# Quasiconvexity and Relatively Hyperbolic Groups that Split

Hadi Bigdely  Dept. of Math & Stats.
McGill University
Montreal, Quebec, Canada H3A 2K6<EMAIL_ADDRESS>and  Daniel T. Wise
<EMAIL_ADDRESS>
###### Abstract.

We explore the combination theorem for a group $G$ 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 $G$ under
certain conditions. We then provide a criterion for the relative
quasiconvexity of a subgroup $H$ depending on the relative quasiconvexity of
the intersection of $H$ with the vertex groups of $G$. We give an application
towards local relative quasiconvexity.

###### Key words and phrases:

relatively hyperbolic, relatively quasiconvex, graphs of groups

###### 2000 Mathematics Subject Classification:

20F06,

Research supported by NSERC

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 $G$
splits as a graph of relatively hyperbolic groups with malnormal relatively
quasiconvex edge groups, then a fine hyperbolic graph for $G$ can be built
from fine hyperbolic graphs for the vertex groups. This leads to short proofs
of the relative hyperbolicity of $G$ as well as to a concise criterion for the
relative quasiconvexity of a subgroup $H$ of $G$.

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 Bestvina-Feighn 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 Bestvina-Feighn 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 Bestvina-
Feighn combination theorem that follows Farb’s approach but uses a generalized
“partial electrocution” [17]. Their result appears to be a far-reaching
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 $G$ 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 $G$ is hyperbolic relative to $\mathbb{Q}=\\{Q_{1},\dots,Q_{j}\\}$ where
each $Q_{i}$ is the stabilizer of a “parabolic tree”. (See Definition 1.3.)

A simplistic example illustrating Theorem A is an amalgamated product
$G=G_{1}*_{C}G_{2}$ where each $G_{i}=\pi_{1}M_{i}$ and $M_{i}$ is a cusped
hyperbolic manifold with a single boundary torus $T_{i}$. And $C$ is an
arbitrary common subgroup of $\pi_{1}T_{1}$ and $\pi_{1}T_{2}$. Then $G$ is
hyperbolic relative to $\pi_{1}T_{1}*_{C}\pi_{1}T_{2}$.

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 Mj-Reeves.

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 $G$ split as a finite graph of groups. Suppose

  * (a)

Each $G_{\nu}$ is hyperbolic relative to $\mathbb{P}_{\nu}$;

  * (b)

Each $G_{e}$ is total and relatively quasiconvex in $G_{\nu}$;

  * (c)

$\\{G_{e}:e~{}\textrm{is attached to }\nu\\}$ is almost malnormal in $G_{\nu}$
for each vertex $\nu$.

Then $G$ is hyperbolic relative to
$\bigcup_{\nu}\mathbb{P}_{\nu}-\\{\text{repeats}\\}$.

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 Mj-Reeves.

######  Definition 0.1.

(Tamely generated) Let $G$ split as a graph of groups with relatively
hyperbolic vertex groups. A subgroup $H$ is _tamely generated_ if the induced
graph of groups $\Gamma_{H}$ has a $\pi_{1}$-isomorphic subgraph of groups
$\Gamma^{\prime}_{H}$ that is a finite graph of groups each of whose vertex
groups is relatively quasiconvex in the corresponding vertex group of $G$.

Note that $H$ is tamely generated when $H$ is finitely generated and there are
finitely many $H$-orbits of vertices $v$ in $T$ with $H_{v}$ nontrivial, and
each such $H_{v}$ is relatively quasiconvex in $G_{v}$. However the above
condition is not necessary. For instance, let $G=F_{2}\times\mathbb{Z}_{2}$,
and consider a splitting where $\Gamma$ is a bouquet of two circles, and each
vertex and edge group is isomorphic to $\mathbb{Z}_{2}$. Then every f.g.
subgroup $H$ of $F_{2}\times\mathbb{Z}_{2}$ is tamely generated, but no
subgroup containing $\mathbb{Z}_{2}$ satisfies the condition that there are
finitely many $H$-orbits of vertices $\omega$ with $H_{\omega}$ nontrivial.

The geometric construction proving Theorem A allows us to give a simple
criterion for quasiconvexity of a subgroup $H$ relative to $\mathbb{Q}$.
Again, coupling this with Yang’s work, we obtain (as Theorem 4.13) the
following criterion for quasiconvexity relative to $\mathbb{P}$:

######  Main Theorem C (Quasiconvexity Criterion).

Let $G$ be hyperbolic relative to $\mathbb{P}$ where each $P\in\mathbb{P}$ is
finitely generated. Suppose $G$ splits as a finite graph of groups. Suppose

  1. (a)

Each $G_{e}$ is total in $G$;

  2. (b)

Each $G_{e}$ is relatively quasiconvex in $G$;

  3. (c)

Each $G_{e}$ is almost malnormal in $G$.

Let $H\leq G$ be tamely generated. Then $H$ is relatively quasiconvex in $G$.

Recall that $G$ is _locally relatively quasiconvex_ if each finitely generated
subgroup $H$ of $G$ is quasiconvex relative to the peripheral structure of
$G$. 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 $P$ is _small_ if there is no embedding $F_{2}\hookrightarrow P$, and
$G$ 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
$\mathbb{P}$ is a collection of free-abelian groups, the following inductive
consequence of Corollary 3.3 generalizes Dahmani’s result.

######  Theorem D.

Let $G$ be hyperbolic relative to a collection of Noetherian subgroups
$\mathbb{P}$ and suppose $G$ has a small hierarchy. Then $G$ is locally
relatively quasiconvex.

Although Theorem D is implicit in Dahmani’s work, we believe Theorem C is new.

Figure 1. A fine graph $K_{G}$ for $G=A*_{C}B$ is built from copies of fine
graphs $K_{A}$ and $K_{B}$ for $A$ and $B$ by gluing new edges together along
vertices stabilized by $C$. The parabolic trees of $T$ are images of trees
formed from the new edges in $K_{G}$. We obtain a fine hyperbolic graph
$\bar{K}_{G}$ with finite edge stabilizers as a quotient
$K_{G}\rightarrow\bar{K}_{G}$.

The main construction and its application: Although we work in somewhat
greater generality, let us focus on the simple case of an amalgamated product
$G=A*_{C}B$ where $A,B$ are relatively hyperbolic and $C$ is parabolic on each
side. The central theme of this paper is a construction that builds a fine
hyperbolic graph $\bar{K}_{G}$ for $G$ from fine hyperbolic graphs $K_{A}$ and
$K_{B}$ for $A,B$. (See Figure 1.) This is done in two steps: Guided by the
Bass-Serre tree, we first construct a graph $K_{G}$ which is a tree of spaces
whose vertex spaces are copies of $K_{A}$ and $K_{B}$, and whose edge spaces
are ordinary edges. Though $K_{G}$ is fine and hyperbolic, its edges have
infinite stabilizers. We remedy this by quotienting these edge spaces to form
the fine hyperbolic graph $\bar{K}_{G}$. The vertices of $\bar{K}_{G}$ are
quotients of “parabolic trees” in $K_{G}$. The fine hyperbolic graph
$\bar{K}_{G}$ quickly proves that $G$ is hyperbolic relative to the collection
$\mathbb{Q}$ 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 $G$ is
hyperbolic relative to
$\mathbb{P}_{G}=\mathbb{P}_{A}\cup\mathbb{P}_{B}-\\{C\\}$ when $C$ is maximal
parabolic on each side and $A,B$ are hyperbolic relative to $\mathbb{P}_{A}$
and $\mathbb{P}_{B}$.

Our primary application is to give an easy criterion to recognize
quasiconvexity. A subgroup $H$ is relatively quasiconvex in $G$ if there is an
$H$-cocompact quasiconvex subgraph $\bar{L}\subset\bar{K}_{G}$ of the fine
hyperbolic $G$-graph. The tree-like nature of our graph $\bar{K}_{G}$, permits
us to naturally build the quasiconvex $H$-graph $\bar{L}$. When $H$ is
relatively quasiconvex, there are finitely many $H$-orbits of nontrivially
$H$-stabilized vertices in the Bass-Serre tree $T$, and each of these
stabilizers is relatively quasiconvex in its vertex group. Choosing finitely
many quasiconvex subgraphs in the corresponding copies of $K_{A}$ and $K_{B}$,
we are able to combine these together to form $L$ in $K_{G}$ and then to form
a quasiconvex $H$-subgraph $\bar{L}$ in $\bar{K}_{G}$.

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 $M$ be a compact irreducible 3-manifold. And let $M_{1},\ldots,M_{r}$
denote the graph manifolds obtained by removing each (open) hyperbolic piece
in the geometric decomposition of $M$. Then $\pi_{1}M$ is hyperbolic relative
to $\\{\pi_{1}M_{1},\ldots,\pi_{1}M_{r}\\}$.

