Quasiconvexity and Relatively Hyperbolic Groups that Split
Abstract.
We explore the combination theorem for a group splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of under certain conditions. We then provide a criterion for the relative quasiconvexity of a subgroup depending on the relative quasiconvexity of the intersection of with the vertex groups of . We give an application towards local relative quasiconvexity.
Key words and phrases:
relatively hyperbolic, relatively quasiconvex, graphs of groups2000 Mathematics Subject Classification:
20F06,The goal of this paper is to examine relative hyperbolicity and quasiconvexity in graphs of relatively hyperbolic vertex groups with almost malnormal quasiconvex edge groups. The paper hinges upon the observation that if splits as a graph of relatively hyperbolic groups with malnormal relatively quasiconvex edge groups, then a fine hyperbolic graph for can be built from fine hyperbolic graphs for the vertex groups. This leads to short proofs of the relative hyperbolicity of as well as to a concise criterion for the relative quasiconvexity of a subgroup of .
Bestvina and Feighn proved a combination theorem that characterized the hyperbolicity of groups splitting as graphs of hyperbolic groups [2]. Their geometric characterization is akin to the flat plane theorem characterization of hyperbolicity for actions on CAT(0) spaces, and leads to explicit positive results, especially in an “acylindrical” scenario where some form of malnormality is imposed on the edge groups. The BestvinaFeighn combination theorem has been revisited multiple times in a hyperbolic setting, and more recently in a relatively hyperbolic context but through diverse methods.
Dahmani proved a combination theorem for relatively hyperbolic groups using the convergence group approach [4]. Later Alibegović proved similar results in [1] using a method generalizing parts of the BestvinaFeighn approach. Osin reproved Dahman’s result in the general context of relative Dehn functions [19]. Most recently, Mj and Reeves gave a generalization of the BestvinaFeighn combination theorem that follows Farb’s approach but uses a generalized “partial electrocution” [17]. Their result appears to be a farreaching generalization at the expense of complex geometric language.
Our own results revisit these relatively hyperbolic generalizations, and we offer a very concrete approach employing Bowditch’s fine hyperbolic graphs. The most natural formulation of our main combination theorem (proven as Theorem 1.4) is as follows:
Theorem A (Combining Relatively Hyperbolic Groups Along Parabolics).
Let split as a finite graph of groups. Suppose each vertex group is relatively hyperbolic and each edge group is parabolic in its vertex groups. Then is hyperbolic relative to where each is the stabilizer of a “parabolic tree”. (See Definition 1.3.)
A simplistic example illustrating Theorem A is an amalgamated product where each and is a cusped hyperbolic manifold with a single boundary torus . And is an arbitrary common subgroup of and . Then is hyperbolic relative to .
We note that Theorem A is more general than results in the same spirit that were obtained by Dahmani, Alibegović, and Osin. In particular, they require that edge groups be maximal parabolic on at least one side, but we do not. We believe that Theorem A could be deduced from the results of MjReeves.
In Section 4, we employ work of Yang [22] on extended peripheral structures, to obtain the following seemingly more natural corollary of Theorem A which is proven as Corollary 4.6:
Corollary B.
Let split as a finite graph of groups. Suppose

Each is hyperbolic relative to ;

Each is total and relatively quasiconvex in ;

is almost malnormal in for each vertex .
Then is hyperbolic relative to .
The “omitted repeats” in the conclusion of Corollary B refer to (some of) the parabolic subgroups of vertex groups that are identified through an edge group.
It is not clear whether Corollary B could be obtained using the method of Dahmani, Alibegović, or Osin. However, we suspect it could be extracted from the result of MjReeves.
Definition 0.1.
(Tamely generated) Let split as a graph of groups with relatively hyperbolic vertex groups. A subgroup is tamely generated if the induced graph of groups has a isomorphic subgraph of groups that is a finite graph of groups each of whose vertex groups is relatively quasiconvex in the corresponding vertex group of .
Note that is tamely generated when is finitely generated and there are finitely many orbits of vertices in with nontrivial, and each such is relatively quasiconvex in . However the above condition is not necessary. For instance, let , and consider a splitting where is a bouquet of two circles, and each vertex and edge group is isomorphic to . Then every f.g. subgroup of is tamely generated, but no subgroup containing satisfies the condition that there are finitely many orbits of vertices with nontrivial.
The geometric construction proving Theorem A allows us to give a simple criterion for quasiconvexity of a subgroup relative to . Again, coupling this with Yang’s work, we obtain (as Theorem 4.13) the following criterion for quasiconvexity relative to :
Main Theorem C (Quasiconvexity Criterion).
Let be hyperbolic relative to where each is finitely generated. Suppose splits as a finite graph of groups. Suppose

