# Local connectedness of boundaries for relatively hyperbolic groups

Ashani Dasgupta  Department of Mathematical Sciences
University of Wisconsin–Milwaukee
P.O. Box 413
Milwaukee, WI 53201
USA<EMAIL_ADDRESS>and  G. Christopher Hruska  Department of Mathematical
Sciences
University of Wisconsin–Milwaukee
P.O. Box 413
Milwaukee, WI 53201
USA<EMAIL_ADDRESS>
###### Abstract.

Let $(\Gamma,\mathbb{P})$ be a relatively hyperbolic group pair that is
relatively one ended. Then the Bowditch boundary of $(\Gamma,\mathbb{P})$ is
locally connected. Bowditch previously established this conclusion under the
additional assumption that all peripheral subgroups are finitely presented,
either one or two ended, and contain no infinite torsion subgroups. We remove
these restrictions; we make no restriction on the cardinality of $\Gamma$ and
no restriction on the peripheral subgroups $P\in\mathbb{P}$.

##  1\. Introduction

One of the earliest major theorems proved about hyperbolic groups is the
result that the Gromov boundary of a one-ended hyperbolic group is locally
connected and without cut points. The proof combines work of Bestvina–Mess,
Levitt, Bowditch, and Swarup [BM91, Lev98, Bow99c, Swa96]. Notable
applications of this result include Geoghegan’s semistability theorem [GS19],
the topological invariance of the JSJ decomposition [Bow98], the one-
dimensional boundary classification theorem of [KK00], and the existence of
quasi-hyperbolic planes in hyperbolic groups [BK05].

Subsequently Bowditch studied the local connectedness of boundaries of
relatively hyperbolic groups. Bowditch introduces a natural boundary of a
relatively hyperbolic pair, the Bowditch boundary, and establishes many of its
fundamental properties in [Bow12]. In a series of articles [Bow99a, Bow99b,
Bow01], he proves that the Bowditch boundary of a finitely presented
relatively hyperbolic group is locally connected if the boundary is connected,
provided certain restrictions are imposed on the subgroups known as peripheral
subgroups. These restrictions require peripheral subgroups to be finitely
presented, to be either one or two ended, and to not contain infinite torsion
subgroups. In this paper we extend Bowditch’s work and remove all
restrictions, proving the following theorem.

######  Theorem 1.1.

Let $(\Gamma,\mathbb{P})$ be a relatively hyperbolic group pair with connected
boundary $M=\partial(\Gamma,\mathbb{P})$. Then $M$ is locally connected and
every cut point of $M$ is the fixed point of a parabolic subgroup.

The boundary $M$ has a cut point if and only if $\Gamma$ splits relative to
$\mathbb{P}$ over a subgroup of a member of $\mathbb{P}$.

In particular this theorem is not restricted to finitely generated groups and
imposes no restrictions on the peripheral subgroups. Osin extends the notion
of relative hyperbolicity to groups that need not be finitely generated in
[Osi06b]. The construction and basic properties of the Bowditch boundary have
been extended to this more general setting by Gerasimov and Potyagailo [Ger12,
GP15, GP16]. The boundary of a relatively hyperbolic group pair is always
compact, and when $\Gamma$ is countable it is metrizable. Thus, in the
countable case, a connected Bowditch boundary is a Peano continuum.

We note that, throughout this paper, we consider only infinite peripheral
subgroups. (If any peripheral subgroup is finite, the boundary contains a
dense set of isolated points and, hence, is not locally connected.) Under this
convention, the boundary of $(\Gamma,\mathbb{P})$ is connected if and only if
$\Gamma$ does not split over a finite subgroup relative to $\mathbb{P}$. A
splitting is _relative to $\mathbb{P}$_ if every member of $\mathbb{P}$ is
contained in a vertex group of the splitting.

In the case that the boundary of $(\Gamma,\mathbb{P})$ is connected and
contains cut points, we obtain the following topological characterization of
the boundary as the inverse limit of a tree of compacta in the sense of
Świątkowski [Świ20].

######  Theorem 1.2 (Cut point tree).

Suppose $(\Gamma,\mathbb{P})$ is relatively hyperbolic with connected boundary
$M=\partial(\Gamma,\mathbb{P})$. Consider the bipartite graph $T$ with vertex
set $V_{0}\sqcup V_{1}$, where $V_{0}$ is the family of nontrivial maximal
subcontinua of $M$ that are not separated by any cut point of $M$ and $V_{1}$
is the set of cut points of $M$. Vertices $v\in V_{1}$ and $B\in V_{0}$ are
joined by an edge if $v\in B$. Then we have the following.

  1. (1)

The graph $T$ is a simplicial tree equal to the canonical JSJ tree of
cylinders for splittings over parabolic subgroups relative to $\mathbb{P}$.

  2. (2)

Each group $\operatorname{Stab}(B)$ with $B\in V_{0}$ is finitely generated
and relatively hyperbolic with boundary $B$, a Peano continuum with no cut
points.

  3. (3)

The boundary $M$ is equal to the inverse limit of a tree of compacta, obtained
by gluing the boundaries $B$ of the subgroups $\operatorname{Stab}(B)$ along
points in the pattern of $T$.

This characterization simplifies related treelike decompositions of boundaries
described in the finitely generated case in [Bow99c, Dah03] and extends this
treelike structure to groups that need not be finitely generated. The
canonical tree of cylinders is called the “maximal peripheral splitting” in
[Bow99c]. See Proposition 2.13 for a description of this canonical tree.

Haulmark–Hruska have used Theorem 1.1 in their proof that the JSJ
decomposition of a one-ended relatively hyperbolic pair over elementary
subgroups depends only on the topology of the boundary [HH]. Hruska–Walsh have
used Theorem 1.1 in their examination of relatively hyperbolic pairs with
planar boundary [HW]. Regarding semistability at infinity, Theorem 1.1,
together with work of Mihalik–Swenson and Haulmark–Mihalik, gives the
following immediate result (see [MS21, HM] for more details).

######  Corollary 1.3.

Suppose $(\Gamma,\mathbb{P})$ is nonelementary relatively hyperbolic and
$\Gamma$ is finitely generated and one ended.

  1. (1)

If each $P\in\mathbb{P}$ is one ended and $\Gamma$ does not split over any
subgroup $Q\leq P$ with $P\in\mathbb{P}$, then $\Gamma$ is semistable at
infinity.

  2. (2)

If each $P\in\mathbb{P}$ is finitely presented and semistable at infinity,
then $\Gamma$ is semistable at infinity.

We also note that the main results of Haulmark [Hau19] apply to any relatively
hyperbolic pair with locally connected boundary. (They are stated there with
more restrictive hypotheses, but only local connectedness is used in the
proofs.) Thus we have the following immediate consequence.

######  Corollary 1.4.

Let $(\Gamma,\mathbb{P})$ be relatively hyperbolic. Suppose $\Gamma$ does not
split relative to $\mathbb{P}$ over an elementary subgroup and each
$P\in\mathbb{P}$ is one ended. If the boundary $\partial(\Gamma,\mathbb{P})$
is one dimensional, then it is homeomorphic to a circle, a Sierpinski carpet,
or a Menger curve.

We remark that de Souza has independently obtained a variation of Bowditch’s
local connectivity theorem in [Sou] using a method that is only valid when the
relatively hyperbolic group $\Gamma$ is finitely presented and one ended.
Theorem 1.1 includes and extends the result of de Souza.

###  1.1. Methods of proof

The proofs of Theorems 1.1 and 1.2 roughly follow the same outline as the
prior proofs of Swarup and Bowditch. However, several steps require new ideas
to extend their conclusions from restricted settings to the broader setting of
this paper.

In [Bow12], Bowditch constructs the boundary of a relatively hyperbolic pair
$(\Gamma,\mathbb{P})$ and proves that it is compact and Hausdorff, provided
that $\Gamma$ is countable. In Section 3, we show that the same elementary
construction extends to the general case with no cardinality restrictions on
$\Gamma$. The resulting Bowditch boundary has a simple, concrete description
and coincides with the canonical boundary introduced by Gerasimov in [Ger12]
as the abstract Cauchy–Samuel completion of a certain uniform space. A reader
who is only interested in countable relatively hyperbolic groups may safely
skip this section, whose only purpose is to extend the reach of this paper
beyond countable groups.

In Section 4, we show that a splitting over parabolic subgroups gives rise to
a treelike decomposition of the boundary. This observation is due to Bowditch
[Bow01] in the finitely generated case. Dahmani uses an elaborate hands-on
construction of a tree of compacta to prove a combination theorem for
relatively hyperbolic group pairs in [Dah03], which is extended to a more
general setting by Tomar [Tom]. We simplify this construction by
characterizing the treelike structure as the inverse limit of a tree system of
compacta in the sense of Świątkowski [Świ20]. The inverse limit construction
can be used in place of the elaborate constructions of [Dah03] and [Tom] and
may lead to a deeper understanding of combination theorems.

This simplification allows us to eliminate all restrictions on the cardinality
of $\Gamma$ and leads to characterizations of connectedness and local
connectedness of boundaries in Section 5. Using an accessibility theorem for
parabolic splittings (see Proposition 2.13), we reduce from the general case
to the more well-behaved setting of finitely generated relatively hyperbolic
groups. After this point, all groups considered are finitely generated.

Section 6 discusses pretrees and establishes results about non-nesting actions
and full quotients of pretrees, which are used in the following two sections.
Section 7 contains a brief summary of the cut point pretree construction of
Swenson [Swe00] and some of its key dynamical properties. We use this pretree
in place of Bowditch’s more technically elaborate “pretree completion” of
[Bow99c] to simplify the analysis in the following section.

Section 8 is a study of a construction due to Bowditch that associates a dual
dendrite $D$ to each continuum $M$ with a preferred family of cut points $C$.
Proposition 8.4 characterizes this dendrite as the maximal Hausdorff quotient
of $M$ in which any two points are identified if they are not separated by a
point of $C$. Bowditch [Bow99c] establishes that this dendrite is nontrivial
if $M$ admits a convergence action by a one-ended group $G$ containing no
infinite torsion group and $M$ has a nonparabolic cut point. The main goal of
Section 8 is to extend this result from one-ended groups to relatively one-
ended groups. This extension is one of the main innovations of the present
proof; it makes essential use of the non-nesting property and the results
about pretrees established in Sections 6 and 7.

In Section 9 we verify that a theorem of Levitt [Lev98] about non-nesting
actions on trees extends from finitely presented groups to groups with a
finite relative presentation. Section 10 contains an accessibility theorem for
relatively hyperbolic groups and splittings over two-ended loxodromic
subgroups. It also contains the proofs of Theorems 1.1 and 1.2, which combine
the ingredients discussed above.

###  1.2. Acknowledgements

We give thanks to Arka Banerjee, Victor Gerasimov, Craig Guilbault, Prayagdeep
Parija, and Kim Ruane for helpful conversations that influenced this paper.
The second author was partially supported by grant #714338 from the Simons
Foundation.

##  2\. Relatively hyperbolic groups and their splittings

###  2.1. Relative hyperbolicity

This section discusses relatively hyperbolic groups and their boundaries. For
more details, see [Bow12, GP15, GP16]. Alternative approaches may be found in
[DS05, Osi06b, GM08].

######  Definition 2.1.

A _graph_ $K$ is a set $V=V(K)$ of _vertices_ , a set $E=E(K)$ of _oriented
edges_ , an _initial vertex_ $\iota(\vec{e}\,)$, _terminal vertex_
$\tau(\vec{e}\,)$, and _inverse edge_ $-\vec{e}$ for each oriented edge
$\vec{e}\in E(K)$ satisfying $-\vec{e}\neq\vec{e}$, $-(-\vec{e}\,)=\vec{e}$,
and $\iota(\vec{e}\,)=\tau(-\vec{e}\,)$. An (unoriented) _edge_ $e$ is a pair
of oriented edges $\\{\vec{e},-\vec{e}\,\\}$.

In a graph $K$, a _path_ is indicated by giving its sequence of oriented edges
or its sequence of vertices (if the graph is simplicial). A path is _simple_
if its vertices are distinct. A _cycle_ is a path from a vertex to itself, and
a _circuit_ is a cycle with all vertices distinct except for the initial and
terminal vertices, which are equal. A _simplicial tree_ is a connected graph
with no circuits.

We treat graphs as combinatorial objects, as above, or as topological or
metric spaces depending on the context.

######  Definition 2.2.

A geodesic space $X$ is _$\delta$ –hyperbolic_ if each side of a geodesic
triangle lies in the $\delta$–neighborhood of the union of the other two
sides.

######  Definition 2.3.

A _group pair_ $(\Gamma,\mathbb{P})$ consists of a group $\Gamma$ together
with a finite collection $\mathbb{P}$ of subgroups of $\Gamma$. Given a group
pair $(\Gamma,\mathbb{P})$, an action of $\Gamma$ on $X$ is _relative to
$\mathbb{P}$_ if each member of $\mathbb{P}$ has a fixed point in $X$.

######  Definition 2.4 (Relatively hyperbolic).

A connected simplicial graph $K$ is _fine_ if for each $n$, each edge in $K$
is contained in only finitely many circuits of length $n$. A group pair
$(\Gamma,\mathbb{P})$ is _relatively hyperbolic_ if $\Gamma$ admits an action
relative to $\mathbb{P}$ on a graph $K$ with the following properties:

  1. (1)

The graph $K$ is fine and $\delta$–hyperbolic for some $\delta<\infty$.

  2. (2)

There are finitely many $\Gamma$–orbits of edges and each edge stabiliser is
finite.

  3. (3)

The elements of $\mathbb{P}$ are representatives of the conjugacy classes of
infinite vertex stabilisers of $K$.

The special case in which $\mathbb{P}$ is the empty collection reduces to the
notion of a word hyperbolic group. As our main results are already known in
this case, we assume throughout that $\mathbb{P}$ is nonempty. In this case,
we may assume, in addition, that every vertex of $K$ has an infinite
stabilizer (see [Bow12]). A graph $K$ with the above properties is a
_$(\Gamma,\mathbb{P})$ –graph_. If $(\Gamma,\mathbb{P})$ is relatively
hyperbolic, a subgroup of $\Gamma$ is _parabolic_ if it is a subgroup of a
conjugate of some $P\in\mathbb{P}$ and is _loxodromic_ if it is $2$–ended and
nonparabolic. A subgroup is _elementary_ if it is either finite, parabolic, or
loxodromic.

######  Definition 2.5 (Convergence groups).

For any compact Hausdorff space $M$, let $\Delta\subset M^{n}$ denote the
subset of points with at least two coordinates equal. The _space of distinct
$n$–tuples_ $\Theta^{n}(M)$ is the quotient of $M^{n}-\Delta$ by the symmetric
group $S_{n}$, which acts by permuting the coordinates. An action of a
discrete group on $M$ is a _convergence group action_ if $M$ contains at least
three points and the action on $\Theta^{3}(M)$ is proper. An infinite subgroup
$P$ of a convergence group is _parabolic_ if it has a unique fixed point.

A _$32$ –action_ is an action that is proper on $\Theta^{3}(M)$ (_i.e._ , a
convergence group) and also cocompact on $\Theta^{2}(M)$. A _$32$ –action of a
group pair_ $(\Gamma,\mathbb{P})$ is a $32$–action of $\Gamma$ such that the
elements of $\mathbb{P}$ are representatives of the conjugacy classes of
maximal parabolic subgroups. We note that this definition is substantially
stronger than the property that $\Gamma$ acts relative to $\mathbb{P}$.

######  Definition 2.6 (Bowditch boundary).

Suppose $(\Gamma,\mathbb{P})$ is relatively hyperbolic and $K$ is a
$(\Gamma,\mathbb{P})$–graph. Using a variation of a construction due to
Bowditch, Gerasimov constructs a compact Hausdorff space $M=M(K)$ on which
$\Gamma$ acts by homeomorphisms relative to $\mathbb{P}$ [Bow12, Ger12]. If
$\Gamma$ is elementary, then $M$ contains exactly zero, one, or two points
depending on whether $(\Gamma,\mathbb{P})$ is finite, parabolic, or
loxodromic. In the loxodromic case, $\mathbb{P}$ is empty and $M$ is
$\Gamma$–equivariantly equivalent to the two-point set of ends of $\Gamma$.
When $\Gamma$ is nonelementary, $M$ is infinite and the action of $\Gamma$ on
$M$ is a $32$–action of the pair $(\Gamma,\mathbb{P})$.

Although a relatively hyperbolic pair $(\Gamma,\mathbb{P})$ admits many
different $(\Gamma,\mathbb{P})$–graphs $K$, the associated compactum $M=M(K)$
is uniquely determined up to $\Gamma$–equivariant homeomorphism [Bow12, GP16].
This unique compactum $M$ is the _Bowditch boundary_ , denoted
$M=\partial(\Gamma,\mathbb{P})$.

