Relative Dehn functions, hyperbolically embedded subgroups and combination theorems

Abstract Consider the following classes of pairs consisting of a group and a finite collection of subgroups: • 
$ \mathcal{C}= \left \{ (G,\mathcal{H}) \mid \text{$\mathcal{H}$ is hyperbolically embedded in $G$} \right \}$
 • 
$ \mathcal{D}= \left \{ (G,\mathcal{H}) \mid \text{the relative Dehn function of $(G,\mathcal{H})$ is well-defined} \right \} .$
 Let 
$G$
 be a group that splits as a finite graph of groups such that each vertex group 
$G_v$
 is assigned a finite collection of subgroups 
$\mathcal{H}_v$
 , and each edge group 
$G_e$
 is conjugate to a subgroup of some 
$H\in \mathcal{H}_v$
 if 
$e$
 is adjacent to 
$v$
 . Then there is a finite collection of subgroups 
$\mathcal{H}$
 of 
$G$
 such that 1. If each 
$(G_v, \mathcal{H}_v)$
 is in 
$\mathcal C$
 , then 
$(G,\mathcal{H})$
 is in 
$\mathcal C$
 . 2. If each 
$(G_v, \mathcal{H}_v)$
 is in 
$\mathcal D$
 , then 
$(G,\mathcal{H})$
 is in 
$\mathcal D$
 . 3. For any vertex 
$v$
 and for any 
$g\in G_v$
 , the element 
$g$
 is conjugate to an element in some 
$Q\in \mathcal{H}_v$
 if and only if 
$g$
 is conjugate to an element in some 
$H\in \mathcal{H}$
 . That edge groups are not assumed to be finitely generated and that they do not necessarily belong to a peripheral collection of subgroups of an adjacent vertex are the main differences between this work and previous results in the literature. The method of proof provides lower and upper bounds of the relative Dehn functions in terms of the relative Dehn functions of the vertex groups. These bounds generalize and improve analogous results in the literature.


Introduction
Consider the following classes of pairs consisting of a group and a finite collection of subgroups: Theorem 1.1.Let G be a group that splits as a finite graph of groups such that each vertex group G v is assigned a finite collection of subgroups H v , and each edge group G e is conjugate to a subgroup of some H ∈ H v if e is adjacent to v. Then there is a finite collection of subgroups H of G such that The theorem is trivial without the third item in the conclusion; indeed, the pair (G, {G}) belongs to both C and D. In comparison with previous results in the literature, our main contribution is that our combination results do not assume that edge groups are finitely generated or contained in H v .
The notion of a hyperbolically embedded collection of subgroups was introduced by Dahmani, Guirardel, and Osin [9].A pair (G, H) in C is called a hyperbolically embedded pair, and we write H → h G. Our combination results for hyperbolically embedded pairs (G, H) generalize analogous results for relatively hyperbolic pairs in [1,7,8,16,17] and for hyperbolically embedded pairs [9,14].
The notions of finite relative presentation and relative Dehn function G,H of a group G with respect to a collection of subgroups H were introduced by Osin [18] generalizing the notions of finite presentation and Dehn function of a group.A pair (G, H) is called finitely presented if G is finitely presented relative to H, and G,H is called the Dehn function of the pair (G, H).While a finitely presented group has a well-defined Dehn function; in contrast, the Dehn function of a finitely presented pair (G, H) is not always well defined, for a characterization see [12, Thm.E (2)].Our result generalizes combination results for pairs (G, H) with well-defined Dehn function by Osin [17,Thms. 1.2 and 1.3].
We prove Theorem 1.1 for the case of graphs of groups with a single edge, since then the general case follows directly by induction on the number of edges of the graph.This particular case splits into three subcases corresponding to the three results stated below.The proofs of these subcases use characterizations of pairs (G, H) being hyperbolically embedded [15,Thm. 5.9] and having a well defined Dehn function [12,Thm. 4.7] in terms of existence of G-graphs with certain properties that relate to Bowditch's fineness [5].These characterizations are discussed in Section 2. The proof of Theorem 1.1 for the case of a graph of groups with a single edge entails the construction of graphs satisfying the conditions of those characterizations for the fundamental group of the graph of groups.We use the existing graphs for the vertex groups as building blocks.
Our method of proof provides lower and upper bounds for the relative Dehn function of the fundamental group of the graph of groups in the terms of the relative Dehn functions of the vertex groups; see Section 6.Specifically, Theorem 1.6 below generalizes results of Brick [6] on bounds for the Dehn functions of free products (see the improvement by Guba and Sapir [11]) and improve the bounds found by Osin for relative Dehn functions in [18, Thms 1.2 and 1.3].
Our main result reduces to the following statements.
Theorem 1.2 (Amalgamated Product).For i ∈ {1, 2}, let (G i , H i ∪ {K i }) be a pair and ∂ i : C → K i a group monomorphism.Let G 1 * C G 2 denote the amalgamated product determined by G 1 − → G 2 , and let H = H 1 ∪ H 2 .Then: In the following statements, for a subgroup K of a group G and an element g ∈ G, the conjugate subgroup gKg −1 is denoted by K g .

Theorem 1.3 (HNN-extension I)
. Let (G, H ∪ {K, L}) be a pair with K = L, C a subgroup of K, and ϕ : C → L a group monomorphism.Let G * ϕ denote the HNN-extension G, t | tct −1 = ϕ(c) for all c ∈ C .Then: Note that the third items of Theorems 1.2 and 1.3 follow directly from standard arguments in combinatorial group theory.This article focuses on proving the other statements.