Each is total in ;

Each is relatively quasiconvex in ;

Each is almost malnormal in .
Let be tamely generated. Then is relatively quasiconvex in .
Recall that is locally relatively quasiconvex if each finitely generated subgroup of is quasiconvex relative to the peripheral structure of . I. Kapovich first recognized that hyperbolic limit groups are locally relatively quasiconvex [12], and subsequently Dahmani proved that all limit groups are locally relatively quasiconvex [4].
A group is small if there is no embedding , and has a small hierarchy if it can be built from small subgroups by a sequence of AFP’s and HNN’s along small subgroups (see Definition 3.4). When is a collection of freeabelian groups, the following inductive consequence of Corollary 3.3 generalizes Dahmani’s result.
Theorem D.
Let be hyperbolic relative to a collection of Noetherian subgroups and suppose has a small hierarchy. Then is locally relatively quasiconvex.
Although Theorem D is implicit in Dahmani’s work, we believe Theorem C is new.
The main construction and its application: Although we work in somewhat greater generality, let us focus on the simple case of an amalgamated product where are relatively hyperbolic and is parabolic on each side. The central theme of this paper is a construction that builds a fine hyperbolic graph for from fine hyperbolic graphs and for . (See Figure 1.) This is done in two steps: Guided by the BassSerre tree, we first construct a graph which is a tree of spaces whose vertex spaces are copies of and , and whose edge spaces are ordinary edges. Though is fine and hyperbolic, its edges have infinite stabilizers. We remedy this by quotienting these edge spaces to form the fine hyperbolic graph . The vertices of are quotients of “parabolic trees” in . The fine hyperbolic graph quickly proves that is hyperbolic relative to the collection of subgroups stabilizing parabolic trees. Variations on the construction, hypotheses on the edge groups, and interplay with previous work on peripheral structures, leads to a variety of relatively hyperbolic conclusions. The simplest and most immediate in the case above, is that is hyperbolic relative to when is maximal parabolic on each side and are hyperbolic relative to and .
Our primary application is to give an easy criterion to recognize quasiconvexity. A subgroup is relatively quasiconvex in if there is an cocompact quasiconvex subgraph of the fine hyperbolic graph. The treelike nature of our graph , permits us to naturally build the quasiconvex graph . When is relatively quasiconvex, there are finitely many orbits of nontrivially stabilized vertices in the BassSerre tree , and each of these stabilizers is relatively quasiconvex in its vertex group. Choosing finitely many quasiconvex subgraphs in the corresponding copies of and , we are able to combine these together to form in and then to form a quasiconvex subgraph in .
We conclude by mentioning the following consequence of Corollary 1.5 that is a natural consequence of the viewpoint developed in this paper.
Corollary E.
Let be a compact irreducible 3manifold. And let denote the graph manifolds obtained by removing each (open) hyperbolic piece in the geometric decomposition of . Then is hyperbolic relative to .
As explained to us by the referee, the relative hyperbolicity of was previously proved by DrutuSapir using work of KapovichLeeb. This previous proof is deep as it uses the structure of the asymptotic cone due to KapovichLeeb together with the technical proof of DrutuSapir that asymptotically tree graded groups are relatively hyperbolic [13, 5].
1. Combining Relatively Hyperbolic Groups along Parabolics
The class of relatively hyperbolic groups was introduced by Gromov [8] as a generalization of the class of fundamental groups of complete finitevolume manifolds of pinched negative sectional curvature. Various approaches to relative hyperbolicity were developed by Farb [6], Bowditch [3] and Osin [20], and as surveyed by Hruska [10], these notions are equivalent for finitely generated groups. We follow Bowditch’s approach:
Definition 1.1 (Relatively Hyperbolic).
A circuit in a graph is an embedded cycle. A graph is fine if each edge of lies in finitely many circuits of length for each .
A group G is hyperbolic relative to a finite collection of subgroups if acts cocompactly(without inversions) on a connected, fine, hyperbolic graph with finite edge stabilizers, such that each element of equals the stabilizer of a vertex of , and moreover, each infinite vertex stabilizer is conjugate to a unique element of . We refer to a connected, fine, hyperbolic graph equipped with such an action as a graph. Subgroups of G that are conjugate into subgroups in are parabolic.
Technical Remark 1.2.
Given a finite collection of parabolic subgroups , we choose so that there is a prescribed choice of parabolic subgroup so that is “declared” to be conjugate into . This is automatic for an infinite parabolic subgroup but for finite subgroups there could be ambiguity. One way to resolve this is to revise the choice of as follows: For any finite collection of parabolic subgroups in , we moreover assume each is conjugate to a subgroup of and we assume that no two (finite) subgroups in are conjugate. We note that finite subgroups can be freely added to or omitted from the peripheral structure of (see e.g. [16]).
Definition 1.3 (Parabolic tree).
Let split as a finite graph of groups where each vertex group is hyperbolic relative to , and where each edge group embeds as a parabolic subgroup of its two vertex groups. Let be the BassSerre tree. Define the parabolic forest by:

A vertex in is a pair where and is a conjugate of an element of .

