# Homomorphisms to $3$–manifold groups

Daniel Groves  Department of Mathematics, Statistics, and Computer Science,
University of Illinois at Chicago, 322 Science and Engineering Offices (M/C
249), 851 S. Morgan St., Chicago, IL 60607-7045<EMAIL_ADDRESS>,
Michael Hull  Department of Mathematics & Statistics, University of North
Carolina at Greensboro, 116 Petty Building, Greensboro, NC 27402-6170
<EMAIL_ADDRESS>and  Hao Liang  School of Mathematics, Sun Yat-sen University,
Guangzhou, Guangdong, 510275, People’s Republic of China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We prove foundational results about the set of homomorphisms from a finitely
generated group to the collection of all fundamental groups of compact
$3$–manifolds and answer questions of Reid–Wang–Zhou [27] and Agol–Liu [3].

The first author is supported in part by NSF grant DMS-1904913. The third
author is supported by NSFC (No.11701581 and No.11521101).

###### Contents

  1. 1 Introduction
  2. 2 Outline of proofs
  3. 3 Reduction to $\mathcal{M}_{\operatorname{Gen}}^{\pi}$
  4. 4 Generalized geometric decompositions
  5. 5 The Collapsed Space
  6. 6 Sequences which are not $\mathcal{C}$–divergent
  7. 7 Limits and $\mathbb{R}$–trees
  8. 8 JSJ-decompositions and modular automorphisms
  9. 9 Resolutions and factoring
  10. A Edge-twisted graphs of groups
  11. B Relative Linnell and JSJ decompositions

##  1\. Introduction

There is a well-developed structure theory for compact $3$–manifolds based on
the Geometrization Theorem of Perelman (conjectured by Thurston). This gives a
decomposition of a compact $3$–manifold into pieces which each admit a unique
geometric structure modeled on one of the eight Thurston geometries. While
many open questions remain, this structure theory provides a powerful starting
point.

In addition to the $3$–manifolds themselves, maps between $3$–manifolds have
also been studied by many people. Often the focus has been on maps of non-zero
degree and on finiteness results for sets of such maps. Wang’s 2002 ICM
address [34] surveys results up to that time. Since then, one of the most
striking results is a theorem of Liu [21]: for any closed, orientable
$3$–manifold $M$ and any $d>0$, there are only finitely many distinct
$3$–manifolds $N$ which admit maps $M\to N$ of degree $d$.

Our focus is on induced homomorphisms between fundamental groups. For
aspherical $3$–manifolds, the set of maps up to homotopy is the same as the
set of homomorphisms between fundamental groups up to conjugacy, so many
questions about maps reduce to questions about the induced homomorphisms on
$\pi_{1}$. If $M$ and $N$ are closed $3$–manifolds and $f\colon\thinspace M\to
N$ has degree $1$ then the induced map
$f_{\ast}\colon\thinspace\pi_{1}(M)\to\pi_{1}(N)$ is surjective. However, it
is not true that any $\pi_{1}$–surjection
$\phi\colon\thinspace\pi_{1}(M)\to\pi_{1}(N)$ arises in this manner (for
example, $\phi$ may factor through a map to a free group).

A $3$–manifold group is the fundamental group of a compact $3$–manifold,
possibly with boundary. Many results about epimorphisms between $3$–manifold
groups were motivated by conjectures of J. Simon about knot groups111A _knot
group_ is the fundamental group of a knot complement in $S^{3}$.  (see Kirby’s
list [1, Problem 1.12 C & D]). Simon conjectured that given a knot group $G$
there is a number $N_{G}$ so any sequence of epimorphisms of knot groups
$G\twoheadrightarrow G_{1}\twoheadrightarrow...\twoheadrightarrow G_{n}$ with
$n\geq N_{G}$ contains an isomorphism. The second Simon conjecture, strictly
stronger, was that given a knot group $G$ there are only finitely many knot
groups $H$ so there is an epimorphism $G\twoheadrightarrow H$. Simon’s
conjectures were proved by Agol–Liu [3]. See the introduction to [3] for more
background and history on the Simon Conjectures.

Kirby [1, Problem 1.12(C), Remarks] asked whether the first conjecture holds
for general $3$–manifold groups. (Since there are $3$–manifold groups which
surject a nonabelian free group, clearly the analogue of the second conjecture
does not hold.) A counterexample to this question was found by Reid–Wang–Zhou
[27, Example 3.3], and they asked about the following weaker version.

######  Question 1.

[27, Question 3.1.(C2)] Let $M_{i}$ be closed orientable aspherical
$3$–manifolds. Does any infinite sequence of epimorphisms
$\pi_{1}(M_{0})\twoheadrightarrow\pi_{1}(M_{1})\twoheadrightarrow\cdots\twoheadrightarrow\pi_{1}(M_{n})\twoheadrightarrow\cdots$
contain an isomorphism?

Reid–Wang–Zhou [27, Theorem 3.4] gave a positive answer in case all $M_{i}$
are (closed, orientable, aspherical) Seifert manifolds, and Soma gave a
positive answer in case all $M_{i}$ are hyperbolic [31, Theorem 1].

Our first result gives a positive answer to Question 1 in full generality. In
fact, we have no assumptions on the $3$–manifolds $M_{i}$, except that they be
compact.

######  Theorem A.

Let $M_{i}$ be compact $3$–manifolds (possibly with boundary). Every infinite
sequence of epimorphisms
$\pi_{1}(M_{1})\twoheadrightarrow\cdots\twoheadrightarrow\pi_{1}(M_{n})\twoheadrightarrow\cdots$
contains an isomorphism.

This result is related to the following partial order on $3$–manifold groups:
$G_{1}\geq G_{2}$ if there is an epimorphism $\phi\colon\thinspace G_{1}\to
G_{2}$. Per Calegari [9], understanding this order is an important question in
$3$–manifold topology. Theorem A proves the descending chain condition for
this order.