Corollary 1.4 (HNN-extension II)
. Let (G, H ∪ {K}) be a pair, C a subgroup of K, s ∈ G, and ϕ : C → K s a group monomorphism.Let G * ϕ denote the HNN-extension G, t | tct −1 = ϕ(c) for all c ∈ C .Then: Proof.First, we prove the statement in the case that s is the identity element of G. Let L be the HNNextension L = K * ϕ .Observe that there is a natural isomorphism between G * ϕ and the amalgamated product G * K L. In this case, the conclusion of the corollary is obtained directly by invoking Theorem 1.2, since the pair (L, {L}) is in both classes C and D. Now we argue in the case that s ∈ G is arbitrary.Let ψ : C → K the composition I s • ϕ where I s is the inner automorphism there is a natural isomorphism G * ϕ → G * ψ which restricts to the identity on the base group G, and the stable letter of G * ψ corresponds to s −1 t in G * ϕ .Since ψ maps C ≤ K into K, we have reduced the case of arbitrary s ∈ G to the case that s is the identity in G and the statement of the corollary follows.
Let us describe the argument proving our main result using the three previous statements.The argument relies on the following observation.Remark 1.5.If a pair (G, H ∪ {L}) belongs to C (respectively D) and g ∈ G then (G, H ∪ {L g }) belongs to C (respectively D).This statement can be seen directly from the original definitions of hyperbolically embedded collection of subgroups [9] and relative Dehn function [18].It can be also deduced directly from Theorems 2.2 and 2.9, respectively, in the main body of the article.
Proof of Theorem 1.1.The case of a tree of groups satisfying the hypothesis of the theorem follows from Theorem 1.2 and Remark 1.5.Then the general case reduces to the case of a graph of groups with a single vertex, where the vertex group corresponds to the fundamental group of a maximal tree of groups.In the case of a graph of groups with a single vertex, each edge corresponds to applying either Theorem 1.

Under the assumptions of
where i denotes the super-additive closure of i .[19]; by a nontrivial collection we mean that it contains a proper infinite subgroup.This alternative approach does not guarantee that the collection H satisfies the third condition of Theorem 1.1.4. Theorems 1.2(1) and 1.3 (1), in the case that G i is hyperbolic relative to H i for i = 1, 2, follow from results of Wise and the first author [7, Thm.A].

Organization
The rest of the article consists of five sections.In Section 2, we review characterizations of pairs (G, H) being hyperbolically embedded and having well-defined Dehn functions in terms of actions on graphs.In Section 3, we reduce the proof of Theorems 1.2 and 1.3 to prove two technical results, Theorems 3.1 and 3.2.Their proofs are the content of Sections 4 and 5, respectively.The last section contains the proof of Theorem 1.6.

Characterizations using fineness
In this section, we describe a characterization of pairs (G, H) being hyperbolically embedded, Theorem 2.2; and a characterization of the pairs having a well-defined Dehn function, Theorem 2.9.These characterizations are in terms of existence of G-graphs with certain properties that relate to Bowditch's fineness [5], a notion that is defined below.The characterizations are re-statements of previous results in the literature [15, Thm.5.9] and [12,Thm. 4.7].This section also includes a couple of lemmas that will be of use in later sections.All graphs = (V, E) considered in this section are simplicial, so we consider the set of edges E to be a collection of subsets of cardinality two of the vertex set V.
Let be a simplicial graph, let v be a vertex of , and let T v denote the set of the vertices adjacent to v.For x, y ∈ T v , the angle metric ∠ v (x, y) is the combinatorial length of the shortest path in the graph − {v} between x and y, with ∠ v (x, y) = ∞ if there is no such path.The graph is fine at v if (T v , ∠ v ) is a locally finite metric space.A graph is fine if it is fine at every vertex.
It is an observation that a graph is fine if and only if for every pair of vertices x, y and every positive integer n, there are finitely many embedded paths between x and y of length at most n; for a proof see [5].

Hyperbolically embedded pairs
In [19, Definition 2.9], Osin defines the notion of a collection of subgroups H being hyperbolically embedded into a group G.This relation is denoted as H → h G and, in this case, we say that the pair (G, H) is a hyperbolically embedded pair.In this article, we use the following characterization of hyperbolically embedded collection proved in [15] as our working definition.Let us observe that in [15], Theorem 2.2 is proved for the case that H consists of a single infinite subgroup, and the authors observe that the argument in the case that H is a finite collection of infinite subgroups (such that no pair of distinct infinite subgroups in H are conjugate in G) follows by the same argument.Then the general case in which H is a finite collection of subgroups follows from the following statement: if H is a collection of subgroups and K a finite subgroup of a group G, then:

H → h G if and only if H ∪ {K} → h G.
2. There is (G, H)-graph if and only if there is a (G, H ∪ {K})-graph.
The first statement is a direct consequence of the definition of hyperbolically embedded collection by Osin [19].The if part of the second statement is trivial, and the only if part follows directly from [2,Thm. 3.4].