An edge in is a pair where is an edge of and is its stabilizer.

The edge is attached to and where and are the initial and terminal vertex of and is the conjugate of an element of that is declared to contain . Likewise for . We arranged for this unique determination in Technical Remark 1.2.
Each component of is a parabolic tree and the map is injective on the set of edges, and in particular each parabolic tree embeds in . Let be representatives of the finitely many orbits of parabolic trees under the action on . Let , for each .
Theorem 1.4 (Combining Relatively Hyperbolic Groups Along Parabolics).
Let split as a finite graph of groups. Suppose each vertex group is relatively hyperbolic and each edge group is parabolic in its vertex groups. Then is hyperbolic relative to .
Proof.
For , let be hyperbolic relative to and let be a graph. For each , following the Technical Remark 1.2, we choose a specific vertex of whose stabilizer equals . Note that, in general there could be more than one possible choice when , but by Technical Remark 1.2 we have a unique choice. Translating determines a “choice” of vertex for conjugates.
We now construct a graph . Let be the tree of spaces whose underlying tree is the BassSerre tree with the following properties:

Vertex spaces of are copies of appropriate elements in . Specifically, is a copy of where is the image of under .

Each edge space is an ordinary edge, denoted as an ordered pair that is attached to the vertices in and that were chosen to contain .
Note that each acts on and there is a equivariant map . Let be the quotient of obtained by contracting each edge space. Observe that acts on and there is a equivariant map . Moreover the preimage of each open edge of is a single open edge of .
We now show that is a graph. Since any embedded cycle lies in some vertex space, the graph is fine and hyperbolic. There are finitely many orbits of vertices in and therefore finitely many orbits of vertices in . Likewise, there are finitely many orbits of edges in . The stabilizer of an (open) edge of equals the stabilizer of the corresponding (open) edge in , and is thus finite. By construction, there is a equivariant embedding where is the parabolic forest associated to and . Finally, the preimage in of a vertex of is precisely a parabolic tree and thus the stabilizer of a vertex of is a conjugate of some . ∎
We now examine some conclusions that arise when the parabolic trees are small. An extreme case arises when the edge groups are isolated from each other as follows:
Corollary 1.5.
Let split as a finite directed graph of groups where each vertex group is hyperbolic relative to . Suppose that:

Each edge group is parabolic in its vertex groups.

Each outgoing infinite edge group is maximal parabolic in its initial vertex group and for each other incoming and outgoing infinite edge group or or , none of its conjugates lie in .
Then is hyperbolic relative to .
Proof.
We can arrange for finitely stabilized edges of to be attached to distinct chosen vertices when they correspond to distinct edges of . Thus, parabolic trees are singletons and/or pods consisting of edges that all terminate at the same vertex where and . Recall that an pod is a tree consisting of edges glued to a central vertex. ∎
Corollary 1.6.
Let split as a finite graph of groups. Suppose each vertex group is hyperbolic relative to . For each assume that the collection is a collection of maximal parabolic subgroups of . Then is hyperbolic relative to . Specifically, we remove an element of if it is conjugate to another one.
Corollary 1.7.

Let and be hyperbolic relative to and . Let where each and is identified with the subgroup of . Then is hyperbolic relative .

Let be hyperbolic relative to . Let be isomorphic to a subgroup of a maximal parabolic subgroup not conjugate to . Let where . Then is hyperbolic relative to .