Since the Borel Conjecture is true in dimension $3$ (see for instance [18,
Section 5]), we obtain:

######  Corollary B.

Let $(M_{i})_{i\in\mathbb{N}}$ be a sequence of closed, aspherical
$3$–manifolds and let $f_{i}\colon M_{i}\to M_{i+1}$ be $\pi_{1}$–surjective
maps. There exists $N$ so $f_{i}$ is homotopic to a homeomorphism for all
$i\geq N$.

Theorem A is a consequence of our main result (Theorem D below), a basic
structural result about the set of _all_ homomorphisms from an _arbitrary_
finitely generated group to the set of _all_ $3$–manifold groups (with no
assumptions, topological or algebraic). For a group $G$ and a set of groups
$\mathcal{G}$, ${\operatorname{Hom}}(G,\mathcal{G})$ is the disjoint union

$\bigcup_{\Gamma\in\mathcal{G}}{\operatorname{Hom}}(G,\Gamma).$

Let $\mathcal{M}^{\pi}$ denote the family of all $3$–manifold groups. At the
end of [3], Agol–Liu discussed approaches to describing
${\operatorname{Hom}}(G,\mathcal{M}^{\pi})$ for an arbitrary finitely
generated group $G$. In order to phrase their question in the terminology of
this paper and [12], we introduce the following definition.

######  Definition C.

[12, Theorem 3.7] A set of groups $\mathcal{G}$ is equationally noetherian if
for any finitely generated group $G$ there is a finitely presented group
$G_{0}$ and a surjection $\alpha\colon\thinspace G_{0}\twoheadrightarrow G$ so
that the map

$\alpha^{*}\colon\thinspace{\operatorname{Hom}}(G,\mathcal{G})\to{\operatorname{Hom}}(G_{0},\mathcal{G})$

induced by precomposition with $\alpha$ is a bijection.

That $\mathcal{G}$ is equationally noetherian is a kind of ‘coherence’ – from
the perspective of maps to $\mathcal{G}$ every finitely generated group is
finitely presented. From the perspective of equations over groups, (a
different view on homomorphisms) $\mathcal{G}$ being equationally noetherian
says the analogue of the Hilbert Basis Theorem holds over $\mathcal{G}$.

In this language, Agol and Liu ask the following question.

######  Question 2.

[3, Question 10.3] Is $\mathcal{M}^{\pi}$ an equationally noetherian family of
groups?

Our main result answers Question 2 in the affirmative.

######  Theorem D.

$\mathcal{M}^{\pi}$ is an equationally noetherian family of groups.

An equationally noetherian family of groups satisfies a descending chain
condition on sequences of epimorphisms, so Theorem A is a consequence of
Theorem D (see Section 2 for more details).

The notion of an equationally noetherian family of groups was introduced in
[12]. It is a natural extension of the well-studied notion of an equationally
noetherian group. However, the property that $\mathcal{G}$ is an equationally
noetherian family is much stronger than the property that each individual
element of $\mathcal{G}$ is equationally noetherian. For example, as noted in
[12, Example 3.14], the set of finite groups is not equationally noetherian,
whereas each individual finite group is. However, if $\mathcal{G}$ is
equationally noetherian then certainly every element of $\mathcal{G}$ is also.

######  Corollary E.

Let $M$ be a compact 3–manifold (possibly with boundary). Then $\pi_{1}(M)$ is
equationally noetherian.

An application of the Hilbert Basis Theorem shows that each linear group is
equationally noetherian. So if all $3$–manifold groups are linear, Corollary E
would follow immediately. This remains an open question (though only for
certain graph manifolds). For Theorem D to be a consequence of linearity
requires a single linear group into which every $3$–manifold group embeds.
This is also an open question, but a positive answer seems out of reach of
current methods.

We now outline the contents of this paper. In Section 2 we outline the proof
of Theorem D and use this to derive Theorem A. In Section 2 Theorem D is
reduced to four results, proved later in the paper. In Section 3 we prove
Theorem 2.2, which allows us to focus on a restricted class of $3$–manifold
groups. In Section 4 we consider various splittings of limit groups. Later
sections are dedicated to the proof of the main technical result, Theorem 2.7.
Our approach is to construct limiting actions on $\mathbb{R}$-trees, so as to
be able to apply the Rips machine and an analogue of Sela’s shortening
argument.

We briefly describe the novelties of this paper. In Section 5 we introduce the
collapsed space, a variant of a construction of Farb from [11]. This is
carefully tuned so that the limiting $\mathbb{R}$-tree built in Section 7
behaves well and so that the shortening argument in Section 9 proceeds
smoothly. In Section 6 we consider the case that actions on the collapsed
space do not diverge, and find another tree upon which to act. In this
section, we introduce the class of _edge-twisted_ graphs of groups (see
Definition 6.5), which have the property that certain vertex groups are
finitely generated (Theorem 6.7, proved in Appendix A), which is key to our
proof.

##  2\. Outline of proofs

In this section we explain the proof of Theorem D. Specifically, we explain
the outline of the proof, and also state four results: Theorem 2.2, Theorem
2.5, Theorem 2.6, and Theorem 2.7 (proved in later sections). Assuming these
results, we give a complete proof of Theorem D. We deduce Theorem A from
Theorem D.

Our first reduction uses the basic structure of compact $3$–manifolds and
$3$–manifold groups to restrict the class of $3$–manifolds we need to
consider. We refer to [4] for background about $3$–manifolds, their
fundamental groups, etc.

######  Definition 2.1.

Let $\mathcal{M}_{\operatorname{Gen}}$ be the collection of $3$–manifolds $M$
so that:

  1. (1)

$M$ is closed, orientable, and irreducible;

  2. (2)

All geometric pieces are hyperbolic or large Seifert-fibered; and

  3. (3)

The base orbifold of each Seifert-fibered piece is orientable.

Let $\mathcal{M}_{\operatorname{Gen}}^{\pi}$ denote the collection of
fundamental groups of elements of $\mathcal{M}_{\operatorname{Gen}}$.