Relative presentations
In [18, Chapter 2], Osin introduces the notions of relative presentation of a group with respect to a collection of subgroups, and relative Dehn functions.We briefly recall these notions below.
Let G be a group and let H be a collection of subgroups.A subset S of G is a relative generating set of G with respect to H if the natural homomorphism is surjective, where F(S) denotes the free group with free generating set S. A relative generating set of G with respect to H is called a generating set of the pair (G, H).A pair that admits a finite generating set is called a finitely generated pair.Let R ⊆ F(S, H) be a subset that normally generates the kernel of the above homomorphism.In this case, we have a short exact sequence of groups and the triple is called a relative presentation of G with respect to H, or just a presentation of the pair (G, H).
Abusing notation, we write G = S, H | R .If both S and R are finite we say that the pair (G, H) is finitely presented.
Lemma 2.4.Let G be a group and let H 0 H be a collection of subgroups.Let P denote the subgroup of G generated by S 0 and the subgroups in where R is the image of R under the natural epimorphism ϕ : Our hypotheses imply that the natural epimorphisms A * B → G and B → P induce short exact sequences: Let us identify P = B/K.The natural epimorphism of the statement of the lemma By the definition of N, we have that ϕ(N) is the normal subgroup of A * P generated by R = ϕ(R).
Therefore, the natural epimorphism A * P → G induces a short exact sequence: which concludes the proof.
The following pair of lemmas allow us to conclude that certain amalgamated products and HNNextensions preserve relative finite presentability.

Lemma 2.5 (Amalgamated products). For
The proof concludes by invoking Lemma 2.4.

Lemma 2.6 (HNN-extension). Let (G, H ∪ {K, L}) be a pair with
where R is the set of relations obtained by taking each element of R and replacing all occurrences of elements k ∈ K by words t −1 k t t.In particular, R and R have the same cardinality.
Proof.Let J denote the subgroup K t , and let ψ : The proof concludes by invoking Lemma 2.4.

Relative Dehn functions
Suppose that S, H | R is a finite relative presentation of the pair (G, H).For a word W over the alphabet S = S H∈H (H − {1}) representing the trivial element in G, there is an expression: where R i ∈ R and f i ∈ F(S).We say a function f : Concrete examples of Cayley-Abels graphs can be exhibited using the following construction introduced by Farb [10]; see also [13].Every coned-off Cayley graph ˆ (G, H, S) with S a finite relative generating set is a Cayley-Abels graph.The following result implies that coned-off Cayley graphs are, up to quasi-isometry, independent of the choice of finite generating set, and we denote them by ˆ (G, H).Observe now that Theorem 2.9 also follows from the following result.Theorem 2.12.[2, Theorem H] If and are Cayley-Abels graphs of the proper pair (G, H), then: 1. and are quasi-isometric, and 2. is fine if and only if is fine.

Combination theorems for graphs
In this section, we state two technical results, Theorems 3.1 and 3.2, which will be proven in the subsequent sections.The section includes how to deduce the main results of the article, Theorems 1.2 and 1.3, from these technical results.
Then there is a G-graph with the following properties: 1. has a vertex z such that the G-stabilizer G z = K 1 , K 2 , and there is a 6.If i has finitely many G i -orbits of vertices (edges) for i = 1, 2, then has finitely many G-orbits of vertices (resp.edges).

If i is fine for
Let us explain how Theorem 1.2 follows from the above result.
Proof of Theorem 1.2.For the first statement, suppose ) is a proper pair by a standard argument using normal forms.Then invoke Theorem 2.2 to obtain that The second statement is proved analogously.Suppose the relative Dehn function of ) is also a proper pair by a standard argument using normal forms.By Theorem 2.9, (G i , H i ∪ {K i }) is finitely presented and admits a fine Cayley-Abels graph i .In particular, there is a vertex x i ∈ i with G i -stabilizer equal to K i .Apply Theorem 3.1 to 1 , 2 and the vertices x 1 , x 2 to obtain a fine Cayley-Abels graph for the pair is finitely presented by Lemma 2.5, then Theorem 2.9 implies that the relative Dehn function of has a vertex z such that G z = K t , L , and there is a G-equivariant inclusion → such that x → t −1 .zand y → z.
Proof of Theorem 1.3.This proof is completely analogous to the proof of Theorem 1.2: invoke Theorem 3.2 and Lemma 2.6 instead of Theorem 3.1 and Lemma 2.5, respectively.

Amalgamated products and graphs
This section describes an argument proving Theorem 3.1.While the statement of this result seems intuitive, we are not aware of a full account of those techniques in a common framework, so this section provides a detailed construction.

Pushouts in the category of G-sets
Let φ : R → S and ψ : R → T be G-maps.The pushout of φ and ψ is defined as follows.Let Z be the G-set obtained as the quotient of the disjoint union of G-sets S T by the equivalence relation generated by all pairs s ∼ t with s ∈ S and t ∈ T satisfying that there is r ∈ R such that φ(r) = s and ψ(r) = t.There are canonical G-maps ı : S → Z and j : T → Z such that ı • φ = j • ψ.This construction satisfies the universal property of pushouts in the category of G-sets.Proof.Since ı and j are G-maps, G s , Let r 0 denote the element r ∈ R in the statement, in particular, s = φ(r 0 ), t = ψ(r 0 ) and R = G.r 0 .Since j (t) = ı(g.s), the definition of Z as a collection of equivalence classes in S T implies that there is a sequence r 0 , r 1 , r 1 . . ., r k , r k of elements of R such that t = ψ(r 0 ), φ(r 0 ) = φ(r 1 ), ψ(r 1 ) = ψ(r 1 ), . . ., ψ(r k ) = ψ(r k ), φ(r k ) = g.s.
First note that since φ and ψ are G-maps a i .si = s i+1 and b j .tj = t j+1 , and hence and analogously Since t 0 = t and s 0 = s, we have that and hence

