is virtually nilpotent. Then we can apply the Homological (n−1)−n−(n+1) Conjecture since Q is virtually nilpotent and obtain that P is of type FPn.□ 4. The Homological 1– 2– 3 Conjecture for finitely presented Q In this section, we prove Theorem E. Theorem 4.1. The Homological 1– 2– 3Conjecture holds if in addition Qis finitely presented. Proof Let A↪G1↠π1Q and B↪G2↠π2Q be short exacts sequences of groups with A finitely generated, G1 and G2 of type FP2 and Q is FP3 and finitely presented. Denote by P the associated fibre product. We aim to show that P is of type FP2. By Corollary 3.6, we can assume that G2 is a free group F with a finite free basis X. Let ⟨X∣R˜⟩,whereR˜={ri(x̲)}i∈I0 be a finite presentation of Q. Then there is a presentation of the group G1 G1=⟨X∪A0∣R1∪R2∪R3⟩, where A0={a1,a2,…,ak} is a finite set of generators of A and R1={ri(x̲)wi(a̲)−1}i∈I0,R2={ajxvj,x(a̲)−1}1≤j≤k,x∈X∪X−1andR3={zj(a̲)}j∈J for some possibly infinite index set J and wi(a̲), vj,x(a̲), zj(a̲) are elements of the free group with a free basis A0. Let R be the normal subgroup of the free group F(X∪A0) with a free basis X∪A0 generated as a normal subgroup by R1∪R2∪R3. Then there is a short exact sequence of groups R↣F(X∪A0)↠G1. Since G1 is FP2, we obtain that R/[R,R] is finitely generated as ZG1-module via conjugation. Hence, there is a finite subset J0 of J such that R=R0[R,R], (4.1) where R0 is the normal closure of R1∪R2∪R3,0 in the free group F(X∪A0) and R3,0={zj(a̲)}j∈J0⊆R3. Consider the group G˜1=F(X∪A0)/R0. Then there is a natural projection π:G˜1=F(X∪A0)/R0→G1=F(X∪A0)/R with kernel S=R/R0. By (4.1), we have S=[S,S]. (4.2) Since R1∪R2 are relations in G˜1, we deduce that the subgroup A˜1 of G˜1 generated by the elements of A0 is a normal subgroup of G˜1 such that G˜1/A˜1≃Q. Thus there is a short exact sequence of groups A˜1↣G˜1↠π˜1Q (4.3) with both G˜1 and Q finitely presented. Denote by P˜1 the fibre product of the short exact sequences (4.3) and B↣F↠π2Q, thus P˜1={(h1,h2)∈G˜1×F∣π˜1(h1)=π2(h2)}. Since Q is FP3 and is finitely presented, it is F3. Then the 1–2–3 Theorem from [6] implies that P˜1 is finitely presented. Recall that P is the fibre product of the original short exact sequences A↣G1↠π1Q and B↣F↠π2Q, that is P={(h1,h2)∈G1×F∣π1(h1)=π2(h2)}. The map π×idF:G˜1×F→G1×F induces an epimorphism μ:P˜1↠P with kernel ker(μ)=S, where S=ker(π). Then P˜1=F(X∪A0)/S1 for some normal subgroup S1 of F(X∪A0) and P=F(X∪A0)/S2 for some normal subgroup S2 of F(X∪A0) such that S1⊆S2 and S2/S1=S=[S,S]. Hence S2=S1[S2,S2], so S2/[S2,S2] is an epimorphic image of the abelianization S1/[S1,S1]. Since P˜1 is finitely presented, it is of type FP2. Then we deduce that S1/[S1,S1] is finitely generated as ZP˜1-module via conjugation. Thus, its epimorphic image S2/[S2,S2] is finitely generated as ZP-module via conjugation, hence P is of type FP2 as claimed.□ Funding The first named author was partially supported by ‘bolsa de produtividade em pesquisa’ 303350/2013-0 from CNPq, Brazil and Grant 2016/05678-3 from FAPESP, Brazil. The second named author was supported by PhD grant from CAPES/CNPq, Brazil. References 1 G. 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Short , On the finite presentation of subdirect products and the nature of residually free groups , Am. J. Math. 135 ( 2013 ), 891 – 933 . Google Scholar CrossRef Search ADS 7 M. R. Bridson , J. Howie , C. F. Miller and H. Short , Subgroups of direct products of limit groups , Ann. Math. 170 ( 2009 ), 1447 – 1467 . Google Scholar CrossRef Search ADS 8 K. S. Brown , Cohomology of Groups , Springer-Verlag, New York , 1982 . Google Scholar CrossRef Search ADS 9 W. Dison , Isoperimetric functions for subdirect products and Bestvina-Brady groups, PhD Thesis, Imperial College London, 2008 . 10 R. Gehrke , The higher geometric invariants for groups with sufficient commutativity , Comm. Algebra 26 ( 1998 ), 1097 – 1115 . Google Scholar CrossRef Search ADS 11 O. Kharlampovich and A. Myasnikov , Elementary theory of free non-abelian groups , J. Algebra 302 ( 2006 ), 451 – 552 . Google Scholar CrossRef Search ADS 12 D. H. Kochloukova , On subdirect products of type FPm of limit groups , J. 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Schütz , On the direct product conjecture for sigma invariants , Bull. Lond. Math. Soc. 40 ( 2008 ), 675 – 684 . Google Scholar CrossRef Search ADS 20 Z. Sela , Diophantine geometry over groups. VI. The elementary theory of a free group , Geom. Funct. Anal . 16 ( 2006 ), 707 – 730 . 21 C. T. C. Wall , Finiteness Conditions for CW-Complexes , Ann. Math. 81 ( 1965 ), 56 – 69 . Google Scholar CrossRef Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

The Quarterly Journal of Mathematics – Oxford University Press

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