Our first reduction is the following, proved in Section 3.

######  Theorem 2.2.

If $\mathcal{M}_{\operatorname{Gen}}^{\pi}$ is equationally noetherian then so
is $\mathcal{M}^{\pi}$.

Throughout this paper, we fix a non-principal ultrafilter $\omega$ and use the
concepts of $\omega$–limits, etc. See [33] for the background and basic
results about ultrafilters, ultralimits, etc. Fix a non-principal ultrafilter
$\omega$ which is a finitely additive probability measure $\omega\colon
2^{\mathbb{N}}\to\\{0,1\\}$ so $\omega(F)=0$ for any finite set $F$. A
statement $P(i)$ depending on an index $i$ holds $\omega$–almost surely if
$\omega(\\{i\;|\;P(i)\text{ holds}\\})=1$.

######  Definition 2.3.

Suppose $\mathcal{G}$ is a family of groups, $G$ is a finitely generated
group, and $(\phi_{i})$ is a sequence of homomorphisms from
${\operatorname{Hom}}(G,\mathcal{G})$. Associated to $(\phi_{i})$ is the
_stable kernel_ $\operatorname{Ker}^{\omega}(\phi_{i})=\\{g\in G\mid\text{
$\omega$--almost surely }\phi_{i}(g)=1\\}$ and a _$\mathcal{G}$ –limit group_
defined by $L:=G/\operatorname{Ker}^{\omega}(\phi_{i})$. Let $\phi_{\infty}$
denote the quotient map $G\twoheadrightarrow L$. The sequence $(\phi_{i})$
_$\omega$ –factors through the limit_ if $\phi_{i}$ $\omega$–almost surely
factors through $\phi_{\infty}$.

######  Theorem 2.4.

[12, Theorem 3.5] The family $\mathcal{G}$ is equationally noetherian if and
only if for every finitely generated group $G$, every sequence from
${\operatorname{Hom}}(G,\mathcal{G})$ $\omega$–factors through the limit.

Our proof of Theorem D proceeds by verifying the criterion from Theorem 2.4.
One feature a sequence $(\phi_{i}\colon\thinspace G\to\Gamma_{i})$ from
${\operatorname{Hom}}(G,\mathcal{M}_{\operatorname{Gen}}^{\pi})$ might have is
being _$\mathcal{T}$ –divergent_ (see Definition 4.4 – roughly speaking the
actions of $\phi_{i}$ on the trees dual to the geometric decompositions
diverge). The main result of [12] implies the following theorem, proved in
Section 4.

######  Theorem 2.5.

Suppose that for every finitely generated group $G$, every sequence from
${\operatorname{Hom}}(G,\mathcal{M}_{\operatorname{Gen}}^{\pi})$ which is not
$\mathcal{T}$–divergent $\omega$–factors through the limit. Then for every
finitely generated group $G$ _every_ sequence from
${\operatorname{Hom}}(G,\mathcal{M}_{\operatorname{Gen}}^{\pi})$
$\omega$–factors through the limit.

Thus, we fix a non-$\mathcal{T}$–divergent sequence $(\phi_{i}\colon\thinspace
G\to\Gamma_{i})$ defining an $\mathcal{M}_{\operatorname{Gen}}^{\pi}$–limit
group $L$. The $G$–actions on the trees associated to the $\Gamma_{i}$
converge to an $L$–action on a (simplicial) tree, inducing the geometric
decomposition of $L$ (Definition 4.15). A key set of subgroups are _stably
parabolic_ subgroups (Definition 4.5). These are abelian (Lemma 4.6) and edge
groups of the geometric decomposition are stably parabolic. The following is
proved in Section 4.

######  Theorem 2.6.

Let $L$ be an $\mathcal{M}_{\operatorname{Gen}}^{\pi}$–limit group defined by
a non–$\mathcal{T}$–divergent sequence $(\phi_{i})$ and suppose the edge
groups of the geometric decomposition of $L$ (wrt $(\phi_{i})$) are finitely
generated. Then $(\phi_{i})$ $\omega$–factors through the limit.

At this point, the following result (together with Theorems 2.2, 2.5 and 2.6)
completes the proof of Theorem D.

######  Theorem 2.7.

Let $L$ be an $\mathcal{M}_{\operatorname{Gen}}^{\pi}$–limit group defined by
a non-$\mathcal{T}$–divergent sequence $(\phi_{i})$. All stably parabolic
subgroups of $L$ are finitely generated.

Our approach to proving Theorem 2.7 is to analyze the geometric decomposition
of $L$ using the _collapsed spaces_ defined in Section 5. This leads to the
notion of a _$\mathcal{C}$ –divergent_ sequence (see Definition 6.1). In
Section 6 we prove Theorem 6.2 (modulo the technical Theorem 6.7 proved in
Appendix A), which deals with the case where the sequence is not
$\mathcal{C}$–divergent. After build-up in Sections 5, 7 and 8, we prove
Theorem 2.7 in Section 9.

We finish by explicitly deducing Theorem D from the results in this section.
By Theorem 2.2 it suffices to prove $\mathcal{M}_{\operatorname{Gen}}^{\pi}$
is equationally noetherian. Since edge groups of the geometric decomposition
are stably parabolic, Theorems 2.7 and 2.6 prove that
non–$\mathcal{T}$–divergent sequences $\omega$–factor through the limit.
Theorems 2.5 and 2.4 complete the proof of Theorem D.

###  2.1. Proof of Theorem A

We now deduce Theorem A from Theorem D. Suppose that
$(\Gamma_{i})_{i=1}^{\infty}$ is a sequence from $\mathcal{M}^{\pi}$, and that
for each $i\geq 1$ there is a surjection
$\tau_{i}\colon\thinspace\Gamma_{i}\to\Gamma_{i+1}$. Define $(\rho_{i})$ from
${\operatorname{Hom}}(\Gamma_{1},\mathcal{M}^{\pi})$ by
$\rho_{i}=\tau_{i}\circ\cdots\circ\tau_{1}\colon\thinspace\Gamma_{1}\to\Gamma_{i+1}$.
By Theorem D $\mathcal{M}^{\pi}$ is equationally noetherian, so by Theorem 2.4
$(\rho_{i})$ $\omega$–factors through the limit map $\rho_{\infty}$.