Extending actions on sets
In the case that K is a subgroup of G and S is a K-set, one can extend the K-action on S to a G-set G × K S that we now describe.Up to isomorphism of K-sets, we can assume that S is a disjoint union of K-sets: is injective.This construction satisfies a number of useful properties that we summarize in the following proposition.
For n a natural number and a set X, let [X] n denote the collection of subsets of X of cardinality n.If X is a G-set, then [X] n is a G-set with action defined as g.{x 1 , . . ., x n } = {g.x 1 , . . ., g.x n }.Proposition 4.2.Let K ≤ G and S a K-set.

In part three, if f induces an injective map S/K → T/G and
Proof.The first four statements are observations.For the fifth statement, suppose f (ı(s 1 )) = f (g.ı(s 2 )).Then f (s 1 ) = g.f(s 2 ).Since the map S/K → T/G induced by f is injective, we have that s 1 and s 2 are in the same K-orbit in S, say s 2 = k.s 1 for k ∈ K.It follows that f (s 1 ) = gk.f(s 1 ), and since K s 1 = G f (s 1 ) , we have that gk ∈ K s 1 .Therefore ı(s 1 ) = ı(gk.s 1 ) = g.ı(ks 1 ) = g.ı(s 2 ).
The sixth statement is proved as follows.The K-map ı : As the reader might have noticed, this construction is an instance of general categorical phenomena; that formulation will have no use in this article so we will not discuss it.

Graphs as 1-dimensional complexes
While the objectives of this section only require us to consider simplicial graphs, the category of simplicial graphs does not have pushouts [20].For this reason, it is convenient to work within the framework of one-dimensional complexes or equivalently graphs in the sense that we describe below.We will only consider a particular class of pushouts of graphs that behaves well over simplicial graphs.A graph is a triple (V, E, r), where V and E are sets, and r : E → [V] 2 is a function where [V] n is the collection of nonempty subsets of V of cardinality at most n.Elements of the set V and E are called vertices and edges, respectively; the function r is called the attaching map.For a graph , we denote V( ) and E( ) its vertex and edge set, respectively.If v ∈ V( ), e ∈ E( ) and v ∈ r(e), then v is incident to e, and v is called an endpoint of e. Vertices incident to the same edge are called adjacent.
The graph (V, E, r) is simplicial if every edge has two distinct endpoints and r is injective.
A graph is a subgraph of a graph if V( ) ⊂ V( ), E( ) ⊂ E( ) and r equals the restriction of r to E( ).Abusing notation, we consider any vertex of a graph as an edgeless subgraph with a single vertex, and any edge e of as the subgraph with vertex set the set of vertices incident to e in and edge set consisting of only e.
For a vertex u of a simplicial graph = (V, E, r), let star (u) denote the subgraph with vertex set V(star(u)) = {u} ∪ {v ∈ V | v is adjacent to u} and edge set E(star(u)) = {e ∈ E | the endpoints of e belong to V(star(u))} and the attaching map is the corresponding restriction of r.
Our notion of morphism allows the collapse of edges to single vertices.Specifically, a morphism of φ : (V, E, r) → (V , E , r ) of graphs is a pair of maps φ 0 : V → V and φ 1 : where the horizontal bottom arrow φ 0 is the natural G-map induced by φ 0 : V → V , and V → [V ] 1 is the natural bijection given by v → {v}.Observe that in general for a morphism φ = (φ 0 , φ 1 ) : → of graphs, the map φ 0 does not determine φ 1 ; however if is simplicial then φ 0 determines φ 1 .A morphism (φ 0 , φ 1 ) is a monomorphism (also called an embedding) if both maps are injective.Given a graph morphism φ = (φ 0 , φ 1 ) : → and a subgraph of , the preimage φ −1 ( ) is the subgraph of with vertex set φ −1 0 (V( )) and edge set φ −1 1 (V( ) ∪ E( )).Let G be a group.A G-graph is a graph (V, E, r) where V and E are G-sets, and r is a G-map with respect to the natural G-action on [V] 2 induced by the G-set V. A morphism (φ 0 , φ 1 ) of G-graphs is a morphism of graphs such that each φ i is a G-map.A G-equivariant embedding is a monomorphisms of G-graphs.A G-action on a graph has no inversions if for every e ∈ E and g ∈ G such that g.e = e, g.v = v for every v ∈ r(e).For a G-action without inversions on a graph and K ≤ G, let K denote subgraph of defined by for all e ∈ K}.

Extending group actions on graphs
Let K be a subgroup of G, and let = (V, E, r) be a K-graph.Define where r is unique G-map induced by the commutative diagram where ı : V → G × K V and j : E → G × K E are the canonical K-maps, see Lemma 4.2(3).Note that there is a canonical K-equivariant embedding induced by ı and j .We consider a K-subgraph of G × K .
Remark 4.3.Proposition 4.2, parts 2 and 4 imply: In particular, if is connected, then every connected component of G × K is isomorphic to .