######  Remark 2.7 (On finite generation).

The definitions of relative hyperbolicity for a group pair
$(\Gamma,\mathbb{P})$ above make no restriction on the cardinality of the
discrete group $\Gamma$. However all likely examples of interest in geometric
group theory involve countable groups $\Gamma$, and it is frequently useful to
further restrict attention to the case when $\Gamma$ is finitely generated. We
note that if $(\Gamma,\mathbb{P})$ is relatively hyperbolic then $\Gamma$ is
countable if and only if the Bowditch boundary $M=\partial(\Gamma,\mathbb{P})$
is metrizable [Ger09].

In the finitely generated setting, $M$ arises as the (metrizable) Gromov
boundary of a locally finite $\delta$–hyperbolic graph, known as the cusped
space, whose geometry is central to many applications in the literature (see
[GM08]). Cusped spaces do not play a significant role in this article, except
for a brief appearance in the proof of Proposition 10.5.

###  2.2. Splittings of relatively hyperbolic groups

This section summarizes the structure of splittings of relatively hyperbolic
groups over parabolic subgroups. The results of this section were first
introduced in the finitely generated setting by Bowditch [Bow01]. This section
extends that study to arbitrary relatively hyperbolic pairs.

Let $\Gamma$ be any group, and let $\mathbb{A}$ be a family of subgroups
closed under conjugation. Suppose $\Gamma$ acts on a simplicial tree $T$
without inversions and with no proper invariant subtree. Then $T$ is an
_$\mathbb{A}$ –tree_ if each edge stabilizer is a member of $\mathbb{A}$.
Suppose $(\Gamma,\mathbb{P})$ is a group pair. An _$(\mathbb{A},\mathbb{P})$
–tree_ is an $\mathbb{A}$–tree $T$ such that each member of $\mathbb{P}$ has a
fixed vertex in $T$. Two $(\mathbb{A},\mathbb{P})$–trees are _equivalent_ if
there are $\Gamma$–equivariant maps $T\to T^{\prime}$ and also $T^{\prime}\to
T$.

######  Definition 2.8 (Tree of cylinders).

Suppose $(\Gamma,\mathbb{P})$ is relatively hyperbolic. Let $\mathbb{E}$ be
either the collection of infinite elementary subgroups or the collection of
infinite parabolic subgroups of $\Gamma$. Given an
$(\mathbb{E},\mathbb{P})$–tree $T$, edges $e$ and $e^{\prime}$ are equivalent,
denoted $e\sim e^{\prime}$, if their stabilizers are _co-elementary_ , which
means that the subgroup generated by $\Gamma_{e}$ and $\Gamma_{e^{\prime}}$ is
elementary. For each equivalence class, the smallest subtree containing that
class is a _cylinder_. The _tree of cylinders_ $T_{c}$ associated to $T$ is
the bipartite graph with vertex set $V_{0}(T_{c})\sqcup V_{1}(T_{c})$, where
$V_{0}(T_{c})$ is the set of all vertices $v$ of $T$ that belong to at least
two cylinders, $V_{1}(T_{c})$ is the set of cylinders $w$ of $T$, and there is
an edge between $v$ and $w$ in $T_{c}$ if and only if $v$ is a vertex of the
subtree $w$. The graph $T_{c}$ is an $(\mathbb{E},\mathbb{P})$–tree equivalent
to $T$, and the trees $T$ and $T_{c}$ have the same family of nonelementary
vertex stabilizers. See [GL11] for more details on trees of cylinders.

######  Proposition 2.9.

Let $(\Gamma,\mathbb{P})$ be relatively hyperbolic. Let $T$ be any tree on
which $\Gamma$ acts relative to $\mathbb{P}$ with finitely many orbits of
edges such that all edge stabilizers are infinite, parabolic, and finitely
generated. Let $T_{c}$ be its tree of cylinders. For each vertex $v\in
V_{0}(T_{c})$, the stabilizer $\Gamma_{v}$ is hyperbolic relative to a family
of subgroups
$\mathbb{Q}_{v}=\operatorname{Inc}_{v}\cup\mathbb{P}_{|\Gamma_{v}}$, defined
as follows:

  1. (1)

$\operatorname{Inc}_{v}$ consists of representatives of the conjugacy classes
in $\Gamma_{v}$ of stabilizers of edges incident to $v$ in $T_{c}$.

  2. (2)

$\mathbb{P}_{|\Gamma_{v}}$ consists of representatives of the conjugacy
classes in $\Gamma_{v}$ of subgroups $gPg^{-1}$ with $g\in\Gamma$ and
$P\in\mathbb{P}$ that stabilize $v$ and do not stabilize any edge incident to
$v$ in $T_{c}$.

###### Proof.

Each parabolic subgroup of $\Gamma$ is relatively quasiconvex in the sense of
Bigdely–Wise [BW13]. In particular, since the stabilizer of each edge of $T$
is relatively quasiconvex, the stabilizer of each vertex is also relatively
quasiconvex (see [BW13, Lem. 4.9]). (We note that, as stated, the cited result
requires that each vertex stabilizer and each edge stabilizer be finitely
generated. However, the condition on vertex groups is never used in the proof
and may be omitted.)

Relative quasiconvexity implies that each $\Gamma_{v}$ with $v\in
V_{0}(T_{c})$ is hyperbolic relative to a family of representatives of the
conjugacy classes in $\Gamma_{v}$ of infinite subgroups of the form
$\Gamma_{v}\cap gPg^{-1}$ for $g\in\Gamma$ and $P\in\mathbb{P}$. Such
subgroups that do not stabilize an edge incident to $v$ must be fully
contained in $\Gamma_{v}$ since the action on $T$ is relative to $\mathbb{P}$.
∎

A $(\Gamma,\mathbb{P})$–graph compatible with a given splitting may be
constructed as follows. (See [BW13, Thm. 1.4] for more details.)

######  Definition 2.10 (Compatible $(\Gamma,\mathbb{P})$–graph).

Suppose $(\Gamma,\mathbb{P})$ is relatively hyperbolic and $\Gamma$ acts on a
tree of cylinders $T_{c}$ as in Proposition 2.9. For each vertex $v\in
V_{0}(T_{c})$, let $K_{v}$ be a $(\Gamma_{v},\mathbb{Q}_{v})$–graph. If $v$
and $v^{\prime}$ are in the same $\Gamma$–orbit, we choose the same graph
$K_{v}=K_{v^{\prime}}$. For each $w\in V_{1}(T_{c})$ let $K_{w}$ be a graph
consisting of a single vertex. For each edge $e$ of $T_{c}$ joining vertices
$v$ and $w$, we glue the vertex $K_{w}$ to the unique vertex of $K_{v}$ fixed
by the parabolic subgroup $\Gamma_{e}$. The $(\Gamma,\mathbb{P})$–graph $K$
resulting from these identifications is said to be _compatible_ with $T_{c}$.

The following theorem due to Osin is often useful when studying splittings
over parabolic subgroups.

######  Theorem 2.11 ([Osi06b], Thm. 2.44 and Prop. 2.29).

Let $(\Gamma,\mathbb{P})$ be relatively hyperbolic. If $\Gamma$ does not split
relative to $\mathbb{P}$ over parabolic subgroups, then $\Gamma$ and the
members of $\mathbb{P}$ are all finitely generated.

######  Definition 2.12.

A group pair $(\Gamma,\mathbb{P})$ is _relatively one ended_ if $\Gamma$ is
infinite, $\Gamma\notin\mathbb{P}$, and $\Gamma$ does not split over a finite
subgroup relative to $\mathbb{P}$.

The tree $T_{c}$ in the following proposition is the _canonical JSJ tree of
cylinders_ for splittings of $\Gamma$ over parabolic subgroups relative to
$\mathbb{P}$. The uniqueness of the tree $T_{c}$ is discussed in [GL11].

######  Proposition 2.13.

Let $(\Gamma,\mathbb{P})$ be relatively hyperbolic and relatively one ended.
There exists a tree $T_{JSJ}$ on which $\Gamma$ acts relative to $\mathbb{P}$
with finitely many orbits of edges having the following properties.

All edge stabilizers of $T_{JSJ}$ are infinite, parabolic, and finitely
generated. In the associated tree of cylinders $T_{c}$, the stabilizer of
every edge is finitely generated. For each $v\in V_{0}(T_{c})$ the stabilizer
$\Gamma_{v}$ is finitely generated, the pair $(\Gamma_{v},\mathbb{Q}_{v})$ is
relatively hyperbolic, and $\Gamma_{v}$ does not split over a finite or
parabolic subgroup relative to $\mathbb{Q}_{v}$.

###### Proof.

The JSJ tree $T_{JSJ}$ for $\Gamma$ relative to $\mathbb{P}$ over parabolic
subgroups is a tree on which $\Gamma$ acts relative to $\mathbb{P}$ with
finitely many edge orbits and all edge stabilizers finitely generated and
parabolic (see [GL17, §3.8] for details). One-endedness implies that the
stabilizer of each edge of $T_{JSJ}$ is infinite. If $T_{c}$ is its tree of
cylinders, Proposition 2.9 implies that the stabilizer $\Gamma_{v}$ is
hyperbolic relative to $\mathbb{Q}_{v}$ for each $v\in V_{0}(T_{c})$.
According to [GL17, Cor. 4.14], the group $\Gamma_{v}$ does not split relative
to $\mathbb{Q}_{v}$ over any finite or parabolic subgroup. Thus, by Theorem
2.11 the group $\Gamma_{v}$ and all members of $\mathbb{Q}_{v}$ are finitely
generated. In particular, every edge incident to $v$ in $T_{c}$ has a finitely
generated stabilizer as well. ∎

##  3\. Constructing the Bowditch boundary

In this section, we study the definition of the Bowditch boundary, proving the
equivalence of two constructions of the boundary due to Bowditch [Bow12] and
Gerasimov [Ger12]. In each case, a fine $\delta$–hyperbolic graph $K$ leads to
a canonical compact Hausdorff space that contains the vertex set $V(K)$ as a
dense subspace. Bowditch constructs a compact Hausdorff space $\Delta K$ whose
underlying set is $V(K)\cup\partial K$. The topology on $\Delta K$ is given by
an explicit neighborhood basis, which is quite simple and easy to work with
directly. However, the proof in [Bow12] that $\Delta K$ is compact is valid
only in the special case that $K$ has a countable vertex set. Gerasimov gives
a different definition of a compactum $M=M(K)$ for arbitrary $K$ by first
defining a uniform structure on $V(K)$ and then letting $M$ be its
Cauchy–Samuel completion, a space whose points are the minimal Cauchy filters
on $V(K)$. The latter construction is theoretically elegant but quite
technically elaborate, and the exact relationship between the compactum $M$
and the set $V(K)\cup\partial K$ is a bit obscured by this approach.

In this section we show—with no restriction on the cardinality of $K$—that the
two constructions lead to spaces that are canonically homeomorphic. In order
to produce this homeomorphism between $\Delta K$ and $M(K)$, the main step is
Proposition 3.3, which establishes the compactness of $\Delta K$ for an
arbitrary fine $\delta$–hyperbolic graph $K$. We begin with a study of the
topology on $\Delta K$ introduced in [Bow12, §8].

######  Definition 3.1 (Topology on $\Delta K$).

Let $K$ be an arbitrary fine $\delta$–hyperbolic graph for some constant
$\delta$. Let $\Delta K$ denote the disjoint union $V(K)\cup\partial K$. For
points $a,b\in\Delta K$, we denote a geodesic from $a$ to $b$ by $[a,b]$. For
each $a\in\Delta K$ and each finite subset $A\subseteq V(K)$, define

$M^{\prime}(a,A)=\bigl{\\{}\,{b\in\Delta K}\bigm{|}{\text{there exists $[a,b]$
such that $[a,b]\cap A-{a}=\varnothing$}}\,\bigr{\\}}.$

It is established in [Bow12, §8] that the sets $M^{\prime}(a,A)$ form a base
for a topology on $\Delta K$ such that for each $a$ the family
$\bigl{\\{}M^{\prime}(a,A)\\}_{A}$ is a neighborhood base at $a$. The space
$\Delta K$ is Hausdorff and contains $V(K)$ as a dense subset.

Suppose $f\colon\mathbb{N}\to\mathbb{N}$ is any function with $f\geq
1_{\mathbb{N}}$ and bounded above by a linear function. A _simple
$f$–quasigeodesic_ is a simple path $\gamma$ such that for each subpath
$\gamma^{\prime}$ with endpoints $x$ and $y$ we have
$\operatorname{length}(\gamma^{\prime})\leq f\bigl{(}d(x,y)\bigr{)}$. For each
$f$ as above, equivalent neighborhood bases for a point $a\in\Delta K$ are
given by the collections

$\bigl{\\{}M^{\prime}_{f}(a,A)\bigr{\\}}_{A}\quad\text{and}\quad\bigl{\\{}P^{\prime}_{f}(a,J)\bigr{\\}}_{J}$

defined below, where $A$ ranges over all finite sets of vertices and $J$
ranges over all finite sets of edges of $K$. (When $a$ is a vertex, it
suffices to let $J$ range over finite sets of edges incident to $a$.)

  1. (1)

The set $M^{\prime}_{f}(a,A)$ contains all $b\in\Delta K$ such that vertex set
of at least one simple $f$–quasigeodesic from $a$ to $b$ is disjoint from
$A-\\{a\\}$.

  2. (2)

The set $P^{\prime}_{f}(a,J)$ contains all $b\in\Delta K$ such that edge set
of at least one simple $f$–quasigeodesic from $a$ to $b$ does not intersect
$J$.

Shortcutting is a process that produces a simple quasigeodesic from a pair of
geodesics as in the following lemma (see [Bow12, §8] for a proof).

######  Lemma 3.2 (Shortcutting).

There exists a function $f_{1}\colon\mathbb{N}\to\mathbb{N}$ with $f_{1}\geq
1$ and bounded above by a linear function and there exists a constant $R_{1}$,
such that $f_{1}$ and $R_{1}$ depend only on the hyperbolicity constant of $K$
and such that the following holds. Suppose $a$ and $b$ are two points in
$\Delta K$ and $c\in V(K)$ such that both $a$ and $b$ are not the same point
of $\partial K$. Let $\alpha$ be a geodesic from $a$ to $c$ and $\beta$ be a
geodesic from $b$ to $c$. Then there exists a simple $f_{1}$–quasigeodesic
$\gamma$ from $a$ to $b$ such that
$\gamma=\alpha^{\prime}\cup\omega\cup\beta^{\prime}$, where the path
$\alpha^{\prime}$ is a subpath of $\alpha$, the path $\beta^{\prime}$ is a
subpath of $\beta$, and the path $\omega$ has length at most $R_{1}$. If $a\in
V(K)$ then the edge of $\gamma$ incident to $a$ is contained in $\alpha$.

We write $M^{\prime}_{f_{1}}(a,A)$ as $M^{\prime}_{1}(a,A)$ and so on.
Bowditch claims the following result for arbitrary $K$ in [Bow12, Prop. 8.6].
However, the proof implicitly assumes that $K$ has a countable vertex set. The
following proposition extends Bowditch’s result to the general case, removing
this restriction on the cardinality of $K$.

######  Proposition 3.3.

The space $\Delta K$ is compact.

###### Proof.

We first note that, for each finite set $A$ of vertices of $K$, the finite
collection $\mathcal{U}_{A}=\bigl{\\{}\,{M^{\prime}(a,A)}\bigm{|}{a\in
A}\,\bigr{\\}}$ is an open cover of $\Delta(K)$. Indeed, for each $x\in\Delta
K$, there exists a geodesic from $x$ to some point $a\in A$ that contains no
points of $A$ in its interior, so that $x\in M^{\prime}(a,A)$.

Let $\mathcal{U}$ be an arbitrary open cover of $\Delta K$. In order to show
that $\mathcal{U}$ has a finite subcover, it suffices to show that for some
finite set $A\subset V(K)$ the finite cover $\mathcal{U}_{A}$ refines
$\mathcal{U}$. Suppose by way of contradiction that for every finite set
$A\subset V(K)$, there is at least one _rowdy_ element $a_{A}\in A$ such that
$M^{\prime}(a_{A},A)$ is not contained in any $U\in\mathcal{U}$. Fix any
vertex $x_{0}$ in $K$. For each finite $A$, choose a geodesic $\alpha_{A}$
from $x_{0}$ to $a_{A}$.

Observe that for each $a\in V(K)$, each $U\in\mathcal{U}$ containing $a$, and
each finite set $B\subset V(K)$, there exists a finite set $A$ containing
$B\cup\\{a\\}$ such that $M^{\prime}(a,A)\subseteq U$. Indeed, since $U$ is
open, we may choose a finite set of vertices $F=F(a,U)$ such that $a\in
M^{\prime}(a,F)\subset U$. But then the finite set $A=F\cup B\cup\\{a\\}$
satisfies $M^{\prime}(a,A)\subseteq M^{\prime}(a,F)\subseteq U$. In
particular, any finite set $B$ of vertices with rowdy element $a_{B}\in B$,
can be increased to a larger finite set $A$ such that $a_{B}$ is no longer
rowdy with respect to $A$.