Now, $\operatorname{Ker}(\rho_{i+1})\subseteq\operatorname{Ker}(\rho_{i})$, so

$\operatorname{Ker}(\rho_{\infty})=\bigcup_{i=1}^{\infty}\operatorname{Ker}(\rho_{i}).$

By Theorem D for $\omega$–almost all $j$ the homomorphism $\rho_{j}$ factors
through $\rho_{\infty}$. Fix such a $j$. Then
$\operatorname{Ker}(\rho_{\infty})\subseteq\operatorname{Ker}(\rho_{j})$, and
since $\operatorname{Ker}(\rho_{j})\subset\operatorname{Ker}(\rho_{\infty})$,
the two are equal. Thus, the ascending sequence of kernels stabilizes after
$\rho_{j}$, and for all $k\geq j$ the map $\tau_{k}$ is an isomorphism,
completing the proof.

###  2.2. Notation and conventions

If $X$ is a metric space and $x,y\in X$ then we denote a geodesic between $x$
and $y$ by $[x,y]$.

A graph of groups is minimal if there are no proper invariant subtrees of the
Bass–Serre tree. Let $G$ be a group, and $\mathcal{A}$ and $\mathcal{H}$
families of subgroups closed under conjugation. An
_$(\mathcal{A},\mathcal{H})$ –splitting_ of $G$ is a splitting of $G$ in which
all edge groups are in $\mathcal{A}$ and all groups in $\mathcal{H}$ are
elliptic.

Throughout this paper a _$\delta$ –hyperbolic space_ is a geodesic metric
space in which all geodesic triangles are _$\delta$ –thin_ in the sense of [8,
Definition III.H.1.16]. We remark that if a geodesic metric space has
$\nu$–slim triangles (in the sense of [8, Definition III.H.1.1]) then it has
$4\nu$–thin triangles.

##  3\. Reduction to $\mathcal{M}_{\operatorname{Gen}}^{\pi}$

The purpose of this section is to record some basic facts about (limit groups
over the family of) $3$–manifold groups, and in particular to prove Theorem
2.2.

###  3.1. Equationally Noetherian families

In this subsection, we record some facts about equationally noetherian
families not particular to $3$–manifold groups. The first follows from [5,
Theorem B1] and [12, Lemma 3.9].

######  Lemma 3.1.

For any field $K$ and $n\geq 1$, if $\mathcal{G}$ is a family of groups so
each $\Gamma\in\mathcal{G}$ is a subgroup of $GL(n,K)$, then $\mathcal{G}$ is
equationally noetherian.

######  Lemma 3.2.

[12, Lemma 3.20] Suppose $(\phi_{i})$ is a sequence from
${\operatorname{Hom}}(G,\mathcal{G})$ and the associated limit group is
finitely presented relative to subgroups $\\{P_{1},...,P_{n}\\}$. Suppose for
each $P_{j}$, there is $\widetilde{P}_{j}\leq G$ so that
$\phi_{\infty}(\widetilde{P}_{j})=P_{j}$ and that
$\phi_{i}|_{\widetilde{P}_{j}}$ $\omega$–almost surely factors through
$\phi_{\infty}|_{\widetilde{P}_{j}}$. Then $(\phi_{i})$ $\omega$–factors
through the limit.

Theorem 1 of [6] states that if $H$ is a finite index subgroup of $G$ and $H$
is equationally noetherian, then $G$ is equationally noetherian. We generalize
this to a family of groups.

######  Definition 3.3.

Let $\mathcal{G}$ be a family of groups and let $n\geq 1$. The class
$\mathcal{G}^{\leq n}$ is the set of groups $G$ containing a subgroup from
$\mathcal{G}$ of index at most $n$.

######  Proposition 3.4.

Suppose that $\mathcal{G}$ is an equationally noetherian family of groups. For
any $n\geq 1$ the family $\mathcal{G}^{\leq n}$ is equationally noetherian.

###### Proof.

Let $G$ be finitely generated and let $(\phi_{i}\colon G\to\Gamma_{i})$ be a
sequence with $\Gamma_{i}\in\mathcal{G}^{\leq n}$. By Theorem 2.4 it suffices
to show that $\omega$–almost surely
$\operatorname{Ker}(\phi_{\infty})\subseteq\operatorname{Ker}(\phi_{i})$. So,
let $\Gamma_{i}^{\prime}\in\mathcal{G}$ be so that
$|\Gamma_{i}:\Gamma_{i}^{\prime}|\leq n$. Let $H$ be the intersection of all
subgroups of $G$ of index at most $n$. Then
$\phi_{i}(H)\subseteq\Gamma_{i}^{\prime}$, so
$\phi_{i}|_{H}\in{\operatorname{Hom}}(H,\mathcal{G})$. Since $\mathcal{G}$ is
equationally noetherian by Theorem 2.4 $\omega$–almost surely
$\operatorname{Ker}(\phi_{\infty}|_{H})\subseteq\operatorname{Ker}(\phi_{i}|_{H})$.