Let be hyperbolic relative to . Let be isomorphic to . Let . Then is hyperbolic relative to .
Remark 1.8.
Note that in this Corollary and some similar results when we say , we mean if then replace by in .
Proof.
(1): In this case, the parabolic trees are either singletons stabilized by a conjugate of an element of , or parabolic trees are pods stabilized by conjugates of .
(2): The proof is similar.
(3): All parabolic trees are singletons except for those that are translates of a copy of the BassSerre tree for . Following the proof of Theorem 1.4, let , if the preimage of in is not attached to an edge space, then is conjugate to an element of , otherwise is conjugate to . ∎
2. Relative Quasiconvexity
Dahmani introduced the notion of relatively quasiconvex subgroup in [4]. This notion was further developed by Osin in [20], and later Hruska investigated several equivalent definitions of relatively quasiconvex subgroups [10]. MartinezPedroza and the second author introduced a definition of relative quasiconvexity in the context of fine hyperbolic graphs and showed this definition is equivalent to Osin’s definition [16]. We will study relatively quasiconvexity using this fine hyperbolic viewpoint. Our aim is to examine the relative quasiconvexity of a certain subgroup which are themselves amalgams, and we note that powerful results in this direction are given in [15].
Definition 2.1 (Relatively Quasiconvex).
Let be hyperbolic relative to . A subgroup of is quasiconvex relative to if for some (and hence any) graph , there is a nonempty connected and quasiisometrically embedded, cocompact subgraph of . In the sequel, we sometimes refer to as a quasiconvex cocompact subgraph of .
Remark 2.2.
It is immediate from the Definition 2.1 that in a relatively hyperbolic group, any parabolic subgroup is relatively quasiconvex, and any relatively quasiconvex subgroup is also relatively hyperbolic. In particular, the relatively quasiconvex subgroup is hyperbolic relative to the collection consisting of representatives of stabilizers of vertices of . Note that a conjugate of a relatively quasiconvex subgroup is also relatively quasiconvex. And the intersection of two relatively quasiconvex subgroups is relatively quasiconvex. Specifically, this last statement was proven when is f.g. in [15], and when is countable in [10].
Relative quasiconvexity has the following transitive property proven by Hruska for countable relatively hyperbolic groups in [10]:
Lemma 2.3.
Let be hyperbolic relative to . Suppose that is relatively quasiconvex in , and note that is then hyperbolic relative to as in Remark 2.2. Then is quasiconvex relative to if and only if is quasiconvex relative to .
Proof.
Let be a graph. As is quasiconvex relative to , there is a cocompact and quasiconvex subgraph . Note that is a graph. Let .
If is quasiconvex in relative to , there is an cocompact quasiconvex subgraph . Since the composition is a quasiisometric embedding, is quasiconvex relative to . Conversely, if is quasiconvex in relative to , then there is an cocompact quasiconvex subgraph . Let and note that is cocompact and hence also quasiconvex, and thus also serves as a fine hyperbolic graph for . Now is quasiconvex since is quasiconvex so is relatively quasiconvex in . ∎
Remark 2.4.
One consequence of Theorem 1.4 and its various Corollaries, is that when splits as a graph of relatively hyperbolic groups with parabolic subgroups, then each of the vertex groups is quasiconvex relative to the peripheral structure of . (For Theorem 1.4 this is , and for Corollary 1.6 this is .) Indeed, is a cocompact quasiconvex subgraph in the fine graph constructed in the proof.
Lemma 2.5.
Let be a f.g. group that split as a finite graph of groups . If each edge group is f.g. then each vertex group is f.g.
Proof.
Let . We regard as of a complex corresponding to . We show that each vertex group equals . Let and consider an expression of as a product of normal forms of the . Then equals some product . There is a disc diagram whose boundary path is . See Figure 2. The region of that lies along shows that equals the product of elements in edge groups adjacent to , together with elements of that lie in the normal forms of . ∎
Theorem 2.6 (Quasiconvexity of a Subgroup in Parabolic Splitting).
Let split as a finite graph of relatively hyperbolic groups such that each edge group is parabolic in its vertex groups. (Note that is hyperbolic relative to by Theorem 1.4.) Let be tamely generated. Then is quasiconvex relative to . Moreover if each in the BassSerre tree is finitely generated then is finitely generated.
Proof.
Since there are finitely many orbits of vertices whose stabilizers are finitely generated, is finitely generated. For each , let be hyperbolic relative to and let be a graph. Let be the graph constructed in the proof of Theorem 1.4 and let be its quotient. We will construct an cocompact quasiconvex, connected subgraph of .
Let be the minimal invariant subgraph of . Recall that each edge of (and hence ) corresponds to an edge of . Let denote the subgraph of that is the union of all edges correspond to edges of . Let be a representatives of orbits of vertices of . For each , let be a cocompact quasiconvex subgraph such that contains . (There are finitely many orbits of such endpoints of edges in .) Let and let be the image of under . Observe that is quasiconvex in since is a “tree union” and each such of is quasiconvex in . And likewise, is quasiconvex in . ∎
Corollary 2.7 (Characterizing Quasiconvexity in Maximal Parabolic Splitting).
Let split as a finite graph of countable groups. For each , let be hyperbolic relative to and let the collection be a collection of maximal parabolic subgroups of . Note that is hyperbolic relative to by Corollary 1.6. Let be the BassSerre tree and let be a subgroup of . The following are equivalent:

is tamely generated and each in the BassSerre tree is f.g.