Pushouts of graphs
Let X and Y be G-graphs, let C ≤ G be a subgroup and suppose X C and Y C are nonempty.Let x ∈ X C and y ∈ Y C be vertices.The C-pushout Z of X and Y with respect to the pair (x, y) is the G-graph Z obtained by taking the disjoint union of X and Y and then identifying the vertex g.x with the vertex g.y for every g ∈ G.
Equivalently, the C-pushout Z of X and Y with respect to the pair (x, y) is the G-graph Z whose vertex set V(Z) is the pushout of the G-maps κ 1 : G/C → V(X) and κ 2 : G/C → V(Y) given by C → x and C → y; and edge set the disjoint union of the G-sets E(X) and E(Y), and attaching map E(Z) → V(Z) 2 defined as the union of the attaching maps for X and Y postcomposed with the maps V(X) → V(Z) and The standard universal property of pushouts holds for this construction: if Remark 4.4.Let Z be the C-pushout of X and Y with respect to a pair (x, y).
For any edge e of X, G e = G ı 1 (e) .
3. If X/G and Y/G both have finitely many vertices (resp.edges), then Z/G has finitely many vertices (resp.edges).Let Z be the C-pushout of X and Y.By parts ( 4) and ( 7) of Proposition 4.5, Z is a connected edgeless G-graph and hence a single vertex.b 3 ) be the coned-off Cayley graphs.Note that X is the Bass-Serre tree of the splitting of A as the graph of groups

Example 4.2. Let
with two vertices and two edges with trivial edge group.
Let G = A * C B be the amalgamated product where C corresponds to the cyclic subgroup By the fourth, fifth and sixth statements of Proposition 4.5 below, Z is a tree, it contains three distinct G-orbits of vertices, two of these Gorbits have all representatives with trivial stabilizer, and there is a vertex z with stabilizer a 1 , and there are four distinct orbits of edges all with representatives having trivial stabilizer.Hence, Z is the Bass-Serre tree of a splitting of G given by the graph of groups The following properties hold: From here on, we consider X and Y as subgraphs of Z via these canonical embeddings.3.For every vertex v (resp.edge e) of Z, there is g ∈ G such that g.v is a vertex (resp. is an edge g.e) of the subgraph X ∪ Y.

For every vertex v of X which is not in the
For every edge e of X (resp.Y), A e = G e (resp.B e = G e ) where G e is the G-stabilizer of e in Z .6.If the complexes X/A and Y/B both have finitely many vertices (resp.edges), then Z/G has finitely many vertices (resp.edges).7. If X and Y are connected, then Z is connected.

8.
There is a G-tree T and a morphism ξ : Z → T of G-graphs with the following properties: The G-orbit of ξ (z) and its complement in the set of vertices of T make T a bipartite graph; the preimage ξ −1 (ξ (z)) is a single vertex; and if a vertex v of T is not in the G-orbit of ξ (z), then the preimage of the star of v is a subgraph of Z isomorphic to X or Y.
Proof.The first item is a direct consequence of Proposition 4.1.For the second item, first note that that the composition X → G × A X ı 1 − → Z is a morphism of A-graphs.Observe that to prove the embedding part is enough to consider only vertices of X that are in the A-orbit of x.Suppose that a.x and x with a ∈ A both map to z ∈ Z.Then, a ∈ G z = A x * C B y and therefore a ∈ A x and hence a.x = x.Item three follows directly from the definition of Z, and items four to six are consequences of Proposition 4.2.
To prove the seventh statement suppose that X and Y are connected graphs.The subgraph X ∪ Y of Z is connected since both X and Y contain the vertex z.On the other hand, any vertex of Z belongs to a translate of X ∪ Y by an element of G. Therefore to prove that Z is connected, it is enough to show that for any g ∈ G there is a path in Z from z to g.z.For any g ∈ G and a ∈ A, there is a path from g.z to ga.z in Z: indeed, there is a path from z to a.z in the connected A-subgraph X of Z, and hence there is a path from g.z to ga.z in Z. Analogously, for any g ∈ G and b ∈ B, there is a path from g.z to gb.z.Since any element of G is of the form a 1 b 1 . . .a n b n with a i ∈ A and b i ∈ B, there is a path from z to g.z for any g ∈ G.
Now we prove the eighth statement.Observe that G splits as G = A * Ax (A x * C B y ) * By B where the subgroups A x , B y and A x * C B y are naturally identified with the G-stabilizers of x ∈ G × A X, y ∈ G × B Y, and z ∈ Z.Let T denote the Bass-Serre tree of this splitting.The vertex and edge sets of T can be described as: respectively.Note that T is a bipartite G-graph, the equivariant bipartition of the vertices given by G/A G/B and G/(A x * C B y ).
Consider the A-map from X to T that maps every vertex of X not in the A-orbit of x to the vertex A, and x → A x * C B y .Since T is simplicial, this induces a unique morphism of G-graphs j 1 : G × A X → T. Analogously, there is B-map Y → T that maps every vertex not in the B-orbit of y to the vertex B and y → A x * C B y ; this induces a unique G-map Consider the G-maps κ 1 : G/C → G × A X and κ 2 : G/C → G × B Y given by C → x and C → y, respectively.Since j 1 • κ 1 = j 2 • κ 2 , the universal property implies that there is a surjective G-map ξ : Z → T.
Note that ξ −1 (ξ (z)) is contained in the orbit G.z. Suppose g.z ∈ ξ −1 (ξ (z)).Then g(A x * C B y ) = A x * C B y and hence g ∈ A x * C B y .Since A x * C B y is the G-stabilizer of z, we have that g.z = z.This shows that ξ −1 (ξ (z)) = {z}.
Let us conclude by proving that if v ∈ V(T) is not in the G-orbit of ξ (z) = A x * c B y then ξ −1 (star T (v)) is a graph isomorphic to either X or Y.Note that such a vertex v is an element of G/A ∪ G/B.By equivariance, it is enough to consider the two symmetric cases, namely v = A or v = B. Let us prove Note that the quotient map: Example 5.1.Let ϕ : A → A be a group automorphism and consider the HNN-extension G = A * ϕ .Let X be the A-set consisting of a single point.Then G × A X is the G-space G/A, and then the ϕ-coalescence of X is again a single point.