We will construct a geodesic ray $\epsilon$ with vertices
$(x_{0},x_{1},x_{2},\dots)$ and a nested sequence of cofinal collections of
finite vertex sets $\Lambda_{0}\subseteq\Lambda_{1}\subseteq\cdots$ such that,
if $A\in\Lambda_{n}$, the geodesic $\alpha_{A}$ shares its first $n$ edges
with $\epsilon$. (A collection $\Lambda$ of finite sets is _cofinal_ if for
each finite set $A$, some member of $\Lambda$ contains $A$.) We construct
$x_{n+1}$ and $\Lambda_{n+1}$ recursively, beginning with the previously
chosen $x_{0}$ and the collection of all finite sets for $\Lambda_{0}$.

Assume that $x_{n}$ and $\Lambda_{n}$ have been previously constructed.
Clearly $x_{n}\in U$ for some $U\in\mathcal{U}$, so that $x_{n}$ has a basic
neighborhood $P_{1}^{\prime}(x_{n},J_{n})\subseteq U$ for some finite set of
edges $J_{n}$ incident to $x_{n}$. Let
$\Lambda^{\prime}_{n}\subseteq\Lambda_{n}$ be the subcollection of all finite
sets of vertices $B$ large enough to contain all endpoints of the edges in
$J_{n}$ and large enough that $x_{n}$ is not rowdy with respect to $B$.

Suppose $B\in\Lambda^{\prime}_{n}$. Then the geodesic $\alpha_{B}$ from
$x_{0}$ to $a_{B}$ shares its first $n$ edges with $\epsilon$ and has length
at least $n+1$. We show that the edge of $\alpha_{B}$ following $x_{n}$ must
be a member of the set $J_{n}$. If not, then by Lemma 3.2, for each $b\in
M^{\prime}(a_{B},B)$ there is a simple $f_{1}$–quasigeodesic $\gamma$ from
$x_{n}$ to $b$ whose first edge is contained in $\alpha_{B}$ and, thus, is not
contained in $J_{n}$. In particular, the quasigeodesic $\gamma$ avoids
$J_{n}$, so that $b\in P^{\prime}_{1}(x_{n},J_{n})$. Consequently,
$M^{\prime}(a_{B},B)\subseteq P^{\prime}_{1}(x_{n},J_{n})\subseteq U$,
contradicting the assumption that $a_{B}$ is rowdy in $B$. Therefore, the edge
of $\alpha_{B}$ following $x_{n}$ must be a member of $J_{n}$ as claimed.
Since $J_{n}$ is finite, we may pass to a cofinal subcollection
$\Lambda_{n+1}\subseteq\Lambda^{\prime}_{n}$ such that all geodesics
$\alpha_{C}$ with $C\in\Lambda_{n+1}$ involve the same edge of $J_{n}$.
Letting $x_{n+1}$ be the other endpoint of this edge (distinct from $x_{n}$)
completes the recursive construction of the ray $\epsilon$.

Let $y\in\partial K$ be the endpoint of $\epsilon$. There is some
$U\in\mathcal{U}$ such that $y\in U$. There exists some finite set $S\subset
V(K)$ and a basic neighborhood $M^{\prime}_{1}(y,S)\subseteq U$. As $S$ is
finite, it lies in the ball of radius $r$ about $x_{0}$ for some $r<\infty$.

Choose $n>r+R_{1}+\delta+1$ where $R_{1}$ is the constant from Lemma 3.2 and
$\delta$ is the hyperbolicity constant of $K$. Choose any $A\in\Lambda_{n}$
large enough that $S\subseteq A$. We claim that $M^{\prime}(a_{A},A)\subseteq
M^{\prime}_{1}(y,S)$.

Let $\alpha$ be a geodesic from $a_{A}$ to $y$. Consider the thin geodesic
triangle $T_{thin}$ with vertices $y,x_{n},a_{A}$ such that the geodesic
$\alpha$ is the side of the triangle from $y$ to $a_{A}$, the side from
$x_{n}$ to $a_{A}$ is contained in the geodesic from $x_{0}$ to $a_{A}$ that
intersects $\epsilon$ up to length $n$ and the geodesic from $x_{n}$ to $y$ is
contained in $\epsilon$. The geodesic $\alpha$ is in the $\delta$–neighborhood
of the union of geodesics $[x_{n},y]$ and $[x_{n},a_{A}]$. In particular if
$z$ is any vertex of $\alpha$ then $d(x_{0},z)\geq n-\delta=r+R_{1}+1$. Hence
$d(x_{0},z)>r$, implying that $z\not\in S$.

Now consider any $x\in M^{\prime}(a_{A},A)$. Since $S\subseteq A$, there
exists a geodesic $\beta$ from $a_{A}$ to $x$ with vertex set disjoint from
$S$. Applying Lemma 3.2, we obtain a simple $f_{1}$–quasigeodesic
$\gamma=\alpha^{\prime}\cup\omega\cup\beta^{\prime}$ from $y$ to $x$ where
$\alpha^{\prime}\subseteq\alpha$ and $\beta^{\prime}\subseteq\beta$ and
$\omega$ has length at most $R_{1}$. Then $\gamma$ misses $S$. Indeed, if
$s\in S$ intersects $\gamma$ then $s\in V(\omega)$. Let $t_{1}$ be the vertex
at which $\alpha^{\prime}$ meets $\omega$. Then
$d(x_{0},t_{1})<d(x_{0},s)+d(s,t_{1})\leq r+R_{1}$, contradicting the fact
that $d(x_{0},t_{1})>r+R_{1}+1$. Hence $V(\gamma)\cap S=\varnothing$.
Therefore $x\in M^{\prime}_{1}(y,S)$, which establishes that
$M^{\prime}(a_{A},A)\subseteq M^{\prime}_{1}(y,S)\subseteq U$. However, this
conclusion contradicts our initial assumption that $a_{A}$ is rowdy in $A$. ∎

Next we define a uniformity $\mathcal{D}$ on $\Delta K$ that induces the given
topology.

######  Definition 3.4.

Let $J$ be any finite set of edges in $K$. Define an entourage
$D_{J}\subseteq\Delta K\times\Delta K$ that contains all pairs $(a,b)$ such
that there exists a geodesic from $a$ to $b$ that does not contain an edge
from $J$. In other words, $(a,b)\in D_{J}$ if and only if $b\in
P^{\prime}(a,J)$. The proof of Proposition 3.3 shows that the family of
entourages $D_{J}$ for all finite edge sets $J$ is a base for the uniform
structure $\mathcal{D}$ on the compact space $\Delta K$ (see [Wil70, §36]).

######  Remark 3.5 (Gerasimov’s definition).

To any fine $\delta$–hyperbolic graph $K$ we have associated a canonical
compact Hausdorff space $\Delta K$ that contains the vertex set $V(K)$ as a
dense subspace. Using a different construction, Gerasimov associates a compact
Hausdorff space $M=M(K)$ to $K$ such that $M$ also contains $V(K)$ as a dense
subspace (see [Ger12] and [GP16, §2]). We note that the space $M$ of Gerasimov
coincides with the space $\Delta K$ constructed above, in the sense that the
identity map on $V(K)$ extends to a unique homeomorphism $M\to\Delta(K)$.

Indeed, the canonical uniform structure $\mathcal{D}$ on $\Delta K$ restricts
to a uniform structure, also denoted $\mathcal{D}$, on $V(K)$. By Proposition
3.3, the space $\Delta K$ is compact and, thus, complete as a uniform space.
As Gerasimov defines $M$ to be the Cauchy–Samuel completion of this same
uniform space $\bigl{(}V(K),\mathcal{D}\bigr{)}$, the claim follows
immediately from the uniqueness of completions.

If $(\Gamma,\mathbb{P})$ is relatively hyperbolic with
$(\Gamma,\mathbb{P})$–graph $K$, then the action of $\Gamma$ on $V$ induces a
$32$–action of the pair $(\Gamma,\mathbb{P})$ on $\Delta K=M(K)$ by [Ger12].
Thus $\Delta K$ may be identified with the Bowditch boundary
$\partial(\Gamma,\mathbb{P})$ using the uniqueness theorem of [Bow12] in the
finitely generated case and of [GP16] in general (see Definition 2.6).

##  4\. Trees of compacta

Suppose $(\Gamma,\mathbb{P})$ is relatively hyperbolic and $\Gamma$ acts on a
simplicial tree $T$ relative to $\mathbb{P}$ with infinite, finitely generated
parabolic edge stabilizers. In this section, we show that the boundary of
$(\Gamma,\mathbb{P})$ may be described as the limit of a tree of compacta
formed by gluing the boundaries of the vertex groups along parabolic points in
the pattern of the tree $T$ and then compactifying. The main results of this
section are inspired by the treelike structures associated to peripheral
splittings in [Bow01] and [Dah03]. The inverse limit construction is a minor
variation of a construction due to Świątkowski [Świ20], which itself is
modeled on the Ancel–Siebenmann characterization of trees of manifolds [AS85,
Jak91].

###  4.1. The limit of a tree system

######  Definition 4.1.

A _tree system_ $\Theta$ of compacta consists of the following data:

  1. (1)

A bipartite tree $T$ with bipartition $V(T)=V_{0}(T)\sqcup V_{1}(T)$.

  2. (2)

To each vertex $v$ of $V_{0}(T)$, we associate a compact space $M_{v}$ called
a _component space_. For each $w\in V_{1}$ we associate a one-point space
$M_{w}$, known as a _peripheral point_.

  3. (3)

To each oriented edge $\vec{e}\in E(T)$, we associate a one-point space
$\Sigma_{\vec{e}}$ and a map $i_{\vec{e}}\colon\Sigma_{\vec{e}}\to
M_{t(\vec{e}\,)}$. We require that in each component space $M_{v}$ the
adjacent edge spaces have distinct images.

Let $\\#\Theta$ denote the quotient space of $\bigsqcup_{v\in V(T)}M_{v}$ by
the equivalence relation generated by $i_{\vec{e}}(\Sigma_{\vec{e}})\sim
i_{-\vec{e}}(\Sigma_{-\vec{e}})$ for all oriented edges $\vec{e}\in E(T)$.

######  Definition 4.2.

Suppose $S$ is any subtree of $T$. Let $V(S)=V_{0}(S)\sqcup V_{1}(S)$ and
$E(S)$ denote the set of vertices and the set of edges respectively of $S$.
Let $\Theta_{S}$ be the restriction of the tree system $\Theta$ to the subtree
$S$.

######  Definition 4.3 (Limit of a tree system).

For each finite subtree $F$ of $T$, the _partial union_ is the compact space
$M_{F}=\\#\Theta_{F}$. For a pair of subtrees $F_{1}\subseteq F_{2}$, the
_collapse map_ $f_{F_{1}F_{2}}\colon M_{F_{2}}\to M_{F_{1}}$ is the quotient
map obtained by collapsing $M_{v}$ to a point for each $v\in
V(F_{2})-V(F_{1})$. Notice that $f_{F_{1}F_{2}}\circ
f_{F_{2}F_{3}}=f_{F_{1}F_{3}}$ whenever $F_{1}\subseteq F_{2}\subseteq F_{3}$.
Hence the system of spaces $M_{F}$ and maps $f_{FF^{\prime}}$ with $F\subseteq
F^{\prime}$ is an inverse system of compacta indexed by the poset of all
finite subtrees $F$ of $T$.

The _limit_ $\lim\Theta$ _of the tree system_ $\Theta$ is the inverse limit of
the above inverse system. Observe that $\lim\Theta$ is compact and Hausdorff,
since it is an inverse limit of compact Hausdorff spaces.

###  4.2. The boundary as a tree system

Suppose $(\Gamma,\mathbb{P})$ is relatively hyperbolic. Let $T$ be any
simplicial tree on which $\Gamma$ acts relative to $\mathbb{P}$ with finitely
many orbits of edges such that all edge stabilizers are infinite, parabolic,
and finitely generated. Let $T_{c}$ be the associated tree of cylinders. Fix a
$(\Gamma,\mathbb{P})$–graph $K$ compatible with this splitting.

######  Definition 4.4.

A _ray_ in a tree is a subgraph isomorphic to $[0,\infty)$ with its natural
structure as an infinite graph. Two rays are _equivalent_ if their
intersection is a ray. An equivalence class of rays is an _end_ of the tree.

######  Definition 4.5.

Let $K^{\prime}$ be the barycentric subdivision of $K$. We consider
$K^{\prime}$ and $T_{c}$ as length metric spaces in which each edge has length
one. Consider the map $f\colon K^{\prime}\to T_{c}$ defined as follows. For
each $v\in V_{0}(T_{c})$ send the barycenter of each edge of $K_{v}$ to the
vertex $v$. We map each vertex $y\in K$ to the unique cylinder $w\in
V_{1}(T_{c})$ fixed by $\Gamma_{w}$ if such a cylinder exists, and otherwise
map $y$ to the unique vertex $v\in V_{0}(T_{c})$ fixed by $\Gamma_{u}$.
Whenever vertices $x,y$ of $K^{\prime}$ are adjacent, their images $f(x)$ and
$f(y)$ coincide or are adjacent. We extend $f$ linearly to the edges of
$K^{\prime}$ in the natural way.

######  Lemma 4.6.

The map $f\colon K^{\prime}\to T_{c}$ induces a bijection $\partial
f\colon\partial_{T}K\to\operatorname{Ends}(T_{c})$, where
$\partial_{T}K\subset\partial K$ is the set of points represented by geodesic
rays that have bounded intersection with each subgraph $K_{v}$ of $K$.

###### Proof.

The map $f$ is Lipschitz, by construction. Since each subgraph $K_{v}$ is
convex and its image in $T_{c}$ has diameter $2$, it follows from
Kapovich–Rafi [KR14, Prop. 2.5] that $f$ is alignment preserving in the sense
of Dowdall–Taylor [DT17]. The result is a direct consequence of [DT17, Thm.
3.2]. ∎

The _tree system associated to $T_{c}$_ is the system $\Theta(T_{c})$ with
underlying tree $T_{c}$ whose vertex spaces are the compacta $M_{v}=\Delta
K_{v}$ for $v\in V(T_{c})$ such that each edge map $i_{\vec{e}}$ has image
equal to the unique point of $\Delta K_{t(\vec{e}\,)}$ fixed by $\Gamma_{e}$.
By the definition of the tree of cylinders, if $\vec{e}_{0}$ and $\vec{e}_{1}$
are distinct oriented edges of $T_{c}$ with terminal vertex $v$, the parabolic
groups $\Gamma_{e_{0}}$ and $\Gamma_{e_{1}}$ are not co-elementary. Therefore,
the vertices of $K_{v}$ fixed by $\Gamma_{e_{0}}$ and $\Gamma_{e_{1}}$ are
distinct.

######  Proposition 4.7.

There is a function
$\rho\colon\\#\Theta(T_{c})\cup\operatorname{Ends}(T_{c})\to\Delta K$ with the
following properties:

  1. (1)

$\rho$ is a bijection.

  2. (2)

$\rho$ is continuous on $\\#\Theta(T_{c})$.

  3. (3)

For each $v\in V(T_{c})$ the restriction of $\rho$ to $\Delta K_{v}$ is the
natural inclusion $\Delta K_{v}\to\Delta K$.

  4. (4)

The restriction of $\rho$ to $\operatorname{Ends}(T_{c})$ is the map $\partial
f^{-1}\colon\operatorname{Ends}(T_{c})\to\partial_{T}K$

###### Proof.

Since the subgraphs $K_{v}$ are convex, a geodesic ray of $K$ having unbounded
intersection with some $K_{v}$ represents a point of $\partial K_{v}$. Thus
the complement in $\Delta K$ of the set $\partial_{T}K$ equals the union of
the sets $\Delta K_{v}$.

For each $v,v^{\prime}\in V_{0}(T_{c})$, the sets $\Delta K_{v}$ and $\Delta
K_{v^{\prime}}$ intersect precisely when the subgraphs $K_{v}$ and
$K_{v^{\prime}}$ intersect, in which case their intersection is a single
vertex $w$ of $K$ and the corresponding vertex of $V_{1}(T_{c})$ is incident
to $v$ and $v^{\prime}$. Thus the system of inclusions $\Delta K_{v}\to\Delta
K$ induces a continuous injection $\\#\Theta(T_{c})\to\Delta K$ with image
$\Delta K-\partial_{T}K$. ∎

######  Definition 4.8 (Halfspaces).

Let $\vec{e}$ be any oriented edge of $T_{c}$. Removing the open edge $e$ from
$T_{c}$ leaves two components $T_{c}(\vec{e}\,)$ and $T_{c}(-\vec{e}\,)$, the
first containing the vertex $t(\vec{e}\,)$ and the second containing
$o(\vec{e}\,)$. The _halfspaces_ of $\partial(G,\mathbb{P})$ corresponding to
$\vec{e}$ are the sets $H(\vec{e}\,)$ and $H(-\vec{e}\,)$, where

$H(\vec{e}\,)=\rho\Biggl{(}\operatorname{Ends}T_{c}(\vec{e}\,)\cup{\bigcup_{v\in
V(T_{c}(\vec{e}\,))}\\!\\!\\!\\!\Delta K_{v}}\Biggr{)}.$

######  Proposition 4.9.

For each oriented edge $\vec{e}$ of $T_{c}$, the halfspaces $H(\vec{e}\,)$ and
$H(-\vec{e}\,)$ are closed sets of $\Delta K$ with union $\Delta K$ and with
intersection the one-point space $K_{w}$, where $w$ is the vertex of
$V_{1}(T_{c})$ incident to $e$.

