Hyperbolicity and bounded-valued cohomology

We generalise a theorem of Gersten on surjectivity of the restriction map in $\ell^{\infty}$-cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and $\ell^{\infty}$-cohomology calculations for some well-known classes of groups. Along the way, we obtain hyperbolicity criteria for groups of type $FP_2(\mathbb Q)$ and for those satisfying a rational homological linear isoperimetric inequality, answering a question of Arora and Mart\'{i}nez-Pedroza.


Introduction
Bounded-valued cohomology, also known as ℓ 8 -cohomology, is a quasi-isometry invariant of finitely generated groups.It was introduced by Gersten as a tool to find lower bounds for the Dehn function of some finitely presented groups [Ger95,Ger98].Later, Gersten found a complete characterisation of hyperbolic groups in terms of ℓ 8 -cohomology [Ger96a].
Theorem 1.1 (Gersten).Suppose that G is a finitely presented group.Then G is hyperbolic if and only if H 2 p8q pG; ℓ 8 pN, Rqq " 0.
Recently, Milizia in [Mil21], extended this characterisation to relatively hyperbolic groups, and also gave a characterisation of amenable groups using the comparison map from ordinary group cohomology to ℓ 8 -cohomology.Another result of Gersten concerns the surjectivity of the restriction homomorphism in ℓ 8 -cohomology of groups [Ger96a, Theorem 6.3].
Theorem 1.2 (Gersten).Let G be the fundamental group of a finite aspherical 3-complex such that cd G " 2. Suppose H is a finitely presented subgroup of G. Then the restriction homomorphism H 2 p8q pG; V q Ñ H 2 p8q pH; V q is surjective for all normed Z-modules V .
By combining Theorems 1.1 and 1.2, Gersten in [Ger96a] obtained a new proof of his remarkable theorem which states that a finitely presented subgroup of a hyperbolic group G with cd G " 2 is hyperbolic [Ger96c].
Our main result is the following generalisation of Theorem 1.2.
Theorem 1.3.Let R be a subring of Q containing Z. Suppose H is a group of type F P n pRq and is a subgroup of G with cd R G " n ă 8. Then the restriction homomorphism H n p8q pG; V q Ñ H n p8q pH; V q is surjective for all normed R-modules V .
Here, cd R G stands for the cohomological dimension of a group G over a ring R. When R " Z, we omit it from the notation and simply write cd G.This notion and properties F , F P pRq, F P n pRq and F P 8 pRq are reviewed is Section 2. Theorem 1.3 generalises Theorem 1.2 in several ways: by allowing torsion in G, by removing the finiteness assumption on G, and by extending to higher cohomological dimensions.We can then apply Theorem 1.3 to subgroups of hyperbolic groups of type F , or more generally of type F P pQq.
A major tool in the proof of Theorem 1.3 is our new characterisation of ℓ 8 -cohomology of a group G as the cohomology of orbitally-bounded cochains on any projective resolution of R over the group ring RG.The freedom to interpolate between different projective resolutions allows us to also generalise Gersten's Theorem 1.1, from finitely presented groups, to groups of type F P 2 pQq.Together, they lead to applications such as Corollary 1.5 below.
Corollary 1.4.Let G be a hyperbolic group which is not virtually free.Suppose H is a subgroup of type F .Then H n p8q pH; V q " 0 for all normed R-modules V , where n " cd Q G.
Corollary 1.5.Let H be a subgroup of a hyperbolic group G such that cd Q G " 2. Then H is hyperbolic if and only if H is of type F P 2 pQq.
Weaker versions of Corollary 1.5 were previously established by Gersten when cd G " 2 and H is of type F P 2 pZq [Ger96c], and more recently by Arora and Martínez-Pedroza [AMP21] when cd Q G " 2 and H is finitely presented.Gersten's result implied that homological coherence and coherence are equivalent for hyperbolic groups of cohomological dimension two.Corollary 1.5 strengthens this both in the direction of the hyperbolic group and the subgroup: for hyperbolic groups of rational cohomological dimension two, rational homological coherence is equivalent to coherence.
In [Rip82], Rips constructed the first examples of finitely generated subgroups of hyperbolic (small cancellation) groups that are not finitely presented and hence not hyperbolic (see Theorem 5.3).Next corollary shows that these subgroups are not even of type F P 2 pQq.
Corollary 1.6.Given any finitely presented group Q, let be the Rips construction [Rip82, Theorem 1], for G being C 1 p1{6q, with N finitely generated.Suppose P is a finitely generated subgroup of Q which is not finitely presented.Then q ´1pP q is finitely generated but not of type F P 2 pQq.
An advantage of using rational cohomological dimension over the integral one is that it allows torsion.Corollary 1.5 can then be directly applied to one-relator groups.
Corollary 1.7.A subgroup of a hyperbolic one-relator group is hyperbolic if and only if it is of type F P 2 pQq.
Corollary 1.7 in particular applies to one-relator groups with torsion, since they are hyperbolic [BFR19].Note that Jaikin-Zapirain and Linton subsequently proved that one-relator groups are coherent [JZL23], from which the above may be alternatively deduced in combination with earlier results (in fact, F P 2 pQq may be relaxed to finitely generated).
Groups of rational cohomological dimension 2 play a prominent role in geometric group theory.Besides aforementioned examples, they also notably include random groups and fundamental groups of graphs of virtually free groups with virtually cyclic edge groups.Both of these classes of groups were shown to satisfy Gromov's Surface Subgroup Conjecture [CW15], [Wil18].
Recall that a finitely presented group is necessarily of type F P 2 pZq, i.e. almost finitely presented, which in turn is stronger than being of type F P 2 pQq.In order to relax the finiteness assumption on the subgroup H in Corollary 1.5, we generalise in Section 4 Gersten's Theorem 1.1 and a result of Mineyev in [Min02] from finitely presented groups to groups of type F P 2 pQq.This also leads us to answer positively a question of Arora and Martínez-Pedroza [AMP21, Question 1.9] on hyperbolicity of groups of type F P 2 pQq satisfying a rational homological linear isoperimetric inequality.
Theorem 1.8.Suppose G is a group.Then G is hyperbolic if and only if it is of type F P 2 pQq and its rational homological filling function of degree 2 is bounded above by a linear function.
A similar result over other rings, but with a notion of an isoperimetric inequality that is not the usual one, appears in [KK21].