As explained to us by the referee, the relative hyperbolicity of $\pi_{1}(M)$
was previously proved by Drutu-Sapir using work of Kapovich-Leeb. This
previous proof is deep as it uses the structure of the asymptotic cone due to
Kapovich-Leeb together with the technical proof of Drutu-Sapir 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 finite-volume
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 $\Gamma$ is fine if each
edge of $\Gamma$ lies in finitely many circuits of length $n$ for each $n$.

A group G is _hyperbolic relative to a finite collection of subgroups_
$\mathbb{P}$ if $G$ acts cocompactly(without inversions) on a connected, fine,
hyperbolic graph $\Gamma$ with finite edge stabilizers, such that each element
of $\mathbb{P}$ equals the stabilizer of a vertex of $\Gamma$, and moreover,
each infinite vertex stabilizer is conjugate to a unique element of
$\mathbb{P}$. We refer to a connected, fine, hyperbolic graph $\Gamma$
equipped with such an action as a _$(G;\mathbb{P})$ -graph_. Subgroups of G
that are conjugate into subgroups in $\mathbb{P}$ are _parabolic_.

######  Technical Remark 1.2.

Given a finite collection of parabolic subgroups $\\{A_{1},\dots,A_{r}\\}$, we
choose $\mathbb{P}$ so that there is a prescribed choice of parabolic subgroup
$P_{i}\in\mathbb{P}$ so that $A_{i}$ is _“declared”_ to be conjugate into
$P_{i}$. This is automatic for an infinite parabolic subgroup $A$ but for
finite subgroups there could be ambiguity. One way to resolve this is to
revise the choice of $\mathbb{P}$ as follows: For any finite collection of
parabolic subgroups $\\{A_{1},\dots,A_{r}\\}$ in $G$, we moreover assume each
$A_{i}$ is conjugate to a subgroup of $\mathbb{P}$ and we assume that no two
(finite) subgroups in $\mathbb{P}$ are conjugate. We note that finite
subgroups can be freely added to or omitted from the peripheral structure of
$G$ (see e.g. [16]).

######  Definition 1.3 (Parabolic tree).

Let $G$ split as a finite graph of groups where each vertex group $G_{\nu}$ is
hyperbolic relative to $\mathbb{P}_{\nu}$, and where each edge group $G_{e}$
embeds as a parabolic subgroup of its two vertex groups. Let $T$ be the Bass-
Serre tree. Define the _parabolic forest_ $F$ by:

  1. (1)

A _vertex_ in $F$ is a pair $(u,P)$ where $u\in T^{0}$ and $P$ is a
$G_{u}$-conjugate of an element of $\mathbb{P}_{u}$.

  2. (2)

An _edge_ in $F$ is a pair $(e,G_{e})$ where $e$ is an edge of $T$ and $G_{e}$
is its stabilizer.

  3. (3)

The edge $(e,G_{e})$ is _attached_ to $(\iota(e),\iota(P_{e}))$ and
$(\tau(e),\tau(P_{e}))$ where $\iota(e)$ and $\tau(e)$ are the initial and
terminal vertex of $e$ and $\iota(P_{e})$ is the $G_{\iota(e)}$-conjugate of
an element of $\mathbb{P}$ that is declared to contain $G_{e}$. Likewise for
$(\tau(e),\tau(P_{e}))$. We arranged for this unique determination in
Technical Remark 1.2.

Each component of $F$ is a _parabolic tree_ and the map $F\rightarrow T$ is
injective on the set of edges, and in particular each parabolic tree embeds in
$T$. Let $S_{1},\dots,S_{j}$ be representatives of the finitely many orbits of
parabolic trees under the $G$ action on $F$. Let $Q_{i}=stab(S_{i})$, for each
$i$.

######  Theorem 1.4 (Combining Relatively Hyperbolic Groups Along Parabolics).

Let $G$ split as a finite graph $\Gamma$ of groups. Suppose each vertex group
is relatively hyperbolic and each edge group is parabolic in its vertex
groups. Then $G$ is hyperbolic relative to
$\mathbb{Q}=\\{Q_{1},\dots,Q_{j}\\}$.

###### Proof.

For $u\in\Gamma^{0}$, let $G_{u}$ be hyperbolic relative to $\mathbb{P}_{u}$
and let $K_{u}$ be a $(G_{u};\mathbb{P}_{u})$-graph. For each
$P\in\mathbb{P}_{u}$, following the Technical Remark 1.2, we choose a specific
vertex of $K_{u}$ whose stabilizer equals $P$. Note that, in general there
could be more than one possible choice when $|P|<\infty$, but by Technical
Remark 1.2 we have a unique choice. Translating determines a “choice” of
vertex for conjugates.

We now construct a $(G;\mathbb{Q})$-graph $\bar{K}$. Let $K$ be the tree of
spaces whose underlying tree is the Bass-Serre tree $T$ with the following
properties:

  1. (1)

Vertex spaces of $K$ are copies of appropriate elements in
$\\{K_{u}:u\in\Gamma^{0}\\}$. Specifically, $K_{\nu}$ is a copy of $K_{u}$
where $u$ is the image of $\nu$ under $T\rightarrow\Gamma$.

  2. (2)

Each edge space $K_{e}$ is an ordinary edge, denoted as an ordered pair
$(e,G_{e})$ that is attached to the vertices in $K_{\iota(e)}$ and
$K_{\tau(e)}$ that were chosen to contain $G_{e}$.

Note that each $G_{\nu}$ acts on $K_{\nu}$ and there is a $G$-equivariant map
$K\rightarrow T$. Let $\bar{K}$ be the quotient of $K$ obtained by contracting
each edge space. Observe that $G$ acts on $\bar{K}$ and there is a
$G$-equivariant map $K\rightarrow\bar{K}$. Moreover the preimage of each open
edge of $\bar{K}$ is a single open edge of $K$.

We now show that $\bar{K}$ is a $(G;\mathbb{Q})$-graph. Since any embedded
cycle lies in some vertex space, the graph $\bar{K}$ is fine and hyperbolic.
There are finitely many orbits of vertices in $K$ and therefore finitely many
orbits of vertices in $\bar{K}$. Likewise, there are finitely many orbits of
edges in $\bar{K}$. The stabilizer of an (open) edge of $\bar{K}$ equals the
stabilizer of the corresponding (open) edge in $K$, and is thus finite. By
construction, there is a $G$-equivariant embedding $F\hookrightarrow K$ where
$F$ is the parabolic forest associated to $G$ and $T$. Finally, the preimage
in $K$ of a vertex of $\bar{K}$ is precisely a parabolic tree and thus the
stabilizer of a vertex of $\bar{K}$ is a conjugate of some $Q_{j}$. ∎

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 $G$ split as a finite directed graph of groups where each vertex group
$G_{\nu}$ is hyperbolic relative to $\mathbb{P}_{\nu}$. Suppose that:

  1. (1)

Each edge group is parabolic in its vertex groups.

  2. (2)

Each outgoing infinite edge group $G_{\vec{e}}$ is maximal parabolic in its
initial vertex group $G_{\nu}$ and for each other incoming and outgoing
infinite edge group $G_{{}\reflectbox{$\vec{\reflectbox{$e$}}$}}$ or
$G_{\vec{d}}$ or $G_{\reflectbox{$\vec{\reflectbox{$d$}}$}}$, none of its
conjugates lie in $G_{\vec{e}}$.

Then $G$ is hyperbolic relative to
$\mathbb{P}=\bigcup_{\nu}\mathbb{P}_{\nu}-\\{\text{outgoing edge groups}\\}$.

###### Proof.

We can arrange for finitely stabilized edges of $F$ to be attached to distinct
chosen vertices when they correspond to distinct edges of $T$. Thus, parabolic
trees are singletons and/or $i$-pods consisting of edges that all terminate at
the same vertex $\\{(\nu,P^{g})\\}$ where $P\in\mathbb{P}_{\nu}$ and $g\in
G_{\nu}$. Recall that an _$i$ -pod_ is a tree consisting of $i$ edges glued to
a central vertex. ∎

######  Corollary 1.6.

Let $G$ split as a finite graph of groups. Suppose each vertex group $G_{\nu}$
is hyperbolic relative to $\mathbb{P}_{\nu}$. For each $G_{\nu}$ assume that
the collection $\\{G_{e}:e\textrm{ is attached to }\nu\\}$ is a collection of
maximal parabolic subgroups of $G_{\nu}$. Then $G$ is hyperbolic relative to
$\mathbb{P}=\bigcup_{\nu}\mathbb{P}_{\nu}-\\{\text{repeats}\\}$. Specifically,
we remove an element of $\bigcup_{\nu}\mathbb{P}_{\nu}$ if it is conjugate to
another one.