Let $g\in G$. If $\phi_{\infty}(g)\notin\phi_{\infty}(H)$, then for any $h\in
H$ $gh\notin\operatorname{Ker}(\phi_{\infty})$. Suppose now that
$\phi_{\infty}(g)\in\phi_{\infty}(H)$. Fix $k\in H$ such that
$\phi_{\infty}(g)=\phi_{\infty}(k)$. Now, $\omega$–almost surely
$gk^{-1}\in\operatorname{Ker}(\phi_{i})$ and
$\operatorname{Ker}(\phi_{\infty}|_{H})\subseteq\operatorname{Ker}(\phi_{i}|_{H})$.
For any such $i$ and for any $h\in H$, if
$gh\in\operatorname{Ker}(\phi_{\infty})$ then
$kh\in\operatorname{Ker}(\phi_{\infty})\subseteq\operatorname{Ker}(\phi_{i})$.
Hence $gh=gk^{-1}kh\in\operatorname{Ker}(\phi_{i})$. Putting together these
two cases, we see that for any $g\in G$ $\omega$–almost surely
$\operatorname{Ker}(\phi_{\infty})\cap
gH\subseteq\operatorname{Ker}(\phi_{i})$. Repeating this argument over a
finite of coset representatives of $H$ in $G$ proves that $\omega$–almost
surely
$\operatorname{Ker}(\phi_{\infty})\subseteq\operatorname{Ker}(\phi_{i})$. ∎

######  Remark 3.5.

The bound on the index in Proposition 3.4 is essential. Indeed, it is
straightforward to see that if one considers the family
$\mathcal{G}^{\mathrm{fin}}$ of groups which have a finite index subgroup
contained in $\mathcal{G}$ then $\mathcal{G}$ may be equationally noetherian
while $\mathcal{G}^{\mathrm{fin}}$ is not. This is true, for example, in case
$\mathcal{G}$ is the family consisting of the trivial group, since then
$\mathcal{G}^{\mathrm{fin}}$ is the family of finite groups, which we have
already seen is not equationally noetherian.

######  Definition 3.6.

If $\mathcal{G}$ is a family of groups, then let $\mathcal{G}^{\ast}$ be the
family of all free products of elements of $\mathcal{G}$.

######  Theorem 3.7.

[12, Cor. C] If $\mathcal{G}$ is equationally noetherian then so is
$\mathcal{G}^{\ast}$.

###  3.2. The geometric decomposition

If $M$ is a closed, orientable, irreducible $3$–manifold then the
Geometrization Theorem (see [24, 25, 26, 10, 17, 22]) gives a (possibly
trivial) decomposition of $M$ along incompressible tori into geometric pieces.
This induces a splitting of $\pi_{1}(M)$ which we call the _geometric
decomposition_. We refer to [4], [28], or [32] for the definition of a
geometric $3$–manifold, and to [28] for details about each of the $8$ Thurston
geometries.

Assume $M$ is a closed, orientable, irreducible 3–manifold and also that $M$
is not a geometric 3–manifold (in particular, $M$ is not a torus bundle over a
circle). Then the each piece of the geometric decomposition of $M$ is either
hyperbolic, large Seifert-fibered, or a twisted $I$–bundle over a Klein
bottle. Replacing $M$ by a double cover as in [37, Lemma 2.4] we may assume
there are no twisted $I$–bundles over a Klein bottle. By a similar argument,
there is a double cover so all the base orbifolds of large Seifert-fibered
pieces are orientable. Thus:

######  Lemma 3.8.

Let $M$ be a closed, orientable, irreducible, non-geometric $3$–manifold. Then
$M$ has a cover of degree at most $4$ which lies in
$\mathcal{M}_{\operatorname{Gen}}$.

An important property of elements of $\mathcal{M}_{\operatorname{Gen}}^{\pi}$
is the following result, which follows immediately from the proof of [37,
Lemma 2.4]. Recall a group action on a tree $T$ is _$k$ –acylindrical_ if the
stabilizer of any segment in $T$ of length at least $k+1$ is trivial.

######  Lemma 3.9.

Suppose $\Gamma\in\mathcal{M}_{\operatorname{Gen}}^{\pi}$ and that
$T_{\Gamma}$ is the Bass-Serre tree dual to the geometric decomposition of
$\Gamma$. The $\Gamma$–action on $T_{\Gamma}$ is $2$–acylindrical.

###  3.3. Reduction to $\mathcal{M}_{\operatorname{Gen}}^{\pi}$

We now identify the some pertinent classes of $3$–manifolds.

######  Definition 3.10.

Let $\mathcal{M}_{\operatorname{Ori}}^{\pi}$ be set of the fundamental groups
of closed, orientable $3$–manifolds. Let
$\mathcal{M}_{\operatorname{Irr}}^{\pi}$ be the set of fundamental groups of
closed, orientable and irreducible $3$–manifolds. Let
$\mathcal{M}_{\operatorname{Geo}}^{\pi}$ be the set of fundamental groups of
compact geometric $3$–manifolds with all boundary components tori.

######  Lemma 3.11.

If $\mathcal{M}_{\operatorname{Ori}}^{\pi}$ is equationally noetherian then so
is $\mathcal{M}^{\pi}$.

###### Proof.

The fundamental group of any compact 3–manifold with boundary is a subgroup
(in fact, a retract) of the fundamental group of a closed 3–manifold. It is an
immediate consequence of [12, Lemma 3.9] that a family of groups $\mathcal{G}$
is equationally noetherian if and only if the family of all subgroups of
elements of $\mathcal{G}$ is equationally noetherian, so it suffices to
consider only closed manifolds.

For each closed 3–manifold $M$, $\pi_{1}(M)$ has an index at most $2$ subgroup
belonging to $\mathcal{M}_{\operatorname{Ori}}^{\pi}$, so the result follows
from Proposition 3.4. ∎

Each closed, orientable 3–manifold $M$ has a unique Milnor–Kneser
decomposition which induces a free product decomposition of $\pi_{1}(M)$:

$\pi_{1}(M)\cong\pi_{1}(M_{1})\ast...\ast\pi_{1}(M_{k})\ast F_{n}$

where each $\pi_{1}(M_{i})\in\mathcal{M}_{\operatorname{Irr}}^{\pi}$ (and
$F_{n}$ is a free group of rank $n$). The following is thus an immediate
consequence of Theorem 3.7

######  Corollary 3.12.

If $\mathcal{M}_{\operatorname{Irr}}^{\pi}$ is equationally noetherian then so
is $\mathcal{M}_{\operatorname{Ori}}^{\pi}$.