As Gersten states in [Ger92], "Calculation of H n p8q pG; Rq poses an interesting but difficult problem".Indeed, there are only a few such calculations in the literature.Also, applications of these calculations have mostly been in degrees n " 1, 2. Next, we state some applications of Theorem 1.3 on (non-)vanishing of H n p8q pG; Rq in the top degree.Corollary 1.9.Let A " Z or R. Suppose Γ is one of the following groups and d " vcd Γ ě 1: (i) Γ is a finitely generated right-angled Artin group; (ii) Γ is the Bestvina-Brady group of any finite flag complex L n with H n pL; Aq " 0; (iii) Γ is the mapping class group of a closed orientable surface of genus 2; (iv) Γ " SLpn, Zq for n ě 2; (v) Γ " OutpF n q for n ě 2.
Then the cokernel of the comparison map ι d : H d pΓ; Aq Ñ H d p8q pΓ; Aq contains a continuum of linearly independent elements.
In contrast, by a result of Wienhard [Wie12], all uniform lattices in connected semisimple Lie groups with finite centre have vanishing ℓ 8 -cohomology with real coefficients in dimensions above the dimension of maximal flats of the associated symmetric space.In Section 5, we obtain an analogue of Wienhard's result for relatively hyperbolic groups (see Corollary 5.14).
Poincaré duality groups have vanishing ℓ 8 -cohomology with real coefficients in the top dimension precisely when they are not amenable (see Lemma 5.5).This can be seen from the fact that ℓ 8 -cohomology of a countable group is dual to the uniformly finite homology of Block and Weinberger and their fundamental characterisation of amenable groups via uniformly finite homology in degree zero [BW92].The following are applications of this result and Theorem 1.3.
Corollary 1.10.Let G be a finitely generated relatively hyperbolic group with cd Q G ă 8. Let Γ be a finitely generated group with 2 ď cd Q Γ ă 8. Suppose that Γ has an amenable Poincaré duality subgroup of dimension cd Corollary 1.10 applies, for example, to groups Γ listed in Corollary 1.9; of note being the case when Γ is a Bestvina-Brady group, as the other cases were previously known.
Corollary 1.11.Let Γ be a finitely generated group with 2 ď cd Q Γ ă 8. Suppose that Γ has an amenable Poincaré duality subgroup of dimension cd Q Γ. Suppose Γ is quasi-isometric to the fundamental group G of a finite graph of hyperbolic groups.Then cd Q G " cd Q Γ " 2.
Corollary 1.12.Let Γ be a finitely generated group satisfying the hypothesis of Corollary 1.9.Suppose Γ is quasi-isometric to the fundamental group of a finite graph of hyperbolic groups.Then cd Γ ď 2, and Γ is either a right-angled Artin group or a Bestvina-Brady group.
Besides our main result, Theorem 1.3 and its applications, in Section 5, we consider a long exact sequence in group cohomology arising from a group pair of a relatively hyperbolic group G and the collection of its peripheral subgroups.We use this sequence for calculations of ℓ 8 -cohomology of G.The following result is an application of such a calculation.Theorem 1.13.Let M be a complete, finite-volume Riemannian manifold of dimension n of negative sectional curvature bounded away from zero.Then H n p8q pM ; Rq " 0. In addition, if M is non-compact, then H n´1 p8q pM ; Rq contains a continuum of linearly independent elements.
Another motivation to study ℓ 8 -cohomology comes from its connection to Gromov's bounded cohomology via the comparison maps: (1) The composition of the comparison maps is known to vanish [Wie12] and Wienhard asks when the sequence is exact [Wie12, Question 8].Blank in [Bla15, Question 6.3.10] also asks if the sequence (1) can be extended.Frigerio and Sisto in [FS20], show that the exactness of (1), when G is finitely presented, V " R and n " 2, is equivalent to a conjecture of Gromov on bounded primitives of differential forms on universal covers of closed Riemannian manifolds [Gro93].Using this reformulation, Gromov's conjecture was recently disproved by Ascari and Milizia [AM22].
In Section 3, we make the following observation.
Theorem 1.14.Let G be a relatively hyperbolic group with amenable peripheral subgroups, then the sequence (1) is exact for all n ě 2 and all dual normed RG-modules V .
We should note that when n " 2 and V " R, Theorem 1.14 follows from Theorem 4 and Proposition 14 of [FS20].
In Section 5, we also define ℓ 8 -cohomological dimension cd p8q G of a group G.It is a quasiisometry invariant of finitely generated groups.As we have alluded to in Corollaries 1.10 and 1.12, cd p8q G can be useful for distinguishing finitely generated groups up to quasi-isometry.By Theorem 1.1 and [Min00, Theorem 20], a finitely presented group G is hyperbolic if and only if cd p8q G ď 1.It is natural to ask what other properties of a group G can be detected by cd p8q G.In Section 6, we end with some questions on cd p8q G and its connections to acylindrical hyperbolicity, the dimension of maximal flats in cocompact CAT(0) complexes, and higher homological filling functions of G.
1.1.A few words on the proof of Theorem 1.3.In Gersten's proof of Theorem 1.2 [Ger96a, Theorem 6.3], he constructs a retraction from the relation module Z 1 pGq to the relation module Z 1 pHq which is a bounded linear map with respect to an ℓ 1 -norm on Z 1 pGq and the standard norm on Z 1 pHq.A key property used by Gersten is that Z 1 pGq can be constructed to be finitely generated and free.This follows from the assumptions that cd G " 2 and that G admits a finite 3-dimensional KpG, 1q-complex.The ℓ 1 -norm on Z 1 pGq is then equivalent to the standard norm, and the retraction becomes bounded with respect to the standard norms.As he states, "When translated into the language of ℓ 8 -cohomology, this means that the restriction homomorphism is surjective".This method of using the standard norm and Gersten's original definition of ℓ 8cohomology does not work in our generality, because we have no finiteness assumptions on G. First, we make the key observation that the ℓ 8 -cohomology of a group can be defined using orbitally-bounded cochains on any projective resolution.For our purposes, this resolution is the truncation of a suitably constructed free RG-resolution F ˚։ R. In order to deal with torsion in G, we work over any ring Z ď R ď Q.The resolution F ˚։ R comes with an embedded RH-resolution P ˚։ R by finitely generated free RH-modules and such that Z n´1 pF ˚q is a free RG-module equipped with an ℓ 1 -norm.We then construct a retraction ρ : Z n´1 pF ˚q Ñ Z n´1 pP ˚q which is also a bounded linear map with respect to the corresponding norms.With this in hand, given an n-cocycle α P Hom R pZ n´1 pF ˚q, V q bounded on H-orbits, the retraction ρ defines an n-cocycle β P Hom R pZ n´1 pF ˚q, V q, which maps to α under the restriction map.The fact that ρ is a bounded linear map allows us to show that β is bounded on G-orbits.This shows that the restriction map in ℓ 8 -cohomology is surjective in degree n.