The first two of the following cases were treated by Dahmani, Alibegović, and
Osin [1, 4, 19]:

######  Corollary 1.7.

  1. (1)

Let $G_{1}$ and $G_{2}$ be hyperbolic relative to $\mathbb{P}_{1}$ and
$\mathbb{P}_{2}$. Let $G=G_{1}\ast_{P_{1}={P_{2}^{\prime}}}G_{2}$ where each
$P_{i}\in\mathbb{P}_{i}$ and $P_{1}$ is identified with the subgroup
${P_{2}}^{\prime}$ of $P_{2}$. Then $G$ is hyperbolic relative
$\mathbb{P}_{1}\cup\mathbb{P}_{2}-\\{P_{1}\\}$.

  2. (2)

Let $G_{1}$ be hyperbolic relative to $\mathbb{P}$. Let $P_{1}\in\mathbb{P}$
be isomorphic to a subgroup ${P_{2}}^{\prime}$ of a maximal parabolic subgroup
$P_{2}$ not conjugate to $P_{1}$. Let
$G=G_{1}*_{{P_{1}}^{t}={P_{2}}^{\prime}}$ where ${P_{1}}^{t}=t^{-1}P_{1}t$.
Then $G$ is hyperbolic relative to $\mathbb{P}-\\{P_{1}\\}$.

  3. (3)

Let $G_{1}$ be hyperbolic relative to $\mathbb{P}$. Let $P\in\mathbb{P}$ be
isomorphic to ${P}^{\prime}\leq P$. Let $G=G_{1}*_{{P}^{t}={P}^{\prime}}$.
Then $G$ is hyperbolic relative to $\mathbb{P}\cup\langle P,t\rangle-\\{P\\}$.

######  Remark 1.8.

Note that in this Corollary and some similar results when we say
$P_{i}\in\mathbb{P}_{i}$, we mean if $P^{g}_{i}\in\mathbb{P}_{i}$ then replace
$P^{g}_{i}$ by $P_{i}$ in $\mathbb{P}_{i}$.

###### Proof.

(1): In this case, the parabolic trees are either singletons stabilized by a
conjugate of an element of $\mathbb{P}_{1}\cup\mathbb{P}_{2}-\\{P_{1}\\}$, or
parabolic trees are $i$-pods stabilized by conjugates of $P_{2}$.

(2): The proof is similar.

(3): All parabolic trees are singletons except for those that are translates
of a copy of the Bass-Serre tree for $P*_{P^{t}=P^{\prime}}$. Following the
proof of Theorem 1.4, let $\nu\in\bar{K}$, if the preimage of $\nu$ in $K$ is
not attached to an edge space, then $G_{\nu}$ is conjugate to an element of
$\mathbb{P}-\\{P\\}$, otherwise $G_{\nu}$ is conjugate to $\langle
P,t\rangle$. ∎

######  Example 1.9.

We encourage the reader to consider the case of Theorem 1.4 and Corollaries
1.6 and 1.7, in the scenario where $G$ splits as a graph of free groups with
cyclic edge groups. A very simple case is: Let $G=\langle
a,b,t~{}|~{}(W^{n})^{t}=W^{m}\rangle$ where $W\in\langle a,b\rangle$ and
$m,n\geq 1$. Then $G$ is hyperbolic relative to $\langle W,t\rangle$.

##  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].
Martinez-Pedroza 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 $G$ be hyperbolic relative to $\mathbb{P}$. A subgroup $H$ of $G$ is
_quasiconvex relative to $\mathbb{P}$_ if for some (and hence any)
$(G;\mathbb{P})$-graph $K$, there is a nonempty connected and quasi-
isometrically embedded, $H$-cocompact subgraph $L$ of $K$. In the sequel, we
sometimes refer to $L$ as a _quasiconvex $H$-cocompact subgraph_ of $K$.

######  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 $H$ is hyperbolic relative to the collection
$\mathbb{P}_{H}$ consisting of representatives of $H$-stabilizers of vertices
of $L\subseteq K$. 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 $G$ is f.g. in [15], and when $G$ 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 $G$ be hyperbolic relative to $\mathbb{P}_{G}$. Suppose that $B$ is
relatively quasiconvex in $G$, and note that $B$ is then hyperbolic relative
to $\mathbb{P}_{B}$ as in Remark 2.2. Then $A\leq B$ is quasiconvex relative
to $\mathbb{P}_{B}$ if and only if $A$ is quasiconvex relative to
$\mathbb{P}_{G}$.

###### Proof.

Let $K$ be a $(G;\mathbb{P}_{G})$-graph. As $B$ is quasiconvex relative to
$\mathbb{P}_{G}$, there is a $B$-cocompact and quasiconvex subgraph $L\subset
K$. Note that $L$ is a $(B;\mathbb{P}_{B})$-graph. Let $A\leq B$.

If $A$ is quasiconvex in $B$ relative to $\mathbb{P}_{B}$, there is an
$A$-cocompact quasiconvex subgraph $M\subset L$. Since the composition
$L_{A}\rightarrow L_{B}\rightarrow K$ is a quasi-isometric embedding, $A$ is
quasiconvex relative to $\mathbb{P}_{G}$. Conversely, if $A$ is quasiconvex in
$G$ relative to $\mathbb{P}_{G}$, then there is an $A$-cocompact quasiconvex
subgraph $M\subset K$. Let $L^{\prime}=L\cup BM$ and note that $L^{\prime}$ is
$B$-cocompact and hence also quasiconvex, and thus $L^{\prime}$ also serves as
a fine hyperbolic graph for $B$. Now $M\subset L^{\prime}$ is quasiconvex
since $M\subset L$ is quasiconvex so $A$ is relatively quasiconvex in $B$. ∎

######  Remark 2.4.

One consequence of Theorem 1.4 and its various Corollaries, is that when $G$
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 $G$. (For Theorem 1.4 this is $\mathbb{Q}$, and for Corollary 1.6
this is $\mathbb{P}-\\{\text{repeats}\\}$.) Indeed, $K_{v}$ is a
$G_{v}$-cocompact quasiconvex subgraph in the fine graph $K$ constructed in
the proof.

######  Lemma 2.5.

Let $G$ be a f.g. group that split as a finite graph of groups $\Gamma$. If
each edge group is f.g. then each vertex group is f.g.

###### Proof.

Let $G=\langle g_{1},\dots,g_{n}\rangle$. We regard $G$ as $\pi_{1}$ of a
$2$-complex corresponding to $\Gamma$. We show that each vertex group $G_{v}$
equals $\langle\\{G_{e}\\}_{\text{$e$ attached to~{}$v$}}\cup\\{g\in
G_{v}:g~{}\text{in normal form of some}~{}g_{i}\\}\rangle$. Let $a\in G_{v}$
and consider an expression of $a$ as a product of normal forms of the
$g_{i}^{\pm 1}$. Then $a$ equals some product
$a_{1}{t_{1}}^{\epsilon_{1}}b_{1}{t_{2}}^{\epsilon_{2}}a_{2}\cdots
a_{n}{t_{m}}^{\epsilon_{m}}b_{k}$. There is a disc diagram $D$ whose boundary
path is
$a^{-1}a_{1}{t_{1}}^{\epsilon_{1}}b_{1}{t_{2}}^{\epsilon_{2}}a_{2}\cdots
a_{n}{t_{m}}^{\epsilon_{m}}b_{k}$. See Figure 2. The region of $D$ that lies
along $a$ shows that $a$ equals the product of elements in edge groups
adjacent to $G_{v}$, together with elements of $G_{v}$ that lie in the normal
forms of $g_{1},\dots,g_{n}$. ∎

Figure 2.

######  Theorem 2.6 (Quasiconvexity of a Subgroup in Parabolic Splitting).

Let $G$ split as a finite graph $\Gamma$ of relatively hyperbolic groups such
that each edge group is parabolic in its vertex groups. (Note that $G$ is
hyperbolic relative to $\mathbb{Q}=\\{Q_{1},\dots,Q_{j}\\}$ by Theorem 1.4.)
Let $H\leq G$ be tamely generated. Then $H$ is quasiconvex relative to
$\mathbb{Q}$. Moreover if each $H_{v}$ in the Bass-Serre tree $T$ is finitely
generated then $H$ is finitely generated.

###### Proof.

Since there are finitely many orbits of vertices whose stabilizers are
finitely generated, $H$ is finitely generated. For each $u\in\Gamma^{0}$, let
$G_{u}$ be hyperbolic relative to $\mathbb{P}_{u}$ and let $K_{u}$ be a
$(G_{u};\mathbb{P}_{u})$-graph. Let $K$ be the $(G;\mathbb{Q})$-graph
constructed in the proof of Theorem 1.4 and let $\bar{K}$ be its quotient. We
will construct an $H$-cocompact quasiconvex, connected subgraph $\bar{L}$ of
$\bar{K}$.

