Cohomology of group theoretic Dehn fillings II

We study the cohomology of group theoretic Dehn fillings. Applying the Cohen-Lyndon property for sufficiently deep Dehn fillings of hyperbolically embedded subgroups $H\hookrightarrow_h G$, obtained by the second named author, we derive a spectral sequence that computes the cohomology of the corresponding Dehn filling quotients $\overline G$. As an application, we establish an isomorphism between the relative cohomology of the group pair $(G, H)$ and its sufficiently deep Dehn filling quotient pair $(\overline G, \overline H)$. This allows us to generalise the results of Fujiwara and Manning on simplicial volume of Dehn fillings of hyperbolic manifolds to Dehn fillings of Poincar\'e duality pairs. We also generalize the results of Olshanskii, Dahmani-Guirardel-Osin and Hull on SQ-universality and common quotients of acylindrically hyperbolic groups by adding cohomological finiteness conditions. We apply these results to obtain hyperbolic and acylindrically hyperbolic quotients with special properties.

1. Introduction 1.1.Dehn surgery in 3-manifolds.In the late 1970's, Thurston dramatically changed the study of 3-manifolds by introducing his Geometrization Conjecture.As supporting evidence, Thurston proved that many non-Haken 3-manifolds satisfy the conjecture [Thu82], using the notion of a Dehn surgery, which is a two-step procedure of modifying a 3-manifold by first cutting off a solid torus and then gluing the torus back in a different way.Another motivation of Dehn surgery comes from the Lickorish-Wallace theorem, which states that every closed connected orientable 3-manifold can be constructed from the 3-sphere by using finitely many Dehn surgeries.
The second step of the surgery, called Dehn filling, starts with a 3-manifold M with toral boundary and constructs a new manifold by gluing a solid torus to M by identifying their boundaries.Topologically distinct ways of gluing a solid torus are parametrized by free homotopy classes of essential simple closed curves on BM (the image of the meridian circle of the solid torus under the identification), called slopes.For a slope s, the new manifold constructed by the corresponding Dehn filling is denoted by M s .A celebrated result of Thurston asserts that most Dehn fillings preserve hyperbolicity.
Theorem 1.1 (Thurston [Thu82]).Let M be a compact orientable 3-manifold with boundary a torus, and with interior admitting a complete finite volume hyperbolic structure.Then for all but finitely many slopes s on BM , M s admits a hyperbolic structure.

Group theoretic Dehn fillings.
There is an analogous construction in group theory, called (group theoretic) Dehn filling, which can be formalized as follows.Given a group G 2010 Mathematics Subject Classification.20F67, 20F10, 20E06.
with a subgroup H and a normal subgroup N of H, the Dehn filling associated with the triple pG, H, N q is the quotient G{xxN yy, where xxN yy is the normal closure of N in G.
The relation between these two versions of Dehn fillings can be seen as follows: under the assumptions of Theorem 1.1, the natural homomorphism π 1 pBM q Ñ π 1 pM q is injective and thus π 1 pBM q can be thought of as a subgroup of π 1 pM q.Let G " π 1 pM q and H " π 1 pBM q.Every slope s on BM generates a normal subgroup N s ⊳H.As s is the image of the meridian circle of the solid torus, which bounds a disc, we have π 1 pM s q " G{xxN s yy.
Dehn filling is a fundamental tool in group theory.It appears, for instance, in the solution of the Virtual Haken Conjecture [Ago13], the study of the Farrell-Jones Conjecture and the isomorphism problem of relatively hyperbolic groups [ACG18,DG18], and the construction of purely pseudo-Anosov normal subgroups of mapping class groups [DGO17].Other applications of Dehn fillings can be found for example in [AGM16,GMS16].
Algebraic analogs of Theorem 1.1 can be proved for groups satisfying certain negative curvature conditions.The first result of this kind was for relatively hyperbolic groups by Osin [Osi07] and independently, by Groves-Manning [GM08].Later, Dahmani-Guirardel-Osin [DGO17] introduced a generalization of relative hyperbolicity based on the notion of a hyperbolically embedded subgroup and proved a generalization of the main results of [Osi07,GM08].We postpone the definition and motivation of hyperbolically embedded subgroups until Section 3.2 and only discuss several examples for the moment.The reader is referred to the survey [Osi18] for other examples.We use H ãÑ h G to indicate that H is a hyperbolically embedded subgroup of G.
Example 1.2.If a group G is hyperbolic relative to its subgroup H, then H ãÑ h G [DGO17, Proposition 2.4].In particular, if M is a compact orientable manifold with one boundary component and M BM admits a complete finite volume hyperbolic structure, then π 1 pBM q ãÑ h π 1 pM q [Bow12,Far98].
Example 1.3.Another typical example arises if a group G acts on a Gromov hyperbolic space S acylindrically by isometries and g P G is a loxodromic element.Then there exists a maximal virtually-cyclic subgroup Epgq ď G containing g such that Epgq ãÑ h G [DGO17, Corollary 2.9].In particular, if G is a word-hyperbolic group (resp.mapping class group of a finite type surface [DGO17, Theorem 2.19], outer automorphism group of a finite rank free group [DGO17, Theorem 2.20]) and g is an infinite order (resp.a pseudo-Anosov, a fully irreducible) element, then Epgq ãÑ h G.
The following is a group theoretic analog of Thurston's Theorem 1.1 due to Dahmani-Guirardel-Osin.
Theorem 1.4 (Dahmani-Guirardel-Osin [DGO17]).Let G be a group with a subgroup H ãÑ h G. Then there exists a finite set F Ď H t1u such that if N ⊳ H and N X F " H, then the natural homomorphism H{N Ñ G{xxN yy maps H{N injectively onto a hyperbolically embedded subgroup of G{xxN yy.
In fact, Theorem 1.4 generalizes Theorem 1.1: let M be as in Theorem 1.1.By Example 1.2, for all but finitely many slopes s on BM , Theorem 1.4 implies that π 1 pM s q is word-hyperbolic and [GM18, Corollary 1.11] implies that π 1 pM s q is one-ended.The Geometrization Conjecture, proved by Perelman, then implies that M s admits a hyperbolic structure.
1.3.Motivation: cohomology of Dehn fillings.Theorem 1.1 asserts that M s is often hyperbolic and thus its universal cover is H 3 .It follows that the cohomology of π 1 pM s q can be understood by studying the cohomology of M s and the action of π 1 pM s q on H 3 .It is therefore natural to investigate the cohomology of G{xxN yy in the more general setting of Theorem 1.4 where one may look for a similar geometric footing.
Question 1.5.For a group G with a subgroup H ãÑ h G and a normal subgroup N ⊳ H, what can be said about the cohomology of G{xxN yy?
The main goal of this series of two papers is to address this question and to illustrate the implications of the results in this direction.
We should point out that even though we consider the general case of hyperbolically embedded subgroups H ãÑ h G, all of our results are new in the special case when G is hyperbolic relative to H.
Acknowledgements.Most of this work was done when the second named author was a graduate student at Vanderbilt University.He would like to thank his supervisor, Denis Osin, for the valuable discussions.This paper would not have been written without the help of Osin.The second author would also like to thank Anna Marie Bohmann for the helpful comments and thank Ian Leary for answering his question and the suggestion of references.The first named author would like to thank Clara Löh and Kevin Li for helpful comments and suggestions.

Statements of main results
2.1.Cohomological properties of Dehn fillings.To simplify the statement, we introduce the following terminology and notation.
Definition 2.1.Let G be a group and H a subgroup of G.We say that a property P holds for sufficiently deep normal subgroups if there is a finite set F Ď H t1u such that P holds whenever N is a normal subgroup of H and N X F " H.
Given a normal subgroup N of H, let G " G{xxN yy and H " H{N .Theorem 1.4 can now be restated as: let G be a group with a subgroup H ãÑ h G. Then for sufficiently deep N ⊳ H, the natural homomorphism H Ñ G maps H injectively onto a hyperbolically embedded subgroup of G.
The following is a summary of our main results on cohomological properties of Dehn fillings.
Theorem A. Let G be a group with a subgroup H ãÑ h G. Then the following hold for all sufficiently deep N ⊳ H and all G-modules A.
(i) There is a spectral sequence E p,q 2 pAq " # H p pH; H q pN ; Aqq for q ą 0 H p pG; Aq for q " 0 ñ H p`q pG; Aq, where the action of G on A factors through G.In particular, the action of N on A fixes A pointwise.
(ii) (Algebraic Excision) For all n ě 0, there is a natural isomorphism induced by the quotient maps G Ñ G and H Ñ H, H n pG, H; Aq -H n pG, H; Aq.
The spectral sequence together with the algebraic excision have the following application.
Corollary 2.2.Let G be a group with a subgroup H ãÑ h G. Then for all sufficiently deep N ⊳ H and all G-modules A, we have (i) For all n ě cdpHq `2, H n pG; Aq -H n pG; Aq ' H n pH; Aq.
(iii) If G is of type F P n for some n P N `Y t8u (resp.F P ), then G is of type F P n (resp.F P ) if and only if H is of type F P n (resp.F P ).
Here, cdpGq stands for the cohomological dimension of a group G, and N `stands for the set of positive integers.This notion, property F P n , property F P , and relative cohomology of group pairs are reviewed in Section 3.1.Analogous homological statements to Theorem A and Corollary 2.2 also hold.The spectral sequence in Theorem A (i) is a refinement of the classical Lyndon-Hochschild-Serre spectral sequence [HS53,Lyn48] in the setting of Dehn fillings.
Let M and M s be as in Theorem 1.1 and let G " π 1 pM q, H " π 1 pBM q, N " xsy.Then Theorem A (ii) is an immediate consequence of excision.Therefore, Theorem A (ii) can be thought of as an algebraic analog of excision.
Instead of proving Theorem A, we will prove more general results (see Section 4), which cover the case of a hyperbolically embedded family of subgroups and will be useful in the proof of Theorems B, C, and E below, and also cover the case of weakly hyperbolically embedded subgroups and can be applied to graph of groups (see Example 3.7 (e)).
Corollary 2.2 was recently used by Arenas [Are22], who gave a variation of the Rips Construction that produces cubulated hyperbolic groups of cohomological dimension bounded above by the cohomological dimension of an associated compact special cube complex.2.2.Poincaré duality and simplicial volume.In [FM10], Fujiwara and Manning generalize Gromov-Thurston's 2π-theorem on Dehn fillings of 3-manifolds to higher dimensional finite volume hyperbolic manifolds M n with toral cusps.The resulting 2π-fillings are pseudomanifolds and are manifolds if and only if all the filling cores have dimension exactly n´2.They prove that every 2π-filling admits a complete locally CAT(0) metric.In [FM11], they show that the simplicial volume of every 2π-filling is positive and bounded above by the relative simplicial volume ||M , BM || of M .
The following two theorems can be seen as natural generalizations and group theoretic analogs of the results of Fujiwara and Manning.
Theorem B. Let G be a group and H " tH i u m i"1 a collection of subgroups such that H ãÑ h G. Suppose, for some integer 2 ď n, pG, Hq is a PDpnq-pair and that there are sufficiently deep tZ -N i ⊳ H i u m i"1 , such that every H i is a PDpn ´2q-group.Then G is a PDpnq-group.
Theorem C. Let G be a group and H " tH i u m i"1 a collection of subgroups such that H ãÑ h G. Suppose, for some integer n ě 2, pG, Hq is a PDpnq-pair and for a sufficiently deep tN i ⊳ H i u m i"1 , cdpH i q ď n ´2 for each 1 ď i ď m.Then, cdpGq " n, H n pG; Zq " Z.In addition, (i) if the group H i is amenable for each 1 ď i ď m, then ||G|| ď ||G, H||, where || ¨|| denotes the simplicial volume; (ii) if G is hyperbolic relative to H, then ||G|| ą 0.
We should point out that conjecturally by Kropholler, amenable groups of finite cohomological dimension such as H i are virtually solvable [Deg16, Question, p. 2].Also, by Poincaré duality group or pair we will always mean orientable ones.Theorem C can for example be applied when G is the fundamental group of a Riemannian manifold with a complete pinched negative sectional curvature and finite volume.We give one such application in Corollary 6.6 which generalises Theorem 1.5 of [FM11].
Corollary 2.3.Let M be a compact oriented n-manifold with nilmanifold boundary components such that the centre of the fundamental group of each boundary component is of rank at least 2. Suppose the interior of M admits a Riemannian manifold with a complete pinched negative sectional curvature and finite volume.If M T is a sufficiently deep Dehn filling manifold of M , then M T is a closed oriented aspherical n-manifold with