Acknowledgements.The authors are thankful to Roberto Frigerio, Ian Leary, Clara Löh and Eduardo Martínez-Pedroza for helpful discussions and comments.We are also grateful to Francesco Milizia for his remarks which led to an improvement of Corollary 5.14, and to Robert Kropholler for spotting unnecessarily strong assumptions in an earlier draft.Finally, we are thankful to the referee for the careful reading of the paper and their suggestions.

Preliminary results
First, we define the necessary finiteness notions and mention their relevant properties.For more details, we refer the reader to Chapter VIII of [Bro82].
Let R denote a commutative ring with unit and G denote a discrete group.The cohomological dimension of G over R can be defined by cd R G " suptn P N | H n pG, M q ‰ 0 for some RG-module M u.
When R " Z, we omit it from the notation and denote the cohomological dimension by cd G.A group G is said to have finite virtual cohomological dimension equal to n, denoted vcd G, if G has a subgroup H of finite index with cd H " n.This is a well-defined notion by [Bro82, VIII.2.4].A group G is said to be of type F P n pRq for some n P Z ě0 Y t8u if there is a projective RG-resolution such that P i is finitely generated and projective as an RG-module for all i ď n.A projective RG-resolution P ˚։ R is said to be of length n, if P n ‰ 0 and P i " 0 for all i ą n.A group G is said to be of type F P pRq, if there is a projective RG-resolution of R by finitely generated projective RG-modules of finite length.It is immediate from the definitions that if G is of type F P pRq, then it is of type F P 8 pRq.
Denote cd R G " n.By [Bro82, VIII.2.1], it follows that G is of type F P pRq if and only if n ă 8 and G is of type F P n pRq.
There are analogous topological finiteness properties which we explain next.An Eilenberg-MacLane space or KpG, 1q-complex is an aspherical CW-complex with fundamental group equal to G. A group G is of type F n if there is a KpG, 1q-complex X with finite n-skeleton and it is of type F if X is finite.If G is of type F n or F , then it is of type F P n pZq or F P , respectively.A group G is finitely generated if and only if it is of type Let H be a subgroup of G. Recall that given an RH-module M , the induced RG-module is defined as Throughout the remainder of this section, R will denote a subring of Q containing Z. Let F be a free RG-module with free RG-basis Λ.Then Ω " tgα | g P G, α P Λu is a free R-basis of F .The ℓ 1 -norm on F with respect to Λ is defined as Definition 2.2.Let ρ : F ։ M be a surjective homomorphism of RG-modules where F is a free RG-module given the ℓ 1 -norm | ¨|1 with respect to a fixed basis.The filling norm on M (with respect to ρ) is defined as When F is finitely generated over RG, we will say that this is the standard norm on M .
Remark 2.3.The filling norm on an RG-module is a pseudo-norm.It is a norm for example if R " Z.
Lemma 2.4 ([MP17, Lemma 4.6]).Let f : M Ñ N be a homomorphism between finitely generated RG-modules with standard norms } ¨}M and } ¨}N , respectively.Then, there is a constant C ą 0 so that }f pmq} N ď C}m} M for all m P M .
Two filling norms } ¨}1 and } ¨}2 on an RG-module M are called equivalent if there a constant C ą 0 such that for all m P M , Lemma 2.4 implies the following result.
Lemma 2.5.Any two standard norms on a finitely generated RG-module are equivalent.By Lemma 2.5, the standard norm on a finitely generated RG-module is well-defined up to linear equivalence.
We will need the following result of Gersten in [Ger96c].
Proposition 2.6 ([Ger96c, Proposition 4.4]).Assume that there is a short exact sequence of RH-modules where M is a finitely generated RH-module and equipped with the standard norm, N is RH-free equipped with the associated ℓ 1 -norm, and P is projective.Then there is an RHretraction ρ : N Ñ M for ι and a constant C ą 0 so that }ρpxq} ď C|x| 1 for all x P N .
We are ready to prove the main result of this section.
Proposition 2.7.Let G be a group with cd R G " n ě 1. Suppose H is a subgroup of G of type F P pRq and P ˚։ R is a resolution by finitely generated free RH-modules.Then, the following hold.
(1) There is a free RG-resolution F ˚։ R where Z n´1 pF ˚q is RG-free equipped with the associated ℓ 1 -norm and an injective RH-morphism ι ˚: P ˚Ñ F

˚.
(2) There is a constant C ą 0, and an RH-retraction ρ : Z n´1 pF ˚q ։ Z n´1 pP ˚q for ι n´1 : Z n´1 pP ˚q ãÑ Z n´1 pF ˚q such that }ρpyq} ď C|y| 1 for all y P Z n´1 pF ˚q.
Proof.The induction of P ˚։ R gives the free RG-resolution R be a free RG-resolution.By the Fundamental Lemma of Homological Algebra [Bro82, Lemma I.7.4], there is an RG-morphism f ˚: for all i ą 0, Cpf q 0 " Ind G H P 0 ' F 1 0 .Since the natural inclusion F 1 ˚ãÑ Cpf q ˚is a chain homotopy equivalence [Wei94, 1.5.4],we have that Cpf q ˚։ R is a free RG-resolution.Since cd R G " n, the submodule Z n´1 pCpf q ˚q of pn ´1q-cycles in Cpf q n´1 is RG-projective.By applying the Eilenberg swindle [Bro82, Lemma VIII.2.7], we can add an RG-free module N as a direct summand to both Cpf q n and Cpf q n´1 , so that the resulting complex F stays exact and Z n´1 pF ˚q -N .The complex P ˚embeds in F ˚by the maps j P i that send P i naturally into the Ind G H P i summand in F i for each i.Denote this embedding by ι ˚: P ˚ãÑ F ˚. Observe that by construction and Lemma 2.1, ι i is an embedding of P i as a direct RH-module summand in F i , for all i ě 0. Hence ι i splits for all i ě 0. We form the quotient complex and consider the short exact sequence of RH-modules: (2) 0 P ˚F˚Q˚0 .
ιS ince ι i splits and F i is RG-free and in particular RH-free, Q i is RH-projective for all i ě 0. The long exact sequence in homology associated to (2), shows that the complex is exact and Q ˚is RH-projective, it follows that Z 1 pQ ˚q is a projective RH-module.Proceeding by induction, and considering the short exact sequence for all i ě 2, we obtain that Z i pQ ˚q is a projective RH-module for all i ě 1.Therefore, we obtain the short exact sequence: (3) 0

ℓ 8 -cohomology of groups
We start this section with preliminary results on ℓ 8 -cohomology of groups and derive our main result, Theorem 1.3.