Let $T_{H}$ be the minimal $H$-invariant subgraph of $T$. Recall that each
edge of $T$ (and hence $T_{H}$) corresponds to an edge of $K$. Let $F_{H}$
denote the subgraph of $K$ that is the union of all edges correspond to edges
of $T_{H}$. Let $\\{\nu_{1},\dots,\nu_{n}\\}$ be a representatives of
$H$-orbits of vertices of $T_{H}$. For each $i$, let $L_{i}\hookrightarrow
K_{\nu_{i}}$ be a $(H\cap G_{\nu_{i}}^{g_{i}})$-cocompact quasiconvex subgraph
such that $L_{i}$ contains $F_{H}\cap K_{\nu_{i}}$. (There are finitely many
$(H\cap G_{\nu_{i}}^{g_{i}})$-orbits of such endpoints of edges in
$K_{\nu_{i}}$.) Let $L=F_{H}\cup\bigcup_{i=1}^{n}HL_{i}$ and let $\bar{L}$ be
the image of $L$ under $K\rightarrow\bar{K}$. Observe that $L$ is quasiconvex
in $K$ since $K$ is a “tree union” and each such $L_{i}$ of $L$ is quasiconvex
in $K_{\nu_{i}}$. And likewise, $\bar{L}$ is quasiconvex in $\bar{K}$. ∎

######  Corollary 2.7 (Characterizing Quasiconvexity in Maximal Parabolic
Splitting).

Let $G$ split as a finite graph of countable groups. For each $\nu$, let
$G_{\nu}$ be hyperbolic relative to $\mathbb{P}_{\nu}$ and let the collection
$\\{G_{e}:e\textrm{ is attached to }\nu\\}$ be a collection of maximal
parabolic subgroups of $G_{\nu}$. $($Note that $G$ is hyperbolic relative to
$\mathbb{P}=\bigcup_{\nu}\mathbb{P}_{\nu}-\\{\text{repeats}\\}$ by Corollary
1.6.$)$ Let $T$ be the Bass-Serre tree and let $H$ be a subgroup of $G$. The
following are equivalent:

  1. (1)

$H$ is tamely generated and each $H_{v}$ in the Bass-Serre tree $T$ is f.g.

  2. (2)

$H$ is f.g. and quasiconvex relative to $\mathbb{P}$.

###### Proof.

(1 $\Rightarrow$ 2): Follows from Theorem 1.4 and Theorem 2.6.

(2 $\Rightarrow$ 1): Since $H$ is f.g., the minimal $H$-subtree $T_{H}$ is
$H$-cocompact, and so $H$ splits as a finite graph of groups $\Gamma_{H}$.
Since $H$ is quasiconvex in $\mathbb{P}$, it is hyperbolic relative to
intersections with conjugates of $\mathbb{P}$. In particular, the infinite
edge groups in the induced splitting of $H$ 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 $\Gamma_{H}$ is f.g. by
Lemma 2.5.

By Remark 2.4, each vertex group of $G$ is quasiconvex relative to
$\mathbb{P}$, and hence each $G_{\nu}$ is relatively quasiconvex by Remark 2.2
since it is a conjugate of a vertex group. Thus $H_{\nu}=H\cap G_{\nu}$ is
quasiconvex relative to $\mathbb{P}$ by Remark 2.2. Finally, $H_{\nu}$ is
quasiconvex in $G_{\nu}$ by Lemma 2.3. ∎

##  3\. Local Relative Quasiconvexity

A relatively hyperbolic group $G$ is _locally relatively quasiconvex_ if each
f.g. subgroup of $G$ 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 $N$ is _Noetherian_ if each subgroup of $N$ 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).

  1. (1)

Let $G_{1}$ and $G_{2}$ be locally relatively quasiconvex relative to
$\mathbb{P}_{1}$ and $\mathbb{P}_{2}$. Let
$G=G_{1}\ast_{P_{1}={P_{2}^{\prime}}}G_{2}$ where each
$P_{i}\in\mathbb{P}_{i}$ and $P_{1}$ is identified with the subgroup
${P_{2}}^{\prime}$ of $P_{2}$. Suppose $P_{1}$ is Noetherian. Then $G$ is
locally quasiconvex relative to
$\mathbb{P}_{1}\cup\mathbb{P}_{2}-\\{P_{1}\\}$.

  2. (2)

Let $G_{1}$ be a locally relatively quasiconvex relative to $\mathbb{P}$. Let
$P_{1}\in\mathbb{P}$ be isomorphic to a subgroup ${P_{2}}^{\prime}$ of a
maximal parabolic subgroup $P_{2}$ not conjugate to $P_{1}$. Let
$G=G_{1}*_{{P_{1}}^{t}={P_{2}}^{\prime}}$. Suppose $P_{1}$ is Noetherian. Then
$G$ is locally quasiconvex relative to $\mathbb{P}-\\{P_{1}\\}$.

  3. (3)

Let $G_{1}$ be a locally quasiconvex relative to $\mathbb{P}$. Let $P$ be a
maximal parabolic subgroup of $G_{1}$, isomorphic to ${P}^{\prime}\leq P$. Let
$G=G_{1}*_{{P}^{t}={P}^{\prime}}$ and suppose $P$ is Noetherian. Then $G$ is
also locally quasiconvex relative to $\mathbb{P}\cup\langle
P,t\rangle-\\{P\\}$.

###### Proof.

(1): By Corollary 1.7, $G$ is hyperbolic relative to
$\mathbb{P}=\mathbb{P}_{1}\cup\mathbb{P}_{2}-\\{P_{1}\\}$. Let $H$ be a
finitely generated subgroup of $G$. We show that $H$ is quasiconvex relative
to $\mathbb{P}$. Let $T$ be the Bass-Serre tree of $G$. Since $H$ is f.g., the
minimal $H$-subtree $T_{H}$ is $H$-cocompact, and so $H$ splits as a finite
graph of groups $\Gamma_{H}$. Moreover, the edge groups of this splitting are
f.g. since the edge groups of $G$ are Noetherian by hypothesis. Thus each
vertex group of $\Gamma_{H}$ is f.g. by Lemma 2.5. Since $G_{1}$ and $G_{2}$
are locally relatively quasiconvex, each vertex group of $T_{H}$ is relatively
quasiconvex in its “image vertex group” under the map $T_{H}\rightarrow T$.
Now by Theorem 2.6, $H$ is quasiconvex relative to $\mathbb{P}$. The proof of
(2) and (3) are similar. ∎

######  Definition 3.2 (Almost Malnormal).

A subgroup $H$ is _malnormal_ in $G$ if $H\cap H^{g}=\\{1\\}$ for $g\notin H$,
and similarly $H$ is _almost malnormal_ if this intersection $H\cap H^{g}$ is
always finite. Likewise, a collection of subgroups $\\{H_{i}\\}$ is _almost
malnormal_ if $H_{i}^{g}\cap H_{j}^{h}$ is finite unless $i=j$ and $gh^{-1}\in
H_{i}$.

######  Corollary 3.3.

Let $G$ split as a finite graph of groups. Suppose

  * a)

Each $G_{\nu}$ is locally relatively quasiconvex;

  * b)

Each $G_{e}$ is Noetherian and maximal parabolic in its vertex groups;

  * c)

$\\{G_{e}:e\textrm{ is attached to }\nu\\}$ is almost malnormal in $G_{\nu}$,
for any vertex $\nu$.

Then $G$ is locally relatively quasiconvex relative to $\mathbb{P}$, see
Corollary 1.6.

###  3.1. Small-hierarchies 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 (Small-Hierarchy).

A group is _small_ if it has no rank 2 free subgroup. Any small group has a
length 0 small-hierarchy. $G$ has a length $n$ small-hierarchy if $G\cong
A*_{C}B$ or $G\cong A*_{C^{t}=C^{\prime}}$, where $A$ and $B$ have length
$(n-1)$ small-hierarchies, and $C$ is small and f.g. We say $G$ has a _small-
hierarchy_ if it has a length $n$ small-hierarchy for some $n$.

We can define $\mathcal{F}$-hierarchy by replacing “small” by a class of
groups $\mathcal{F}$ closed under subgroups and isomorphisms. For instance,
when $\mathcal{F}$ is the class of finite groups, the class of groups with an
$\mathcal{F}$-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 $F_{2}$. 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 $H$ of a f.g. relatively
hyperbolic group $G$ is relatively quasiconvex.

######  Theorem 3.6.

Let $G$ be f.g. and hyperbolic relative to $\mathbb{P}$ where each element of
$\mathbb{P}$ is Noetherian. Suppose $G$ has a small-hierarchy. Then $G$ 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
$G=A*_{C_{1}}B$ or $G=A*_{C_{1}^{t}=C_{2}}$.

Since $C_{1}$ and $G$ are f.g. the vertex groups are f.g. by Lemma 2.5. Thus,
since $C_{1}$ is relatively quasiconvex the vertex groups are relatively
quasiconvex by Lemma 4.9.

When $C_{1}$ is finite the conclusion follows in each subcase from Theorem
3.1.

When $C_{1}$ is virtually cyclic but not parabolic, then $C_{1}$ lies in a
unique maximal virtually cyclic subgroup $Z$ that is almost malnormal and
relatively quasiconvex by [18]. Thus $G$ is hyperbolic relative to
$\mathbb{P}^{\prime}=\mathbb{P}\cup\\{Z\\}$ by Theorem 4.2.