###### Proof.

According to Proposition 4.7, it suffices to show the following. Suppose $K$
is a fine $\delta$–hyperbolic graph that is the union of two fine
$\delta$–hyperbolic graphs $K_{1}$ and $K_{2}$ whose intersection is a single
vertex $x$. Then $\Delta K_{1}$ and $\Delta K_{2}$ are closed sets with union
$\Delta K$ and with intersection $\\{x\\}$. The assertions about the union and
intersection are evident since, as a set, $\Delta K$ is the union of the
vertex set and the Gromov boundary of $K$.

For each $y\in\Delta K_{2}-\\{x\\}$, the basic open neighborhood
$M^{\prime}\bigl{(}y,\\{x\\}\bigr{)}$ is contained in $\Delta K_{2}-\\{x\\}$.
Thus $\Delta K_{2}-\\{x\\}$ is open in $\Delta K$, and its complement $\Delta
K_{1}$ is closed. By symmetry, $\Delta K_{2}$ is also closed. ∎

A family of subsets $\mathcal{C}$ of a topological space $X$ is _null_ if for
each open cover $\mathcal{U}$ of $X$, all but finitely many members of
$\mathcal{C}$ are $\mathcal{U}$–small. A set is _$\mathcal{U}$ –small_ if it
is contained in some $U\in\mathcal{U}$. If $\mathcal{U}$ is an open cover of
$X$, the _star of a set $A\subset X$ respect to $\mathcal{U}$_, denoted
$\operatorname{Star}(A,\mathcal{U})$, is the union of all members of
$\mathcal{U}$ that intersect $A$. An open cover $\mathcal{V}$ of $X$ is a
_star refinement_ of an open cover $\mathcal{U}$ if for each $V\in\mathcal{V}$
there exists $U\in\mathcal{U}$ such that
$\operatorname{Star}(V,\mathcal{V})\subseteq U$. Every open cover of a compact
Hausdorff space has an open star refinement by [Eng89, Thm. 5.1.12]. The
following is a variation of a well-known result in decomposition space theory.

######  Proposition 4.10.

If $\mathcal{C}$ is a null family of closed, pairwise disjoint subsets of a
compact Hausdorff space $X$, the quotient space $X/\mathcal{C}$ is Hausdorff.

###### Proof.

Let $\hat{\mathcal{C}}$ be the partition of $X$ consisting of the members of
$\mathcal{C}$ and all singleton sets $\\{x\\}$ such that $x$ is not contained
in any member of $\mathcal{C}$. We first show that for each open set $U$ of
$X$, the union $U^{*}$ of all members of $\hat{\mathcal{C}}$ contained in $U$
is an open set. Suppose $U$ is open and $x\in U^{*}$. Then $x\in C$ for some
$C\in\hat{\mathcal{C}}$ with $C\subseteq U$. Let $\mathcal{V}$ be an open star
refinement of the open cover $\mathcal{U}=\\{U,X-C\\}$ and let
$C_{1},\dots,C_{k}$ be the members of $\mathcal{C}-\\{C\\}$ that are not
$\mathcal{V}$–small. Then the set
$W=\operatorname{Star}(C,\mathcal{V})-(C_{1}\cup\cdots\cup C_{k})$ is an open
neighborhood of $x$ contained in $U^{*}$, so that $U^{*}$ is open. Since $X$
is normal, it follows immediately that $X/\mathcal{C}=X/\hat{\mathcal{C}}$ is
Hausdorff. ∎

######  Proposition 4.11.

Let $\mathcal{U}$ be any open cover of $\Delta K$. For all but finitely many
oriented edges $\vec{e}$ of $T_{c}$, at least one of $H(\vec{e}\,)$ and
$H(-\vec{e}\,)$ is $\mathcal{U}$–small.

###### Proof.

Let $J$ be a finite set of edges of $K$, and let $D_{J}\subset\Delta
K\times\Delta K$ be the associated entourage (see Definition 3.4). A set
$Y\subset\Delta K$ has _diameter less than $D_{J}$_ if $(x,y)\in D_{J}$ for
all $x,y\in Y$. By compactness, it suffices to show the following holds for
any $J$. For all but finitely many edges $\vec{e}$ of $T_{c}$, one of the
halfspaces $H(\vec{e}\,)$ and $H(-\vec{e}\,)$ has diameter less than $D_{J}$.

Let $S$ be the finite set of vertices $s\in V_{0}(T_{c})$ such that $K_{s}$
contains an edge of $J$. If an edge ${e}$ of $T_{c}$ is not contained in the
convex hull $\bar{S}$ of $S$ then $e$ may be oriented so that the halftree
$T_{c}(\vec{e}\,)$ is disjoint from $\bar{S}$. For each such edge, the
halfspace $H(\vec{e}\,)$ has diameter less than $D_{J}$, completing the proof
since $\bar{S}$ has only finitely many edges. ∎

######  Proposition 4.12.

The Bowditch boundary $\Delta K$ is homeomorphic to the limit
$\lim\Theta(T_{c})$ of the tree system associated to $T_{c}$.

###### Proof.

For each finite subtree $F$ of $T_{c}$, we define a map $\mu_{F}\colon\Delta
K\to M_{F}$, where $M_{F}$ is the corresponding partial union. Let $E_{F}$ be
the set of all oriented edges with $t(\vec{e}\,)$ in $T_{c}-F$ and with
$o(\vec{e}\,)$ in $F$. We first show that the family of halfspaces
$H(\vec{e}\,)$ with $\vec{e}\in E_{F}$ is null. Let $\mathcal{U}$ be an
arbitrary open cover of $\Delta K$. Form a new cover $\mathcal{U}^{\prime}$ as
follows. Let $a\neq b$ be two points of the partial union $M_{F}$. Define
$\mathcal{U}^{\prime}$ to contain all open sets of $\mathcal{U}$, except that
each set $U\in\mathcal{U}$ with $M_{F}\subset U$ is replaced by the pair of
sets $U-\\{a\\}$ and $U-\\{b\\}$. Then the cover $\mathcal{U}^{\prime}$
refines $\mathcal{U}$, and $M_{F}$ is not $\mathcal{U}^{\prime}$–small.
Nullity now follows by applying Proposition 4.11 to the family of edges
$E_{F}$.

Let $\Delta K/{\sim}$ be the quotient in which each halfspace $H(\vec{e}\,)$
with $\vec{e}\in E_{F}$ is collapsed to a point. The inclusion $M_{F}\to\Delta
K$ induces a continuous bijection $M_{F}\to\Delta K/{\sim}$. According to
Proposition 4.10, the quotient $\Delta K/{\sim}$ is Hausdorff, so this
bijection is a homeomorphism. We define $\mu_{F}$ to be the continuous
retraction $\Delta K\to\Delta K/{\sim}\to M_{F}$. Since the maps $\mu_{F}$
commute with the maps of the inverse system, they induce a continuous map
$\mu\colon\Delta K\to\lim\Theta(T_{c})$. As each $\mu_{F}$ is a surjective map
from a compact space to a Hausdorff space, $\mu$ is closed and surjective
[Eng89, Thm. 3.2.14].

To see that $\mu$ is injective, it suffices to show that

$\mu\circ\rho\colon\\#\Theta(T_{c})\cup\operatorname{Ends}(T_{c})\to\lim\Theta$

is injective, where $\rho$ is the bijection from Proposition 4.7. If $x\neq y$
lie in $\\#\Theta$ then there exists a finite subtree $F$ of $T_{c}$ such that
$x,y\in M_{F}$. Since $\mu_{F}$ is a retraction, we have
$\mu_{f}\circ\rho(x)=x\neq y=\mu_{f}\circ\rho(y)$. Thus
$\mu\circ\rho(x)\neq\mu\circ\rho(y)$.

If $x\neq y$ lie in $\operatorname{Ends}(T_{c})$, let $v\in V_{0}(T_{c})$ be a
vertex on the geodesic from $x$ to $y$. Since the images of distinct edge
spaces are distinct in $\Delta K_{v}$, it follows that $\mu_{v}\circ\rho$ maps
$x$ and $y$ to distinct points of $M_{v}=\Delta K_{v}$. The case when
$x\in\\#\Theta$ and $y\in\operatorname{Ends}(T_{c})$ is similar. In either
case, we see that $\mu\circ\rho(x)\neq\mu\circ\rho(y)$. Thus $\mu\circ\rho$ is
injective. ∎

##  5\. Connectedness properties of boundaries

This section discusses criteria for the connectedness and local connectedness
of the Bowditch boundary.

######  Proposition 5.1.

Let $(\Gamma,\mathbb{P})$ be relatively hyperbolic. The boundary
$M=\partial(\Gamma,\mathbb{P})$ is connected if and only if $\Gamma$ is one
ended relative to $\mathbb{P}$.

This proposition was established in the finitely generated case by Bowditch
[Bow12, Thm. 10.1]. The proof of the forward implication is similar to that
given by Bowditch. Bowditch’s argument for the reverse implication is valid
only when $\Gamma$ is finitely generated. The general case is easily reduced
to the finitely generated case using Propositions 2.13 and 4.12.

###### Proof.

We first assume that $\Gamma$ admits a nontrivial action relative to
$\mathbb{P}$ on a simplicial tree $T$ with finite edge stabilizers. If $K$ is
any $(\Gamma,\mathbb{P})$–graph, we may map the vertex set $V(K)$
equivariantly to the vertex set $V(T)$ of $T$. Each edge $e$ of $T$, separates
$T$ into two halftrees $T_{1}$ and $T_{2}$. Let $A_{i}$ be the set of vertices
of $K$ mapping to the halftree $T_{i}$. Then the set $J$ of edges of $K$
connecting a vertex of $A_{1}$ to a vertex of $A_{2}$ is a finite set. If
$a_{1}$ and $a_{2}$ are vertices of $A_{1}$ and $A_{2}$ respectively, then the
basic open sets $P^{\prime}(a_{1},J)$ and $P^{\prime}(a_{2},J)$ are disjoint
and partition $\Delta K=M$. Thus $M$ is disconnected.

For the converse, assume that $\Gamma$ is one ended relative to $\mathbb{P}$.
Let $T_{c}$ be the JSJ tree of cylinders over parabolic subgroups given by
Proposition 2.13. Then each vertex group $\Gamma_{v}$ with $v\in V_{0}(T_{c})$
is finitely generated and the pair $(\Gamma_{v},\mathbb{Q}_{v})$ is relatively
hyperbolic. Recall that
$\mathbb{Q}_{v}=\operatorname{Inc}_{v}\cup\mathbb{P}_{|\Gamma_{v}}$ is the
peripheral structure given by Proposition 2.9. Furthermore, $\Gamma_{v}$ is
one ended relative to $\mathbb{Q}$.

As $\Gamma_{v}$ is finitely generated, one-endedness of the relatively
hyperbolic pair $(\Gamma_{v},\mathbb{Q}_{v})$ implies that its Bowditch
boundary $\partial(\Gamma_{v},\mathbb{Q}_{v})$ is connected by [Bow12, Prop.
10.1]. According to Proposition 4.12, the boundary
$\partial(\Gamma,\mathbb{P})$ is the inverse limit of a tree system whose
vertex spaces are all connected. In particular, the factor spaces $M_{F}$ of
the inverse system are connected. We conclude that the inverse limit is
connected, since an inverse limit of compact, connected Hausdorff spaces is
always connected [Eng89, Thm. 6.1.20]. ∎

The following result of Capel provides a criterion for the local connectedness
of an inverse limit of compacta.

######  Theorem 5.2 ([Cap54]).

Let $\\{X_{\alpha}\\}$ be an inverse system such that each bonding map
$X_{\alpha}\to X_{\alpha^{\prime}}$ is monotone. If each factor space
$X_{\alpha}$ is compact and locally connected then the inverse limit
$\varprojlim X_{\alpha}$ is locally connected.

Recall that a map $f\colon Y\to Z$ is _monotone_ if $f$ is surjective and for
each $z\in Z$ the set $f^{-1}(z)$ is compact and connected.

######  Theorem 5.3.

Let $(\Gamma,\mathbb{P})$ be a relatively hyperbolic group pair, and suppose
the boundary $M=\partial(\Gamma,\mathbb{P})$ is connected. Let $T_{c}$ be the
tree of cylinders for a splitting of $\Gamma$ relative to $\mathbb{P}$ over
infinite parabolic subgroups. Suppose for each $v\in V_{0}(T_{c})$ that the
Bowditch boundary $\partial(\Gamma_{v},\mathbb{Q}_{v})$ is locally connected.
Then $M$ is also locally connected.

###### Proof.

By Proposition 4.12, the boundary $M$ is an inverse limit of a tree system
$\Theta$ corresponding to the splitting $T_{c}$. For each finite subtree $F$
of $T_{c}$, the partial union $K_{F}$ is formed by gluing finitely many
component spaces $K_{v}$ in the pattern of $F$. Since each $K_{v}$ is
connected, $K_{F}$ is again connected. As a quotient of a compact, locally
connected space, $K_{F}$ is also compact and locally connected.

To apply Theorem 5.2, we only need to show that the bonding map $K_{F_{2}}\to
K_{F_{1}}$ is monotone whenever $F_{1}$ is a subtree of $F_{2}$. But,
according to Definition 4.3, each nontrivial point preimage has the form
$K_{F}$ for some finite subtree $F$, and all such sets are compact and
connected. ∎

##  6\. Pretrees and non-nesting actions

This section reviews the definition and basic properties of pretrees, a
general treelike structure explored extensively by Bowditch in [Bow99c]. A key
result of this section is Lemma 6.5, which states that a non-nesting action on
a pretree induces a non-nesting action on its quotient by any equivariant full
relation. The results of this section lay the foundations for applications to
the cut point structure of boundaries in Sections 7 and 8.

######  Definition 6.1.

Let $K$ be a set with a ternary relation called _betweenness_. If $y$ is
between $x$ and $z$, we write $xyz$. Such a structure is a _pretree_ if it
satisfies the following:

  1. (1)

If $y$ is between $x$ and $z$, then $x\neq z$.

  2. (2)

$y$ is between $x$ and $z$ if and only if $y$ is between $z$ and $x$.

  3. (3)

If $y$ is between $x$ and $z$, then $z$ is not between $x$ and $y$.

  4. (4)

If $xyz$ and $y\neq w$ then either $xyw$ or $wyz$.

For points $x,y$ in a pretree $K$, the _open interval_ $(x,y)$ is the set of
all $z$ between $x$ and $y$. The _closed interval_ $[x,y]$ is the set
$(x,y)\cup\\{x,y\\}$. Half-open intervals are defined similarly. Distinct
points $x\neq y$ are _adjacent_ if the interval $(x,y)$ is empty. A pretree is
_discrete_ if every interval $(x,y)$ is finite. A subset $Q$ of a pretree $K$
is _dense_ in $K$ if for all distinct $x,y\in K$ the interval $(x,y)$ contains
at least one point of $Q$. We say $K$ is _dense_ if it is dense in itself.

A subset $A$ of a pretree $K$ is _full_ if $[x,y]\subseteq A$ for all $x,y\in
A$. An $arc$ of a pretree is a nonempty full subset $A$ such that for all
distinct $x,y,z\in A$, we have $xyz$ or $yzx$ or $zxy$. A pretree is
_complete_ if every arc is an interval. A pretree is _interval complete_ if
every arc contained in an interval is itself an interval. We note that if $K$
is a complete pretree, then any full subpretree of $K$ is interval complete.

Suppose $K$ is a pretree. The _median_ $c=\text{med}(x,y,z)$ of a triple of
points $x,y,z\in K$ is the unique point $c\in[x,y]\cap[y,z]\cap[z,x]$. A
_median pretree_ is a pretree in which every triple has a median.

######  Definition 6.2 (Connector points).

Let $K$ be any complete median pretree. Let $X$ and $Y$ be nonempty, full,
disjoint subsets of $K$ such that $X\cup Y$ is also full. A point $p(Y,X)\in
Y$ is a _connector point_ of the ordered pair $(Y,X)$ in $K$ if we have
$[x,p]\cap Y=\\{p\\}$ for all $x\in X$.

Existence and uniqueness of connector points is given by the following lemma
due to Bowditch.

######  Lemma 6.3 ([Bow99c], Lem. 6.13).

Let $K$ be a complete median pretree, and let $X$ and $Y$ be disjoint, full,
nonempty subsets such that $X\cup Y$ is full. If the connector point $p(Y,X)$
exists, it is unique. Furthermore, at least one of the pairs $(X,Y)$ and
$(Y,X)$ has a connector point.