Throughout, let R be a subring of R containing Z.We equip R with the norm which is the restriction of the usual norm on R. A norm on an R-module V is a function } ¨}V : V Ñ R, so that for any m, m 1 P V and r P R, (1) }m} V ě 0 and }m} V " 0 implies m " 0, (2) }m `m1 } V ď }m} V `}m 1 } V , (3) }rm} V ď |r|}m} V .Let G be a group.A normed RG-module is a normed R-module with an isometric R-linear action of G. Let V be a normed RG-module.We denote by the RG-module of bounded functions defined on G with values in V .The ℓ 8 -cohomology of a group G can be defined as the ordinary (integral) cohomology of G with coefficients in ℓ 8 pG, V q, H p8q pG; V q :" H ˚pG; ℓ 8 pG, V qq " Ext ZG pZ, ℓ 8 pG, V qq.
This definition of ℓ 8 -cohomology in terms of ordinary group cohomology is due to Wienhard [Wie12].It reduces to Gersten's definition when G is assumed to be of type F n , a necessary assumption in the original definition.
Remark 3.1.As noted in [Mil21], two finitely generated quasi-isometric groups have isomorphic ℓ 8 -cohomologies.This can be seen by the fact that ℓ 8 -cohomology of a finitely generated group coincides with Elek's cohomology theory [Ele98] of its Cayley graph.The latter is invariant under quasi-isometries of graphs [Ele98, Theorem 2.1].In fact, Elek defines ℓ 8 -cohomology only with trivial coefficients V " R, but both [Ele98, Definition 2.1] and [Ele98, Theorem 2.1] generalise verbatim to any normed R-module coefficients V .Alternatively, for general coefficients V , the quasi-isometry invariance of ℓ 8 -cohomology follows from Corollary 3.39 of [Li18].When G is of type F n , quasi-isometric invariance of H n p8q pG; Rq was already established in [Ger92].Definition 3.2.Suppose P ˚be a projective RG-resolution of R. We say a cochain β P Hom R pP i , V q is bounded on G-orbits if for all σ P P i , the subset tβpg ¨σq | g P Gu Ă V is bounded.We denote by Hom bG R pP i , V q the subspace containing all such co-chains.
The following observation is essential for our purposes.
Lemma 3.3.Hom bG R pP ˚, V q is a co-chain complex whose cohomology is naturally isomorphic to H p8q pG; V q.
Proof.The boundary operator on P ˚gives a co-boundary operator on Hom R pP ˚, V q which restricts to Hom bG R pP ˚, V q.We claim that Hom bG R pP i , V q is naturally isomorphic to Hom RG pP i , ℓ 8 pG, V qq.For each i ě 0, define One can check that Φ i and Ψ i are well-defined co-chain maps and are inverses of each other.This proves the claim.Thus, the cohomology of Hom bG R pP i , V q is naturally isomorphic to H RpG; ℓ 8 pG, V qq " Ext RG pR, ℓ 8 pG, V qq.
Let F ˚be a free ZG-resolution of Z.By [Bro82, III.3.3],we get the natural isomorphism which shows that H RpG; ℓ 8 pG, V qq is naturally isomorphic to H ˚pG; ℓ 8 pG, V qq.
Remark 3.4.It follows from Lemma 3.3 that the G-action on the coefficient module V has no bearing on the ℓ 8 -cohomology of G, i.e. two different G-actions on V induce isomorphic cohomology groups, a fact that can also be seen directly from the definition.It therefore suffices to consider coefficients that are simply normed R-modules with trivial G-action.
We are now ready to prove our main result.
Theorem 3.5.Let R be a subring of Q containing Z. Suppose H is a group of type F P n pRq and is a subgroup of G with cd R G " n ă 8. Then the restriction homomorphism H n p8q pG; V q Ñ H n p8q pH; V q is surjective for all normed R-modules V .Remark 3.6.Note that since H is a subgroup of G, we have cd R H ď cd R G " n.By Section 2, it follows that H is of type F P n pRq if and only if it is of type F P pRq.
Proof of Theorem 3.5.Since the n " 0 case is trivially satisfied, we assume n ě 1.Since H is of type of type F P pRq, it is of type F P 8 pRq.So, there is a free RH-resolution P ˚։ R by finitely generated free RH-modules (possibly of infinite length) [Bro82, VIII.4.5].We use the notation of Proposition 2.7.Observe that the truncation of the resolutions are RH-and RG-projective resolutions of R, respectively.Let α P Hom bH R pZ n´1 pP ˚q, V q be a cocycle which is bounded on H-orbits.Then β :" α ˝ρ P Hom R pZ n´1 pF ˚q, V q.Since ι n´1 pβq " α, it is left to show that β is bounded on G-orbits.
Now, let ω 1 , . . ., ω k be the standard generators of P n as a free RH-module.Consider λ P P n so that B P n pλq " ρpg ¨σq.Then λ " q 1 ω 1 `¨¨¨`q k ω k with q i P RH.We can write each q i " ř j r ij h ij where h ij P H and r ij P R. Then By the above inequality and the definition of the filling norm on ρpg ¨σq, it follows that for any ε ą 0, there is such λ so that It follows that Since α is bounded on H-orbits, L ă 8.We have established that }βpg ¨σq} V ď LC|σ| 1 for all g P G.This shows that β is bounded on G-orbits.
Remark 3.7.The first example of a non-hyperbolic subgroup H of type F of a hyperbolic group G was given in [IMM22].By Theorem 1.1, we know that H 2 p8q pH; ℓ 8 pN, Rqq ‰ 0. Yet, since G is hyperbolic, H 2 p8q pG; ℓ 8 pN, Rqq " 0. This shows that Theorem 3.5 does not generalise to degrees below the top degree, which in this example is cd G " 5.
We conclude this section with an observation regarding relatively hyperbolic groups.
Theorem 3.8.Let G be a relatively hyperbolic group with amenable peripheral subgroups, then the sequence Ý Ñ H n p8q pG; V q is exact for all n ě 2 and all dual normed RG-modules V .Proof.Denote by H " tH i u iPI the collection of peripheral subgroups of G.There is a commutative diagram with exact middle row: (4) We know that the homomorphism c 1 is surjective [Fra18] and ι 2 is injective [Mil21, Theorem 1.1].The result now follows by a diagram chase.

Hyperbolicity criteria
In [Min02], numerous equivalent conditions for a finitely presented group G to be hyperbolic are listed.In this section, we make the observation that (with a suitable restatement of the results) the finite presentability of G may be relaxed to F P 2 pQq.