Observe that $C_{1}$ is maximal infinite cyclic on at least one side, since
otherwise there would be a nontrivial splitting of $Z$ as an amalgamated free
product over $C_{1}$. We equip the (relatively quasiconvex) vertex groups with
their induced peripheral structures. Note that $C_{1}$ is maximal parabolic on
at least one side and so $G$ is locally relatively quasiconvex relative to
$\mathbb{P}^{\prime}$ by Theorem 3.1. Finally, by Theorem 4.7, any subgroup
$H$ is quasiconvex relative to the original peripheral structure $\mathbb{P}$
since intersections between $H$ and conjugates of $Z$ are quasiconvex relative
to $\mathbb{P}$.

When $C_{1}$ is infinite parabolic, we will first produce a new splitting
before verifying local relative quasiconvexity.

When $G=A*_{C_{1}}B$. Let $D_{a},D_{b}$ be the maximal parabolic subgroups of
$A,B$ containing $C_{1}$, and refine the splitting to:

$A*_{D_{a}}(D_{a}*_{C_{1}}D_{b})*_{D_{b}}B$

The two outer splittings are along a parabolic that is maximal on the outside
vertex group. The inner vertex group $D_{a}*_{C_{1}}D_{b}$ is a single
parabolic subgroup of $G$. Indeed, as $C_{1}$ is infinite,
$D_{a}\supset{C_{1}}\subset D_{b}$ must all lie in the same parabolic subgroup
of $G$. It is obvious that $D_{a}*_{C_{1}}D_{b}$ is locally relatively
quasiconvex with respect to its induced peripheral structure since it is
itself parabolic in $G$. Consequently $(D_{a}*_{C_{1}}D_{b})*_{D_{b}}B$ is
locally relatively quasiconvex by Theorem 3.1, therefore
$G=A*_{D_{a}}\big{(}(D_{a}*_{C_{1}}D_{b})*_{D_{b}}B\big{)}$ is locally
relatively quasiconvex by Theorem 3.1.

When $G\cong A*_{{C_{1}}^{t}=C_{2}}$, let $M_{i}$ be the maximal parabolic
subgroup of $G$ containing $C_{i}$. There are two subsubcases:

[$t\in M_{1}$] Then $C_{2}\leq M_{1}$ and we revise the splitting to $G\cong
A*_{D_{1}}M_{1}$ where $D_{1}=M_{1}\cap A$. And in this splitting the edge
group is maximal parabolic at $D_{1}\subset A$, and $M_{1}$ is parabolic.

[$t\notin M_{1}$] Let $D_{i}$ denote the maximal parabolic subgroup of $A$
containing $C_{i}$. Observe that $\\{D_{1},D_{2}\\}$ is almost malnormal since
$D_{i}=M_{i}\cap A$. We revise the HNN extension to the following:

$\bigg{(}D_{1}^{t}*_{C_{1}^{t}=C_{2}}A\bigg{)}*_{{D_{1}}^{t}=D_{1}}$

where the conjugated copies of $D_{1}$ in the HNN extension embed in the first
and second factor of the AFP.

In both cases, the local relative quasiconvexity of $G$ 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 $G$ is
hyperbolic relative to an almost malnormal quasiconvex subgroup. Generalizing
work of Martinez-Pedroza [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 $\mathbb{P}$ of
subgroups of a group $G$. Each element $P\in\mathbb{P}$ is a peripheral
subgroup of $G$. The peripheral structure $\mathbb{E}=\\{E_{j}\\}_{j\in J}$
_extends_ $\mathbb{P}=\\{P_{i}\\}_{i\in I}$ if for each $i\in I$, there exists
$j\in J$ such that $P_{i}\subseteq E_{j}$. For $E\in\mathbb{E}$, we let
$\mathbb{P}_{E}=\\{P_{i}\,:\,P_{i}\subseteq E,P_{i}\in\mathbb{P},i\in I\\}$.

We will use the following result of Yang [22].

######  Theorem 4.2 (Hyperbolicity of Extended Peripheral Structure).

Let $G$ be hyperbolic relative to $\mathbb{P}$ and let the peripheral
structure $\mathbb{E}$ extend $\mathbb{P}$. Then $G$ is hyperbolic relative to
$\mathbb{E}$ if and only if the following hold:

  1. (1)

$\mathbb{E}$ is almost malnormal;

  2. (2)

Each $E\in\mathbb{E}$ is quasiconvex in $G$ relative to $\mathbb{P}$.

######  Definition 4.3 (Total).

Let $G$ be hyperbolic relative to $\mathbb{P}$. The subgroup $H$ of $G$ is
total relative to $\mathbb{P}$ if: either $H\cap P^{g}=P^{g}$ or $H\cap P^{g}$
is finite for each $P\in\mathbb{P}$ and $g\in G$.

The following is proven in [5]:

######  Lemma 4.4.

If $G$ is f.g. and hyperbolic relative to $\mathbb{P}=\\{P_{1},\dots,P_{n}\\}$
and each $P_{i}$ is hyperbolic relative to
$\mathbb{H}_{i}=\\{H_{i1},\dots,H_{i{m_{i}}}\\}$, then $G$ is hyperbolic
relative to $\bigcup_{1\leq i\leq n}\mathbb{H}_{i}$.

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).

  1. (1)

Let $G_{i}$ be hyperbolic relative to $\mathbb{P}_{i}$ for $i=1,2$. Let
$C_{i}\leq G_{i}$ be almost malnormal, total and relatively quasiconvex. Let
${C_{1}}^{\prime}\leq C_{1}$. Then $G=G_{1}*_{{C_{1}}^{\prime}=C_{2}}G_{2}$ is
hyperbolic relative to
$\mathbb{P}=\mathbb{P}_{1}\cup\mathbb{P}_{2}-\\{P_{2}\in\mathbb{P}_{2}~{}:~{}P^{g}_{2}\subseteq
C_{2},~{}\text{for some}~{}g\in G_{2}\\}$.

  2. (2)

Let $G_{1}$ be hyperbolic relative to $\mathbb{P}$. Let $\\{C_{1},C_{2}\\}$ be
almost malnormal and assume each $C_{i}$ is total and relatively quasiconvex.
Let ${C_{1}}^{\prime}\leq C_{1}$. Then
$G=G_{1}*_{{{C_{1}}^{\prime}={C_{2}}^{t}}}$ is hyperbolic relative to
$\mathbb{P}=\mathbb{P}-\\{P_{2}\in\mathbb{P}_{2}~{}:~{}P^{g}_{2}\subseteq
C_{2},~{}\text{for some}~{}g\in G_{2}\\}$.

###### Proof.

(1): For each $i$, let

$\mathbb{E}_{i}=\mathbb{P}_{i}-\\{P\in\mathbb{P}_{i}\colon
P^{g}\leq{C_{i}},~{}\text{for some}~{}g\in G_{i}\\}\cup\\{C_{i}\\}$

Without loss of generality, we can assume that $\mathbb{E}_{i}$ extends
$\mathbb{P}_{i}$, since we can replace an element of $\mathbb{P}_{i}$ by its
conjugate. We now show that $G_{i}$ is hyperbolic relative to $\mathbb{E}_{i}$
by verifying the two conditions of Theorem 4.2: $\mathbb{E}_{i}$ is malnormal
in $G_{i}$, since $\mathbb{P}_{i}$ is almost malnormal and $C_{i}$ is total
and almost malnormal. Each element of $\mathbb{E}_{i}$ is relatively
quasiconvex, since $C_{i}$ is relatively quasiconvex by hypothesis and each
element of $\mathbb{P}_{i}$ is relatively quasiconvex by Remark 2.2.

We now regard each $G_{i}$ as hyperbolic relative to $\mathbb{E}_{i}$.
Therefore since the edge group $C_{2}={C_{1}}^{\prime}$ is maximal on one
side, by Corollary 1.7, $G$ is hyperbolic relative to
$\mathbb{E}=\mathbb{E}_{1}\cup\mathbb{E}_{2}-\\{C_{2}\\}$.

We now apply Lemma 4.4 to show that $G$ is hyperbolic relative to
$\mathbb{P}$. We showed that $G$ is hyperbolic relative to $\mathbb{E}$. But
each element of $\mathbb{E}$ is hyperbolic relative to $\mathbb{P}$ that it
contains. Thus by Lemma 4.4, we obtain the result.

(2): The proof is analogous to the proof of $(1)$. ∎

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 $G$ split as a finite graph of groups. Suppose

  * (a)

Each $G_{\nu}$ is hyperbolic relative to $\mathbb{P}_{\nu}$;

  * (b)

Each $G_{e}$ is total and relatively quasiconvex in $G_{\nu}$;

  * (c)

$\\{G_{e}:e~{}\textrm{is attached to }\nu\\}$ is almost malnormal in $G_{\nu}$
for each vertex $\nu$.