is f.g. and quasiconvex relative to .
Proof.
(2 1): Since is f.g., the minimal subtree is cocompact, and so splits as a finite graph of groups . Since is quasiconvex in , it is hyperbolic relative to intersections with conjugates of . In particular, the infinite edge groups in the induced splitting of are maximal parabolic, and are thus f.g. since the maximal parabolic subgroups of a f.g. relatively hyperbolic group are f.g. [20]. Each vertex group of is f.g. by Lemma 2.5.
3. Local Relative Quasiconvexity
A relatively hyperbolic group is locally relatively quasiconvex if each f.g. subgroup of is relatively quasiconvex. The focus of this section is the following criterion for showing that the combination of locally relatively quasiconvex groups is again locally relatively quasiconvex.
Recall that is Noetherian if each subgroup of is f.g. We now give a criterion for local quasiconvexity of a group that splits along parabolic subgroups.
Theorem 3.1 (A Criterion for Locally Relatively Quasiconvexity).

Let and be locally relatively quasiconvex relative to and . Let where each and is identified with the subgroup of . Suppose is Noetherian. Then is locally quasiconvex relative to .

Let be a locally relatively quasiconvex relative to . Let be isomorphic to a subgroup of a maximal parabolic subgroup not conjugate to . Let . Suppose is Noetherian. Then is locally quasiconvex relative to .

Let be a locally quasiconvex relative to . Let be a maximal parabolic subgroup of , isomorphic to . Let and suppose is Noetherian. Then is also locally quasiconvex relative to .
Proof.
(1): By Corollary 1.7, is hyperbolic relative to . Let be a finitely generated subgroup of . We show that is quasiconvex relative to . Let be the BassSerre tree of . Since is f.g., the minimal subtree is cocompact, and so splits as a finite graph of groups . Moreover, the edge groups of this splitting are f.g. since the edge groups of are Noetherian by hypothesis. Thus each vertex group of is f.g. by Lemma 2.5. Since and are locally relatively quasiconvex, each vertex group of is relatively quasiconvex in its “image vertex group” under the map . Now by Theorem 2.6, is quasiconvex relative to . The proof of (2) and (3) are similar. ∎
Definition 3.2 (Almost Malnormal).
A subgroup is malnormal in if for , and similarly is almost malnormal if this intersection is always finite. Likewise, a collection of subgroups is almost malnormal if is finite unless and .
Corollary 3.3.
Let split as a finite graph of groups. Suppose

Each is locally relatively quasiconvex;

Each is Noetherian and maximal parabolic in its vertex groups;