A group G of type F P 2 pQq is finitely generated, so we may fix a finite generating set and form the Cayley graph Γ.The property F P 2 pQq now implies that Z 1 pΓ, Qq is finitely generated as a QG-module.We choose a finite generating set tα i u iPI for Z 1 pΓ, Qq (as a QGmodule) where each generator α i lies in C 1 pΓ, Zq and hence is represented by a circuit w i in Γ.Without loss of generality we may assume that there are no superfluous generators in this generating set.We attach 2-cells with attaching maps gw i for each g P G and i P I and make the corresponding 2-complex X (where X p1q " Γ).By construction, H 1 pX; Qq " 0 and G acts freely and cocompactly on X.
To distinguish filling norms with different coefficients, we adopt notation similar to [Min02].For a 1-boundary with coefficients in A " Z, Q, R, we define: }b} A :" inf t|a| 1 : a P C 2 pX, Aq, B 2 paq " bu.
We also have the norm | ¨|1 on 1-boundaries coming from the ℓ 1 -norm on 1-chains.
The following result extends Theorem 7 of [Min02].
Theorem 4.1.Suppose the group G is of type F P 2 pQq, and X is the complex above.Then the following are equivalent: (1) G is hyperbolic.
(2) There exists K ě 0 such that for any b P B 1 pX, Zq, we have the inequality: (3) There exists K ě 0 such that for any b P B 1 pX, Rq, we have the inequality: Note that since X is not necessarily simply-connected even when G is finitely presented, results such as Theorem 7 in [Min02] do not apply verbatim.Therefore we give a proof, although the arguments used are almost the same as in the proofs of the original results in the literature.
Proof.Note that by construction of X, any circuit in Γ lies in B 1 pX, Qq. p1q ùñ p2q: Without loss of generality, Γ is δ-hyperbolic for some positive integer δ.Since b is an integral 1-cycle, it can be represented by a circuit γ b in X p1q .We use induction on |b| 1 .Since X p1q is locally finite, there are finitely many G-orbits of circuits of a given length, thus for the base case of 0 ď |b| 1 ď 16δ we may take: For the inductive step, suppose we have |b| 1 ą 16δ.If γ b self-intersects, we may split it into two shorter circuits and the result follows by induction.Otherwise, we may invoke Lemma 2.6 in [BH99] (III.H) to represent γ b " γ 1 `γ2 as a sum in C 1 pX, Zq of a shorter circuit γ a " γ 1 `γ3 and a circuit γ c " γ 2 ´γ3 of length at most 16δ (see Figure 1, where dotted lines denote length).By induction, whence by induction the result holds.p2q ùñ p3q: Suppose that b P B 1 pX, Rq.Applying Theorem 4 and the second part of Theorem 6 from [Min02], there exist p 1 , p 2 , . . ., p n P B 1 pX, Zq and positive from which the required inequality follows (the same value for K works).p3q ùñ p1q: There are finitely many orbits of 2-cells in X, therefore there exists a constant M such that for all a P C 2 pX, Rq, |B 2 paq| 1 ď M |a| 1 holds.Arguing by contradiction, suppose that G is not hyperbolic, but p3q holds.Theorem 1.4 of Papasoglu in [Pap95] and its corollary, Lemma 5.2 of Gersten in [Ger96a], only rely on the group being finitely generated as they are results concerning the Cayley graph.We may therefore find arbitrarily long and thick quadrilaterals w in X p1q , just like in the proof of Proposition 8 of Mineyev in [Min02].Following the argument there, which we can do since the first real homology of X vanishes, we can get: Choosing |w| 1 large enough, this contradicts p3q.Definition 4.2 (Integral part).We say that a ZG-submodule I of a finitely generated QG-module M is an integral part of M if I is finitely generated as a ZG-module as well as generating M as a QG-module.
Note that for a group G which is of type F P 2 pQq but not of type F P 2 pZq, the relation module Z 1 pΓ, Zq is not finitely generated as a ZG-module, thus is not an integral part of Z 1 pΓ, Qq.Proof.This is an extension of the second statement of Lemma 3.1 in [Ger96a], where X was assumed to be simply-connected.Recall that the generating set for 1-cycles of Γ does not have superfluous generators.Over the rationals, |ra| 1 " |r||a| 1 holds in C 2 pX, Qq.Definition 4.4 (Rational homological filling function, [AMP21]).Suppose that G is a group of type F P n`1 pQq.Let be a partial projective resolution via finitely generated QG-modules.Note that by [Bro82, VIII.4.3], ker B n is finitely generated as a QG-module.Let K n be an integral part of ker B n .For γ P K n , define }γ} B n`1 :" inf t}µ} : µ P P n`1 , B n`1 pµq " γu.
Finally, define the n th -filling function where } ¨} denote the appropriate filling norms.
It is Theorem 3.3 in [AMP21] that F V n`1 G,Q is well-defined up to linear equivalence.In particular, being bounded above by a linear function is invariant under choice of a partial resolution P ˚by finitely generated projective QG-modules, integral part or filling norms.
The following question was asked by Arora and Martínez-Pedroza.Question 4.5 ([AMP21, Question 1.9]).Let G be a group of type F P 2 pQq and suppose F V 2 G,Q is bounded above by a linear function.Is G a hyperbolic group?
We give a positive answer to this question below.
Theorem 4.6.Suppose that G is a group of type F P 2 pQq and that F V 2 G,Q is bounded above by a linear function.Then G is hyperbolic.
Proof.By the construction of X above, we have that H 1 pX; Qq " 0. Suppose that G is not hyperbolic.Then by Theorem 4.1, for any K ě 0 there exists some b By considering K " 1, 2, 3, . . ., we form a sequence pb i q iPN .Since each b i is in B 1 pX, Zq, which is contained in Z 1 pX, Qq, we can think of them as rational 1-cycles.Let z 1 , z 2 , . . ., z m be the generators of Z 1 pX, Qq from the construction of X; recall that these are integral 1-cycles.Consider the free QG-module N with generating set te 1 , e 2 , . . ., e m u and the epimorphism φ : N Ñ Z 1 pX, Qq, defined by φpe i q " z i for i " 1, 2, . . ., m.Let f i P N be such that φpf i q " b i .
Consider the free ZG-module M with generating set te 1 , e 2 , . . ., e m u.We have a natural inclusion M Ă N .For each i, there exists some positive integer r i such that r i f i P M and φpr i f i q " r i b i .
The image I " φpM q inside Z 1 pX, Qq is finitely generated as a ZG-module by φpe i q " z i , but it also generates all of Z 1 pX, Qq as a QG-module.Therefore I is an integral part for Z 1 pX, Qq.Furthermore, it contains every r i b i .