2.3.
Quotients of acylindrically hyperbolic groups.The notion of an acylindrically hyperbolic group was introduced by Osin [Osi16] as a generalization of non-elementary hyperbolic and non-elementary relatively hyperbolic groups.Examples of acylindrically hyperbolic groups can be found in many classes of groups that attracted group theorists for years, e.g., mapping class groups of surfaces [MM99,Bow08], outer automorphism groups of free groups [BF10], small cancellation groups [GS18], convergence groups [Sun19], the Cremona group (see [DGO17] and references therein; the main contribution towards showing the acylindrical hyperbolicity of the Cremona group is due to [CL13]), and tame automorphism groups of 3-dimensional affine spaces [LP19].We refer to [Osi18] for details and other examples.
Every acylindrically hyperbolic group G contains hyperbolically embedded subgroups [DGO17, Theorem 6.14] and Dehn fillings can often be applied to construct useful quotients of G.We use Theorem A to study homological properties of those quotients.
Recall that every acylindrically hyperbolic group G has a maximal finite normal subgroup denoted by KpGq [DGO17, Theorem 6.14].
Theorem D. Let G be an acylindrically hyperbolic group, and let C be any countable group.Then C embeds into a quotient G of G{KpGq (in particular, G is a quotient of G) such that (i) G is acylindrically hyperbolic; (ii) if C is finitely generated, then C ãÑ h G; (iii) if G and C are torsion-free, then so is G; (iv) for all n ě 3 and every G-module A, we have where the action of G{KpGq (resp.C) on A is induced by the quotient map G{KpGq Ñ G (resp. the embedding C ãÑ G); (v) cdpGq ď maxtcdpGq, cdpCqu; (vi) if C is finitely generated and G is of type F P n for some n P N `Y t8u (resp.F P ), then G is of type F P n (resp.F P ) if and only if C is of type F P n (resp.F P ).
As an application, we strengthen SQ-universality of hyperbolic groups given by Olshanskii [Ols95] and independently by Delzant [Del96] by adding cohomological conditions.
Corollary 2.4.Let G be a non-elementary hyperbolic group and C any hyperbolic group.Then there is a hyperbolic quotient G of G{KpGq (in particular, G is a quotient of G), where KpGq is the maximal finite normal subgroup of G, such that C embeds into G and the following hold.
(i) For all n ě 3 and every G-module A, we have where the action of G{KpGq (resp.C) on A is induced by the quotient map G{KpGq Ñ G (resp. the embedding C ãÑ G).
Theorem E. Let G 1 and G 2 be finitely generated acylindrically hyperbolic groups.Then there exists a common quotient G of G 1 {KpG 1 q and G 2 {KpG 2 q (in particular, G is a common quotient of G 1 and G 2 ) such that (i) G is acylindrically hyperbolic; (ii) for all n ě 3 and every G-module A, we have where the actions of G 1 {KpG 1 q and G 2 {KpG 2 q on A factor through G; (iii) cdpGq ď maxtcdpG 1 q, cdpG 2 qu; (iv) if G 1 and G 2 are of type F P n for some n P t2, 3, ..., 8u (resp.F P ), then so is G.
Homological analogs of Theorem D (iv), (v) and Theorem E (ii), (iii) also hold (see Remarks 7.3 and 7.11).Theorem D (i), (ii) are proved in [DGO17, Theorem 2.33] and Theorem E (i) is proved in [Hul16,Corollary 7.4].The benefit of Theorems D and E is that they allow one to control the cohomology of the resulting acylindrically hyperbolic quotients.As we illustrate in Section 8, this facilitates the constructions of various acylindrically hyperbolic groups satisfying certain cohomological properties.We list some of them below.Corollary 2.6.Let n ě 5 be an integer.Every non-elementary hyperbolic group G with cdpGq ď n has a hyperbolic quotient G with cdpGq " n such that G contains the Italiano-Martelli-Migliorini group.In particular, there is a type F non-hyperbolic subgroup H ă G.
The next two corollaries strengthen a result of [Hul16, Corllary 1.7] stating that every acylindrically hyperbolic group has an acylindrically hyperbolic quotient with Kazhdan's Property (T).
Corollary 2.8.Let G be any acylindrically hyperbolic group of type F P 8 .Then G has a family of acylindrically hyperbolic quotients tG k u 8 k"2 such that for each k, G k has Kazhdan's Property (T), is of type F P k´1 but not of type F P k .
In particular, since mapping class groups of surfaces of finite type, outer automorphism groups of free groups of finite rank and most 3-manifold groups are acylindrically hyperbolic and of type F P , they all exhibit such quotients.

2.4.
A few words on the proofs of the main results.The first step of the proof of Theorem A is to establish, under the assumptions of the theorem, the isomorphism (1) H n pxxN yy; Aq -CoInd G H H n pN ; Aq for all n ą 0. Here, CoInd G H stands for the co-induction from ZH to ZG.The Lyndon-Hochschild-Serre spectral sequence associated to the triple pG, xxN yy, Aq takes the form E p,q 2 pAq " H p pG; H q pxxN yy; Aqq ñ H p`q pG; Aq.Shapiro's lemma together with (1) yields Theorem A (i).In fact, we will establish a more precise result which involves a morphism between the Lyndon-Hochschild-Serre spectral sequences associated to pG, xxN yy, Aq and pH, N, Aq.Parts (ii) of Theorem A will be proved by an inspection of this morphism.
To prove Theorem B, we analyse the spectral sequence of Theorem A (i) and apply parts (ii) and (v).To show that G is a Poincaré duality group, we use Johnson-Wall characterisation [JW72]; namely, a group Γ is a Poincaré duality group of dimension n if and only if Γ is of type F P , H i pΓ, ZΓq " 0 for i ‰ n and H n pΓ, ZΓq " Z.The proof of Theorem D is a modification of the proof of [DGO17, Theorem 2.33].Given any acylindrically hyperbolic group G, one can find a non-cyclic free group F ãÑ h G 0 " G{KpGq.For any countable group C and any finite subset F Ď F t1u, we will use small cancellation theory to construct a normal subgroup N ⊳ F such that N X F " H, C embeds into F {N , and F {N has the desired cohomological properties.Theorem 1.4 then implies that C embeds into G 0 {xxN yy and Theorem A applied to N ⊳ F ãÑ h G 0 yields the desired cohomological results.

The algebraic excision
The proof of [Hul16, Corollary 7.4] uses small cancellation theory instead of Dehn filling.In order to apply our main result, we carry out an alternative approach.Given finitely generated acylindrically hyperbolic groups G 1 and G 2 , we construct subgroups 2 , where G 1 1 " G 1 {KpG 1 q and G 1 2 " G 2 {KpG 2 q, such that the family tH 1 , H 2 u hyperbolically embeds into r G.For any finite sets tF i Ď H i t1uu i"1,2 , we will use small cancellation theory to construct normal subgroups 2 .Theorem E is then proved by applying general versions of Theorems 1.4 and A. The main difficulty of this argument is the construction of H 1 and H 2 , which is presented in Section 7.2, using a technical tool provided by [DGO17] (see also [Osi07]) called isolated components.
2.5.Organization of the paper.We will start with preliminaries in Section 3, recalling basic definitions of group cohomology, the notion of (weakly) hyperbolically embedded subgroups, isolated components, acylindrically hyperbolic groups, and the structural result of [Sun20] called the Cohen-Lyndon property.The proof of (the general version of) Theorem A is given in Section 4. Theorems B and C are proved in Sections 5 and 6, respectively.Theorems D and E are proved in Section 7 with applications given in Section 8.

Cohomology of groups.
Let G be a group.Recall that the homological and cohomological dimension of G can be defined by hdpGq " suptn P N | H n pG, Aq ‰ 0 for some ZG-module Au, cdpGq " suptn P N | H n pG, Aq ‰ 0 for some ZG-module Au, respectively.
G is of type F P n for some n P N `Y t8u if there is a projective resolution ¨¨¨Ñ P 2 Ñ P 1 Ñ P 0 Ñ Z over ZG such that P k are finitely generated G-modules for all k ď n.G is of type F P if cdpGq ă 8 and G is of type F P 8 .Property F P n can be characterized by the cohomology functor.The following will be useful in the proof of Theorem 4.7.(a) G is of type F P n for some n P N `Y t8u.(b) For every k ď n and every direct system tA i u iPI of G-modules such that lim Ý Ñ A i " 0, we have lim Ý Ñ H k pG; A i q " 0.
Given a family tH λ u λPΛ of subgroups of G, [BE78] defined the relative (co)homology of the group pair pG, tH λ u λPΛ q.We briefly recall the definition.Let ∆ be the kernel of the augmentation À λPΛ ZrG{H λ s ։ Z which sends every left H λ -coset to 1.By definition, (2) H n pG, tH λ u λPΛ ; Aq " Tor G n´1 p∆, Aq, H n pG, tH λ u λPΛ ; Aq " Ext n´1 G p∆, Aq for any G-module A. The dimension shift in the above definition ensures a long exact sequence between the absolute and relative (co)homology (see [BE78, Proposition 1.1]).
3.2.(Weakly) hyperbolically embedded subgroups.The notion of (weakly) hyperbolically embedded subgroups was introduced by [DGO17], which is our main reference for Sections 3.2, 3.3, and 3.4.We first recall the definition and present some examples.The motivation will be discussed afterwards.
Let G be a group, tH λ u λPΛ a family of subgroups of G, X a subset of G such that G is generated by X together with the union of all H λ (in which case X is called a relative generating set of G with respect to tH λ u λPΛ ), and H " Ů λPΛ H λ .Consider the Cayley graph ΓpG, X \ Hq.Note that, for λ P Λ there is a natural embedding ΓpH λ , H λ q ãÑ ΓpG, X \ Hq whose image is the subgraph of ΓpG, X \ Hq with vertices and edges labeled by elements of H λ .Remark 3.2.We do allow X X H λ ‰ H and H λ X H µ ‰ t1u for distinct λ, µ P Λ, in which case there will be multiple edges between some pairs of vertices of ΓpG, X \ Hq.
For λ P Λ, an edge path in ΓpG, X \ Hq between vertices of ΓpH λ , H λ q is called H λadmissible if it does not contain any edge of ΓpH λ , H λ q.Note that an H λ -admissible path is allowed to pass through vertices of ΓpH λ , H λ q.
For example, consider the simple case where tH λ u λPΛ " tHu consists of only one subgroup H ď G.The Cayley graph ΓpG, X \Hq is displayed in Figure 1.The blue path is admissible.The red path is an edge from 1 to h labeled by h P H, and thus is inadmissible.If h happens to be an element of X, i.e., there exists x P X with x " h, and the red path were labeled by x instead of h, then the red path would be admissible.Definition 3.3.For every pair of elements h, k P H λ , let p d λ ph, kq P r0, 8s be the length of a shortest H λ -admissible path connecting the vertices labeled by h and k.If no such path exists, set p d λ ph, kq " 8.The laws of summation on r0, 8q extend naturally to r0, 8s and it is easy to verify that p d λ : H λ ˆHλ Ñ r0, `8s defines a metric on H λ , which is called the relative metric on H λ with respect to X. Definition 3.4.We say that the family tH λ u λPΛ weakly hyperbolically embeds into pG, Xq (denoted by tH λ u λPΛ ãÑ wh pG, Xq) if G is generated by the set X together with union of all H λ , λ P Λ, and the Cayley graph ΓpG, X \ Hq is a Gromov hyperbolic space.
If tH λ u λPΛ ãÑ wh pG, Xq and for each λ P Λ, the metric space pH λ , p d λ q is proper, i.e., every ball of finite radius contains only finitely many elements, then tH λ u λPΛ hyperbolically embeds into pG, Xq (denoted by tH λ u λPΛ ãÑ h pG, Xq).If in addition, X and Λ are finite, then we say that G is hyperbolic relative to tH λ u λPΛ .
Further, we say that the family tH λ u λPΛ hyperbolically embeds into G, denoted by tH λ u λPΛ ãÑ h G, if there exists some subset X Ď G such that tH λ u λPΛ ãÑ h pG, Xq.Notation 3.5.In case tH λ u λPΛ " tHu is a singleton, we will drop the braces and write H ãÑ wh pG, Xq and H ãÑ h G.
We refer to Figure 1 for an illustration of the situation H ãÑ h G.The grey discs represent cosets of H in G.The black edges are labeled by elements of X.The edges and discs appear in a tree-like pattern as ΓpG, X \ Hq is Gromov hyperbolic.