is almost malnormal in , for any vertex .
Then is locally relatively quasiconvex relative to , see Corollary 1.6.
3.1. Smallhierarchies and local quasiconvexity
The main result in this subsection is a consequence of Theorem 3.1 that employs results of Yang [22] stated in Theorems 4.7 and 4.2, and also depends on Lemma 4.9 which is independent of Section 4. The reader may choose to read this subsection and refer ahead to those results, or return to this subsection after reading Section 4.
Definition 3.4 (SmallHierarchy).
A group is small if it has no rank 2 free subgroup. Any small group has a length 0 smallhierarchy. has a length smallhierarchy if or , where and have length smallhierarchies, and is small and f.g. We say has a smallhierarchy if it has a length smallhierarchy for some .
We can define hierarchy by replacing “small” by a class of groups closed under subgroups and isomorphisms. For instance, when is the class of finite groups, the class of groups with an hierarchy is precisely the class of virtually free groups.
Remark 3.5.
The Tits alternative for relatively hyperbolic groups states that every f.g. subgroup is either: elementary, parabolic, or contains a subgroup isomorphic to . The Tits alternative is proven for countable relatively hyperbolic groups in [8, Thm 8.2.F]. A proof is given for convergence groups in [21]. It is shown in [20] that every cyclic subgroup of a f.g. relatively hyperbolic group is relatively quasiconvex.
Theorem 3.6.
Let be f.g. and hyperbolic relative to where each element of is Noetherian. Suppose has a smallhierarchy. Then is locally relatively quasiconvex.
Proof.
The proof is by induction on the length of the hierarchy. Since edge groups are f.g., the Tits alternative shows that there are three cases according to whether the edge group is finite, virtually cyclic, or infinite parabolic, and we note that the edge group is relatively quasiconvex in each case. These three cases are each divided into two subcases according to whether or .
Since and are f.g. the vertex groups are f.g. by Lemma 2.5. Thus, since is relatively quasiconvex the vertex groups are relatively quasiconvex by Lemma 4.9.
When is finite the conclusion follows in each subcase from Theorem 3.1.
When is virtually cyclic but not parabolic, then lies in a unique maximal virtually cyclic subgroup that is almost malnormal and relatively quasiconvex by [18]. Thus is hyperbolic relative to by Theorem 4.2.
Observe that is maximal infinite cyclic on at least one side, since otherwise there would be a nontrivial splitting of as an amalgamated free product over . We equip the (relatively quasiconvex) vertex groups with their induced peripheral structures. Note that is maximal parabolic on at least one side and so is locally relatively quasiconvex relative to by Theorem 3.1. Finally, by Theorem 4.7, any subgroup is quasiconvex relative to the original peripheral structure since intersections between and conjugates of are quasiconvex relative to .
When is infinite parabolic, we will first produce a new splitting before verifying local relative quasiconvexity.
When . Let be the maximal parabolic subgroups of containing , and refine the splitting to:
The two outer splittings are along a parabolic that is maximal on the outside vertex group. The inner vertex group is a single parabolic subgroup of . Indeed, as is infinite, must all lie in the same parabolic subgroup of . It is obvious that is locally relatively quasiconvex with respect to its induced peripheral structure since it is itself parabolic in . Consequently is locally relatively quasiconvex by Theorem 3.1, therefore is locally relatively quasiconvex by Theorem 3.1.
When , let be the maximal parabolic subgroup of containing . There are two subsubcases:
[] Then and we revise the splitting to where . And in this splitting the edge group is maximal parabolic at , and is parabolic.
[] Let denote the maximal parabolic subgroup of containing . Observe that is almost malnormal since . We revise the HNN extension to the following:
where the conjugated copies of in the HNN extension embed in the first and second factor of the AFP.
In both cases, the local relative quasiconvexity of now holds by Theorem 3.1 as before. ∎
4. Relative Quasiconvexity in Graphs of Groups
Gersten [7] and then Bowditch [3] showed that a hyperbolic group is hyperbolic relative to an almost malnormal quasiconvex subgroup. Generalizing work of MartinezPedroza [14], Yang introduced and characterized a class of parabolically extended structures for countable relatively hyperbolic groups [22]. We use his results to generalize our previous results. The following was defined in [22] for countable groups.
Definition 4.1 (Extended Peripheral Structure).
A peripheral structure consists of a finite collection of subgroups of a group . Each element is a peripheral subgroup of . The peripheral structure extends if for each , there exists such that . For , we let .
We will use the following result of Yang [22].
Theorem 4.2 (Hyperbolicity of Extended Peripheral Structure).
Let be hyperbolic relative to and let the peripheral structure extend . Then is hyperbolic relative to if and only if the following hold:

is almost malnormal;

Each is quasiconvex in relative to .
Definition 4.3 (Total).
Let be hyperbolic relative to . The subgroup of is total relative to if: either or is finite for each and .
The following is proven in [5]:
Lemma 4.4.
If is f.g. and hyperbolic relative to and each is hyperbolic relative to , then is hyperbolic relative to .
As an application of Theorem 4.2, we now generalize Corollary 1.7 to handle the case where edge groups are quasiconvex and not merely parabolic.
Theorem 4.5 (Combination along Total, Malnormal and Quasiconvex Subgroups).

Let be hyperbolic relative to for . Let be almost malnormal, total and relatively quasiconvex. Let . Then is hyperbolic relative to .

Let be hyperbolic relative to . Let be almost malnormal and assume each is total and relatively quasiconvex. Let . Then is hyperbolic relative to .
Proof.
(1): For each , let
Without loss of generality, we can assume that extends , since we can replace an element of by its conjugate. We now show that is hyperbolic relative to by verifying the two conditions of Theorem 4.2: is malnormal in , since is almost malnormal and is total and almost malnormal. Each element of is relatively quasiconvex, since is relatively quasiconvex by hypothesis and each element of is relatively quasiconvex by Remark 2.2.
We now regard each as hyperbolic relative to . Therefore since the edge group is maximal on one side, by Corollary 1.7, is hyperbolic relative to .
We now apply Lemma 4.4 to show that is hyperbolic relative to . We showed that is hyperbolic relative to . But each element of is hyperbolic relative to that it contains. Thus by Lemma 4.4, we obtain the result.
(2): The proof is analogous to the proof of . ∎
The following can be obtained by induction using Theorem 4.5 or can be proven directly using the same mode of proof.
Corollary 4.6.
Let split as a finite graph of groups. Suppose

Each is hyperbolic relative to ;

Each is total and relatively quasiconvex in ;

is almost malnormal in for each vertex .
Then is hyperbolic relative to .
Yang characterized relative quasiconvexity with respect to extensions in [22] as follows:
Theorem 4.7 (Quasiconvexity in Extended Peripheral Structure).
Let be hyperbolic relative to and relative to . Suppose that extends . Then

If is quasiconvex relative to , then is quasiconvex relative to .