We compute F V 2 G,Q pnq using the rational cellular structure on X and take I as the required integral part.By our assumptions, there exists a constant C such that F V 2 G,Q pnq ď Cn @n P N.This means that for all positive integers n, we have This implies that for all b P I such that |b| C 1 pX,Qq ď m, we have where |¨| C 1 pX,Qq denotes the ℓ 1 -norm on C 1 pX, Qq.This is a filling norm because the module C 1 pX, Qq is free as a QG-module.
Now choose i such that i ą C. For this i, let n " |b i | 1 .By construction, we have On the other hand, |b i | C 1 pX,Qq " n.Consider r i b i P I. We have so by setting m " |r i |n in (5), we get Applying Lemma 4.3, we get which contradicts (6).Hence G is hyperbolic.
This proves Theorem 1.8, since the converse follows from hyperbolic groups being finitely presented and Theorem 1.8 in [AMP21].
We end this section by giving a generalisation of a theorem of Gersten [Ger96a] from finitely presented groups to groups of type F P 2 pQq.It will be needed for applications to subgroups of hyperbolic groups in Section 5.
We use the shorthand ℓ 8 :" ℓ 8 pN, Rq to denote the injective Banach space of bounded real sequences, which can be thought of as a normed RG-module for any group G with trivial G-action.
Theorem 4.7.Suppose that G is a group of type F P 2 pQq with H 2 p8q pG; ℓ 8 q " 0. Then G is hyperbolic.
Proof.Consider the two-dimensional X above.We can use its cellular complex, extended abstractly to an RG-resolution F ˚։ R (possibly infinite) where F i " C i pX, Rq for i ď 2 and Lemma 3.3 to compute the ℓ 8 -cohomology in lower degrees of G. Thus, our assumption implies that the second cohomology of the co-chain complex For i P t1, 2u, since X has finitely many G-orbits of both 1-cells and 2-cells, we may define the following norm on i-cochains bounded on G-orbits Hom bG R pC i pX, Rq, Rq: }f } 8 :" sup t|f pxq| : x is an i-cell of Xu .
We claim that there exists M ą 0 with the property that for all z P Hom bG R pC 2 pX, Rq, Rq with δpzq " 0, there exists some c P Hom bG R pC 1 pX, Rq, Rq with δpcq " z and }c} 8 ď M }z} 8 .If not, then there exist z i P Hom bG R pC 2 pX, Rq, Rq for i P N such that δpz i q " 0 and }z i } 8 " 1 for all i, with the property that We can now combine the z i to form a 2-cocycle z P Hom bG R pC 2 pX, Rq, ℓ 8 q via zpxqpiq " z i pxq which is bounded on G-orbits, but which is not the image of any 1-cochain bounded on Gorbits.This contradicts the assumption that H 2 p8q pG; ℓ 8 q " 0, which proves the claim.Note that this is similar to the proof of Proposition 4.6 in [Ger96a], where this property was referred to as strong vanishing.
By the same argument as Proposition 4.7 in [Ger96a], it follows that the second cohomology of the co-chain complex Hom bG R pF i , ℓ 8 pS, Rqq vanishes for all discrete sets S. We now follow the argument in the proof of Theorem 4.3 in [Ger96a].By setting S to be the set of all bounded linear maps of norm 1, we can deduce that the norms } ¨}R (denoted N R in [Ger96a]) and | ¨|1 are equivalent on B 1 pX, Rq, which means there exists a constant K such that } ¨}R ď K| ¨|1 .Finally, by Theorem 4.1 we deduce that G is hyperbolic.

Applications
In this section, we give some applications of Section 3 and Theorem 3.5.
5.1.Subgroups of hyperbolic groups.Subgroups of hyperbolic groups inherit some of the properties of hyperbolic groups, but may not inherit hyperbolicity even when these subgroups have strong finiteness conditions [Rip82, Bra99, IMM22].We consider subgroups of type F P pQq.
Corollary 5.1.Let G be a hyperbolic group with cd Q G " n, n ě 2. Suppose H is a subgroup of type F P pQq.Then H n p8q pH; V q " 0 for all normed R-modules V .
When the hyperbolic group has rational cohomological dimension 2, we can say more.
Corollary 5.2.Let G be a hyperbolic group such that cd Q G " 2. Suppose H is a subgroup of G of type F P 2 pQq.Then H is hyperbolic.
Rips constructed subgroups of hyperbolic small cancellation groups that are finitely generated but not finitely presented [Rip82, Theorem 1].Corollary 5.2 shows that those subgroups are not of type F P 2 pQq.
Theorem 5.3 (Rips).Given any finitely presented group Q and any real number λ ą 0, there is a C 1 pλq group G and a short exact sequence Proof of Corollary 1.6.Suppose Q is a finitely presented group that contains a finitely generated subgroup P which is not finitely presented.Consider the preimage H " q ´1pP q.Then H is finitely generated but it is not finitely presented, for otherwise P would be finitely presented.Set λ " 1{6.Then G is hyperbolic and moreover cd Q G ď 2 [Ols91].Since H is not finitely presented, Corollary 5.2 shows that H is not of type F P 2 pQq.

Non-vanishing in top dimension.
The following result was proven by Gersten in [Ger96b, Prop.7.2] for groups of type F n .The proof generalises verbatim to any group.Moreover, H 0 pΓ; ℓ 8 pΓ, Rqq is isomorphic to uniformly finite homology of G in degree 0 [BNW12].By the fundamental result of Block and Weinberger [BW92], amenable groups are exactly those groups for which uniformly finite homology in degree 0 does not vanish.If Γ amenable, then combining Theorem 6.6.1 and Proposition 6.7.2 of [Bla15], it follows that ℓ 8 pΓ, Rq Γ has dimension at least 2 2 |Γ| .

Lemma 5.4 (Gersten
The next proposition is our main tool for establishing non-vanishing of ℓ 8 -cohomology in the top degree.
Proposition 5.6.Let A " Z or R. Let Γ be a group that has an amenable Poincaré duality subgroup N of dimension d ě 1 and d " vcd Γ ă 8 or d " cd Q Γ ă 8. Then H d p8q pΓ, Aq contains a continuum of linearly independent elements.Proof.By Lemma 5.4, it suffices to consider A " R. Recall that ℓ 8 -cohomology is quasiisometry invariant and a finite index subgroup of a Poincaré duality group is again a Poincaré duality group [JW72].Passing to a finite index subgroup of Γ, we can either assume that d " cd Γ ă 8 or d " cd Q Γ ă 8.The result now follows from Theorem 3.5 and Lemma 5.5.
It turns out that many well-known classes of groups contain free abelian or nilpotent subgroups of large Hirsch length.