Consider the case $\mathcal{M}_{\operatorname{LSF}}^{\pi}$ of large
Seifert–fibered manifolds with orientable base orbifold. We prove some basic
results, and also some results we need later. Let
$\mathcal{M}_{\operatorname{Orb}}^{\pi}$ be the set of fundamental groups of
orientable hyperbolic $2$–orbifolds of finite-type. Each
$\Gamma\in\mathcal{M}_{\operatorname{LSF}}^{\pi}$ admits a short exact
sequence which is a central extension:

$1\to\mathbb{Z}\to\Gamma\to\Gamma_{B}\to 1,$

for some $\Gamma_{B}\in\mathcal{M}_{\operatorname{Orb}}^{\pi}$ (the
fundamental group of the base orbifold). From this, the following result
follows quickly.

######  Lemma 3.13.

Let $L$ be an $\mathcal{M}_{\operatorname{LSF}}^{\pi}$–limit group. Then there
is a central extension

$1\to A\to L\to B\to 1,$

where $A$ is abelian and $B$ is an
$\mathcal{M}_{\operatorname{Orb}}^{\pi}$–limit group.

######  Definition 3.14.

A group $G$ is $\mathcal{A}$–slender if every abelian subgroup of $G$ is
finitely generated.

Our analysis of $\mathcal{M}_{\operatorname{LSF}}^{\pi}$–limit groups is based
on combining Lemma 3.13 with the following theorem of the third author.

######  Theorem 3.15.

[19, Theorem 1.2] Let $B$ be an $\mathcal{M}_{\operatorname{Orb}}^{\pi}$–limit
group. Then $B$ is finitely presented and $\mathcal{A}$–slender.

######  Corollary 3.16.

Let $L$ be an $\mathcal{M}_{\operatorname{LSF}}^{\pi}$–limit group. Then $L$
is finitely presented and $\mathcal{A}$–slender.

###### Proof.

Consider the central extension from Lemma 3.13. Since $B$ is finitely
presented by Theorem 3.15, $A$ is finitely generated, for example by [13,
Lemma 12.1]. It is now easy to see $L$ is finitely presented. The fact that
$A$ is finitely generated and abelian subgroups of $B$ are finitely generated
implies that all abelian subgroups of $L$ are finitely generated. ∎

######  Lemma 3.17.

Let $M$ be a large Seifert fibered $3$–manifold with orientable base orbifold
$O$. Let $K$ be the kernel of the natural quotient map
$\pi_{1}(M)\to\pi_{1}(O)$. Let $g,h\in\pi_{1}(M)$. If $[g,h]\in K$, then
$[g,h]=1\in\pi_{1}(M)$.

###### Proof.

All abelian subgroups of $\pi_{1}(O)$ are cyclic, so the images of $g$ and $h$
in $\pi_{1}(O)$ generate a cyclic subgroup. A central extension of a cyclic
group is abelian. ∎

The following result follows quickly from the fact that finite subgroups of
elements of $\mathcal{M}_{\operatorname{Orb}}^{\pi}$ are cyclic.

######  Lemma 3.18.

Let $B$ be an $\mathcal{M}_{\operatorname{Orb}}^{\pi}$–limit group. Any finite
subgroup of $B$ is cyclic.

The following result is now immediate from Lemmas 3.17 and 3.18.

######  Corollary 3.19.

Suppose that $L$ is an $\mathcal{M}_{\operatorname{LSF}}^{\pi}$–limit group,
and let $1\to A\to L\to B\to 1$ be as in Lemma 3.13. Suppose that $E\leq L$
has finite image in $B$. Then $E$ is abelian.

######  Lemma 3.20.

Let $\Gamma\in\mathcal{M}_{\operatorname{Gen}}^{\pi}$, and let $\Gamma_{e}$ be
an edge group of the geometric decomposition of $\Gamma$, with adjacent vertex
groups $\Gamma_{v}$, $\Gamma_{w}$. The pre-images in $\Gamma_{e}$ of the
centers $Z(\Gamma_{v})$ and $Z(\Gamma_{w})$ intersect trivially.

###### Proof.

The only way that $Z(\Gamma_{v})$ and $Z(\Gamma_{w})$ can both be nontrivial
is if the corresponding pieces are both Seifert–fibered. The centers of
adjacent Seifert-fibered pieces intersect trivially, since otherwise the
connecting torus is not part of the characteristic sub-manifold. ∎

######  Theorem 3.21.

$\mathcal{M}_{\operatorname{Geo}}^{\pi}$ is an equationally noetherian family
of groups.

###### Proof.

It is a straightforward observation that any finite union of equationally
noetherian families is equationally noetherian [12, Lemma 3.8]. Hence it
suffices to deal with each geometry individually.

Suppose first that $M$ supports one of the following geometries:
$\mathbb{S}^{3}$, $\mathbb{S}^{2}\times\mathbb{R}$, $\mathbb{E}^{3}$,
$\operatorname{NIL}$ or $\operatorname{SOL}$. Then (as, for example, explained
in [13, $\S 7-11$]) $\pi_{1}(M)$ has a subgroup of index at most $240$ which
is either cyclic (finite or infinite), $\mathbb{Z}^{3}$, or some semi-direct
product $\mathbb{Z}\rtimes\mathbb{Z}$. Each of these groups embeds in
$\operatorname{SL}_{3}(\mathbb{Z})$. Therefore, the union of the set of
fundamental groups of manifolds supporting these five geometries in
equationally noetherian by Proposition 3.4 and Lemma 3.1.

The fundamental group of any hyperbolic $3$–manifold is (isomorphic to) a
subgroup of $\operatorname{PSL}(2,\mathbb{C})$ (and hence also contained in
$\operatorname{SL}(3,\mathbb{C})$ – see [13, Lemma 12.9]). Thus, the family of
fundamental groups of hyperbolic 3–manifolds is equationally noetherian by
Lemma 3.1.