A _full relation_ on a pretree $K$ is an equivalence relation for which every
equivalence class is full. If $\sim$ is a full equivalence relation,
$[x]_{\sim}$ denotes the equivalence class of the point $x$. The set
$K/{\sim}$ has a natural induced pretree structure such that $Y$ is between
$X$ and $Z$ if and only if there exists $y_{0}\in Y$ such that $xy_{0}z$ for
all $x\in X$ and $z\in Z$. If a pretree $K$ is median or complete, then so is
$K/{\sim}$ (see [Bow99c, Lem. 4.2] for details). A point $x$ in a pretree $K$
is _terminal_ if there exists no $y,z\in K$ such that $yxz$. The interior
$K^{o}$ of a pretree $K$ is the subpretree obtained by removing all terminal
points.

######  Lemma 6.4.

Let ${K}$ be any pretree with a full relation $\sim$. There exists a full
subpretree $K^{\prime}$ contained in the interior $K^{o}$ that is invariant
under all automorphisms of $K$ and maps onto the interior of $K/{\sim}$.

###### Proof.

If $X\in{K}/{\sim}$ lies between points $Y$ and $Z$, then by definition, there
exists $x\in X$ lying between every pair $y,z\in{K}$ with $y\in Y$ and $z\in
Z$. Thus, $x$ is an interior point of ${K}$ mapping to $\bar{x}$. Let
$K^{\prime}$ be the smallest full subset of $K$ containing all such points $x$
and invariant under all automorphisms of $K$. Then $K^{\prime}$ has the
desired properties. ∎

An action of a group $\Gamma$ on a pretree $K$ is _non-nesting_ if no
$g\in\Gamma$ maps an interval of $K$ properly inside itself.

######  Lemma 6.5.

If $\Gamma$ has a non-nesting action on a complete median pretree ${K}$, then
the induced action on ${K}/{\sim}$ is non-nesting for any equivariant full
relation $\sim$.

###### Proof.

By way of contradiction, suppose $g\bigl{(}[X,Y]\bigr{)}\subsetneq[X,Y]$ for
some $g\in\Gamma$ and some interval $[X,Y]\subseteq K/{\sim}$. Replacing $g$
with $g^{2}$, we may assume that the restriction $g\big{|}[X,Y]$ is
orientation preserving. We consider two cases, depending on whether $g$ fixes
an endpoint of $[X,Y]$.

Suppose $g(X)\in\bigl{(}X,g(Y)\bigr{)}$ and $g(Y)\in\bigl{(}g(X),Y\bigr{)}$.
Then there exist $a\in g(X)$ and $b\in g(Y)$ with
$a\in\bigl{(}g^{-1}(a),b\bigr{)}$ and $b\in\bigl{(}a,g^{-1}(b)\bigr{)}$. But
then $(a,b)\subsetneq\bigl{(}g^{-1}(a),g^{-1}(b)\bigr{)}$, contradicting that
$\Gamma$ is non-nesting on $K$.

On the other hand, suppose $g(X)=X$. Then $g(Y)\in(X,Y)$, so we may choose
$b\in Y$ so that $g(b)$ is between $x$ and $y$ for any $x\in X$ and $y\in Y$.
Let

$T=\bigl{\\{}\,{t\in K-(X\sqcup Y)}\bigm{|}{\text{$t\in(x,y)$, for some $x\in
X$ and $y\in Y$}}\,\bigr{\\}}.$

We show that $T$ is full. Suppose $t_{1},t_{2}\in T$ and choose
$r\in(t_{1},t_{2})$. Since $t_{i}\in T$ there exist points $x_{i}\in X$ and
$y_{i}\in Y$ such that $x_{i}t_{i}y_{i}$ holds. Suppose
$m_{1}=\text{med}(x_{1},x_{2},t_{1})$ and
$m_{2}=\text{med}(y_{1},y_{2},t_{2})$. Since $X$ and $Y$ are full, we have
$m_{1}\in X$ and $m_{2}\in Y$. Since we have $x_{2}t_{2}y_{2}$,
$t_{2}m_{2}y_{2}$, and $t_{2}m_{2}y_{1}$, we also have $x_{2}t_{2}m_{2}$ and,
thus, $t_{2}\in(x_{2},y_{1})$. A similar argument establishes that
$t_{1}\in(x_{2},y_{1})$, so that $r\in(x_{2},y_{1})$. Since $t_{1}$ and
$t_{2}$ are not members of $X$, it follows that $r\notin X$. Similarly
$r\not\in Y$. Hence $r\in T$, so that $T$ is full.

Notice that $T$ is non-empty as it contains $g(b)$. To see that $X\cup T$ is
full, choose $x\in X$ and $t\in T$ and $s\in(x,t)$. Since $t\in(x,y)$ for some
$y\in Y$, we have $s\in(x,y)$ by [Bow99c, Lem. 2.3]. Clearly if $s$ is not a
member of $X$ then $s$ is a member of $T$ by definition of $T$. Since the
members of $T$ belong to different equivalence classes (possibly multiple
equivalence classes) than the members of $X$, the connector point $p$ of $X,T$
is unique. Since $X$ is stabilized as a set by $g$, the connector point $p$ is
fixed by $g$. This leads to a contradiction, since $\bigl{[}g(p),g(b)\bigr{]}$
is a proper subset of $[p,b]$ in $K$. ∎

Let $\Gamma$ be a group acting on a pretree $K$. The action is _trivial_ if
some $x\in K$ is fixed by the whole group. An automorphism $g$ of $K$ is
_elliptic_ if the fixed point set $Fix(g)$ is nonempty, it is an _inversion_
if some pair of adjacent points is transposed by $g$.

######  Lemma 6.6.

Suppose $P$ is a finitely generated group with a non-nesting action on an
interval complete, median pretree $K$. If $P$ fixes a point of $K$ and
$K^{\prime}$ is any $P$–invariant, full subpretree of $K$, then $P$ fixes a
point of $K^{\prime}$.

###### Proof.

Choose $p\in P$. Since the action is non-nesting, $\operatorname{Fix}(p)$ is
full and for any $x\in K$ the interval $[x,px]$ intersects
$\operatorname{Fix}(p)$ in precisely one point by [BC01, Lemma 3.7]. If we
choose $x\in K^{\prime}$, then the fixed point $y$ is also in $K^{\prime}$,
since $K^{\prime}$ is full. Hence, each element of $P$ has a fixed point in
$K^{\prime}$. As $K^{\prime}$ is also interval complete and median and the
action of $P$ on $K^{\prime}$ is non-nesting, the proof of [BC01, Lemma 5.2]
implies that $P$ has a fixed point in $K^{\prime}$. ∎

##  7\. The cut point pretree

In this section, we review a construction due to Swenson [Swe00] of a cut
point pretree and cut point tree dual to a family of cut points in a
continuum. We establish Proposition 7.8, which describes the structure of this
pretree when the continuum admits a convergence group action.

We recall that a _convergence group action_ is an action on a compact
Hausdorff space $M$ containing at least three points such that the induced
action on the space of distinct triples $\Theta^{3}(M)$ is proper. The action
is _minimal_ if $M$ has no proper subset that is nonempty, closed, and
$\Gamma$–invariant.

The main application of the results below is to the Bowditch boundary of a
finitely generated relatively hyperbolic group. The compact Hausdorff topology
on the boundary $\partial(\Gamma,\mathbb{P})=\Delta K$ defined in Section 3
has a countable basis whenever $\Gamma$ is countable, so the boundary is
metrizable in this setting.

In the sequel, we study convergence actions only on metrizable spaces, usually
requiring connectedness as well. We refer to Tukia [Tuk94] for the basic
structure theory of convergence groups on metrizable spaces.

######  Definition 7.1 (Continua).

For the purposes of this paper, a _continuum_ is a nonempty, compact,
connected, metrizable topological space. A continuum $M$ is _nontrivial_ if it
contains more than one point, in which case it has the cardinality
$\mathfrak{c}$ of the real line [Eng89, 4.5.5]. Any continuum $M$ containing a
cut point must contain at least three points, so that $M$ is nontrivial.

######  Definition 7.2 ($\mathbb{R}$–trees).

An _$\mathbb{R}$ –tree_ is a uniquely arc-connected geodesic metric space, or
equivalently a geodesic space with no subspace homeomorphic to the circle. A
_topological $\mathbb{R}$–tree_ is the underlying topological space of an
$\mathbb{R}$–tree. A _dendrite_ is a compact topological $\mathbb{R}$–tree.

For each continuum $M$ equipped with a preferred family of cut points $C$, a
pretree ${K}$ dual to $C$ is constructed in [Swe00] and described briefly
below.

######  Definition 7.3 (Cut point pretree).

Let $M$ be a continuum, and let $C$ be a nonempty family of cut points of $M$.
A pretree structure on $M$ is given by the following betweenness relation: $c$
is _between_ $a$ and $b$ if $c\in C$ and $M-\\{c\\}$ has a separation into
disjoint nonempty open sets $U$ and $V$ with $a\in U$ and $b\in V$. Consider
the following equivalence relation on $M$.

  1. (1)

If $c\in C$, then $c$ is equivalent to only itself.

  2. (2)

If $a,b\in M-C$ then $a\sim b$ if $(a,b)$ is empty.

Let ${K}$ denote the set of equivalence classes of this relation. The set
${K}$ has an induced pretree structure given by the following betweenness
relation:

  1. (1)

If $x,y\in K$ and $c\in C$, we say that $\\{c\\}$ is between $x$ and $y$ if
for any $a\in x$ and $b\in y$ we have $c\in(a,b)$.

  2. (2)

If $x,y,z\in{K}$ with $z$ not a cut point, we say that $z$ is between $x$ and
$y$ if for all $a\in x$, $b\in y$, and $c\in z$ we have
$[a,c)\cap(c,b]=\varnothing$.

The pretree ${K}$ is a complete median pretree (see [Swe00]), called the _cut
point pretree of $M$ dual to $C$_.

######  Lemma 7.4 ([Swe00], Lem. 4).

  1. (1)

If $M^{\prime}$ is a nontrivial subcontinuum of $M$ not separated by any point
of $C$, then the set $M^{\prime}-C$ is nonempty, dense in $M^{\prime}$, and
contained in a single equivalence class of $K$.

  2. (2)

If $x$ and $y$ are adjacent in $K$ then exactly one of them is a singleton set
$\\{c\\}$ with $c\in C$ and the other of them is a non-singleton equivalence
class whose closure contains the point $c$.

######  Corollary 7.5.

Any homeomorphism of $M$ leaving $C$ invariant induces a pretree automorphism
of $K$ that is not an inversion.

The next result is slightly stronger than the statement of [Swe00, Lem. 7],
but follows immediately from its proof.

######  Lemma 7.6 ([Swe00], Lem. 7).

Let $\Gamma$ have a convergence group action on $M$ and let $g\in\Gamma$ be
loxodromic with fixed points $a$ and $b$ in $M$. Let $K$ be the pretree of $M$
dual to a $\Gamma$–invariant family of cut points $C$.

  1. (1)

If $a$ and $b$ are separated by an element of $C$, then the equivalence
classes $[a]$ and $[b]$ in $K$ are terminal, and $[a]$ and $[b]$ are the only
fixed points of $g$ in $K$.

  2. (2)

If $a$ and $b$ are not separated by an element of $C$, then the set of fixed
points of $g$ in $K$ is equal to the closed interval
$\bigl{[}[a],[b]\bigr{]}$. This closed interval contains at most one class
$[x]$ with $x\notin C$. If the open interval $\bigl{(}[a],[b]\bigr{)}$ is
nonempty, then it contains only one element $[x]$ and we have $a,b\in C$ and
$x\notin C$.

When $C$ is countable, the following construction due to Swenson shows that
the cut point pretree may be extended to form a tree by filling the gaps
between adjacent elements. (See [Swe00, BC01, PS06] for more details.)

######  Definition 7.7 (Cut point tree).

Define

$\widehat{K}={K}\cup\bigcup_{\textrm{$x,y$ adjacent}}I(x,y)$

where $I(x,y)$ is a copy of the real open interval $(0,1)$. We take $I(y,x)$
to be the same arc with opposite orientation. There exists a canonical
extension of the interval relation to $\widehat{K}$ that endows $\widehat{K}$
with the structure of a dense, complete pretree such that each $I(x,y)=(x,y)$.
Each homeomorphism of $M$ leaving $C$ invariant induces an automorphism of
$\widehat{K}$ by extending linearly to the intervals $I(x,y)$. Furthermore,
every interval of $\widehat{K}$ is order-isomorphic to an interval of the real
line [PS06, Thm. 13]. The pretree $\widehat{K}$ is called the _cut point tree_
of $M$ dual to $C$ because it admits a canonical compact topology as a real
tree (see [Bow99c, Prop. 7.13]). We will not discuss the topological structure
here, since it is not used in this paper.

If $\Gamma$ has a convergence group action on $M$ and $C$ is an equivariant
family of cut points, the induced action on the cut point pretree inherits
many dynamical properties, as explained in the following proposition.

######  Proposition 7.8 (_cf._ [Swe00], Thm. 8).

Let $(\Gamma,\mathbb{P})$ be a group pair. Suppose $\Gamma$ has a minimal
convergence group action relative to $\mathbb{P}$ on a continuum $M$. Let $C$
be a $\Gamma$–invariant family of cut points in $M$. Suppose ${K}$ is the cut
point pretree of $M$ dual to $C$. Then the action of $\Gamma$ on $K$ is
nontrivial and without inversions. The action on the interior ${K}^{o}$ of
${K}$ is a non-nesting action such that the stabilizer of each interval of
${K}^{o}$ is elementary in the convergence group $\Gamma$. If each
$P\in\mathbb{P}$ is finitely generated, the action on ${K}^{o}$ is relative to
$\mathbb{P}$.

###### Proof.

Let $\widehat{K}$ be the cut point tree of $M$ dual to $C$. Let

$R=\widehat{K}-\\{\text{terminal points}\\}.$

The proof of [Swe00, Thm. 8] implies that the action of $\Gamma$ on $R$ is
non-nesting and the stabilizer of each interval of $R$ is an elementary
subgroup of $\Gamma$. The action of $\Gamma$ on $K$ is without inversions by
Corollary 7.5.

Next we show that the action of $\Gamma$ on ${K}^{o}$ is relative to
$\mathbb{P}$. Since the action on ${K}^{o}$ is without inversions, it suffices
to show that the action on $R$ is relative to $\mathbb{P}$. It is clear that
each $P\in\mathbb{P}$ has a fixed point in $\widehat{K}$. To complete the
proof, we need to find such a fixed point that lies in the subtree $R$.

Each $P\in\mathbb{P}$ is finitely generated with a non-nesting action on $R$.
If no point of $R$ is fixed by $P$, then by [Lev98, Thm. 3] some infinite
order element $p\in P$ acts hyperbolically on ${R}$ with axis $L$; _i.e._ ,
$p$ fixes no point in $R$, and there exists a proper embedding of the real
line in $R$ whose image $L$ is $p$–invariant and the action of $p$ on $L$ is
topologically conjugate to a translation. The closure of $L$ in $\widehat{K}$
is an interval $[L_{-\infty},L_{\infty}]$ whose endpoints are distinct
${K}$–equivalence classes of $M$ stabilized by $p$. Neither endpoint is a cut
point, since they are terminal in $\widehat{K}$, so there is a cut point
$c\in(L_{-\infty},L_{\infty})$. As $p(c)$ is another such cut point distinct
from $c$, the sets $L_{-\infty}$ and $L_{\infty}$ have disjoint closures. But
$p$ is a parabolic element of the convergence group, so it has a unique fixed
point in $M$, which is contained in any closed $p$–invariant subset [Tuk94,
§2], a contradiction. We conclude that $P$ has a fixed point in $R$.

To see that $\Gamma$ has no fixed point in $\widehat{K}$, we use the
hypothesis that $\Gamma$ acts on $M$ minimally. For each pair of open sets
$U,V\subset M$ separated by a cut point $c\in C$, the density of loxodromic
fixed point pairs [Tuk94, Thm. 2R] implies that there exists a loxodromic
element $h\in\Gamma$ with one fixed point in $U$ and one in $V$. Since its
fixed points are separated by $c$, such an element has no fixed point in $R$
by Lemma 7.6. Thus $h$ acts hyperbolically on $R$ with an axis [Lev98, Thm.
3]. Since the orbit of $c$ is dense in $M$, there exists a pair of such
loxodromic elements whose axes in $R$ do not share a common endpoint. It
follows that $\Gamma$ has no fixed point in $\widehat{K}$. ∎

The cut point tree $\widehat{K}$ has several useful dynamical properties.
However, in the next section we discuss a construction that leads to a
dendrite $D$ with much stronger properties. For instance, the action on
$\widehat{K}$ has elementary interval stabilizers, but the action on $D$ has
finite interval stabilizers (see Proposition 10.4). Lemma 10.2 discusses a
critical collapsing property of $D$ that is not easily established for
$\widehat{K}$. Moreover, $D$ is a topological quotient of $M$, which is not
the case for $\widehat{K}$.