an isomorphism, and
The ϕ-coalescence Z of X with respect to the pair (H 1 , H 2 ) is the quotient G × A X by identifying gtH 1 and gH 2 for every g ∈ G. Hence, Z is naturally isomorphic as a G-set to G/H 1 .Observe that the A-map X → Z given by H 1 → H 1 and ϕ(H 1 ) → tH 1 is an injective A-map.Lemma 5.2.Let H be a subgroup of a group A, let ϕ : H → A be a monomorphism and let G = A * ϕ .Let X be an A-set, let x, y ∈ X be in different A-orbits such that A x = H, A y = ϕ(H).If Z is the ϕ-coalescence of X with respect to (x, y), and z = ρ(y), then: Proof.The inclusion ϕ(H) ⊆ G z is a consequence of ρ being G-equivariant, ϕ(H) = A y and ρ(y) = z.Conversely, let g ∈ G z .Then g.y ∼ y in G × A X, and it follows that there is an integer n ≥ 1 and a sequence w 1 , w 2 , . . ., w n of elements of G × A X such that g.y = w 1 , w n = y and w i and w i+1 are the components of a basic relation (see the definition of coalescence).Since x and y are in different A-orbits in X, they represent different G-orbits in G × A X, see the first item of Proposition 4.2.Hence, we have that w i = g i .xif i is even, and w i = g i .yif i is odd, for some elements g i of G where g 1 = g and g n ∈ φ(H).Note that the integer n is odd, and the chain of basic relations between the w i 's in G × A X can be expressed as:

By definition of basic relation, tg
, and so on until tg −1 n−1 g n ∈ ϕ(H).Since n is odd, we have that which implies g = g 1 ∈ ϕ(H).This shows that G z = ϕ(H) Now we prove the second statement.By Lemma 4.2, the natural A-map X → G × A X is injective.Observe that Z is obtained as a quotient of G × A X by the G-equivariant equivalence relation generated by the basic relation t.x ∼ y.Hence to prove injectivity of X → G × A X → Z, it is enough to show that the restriction to A.x ∪ A.y is injective.Assume there are a 1 , a 2 ∈ A such that a 1 .xand a 2 .xmap to the same element in Z. Then letting a = a −1 2 a 1 , both a.x and x map to the same element in Z.Hence, a.x ∼ x which implies that at −1 .y∼ t −1 .y. Therefore (ta −1 t −1 ).y ∼ y and thus by the first statement, ta −1 t −1 ∈ ϕ(H), and hence a −1 ∈ t −1 ϕ(H)t = H.This results in a ∈ H. Therefore, a −1 2 a 1 ∈ H and a 1 .x= a 2 .x.We have shown As previously observed, the coned-off Cayley graphs of a finitely generated pair (G, H) are Cayley-Abels graphs of the pair.Theorem 2.12 states that all Cayley-Abels graphs of a finitely generated pair are quasi-isometric, and if one of them is fine then all of them are fine.Moreover, fillable and the class of coarse isoperimetric functions are quasi-isometry invariants of graphs by Proposition 6.2.Putting these results together with the two results above, and Theorem 2.11, one obtains the following corollary.Corollary 6.4.Let be a Cayley-Abels graph of finitely generated proper pair (G, H).
1.If G,H is well defined, then is fine and fillable, and G,H f k for all sufficiently large integers k. 2. If is fine and fillable, then (G, H) is finitely presented and G,H is well defined.