Then $G$ is hyperbolic relative to
$\bigcup_{\nu}\mathbb{P}_{\nu}-\\{\text{repeats}\\}$.

Yang characterized relative quasiconvexity with respect to extensions in [22]
as follows:

######  Theorem 4.7 (Quasiconvexity in Extended Peripheral Structure).

Let $G$ be hyperbolic relative to $\mathbb{P}$ and relative to $\mathbb{E}$.
Suppose that $\mathbb{E}$ extends $\mathbb{P}$. Then

  1. (1)

If $H\leq G$ is quasiconvex relative to $\mathbb{P}$, then $H$ is quasiconvex
relative to $\mathbb{E}$.

  2. (2)

Conversely, if $H\leq G$ is quasiconvex relative to $\mathbb{E}$, then $H$ is
quasiconvex relative to $\mathbb{P}$ if and only if $H\cap{E}^{g}$ is
quasiconvex relative to $\mathbb{P}$ for all $g\in G$ and $E\in\mathbb{E}$.

We recall the following observation of Bowditch (see [16, Lem 2.7 and 2.9]).

######  Lemma 4.8 ($G$-attachment).

Let $G$ act on a graph $K$. Let $p,q\in K^{0}$ and $e$ be a new edge whose
endpoints are $p$ and $q$. The _$G$ -attachment_ of $e$ is the new graph
$K^{\prime}=K\cup Ge$ which consists of the union of $K$ and copies $ge$ of
$e$ attached at $gp$ and $gq$ for any $g\in G$. Note that $K^{\prime}$ is
$G$-cocompact/fine/hyperbolic if $K$ is.

In the following lemma, we prove that when a relatively hyperbolic group $G$
splits then relative quasiconvexity of vertex groups is equivalent to relative
quasiconvexity of the edge groups.

######  Lemma 4.9 (Quasiconvex Edges $\Longleftrightarrow$ Quasiconvex
Vertices).

Let $G$ be hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as a finite
graph of groups whose vertex groups and edge groups are finitely generated.
Then the edge groups are quasiconvex relative to $\mathbb{P}$ if and only if
the vertex groups are quasiconvex relative to $\mathbb{P}$.

###### Proof.

If the vertex groups are quasiconvex relative to $\mathbb{P}$ then so are the
edge groups, since relative quasiconvexity is preserved by intersection (see
[10, 15]) in the f.g. group $G$. Assume the edge groups are quasiconvex
relative to $\mathbb{P}$. Let $K$ be a $(G;\mathbb{P})$ graph and let $T$ be
the Bass-Serre tree for $G$. Let $f\colon K\rightarrow T$ be a $G$-equivariant
map that sends vertices to vertices and edges to geodesics. Subdivide $K$ and
$T$, so that each edge is the union of two length $\frac{1}{2}$  _halfedges_.
Let $\nu$ be a vertex in $T$. It suffices to find a $G_{\nu}$-cocompact
quasiconvex subgraph $L$ of $K$.

Let $\\{e_{1},\dots,e_{m}\\}$ be representatives of the $G_{\nu}$-orbits of
halfedges attached to $\nu$. Let $\omega_{i}$ be the other vertex of $e_{i}$
for $1\leq i\leq m$. Since each $G_{\omega_{i}}=G_{e_{i}}$ is f.g. by
hypothesis, we can perform finitely many $G_{\omega_{i}}$-attachments of arcs
so that the preimage of $\omega_{i}$ is connected for each $i$. This leads to
finitely many $G$-attachments to $K$ to obtain a new fine hyperbolic graph
$K^{\prime}$. By mapping the newly attached edges to their associated vertices
in $T$, we thus obtain a $G$-equivariant map $f^{\prime}\colon
K^{\prime}\rightarrow T$ such that $M^{\prime}_{i}=f^{\prime-1}(\omega_{i})$
is connected and $G_{\omega_{i}}$-cocompact for each $i$.

Consider $L^{\prime}=f^{\prime-1}(N_{\frac{1}{2}}(\nu))$ where
$N_{\frac{1}{2}}(\nu)$ is the closed $\frac{1}{2}$-neighborhood of $\nu$. To
see that $L^{\prime}$ is connected, consider a path $\sigma$ in $K^{\prime}$
between distinct components of $L^{\prime}$. Moreover choose $\sigma$ so that
its image in $T$ is minimal among all such choices. Then $\sigma$ must leave
and enter $L^{\prime}$ through the same $g_{\nu}M^{\prime}_{i}$ which is
connected by construction.

We now show that $L^{\prime}$ is quasiconvex. Consider a geodesic $\gamma$
that intersects $L^{\prime}$ exactly at its endpoints. As before the endpoints
of $\gamma$ lie in the same $g_{\nu}M^{\prime}_{i}$. Since
$g_{\nu}M^{\prime}_{i}$ is $\kappa_{i}$-quasiconvex for some $\kappa_{i}$, we
see that $\gamma$ lies in $\kappa$-neighborhood of $g_{\nu}M^{\prime}_{i}$ and
hence in the $\kappa$-neighborhood of $L^{\prime}$. ∎

######  Lemma 4.10 (Total Edges $\Longleftrightarrow$ Total Vertices).

Let $G$ be hyperbolic relative to $\mathbb{P}$. Let $G$ act on a tree $T$. For
each $P\in\mathbb{P}$ let $T_{P}$ be a minimal $P$-subtree. Assume that no
$T_{P}$ has a finite edge stabilizer in the $P$-action. Then edge groups of
$T$ are total in $G$ iff vertex groups are total in $G$.

###### 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 $G_{\nu}$ be a vertex group and $P\in\mathbb{P}$ such
that $P^{g}\cap G_{\nu}$ is infinite for some $g\in G$. If $|P^{g}\cap
G_{e}|=\infty$ for some edge $e$ attached to $\nu$, then $P\subseteq G_{e}$,
thus $P\subseteq G_{e}\subseteq G_{\nu}$. Now suppose that $|P^{g}\cap
G_{e}|<\infty$ for each $e$ attached to $\nu$. If $P^{g}\nleq G_{\nu}$ then
the action of $P^{g}$ on $gT$ violates our hypothesis. ∎

######  Remark 4.11.

Suppose $G$ is f.g. and $G$ is hyperbolic relative to $\mathbb{P}$. Let
$P\in\mathbb{P}$ such that $P=A*_{C}B$ [$P=A*_{C={C^{\prime}}^{t}}$] where $C$
is a finite group. Since $P$ is hyperbolic relative to $\\{A,B\\}$
[$\\{A\\}$], by Lemma 4.4, $G$ is hyperbolic relative to
$\mathbb{P}^{\prime}=\mathbb{P}-\\{P\\}\cup\\{A,B\\}$
[$\mathbb{P}^{\prime}=\mathbb{P}-\\{P\\}\cup\\{A\\}$].

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 $G$ be f.g. and hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as
a finite graph of groups. Suppose

  1. (a)

Each $G_{e}$ is total in $G$;

  2. (b)

Each $G_{e}$ is relatively quasiconvex in $G$;

  3. (c)

$\\{G_{e}:e\textrm{ is attached to }\nu\\}$ is almost malnormal in $G_{\nu}$
for each vertex $\nu$.

Let $H\leq G$ be tamely generated subgroup of $G$. Then $H$ is relatively
quasiconvex in $G$.

###### Proof.

_Technical Point:_ By splitting certain elements of $\mathbb{P}$ to obtain
$\mathbb{P}^{\prime}$ as in Remark 4.11, we can assume that $G$ is hyperbolic
relative to $\mathbb{P}^{\prime}$ and each $G_{\nu}$ is hyperbolic relative to
the conjugates of elements of $\mathbb{P}^{\prime}$ that it contains.

Indeed for any $P\in\mathbb{P}$, if the action of $P$ on a minimal subtree
$T_{P}$ of the Bass-Serre tree $T$, yields a finite graph $\Gamma$ of groups
some of whose edge groups are finite, then following Remark 4.11, we can
replace $\mathbb{P}$ by the groups that complement these finite edge groups,
(i.e. the fundamental groups of the subgraphs obtained by deleting these edges
from $\Gamma$.) Therefore $G$ is hyperbolic relative to $\mathbb{P}^{\prime}$.

No $P\in\mathbb{P}^{\prime}$ has a nontrivial induced splitting as a graph of
groups with a finite edge group.The edge groups are total relative to
$\mathbb{P}^{\prime}$ since they are total relative to $\mathbb{P}$. Therefore
by Lemma 4.10 the vertex groups are total in $G$ relative to
$\mathbb{P}^{\prime}$. By Lemma 4.9, each vertex group $G_{\nu}$ is relatively
quasiconvex in $G$ relative to $\mathbb{P}$, therefore by Theorem 4.7 each
$G_{\nu}$ is quasiconvex in $G$ relative to $\mathbb{P}^{\prime}$. Thus
$G_{\nu}$ has an induced relatively hyperbolic structure
$\mathbb{P}^{\prime}_{\nu}$ as in Remark 2.2. By totality of $G_{\nu}$, we can
assume each element of $\mathbb{P}^{\prime}_{\nu}$ is a conjugate of an
element of $\mathbb{P}^{\prime}$. And as usual we may omit the finite
subgroups in $\mathbb{P}^{\prime}_{\nu}$.