ΓpH, Hq
Remark 3.6.Notice that the above definition of relative hyperbolicity is not commonly used in literature.One of the most commonly used definitions for relative hyperbolicity which we will use later is the following: a group G is hyperbolic relative to a family tH λ u λPΛ of its subgroups if G has a finite relative presentation with respect to tH λ u λPΛ with a linear relative isoperimetric function.The equivalence of these two definitions is proved in [DGO17, Remark 4.41 and Theorem 4.42].
(a) H ãÑ wh pG, Gq for every subgroup H ď G.
(b) H ãÑ h pG, Gq for every finite subgroup H ď G.
(d) If G can be decomposed as a free product of its subgroups tG λ u λPΛ (denoted by G " ˚λPΛ G λ ), then tG λ u λPΛ ãÑ h pG, Hq [DGO17, Example 4.12].(e) More generally, suppose that G " π 1 pGq, where G is a graph of groups.Let tG v u vPV G be the collection of vertex subgroups and tG e u ePEG the collection of edge subgroups.By [DGO17, Example 4.12], tG v u vPV G ãÑ wh pπ 1 pGq, Xq for any set X Ď G consisting of stable letters (i.e., generators corresponding to edges of G T G, where T G is a spanning tree of G).
Recall that a group is word-hyperbolic if it is finitely generated and for some (equivalently, any) finite generating set the corresponding Cayley graph is Gromov hyperbolic.The notion of weak hyperbolic embedding is thus an attempt to study possibly non-word-hyperbolic groups via Gromov hyperbolic spaces.Example 3.7 (a) illustrates a triviality of this notion, in the sense that the weak hyperbolic embedding does not provide any information about the group G. Notice that in that example, the corresponding relative metric is bounded.One might therefore refine the notion by imposing additional conditions on the relative metric, for example, requiring local finiteness of the relative metric and obtaining the notion of hyperbolic embedding.We note that Examples 3.7 (b) and (c) exhibit two kinds of trivialities of hyperbolic embedding, and a further refinement is given by the notion of an acylindrically hyperbolic group (see Section 3.4).
The next lemma tells us how to find hyperbolically embedded subgroups.
(C 1 ) Every H λ acts on S properly.(C 2 ) There exists s P S such that for every λ P Λ, the H λ -orbit H λ psq of s is quasi-convex in S. (C 3 ) For every ǫ ą 0 and some s P S, there exists R ą 0 such that the following holds.
Suppose that for some g P G and λ, µ P Λ, we have then λ " µ and g P H λ , where pgH λ psqq `ǫ denotes the ǫ-neighborhood of gH λ psq in S.
The following proposition, which roughly says that being a hyperbolically embedded subgroup is a transitive property, will be used later.Proposition 3.9 (Dahmani-Guirardel-Osin [DGO17, Proposition 4.35]).Let G be a group, let tH λ u λPΛ be a finite family of subgroups of G, let X Ď G, and let Y λ Ď H λ for every λ P Λ. Suppose that tH λ u λPΛ ãÑ h pG, Xq and for every λ P Λ, there is a family of subgroups

Isolated components.
In the proof of Theorem E, we need to construct specific hyperbolically embedded subgroups.A tool to do this is the notion of an isolated component, which was introduced by [Osi07] for relatively hyperbolic groups and generalized to hyperbolically embedded subgroups by [DGO17].In this section, we recall the definition and collect several results.We start with conventions.Let G be a group and X a generating set of G. Consider the Cayley graph ΓpG, Xq.If p is a path in ΓpG, Xq, then the label of p is denoted by Labppq, the length p is denoted by ℓ X ppq, and the initial (resp.terminal) vertex of p is denoted by p ´(resp.p `).
Now suppose that tH λ u λPΛ is a family of subgroups of G. Let H " Ů λPΛ H λ , and let X be a relative generating set of G with respect to tH λ u λPΛ .For λ P Λ, let p d λ be the relative metric on H λ with respect to X.The following terminology goes back to [Osi06].
Definition 3.10.Let p be a path in ΓpG, X \ Hq.For every λ P Λ, an H λ -subpath q of p is a nontrivial subpath of p such that Labpqq is a word over the alphabet H λ (if p is a cycle, we allow q to be a subpath of some cyclic shift of p).An H λ -subpath q of p is an H λ -component if q is not properly contained in any other H λ -subpath.Two H λ -components q 1 and q 2 of p are connected if there exists a path t in ΓpG, X \ Hq such that t connects a vertex of q 1 to a vertex of q 2 , and that Labptq is a letter from Suppose that q is an H λ -component of a path p Ď ΓpG, X \ Hq.Then q ´(resp.q `) is labeled by an element g P G (resp h P G) and we have g ´1h P H λ .In this case, let A nice property of isolated components is that in a geodesic polygon p, the total p ℓ-length of isolated components of p is bounded linearly by the number of sides of p.More precisely: Proposition 3.11 (Dahmani-Guirardel-Osin [DGO17, Proposition 4.14] (see also [Osi07, Proposition 3.2])).If tH λ u λPΛ ãÑ wh pG, Xq, then there exists a number D ą 0 satisfying the following property: Let p be an n-gon in ΓpG, X \ Hq with geodesic sides p 1 , ..., p n and let I be a subset of the set of sides of p such that every side p i P I is an isolated H λ i -component of p for some λ i P Λ.Then ÿ In fact, the above proposition is the reason why certain properties (for example, Theorems 3.20 and 3.22 below) hold for sufficiently deep Dehn fillings (see Definition 3.16).
The technical lemma below will be used in Section 7.2 along with Lemma 3.8 to construct hyperbolically embedded subgroups.Lemma 3.12 (Dahmani-Guirardel-Osin [DGO17, Lemma 4.21]).Suppose tH λ u λPΛ ãÑ wh pG, Xq.Let W be the set consisting of all words w over X \ H such that (W1) w contains no subwords of type xy, where x, y P X; (W2) if w contains a letter h P H λ for some λ P Λ, then p d λ p1, hq ą 50D, where D is given by Proposition 3.11; is a subword of w, where x P X, h 1 P H λ , h 2 P H µ , then either λ ‰ µ or the element represented by x in G does not belong to H λ (resp.λ ‰ µ).
Then the following hold.
(a) Every path in the Cayley graph ΓpG, X \ Hq labeled by a word from W is a p4, 1qquasi-geodesic.(b) If p is a path in ΓpG, X \ Hq labeled by a word from W , then for every λ P Λ, every H λ -component of p is isolated.(c) For every ǫ ą 0, there exists R ą 0 satisfying the following condition.Let p, q be two paths in ΓpG, X \ Hq such that ℓ X\H ppq ě R, Labppq, Labpqq P W, and p, q are oriented ǫ-close, i.e., where d X\H is the combinatorial metric of ΓpG, X \ Hq.Then there exist five consecutive components of p which are respectively connected to five consecutive components of q.In other words, such that the following hold.
(i) r (resp.t) is a subpath of p (resp.q) whose label does not end with a letter from H. (ii) s (resp.u) is a subpaht of p (resp.q) whose label does not start with a letter from H. (iii) For i " 1, ..., 4, x i and y i are either trivial subpaths or subpaths labeled by a word over X; (iv) For i " 1, ..., 5, a i and b i are connected H λ i -components.Moreover, if x i or y i is the trivial path for some i P t1, ..., 4u, then λ i ‰ λ i`1 .
Remark For the definition an acylindrical action, the reader is referred to [Osi18].Intuitively, one can think of acylindricity as an analog of properness.An acylindrical action of a group G is non-elementary if its orbits are unbounded and G is not virtually-cyclic [Osi16, Theorem 1.1].
Techniques of hyperbolically embedded subgroups are mainly applied to acylindrically hyperbolic groups, because every acylindrically hyperbolic group contains infinitely many hyperbolically embedded virtually free subgroups.
Theorem 3.15 (Dahmani-Guirardel-Osin [DGO17, Theorem 6.14]).Let G be an acylindrically hyperbolic group.Then G has a maximal finite normal subgroup, denoted by KpGq.Moreover, for every n P N `, there exists a free group F of rank n such that F ˆKpGq ãÑ h G.
3.5.Sufficient deepness and Cohen-Lyndon triples.One consequence of Proposition 3.11 is that acylindrical hyperbolicity is preserved by Dehn fillings, provided that the Dehn fillings are sufficiently deep and done on hyperbolically embedded subgroups (see Theorem 3.20).Definition 3.16.Let G be a group with a family of subgroups tH λ u λPΛ ãÑ wh pG, Xq for some subset X Ď G.For every λ P Λ, let p d λ be the relative metric on H λ with respect to X.A property P holds for all sufficiently deep normal subgroups if there exists a constant C ą 0 such that P holds for every family of normal subgroups tN λ ⊳ H λ u λPΛ with p d λ p1, nq ą C for all n P N λ t1u.
Example 3.17.If G " H 1 ˚A H 2 is a free product with amalgamation, then tH 1 , H 2 u ãÑ wh pG, Hq by Example 3.7 (e).For normal subgroups tN i ⊳ H i u i"1,2 , consider the property P: The quotient G{xxN 1 YN 2 yy splits as an amalgamated free product, where xxN 1 YN 2 yy denotes the normal closure of N 1 Y N 2 in G.
Then P holds for all sufficiently deep normal subgroup tN i ⊳ H i u i"1,2 , because P holds whenever N i X A " t1u, which amounts to saying that p d i pnq ą 1 for all n P N i t1u, where p d i is the relative metric on H i corresponding to the weak hyperbolic embedding tH 1 , H 2 u ãÑ wh pG, Hq.Let G be a group with a family of subgroups tH λ u λPΛ ãÑ h G. Suppose that a property P holds for all sufficiently deep normal subgroups tN λ ⊳ H λ u λPΛ .Then there exist finite sets tF λ Ď H λ t1uu λPΛ such that P holds whenever N λ X F λ " H for all λ P Λ.
To simplify statements, we use the following notation.Notation 3.19.Let G be a group and S a subset of G. Then xxSyy denotes the normal closure of S in G. Suppose that tH λ u λPΛ is a family of subgroups of G and tN λ ⊳ H λ u λPΛ is a family of normal subgroups.We call pG, tH λ u λPΛ , tN λ u λPΛ q a group triple.We also let Theorem 3.20 (Dahmani-Guirardel-Osin [DGO17, Theorem 7.19], Osin [Osi16, Theorem 1.2]).Let G be a group with a family of subgroups tH λ u λPΛ ãÑ h G. Then for all sufficiently deep normal subgroups tN λ ⊳ H λ u λPΛ , the natural homomorphism H λ Ñ G is injective for λ P Λ and we have tH λ u λPΛ ãÑ h G.Moreover, if for some λ P Λ, cardpH λ q " 8 and H λ is a proper subgroup of G, then G is acylindrically hyperbolic.
The normal subgroup xxN yy in the above theorem can be described more precisely: it has a particular free product structure.Definition 3.21.A group triple pG, tH λ u λPΛ , tN λ u λPΛ q is called a Cohen-Lyndon triple if there exist left transversals T λ of H λ xxN yy in G such that xxN yy " ˚λPΛ,tPT λ tN λ t ´1.
The above free product structure was first proved by Cohen-Lyndon [CL63] for free groups, hence the name.The following theorem was partially proved by [DGO17, Theorem 7.19], which was later improved by [Sun20].
Theorem 3.22 (Sun [Sun20, Theorem 5.1]).Let G be a group with a family of subgroups tH λ u λPΛ ãÑ wh pG, Xq for some X Ď G. Then for all sufficiently deep normal subgroups tN λ ⊳ H λ u λPΛ , pG, tH λ u λPΛ , tN λ u λPΛ q is a Cohen-Lyndon triple.