Relative Dehn functions and splittings
Let g : N → N be a function.Then g is superadditive if g(m) + g(n) ≤ g(m + n).If g(0) = 0 then the super-additive closure g : N → N of g is the function: and it is an observation that ḡ is the least super-additive function such that g(n) ≤ g(n) for all n.Note that the requirement g(0) = 0 is necessary in order for ḡ to be well defined.An outstanding open question raised by Mark Sapir is whether the Dehn function of any finite presentation is asymptotically equivalent to a superadditive function [11].Proposition 6.5.Let r : → be a retraction of graphs.If is k-fillable, then is k-fillable and Proof.Let c : I → is a singular combinatorial loop.If (P, ) is a k-filling of c in then it is an observation that (P, r • ) is a k-filling of c in .Therefore, area k (c) ≤ area k (c) and the result follows.
The following proposition is the main technical result of the section.Proposition 6.6.Let ξ : → T be a morphism of graphs where T is a bipartite tree, say If there is k > 0 such that each ∈ is a k-fillable graph and then is k-fillable and where g denotes the super-additive closure of the function g : N → N.
Proof.It is an observation that if c 1 and c 2 are singular combinatorial loops in with the same initial point, and both admit k-fillings, then the concatenated loop c 1 • c 2 admits a k-filling and Len(c 1 • c 2 ) = Len(c 1 ) + Len(c 2 ) and area k (c 1 • c 2 ) ≤ area k (c 1 ) + area k (c 2 ).
To prove that f k (n) ≤ g(n), we prove that if c : I → is a singular combinatorial loop in then area k (c) ≤ g(Len(c)).
Let c : I → be a singular combinatorial loop in .Consider the loop ξ • c in the tree T. The image of ξ • c is a finite subtree T c of T. Let #T c ∩ K denote the number of vertices of T c that belong to K. To the loop c assign the complexity |c| = (#T c ∩ K, Len(c)) ∈ N × N. Consider the lexicographical order on N × N, and recall that this is well-ordered set.We prove by induction on (#T c ∩ K, Len(c)) that area k (c) ≤ g(Len(c)).
Base case |c| = (0, m).Suppose that T c does not contain vertices in K.In this case, the bipartite assumption on T implies that T c consists of a single vertex v in L. Since ξ −1 (v) is a single vertex of , it follows that c is constant path and hence Len(c) = 0 and area k (c) = 0 ≤ g(0).
Base case |c| = (1, m).Suppose that the vertex set of T c contains a single vertex in K, say v. Then the bipartite assumption on T implies that T c is a subgraph of star(v) and hence the image of c is contained in the subgraph = ξ −1 (star(v)).By assumption, is k-fillable, and hence there is k-filling of c in which is trivially also a k-filling in .Hence, area k (c) ≤ area k (c) ≤ f k (Len(c)) ≤ g(Len(c)) ≤ g(Len(c)).
General case |c| = (n, m) with n ≥ 2. For the inductive step, suppose that T c ∩ K has at least two vertices in K. Without loss of generality, we can identify the domain I of c with the closed interval [0, 1] (with some CW-structure).Since T c is connected, the bipartite assumption on T, implies that T c contains a vertex v ∈ L such that v is not a leaf of T c , in particular, T c − {v} has at least two connected components.Then where the first inequality follows from the observation in the first paragraph of this proof, the second inequality uses the induction hypothesis, and the third one uses that g is superadditive.

Proof of Theorem 1.6
Proof.The proofs of the first two statements are analogous, and the argument goes back to the method of proof of the corresponding theorems in the introduction.The third statement is a consequence of the second one; see the proof of Corollary 1.4(2).We prove the first statement and leave the proof of the second statement to the reader.We remark that the argument essentially reproves Theorem 1.2 (2).
Let i be a Cayley-Abels graph of (G i , H i ∪ {K i }) for i = 1, 2. Then i has a vertex x i with G i -stabilizer K i .Let be the C-pushout of G × G 1 1 and G × G 2 2 with respect to (x 1 , x 2 ).Theorem 3.1 implies that is a Cayley-Abels graph of (G 1 * C G 2 , H ∪ { K 1 , K 2 }).By Proposition 4.5 (8), there is a morphism of graphs ξ : → T that satisfies the hypothesis of Proposition 6.6, namely, T is a bipartite tree with V(T) = K ∪ L such that ξ −1 (v) is a single vertex for each v ∈ L, and ξ −1 (star(v)) is isomorphic to i for some i = 1, 2 for every v ∈ K. Corollary 6.4 implies that 1 and 2 are both k-fillable for some k.Then Proposition 6.6 implies that is k-fillable and Then Corollary 6.4 implies that max 1 , 2 .
On the other hand, the properties of the morphism → T imply that there is a retraction → i and hence Proposition 6.5 implies that f i k f k and therefore i .
3 or Corollary 1.4 together with Remark 1.5.The following theorem generalizes results of Brick [6, Proposition 3.2] on bounds on Dehn functions of free products and improve bounds for relative Dehn functions found by Osin [18, Theorems 1

Theorem 3 . 2 .
Let (G, H ∪ {K, L}) be a pair with K = L, C ≤ K, and ϕ : C → L a group monomorphism.Let G * ϕ denote the HNN-extension G, t | tct −1 = ϕ(c) for all c ∈ C .Let be a G-graph that has vertices x and y such that their G-stabilizers are K and L, respectively, and their G-orbits are disjoint.Then there is a G * ϕ -graph with the following properties:1.

Proposition 4 . 1 .
Let φ : R → S and ψ : R → T be G-maps.Consider the pushout φ and ψ.Suppose there is r ∈ R such that R = G.r.If s = φ(r), t = ψ(r) and z = ı(s), then the Gstabilizer G z equals the subgroup G s , G t .

Example 4 . 1 .
Let G = A * C B where A and B are free abelian groups of rank two, and C is maximal cyclic subgroup of A and B. Let X be the A-graph consisting of a single vertex with the trivial A-action and define Y analogously for B. Then the graph G × A X is the edgeless G-graph with vertex set the collection of left cosets of G/A; and analogously G × B Y is the edgeless graph with vertex set G/B.