_Step 1:_ We now extend the peripheral structure of each $G_{\nu}$ from
$\mathbb{P}^{\prime}_{\nu}$ to $\mathbb{E}_{\nu}$ where

$\mathbb{E}_{\nu}=\\{G_{e}:e\textrm{ is attached to
}\nu\\}\cup\\{P\in\mathbb{P}^{\prime}_{\nu}:P^{g}\nleq G_{e}\textrm{ for any
$g\in G_{\nu}$}\\}$

Almost malnormality of $\mathbb{E}_{\nu}$ 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 $G$, also relative quasiconvexity of the new
elements $G_{e}$ is Condition (b). Thus by $G_{\nu}$ is hyperbolic relative to
$\mathbb{E}_{\nu}$ by Theorem 4.7.

_Step 2:_ For each $\tilde{\nu}$ in the Bass-Serre tree, its $H$-stabilizer
$H_{\tilde{\nu}}$ lies in $G_{\tilde{\nu}}$ which we identify (by a conjugacy
isomorphism) with the chosen vertex stabilizer $G_{\nu}$ in the graph of group
decomposition. Then $H_{\tilde{\nu}}$ is quasiconvex in $G_{\nu}$ relative to
$\mathbb{E}_{\nu}$ for each $\nu$ by Theorem 4.7, since $\mathbb{E}_{\nu}$
extends $\mathbb{P}^{\prime}_{\nu}$ and each $H_{\tilde{\nu}}$ is quasiconvex
in $G_{\nu}$ relative to $\mathbb{P}^{\prime}_{\nu}$. Therefore $H$ is
quasiconvex relative to $\bigcup\mathbb{E}_{\nu}$ by Corollary 2.7.

_Step 3:_ $H$ is quasiconvex relative to
$\mathbb{P}^{\prime}=\bigcup\mathbb{P}^{\prime}_{\nu}$. Since
$\bigcup\mathbb{E}_{\nu}$ extends
$\mathbb{P}=\bigcup\mathbb{P}^{\prime}_{\nu}$, by Theorem 4.7, it suffices to
show that $H\cap K^{g}$ is quasiconvex relative to $\mathbb{P}^{\prime}$ for
all $K\in\bigcup\mathbb{E}_{\nu}$ and $g\in G$. There are two cases:

Case 1: $K\in\mathbb{P}^{\prime}_{\nu}$ for some $\nu$. Now $H\cap K^{g}$ is a
parabolic subgroup of $G$ relative to $\mathbb{P}^{\prime}$ and is thus
quasiconvex relative to $\mathbb{P}^{\prime}$.

Case 2: $K=G_{e}$ for some $e$ attached to some $\nu$. The group $K$ is
relatively quasiconvex in $G_{\nu}$, therefore by Remark 2.2, $K^{g}$ is also
relatively quasiconvex but in $G_{g\nu}$. Now since $K^{g}\cap H=K^{g}\cap
H_{g\nu}$ and $K^{g}$ and $H_{g\nu}$ are both relatively quasiconvex in
$G_{g\nu}$, the group $K^{g}\cap H$ is relatively quasiconvex in $G_{g\nu}$.
Since by Lemma 4.9, $G_{g\nu}$ is quasiconvex relative to
$\mathbb{P}^{\prime}$, Lemma 2.3 implies that $K^{g}\cap H$ is quasiconvex
relative to $\mathbb{P}^{\prime}$.

Now $H$ is quasiconvex relative to $\mathbb{P}$ by Theorem 4.7, since
$\mathbb{P}$ extends $\mathbb{P}^{\prime}$. ∎

The following result strengthens Theorem 4.12, by relaxing Condition (c).

######  Theorem 4.13 (Quasiconvexity Criterion for Relatively Hyperbolic
Groups that Split).

Let $G$ be f.g. and hyperbolic relative to $\mathbb{P}$ such that $G$ splits
as a finite graph of groups. Suppose

  1. (a)

Each $G_{e}$ is total in $G$;

  2. (b)

Each $G_{e}$ is relatively quasiconvex in $G$;

  3. (c)

Each $G_{e}$ is almost malnormal in $G$.

Let $H\leq G$ be tamely generated. Then $H$ is relatively quasiconvex in $G$.

######  Remark 4.14.

By Lemma 2.3 and Remark 2.4, Condition (b) is equivalent to requiring that
each $G_{e}$ is quasiconvex in $G_{\nu}$. Also we can replace Condition (a) by
requiring $G_{e}$ to be total in $G_{\nu}$.

###### Proof.

We prove the result by induction on the number of edges of the graph of groups
$\Gamma$. The base case where $\Gamma$ has no edge is contained in the
hypothesis. Suppose that $\Gamma$ has at least one edge $e$ (regarded as an
open edge). If $e$ is nonseparating, then $G=A*_{C^{t}=D}$ where $A$ is the
graph of groups over $\Gamma-e$, and $C,D$ are the two images of $G_{e}$.
Condition (c) ensures that $\\{C,D\\}$ is almost malnormal in $A$, and by
induction, the various nontrivial intersections $H\cap A^{g}$ are relatively
quasiconvex in $A^{g}$, and thus $H$ is relatively quasiconvex in $G$ by
Theorem 4.12. A similar argument concludes the separating case. ∎

######  Corollary 4.15.

Let $G$ be f.g. and hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as
a finite graph of groups. Assume:

  * (a)

Each $G_{\nu}$ is locally relatively quasiconvex;

  * (b)

Each $G_{e}$ is Noetherian, total and relatively quasiconvex in $G$;

  * (c)

Each $G_{e}$ is almost malnormal in $G$.

Then $G$ is locally relatively quasiconvex relative to $\mathbb{P}$.

######  Theorem 4.16.

Let $G$ be hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as a graph
$\Gamma$ of groups with relatively quasiconvex edge groups. Suppose $\Gamma$
is bipartite with $\Gamma^{0}=V\sqcup U$ and each edge joins vertices of $V$
and $U$. Suppose each $G_{v}$ is maximal parabolic for $v\in V$, and for each
$P\in\mathbb{P}$ there is at most one $v$ with $P$ conjugate to $G_{v}$. Let
$H\leq G$ be tamely generated. Then $H$ is quasiconvex relative to
$\mathbb{P}$.

The scenario of Theorem 4.16 arises when $M$ is a compact aspherical
3-manifold, from its JSJ decomposition. The manifold $M$ decomposes as a
bipartite graph $\Gamma$ of spaces with $\Gamma^{0}=U\sqcup V$. The
submanifold $M_{v}$ is hyperbolic for each $v\in V$, and $M_{u}$ is a graph
manifold for each $u\in U$. The edges of $\Gamma$ 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 $I$-bundles over Klein bottles where a hyperbolic part
terminates. Hence $\pi_{1}M$ decomposes accordingly as a graph $\Gamma$ of
groups, and $\pi_{1}M$ is hyperbolic relative to $\\{\pi_{1}M_{u}:u\in U\\}$
by Theorem 1.4 or indeed, Corollary 1.5.

###### Proof.

Let $K_{o}$ be a fine hyperbolic graph for $G$. Each vertex group is
quasiconvex in $G$ by Lemma 4.9, and so for each $u\in U$ let $K_{u}$ be a
$G_{u}$-quasiconvex subgraph, and in this way we obtain finite hyperbolic
$G_{u}$-graphs, and for $v\in V$, we let $K_{v}$ be a singleton. We apply the
Construction in the proof of Theorem 1.4 to obtain a fine hyperbolic $G$-graph
$K$ and quotient $\bar{K}$. Note that the parabolic trees are $i$-pods. We
form the $H$-cocompact quasiconvex subgraph $L$ by combining
$H_{\omega}$-cocompact quasiconvex subgraphs $K_{\omega}$ as in the proof of
Theorem 2.6. ∎

######  Theorem 4.17.

Let $G$ be f.g. and hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as
graph $\Gamma$ of groups with relatively quasiconvex edge groups. Suppose
$\Gamma$ is bipartite with $\Gamma^{0}=V\sqcup U$ and each edge joins vertices
of $V$ and $U$. Suppose each $G_{v}$ is almost malnormal and total in $G$ for
$v\in V$. Let $H\leq G$ be tamely generated. Then $H$ is quasiconvex relative
to $\mathbb{P}$.

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 $V$.

Another special case where Theorem 4.17 applies is where
$G=G_{1}*_{{C_{1}}^{\prime}=C_{2}}G_{2}$ is hyperbolic relative to
$\mathbb{P}$, and $C_{2}\leq G_{2}$ is total and relatively quasiconvex in $G$
and almost malnormal in $G_{2}$.