Cohomology of Dehn fillings
In this section, we prove Theorem A. To simplify the notation, we use 3.19.
Proposition 4.1.Let pG, tH λ u λPΛ , tN λ u λPΛ q be a Cohen-Lyndon triple.Then for any G-module A and q ą 1 (also for any G-module A and q ą 0) there are isomorphisms induced by the inclusions N λ ãÑ xxN yy.
We remark that an easy consequence of pG, tH λ u λPΛ , tN λ u λPΛ q being a Cohen-Lyndon triple is that the natural maps H λ Ñ G are injective [Sun20, Lemma 6.4], and thus it makes sense to consider the (co)inductions Ind G H λ and CoInd G H λ .
Proof.We will prove the homological version.The proof of the cohomological version will be analogous.
Let tT λ u λPΛ be the family of left transversals associated to the Cohen-Lyndon triple pG, tH λ u λPΛ , tN λ u λPΛ q, given by Definition 3.21.Then xxN yy " ˚λPΛ,tPT λ tN λ t ´1.Consider the Bass-Serre tree associated to this free product decomposition with vertex set V and edge set E. We then get a short exact sequence of xxN yy-modules The short exact sequence (3) then transforms to a short exact sequence of chain complexes The long exact sequence corresponding to (4) yields Since ZrEs is a free xxN yy-module, H q pxxN yy; ZrEs b Aq " 0 for q ą 0 and f is injective if xxN yy acts trivially on A. This yields an isomorphism (5) θ ˚: à λPΛ,tPT λ H q ptN λ t ´1; Aq Ñ H q pxxN yy; Aq for q ą 1 and also for q " 1 if xxN yy acts trivially on A. The action of G on H q pxxN yy; Aq and the isomorphism θ ˚endow À λPΛ,tPT λ H q ptN λ t ´1; Aq with a G-action.Next, we will show that this action is a direct sum of permutation actions.
The G-action on H q pxxN yy; Aq comes from the G-action on A and conjugation of G on xxN yy which is induced by the diagonal G-action on F ˚bxxN yy A. More explicitly, each g P G acts as For each λ P Λ and t P T λ , the group tH λ t ´1 acts on F i b tN λ t ´1 A by rx, as Þ Ñ rgx, gas, @g P tH λ t ´1.
This gives us a tH λ t ´1-equivariant restriction of θ θ : which in turn induces the tH λ t ´1-equivariant inclusion θ ˚: H q ptN λ t ´1; Aq ãÑ H q pxxN yy; Aq.Now, for each λ P Λ and t P T λ , the action of t on H q pxxN yy; Aq induces a map between two summands on the left-hand side of the isomorphism (5).To see this, note that the map defined by satisfies θ ˝σt " τ t ˝θ.In homology, we then have pσ t q ˚" θ ´1 ˚˝pτ t q ˚˝θ ˚as claimed.We have shown that for each λ P Λ, the G-action on À tPT λ H q ptN λ t ´1; Aq restricts to the tH λ t ´1-action on the summand H q ptN λ t ´1; Aq and that it permutes H q pN λ ; Aq and H q ptN λ t ´1; Aq.It is not difficult to show now, see for example [Bro94, Proposition 5.3], that à Let pG, tH λ u λPΛ , tN λ u λPΛ q be a Cohen-Lyndon triple and A any G-module.There are Lyndon-Hoschild-Serre spectral sequences E λ,2 p,q " H p pH λ ; H q pN λ ; Aqq ñ H p`q pH λ ; Aq, F 2 p,q " H p pG; H q pxxN yy; Aqq ñ H p`q pG; Aq associated with the triples pH λ , N λ , Aq and pG, xxN yy, Aq, respectively (see for example The inclusions H λ ãÑ G and N λ ãÑ xxN yy induce a morphism φ : E p,q Ñ F p,q .By Proposition 4.1 and Shapiro's lemma, the restriction φ : E 2 p,q Ñ F 2 p,q is an isomorphism for all q ą 1 and also for q " 1 if xxN yy acts trivially on A. In short: Theorem 4.2.Let pG, tH λ u λPΛ , tN λ u λPΛ q be a Cohen-Lyndon triple.Then for every Gmodule A, there is a natural morphism of homological spectral sequences φ : E p,q Ñ F p,q such that p,q " H p pG; H q pxxN yy; Aqq ñ H p`q pG; Aq, φ is induced by the inclusions H λ ãÑ G, N λ ãÑ xxN yy, and φ restricts to an isomorphism φ : E 2 p,q -Ý Ñ F 2 p,q for all q ą 1 and also for q " 1 if xxN yy acts trivially on A. Moreover, the analogous statement holds for cohomology as well.
We further investigate the morphism φ of the above theorem.Let P i ։ Z be a free resolution of Z over ZG and S i ։ Z a free resolution of Z over ZG.The spectral sequence E p,q is induced by the double complex C p,q " À λPΛ pP p b S q q b H λ A and F p,q is induced by D p,q " pP p b S q q b G A. The surjections induce a surjection C p,q ։ D p,q , which in turn induces the morphism φ.Let R p,q be the kernel of the surjection C p,q ։ D p,q and E p,q pRq the spectral sequence associated with R p,q .It turns out that E p,q pRq ñ H p`q`1 pG, tH λ u λPΛ ; Aq and we have the following commutative diagram of long exact sequences.
Proposition 4.3.Let pG, tH λ u λPΛ , tN λ u λPΛ q be a Cohen-Lyndon triple and A a G-module.Then there is a commutative diagram of exact sequences where ι ˚and ι ˚are induced by the inclusions H λ ãÑ G and H λ ãÑ G, respectively.Moreover, the cohomological analog of the above statement holds as well.
Proof.We will only prove the homological version since the cohomological version is similar.First, we compute the limit of the spectral sequence E p,q pRq.Claim 4.3.1.E p,q pRq ñ H p`q`1 pG, tH λ u λPΛ ; Aq.
Proof of the claim.Let ∆ be the kernel of the augmentation À λPΛ ZrG{H λ s Ñ Z. Then we have a short exact sequence Thus, tensoring (6) by ´bG A yields Let T n " À p`q"n P p b S q .Notice that T n b ∆ ։ ∆ is a free resolution of ∆ over ZG.Therefore, to prove the claim, it suffices to show that which will be established once we show that (7) is exact.It is easy to check that ZrG{H λ s ¸Ñ A Ñ 0 is exact.Since P p is free abelian and S q is a free G-module, we have an exact sequence which, by basic tensor identities, transforms to the desired one.
Let us return to the proof of the proposition.We have a short exact sequence 0 Ñ R p,q Ñ C p,q Ñ D p,q Ñ 0, whose vertical homology gives the following long exact sequence of the E 1 -terms of the associated spectral sequences ¨¨¨Ñ H q pR p,˚q Ñ H q pC p,˚q Ñ H q pD p,˚q Ñ H q´1 pR p,˚q Ñ ¨¨B y Proposition 4.1 and Shapiro's lemma, This shows that E 1 p,q pRq " H q pR p,˚q " 0 for all q ą 0 and hence E 2 p,0 pRq " H p`1 pG, tH λ u λPΛ ; Aq for all p.Also, since Z b xxN yy A " A, we have E 1 p,0 pCq " H 0 pC p,˚q " So, we get a short exact sequence whose long exact sequence is the bottom row of the desired diagram.Similar to the proof of the claim, one can show that there is a short exact sequence There is a G-module homomorphism from the resolution S q ։ Z to P p ։ Z, which gives rise to a map from (9) to (8), which in turn yields the desired commutative diagram of the associated long exact sequences.
Remark 4.4.Similar to the proof of Proposition 4.3, one can show that there is a short exact sequence where P p and A are as in the proof of Proposition 4.3, and ∆ is the kernel of the augmentation À λPΛ ZrG{H λ s ։ Z.The natural map from (8) to (10) gives rise to a commutative diagram of the corresponding long exact sequences, which together with the five lemma yields an isomorphism H ˚pG, tH λ u λPΛ ; Aq -H ˚pG, tH λ u λPΛ ; Aq (of course, under the assumption that pG, tH λ u λPΛ , tN λ u λPΛ q is a Cohen-Lyndon triple and A is a G-module).Moreover, The analogous isomorphism for cohomology holds as well.
An easy consequence of Proposition 4.3 is a direct sum decomposition of (co)homology.
Corollary 4.5.Let pG, tH λ u λPΛ , tN λ u λPΛ q be a Cohen-Lyndon triple and A a G-module.
(a) If for some p P N, À λPΛ H p pH λ ; Aq " 0 and the natural map where the horizontal maps come from the exact sequences for the pairs pG, Hq, pG, Hq and ψ ˚, θ ˚are natural maps induced by the surjections G ։ G, H λ ։ H λ , respectively.The hypothesis imply that η p : H p pG, Hq Ñ H p pGq is an isomorphism which in turn shows that the map ξ p is injective.Since H p pHq " 0, η p`1 is injective.So ξ p`1 is also injective.This shows λ p " 0 and hence φ p is onto.Since pη p q ´1 ˝ψp is a section for ξ p , the result follows.
An immediate corollary to the above is an estimate of the (co)homological dimension.To shorten the notation, if pG, tH λ u λPΛ , tN λ u λPΛ q is a Cohen-Lyndon triple, let hdpHq " sup Corollary 4.6.Let pG, tH λ u λPΛ , tN λ u λPΛ q be a Cohen-Lyndon triple.Then hdpGq ď maxthdpGq, hdpHq `1, hdpHqu, cdpGq ď maxtcdpGq, cdpHq `1, cdpHqu.By Theorem 3.1, the commutativity of colimits of coefficients with the cohomology functor can be used to characterize property F P n , which is our next goal.
Theorem 4.7.Let G be a group with a finite family of subgroups tH i u m i"1 ãÑ h G. Suppose that G is of type F P n for some n P N `Y t8u.Then for sufficiently deep  ś m i"1 H ˚pH i ; ´q.Let tA j u jPJ be a direct system of G-modules such that lim Ý Ñ A j " 0. By Proposition 4.3, we have a commutative diagram of exact sequences for each j P J: For k ď n, after taking direct limits, we have lim Ý Ñ H k pG; A j q " 0 by Theorem 3.1.By [DGO17, Remark 4.26 and Corollary 4.32], we also have lim Ý Ñ H k pH; A j q " 0, and thus lim Ý Ñ H k pG, H; A j q " 0, which implies that lim Ý Ñ H k pG; A j q " lim Ý Ñ H k pH; A j q " 0 except for k " n.Since lim Ý Ñ H n pH; A j q " 0, we have lim Ý Ñ H n pG; A j q " 0 as well, and thus G is of type F P n , again by Theorem 3.1.
We emphasize that the finiteness of Λ is needed in the above proof to guarantee that lim Ý Ñ H k pH; A j q " lim Ý Ñ H k pH; A j q " 0 for k ď n.
Proof of Theorem A. By Theorem 3.22, pG, H, N q is a Cohen-Lyndon triple for sufficiently deep N ⊳ H. Items (ii) follows directly from Propositions 4.3 and Remark 4.4.
Theorem 4.2 provides us a spectral sequences E p,q 2 ñ H p`q pG; Aq and isomorphisms E p,q 2 -H p pH; H q pN ; Aqq for q ą 0. Theorem A (i) then follows by observing that H p pG; H 0 pxxN yy; Aqq -H p pG; Aq.Similarly, one can prove the homological version.
We collect the results of this section and state the full version of Theorem A.
Theorem 4.9.Let G be a group with a family of hyperbolically embedded subgroups tH λ u λPΛ ãÑ h G. Then the following hold for all sufficiently deep N λ ⊳ H λ , λ P Λ and all G-modules A, where G " G{xxY λPΛ N λ yy.
(i) There is a spectral sequence E p,q 2 pAq " # ś λPΛ H p pH λ ; H q pN λ ; Aqq for q ą 0 H p pG; Aq for q " 0 ñ H p`q pG; Aq, where H λ " H λ{N λ for all λ and the action of G on A factors through G.In particular, the action of xxY λPΛ N λ yy on A fixes A pointwise.
(ii) (Algebraic Excision) For all n ě 0 and λ P Λ, there is a natural isomorphism induced by the quotient maps G Ñ G and H λ Ñ H λ , H n pG, H λ ; Aq -H n pG, H λ ; Aq.