##  8\. A dendrite quotient of the boundary

In [Bow99c], Bowditch constructs a natural dendrite quotient of any continuum
dual to a chosen family of cut points. If $M$ admits a convergence group
action by a one-ended group $\Gamma$ and the chosen cut points are
nonparabolic, then Bowditch shows that the dendrite quotient is nontrivial.
The main goal of this section is to prove Proposition 8.11, which extends
Bowditch’s result to the more general setting in which the group $\Gamma$ acts
relative to $\mathbb{P}$ and does not split relative to $\mathbb{P}$ over a
torsion subgroup.

######  Definition 8.1 (Dendrite quotient).

Let $M$ be a continuum and $C$ a set of cut points of $M$. For each pair
$x,y\in M$, the set of cut points of $M$ separating $x$ from $y$ has a natural
linear order. The _dendrite quotient of $M$ dual to $C$_, denoted
$M/{\approx}$ or $D(M,C)$, is formed from $M$ by identifying two points $x,y$
if the collection of points of $C$ separating them does not contain a subset
order-isomorphic to the rational numbers. When endowed with the quotient
topology, $D(M,C)$ is a dendrite [Bow99c].

In order to better understand the dendrite quotient, we examine the following
transfinite sequence of quotient pretrees.

######  Definition 8.2 (A transfinite sequence of quotients).

Let $M$ be a continuum with a set of cut points $C$. Let $K$ denote the cut
point pretree of $M$ dual to $C$. We recursively define a full equivalence
relation $\sim_{\alpha}$ on $K$ with quotient $\pi_{\alpha}\colon K\to
K_{\alpha}$ for each ordinal $\alpha$. The pretree $K_{0}$ is equal to $K$
itself. If $K_{\alpha}$ has been defined for some ordinal $\alpha$, we define
$K_{\alpha+1}$ by identifying each pair of points $x,y\in K$ such that the
interval between $\pi_{\alpha}(x)$ and $\pi_{\alpha}(y)$ in $K_{\alpha}$ is
finite. If $\alpha$ is a limit ordinal, we define $K_{\alpha}$ by identifying
a pair $x,y\in K$ if $\pi_{\beta}(x)=\pi_{\beta}(y)$ for some $\beta<\alpha$.

######  Proposition 8.3.

Let $M$ be a continuum, let $C$ be a set of cut points of $M$, and let $K$ be
the cut point tree. The quotient map $M\to K$ induces a bijection from the
dendrite $D(M,C)=M/{\approx}$ to the pretree $K_{\alpha_{0}}$ for some ordinal
$\alpha_{0}$. This bijection is also a pretree automorphism.

###### Proof.

For each $\alpha$ such that the natural map $K_{\alpha}\to K_{\alpha+1}$ is
not a bijection, some pair of points of $K$ is identified in $K_{\alpha+1}$
but not in $K_{\alpha}$. Thus the recursive sequence stabilizes at some
ordinal $\alpha_{0}$ of cardinality at most $\left|{K\times
K}\right|\leq\left|{M\times M}\right|=\mathfrak{c}$, where $\mathfrak{c}$ is
the cardinality of the real line.

According to [Bow99c, Lem. 4.4], the stable quotient $K_{\alpha_{0}}$ may also
be described as the quotient $K/{\approx}$ such that $x\approx y$ if the
pretree interval $(x,y)$ in $K$ does not contain a subset order-isomorphic to
the rational numbers. By Definition 7.3, the members of $K$ have two types,
singleton sets $\\{c\\}$ with $c\in C$ and equivalence classes of points of
$M$ not contained in $C$ under the relation $a\mathcal{R}b$ if $a$ and $b$ are
separated in $M$ by no point of $C$. Any two equivalence classes of the second
type are separated by a cut point from $C$, by definition. If an interval of
$K$ contains a subset $R$ order-isomorphic to the rationals, then $R\cap C$ is
also isomorphic to the rationals. Thus, $x\approx y$ holds if and only if the
members of $C$ between $x$ and $y$ do not contain a subset order isomorphic to
the rational numbers.

The relation $\approx$ on $K$ pulls back to the relation $\approx$ on $M$ such
that $x\approx y$ if the set of points of $C$ separating $x$ from $y$ does not
contain a subset order isomorphic to the rationals. Therefore, the map $M\to
K$ induces a pretree automorphism from the dendrite $D(M,C)$ to the stable
pretree $K_{\alpha_{0}}$. ∎

######  Proposition 8.4.

Let $D(M,C)$ be the dendrite quotient of $M$ dual to $C$. Let $\mathcal{R}$ be
the relation on $M$ such that $x\mathcal{R}y$ holds if $x$ and $y$ are points
of $M-C$ that are not separated by any point of $C$. If $M/\mathcal{R}$ is
endowed with the quotient topology, then $D(M,C)$ is equal to the maximal
Hausdorff quotient of $M/\mathcal{R}$.

The _maximal Hausdorff quotient_ of a space $X$ is the universal Hausdorff
quotient $HX$ of $X$ such that every map from $X$ to a Hausdorff space factors
through $HX$ (see [ML98, §V.9]). The proof of Proposition 8.4 uses the notion
of connector points and the following lemma.

######  Lemma 8.5.

Let $X$ and $Y$ be adjacent points of $K_{\alpha}$ for $\alpha\geq 1$. Then
one of the pairs $(X,Y)$ and $(Y,X)$ has a unique connector point and the
other pair has no connector point.

###### Proof.

Recall that $K$ is complete and median by Definition 7.3. When considered as
subsets of $K$, the sets $X$ and $Y$ are nonempty, disjoint, and full with
$X\cup Y$ also full. Under those circumstances, Lemma 6.3 shows that at least
one of the pairs has a unique connector point.

If the pairs $(X,Y)$ and $(Y,X)$ both had connector points, then the two
connectors $p$ and $p^{\prime}$ would be adjacent in ${K}$ and in the same
$\sim_{1}$–class, contradicting the fact that $p$ and $p^{\prime}$ are not
$\sim_{\alpha}$–equivalent. ∎

###### Proof of Proposition 8.4.

As the dendrite $D(M,C)$ is Hausdorff and equal to $K_{\alpha_{0}}$ for some
ordinal $\alpha_{0}$, it suffices to show that any continuous map from
$M/\mathcal{R}$ to a Hausdorff space factors through the quotient
$M/\mathcal{R}=K\to K_{\alpha}$.

We proceed by induction on $\alpha$. The claim is trivial for $\alpha=0$ since
$K=K_{0}$. Recall that $K_{1}$ is formed from $K$ by identifying adjacent
pairs of points $X,Y$ in $K$. If $X$ and $Y$ are adjacent in $K$,
then—considered as subsets of $M$—one is contained in the closure of the other
by Lemma 7.4. Thus $X$ and $Y$ have the same image in any Hausdorff space, and
any map from $M/\mathcal{R}$ to a Hausdorff space factors through $K_{1}$. The
inductive claim is immediate for limit ordinals, so we only need to consider
the successor case.

Suppose any map from $M/\mathcal{R}$ to a Hausdorff space factors through
$K_{\beta}$ for some $\beta\geq 1$, and let $\alpha=\beta+1$. Then
$K_{\alpha}$ is formed from $K_{\beta}$ by identifying all adjacent pairs of
points in $K_{\beta}$. Let $X$ and $Y$ be adjacent elements of $K_{\beta}$,
which we consider as full disjoint subsets of $K$ whose union is full. Without
loss of generality, assume that the connector point $p=p(Y,X)$ exists. Select
any $x\in X$, and let $\hat{C}$ be the set of all $c\in C$ separating $x$ from
$p$. Then $\\{c\\}\in X$ for each $c\in\hat{C}$, since $[x,p]\cap Y=\\{p\\}$.
Furthermore, no point of $X$ is adjacent to $p$ by Lemma 8.5.

We claim that $\hat{C}$ has a limit point in common with the set $p$. Consider
the points $x,p\in K$ as subsets of $M$, and choose $x_{0}\in x$ and $p_{0}\in
p$. Let $I$ be an irreducible subcontinuum of $M$ from $x_{0}$ to $p_{0}$,
_i.e_ , a subcontinuum of $M$ containing $x_{0}$ and $p_{0}$ such that no
proper subcontinuum of $I$ contains both $x_{0}$ and $p_{0}$. By
irreducibility, the points of $\hat{C}$ are precisely the members of $C$ that
separate distinct points of $I$. For each $c\in\hat{C}$, let $I_{c}$ be the
subcontinuum of $I$ consisting of $c$ together with the component of
$I-\\{c\\}$ containing $p_{0}$. Then $I_{c^{\prime}}\subseteq I_{c}-\\{c\\}$,
whenever $c^{\prime}\in\hat{C}$ lies between $c$ and $p_{0}$. If we let
$I_{\omega}$ be the intersection of all $I_{c}$ for $c\in\hat{C}$, then
$I_{\omega}$ has empty intersection with $\hat{C}$ since for each
$c\in\hat{C}$ there exists $c^{\prime}\in\hat{C}$ between $c$ and $p_{0}$.

By compactness of $M$, any neighborhood $U$ of $I_{\omega}$ contains $I_{c}$
for some $c\in\hat{C}$. Thus the closure of $\hat{C}$ intersects $I_{\omega}$.
In order to establish the claim, it suffices to show that $I_{\omega}$ is
contained in the closure of $p$. If $p_{0}\notin C$, the intersection
$I_{\omega}$ is a subcontinuum of $I$ not separated by any member of $C$. By
Lemma 7.4, the set $I_{\omega}-C$ is dense in $I_{\omega}$ and contained in
the $\mathcal{R}$–equivalence class $p$. We conclude that $I_{\omega}$ is in
the closure of $p$. On the other hand, suppose $p_{0}\in C$. If $I_{\omega}$
contained more than the one point $p_{0}$, then $I_{\omega}-C$ would be a
single $\mathcal{R}$–equivalence class contained in $X$ and adjacent to
$\\{p_{0}\\}$, contradicting Lemma 8.5. Thus we must have
$I_{\omega}=\\{p_{0}\\}=p$.

Since $\hat{C}$ has a limit point in common with $p$, the closures of $X$ and
$Y$ intersect in $K$. In particular, $X$ and $Y$ must have the same image in
any Hausdorff space, and any map from $M/\mathcal{R}$ to a Hausdorff space
factors through $K_{\alpha}$. ∎

Throughout the rest of this section we work under the following assumptions.
Let $M$ denote a continuum with a minimal convergence group action by a
finitely generated group $\Gamma$ relative to a family of subgroups
$\mathbb{P}$. We assume that $\Gamma$ does not split relative to $\mathbb{P}$
over torsion subgroups. Assume $M$ has a nonparabolic cut point $c$ with orbit
$C=\Gamma(c)$. We let ${K}$ be the cut point pretree dual to $C$, and let
$(K_{\alpha})$ be the transfinite sequence of quotients of $K$ given by
Definition 8.2.

######  Lemma 8.6.

For each ordinal $\alpha$, the action of $\Gamma$ on the interior of
$K_{\alpha}$ is non-nesting, without inversions, and relative to $\mathbb{P}$.

###### Proof.

By Proposition 7.8, the action of $\Gamma$ on ${K}^{o}$ is non-nesting and
relative to $\mathbb{P}$. Since $K$ is complete and median, its interior is
interval complete and median. Let $K^{\prime}$ be the
$\operatorname{Aut}(K)$–invariant full subpretree of $K$ given by Lemma 6.4.
Since each $P\in\mathbb{P}$ has a fixed point $x$ in $K^{o}$, it also has a
fixed point $x^{\prime}$ in $K^{\prime}$ by Lemma 6.6. Since $\sim_{\alpha}$
is a full relation on $K$, the image $[x^{\prime}]$ in $K_{\alpha}$ lies in
the interior of $K_{\alpha}$ and is also fixed by $P$. Similarly, since the
action of $\Gamma$ on $K^{\prime}$ is non-nesting, the induced action on the
interior of $K_{\alpha}$ is non-nesting by Lemma 6.5. By Corollary 7.5, the
action on $K=K_{0}$ is without inversions. Lemma 8.5 implies the same
conclusion for $\alpha\geq 1$. ∎

######  Remark 8.7 (Collapsing simplicial trees).

Note that, in the successor case, $K_{\beta+1}$ may naturally be considered as
a quotient of $K_{\beta}$ by the relation that two points $x,y\in K_{\beta}$
are equivalent if only finitely many points of $K_{\beta}$ lie between $x$ and
$y$. Each equivalence class in $K_{\beta}$ is a discrete, full, subpretree of
$K_{\beta}$, which may be considered as the vertex set of a simplicial tree
(see [Bow99c, Lem. 3.34]). Two vertices are joined by an edge if and only if
they are adjacent in $K_{\beta}$.

According to Proposition 7.8, the action of $\Gamma$ on $K_{0}=K$ has no fixed
point. We now consider $K_{1}$. The following lemma depends on our assumption
that the family $C$ of cut points of $M$ contains no parabolic points.

######  Lemma 8.8.

The action of $\Gamma$ on $K_{1}$ does not have a fixed point.

###### Proof.

Suppose $\Gamma$ has a fixed point $X$ in $K_{1}$. Since $K_{1}$ is the
quotient of $K$ by the finite interval relation, the equivalence class $X$ is
a $\Gamma$–invariant discrete, full subpretree of $K$ that may be identified
with the set of vertices of a simplicial tree (see Remark 8.7). Observe that
$X$ must contain at least three distinct points. Indeed, as $\Gamma$ has no
fixed point, $X$ contains more than one point. If $X$ had exactly two points
$a$ and $b$, then they would be adjacent, since $X$ is full. But $\Gamma$ acts
without inversions, so it would fix each of $a$ and $b$, which is impossible.
Since $X$ has three distinct points, $a$, $b$, and $c$, it also contains their
median, which lies in the interior $K^{o}$ of $K$.

We now consider the nonempty subpretree $X^{o}$ obtained from $X$ by removing
all terminal points, and we consider $X^{o}$ to be the set of vertices of a
simplicial tree $T$. The action of $\Gamma$ on $K^{o}$ is non-nesting,
relative to $\mathbb{P}$, and without inversions by Proposition 7.8. Thus, the
action of $\Gamma$ on $T$ is relative to $\mathbb{P}$ by Lemma 6.6 and is also
clearly without inversions.

By the construction of $K$, the tree $T$ is bipartite with bipartition
$V(T)=V_{0}(T)\sqcup V_{1}(T)$. The set $V_{0}(T)$ consists of equivalence
classes of points in $M-C$, and the set $V_{1}(T)$ consists of singleton sets
$\\{c\\}$ with $c\in C$. Since the points of $C$ are nonparabolic in the
convergence group action on $M$, the stabilizer of a vertex $v\in V_{1}(T)$ is
either finite or loxodromic in the convergence group $\Gamma$. Any two
loxodromic elements fixing $v$ have a common power $g$ that fixes the same
subset of $K$, so Lemma 7.6 implies that $v$ is incident to at most one edge
of $T$ with loxodromic stabilizer.

Consider the tree $\bar{T}$ formed from $T$ by collapsing to a point each edge
whose stabilizer is loxodromic. Then the preimage of each vertex under the
collapse $T\to\bar{T}$ is a subtree of diameter at most two, since it contains
at most one vertex from $V_{0}(T)$. In particular, $T\to\bar{T}$ is an
equivariant quasi-isometry, so any subgroup of $\Gamma$ fixing a vertex of
$\bar{T}$ also fixes a vertex of $T$. But $\Gamma$ acts on $\bar{T}$ without
inversions, relative to $\mathbb{P}$, and with only finite edge stabilizers,
so $\Gamma$ fixes a vertex of $\bar{T}$ by our hypothesis that $\Gamma$ does
not split relative to $\mathbb{P}$ over finite subgroups. Hence, $\Gamma$ also
fixes a vertex of $T$, contradicting the fact that $\Gamma$ has no fixed point
in $K$. ∎

The next lemma follows by the same reasoning as a portion of the proof of
[Bow99c, Thm. 6.1], using Lemma 7.6 in place of [Bow99c, Lem. 6.9].

######  Lemma 8.9.

Let $X,Y$ be two adjacent points in the interior ${K}^{o}_{\alpha}$ of
${K}_{\alpha}$ for any ordinal $\alpha\geq 1$. Then the stabilizer of
$\\{X,Y\\}$ is a torsion group. ∎

######  Lemma 8.10.

For each $\alpha$, the action of $\Gamma$ on ${K}_{\alpha}$ does not have a
fixed point. The action of $\Gamma$ on the interior ${K}_{\alpha}^{o}$ is non-
nesting and relative to $\mathbb{P}$.

###### Proof.

It follows directly from Lemma 8.6 that the action on the interior of
$K_{\alpha}$ is non-nesting, relative to $\mathbb{P}$, and without inversions
for each $\alpha$. We prove by transfinite induction on $\alpha$ that $\Gamma$
acts on $K_{\alpha}$ without a fixed point for all $\alpha$. Observe that
nontriviality is known for $K_{1}$ by Lemma 8.8.