Proposition 4 . 5 .
and four edges.In particular, Z is the coned-off Cayley graph of ˆ (G, G y , {a 3 , b 3 }).Let G be the amalgamated free product group A * C B, let X be a A-graph, and let Y be a B-graph.Let x ∈ X C and y (ξ • c) −1 (T c − {v}) is a disconnected open subset of [0, 1].Let J 1 be the closure of a connected component of (ξ • c) −1 (T c − {v}).By changing the initial point of the loop c : I → , we can assume that J 1 = [0, α] for some α < 1.Let J 2 = [α, 1], and let c i be the restriction of c to the interval J i .Then c i is singular combinatorial loop, and c is the concatenationc 1 • c 2 .Since T c − {v} is disconnected, it follows that 0 < Len(c i ) < Len(c).Since #T c i ∩ K ≤ #T c ∩ K, it follows that |c i | < |c|.Hence by induction area k (c) ≤ area k (c 1 ) + area k (c 2 )≤ g(Len(c 1 )) + g(Len(c 2 ))≤ g(Len(c 1 ) + Len(c 2 )) = g(Len(c)) Under the assumptions of Corollary 1.4(2), if is a relative Dehn function of (G * ϕ , H ∪ { K, s −1 t }) and 0 is a relative Dehn function of (G, H ∪ {K}) then We conclude the introduction with a more detailed comparison of our results with previous results in the literature.1. Dahmani, Guirardel, and Osin proved Theorem 1.2(1) in the case that ∂ 1 : C → K 1 is an isomorphism and K 1 is finitely generated [9, Thm 6.20]; and Theorem 1.3(1) in the case that C = K and K is finitely generated [9, Thm 6.19].2. Osin proved Theorem 1.2(2) in the case that ∂ 1 : C → K 1 is an isomorphism and K 1 is finitely generated, see [17, Thm 1.3]; and Theorem 1.3(2) in the case that C = K and K is finitely generated, see [17, Thm 1.2].3.Under the assumptions of Theorem 1.1, if each (G v , H v ) ∈ C for every vertex v, and there is at least one v such that H v is nontrivial in G 1.3(2), if is a relative Dehn function of (G * ϕ , H ∪ { K t , L }) and 0 is a relative Dehn function of (G, H ∪ {K, L}) then 0 0 , where 0 is the super-additive closure of 0 3. v , the existence of a nontrivial collection H such that (G, H) ∈ C follows from results of Minasyan and Osin [14, Cor.2.2 and 2.3] and the characterization of acylindrical hyperbolicity in terms of existence of proper infinite hyperbolically embedded subgroups by Osin

Definition 2.3. We
refer to a graph satisfying the conditions of Theorem 2.2 as a (G, H)-graph [18,rem 2.7.[18,Theorem 2.34] Let G be a finitely presented group relative to the collection of subgroups H. Let 1 and 2 be the relative Dehn functions associated with two finite relative presentations.If 1 takes only finite values, then 2 takes only finite values, and 1 2 .The Dehn function of a pair (G, H) is well defined if it takes only finite values.This can be characterized in terms of fine graphs as follows.
a relative isoperimetric function of the relative presentation S, H | R if, for any n ∈ N, and any word W over the alphabet S of length ≤ n representing the trivial element in G, one can write W as in (3) with k ≤ f (n).The smallest relative isoperimetric function of a finite relative presentation S, H | R is called the relative Dehn function of G with respect to H, or the Dehn function of the pair (G, H).This function is denoted by G,H .Theorem 2.7 below justifies the notation G,H for the Dehn function of a finitely presented pair (G, H).For functions f , g : N → N, we write f g if there exist constants C, K, L ∈ N such that f (n) ≤ Cg(Kn) + Ln for every n.We say f and g are asymptotically equivalent, denoted as f g, if f g and g f .1.edgeG-stabilizersare finite, 2. vertex G-stabilizers are either finite or conjugates of subgroups in H, 3. every H ∈ H is the G-stabilizer of a vertex of , and 4. any pair of vertices of with the same G-stabilizer H ∈ H are in the same G-orbit if H is infinite.Theorem 2.9.Let (G, H) be a proper pair.The following statements are equivalent.1.The Dehn function G,H is well defined.2. (G, H) is finitely presented and there is a fine Cayley-Abels graph of (G, H). 3. (G, H) is finitely presented and every Cayley-Abels graph of (G, H) is fine.
Definition 2.10 (Coned-off Cayley graph).Let (G, H) be a pair, and let S be a finite relative generating set of G with respect to H. Denote by G/H the set of all cosets gH with g ∈ G and P ∈ H.The coned-off Cayley graph ˆ (G, H, S) is the graph with vertex set G ∪ G/H and edges of the following type • {g, gs} for s ∈ S and g ∈ G, • {x, gH} for g ∈ G, H ∈ H and x ∈ gH.That a pair (G, H) has a well-defined function is characterized in terms of fineness of coned-off Cayley graphs.
Theorem 2.11.[12, Theorem E] Let (G, H) be a finitely presented pair with a finite generating set S. The Dehn function G,H is well defined if and only if the coned-off Cayley graph ˆ (G, H, S) is fine.
stabilizer of a vertex of .4. If vertex G-stabilizers in are finite or conjugates of subgroups in H ∪ {K, L}, then vertex G * ϕ -stabilizers in are finite or conjugates of subgroups in H ∪ { K t , L }. 5.If has finite edge G-stabilizers, then has finite edge G * ϕ -stabilizers.6.If has finitely many G-orbits of vertices (edges), then has finitely many G * ϕ -orbits of vertices (resp.edges).7. If is fine, then is fine.