Dehn fillings and duality
Let G a group and H " tH i u m i"1 a finite collection of subgroups.Following Bieri-Eckmann [BE78], we say that pG, Hq is a duality pair of dimension n, with dualizing module C, if for all k P Z and all G-modules A, one has given by the cap product by the fundamental class e P H n pG, H; Cq.In which case, it follows that C -H n pG, H; ZGq.If C " Z with trivial G-action, the pair is called an (orientable) Poincaré duality pair, in short, a PDpnq-pair.
Let pG, Hq be a duality pair of dimension n with dualizing module C and let ∆ be the kernel of the augmentation G is a duality group of dimension n ´1 with dualizing module ∆ b C and each H i is a duality group of dimension n ´1 with dualizing module C (thinking as an H i -module).
The following result generalises [Wan18, Corollary 1.5] which deals with the case where pG, Hq is a type F 8 relatively hyperbolic group pair.Theorem 5.2.Let G be a group and H " tH i u m i"1 a collection of subgroups such that H ãÑ h G. Suppose, for some integer 0 ă n, pG, Hq is a PDpnq-pair and that there are sufficiently deep tZ -N i ⊳ H i u m i"1 such that each member of H " tH i u m i"1 is a PDpn ´2qgroup.Then G is a PDpnq-group.
Proof.For each 1 ď i ď m, H k pH i ; ZGq " 0 if k ‰ n ´2 and by Corollary 5.1, H k pG, H; ZGq " 0 if k ‰ n ´1, n.The long exact sequence in cohomology for the pair pG, Hq shows that H k pG; ZGq " 0 if k ‰ n ´2, n ´1, n and for k " n it gives H n pG; ZGq -H n pG, H; ZGq -Z.
Next, we consider the spectral sequence of Theorem 4.2 # ś m i"1 H p pH i ; H q pN i ; ZGqq for q ą 0 H p pG; ZGq for q " 0 ñ H p`q pG; ZGq.
Since each N i -Z, E p,q 2 " 0 for q ą 1 and E p,1 2 -ś m i"1 H p pH i ; ZGq.Since there are only two nontrivial rows on the E p,q 2 -page of the spectral sequence, we obtain a Gysin long exact sequence (11) For k " n ´1, (11) and the morphism between the Lyndon-Hochschild-Serre spectral sequences associated to the inclusions pH λ , N λ q ãÑ pG, xxN

Simplicial volume of Dehn fillings
For detailed background on bounded cohomology and simplicial volume, we refer to [Gro82], [Iva85] and [Fri17].
Let G be a group.Consider the singular chain complex C ˚pBG; Rq endowed with the ℓ 1 -norm The cochain complex of bounded cochains C b pBG; Rq coincides with the normed dual complex of C ˚pBG; Rq.The norm on chains induces a ℓ 1 -semi-norm || ¨|| 1 on H ˚pG; Rq " H ˚pC ˚pBG; Rqq and ℓ 8 -semi-norm || ¨|| 8 on H b pG; Rq " H ˚pC b pBG; Rqq.When H " tH i u m i"1 is a collection of subgroups of G, H ˚pG, H; Rq and H b pG, H; Rq and their seminorms are defined analogously [MY07, §9.2].
As before, when N i ⊳ H i , we let N " Ť m i"1 N i , H " tH i u m i"1 and G " G{xxN yy.Lemma 6.1.Let G be a group and H " tH i u m i"1 a collection of subgroups such that H ãÑ h G. Suppose, for some integer n ě 2, pG, Hq is a PDpnq-pair and for a sufficiently deep tN i ⊳ H i u m i"1 , cdpH i q ď n ´2 for each 1 ď i ď m.Then H n pG; Zq " Z.
Proof.The result follows from the long exact sequence in homology of the pair pG, Hq and the isomorphism H n pG, H; Zq -H n pG, H; Zq " Z. BH i q; Rq, rf s " rG, Hs P H n pG, H; Rqu.
The simplicial volume of pG, Hq is define analogously.
The following result is a generalisation of [FM11, Theorem 1.5].
Theorem 6.3.Let G be a group and H " tH i u m i"1 a collection of subgroups such that H ãÑ h G. Suppose, for some integer n ě 2, pG, Hq is a PDpnq-pair and for a sufficiently deep tN i ⊳ H i u m i"1 , cdpH i q ď n ´2 for each 1 ď i ď m.Then, cdpGq " n, H n pG; Zq " Z.In addition, Proof.The first two claims follow from Corollary 2.2 (ii) and Lemma 6.1.6.1.Dehn fillings of pinched negatively curved manifolds.In this section, we illustrate how the results of the previous sections apply in a geometric setting.First, we need a lemma.Lemma 6.4.Let G be a group and H " tH i u m i"1 a collection of finitely generated torsionfree nilpotent subgroups with rkpZpH i qq ě 2, for all i.Then there are infinitely many collections of subgroups tN i u m

For part (i), by
i"1 such that Z -N i ⊳ ZpH i q, H i is torsion-free and j k : H k pG; Zq Ñ H k pG, H; Zq is an isomorphism for all k ą maxthpH i q | 1 ď i ď mu where hpH i q is the Hirsch length of H i .
Proof.For each i, the center ZpH i q contains a factor xx i , y i y -Z 2 .Let N s,t i " xx s i y t i y ⊳ H i where s, t P Z are co-prime.Then the quotient H i " H i {N s,t i is a torsion-free nilpotent group.The long exact sequence in homology for the pair pG, Hq establishes the isomorphism The collections tN s,t i u m i"1 enumerated by the co-prime pairs ps, tq have the required properties.Definition 6.5.Let M be a Riemannian n-manifold with a complete pinched negative sectional curvature and finite volume.All the cuspidal ends of M are almost-flat manifolds and hence by the results of Gromov and Ruh are infra-nil [Gro78,Ruh82].In particular, they are finitely covered by nilmanifolds.Let us assume here that all the cuspidal ends tL i u m i"1 of M are nilmanifolds such that rkpZpπ 1 pL i qq ě 2. Let M be the natural compactification of M with boundary components tL i u m i"1 .Let G " π 1 pM q and H i " π 1 pL i q.Then G is hyperbolic relative to the fundamental groups of the cuspidal ends and in particular tH i u m i"1 ãÑ h G [Bow12,Far98].By Theorem A (ii) and Lemma 6.4, there are infinitely many collections of sufficiently deep normal subgroups tN i u m i"1 satisfying the conclusions of both such that j n : H n pG; Zq -Ý Ñ H n pG, H; Zq.Let tN i u m i"1 be one such collection.By [Osi07, Theorem 1.1], we can also assume that tN i u m i"1 are sufficiently deep so that G is hyperbolic relative to H. Let T i ãÑ L i π i Ý Ñ B i be the circle bundle with a nilmanifold base corresponding to the short exact sequence N i ãÑ H i ։ H i and denote by CpL i , T i q the mapping cylinder of π i for each i (see Figure 2 for an illustration).Note that CpL i , T i q is manifold with boundary L i .We define where each φ i is the canonical identification of BCpL i , T i q with L i and call it a sufficiently deep Dehn filling of M .
Given a closed oriented n-manifold M possibly with boundary BM , the simplicial volume of pM, BM q is defined as where rM, BM s P H n pM, BM ; Rq is the image of the fundamental class under the change of coefficients map H n pM, BM ; Zq Ñ H n pM, BM ; Rq.The following result is again a generalisation of [FM11, Theorem 1.5].
Corollary 6.6.Let M be a compact oriented n-manifold with nilmanifold boundary components tL i u m i"1 as above.Suppose M T is a sufficiently deep Dehn filling of M .Then, M T is a closed oriented aspherical n-manifold with fundamental group G and Proof.We first show that M T is aspherical with fundamental group G.
Let π : Ă M Ñ M be the universal covering map and for each 1 ď i ď m, denote by L i the universal cover of L i which is a simply connected nilpotent Lie group containing H i as a uniform lattice.The universal cover Y of M is a subspace of Ă M with boundary a G-orbit of almost-flat totally geodesic submanifolds homeomorphic to L i for each 1 ď i ď m.Let K " Y {xxN yy, which is a cover of M with fundamental group xxN yy.The boundary of K is a disjoint union of G-orbits of submanifolds homeomorphic to L 1 i " L i {N i for each 1 ď i ď m.The circle bundle T i ãÑ L i π i Ý Ñ B i is the quotient of the short exact sequence R ãÑ L i ։ B i of simply connected nilpotent Lie groups by the action of H i .Hence, it lifts to the circle bundle where each ψ i is the canonical identification of GˆH i BCpL 1 i , T i q with GL 1 i .The manifold M 1 is simply connected by constuction.By Mayer-Vietoris homology sequence and the Cohen-Lyndon property of Theorem 3.22, it follows M 1 has also trivial homology and is therefore contractible.The group G acts freely on M 1 and the quotient M 1 {G is homeomorphic to M T .
Since pM , BM q » pBG, Ů m i"1 BH i q, we have rM , BM s P H n pG, H; Rq.Similarly, since M T » BG, we have rM T s P H n pG; Rq.Applying Theorem 6.3, we obtain Remark 6.7.In the geometric setting of Corollary 6.6, the isomorphism p n : H n pG, H; Zq -Ý Ñ H n pG, H; Zq used in the proof of Theorem 6.3 also follows from excising the interiors of the attached submanifolds tCpL i , T i qu m i"1 from M T which gives an isomorphism between the (co)homologies of pM , BM q and pM T , tCpL i , T i qu m i"1 q.So, one can think of the isomorphism in Theorem A (ii) as a group theoretic analog of topological excision.
We should also remark that the right-hand-side inequality of Corollary 6.6 can be deduced from Gromov's Additivity Theorem [Fri17, Theorem 7.6].
7. Quotients of acylindrically hyperbolic groups 7.1.Cohomology and embedding theorems.We prove Theorem D in this subsection.The reader is referred to Section 2.4 for an outline of the proof.In the sequel, we employ the convention that if X is a set of alphabets and w is a word over X, then }w} denotes the length of w.In certain cases, it might be possible to view w as a word over another alphabet Y .In such a case, we will use }w} X (resp.}w} Y ) to denote the number of letters of X (resp.Y ) in w.For two words w and v, we write w " v to indicate that there is a letter-by-letter equality between w and v.
Lemma 7.1.Let F 4 be a free group of rank 4, F Ď F 4 a finite set, and C a countable group with cdpCq ě 2. Then there exists a quotient R of F 4 such that the following hold.
(5) For every n ě 3 and every R-module A, we have where the action of C on A is induced by the embedding C ãÑ R. (6) If C is finitely generated, then R 0 is hyperbolic relative to C. (7) If C is of type F P n for some n P N `Y t8u, then so is R.
Remark 7.2.Lemma 7.1 (1) (6) are proved in [DGO17,Lemma 8.4].We refine the method therein so as to impose (co)homological conditions.The proof of Lemma 7.1 relies on small cancellation theory, the reader is referred to [LS01, Chapter V] for a treatment.
Proof.Let tx, y, z, tu be a free basis of F 4 , let F 3 ă F 4 be the subgroup generated by x, y, t, and let tc i u iPI be a generating set of C.There exist freely reduced words tw i u iPI , tv i u iPI over the alphabet tx, yu such that (a) the words tc i w i u iPI satisfy the C 1 p1{2q small cancellation condition over the free product xxy ˚xyy ˚C; (b) the words tv i u iPI satisfy the C 1 p1{2q small cancellation condition over the alphabet tx, yu; (c) the words ttc i w i t ´1v i u iPI satisfy the C 1 p1{6q small cancellation condition over the free product xxy ˚xyy ˚xty ˚C.
Indeed, we can first construct words w i satisfying condition (a), and then pick sufficiently long words v i to ensure conditions (b) and (c).
Let N (resp.N 0 ) be the normal subgroup of F 4 ˚C (resp.F 3 ˚C) generated by ttc i w i t ´1v i u iPI , and let R 0 " pF 3 ˚Cq{N 0 , R " pF 4 ˚Cq{N.
For i P I, let t (resp.c i , w i , v i , z) be the image of t (resp.c i , w i , v i , z) under the quotient map F 4 ˚C Ñ R. Then we have R " xzy ˚R0 " Z ˚R0 .
Note that tc i w i t ´1v i " 1 and we can rewrite this equation as c i " t ´1v ´1 i tw ´1 i .Thus, R is generated by t, z, w i , v i , i P I, and hence is a quotient of F 4 .