Corollary 5.7.Let A " Z or R. Suppose Γ is one of the following groups and d " vcd Γ ě 1: (i) Γ is a finitely generated right-angled Artin group; (ii) Γ is the Bestvina-Brady group of any finite flag complex L n with H n pL; Aq " 0; (iii) Γ is the mapping class group ModpS 2 q of a closed orientable surface of genus 2; (iv) Γ " SLpn, Zq for n ě 2; (v) Γ " OutpF n q for n ě 2.
Then the cokernel of the comparison map ι d : H d pΓ; Aq Ñ H d p8q pΓ; Aq contains a continuum of linearly independent elements.Proof.We first check that each case satisfies the hypothesis of Proposition 5.6 where N can be taken either abelian or nilpotent.
(i) The dimension of the flag complex L for Γ equals to d ´1 (see for example [LS11,Proposition 18]) and hence Γ contains a subgroup isomorphic to Z d .(ii) Γ contains a free abelian subgroups of rank equal to the dimension of L which is n.By our hypothesis and [LS11, Theorem 22], cd Γ " n. (iii) In this case d " 3 and ModpS 2 q contains a group isomorphic to Z 3 generated by Dehn twists [Har86].(iv) It is known that vcd Γ " npn ´1q{2 [Bro82].The unipotent subgroup of Γ of upper triangular matrices is a nilpotent subgroup of Hirsch length (and hence rational cohomological dimension) equal to npn ´1q{2.
(v) It is shown in [CV86] that OutpF n q contains a free abelian subgroup of rank d.
Proposition 5.6 implies that H d p8q pΓ; Aq contains a continuum of linearly independent elements.Our stronger assertion follows, since in each of the cases (i) and (iii)-(v), the groups are of type V F , in which case H d pΓ; Aq is a finitely generated A-module.In case (ii), it follows by Corollary 12 of [LS11]  Proof.For any subgroup H ď G, there is an RH-module isomorphism ℓ 8 pG, W qℓ 8 pH, ℓ 8 pHzG, W qq.
The result now follows from applying this isomorphism to the Mayer-Vietoris long exact sequence of G in group cohomology with ℓ 8 pG, W q coefficients.Corollary 5.9.Let Γ be a finitely generated group with 2 ď cd Q Γ ă 8. Suppose that Γ has an amenable Poincaré duality subgroup of dimension cd Q Γ. Suppose Γ is quasi-isometric to the fundamental group G of a finite graph of hyperbolic groups.Then cd Q G " cd Q Γ " 2.
Proof.Since rational cohomological dimension is a quasi-isometry invariant amongst groups with finite rational cohomological dimension [Sau06, Theorem 1.2], denote Suppose n ě 3. Let tG v u vPV be the vertex groups and tG e u ePE be the edge groups of the finite graph of groups.We apply the Mayer-Vietoris sequence of Lemma 5. We recall that by a result of Dunwoody, finitely generated groups of rational cohomological dimension at most one are virtually free [Dun79].Thus the assumption of Corollary 5.9 that cd Q Γ ě 2 is in fact equivalent to G not being virtually free.
Corollary 5.10.Let Γ be a finitely generated group satisfying the hypothesis of Corollary 1.9.Suppose Γ is quasi-isometric to the fundamental group of a finite graph of hyperbolic groups.Then either Γ is a right angled Artin group or a Bestvina-Brady groups of cohomological dimension at most 2.
Proof.In the proof of Corollary 5.7, we have established that Γ has either an abelian or a nilpotent Poincaré duality subgroup of dimension equal to cd Q Γ.By Corollary 5.9, cd Q Γ " 2, which rules out ((iii))-((v)).
5.3.ℓ 8 -cohomological dimension.Suppose G is a group.Define the ℓ 8 -cohomological dimension of G by cd p8q G :" suptn | H n p8q pG; V q ‰ 0 for some normed R-module V u.Let V be a normed R-module such that H n p8q pG; V q ‰ 0. Viewing V as a normed Q-module, by the last claim of the proof of Lemma 3.3, we obtain H n Q pG; ℓ 8 pG, V qq " H n p8q pG; V q.This shows that cd Q G ě n and hence cd Q G ě cd p8q G.
Note also that when G is finitely generated, then by Remark 3.1, cd p8q G is a quasiisometry invariant of G.We end this section with some applications to cd p8q G.
Lemma 5.11.Let G be a group of type F P 2 pQq.Then G is an infinite hyperbolic group if and only if cd p8q G " 1.
Here is another reformulation of Corollary 5.1.
Corollary 5.12.Let G be a hyperbolic group with cd Q G " n, n ě 2. Suppose H is a subgroup of type F P pQq.Then cd p8q H ď n ´1.
Before stating the next result, we need the following lemma.Lemma 5.13.Let H be a subgroup of G and V be a normed R-module.Then ℓ 8 pH, V q is a direct summand of ℓ 8 pG, V q as an RH-module.
Proof.Let T be a right transversal of H in G.We define ϕ : ℓ 8 pH, V q Ñ ℓ 8 pG, V q, f Þ Ñ pht Þ Ñ f phqq, @h P H, @t P T.
One can check that ϕ is an H-equivariant section of the restriction morphism from ℓ 8 pG, V q to ℓ 8 pH, V q.
Corollary 5.14.Let G be a finitely generated infinite group, and hyperbolic relative to H " tH i u iPI .Then cd p8q G " maxt1, cd p8q H i | i P Iu.
Proof.Denote m :" maxt1, cd p8q H i | i P Iu, which may be infinity.If G is hyperbolic, then so are all peripheral subgroups they are undistorted.In this case, cd p8q G " m " 1.
Suppose cd p8q G ě 2. Using the long exact sequence in group cohomology for the group pair pG, Hq with coefficients in ℓ 8 pG, V q, we have , we know that H j p8q pG, H; V q " 0 for all j ě 2. So, the long exact sequence shows that By Lemma 5.13, ℓ 8 pH i , V q is a direct summand of ℓ 8 pG, V q; therefore, it follows that H j p8q pH i ; V q is isomorphic to a direct summand of H j p8q pG; V q, for each i P I.This gives us cd p8q G ě m.
On the other hand, since H j p8q pG; V q -ź iPI H j p8q pH i ; ℓ 8 pH i zG, V qq, @j ě 2, we obtain cd p8q G " m.
Corollary 5.15.Suppose Γ is the fundamental group of a non-compact complete finitevolume Riemannian n-dimensional manifold M of negative sectional curvature bounded away from zero.Then cd p8q Γ " n ´1 and H n´1 p8q pΓ; Rq contains a continuum of linearly independent elements.Proof.We know that Γ is hyperbolic relative to the fundamental groups of the cusps, which are infra-nilmanifolds of dimension pn ´1q [Far98].Therefore, peripheral subgroups are torsion-free virtually nilpotent groups of Hirsch length pn ´1q.The first claim now follows from Corollary 5.14 and Lemma 5.5.