The two remaining geometries correspond to _large Seifert fibered manifolds_.
Since we can pass to an index $2$ subgroup to lie in
$\mathcal{M}_{\operatorname{LSF}}^{\pi}$, by Proposition 3.4 it suffices to
consider $\mathcal{M}_{\operatorname{LSF}}^{\pi}$–limit groups. If $L$ is an
$\mathcal{M}_{\operatorname{LSF}}^{\pi}$–limit group then $L$ is finitely
presented by 3.16. It follows that the sequence of maps defining $L$
$\omega$–factors through the limit (see Lemma 3.2). Thus the family of
fundamental groups of large Seifert-fibered manifolds is equationally
noetherian by Theorem 2.4. ∎

Summarizing the above discussion and the previous results in the section, we
have the following result from Section 2.

See 2.2

###### Proof.

By Lemma 3.11 and Corollary 3.12 it is enough to prove that
$\mathcal{M}_{\operatorname{Irr}}^{\pi}$ is equationally noetherian. By
Theorem 3.21 $\mathcal{M}_{\operatorname{Geo}}^{\pi}$ is equationally
noetherian, so it is enough to consider closed, orientable, irreducible, non-
geometric $3$–manifolds. By Lemma 3.8 any such $3$–manifold has a degree at
most $4$ cover which lies in $\mathcal{M}_{\operatorname{Gen}}$, so by
Proposition 3.4 in suffices to prove that
$\mathcal{M}_{\operatorname{Gen}}^{\pi}$ is equationally noetherian. ∎

##  4\. Generalized geometric decompositions

In order to prove Theorem D, it remains to prove Theorems 2.5, 2.6 and 2.7. In
this section, we provide the setting and some basic definitions, and prove
Theorems 2.5 and 2.6.

Let $G=\langle S\rangle$ be a finitely generated group, and let
$\left(\phi_{i}\colon G\to\Gamma_{i}\right)$ be a sequence from
${\operatorname{Hom}}(G,\mathcal{M}_{\operatorname{Gen}}^{\pi})$. Let
$L=G/\operatorname{Ker}^{\omega}(\phi_{i})$ be the
$\mathcal{M}_{\operatorname{Gen}}^{\pi}$–limit group associated to
$(\phi_{i})$, with associated limit map $\phi_{\infty}\colon\thinspace
G\twoheadrightarrow L$ (see Definition 2.3).

######  Definition 4.1.

Suppose that $G,(\phi_{i})$ and $L$ are as above, and that $g\in L$. An
_$\omega$ –approximation _to $g$ is a sequence $(g_{i})$ so
$g_{i}\in\Gamma_{i}$ and there is an element $\widetilde{g}\in G$ so that
$\phi_{\infty}(\widetilde{g})=g$ and for each $i$ we have
$\phi_{i}(\widetilde{g})=g_{i}$.

Similarly, if $F\subset L$ is a finite set then an _$\omega$ –approximation
_to $F$ is a sequence of (ordered) tuples $(F_{i})$ from $\Gamma_{i}$ so that
for some lift $\widetilde{F}$ of $F$ to $G$ (so $\phi_{\infty}$ restricts to
an ordered bijection from $\widetilde{F}$ to $F$) we have
$\phi_{i}(\widetilde{F})=F_{i}$ (again, as an ordered bijection).

The following lemmas are immediate.

######  Lemma 4.2.

Suppose that $g\in L$ and that $(g_{i})$ and $(g_{i}^{\prime})$ are
$\omega$–approximations to $g$. Then $\omega$–almost surely
$g_{i}=g_{i}^{\prime}$. Similarly, if $F\subset L$ is finite and $(F_{i})$,
$(F_{i}^{\prime})$ are $\omega$–approximations to $F$ then $\omega$–almost
surely $F_{i}=F_{i}^{\prime}$.

######  Lemma 4.3.

Suppose $g^{1},\dots,g^{s}\in L$ and for each $1\leq j\leq s$, $(g_{i}^{j})$
is an $\omega$–approximation to $g^{j}$. For any word $w(x_{1},\dots,x_{s})$
in $\\{x_{i}^{\pm}\\}$ we have $w(g^{1},\ldots,g^{s})=1$ in $L$ if and only if
for $\omega$–almost all $i$ $w(g_{i}^{1},\ldots,g_{i}^{s})=1$ in $\Gamma_{i}$.

We are only concerned with properties of $\omega$–approximations that hold
$\omega$–almost surely for terms in the sequence, and so it is always
irrelevant which $\omega$–approximation we choose for an element (or finite
subset) of $L$. We use this observation frequently without mention.

Let $S$ be a finite generating set for $G$, $(S_{i})$ an
$\omega$–approximation to $S$, and $T_{i}$ the tree dual to the geometric
decomposition of $\Gamma_{i}$. Let

$\|\phi_{i}\|_{T_{i}}=\inf_{t\in T_{i}}\max_{s\in
S_{i}}d_{T_{i}}\left(t,s.t\right).$

######  Definition 4.4.

The sequence $\left(\phi_{i}\right)$ is _$\mathcal{T}$ –divergent_ if
${\lim}^{\omega}\|\phi_{i}\|_{T_{i}}=\infty$.

We are now ready to prove the following theorem from Section 2.

See 2.5

###### Proof.

The action of any $\Gamma\in\mathcal{M}_{\operatorname{Gen}}^{\pi}$ on its
geometric tree is $2$–acylindrical (see Lemma 3.9). In particular,
$\mathcal{M}_{\operatorname{Gen}}^{\pi}$ is (in the terminology of [12]) a
uniformly acylindrical family of groups (see, for example, [12, p.7142]).
Moreover, any sequence from
${\operatorname{Hom}}(G,\mathcal{M}_{\operatorname{Gen}}^{\pi})$ which is not
$\mathcal{T}$–divergent is ‘non-divergent’ in the sense of [12]. Thus, Theorem
2.5 is an immediate consequence of [12, Theorem B]. ∎

As explained in Section 2, by Theorem 2.5 we may assume the sequence
$(\phi_{i})$ is not $\mathcal{T}$–divergent.