Let
α : F 4 Ñ R be the corresponding quotient map, which is the restriction of the quotient map F 4 ˚C Ñ R to F 4 .It follows from the Greendlinger's lemma for free products [LS01, Chapter V Theorem 9.3] that if }w i }, }v i }, i P I, are sufficiently large, then α is injective on F and thus statement (2) is guaranteed.
Let L " xxy ˚xyy ˚C, let W ď L be the subgroup generated by tc i w i u iPI , and let V ď L be the subgroup generated by tv i u iPI .
Claim 7.2.1.W (resp. V ) is a free group with basis tc i w i u iPI , (resp.tv i u iPI ).In particular, W and V are both of rank cardpIq.
Proof of Claim 7.2.1.We prove the claim for W .The proof for V is similar.Let be a nonempty freely reduced word over the alphabet tc i w i u iPI , where i k P I and ǫ k " ˘1 for k " 1, ..., ℓ.Think of u as a word over the alphabet xxy Y xyy Y C and then reduce u to its normal form u corresponding to the free product xxy ˚xyy ˚C (see [LS01,Chapter IV] for the definition of normal forms).By condition (a) and that the words w i do not involve inverses of the generators x, y, for each factor pc i k w i k q ǫ k of u, a non-empty subword of pc i k w i k q ǫ k survives in u.In particular, u is nonempty and thus u does not represent 1 in L.
Note that the relations tc i w i t ´1v i " 1 can be rewritten as tc i w i t ´1 " v ´1 i .Thus, R 0 is the HNN-extension of L with associated subgroups W and V .In particular, L embeds into R 0 .As cardpLq " 8, we have cardpR 0 q " 8. Since C embeds into L, C embeds into R 0 .Thus, statement (1) holds.
If C is torsion-free, then so is L. Being an HNN-extension of L, R 0 is also torsion-free, and thus so is R " Z ˚R0 , which is statement (3).
By [Bie75, Theorem 3.1], there is a long exact sequence for any R 0 -module A, As W is free, exact sequence (12) implies for n ě 3, As R " Z ˚R0 , the cohomology part of statement (5) holds.Similarly, one can prove the homology part of statement (5).Statement (4) follows from statement (5) and cdpCq ě 2. If C is finitely generated, then we can construct R using a finite generating set of C. Then R 0 is the quotient of F 3 ˚C by adding finitely many relations tc i w i t ´1v i , and thus has a finite relative presentation over C. The Greendlinger's lemma for free products implies that the relative isoperimetric function of R 0 with respect to C is linear.Thus, R 0 is hyperbolic relative to C (see Remark 3.6), which is statement (6).
If C is of type F P n for some n P N `Y t8u, then C is finitely generated and we can construct R using a finite generating set of C, that is, cardpIq ă 8.Note that the rank of the free group W is cardpIq.Thus, W is of type F P 8 .Note also that L is the free product of a finite rank free group with C and thus is of type F P n .Let tA i u iPI be a direct system of R 0 -modules such that lim Ý Ñ A i " 0. For every k ď n, Theorem 3.1 implies where the actions of W and L on A i are induced by their embeddings into R 0 .It follows from the five lemma and exact sequence (12) that lim Ý Ñ H k pR 0 ; A i q " 0. Therefore, R 0 is of type F P n by Theorem 3.1.As R " Z ˚R0 , R is of type F P n , which is statement (7).
If cdpCq " 0, then C " t1u.Let G " G 0 .By Theorem 3.15, C ãÑ h G. First consider statement (vi).As G and G are quasi-isometric (note that the assumption of (vi) implies that G and G are finitely generated), [Alo94, Corollary 9] implies (vi).Other conclusions of Theorem D hold trivially.
If cdpCq " 1, then by the Stallings-Swan theorem [Swa69, corollary to Theorem 1], C is free.By Theorem 3.15, there exists a finitely generated non-cyclic free group F ãÑ h G 0 .Let G " G 0 .It is well-known that the free group C embeds into F .Thus, C also embeds into G.All conclusions except for (ii) hold trivially.If in addition, C is finitely generated, then C is a finite rank free group and we can let F " C. Thus, (ii) also holds.
Let us assume cdpCq ě 2. By Theorem 3.15, there exists X Ď G 0 and a free subgroup F 4 ď G 0 of rank 4 such that F 4 ãÑ h pG 0 , Xq.There exists a finite set F Ď F 4 t1u such that if N ⊳ F 4 satisfies N X F " H, then the conclusions of Theorems A and 3.20 and [DGO17, Theorem 7.15] hold.By Lemma 7.1, C embeds into an infinite quotient R " Z ˚R0 of F 4 such that the conclusions of Lemma 7.1 hold and the quotient map As R " Z ˚R0 , R 0 is a proper subgroup of R and in particular, R 0 is a proper subgroup of G.By Example 3.7 (d), R 0 ãÑ h R. Proposition 3.9 then implies that R 0 ãÑ h G.As cardpR 0 q " 8, Theorem 3.20 implies that G is acylindrically hyperbolic, that is, statement (i) holds.As C embeds into R 0 , C also embeds into G.
If C is finitely generated, then Lemma 7.1 implies that R 0 is hyperbolic relative to C, in particular, C ãÑ h R 0 .As R 0 ãÑ h G, we have C ãÑ h G by Proposition 3.9.Thus, statement (ii) holds.
Suppose that G and C are torsion-free and there is a non-trivial finite-order element g P G. Then G 0 " G. Denote the image of X under the quotient map G ։ G by X.Then R ãÑ h pG, Xq [DGO17, Theorem 7.15 (b)].As g has finite order, it acts elliptically on the Cayley graph ΓpG, X \ Rq.By [DGO17, Theorem 7.15 (f)], there is an element g P G such that g is mapped to g under the quotient map G ։ G and g acts elliptically on ΓpG, X \ F 4 q.As g has finite order, g n P xxN yy for some n ą 0. Since g n is elliptic as g is, there is some h P G such that hg n h ´1 P N ď F 4 [DGO17, Theorem 7.15 (d)].By Lemma 7.1 (7), R is torsion free and thus hgh ´1 R F 4 .But which is in contradiction with the almost malnormality of F 4 in G [DGO17, Proposition 2.10].We have proved statement (iii).
If G and C are of type F P n for some n P N `Y t8u, then Lemma 7.1 implies that so is R.As G and G 0 are quasi-isometric, G 0 is of type F P n [Alo94, Corollary 9].As F 4 is of type F P 8 , Corollary 2.2 (iii) implies that G is of type F P n .Conversely, if C is finitely generated and G and G are of type F P n for some n P N `Y t8u, then C ãÑ h G by the previously proved statement (ii).Thus, C is of type F P n [DGO17, Theorem 2.11].The previously proved statement (v) implies that cdpGq ă 8 if and only if maxtcdpGq, cdpCqu ă 8, which further implies the statement about type F P .This finishes the proof of statement (vi).
Remark 7.3.Analogous to the above proof, in the setting of Theorem D, we have hdpGq ď maxthdpGq, hdpCq, 2u.7.2.Constructing hyperbolically embedded subgroups.We will prove Theorem E in the next subsection, where the following technical result will be needed.The reader is referred to Section 2.4 for an outline of the proof.
Proposition 7.4.Suppose that G is a group, Λ is a finite set and tk λ u λPΛ is a set of positive integers.Let a λ,i P G for λ P Λ and i " 1, 2, ¨¨¨, k λ .Also suppose that there is a family of subgroups tF λ u λPΛ ãÑ h G such that (a) each F λ is free of rank 2k λ with basis tf λ,i , g λ,i u k λ i"1 ; and (b) a λ,i R F λ t1u for all λ P Λ and i " 1, 2, ¨¨¨, k λ .
Then for sufficiently large n P N `, the set tf n λ,i a λ,i g n λ,i u k λ i"1 Ď G freely generates a subgroup H λ ď G and (13) tH λ u λPΛ ãÑ h G.
The rest of this subsection is devoted to the proof of the above proposition.We first find a "sufficiently large" n P N `.As F λ " ˚kλ i"1 pxf λ,i y˚xg λ,i yq, we have txf λ,i y, xg λ,i yu k λ i"1 ãÑ h F λ by Example 3.7.It follows from Proposition 3.9 that there exists a set X Ď G such that (14) txf λ,i y, xg λ,i yu λPΛ,iPt1,...,k λ u ãÑ h pG, Xq.
By [DGO17, Corollary 4.27], we may assume that a λ,i P X for all λ P Λ and i " 1, 2, ¨¨¨, k λ , as Λ and k λ are finite.For λ P Λ and i P t1, ..., k λ u, let p d λ,i,f : xf λ,i y ˆxf λ,i y Ñ r0, `8s, p d λ,i,g : xg λ,i y ˆxg λ,i y Ñ r0, `8s be the relative metrics corresponding to (14).The metrics p d λ,i,f and p d λ,i,g are locally finite.As cardpΛq, k λ ă 8, for sufficiently large n, we will have p d λ,i,f p1, f n λ,i q, p d λ,i,g p1, g n λ,i q ą 50D for all λ and i, where D ą 0 is given by Lemma 3.11.We fix one such n and let H λ ď G be as in Proposition 7.4.For simplicity, denote the set tf n λ,i a λ,i g n λ,i u k λ i"1 by U λ .Notice that F λ X F µ " t1u whenever λ ‰ µ [DGO17, Proposition 4.33].The following are easy consequences of this fact.
For simplicity, let Remark 7.5.For every λ P Λ and every freely reduced word w over U λ , we can think of w as a word over the alphabet X \ K, i.e., regard every f n λ,i (resp.g n λ,i ) as a letter from xf λ,i y (resp.xg λ,i y) and regard every a λ,i as a letter from X.In this sense, w satisfies the conditions (W1), (W2), and (W3) of Lemma 3.12.
Proof.We need to show that for every λ P Λ, every nonempty freely reduced word over U λ does not represent 1 in G. Suppose that is a freely reduced word over U λ for some λ P Λ such that w represents 1 in G, where ǫ k " ˘1 for k " 1, ..., ℓ.As Remark 7.5, we think of w as a word over X \ K. Then w labels a 3ℓ-gon p in ΓpG, X \ Kq with geodesic sides.Notice that p has 2ℓ components.By Lemma 3.12, each of these components is isolated.Proposition 3.11 then implies 2n ¨50D ď 3nD, which is absurd.Therefore, such a word w does not exist.
Consider the action G ñ ΓpG, X \Kq.Let d X\K be the combinatorial metric of ΓpG, X \ Kq.We verify that, with respect to this action, the family tH λ u λPΛ satisfies the conditions (C 1 ), (C 2 ), and (C 3 ) of Lemma 3.8, which then implies tH λ u λPΛ ãÑ h G.The verification is divided into the following Lemmas 7.7, 7.8, and 7.9.
Proof.Fix λ P Λ.It suffices to prove that for every R ą 0, there are only finitely many h P H λ such that d X\K p1, hq ď R. Let h P H λ such that d X\K p1, hq ď R and let w be a freely reduced word over U λ representing h in G.As in Remark 7.5, think of w as a word over X \ K.By Lemma 3.12, w labels a p4, 1q-quasi-geodesic in ΓpG, X \ Kq.Thus, There are only finitely many words w satisfying the above inequality.It follows that the number of h P H λ such that d X\K p1, hq ď R is finite.
For every λ P Λ, we identify H λ with the subset of ΓpG, X \ Kq labeled by elements of H λ .Equivalently, we identify H λ with the H λ -orbit of the identity vertex of ΓpG, X \ Kq.
Proof.Fix λ P Λ.Let h P H λ and let γ be a geodesic in ΓpG, X \ Kq from the vertex 1 to the vertex h.As ΓpG, X \ Kq is a Gromov hyperbolic space, there exists R ą 0 such that if α and β are p4, 1q-quasi-geodesics with the same endpoint, then d Hau pα, βq ď R, where d Hau is the Hausdorff metric with respect to d X\K .
Let w be a freely reduced word over U λ representing h in G.As in Remark 7.5, think of w as a word over X \ K.By Lemma 3.12, w labels a p4, 1q-quasi-geodesic α in ΓpG, X \ Kq.Note that α lies in the 2-neighborhood of the orbit H λ , and γ lies in the R-neighborhood of α.Thus, γ lies in the pR `2q-neighborhood of H λ .
For λ, µ P Λ, the orbits H λ and H µ are subsets of ΓpG, X\Kq.Thus, it makes sense to talk about the diameter of H µ XpgH λ q `ǫ in ΓpX \Kq, which is denoted by diam `Hµ X pgH λ q `ǫ˘.
Lemma 7.9.For every ǫ ą 0, there exists R ą 0 such that the following holds.Suppose that for some g P G and λ, µ P Λ, we have diam `Hµ X pgH λ q `ǫ˘ě R.
Let p (resp.q) be a path from v 1 (resp.v 3 ) to v 2 (resp.v 4 ) such that Labppq (resp.Labpqq) is a freely reduced word over U µ (resp.U λ ).Then ℓppq ě R and p, q are oriented ǫ-close.By Lemma 3.12, there exist five consecutive components of p which are connected to five consecutive components of q.In particular, there exist two pairs of adjacent components of p which are connected to four consecutive components of q.Some of possible configurations of these two pairs of adjacent components are shown by Figure 3, where each horizontal line represents one possible configuration, the red and blue segments represent the two pairs of adjacent components of p, and the corresponding labels are written on top of the subpaths.
Below, we assume, without loss of generality, that these two pairs of adjacent components are of the form f ´n µ,i g ´n µ,j , f ´n µ,j g ´n µ,r .Other possible configurations can be analyzed similarly.We distinguish two cases.
Case 1.The first pair of adjacent components of p are respectively connected to a pair of adjacent components of q. g ´n µ,r Case 1 is displayed by Figure 4, where the red (resp.blue) dashed line represents a path with label in xf n µ,i y (resp.xg n µ,j y) connecting the corresponding red (resp.blue) components.Equations ( 15), (16), and (17) imply that λ " µ and the red (resp.blue) component of q is labeled by f ´n µ,i (resp.g ´n µ,i ).As the red and blue dashed lines form a loop, another consequence of (15) is that both of these dashed lines are labeled by 1.
Let p 1 (resp.q 1 ) be the subpath of p (resp.q) labeled by uf ´n µ,i (resp.vf ´n µ,i ).Then p 1 " q 1 .By the structure of U µ , Labpp 1 q and Labpq 1 q are words over U µ and thus represent elements in H µ .Therefore, p 1 P H µ X gH µ .It follows that g P H µ .
Case 2. The first pair of adjacent components of p are respectively connected to two consecutive, but not adjacent, components of q.
Case 2 is displayed by Figure 5. Once again, Equations ( 15), (16), and (17) imply λ " µ.The structures of U µ imply i " j " r.The red (resp.blue) dashed line on the left is labeled by an element in xf n µ,i y (resp.xg n µ,i y).As these dashed lines and the yellow segment labeled by a µ,i form a loop, assumption (b) of Proposition 7.4 implies that both of these dashed lines are labeled by 1.Similarly, the red and blue dashed lines on the right are both labeled by 1.
Therefore, the word g 2n µ,i f 2n µ,i a µ,i labels a loop in ΓpG, X \ Kq and thus represents 1 in G, which is in contradiction with assumption (b) of Proposition 7.4.Hence, Case 2 is in fact impossible.
The idea is to consider r The quotient G is constructed by adding particular relations (which will be done by Dehn filling) to r G which identify elements of A (resp.B) with certain elements of G 1 2 (resp.G 1 1 ).There exists a finite generating set A " ta 1 , ..., a k u (resp.B " tb 1 , .. 2 ) for some k P N `.To simplify the argument, let a k`1 " a k`2 " b k`1 " b k`2 " 1.
To perform Dehn filling on r G, the first step is to find hyperbolically embedded subgroups.By [Hul16, Lemma 5.10], G 1 1 and G 1 2 are acylindrically hyperbolic with KpG 1 1 q " KpG 1 2 q " t1u.Thus, Theorem 3.15 implies that there exist free groups 2 , each of which has rank 2k `4.By Example 3.7, we have tG 1 1 , G 1 2 u ãÑ h r G. Thus, Proposition 3.9 implies tF 1 , F 2 u ãÑ h r G.
In fact, F 1 , F 2 are not quite the subgroups that we want, and we will apply Proposition 7.4 to construct other hyperbolically embedded subgroups from A, B, F 1 , F 2 .Note that for i"1 (resp.tf 2,i , g 2,i u k`2 i"1 ) be a basis of the free group F 1 (resp.F 2 ).Then Proposition 7.4 implies the following.
Lemma 7.10.For sufficiently large ℓ Proof of Theorem E. As G 1 1 and G 1 2 are acylindrically hyperbolic, we have |G 1 1 | " |G 1 2 | " 8 and thus cdpG 1 1 q, cdpG 1 2 q ě 1. Suppose cdpG 1 1 q " cdpG 1 2 q " 1.Then G 1 1 and G 1 2 are free by the Stallings-Swan theorem [Swa69, corollary to Theorem 1].Without loss of generality, we may assume that the rank of G 1 1 is greater than or equal to the rank of , and (iii) follow trivially.Statement (iv) also holds because if G 2 is of type F P n for some n P t2, 3, ..., 8u, then G 1 2 is a finite rank free group and thus of type F P 8 .
Thus, let us assume maxtcdpG 1 1 q, cdpG 1 2 qu ě 2. Fix a sufficiently large ℓ P N `and let H 1 and H 2 be the subgroups given by Lemma 7.10.By Remark 3.18, Lemma 7.10, and Theorems 3.20, 3.22, there exist finite sets uq is a Cohen-Lyndon triple and thus Theorems 4.5 and 4.7 and Corollary 4.6 can be applied to it.
Let tu i u k i"1 (resp.tv i u k i"1 ) be freely reduced words over the alphabet ).By Lemma 7.10, H 1 and H 2 are freely generated by tf ℓ 1,i b i g ℓ 1,i u k`2 i"1 and tf ℓ 2,i a i g ℓ 2,i u k`2 i"1 , respectively.Thus, H 1 {N 1 and H 2 {N 2 can be presented as where the last equality of (18) (resp.( 19)) follows from eliminating f ℓ 1,i b i g ℓ 1,i (resp.f ℓ 2,i a i g ℓ 2,i ) for i " 1, ..., k using Tietze transformations (see [LS01, Chapter II]).
Finally, suppose that G 1 and G 2 are of type F P n for some n P t2, 3, ..., 8u.Then so are G 1 1 and G 1 2 [Alo94, Corollary 9].As H 1 {N 1 and H 2 {N 2 are free groups of finite rank, they are of type F P 8 .Therefore, Theorem 4.7 implies that G is of type F P n and thus statement (iv) holds.