Let $\alpha$ be a limit ordinal. Assume that $\Gamma$ is known to act
nontrivially on $K_{\beta}$ for all $\beta<\alpha$. Suppose, by way of
contradiction that $\Gamma$ fixes a point $X$ in $K_{\alpha}$. Then $X$ is an
equivalence class in ${K}$ stabilized by $\Gamma$. Choose any $x\in X$ and a
finite generating set $\\{s_{1},\dots,s_{n}\\}$ of $\Gamma$. Clearly as
$s_{i}(x)\sim_{\alpha}x$, there exists $\beta_{i}<\alpha$ such that
$x\sim_{\beta_{i}}s_{i}(x)$ for each $i$. Taking
$\beta=\max\\{\beta_{1},\dots,\beta_{n}\\}$, we see that $\Gamma$ fixes a
point in ${K}_{\beta}$, contradicting our hypothesis since $\beta<\alpha$.

For the successor step, suppose $\Gamma$ is known to act nontrivially on
$K_{\beta}$ for some $\beta\geq 1$. We show that $\Gamma$ acts nontrivially on
$K_{\alpha}$ for $\alpha=\beta+1$. By way of contradiction, suppose $\Gamma$
fixes a point $x$ in $K_{\alpha}$. As explained in Remark 8.7, since
$K_{\alpha}$ is the quotient of ${K}_{\beta}$ by the finite interval relation,
the preimage of $\\{x\\}$ in $K_{\beta}$ is a $\Gamma$–invariant, discrete,
full subpretree $X$. As in the proof of Lemma 8.8, the interior $X^{o}$ is a
nonempty discrete full subpretree of $K^{o}_{\beta}$ on which $\Gamma$ acts
relative to $\mathbb{P}$ without inversions.

The discrete median pretree $X^{o}$ may be considered as the set of vertices
of a simplicial tree $T$ on which $\Gamma$ acts without inversions relative to
$\mathbb{P}$. Moreover, by Lemma 8.9 the stabilizer of each edge of $T$ is a
torsion group. As we are working under the hypothesis that $\Gamma$ does not
split over a torsion group relative to $\mathbb{P}$, there is a vertex of $T$
fixed by $\Gamma$, contradicting the inductive hypothesis that $\Gamma$ has no
fixed point in $K_{\beta}$. ∎

Combining the results established above gives the following conclusion.

######  Proposition 8.11.

Let $M$ be a continuum with a minimal convergence group action by a finitely
generated group $\Gamma$ relative to a family of subgroups $\mathbb{P}$.
Assume that $\Gamma$ does not split relative to $\mathbb{P}$ over torsion
subgroups.

If $M$ has a nonparabolic cut point $c$ with orbit $C=\Gamma(c)$, then the
induced action of $\Gamma$ on the dendrite quotient $D(M,C)$ is a convergence
group action without a fixed point. The action on the interior of $D(M,C)$ is
non-nesting and relative to $\mathbb{P}$ with finite interval stabilizers.

We now use the fact that the quotient $D(M,C)$ of $M$ is Hausdorff.

###### Proof.

As explained in Proposition 8.3, the dendrite quotient $D=D(M,C)$ is
isomorphic as a pretree to the stable pretree $K_{\alpha_{0}}$. Therefore, the
action of $\Gamma$ on $D$ is nontrivial, and the action on its interior is
non-nesting and relative to $\mathbb{P}$ by Lemma 8.10. Since $D$ is
Hausdorff, the quotient $M\to D$ is a closed map. As $\Gamma$ has a minimal
convergence group action on $M$, the induced action on $D$ is also a minimal
convergence group action.

Let $I=[x,y]$ be a nondegenerate interval in the interior of $D$. The
stabilizer of $I$ has a subgroup $H$ of index at most two fixing $x$ and $y$.
By non-nesting, $H$ fixes $I$ pointwise. As $H$ is a convergence group on $M$
with at least three fixed points, $H$ is finite, so that
$\operatorname{Stab}(I)$ is finite as well. ∎

######  Corollary 8.12.

Proposition 8.11 holds if we assume, instead, that $(\Gamma,\mathbb{P})$ is
relatively hyperbolic with boundary $M=\partial(\Gamma,\mathbb{P})$ and
$\Gamma$ does not split relative to $\mathbb{P}$ over a finite or parabolic
subgroup.

###### Proof.

We assume $(\Gamma,\mathbb{P})$ is nonelementary, since otherwise the result
is trivial. Thus the action of $\Gamma$ on the compactum $M$ is a minimal
convergence group action. Since $\Gamma$ does not split over a finite or
parabolic subgroup relative to $\mathbb{P}$, the group $\Gamma$ is finitely
generated (Theorem 2.11) and the space $M$ is connected (Proposition 5.1) and
metrizable ([Ger09]).

The Tits Alternative for convergence groups on metrizable compacta states that
every subgroup of $\Gamma$ not containing a free group of rank two is
elementary [Tuk94, Thm. 2U]. In particular, any infinite torsion subgroup must
be parabolic, so $\Gamma$ does not split over a torsion group relative to
$\mathbb{P}$. ∎

##  9\. From topological trees to metric and simplicial trees

In this section, we extend a theorem of Levitt from the setting of finitely
presented groups to groups with a finite relative presentation. The main
result of the section is Corollary 9.3, which produces an action on a
simplicial tree from a non-nesting action on a topological $\mathbb{R}$–tree.

######  Definition 9.1.

Given a group pair $(\Gamma,\mathbb{P})$, we say that $\Gamma$ is _finitely
presented relative to $\mathbb{P}=\\{P_{1},\dots,P_{k}\\}$_ if there exists a
finitely generated free group $F$ such that $\Gamma$ is the quotient of
$P_{1}*\cdots*P_{k}*F$ by the normal closure of a finite subset.

The following result was established for finitely presented groups by Levitt
[Lev98]. We extend that result to certain relatively finitely presented
groups.

######  Proposition 9.2.

Suppose the finitely generated group $\Gamma$ is also finitely presented
relative to a finite family of finitely generated subgroups $\mathbb{P}$. If
$\Gamma$ admits a nontrivial non-nesting action on a topological
$\mathbb{R}$–tree $R$ relative to $\mathbb{P}$, then it admits a nontrivial
isometric action on some $\mathbb{R}$–tree $R_{0}$ relative to $\mathbb{P}$
such that every subgroup fixing an interval in $R_{0}$ also fixes an interval
in $R$.

###### Proof.

The first step of the proof of [Lev98, Thm. 1] is to construct a certain
resolution of the action on $R$ whenever $\Gamma$ is finitely presented. As in
Guirardel–Levitt [GL15], we show that such a resolution exists under the
weaker hypothesis that $(\Gamma,\mathbb{P})$ has a finite relative
presentation and $\Gamma$ and each $P\in\mathbb{P}$ are finitely generated.
For each $P\in\mathbb{P}$, choose a presentation
$P=\langle\,{S_{P}}\mid{\mathcal{W}_{P}}\,\rangle$ with $S_{P}$ finite. Fix a
presentation $\Gamma=\langle\,{S}\mid{\mathcal{W}}\,\rangle$ such that $S$ is
a finite set containing each $S_{P}$ and such that $\mathcal{W}$ contains each
$\mathcal{W}_{P}$ together with a finite set $\mathcal{W}_{0}$ of additional
relators.

Let $K$ be a subtree of $R$ that is the convex hull of a finite set of points.
For each $s\in S$, let $K_{s}=K\cap s^{-1}K$ and consider the restriction
$\phi_{s}\colon K_{s}\to sK_{s}$ of $s$. Choose $K$ large enough that each set
$K_{s}$ contains more than one point, that for each relator
$r=s_{1}^{\epsilon_{1}}\cdots s_{n}^{\epsilon_{n}}$ in $\mathcal{W}_{0}$ the
domain of the map
$\phi_{s_{1}}^{\epsilon_{1}}\cdots\phi_{s_{n}}^{\epsilon_{n}}$ is nonempty,
and that $K$ contains a point $x_{P}$ fixed by $P$ for each $P\in\mathbb{P}$.

Build a foliated $2$–complex $\Sigma$ by starting with $K$ (foliated by
points) and gluing bands $K_{s}\times[0,1]$ (foliated by $\\{*\\}\times[0,1]$)
to $K$ so that $(x,0)$ is glued to $x$ and $(x,1)$ is glued to $\phi_{s}(x)$.
Then $\pi_{1}(\Sigma)$ may be naturally identified with the free group on $S$.
By construction, each relator of $\mathcal{W}$ is represented (up to free
homotopy) by a loop contained in a leaf of $\Sigma$. The natural homomorphism
$\rho\colon\pi_{1}(\Sigma)\to\Gamma$ gives rise to a foliated covering space
$p\colon\bar{\Sigma}_{\rho}\to\Sigma$. Furthermore, there exists a unique
$\Gamma$–equivariant _resolution map_ $f\colon\bar{\Sigma}_{\rho}\to R$ that
restricts to the covering map $p$ on some fixed component of $p^{-1}(K)$ and
is constant on each leaf of the foliation.

The rest of the argument of [Lev98] follows without change using this
resolution map. The finite system of maps $\\{\phi_{s}\\}$ is closed and non-
nesting in the sense of [Lev98] since the action on $R$ is non-nesting. Given
such a system, Levitt shows that $K$ admits a probability measure $\mu$ with
no atom such that each $\phi_{s}$ maps the restriction $\mu|K_{s}$ onto
$\mu|sK_{s}$. The measure on $K$ induces a transverse measure (perhaps not of
full support) on the foliated cover $\bar{\Sigma}_{\rho}$. Integration of this
measure along paths gives a length pseudometric on $\bar{\Sigma}_{\rho}$.
Identifying points $u,v$ with $d(u,v)=0$ produces a metric $\mathbb{R}$–tree
$R_{0}$ on which $\Gamma$ acts nontrivially and isometrically relative to
$\mathbb{P}$ such that every subgroup fixing an interval in $R_{0}$ also fixes
an interval in $R$. ∎

The fundamental structure theorem for stable actions of finitely presented
groups on $\mathbb{R}$–trees is the splitting theorem of Bestvina–Feighn
[BF95]. A theorem of Osin [Osi06b] states that every relatively hyperbolic
pair $(\Gamma,\mathbb{P})$ is relatively finitely presented, and if $\Gamma$
is finitely generated then so is each $P\in\mathbb{P}$. A relatively
hyperbolic analogue of the Bestvina–Feighn structure theorem due to
Guirardel–Levitt [GL15] gives the following corollary.

######  Corollary 9.3.

Let $(\Gamma,\mathbb{P})$ be relatively hyperbolic with $\Gamma$ finitely
generated. If $\Gamma$ acts nontrivially by non-nesting homeomorphisms on a
topological $\mathbb{R}$–tree $R$ relative to $\mathbb{P}$ with elementary arc
stabilizers, then $\Gamma$ splits over an elementary subgroup relative to
$\mathbb{P}$. ∎

##  10\. Cut points and local connectedness

This section contains the proof of the main theorem using the ingredients
established above. Broadly speaking, we follow a strategy analogous to that of
Swarup and Bowditch [Swa96, Bow99a, Bow99b, Bow01]. Various components of
Swarup and Bowditch’s proofs require significant extension to apply in the
general relatively hyperbolic setting. In this section, we prove Proposition
10.1, a key accessibility result for splittings over two-ended groups. Many
other such extensions are established in the sections above. Once we have
these extensions in place, the local connectedness proof follows the overall
plan of the original proofs due to Swarup and Bowditch.

If $(\Gamma,\mathbb{P})$ is relatively hyperbolic, let $\mathbb{L}$ be the
family of all loxodromic subgroups, _i.e._ , subgroups that are virtually
infinite cyclic and not parabolic.

######  Proposition 10.1.

Let $(\Gamma,\mathbb{P})$ be relatively hyperbolic.

  1. (1)

Given any $(\mathbb{L},\mathbb{P})$–tree, each vertex stabilizer $\Gamma_{v}$
is hyperbolic relative to $\mathbb{P}_{|\Gamma_{v}}$ and is also hyperbolic
relative to
$\mathbb{Q}_{v}=\operatorname{Inc}_{v}\cup\mathbb{P}_{|\Gamma_{v}}$, as
defined in Proposition 2.9.

  2. (2)

Suppose $\Gamma$ does not split over finite or parabolic subgroups relative to
$\mathbb{P}$. Then there exists an $(\mathbb{L},\mathbb{P})$–tree $T_{\ell}$
with the property that each vertex stabilizer $\Gamma_{v}$ is finitely
generated and does not split over loxodromic subgroups relative to
$\mathbb{Q}_{v}$.

###### Proof.

As in the proof of Proposition 2.9, the vertex stabilizers of an
$(\mathbb{L},\mathbb{P})$–tree are relatively quasiconvex, since all edge
stabilizers are relatively quasiconvex [BW13]. It follows that each
$\Gamma_{v}$ is relatively hyperbolic with respect to
$\mathbb{P}_{|\Gamma_{v}}$. Since the subgroups in the family
$\operatorname{Inc}_{v}$ are maximal loxodromic in
$(\Gamma_{v},\mathbb{P}_{|\Gamma_{v}})$, a theorem of Osin [Osi06a] implies
that $\Gamma_{v}$ is also hyperbolic relative to
$\mathbb{Q}_{v}=\operatorname{Inc}_{v}\cup\mathbb{P}_{|\Gamma_{v}}$.

We now consider the second assertion. We may assume $\Gamma$ is nonelementary,
since the result is trivial in the elementary case. Let $\mathbb{E}$ be the
family of all elementary subgroups of $(\Gamma,\mathbb{P})$. By hypothesis,
each $(\mathbb{E},\mathbb{P})$–tree involves only loxodromic edge groups.
Furthermore, $\Gamma$ and all members of $\mathbb{P}$ are finitely generated
by Theorem 2.11. The JSJ tree of cylinders over $\mathbb{E}$ relative to
$\mathbb{P}$ given by Guirardel–Levitt [GL17, Cor. 9.20] is an
$(\mathbb{L},\mathbb{P})$–tree $T$ with the following property. If $v$ is any
vertex of $T$, either $\Gamma_{v}$ has no splittings over loxodromic subgroups
relative to $\mathbb{Q}_{v}$ or $\Gamma_{v}$ is an extension with finite
kernel of the fundamental group of a compact hyperbolic $2$–orbifold
$\Sigma_{v}$ with geodesic boundary and the subgroups $\mathbb{Q}_{v}$ are the
lifts to $\Gamma_{v}$ of the fundamental groups of the boundary components of
$\Sigma_{v}$. In the orbifold case, every loxodromic splitting of $\Gamma_{v}$
relative to $\mathbb{Q}_{v}$ is induced by a splitting of $\pi_{1}(\Sigma)$
dual to a family of disjoint, essential simple closed geodesics of
$\Sigma_{v}$ by Morgan–Shalen [MS84, Thm. III.2.6]. A standard argument
involving reduction of Euler characteristic shows that $\Sigma_{v}$ may be cut
along some finite family of geodesics into pieces containing no essential
simple closed geodesic. Thus, we may refine $T$ using splittings of its vertex
groups to produce an $(\mathbb{L},\mathbb{P})$–tree $T_{\ell}$ each of whose
vertex groups $\Gamma_{v}$ does not split relative to $\mathbb{Q}_{v}$.

Finally we observe that each vertex stabilizer $\Gamma_{v}$ of $T_{\ell}$ is
finitely generated. Indeed, $\Gamma_{v}$ is hyperbolic relative to the family
$\mathbb{P}_{|\Gamma_{v}}$, but each member of $\mathbb{P}_{|\Gamma_{v}}$ is
conjugate in $\Gamma$ to a member of $\mathbb{P}$. In particular, $\Gamma_{v}$
is hyperbolic relative to finitely generated subgroups, which implies that
$\Gamma_{v}$ is finitely generated (see [Osi06b, GP15]). ∎

For the next two lemmas, we work under the following assumptions. Let
$(\Gamma,\mathbb{P})$ be relatively hyperbolic. Assume $\Gamma$ does not split
relative to $\mathbb{P}$ over a finite or parabolic subgroup. We note that, by
Proposition 5.1 the boundary $M=\partial(\Gamma,\mathbb{P})$ is connected.
Furthermore, by Theorem 2.11 the group $\Gamma$ is finitely generated. Suppose
the boundary $M$ has a nonparabolic cut point $c$ with orbit $C=\Gamma(c)$ in
$M$. Let $R$ be the topological $\mathbb{R}$–tree obtained by removing all
terminal points from the dendrite $D(M,C)$.

######  Lemma 10.2.

Suppose $\Gamma$ splits over a loxodromic subgroup $H$ relative to
$\mathbb{P}$. Then the induced action of $H$ on the topological
$\mathbb{R}$–tree $R$ has a fixed point.