Let H " tH i u iPI denote the collection of the fundamental groups of the cusps.For the second claim, observe that the long exact sequence in group cohomology for the group pair pΓ, Hq with coefficients in ℓ 8 pΓ, V q gives a surjection from H n´1 p8q pΓ, Rq to ś iPI H n´1 pH i ; ℓ 8 pΓ, Rqq.This in turn surjects onto ś iPI H n´1 p8q pH i ; Rq by Lemma 5.13.By Lemma 5.5, we know that each H n´1 p8q pH i ; Rq has dimension at least 2 2 |H i | , hence the claim follows.
Proof of Theorem 1.13.There is an isomorphism H p8q pM ; V q -H p8q pπ 1 pM q; V q [Mil21, Remark 2.3].If M is non-compact, then by Corollary 5.15, we obtain that H n p8q pM ; Rq " 0 and H n´1 p8q pM ; Rq contains a continuum of linearly independent elements.If M is compact, then π 1 pM q is a Poincaré duality group.It is non-amenable, since π 1 pM q is a non-elementary hyperbolic group.By Lemma 5.5, it follows that H n p8q pM ; Rq " 0. Corollary 5.16.Let G be a finitely generated relatively hyperbolic group with cd Q G ă 8. Let Γ be a finitely generated group with 2 ď cd Q Γ ă 8. Suppose that Γ has an amenable Poincaré duality subgroup of dimension cd Q Γ.If G is quasi-isometric to Γ, then there is a peripheral subgroup H of G such that cd Q H " cd Q G.
Proof.Suppose each peripheral subgroup H of G satisfies cd Q H ă cd Q G. Since rational cohomological dimension is a quasi-isometry invariant amongst group with finite rational cohomological dimension [Sau06, Theorem 1.2], n :" cd Q G " cd Q Γ.By Corollary 5.14, cd p8q G ď n ´1, implying H n p8q pG; Rq " 0. Since G and Γ are quasi-isometric, we have H n p8q pΓ; Rq " 0. On the other hand, by Proposition 5.6, H n p8q pΓ; Rq ‰ 0, which is a contradiction.
Corollary 5.16 applies to groups Γ satisfying the hypothesis of Corollary 5.7.We should mention that a finitely generated group which is quasi-isometric to a finitely generated relatively hyperbolic group is itself relatively hyperbolic [Dru09].It is known that, apart from perhaps some right-angled Artin groups and Bestvina-Brady groups, that the other classes of groups listed in Corollary 1.9 are not relatively hyperbolic.In the case of rightangled Artin groups, they are non-virtually cyclic relatively hyperbolic if and only if the associated flag complex is disconnected [BDM09].

Some questions
We have only scratched the surface and many questions on the relationship between ℓ 8cohomology of a group and its geometric properties remain unanswered.We state some of these next.Question 6.1.Let H be a hyperbolically embedded subgroup of a finitely generated group G.Is cd p8q H ď cd p8q G?
For the definition of a hyperbolically embedded subgroup, we refer to [DGO17].There are many examples of such subgroups, such as parabolic subgroups of a relatively hyperbolic group; or the fact that a group is acylindrically hyperbolic if and only if it contains a nontrivial hyperbolically embedded virtually cyclic subgroup [Osi16].In the former case, the question has a positive answer by Corollary 5.14.Question 6.2.Let G be a group acting properly and cocompactly by simplicial isometries on a CAT(0)-simplicial complex X.Does cd p8q G equal to the maximum dimension of flats in X?
By [Ger98,Theorem 7.1], it follows that H 2 p8q pG; ℓ 8 q ‰ 0 if and only if X contains a flat plane.Question 6.3.Let G be the outer automorphism group of a finitely generated right-angled Artin group with vcd G " d ě 2. Is H d 8 pG, Rq ‰ 0? By Corollary 5.7, we know that Question 6.3 has a positive answer in the extremal cases when the associated graph is either a set of points or it is complete.In [DSW21], it is conjectured that G contains a poly-Z subgroup of Hirsch length equal to vcd G.If this conjecture holds, then Question 6.3 can be answered positively by Proposition 5.6.
Question 6.4.Let G be a group of type F P n pQq for n ě 2. Is cd p8q G ď n ´1 if and only if the filling function F V n G,Q is bounded above by a linear function?
Question 6.4 has a positive answer for n " 2 by Theorems 4.7 and 4.6, and for all n ě 2 when G is hyperbolic by [Min00].We note that Question 3.9 of [AMP21] asks if F V n G,Q pkq ă 8 for each k P N when G is of type F P n pQq.

Figure 1 .
Figure 1.Splitting up a filling of γ b .
Lemma 4.3.For b P B 1 pX, Qq and any r P Q, we have }rb} Q " |r|}b} Q .
n´1 pP ˚q Z n´1 pF ˚q Z n´1 pQ ˚q 0 ι n´1 of projective RH-modules.Since H is of type F P pRq, by [Bro82, VIII.4.3],Z n´1 pP ˚q is a finitely generated RH-module.Since Z n´1 pF ˚q is a free RH-module, by Proposition 2.6, there is a constant C ą 0, and an RH-retraction ρ : Z n´1 pF ˚q ։ Z n´1 pP ˚q of ι n´1 such that }ρpyq} ď C|y| 1 for all y P Z n´1 pP ˚q.
).Let G be a group.Then H i p8q pG; Zq " H i p8q pG; Rq for all i ě 2. Lemma 5.5.Let Γ be a Poincaré duality group of dimension d ě 1.Then Γ is amenable if and only if H d p8q pΓ, Rq ‰ 0 if and only if H d p8q pΓ, Rq has uncountably infinite dimension over R. Proof.Recall that ℓ 8 -cohomology is quasi-isometry invariant and a finite index subgroup of a Poincaré duality group is again a Poincaré duality group [JW72].Passing to an index at most 2 subgroup of Γ, we can assume it is an orientable Poincaré duality group.By duality, we have H d p8q pΓ; Rq " H d pΓ; ℓ 8 pΓ, Rqq " H 0 pΓ; ℓ 8 pΓ, Rqq " ℓ 8 pΓ, Rq Γ .
that H d pΓ; Aq is countably generated as an Amodule.Lemma 5.8.Let G be the fundamental group of a graph of groups with vertex groups tG v u vPV and edge groups tG e u ePE .Then there is a long exact Mayer-Vietoris sequence: e ; ℓ 8 pG e zG, W qq Ñ . . .for all i ą 0 and normed RG-modules W .