Some applications
In this section, we give some applications of Theorems D and E. 8.1.Infinite dimension torsion-free F P 8 groups.If a group G has cdpGq ă 8, then G is necessarily torsion-free.Of course, the converse is not true, e.g., cdpZ 8 q " 8. Observe however that Z 8 is not of type F P 8 .In fact, Bieri asked if every torsion-free group of type F P 8 is also of type F P or if there is an F P 8 group with a non-finitely generated free abelian subgroup [Wal79,Problem F11].In [BG84], Brown and Geoghegan settled both questions, by showing that Thompson's group F is of type F P 8 .
For every torsion-free acylindrically hyperbolic group G, Theorem D embeds F into an acylindrically hyperbolic quotient G of G, which is a torsion-free F P 8 group of infinite cohomological dimension.We thus have the following.
Corollary 8.1.Every torsion-free acylindrically hyperbolic group G of type F P 8 has a torsion-free acylindrically hyperbolic quotient G of type F P 8 which contains the Thompson group F .In particular, cdpGq " 8. 8.2.Quotients of hyperbolic groups.SQ-universality of non-elementary hyperbolic groups was proved by Olshanskii [Ols95] and independently by Delzant [Del96].In Theorem D, if G and C are word-hyperbolic, a hyperbolic quotient G of G can be constructed so that the conclusions hold.The next corollary is thus a strengthening of the aforementioned result.
Corollary 8.2.Let G be a non-elementary hyperbolic group and C any hyperbolic group.Then there is a hyperbolic quotient G of G{KpGq (in particular, G is a quotient of G), where KpGq is the maximal finite normal subgroup of G, such that C embeds into G and the following hold.(ii) cdpGq ď maxtcdpGq, cdpCqu.
Proof.By passing to G{KpGq, we may assume that KpGq " t1u.By [Osi21, Corollary 4.21], there is a free subgroup F 2 ă G of rank 2 such that G is hyperbolic relative to F 2 .By [Jit02], four random elements of F 2 freely generate a free subgroup F 4 ď F 2 that is malnormal in F 2 .It is easy to see that F 4 is also quasi-convex in F 2 , and thus [Bow12, Theorem 7.11] implies that F 2 is hyperbolic relative to F 4 .Thus, G is hyperbolic relative to F 4 [Osi06].By Lemma 7.1, C embeds into a quotient R of F 4 such that R is hyperbolic relative to C and the quotient map F 4 ։ R is injective on any given finite set F Ď F 4 t1u.subgroup of type F that is not hyperbolic.Their group is the fundamental group of 5dimensional hyperbolic pseudo-manifold that fibers over the circle.The fundamental group of the fiber is of type F but not hyperbolic.
Corollary 8.3.Let n ě 5 be an integer.Every non-elementary hyperbolic group G with cdpGq ď n has a hyperbolic quotient G with cdpGq " n such that G contains the Italiano-Martelli-Migliorini group.In particular, there is a type F non-hyperbolic subgroup H ă G.
Proof.By Kazhdan's theorem, the Lie group Spp2, 1q contains a acylindrically hyperbolic non-uniform arithmetic lattice Γ with Property (T).By [ADMPS17, Proposition 4.1], vcdpΓq " 7. Since Γ is a lattice in Spp2, 1q, it is of type F and in particular of type F P [Ash84].Passing to a torsion-free finite index subgroup, we can assume that Γ is torsion-free.The result now follows from Theorem E applied to G and Γ.
The above result can for example be applied to mapping class groups of surfaces of finite type, outer automorphism groups of free groups of finite rank and (non-virtually polycyclic) fundamental groups of compact orientable 3-manifolds which all exhibit nice cohomological finiteness conditions.8.4.Acylindrically hyperbolic quotients distinguishable by cohomology.Let G be any acylindrically hyperbolic group.For each k ě 3, Theorem D implies that one can embed Z k into a quotient G k of G such that H n pG k ; Zq -H n pG; Zq ' H n pZ k ; Zq for all n ě 3. Suppose cdpGq ă 8. Then H ˚pG k ; Zq ‰ H ˚pG ℓ ; Zq for k, ℓ ą cdpGq.Suppose that G is of type F P 8 instead.Then H k pG; Zq is finitely generated, and thus H k pG k ; Zq fl H k pG ℓ ; Zq for k ‰ ℓ.This establishes the following.
Corollary 8.5.Let G be any acylindrically hyperbolic group.Then there exists an infinite family tG k u 8 k"3 of acylindrically hyperbolic quotients of G such that H n pG k ; Zq -H n pG; Zq'H n pZ k ; Zq for all n, k ě 3.In particular, if cdpGq ă 8 or G is of type F P 8 , then each pair of elements of tG k u 8 k"l have non-isomorphic integral cohomology for l " cdpGq `1 and l " 3, respectively.
If instead of embedding Z k , we embed the family of Houghton's groups or Abel's groups, then we will obtain quotients that are separated by homological finiteness properties.
Corollary 8.6.Let G be any acylindrically hyperbolic group of type F P 8 .Then G has a family of acylindrically hyperbolic quotients tG k u 8 k"2 such that for each k, G k has Kazhdan's Property (T), is of type F P k´1 but not of type F P k .
We remark that being of type F P n is a quasi-isometric invariant [Alo94, Corollary 9].Therefore, the quotients tG k u kě1 in Corollary 8.6 are pairwise non-quasi-isometric.
Proof.By Corollary 8.4, we can pass to a quotient G of G that is acylindrically hyperbolic and of type F P 8 and has property (T).There exists a family of groups tH k u 8 k"2 such that for all k, H k is of type F P k´1 but not of type F P k .For example, one can let tH k u 8 k"2 be the family of Houghton's groups or Abel's groups [Bro87, Theorems 5.1 and 6.1].Theorem D then embeds each H k to a quotient G k of G with the desired properties.Question 8.7 (Osin, [Osi08, Problem 2.3]).Let G be a Poincaré duality group of dimension 3 such that G is hyperbolic relative to its subgroup H.If G has positive virtual first Betti number, is it true that for most N ⊳ H, G{xxN yy also has positive virtual first Betti number?Question 8.7 is only about virtual Betti numbers.For actual Betti numbers there are counterexamples.For example, let S be an orientable surface of genus two.Take a suitable pseudo-Anosov homeomorphism φ on S such that the mapping torus T φ has b 1 pT φ q " 1.Then π 1 pT φ q is hyperbolic and is a semi-direct product π 1 pT φ q " π 1 pSq ¸xty.π 1 pT φ q is hyperbolic relative to xty [Bow12], and for every n ě 1, b 1 pπ 1 pT φ q{xxt n yyq " 0 since the image of t generates H 1 pπ 1 pT φ q; Qq.Using this example, we prove the following.
Corollary 8.8.Let G be any acylindrically hyperbolic group.Then G has an acylindrically hyperbolic quotient G such that b 1 pGq " 0, cdpGq ď maxtcdpGq, 3u, and H n pG; Aq -H n pG; Aq for all n ě 4 and any G-module A.
Proof.The idea is to construct an acylindrically hyperbolic group G 1 such that b 1 pG 1 q " 0 and cdpG 1 q ď 3. Once such a G 1 has been constructed, Theorem E will produce a common quotient G of G and G 1 with the desired properties.
So it remains to construct G 1 .Let T φ , t be as above.Take eight elements a 1 , a 2 , ¨¨¨, a 8 that generate π 1 pSq as a group.Then a 1 , a 2 , ¨¨¨, a 8 , t generate π 1 pT φ q as a group.Let X be the set consisting of a 1 , a 2 , ¨¨¨, a 8 , t and their inverses.Randomly generate two words w 1 , w 2 over X of length n and identify w 1 , w 2 with the elements of π 1 pT φ q they represent.
By [MS19, Theorem 1], as n Ñ 8, with probability 1 we have that xw 1 , w 2 y ď π 1 pT φ q is free of rank 2 and xw 1 , w 2 y ãÑ h π 1 pT φ q.Moreover, as n Ñ 8, with positive probability we have that both w 1 and w 2 have non-zero t-exponents.Thus, there exist w 1 , w 2 P π 1 pT φ q such that (a) xw 1 , w 2 y ď π 1 pT φ q is free of rank 2, (b) xw 1 , w 2 y ãÑ h π 1 pT φ q, and (c) w 1 and w 2 have non-zero t-exponents.
Let w be a word over w 1 , w 2 such that w is not a proper power, satisfies the C 1 p1{6q small cancellation condition, and w has positive t-exponent.Let N be the normal subgroup of xw 1 , w 2 y generated by w.By [Lyn50, Theorem 11.1], xw 1 , w 2 y{N has cdpxw 1 , w 2 y{N q ď 2. Let xxN yy be the normal closure of N in π 1 pT φ q.By making w sufficiently long, we can guarantee that N avoids any given finite subset of xw 1 , w 2 y t1u, and therefore guarantee that π 1 pT φ q{xxN yy is acylindrically hyperbolic and cdpπ 1 pT φ q{xxN yyq ď 3 by Theorems 3.20 and A. Since w has positive t-exponent, we have b 1 pπ 1 pT φ q{xxN yyq " 0. So it suffices to let G 1 " π 1 pT φ q{xxN yy.