###### Proof.

Recall that the dendrite $D$ is formed from $M$ by identifying two points $x$
and $y$ if the collection of points of $C$ separating them does not contain a
subset order-isomorphic to the rational numbers. The action of $\Gamma$ on $M$
is a convergence group action, and, by Corollary 8.12, the induced action of
$\Gamma$ on $D$ is also a convergence group action.

By a result of Haulmark–Hruska about splittings of relatively hyperbolic
groups over elementary subgroups [HH, Cor. 6.8], the two-point limit set
$\Lambda H$ is not separated by any cut point of $M$. In particular, the
quotient map $M\to D$ sends $\Lambda H$ to a single point $z$ fixed by
$\Gamma_{v}$. Let $h\in H$ be an infinite order element. If $h$ had another
fixed point $w\in D$, its preimage in $M$ would be a closed $h$–invariant set
not intersecting the limit set of $\langle h\rangle$ in $M$, which is
impossible in a convergence group action (see [Tuk94, §2]). Here, we use that
$D$ is a $T_{1}$–space, so that the preimage of $w$ in $M$ is closed. Thus $h$
has $z$ as its unique fixed point in $D$.

We claim that $z$ lies in the interior $R$. Recall that the action of $\Gamma$
on $R$ is non-nesting by Corollary 8.12. As in the proof of Proposition 7.8,
if the fixed point $z$ of $h$ does not lie in $R$, then [Lev98, Thm. 3]
implies that $h$ fixes a pair of terminal points of $D$ that are the endpoints
of an axis. Since $z$ is the unique fixed point of $h$ in $D$, we conclude
that $z$ must lie in $R$. ∎

######  Lemma 10.3.

Let $T_{\ell}$ be an $(\mathbb{L},\mathbb{P})$–tree as in Proposition 10.1
such that each vertex group $\Gamma_{v}$ does not split over loxodromic
subgroups relative to $\mathbb{Q}_{v}$. Then the induced action of each
$\Gamma_{v}$ on the topological $\mathbb{R}$–tree $R$ has a fixed point.

###### Proof.

Recall that $\Gamma_{v}$ is finitely generated and hyperbolic relative to
$\mathbb{Q}_{v}=\operatorname{Inc}_{v}\cup\mathbb{P}_{|\Gamma_{v}}$. By
Corollary 8.12, the action of $\Gamma$ on $R$ is relative to $\mathbb{P}$, so
that the restricted action of $\Gamma_{v}$ on $R$ is relative to
$\mathbb{P}_{|\Gamma_{v}}$. The subgroups $\operatorname{Inc}_{v}$ are also
finitely generated, since they are virtually cyclic. By Lemma 10.2, the action
on $R$ is also relative to the family of loxodromic edge stabilizers
$\operatorname{Inc}_{v}$.

If the action of $\Gamma_{v}$ on $R$ had no fixed point, it would be a
nontrivial non-nesting action relative to $\mathbb{Q}_{v}$ such that every
interval stabilized by $\Gamma_{v}$ has a finite stabilizer by Corollary 8.12.
But then $\Gamma_{v}$ would split over an elementary subgroup relative to
$\mathbb{Q}_{v}$ by Corollary 9.3. Any splitting of $\Gamma_{v}$ relative to
$\mathbb{Q}_{v}$ over a finite or parabolic subgroup extends to a splitting of
$\Gamma$ relative to $\mathbb{P}$ over finite or parabolic subgroups, so
$\Gamma_{v}$ has no such splitting. Thus $\Gamma_{v}$ must split over a
loxodromic subgroup relative to $\mathbb{Q}_{v}$, contradicting the choice of
$T_{\ell}$. ∎

######  Proposition 10.4.

Let $(\Gamma,\mathbb{P})$ be relatively hyperbolic. Suppose $\Gamma$ does not
split over a finite or parabolic subgroup relative to $\mathbb{P}$. Then
$M=\partial(\Gamma,\mathbb{P})$ is connected and every cut point of $M$ is a
parabolic point.

###### Proof.

By Proposition 5.1 and Theorem 2.11, the space $M$ is connected and the group
$\Gamma$ is finitely generated. We suppose, by way of contradiction that $M$
has a nonparabolic cut point $c$ with orbit $C=\Gamma(c)$. Let $D=D(M,C)$ be
the dendrite quotient of $M$ given by Corollary 8.12. If $R$ is the
topological $\mathbb{R}$–tree obtained from $D$ by removing all terminal
points, then each interval of $R$ is stabilized by a finite subgroup of
$\Gamma$. In particular, an infinite subgroup of $\Gamma$ cannot fix more than
one point of $R$.

Consider the simplicial $(\mathbb{L},\mathbb{P})$–tree $T_{\ell}$ given by
Proposition 10.1. For each vertex $v$ of $T_{\ell}$, the stabilizer
$\Gamma_{v}$ has a fixed point $p_{v}$ in $R$. Since $\Gamma_{v}$ is infinite,
its fixed point $p_{v}$ must be unique. For any edge $e$ of $T_{\ell}$ joining
vertices $w$ and $w$, the group $\Gamma_{e}$ stabilizes the arc
$[p_{v},p_{w}]$ in $R$. Since $\Gamma_{e}$ is also infinite, it follows that
the points $p_{v}$ and $p_{w}$ fixed by $\Gamma_{v}$ and $\Gamma_{w}$ must be
equal, so all vertex groups have the same unique fixed point in $R$.
Therefore, the entire group $\Gamma$ fixes a single point of $R$. This
conclusion contradicts Corollary 8.12, which states that $\Gamma$ does not
have a fixed point in the dendrite $D$. ∎

Recall that a _Peano continuum_ is a compact, connected, locally connected,
metrizable topological space. The next result is due to Bowditch [Bow99a],
under the hypothesis that all cut points are parabolic. We provide details for
the benefit of the reader.

######  Proposition 10.5 ([Bow99a]).

Let $(\Gamma,\mathbb{P})$ be relatively hyperbolic. Suppose $\Gamma$ does not
split over a finite or parabolic subgroup relative to $\mathbb{P}$. Then
$M=\partial(\Gamma,\mathbb{P})$ is a Peano continuum without cut points.

###### Proof.

Since $\Gamma$ is finitely generated, $M$ is the Gromov boundary of a certain
$\delta$–hyperbolic space $X$ with constant horospherical distortion in the
sense of Bowditch [Bow12, Bow99a]. In particular, $M$ is metrizable.

Suppose $M$ is not locally connected. Since $M$ is connected, [Bow99a, Thm.
1.1] implies that the cusped space $X$ has a separating horoball based at a
point $\eta\in M$. The proof of [BM91, Thm. 3.3] shows that the base point of
any separating horoball in $X$ is a cut point of $M=\partial X$. By
Proposition 10.4, the cut point $\eta$ is a parabolic point of $M$. However,
since $\Gamma$ does not split over a parabolic subgroup relative to
$\mathbb{P}$, the space $X$ cannot have a separating horoball based at a
parabolic point [Bow99a, Thm. 1.2]. We conclude that $M$ is locally connected,
so that $M$ is a Peano continuum.

Finally, if the Peano continuum $M$ had a parabolic cut point, then by
[Bow99a, Prop. 5.1] the space $X$ would contain a separating horoball, which
is impossible. ∎

The proof of Theorem 1.1 combines Proposition 10.5 and Theorem 5.3.

###### Proof of Theorem 1.1.

One-endedness implies that $M$ is connected. Consider the JSJ tree of
cylinders $T_{c}$ for $\Gamma$ over parabolic subgroups relative to
$\mathbb{P}$ given by Proposition 2.13. For each vertex $v\in V_{0}(T_{c})$,
the stabilizer $\Gamma_{v}$ is hyperbolic relative to $\mathbb{Q}_{v}$ and
does not split relative to $\mathbb{P}$ over any finite or parabolic subgroup.
By Proposition 10.5, the boundary $\partial(\Gamma_{v},\mathbb{Q}_{v})$ is a
Peano continuum with no cut points. Thus, by Theorem 5.3, the boundary $M$ is
locally connected. We also conclude that $M$ has a cut point if and only if
the tree $T_{c}$ is nontrivial, which happens if and only if $\Gamma$ splits
relative to $\mathbb{P}$ over a parabolic subgroup.

Recall that $M$ is the limit of the tree system $\Theta(T_{c})$ associated to
$T_{c}$ by Proposition 4.12. We show that a nonparabolic point of $M$ cannot
be a cut point, using the notation from Section 4.2. For each $v\in
V_{0}(T_{c})$, the boundary $M$ maps onto
$M_{v}=\partial(\Gamma_{v},\mathbb{Q}_{v})$ by a retraction that collapses to
a point each halfspace $H(\vec{e}\,)$ not containing $M_{v}$. It follows that
any nonparabolic cut point of $M$ contained in $M_{v}$ would also be a cut
point of $M_{v}$. Thus no such cut point exists. The subspace
$\\#\Theta=\bigcup M_{v}$ is a dense connected subspace of $M$, since it
contains the vertex set $V(K)$ of some $(\Gamma,\mathbb{P})$–graph $K$
compatible with the splitting $T_{c}$. If $\xi$ is in its complement, then

$\\#\Theta\subset M-\\{\xi\\}\subset\overline{\\#\Theta}=M.$

Thus $M-\\{\xi\\}$ is connected and $\xi$ is also not a cut point of $M$. ∎

We conclude by observing that the proof of Theorem 1.1 also establishes
Theorem 1.2.

## References

  * [AS85] F.D. Ancel and L. Siebenmann.  The construction of homogeneous homology manifolds.  Abstracts Amer. Math. Soc., 6, 1985.  Abstract number 816-57-72.
  * [BC01] B.H. Bowditch and J. Crisp.  Archimedean actions on median pretrees.  Math. Proc. Cambridge Philos. Soc., 130(3):383–400, 2001.
  * [BF95] M. Bestvina and M. Feighn.  Stable actions of groups on real trees.  Invent. Math., 121(2):287–321, 1995.
  * [BK05] M. Bonk and B. Kleiner.  Quasi-hyperbolic planes in hyperbolic groups.  Proc. Amer. Math. Soc., 133(9):2491–2494, 2005.
  * [BM91] M. Bestvina and G. Mess.  The boundary of negatively curved groups.  J. Amer. Math. Soc., 4(3):469–481, 1991.
  * [Bow98] B.H. Bowditch.  Cut points and canonical splittings of hyperbolic groups.  Acta Math., 180(2):145–186, 1998.
  * [Bow99a] B.H. Bowditch.  Boundaries of geometrically finite groups.  Math. Z., 230(3):509–527, 1999.
  * [Bow99b] B.H. Bowditch.  Connectedness properties of limit sets.  Trans. Amer. Math. Soc., 351(9):3673–3686, 1999.
  * [Bow99c] B.H. Bowditch.  Treelike structures arising from continua and convergence groups.  Mem. Amer. Math. Soc., 139(662):1–86, 1999.
  * [Bow01] B.H. Bowditch.  Peripheral splittings of groups.  Trans. Amer. Math. Soc., 353(10):4057–4082, 2001.
  * [Bow12] B.H. Bowditch.  Relatively hyperbolic groups.  Internat. J. Algebra Comput., 22(3):1250016, 66, 2012.
  * [BW13] H. Bigdely and D.T. Wise.  Quasiconvexity and relatively hyperbolic groups that split.  Michigan Math. J., 62(2):387–406, 2013.
  * [Cap54] C.E. Capel.  Inverse limit spaces.  Duke Math. J., 21:233–245, 1954.
  * [Dah03] F. Dahmani.  Combination of convergence groups.  Geom. Topol., 7:933–963, 2003.
  * [DS05] C. Drutu and M. Sapir.  Tree-graded spaces and asymptotic cones of groups.  Topology, 44(5):959–1058, 2005.  With an appendix by D. Osin and M. Sapir.
  * [DT17] S. Dowdall and S.J. Taylor.  The co-surface graph and the geometry of hyperbolic free group extensions.  J. Topol., 10(2):447–482, 2017.
  * [Eng89] R. Engelking.  General topology, volume 6 of Sigma Series in Pure Mathematics.  Heldermann Verlag, Berlin, 2nd edition, 1989.
  * [Ger09] V. Gerasimov.  Expansive convergence groups are relatively hyperbolic.  Geom. Funct. Anal., 19(1):137–169, 2009.
  * [Ger12] V. Gerasimov.  Floyd maps for relatively hyperbolic groups.  Geom. Funct. Anal., 22(5):1361–1399, 2012.
  * [GL11] V. Guirardel and G. Levitt.  Trees of cylinders and canonical splittings.  Geom. Topol., 15(2):977–1012, 2011.
  * [GL15] V. Guirardel and G. Levitt.  Splittings and automorphisms of relatively hyperbolic groups.  Groups Geom. Dyn., 9(2):599–663, 2015.
  * [GL17] V. Guirardel and G. Levitt.  JSJ decompositions of groups, volume 395 of Astérisque.  Société Mathématique de France, Paris, 2017.
  * [GM08] D. Groves and J.F. Manning.  Dehn filling in relatively hyperbolic groups.  Israel J. Math., 168:317–429, 2008.
  * [GP15] V. Gerasimov and L. Potyagailo.  Non–finitely generated relatively hyperbolic groups and Floyd quasiconvexity.  Groups Geom. Dyn., 9(2):369–434, 2015.
  * [GP16] V. Gerasimov and L. Potyagailo.  Similar relatively hyperbolic actions of a group.  Int. Math. Res. Not. IMRN, 2016(7):2068–2103, 2016.
  * [GS19] R. Geoghegan and E. Swenson.  On semistability of $\operatorname{CAT}(0)$ groups.  Groups Geom. Dyn., 13(2):695–705, 2019.
  * [Hau19] M. Haulmark.  Local cut points and splittings of relatively hyperbolic groups.  Algebr. Geom. Topol., 19(6):2795–2836, 2019.
  * [HH] M. Haulmark and G.C. Hruska.  On canonical splittings of relatively hyperbolic groups.  To appear in _Israel J. Math._ arXiv:1912.00886.
  * [HM] M. Haulmark and M. Mihalik.  Relatively hyperbolic groups with semistable peripheral subgroups.  arXiv:2101.05923.
  * [HW] G.C. Hruska and G.S. Walsh.  Planar boundaries and parabolic subgroups.  arXiv:2008.07639.
  * [Jak91] W. Jakobsche.  Homogeneous cohomology manifolds which are inverse limits.  Fund. Math., 137(2):81–95, 1991.
  * [KK00] M. Kapovich and B. Kleiner.  Hyperbolic groups with low-dimensional boundary.  Ann. Sci. École Norm. Sup. (4), 33(5):647–669, 2000.
  * [KR14] I. Kapovich and K. Rafi.  On hyperbolicity of free splitting and free factor complexes.  Groups Geom. Dyn., 8(2):391–414, 2014.
  * [Lev98] G. Levitt.  Non-nesting actions on real trees.  Bull. London Math. Soc., 30(1):46–54, 1998.
  * [ML98] S. Mac Lane.  Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics.  Springer-Verlag, New York, second edition, 1998.
  * [MS84] J.W. Morgan and P.B. Shalen.  Valuations, trees, and degenerations of hyperbolic structures. I.  Ann. of Math. (2), 120(3):401–476, 1984.
  * [MS21] M. Mihalik and E. Swenson.  Relatively hyperbolic groups with semistable fundamental group at infinity.  J. Topol., 14(1):39–61, 2021.
  * [Osi06a] D.V. Osin.  Elementary subgroups of relatively hyperbolic groups and bounded generation.  Internat. J. Algebra Comput., 16(1):99–118, 2006.
  * [Osi06b] D.V. Osin.  Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems.  Mem. Amer. Math. Soc., 179(843):1–100, 2006.
  * [PS06] P. Papasoglu and E. Swenson.  From continua to $\mathbb{R}$–trees.  Algebr. Geom. Topol., 6:1759–1784, 2006.
  * [Sou] L.H.R. de Souza.  Equivariant blowups of bounded parabolic points.  arXiv:2008.05822.
  * [Swa96] G.A. Swarup.  On the cut point conjecture.  Electron. Res. Announc. Amer. Math. Soc., 2(2):98–100, 1996.
  * [Swe00] E.L. Swenson.  A cutpoint tree for a continuum.  In M. Atkinson, N. Gilbert, J. Howie, S. Linton, and E. Robertson, editors, Computational and geometric aspects of modern algebra, volume 275 of London Math. Soc. Lecture Note Ser., pages 254–265. Cambridge Univ. Press, Cambridge, 2000.
  * [Świ20] J. Świątkowski.  Trees of metric compacta and trees of manifolds.  Geom. Topol., 24(2):533–592, 2020.
  * [Tom] R. Tomar.  Boundaries of graphs of relatively hyperbolic groups with cyclic edge groups.  arXiv:2104.08843.
  * [Tuk94] P. Tukia.  Convergence groups and Gromov’s metric hyperbolic spaces.  New Zealand J. Math., 23(2):157–187, 1994.
  * [Wil70] S. Willard.  General topology.  Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970\.