Conversely, if is quasiconvex relative to , then is quasiconvex relative to if and only if is quasiconvex relative to for all and .
We recall the following observation of Bowditch (see [16, Lem 2.7 and 2.9]).
Lemma 4.8 (attachment).
Let act on a graph . Let and be a new edge whose endpoints are and . The attachment of is the new graph which consists of the union of and copies of attached at and for any . Note that is cocompact/fine/hyperbolic if is.
In the following lemma, we prove that when a relatively hyperbolic group splits then relative quasiconvexity of vertex groups is equivalent to relative quasiconvexity of the edge groups.
Lemma 4.9 (Quasiconvex Edges Quasiconvex Vertices).
Let be hyperbolic relative to . Suppose splits as a finite graph of groups whose vertex groups and edge groups are finitely generated. Then the edge groups are quasiconvex relative to if and only if the vertex groups are quasiconvex relative to .
Proof.
If the vertex groups are quasiconvex relative to then so are the edge groups, since relative quasiconvexity is preserved by intersection (see [10, 15]) in the f.g. group . Assume the edge groups are quasiconvex relative to . Let be a graph and let be the BassSerre tree for . Let be a equivariant map that sends vertices to vertices and edges to geodesics. Subdivide and , so that each edge is the union of two length halfedges. Let be a vertex in . It suffices to find a cocompact quasiconvex subgraph of .
Let be representatives of the orbits of halfedges attached to . Let be the other vertex of for . Since each is f.g. by hypothesis, we can perform finitely many attachments of arcs so that the preimage of is connected for each . This leads to finitely many attachments to to obtain a new fine hyperbolic graph . By mapping the newly attached edges to their associated vertices in , we thus obtain a equivariant map such that is connected and cocompact for each .
Consider where is the closed neighborhood of . To see that is connected, consider a path in between distinct components of . Moreover choose so that its image in is minimal among all such choices. Then must leave and enter through the same which is connected by construction.
We now show that is quasiconvex. Consider a geodesic that intersects exactly at its endpoints. As before the endpoints of lie in the same . Since is quasiconvex for some , we see that lies in neighborhood of and hence in the neighborhood of . ∎
Lemma 4.10 (Total Edges Total Vertices).
Let be hyperbolic relative to . Let act on a tree . For each let be a minimal subtree. Assume that no has a finite edge stabilizer in the action. Then edge groups of are total in iff vertex groups are total in .
Proof.
Since the intersection of two total subgroups is total, if the vertex groups are total then the edge groups are also total. We now assume that the edge groups are total. Let be a vertex group and such that is infinite for some . If for some edge attached to , then , thus . Now suppose that for each attached to . If then the action of on violates our hypothesis. ∎
Remark 4.11.
Suppose is f.g. and is hyperbolic relative to . Let such that [] where is a finite group. Since is hyperbolic relative to [], by Lemma 4.4, is hyperbolic relative to [].
We now describe a more general criterion for relative quasiconvexity which is proven by combining Corollary 2.7 with Theorem 4.7.
Theorem 4.12.
Let be f.g. and hyperbolic relative to . Suppose splits as a finite graph of groups. Suppose

Each is total in ;

Each is relatively quasiconvex in ;