Corollary 2. 5 .
Every torsion-free acylindrically hyperbolic group G of type F P 8 has a torsion-free acylindrically hyperbolic quotient G of type F P 8 which contains the Thompson group F .The existence of a pair H ă G, where G is hyperbolic and H is of type F but not hyperbolic, was a well-known open problem, raised in particular by Bestvina [Besb, Question 2.1], Brady [Bra99, Question 7.2], Bridson [Besa, Question 4.1], and Jankiewicz-Norin-Wise [JNW21, Section 7].The first example of such a pair was constructed recently by Italiano-Martelli-Migliorini [IMM21, Corollary 2].By Corollary 2.4, we obtain: Theorem 3.1 (Bieri [Bie81, Theorem 1.3], Brown [Bro75, Theorem 2]).For a group G, the following are equivalent.
Remark 3.18.In Definition 3.16, if tH λ u λPΛ ãÑ h pG, Xq, then the relative metrics p d λ are locally finite.Thus, card ´th P H λ | p d λ p1, hq ď Cu ¯ă 8 for all C ą 0. Therefore: Tensoring the above sequence by pF i b Aq b xxN yy ´, we obtain a short exact sequence (3) 0 Ñ pF i b Aq b xxN yy ZrEs Ñ pF i b Aq b xxN yy ZrV s Ñ F i b xxN yy A Ñ 0. Note that there is an isomorphism of xxN yy-modules ZrV s -à λPΛ,tPT λ ZrxxN yy{tN λ t ´1s.

Figure 3 .
Figure 3.Some possible configurations of the two pairs of adjacent components of p

( i )
For all n ě 3 and every G-module A, we haveH n pG; Aq -H n pG{KpGq; Aq ' H n pC; Aq,where the action of G{KpGq (resp.C) on A is induced by the quotient map G{KpGq Ñ G (resp. the embedding C ãÑ G).
Since C is hyperbolic, so is R [Osi06, Corollary 2.41].Let G be the quotient of G by the Dehn filling F 4 ։ R. By avoiding a suitable finite set F, we may assume that G is hyperbolic relative to R [Osi07, Theorem 1.1].Therefore, G is hyperbolic [Osi06, Corollary 2.41].Items (i) and (ii) are immediate consequences of Theorem D. Recently in [IMM21, Corollary 2], Italiano-Martelli-Migliorini settled a long-standing open problem by constructing the first example of hyperbolic group which contains a

8. 3 .
Property (T) quotients.A group G has Kazhdan's property (T) if every affine isometric action of G on a Hilbert space has a global fixed point.The next result strengthens [Hul16, Corllary 1.7].

8. 5 .
First Betti number and a question of Osin.Motivated by the Virtual Positive First Betti Number Conjecture which has now been settled by Agol [Ago13], Osin posed the following question.
Theorem A (ii) is again key in proving Theorem C. Since all H i are amenable, the natural map in bounded cohomology H n b pG, H; Rq Ñ H n b pG; Rq is an isometric isomorphism.The duality pairing between bounded cohomology and ordinary homology leads to the inequality ||G|| ď ||G, H||.When G is hyperbolic relative to H, then a sufficiently deep Dehn filling quotient G is hyperbolic relative to H [Osi07, Theorem 1.1].This allows us to show that ||G|| ą 0.
¨¨¨À λPΛ H i pH λ ; Aq H i pG; Aq H i pG, tH λ u λPΛ ; Aq À λPΛ H i´1 pH λ ; Aq ¨¨À λPΛ H i pH λ ; Aq H i pG; Aq H i pG, tH λ u λPΛ ; Aq À λPΛ H i´1 pH λ ; Aq ¨¨ϊ ˚id ι˚ Aq is injective, then H p pG; Aq -H p pG; Aq ' ˜à λPΛ If for some p P N, ś λPΛ H p pH λ ; Aq " 0 and the natural map H p´1 pG; Aq Ñ ś λPΛ H p´1 pH λ ; Aq is surjective, then H p pG; Aq -H p pG; Aq ' ˜ź Proof.We only prove the cohomological version (b) and point out that the proof of (a) is analogous.To shorten the notation, we denote H ˚p´; Aq by H ˚p´q, ś λPΛ H ˚pH λ ; Aq by H ˚pHq, ś λPΛ H ˚pH λ ; Aq by H ˚pHq, and H ˚pG, tH λ u λPΛ ; Aq by H ˚pG, Hq.Use Proposition 4.3 and consider the commutative diagrams of exact sequences λPΛ H p pH λ ; Aq ¸.
and only if all H i are of type F P n .Combining the above with Corollary 4.6, we obtain the following.Corollary 4.8.Let G be a group with a finite family of subgroupstH i u m i"1 ãÑ h G. Suppose that G is of type F P .Then for sufficiently deep tN i ⊳ H i u m i"1 , Gis of type F P if and only if all H i are of type F P .Proof of Theorem 4.7.Suppose that G is of type F P n .By Theorem 3.20, we may assume that tH i u m i"1 ãÑ h G. Then H i are of type F P n as G is [DGO17, Remark 4.26 and Corollary 4.32].Conversely, suppose that all H i are of type F P n .To shorten notations, we write H ˚pG, H; ´q for H ˚pG, tH i u m i"1 ; ´q, H ˚pH; ´q for ś m i"1 H ˚pH i ; ´q, and H ˚pH; ´q for is of type F P n for some n P N `Y t8u, then G is of type F P n if and only if every H λ is of type F P n .(vi) If |Λ| ă 8 and G is of type F P , then G is of type F P if and only if every H λ is of type F P .Example 4.10.Let G be a free group with basis tx, yu and let H " xhy ď G where h " xyx ´1y ´1.Then H ãÑ h G by Example 1.3 and cdpHq `1 " 2. Let N " xh k y ⊳ H.Note that we can pick k large enough so that N avoids any given finite subset of H t1u. By [Lyn50, Theorem 11.1], H 2 pG; Zq -Z, and it is well-known that H 2 pG; Zq " 0 and H 2 pH; Zq -Z{kZ.Thus, H 2 pG; Zq fl H 2 pG; Zq ' H 2 pH; Zq.Similarly, hdpHq `1 " 2 and one can show that H 2 pG; Zq fl H 2 pG; Zq ' H 2 pH; Zq.
Corollary 5.1.Let G be a group and H " tH i u m i"1 a collection of subgroups such that H ãÑ h G. Suppose pG, Hq is a duality pair of dimension n, with dualizing module C. Then for all sufficiently deep tN i ⊳ H i u m i"1 and k P Z ZGq -H n´k pG; C b ZGq -H n´k pG; Ind G xxN yy Res G xxN yy Cq -H n´k pxxN yy; Cq, i H n´k pN i ; Cq for k ‰ n, Cfor k " n.
H n´2 pH i ; H 1 pN i ; ZGqq H n´2 pH i ; H 1 pN i ; ZGqq Band the fact the kernel of B is H 1 pG, ZGq " 0. This finishes the proof of H n´1 pG; ZGq " 0. By [BE78, Theorem 6.2], G and each H i are of type F P .So, by Corollary 4.8, G is of type F P .Again, by [BE78, Theorem 6.2], G is a PDpnq-group.
n : H n b pG, H; Rq Ñ H n b pG; n pH i ; Rq c Since cdpH i q ď n free group and thus is acylindrically hyperbolic.If H 1 {N 1 is a proper subgroup of G, then equation (20), statement (1), and Theorem 3.20 imply that G is acylindrically hyperbolic.Statement (i) is proved.
Consider statement (ii).For every n ě 3 and every G-module A, we haveH n pG; Aq -H n p r G; Aq ' H n pH 1 {N 1 ; Aq ' H n pH2 {N 2 ; Aq by Theorem 4.5 -H n p r G; Aq as H 1 {N 1 and H 2 {N 2 are free groups