###### Proof.

Following the Technical Point in the proof of Theorem 4.12, by splitting
certain elements of $\mathbb{P}$ to obtain $\mathbb{P}^{\prime}$ as in Remark
4.11, we can assume that $G$ is hyperbolic relative to $\mathbb{P}^{\prime}$
where each $P^{\prime}\in\mathbb{P}^{\prime}$ is elliptic with respect to the
action of $G$ on the Bass-Serre tree $T$. Since $\mathbb{P}$ extends
$\mathbb{P}^{\prime}$ and each $G_{v}\cap P^{g}$ is conjugate to an element of
$\mathbb{P}^{\prime}$, we see that each $G_{v}$ is quasiconvex in $G$ relative
to $\mathbb{P}^{\prime}$ by Theorem 4.7, and moreover, since elements of
$\mathbb{P}^{\prime}$ are vertex groups of elements of $\mathbb{P}$, each
$G_{v}$ is total relative to $\mathbb{P}^{\prime}$. Therefore each $G_{v}$ is
hyperbolic relative to a collection $\mathbb{P}^{\prime}_{v}$ of conjugates of
elements of $\mathbb{P}^{\prime}$.

We argue by induction on the number of edges of $\Gamma$. If $\Gamma$ has no
edge the result is contained in the hypothesis. Suppose $\Gamma$ has at least
one edge $e$. If $e$ is separating and $\Gamma=\Gamma_{1}\sqcup
e\sqcup\Gamma_{2}$ where $e$ attaches $v\in\Gamma^{0}_{1}$ to
$u\in\Gamma^{0}_{2}$ then $G=G_{1}*_{G_{e}}G_{2}$ where
$G_{i}=\pi_{1}(\Gamma_{i})$. Each $G_{e}$ is the intersection of vertex groups
and hence quasiconvex relative to $\mathbb{P}^{\prime}$. By Lemma 4.9, the
groups $G_{1}$ and $G_{2}$ are quasiconvex in $G$ relative to
$\mathbb{P}^{\prime}$. Thus $G_{i}$ is hyperbolic relative to
$\mathbb{P}^{\prime}_{i}$ by Remark 2.2.

Observe that $T$ contains subtrees $T_{1}$ and $T_{2}$ that are the Bass-Serre
trees of $\Gamma_{1}$ and $\Gamma_{2}$, and $T-G\tilde{e}=\\{gT_{1}\cup
gT_{2}\ :\ g\in G\\}$. The Bass-Serre tree $\bar{T}$ of $G_{1}*_{G_{e}}G_{2}$
is the quotient of $T$ obtained by identifying each $gT_{i}$ to a vertex.

Since $H$ is relatively finitely generated, there is a finite graph of groups
$\Gamma_{H}$ for $H$, and a map $\Gamma_{H}\rightarrow\Gamma$. Removing the
edges mapping to $e$ from $\Gamma_{H}$, we obtain a collection of finitely
many graphs of groups - some over $\Gamma_{1}$ and some over $\Gamma_{2}$.
Each component of $\Gamma_{H}$ corresponds to the stabilizer of some $gT_{i}$
and is denoted by $H_{gT_{i}}$, and since that component is a finite graph
with relatively quasiconvex vertex stabilizers, we see that each $H_{gT_{i}}$
is relatively quasiconvex in $G_{i}$ relative to $\mathbb{P}^{\prime}_{i}$ by
induction on the number of edges of $\Gamma_{H}$.

We extend the peripheral structure $\mathbb{P}^{\prime}_{1}$ of $G_{1}$ to
$\mathbb{E}_{1}=\\{G_{1}\\}$. Note that now each $H_{gT_{1}}$ is quasiconvex
in $G_{1}$ relative to $\mathbb{E}_{1}$ by Theorem 4.7. Let

$\mathbb{E}=\mathbb{E}_{1}\cup\mathbb{P}^{\prime}_{2}-\\{P\in\mathbb{P}^{\prime}_{2}\colon
P^{g}\leq{G_{e}},~{}\text{for some}~{}g\in G_{2}\\}.$

Observe that $\mathbb{E}$ extends $\mathbb{P}^{\prime}$. Since $G_{v}$ is
total and quasiconvex in $G$ relative to $\mathbb{P}^{\prime}$ and
$\mathbb{E}$ extends $\mathbb{P}^{\prime}$, the group $G_{1}$ is total and
quasiconvex in $G$ relative to $\mathbb{E}$ by Theorem 4.7. Therefore $G$ is
hyperbolic relative to $\mathbb{E}$ by Theorem 4.2.

Since $G_{1}$ is maximal parabolic in $G$, by Theorem 4.16 $H$ is quasiconvex
in $G$ relative to $\mathbb{E}$. The graph $\Gamma_{H}$ shows that $H$ is
generated by finitely many hyperbolic elements and vertex stabilizers
$H_{g\bar{T}_{i}}$ and each $H_{g\bar{T}_{i}}=H_{gT_{i}}$ which we explained
above is relatively quasiconvex in $G_{i}$.

We now show that $H$ is quasiconvex relative to $\mathbb{P}^{\prime}$ and
therefore relative to $\mathbb{P}$ by Theorem 4.7. Since $\mathbb{E}$ extends
$\mathbb{P}^{\prime}$, by Theorem 4.7, it suffices to show that $H\cap E^{g}$
is quasiconvex relative to $\mathbb{P}^{\prime}$ for all $E\in\mathbb{E}$ and
$g\in G$. There are two cases:

Case 1: $E\in\mathbb{P}^{\prime}_{2}$. Now $H\cap E^{g}$ is a parabolic
subgroup of $G$ relative to $\mathbb{P}^{\prime}$ and is thus quasiconvex
relative to $\mathbb{P}^{\prime}$.

Case 2: $E=G_{1}$. Then $H\cap E^{g}$ is quasiconvex relative to
$\mathbb{P}^{\prime}_{1}$ since $(H\cap E^{g})=H_{gT_{1}}$ is quasiconvex in
$G_{1}^{g}$ relative to $\mathbb{E}_{1}^{g}=\\{G_{1}^{g}\\}$. Since
$E^{g}=G_{1}^{g}$ is quasiconvex relative to $\mathbb{P}^{\prime}$, Lemma 2.3
implies that $H\cap E^{g}$ is quasiconvex relative to $\mathbb{P}^{\prime}$.

Now assume that $e$ is nonseparating. Let $u\in U$ and $v\in V$ be the
endpoints of $e$. Then $G=G_{1}*_{C^{t}=D}$ where $G_{1}$ is the graph of
groups over $\Gamma-e$, and $C$ and $D$ are the images of $G_{e}$ in $G_{v}$
and $G_{u}$ respectively. We first reduce the peripheral structure of $G$ from
$\mathbb{P}$ to $\mathbb{P}^{\prime}$, and we then extend from
$\mathbb{P}^{\prime}$ to $\mathbb{E}$ with:

$\mathbb{E}=\\{G_{v}\\}\cup\mathbb{P}^{\prime}-\\{P\in\mathbb{P}^{\prime}\colon
P^{g}\leq{G_{v}},~{}\text{for some}~{}g\in G\\}.$

$G$ is hyperbolic relative to $\mathbb{E}$ by Theorem 4.2 as $G_{v}$ is almost
malnormal, total, and quasiconvex relative to $\mathbb{P}$ . The argument
follows by induction and Theorem 4.16 as in the separating case. ∎

Theorem 4.13 suggests the following criterion for relative quasiconvexity:

######  Conjecture 4.18.

Let $G$ be hyperbolic relative to $\mathbb{P}$. Suppose $G$ splits as a finite
graph of groups with f.g. relatively quasiconvex edge groups. Suppose $H\leq
G$ is tamely generated such that each $H_{v}$ is f.g. for each $v$ in the
Bass-Serre tree. Then $H$ is relatively quasiconvex in $G$.

When the edge groups are separable in $G$, there is a finite index subgroup
$G^{\prime}$ whose splitting has relatively malnormal edge groups (see e.g.
[11, 9]). Consequently, if moreover, the edge groups of $G$ are total, then
the induced splitting of $G^{\prime}$ satisfies the criterion of Theorem 4.13,
and we see that Conjecture 4.18 holds in this case. In particular, Conjecture
4.18 holds when $G$ is virtually special and hyperbolic relative to virtually
abelian subgroups, provided that edge groups are also total. We suspect the
totalness assumption can be dropped totally.

As a closing thought, consider a hyperbolic $3$-manifold $M$ virtually having
a malnormal quasiconvex hierarchy (conjecturally all closed $M$). Theorem 4.13
suggests an alternate approach to the tameness theorem, which could be
reproven by verifying:

_If the intersection of a f.g. $H$ with a malnormal quasiconvex edge group is
infinitely generated then $H$ is a virtual fiber._

Acknowledgement: We are extremely grateful to the anonymous f.g. referee whose
very helpful corrections and adjustments improved the results and exposition
of this paper.

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