is almost malnormal in for each vertex .
Let be tamely generated subgroup of . Then is relatively quasiconvex in .
Proof.
Technical Point: By splitting certain elements of to obtain as in Remark 4.11, we can assume that is hyperbolic relative to and each is hyperbolic relative to the conjugates of elements of that it contains.
Indeed for any , if the action of on a minimal subtree of the BassSerre tree , yields a finite graph of groups some of whose edge groups are finite, then following Remark 4.11, we can replace by the groups that complement these finite edge groups, (i.e. the fundamental groups of the subgraphs obtained by deleting these edges from .) Therefore is hyperbolic relative to .
No has a nontrivial induced splitting as a graph of groups with a finite edge group.The edge groups are total relative to since they are total relative to . Therefore by Lemma 4.10 the vertex groups are total in relative to . By Lemma 4.9, each vertex group is relatively quasiconvex in relative to , therefore by Theorem 4.7 each is quasiconvex in relative to . Thus has an induced relatively hyperbolic structure as in Remark 2.2. By totality of , we can assume each element of is a conjugate of an element of . And as usual we may omit the finite subgroups in .
Step 1: We now extend the peripheral structure of each from to where
Almost malnormality of follows from Condition (c) and the totality of the edge groups in their vertex groups which follows by the totality of the edge groups in , also relative quasiconvexity of the new elements is Condition (b). Thus by is hyperbolic relative to by Theorem 4.7.
Step 2: For each in the BassSerre tree, its stabilizer lies in which we identify (by a conjugacy isomorphism) with the chosen vertex stabilizer in the graph of group decomposition. Then is quasiconvex in relative to for each by Theorem 4.7, since extends and each is quasiconvex in relative to . Therefore is quasiconvex relative to by Corollary 2.7.
Step 3: is quasiconvex relative to . Since extends , by Theorem 4.7, it suffices to show that is quasiconvex relative to for all and . There are two cases:
Case 1: for some . Now is a parabolic subgroup of relative to and is thus quasiconvex relative to .
Case 2: for some attached to some . The group is relatively quasiconvex in , therefore by Remark 2.2, is also relatively quasiconvex but in . Now since and and are both relatively quasiconvex in , the group is relatively quasiconvex in . Since by Lemma 4.9, is quasiconvex relative to , Lemma 2.3 implies that is quasiconvex relative to .
Now is quasiconvex relative to by Theorem 4.7, since extends . ∎
Theorem 4.13 (Quasiconvexity Criterion for Relatively Hyperbolic Groups that Split).
Let be f.g. and hyperbolic relative to such that splits as a finite graph of groups. Suppose

Each is total in ;

Each is relatively quasiconvex in ;

Each is almost malnormal in .
Let be tamely generated. Then is relatively quasiconvex in .
Remark 4.14.
Proof.
We prove the result by induction on the number of edges of the graph of groups . The base case where has no edge is contained in the hypothesis. Suppose that has at least one edge (regarded as an open edge). If is nonseparating, then where is the graph of groups over , and are the two images of . Condition (c) ensures that is almost malnormal in , and by induction, the various nontrivial intersections are relatively quasiconvex in , and thus is relatively quasiconvex in by Theorem 4.12. A similar argument concludes the separating case. ∎
Corollary 4.15.
Let be f.g. and hyperbolic relative to . Suppose splits as a finite graph of groups. Assume:

Each is locally relatively quasiconvex;

Each is Noetherian, total and relatively quasiconvex in ;

Each is almost malnormal in .
Then is locally relatively quasiconvex relative to .
Theorem 4.16.
Let be hyperbolic relative to . Suppose splits as a graph of groups with relatively quasiconvex edge groups. Suppose is bipartite with and each edge joins vertices of and . Suppose each is maximal parabolic for , and for each there is at most one with conjugate to . Let be tamely generated. Then is quasiconvex relative to .
The scenario of Theorem 4.16 arises when is a compact aspherical 3manifold, from its JSJ decomposition. The manifold decomposes as a bipartite graph of spaces with . The submanifold is hyperbolic for each , and is a graph manifold for each . The edges of correspond to the “transitional tori” between these hyperbolic and complementary graph manifold parts. Some of the graph manifolds are complex but others are simpler Seifert fibered spaces; in the simplest cases, thickened tori between adjacent hyperbolic parts or bundles over Klein bottles where a hyperbolic part terminates. Hence decomposes accordingly as a graph of groups, and is hyperbolic relative to by Theorem 1.4 or indeed, Corollary 1.5.
Proof.
Let be a fine hyperbolic graph for . Each vertex group is quasiconvex in by Lemma 4.9, and so for each let be a quasiconvex subgraph, and in this way we obtain finite hyperbolic graphs, and for , we let be a singleton. We apply the Construction in the proof of Theorem 1.4 to obtain a fine hyperbolic graph and quotient . Note that the parabolic trees are pods. We form the cocompact quasiconvex subgraph by combining cocompact quasiconvex subgraphs as in the proof of Theorem 2.6. ∎
Theorem 4.17.
Let be f.g. and hyperbolic relative to . Suppose splits as graph of groups with relatively quasiconvex edge groups. Suppose is bipartite with and each edge joins vertices of and . Suppose each is almost malnormal and total in for . Let be tamely generated. Then is quasiconvex relative to .
Theorem 4.17 covers the case where edge groups are almost malnormal on both sides since we can subdivide to put barycenters of edges in .
Another special case where Theorem 4.17 applies is where is hyperbolic relative to , and