######  Definition 4.5.

Let $g\in L$, and let $(g_{i})$ be an $\omega$–approximation to $g$. Then $g$
is _stably parabolic with respect to $(\phi_{i})$_ if $\omega$–almost surely
$g_{i}$ fixes an edge in $T_{i}$. A subgroup $H\leq L$ is _stably parabolic
with respect to $(\phi_{i})$_ if for any finite $F\subset H$ and any
$\omega$–approximation $(F_{i})$ of $F$ $\omega$–almost surely there is an
edge in $T_{i}$ fixed by each $f\in F_{i}$. The set of stably parabolic
subgroups of $L$ with respect to $(\phi_{i})$ is $\mathcal{H}_{L,(\phi_{i})}$,
or just $\mathcal{H}_{L}$ when $(\phi_{i})$ is implied.

Since each subgroup of $\Gamma_{i}$ which stabilizes an edge in $T_{i}$ is
isomorphic to $\mathbb{Z}^{2}$, the following result is immediate from Lemma
4.3.

######  Lemma 4.6.

Any stably parabolic subgroup of $L$ is abelian.

######  Definition 4.7.

A subgroup $H$ of $L$ is _$\omega$ –geometric_ if for any finite subset
$F\subset H$ and any $\omega$–approximation $(F_{i})$ of $F$, $\omega$–almost
surely there is a vertex $v_{i}$ of $T_{i}$ so that $F_{i}$ fixes $v_{i}$.

######  Terminology 4.8.

Let $H$ be an $\omega$–geometric subgroup of $L$ and $v_{i}$ be as in
Definition 4.7. Then either

  1. (1)

$\omega$–almost surely $v_{i}$ is of hyperbolic type; or

  2. (2)

$\omega$–almost surely $v_{i}$ is of LSF-type.

In the first case $H$ is _hyperbolic-type_ and in the second $H$ is _LSF-
type_.

######  Definition 4.9.

Let $H$ be an $\omega$–geometric subgroup of $L$, with associated sequence
$v_{i}$ of vertices of $T_{i}$, and let $\Gamma_{i,v_{i}}$ be the associated
vertex group of $\Gamma_{i}$. The stable center of $H$, denoted
$Z^{\omega}(H)$, is the set of $g\in H$ so that for any $\omega$–approximation
$(g_{i})$ of $g$, $\omega$–almost surely $g_{i}\in Z(\Gamma_{i,v_{i}})$

Observe that if $H$ is a hyperbolic-type $\omega$–geometric subgroup of $L$
then $Z^{\omega}(H)=\\{1\\}$.

As we see in Definition 4.15 below, $L$ admits a splitting arising from a
limiting action on the trees $T_{i}$. This geometric decomposition is the
natural splitting associated to $L$, but for technical reasons in Section 9 we
need a more general class of splittings of $L$.

######  Definition 4.10.

A minimal abelian splitting ${{\mathbb{G}}}$ of $L$ is a generalized geometric
decomposition (or GGD) with respect to $(\phi_{i})$ if

  1. (1)

Each vertex group of ${{\mathbb{G}}}$ is $\omega$–geometric

  2. (2)

Let $v$ and $w$ be adjacent vertices of ${{\mathbb{G}}}$ with sequences
$(v_{i})$ and $(w_{i})$ as in the definition of $\omega$–geometric. Then
$\omega$–almost surely $v_{i}$ and $w_{i}$ are adjacent in $T_{i}$;

  3. (3)

${{\mathbb{G}}}$ is an
$\left(\mathcal{H}_{L,(\phi_{i})},\mathcal{H}_{L,(\phi_{i})}\right)$–splitting.

  4. (4)

For each LSF–type vertex group ${{\mathbb{G}}}_{v}$ of ${{\mathbb{G}}}$,
$Z^{\omega}({{\mathbb{G}}}_{v})$ is contained in each edge group of
${{\mathbb{G}}}$ adjacent to ${{\mathbb{G}}}_{v}$.

For the rest of this subsection, fix the above notation, and let
${{\mathbb{G}}}$ be a GGD of $L$ with respect to $(\phi_{i})$. The sequences
$(v_{i})$ and $(w_{i})$ in Definition 4.10 are implicitly fixed.

######  Lemma 4.11.

If ${{\mathbb{G}}}_{v}$ is an LSF-type vertex group of ${{\mathbb{G}}}$ then
$Z^{\omega}({{\mathbb{G}}}_{v})$ is central in ${{\mathbb{G}}}_{v}$.

###### Proof.

Let $g\in Z^{\omega}({{\mathbb{G}}}_{v})$ and $h\in{{\mathbb{G}}}_{v}$. If
$(g_{i})$ and $(h_{i})$ are $\omega$–approximations to $g$ and $h$,
respectively then $\omega$–almost surely there is an LSF-type vertex group
$\Gamma_{i,v_{i}}$ of $\Gamma_{i}$ so that $g_{i}\in Z(\Gamma_{i,v_{i}})$ and
$h_{i}\in\Gamma_{i,v_{i}}$. Therefore $\omega$–almost surely $[g_{i},h_{i}]=1$
in $\Gamma_{i}$. By Lemma 4.3 $[g,h]=1$ in $L$. ∎

Let $V$ be a vertex group of ${{\mathbb{G}}}$. By Lemma 4.11,
$Z^{\omega}(V)\unlhd V$, and we define $\overline{V}=V/Z^{\omega}(V)$. Let
$\pi:V\rightarrow\overline{V}$ be the quotient map.

######  Lemma 4.12.

Let $V$ be an LSF type vertex group of ${{\mathbb{G}}}$. If $H$ is an abelian
subgroup of $\overline{V}$ then $\pi^{-1}(H)$ is an abelian subgroup of $V$.

###### Proof.

Let $\bar{g},\bar{h}\in H$, let $g\in\pi^{-1}(\bar{g})$ and
$h\in\pi^{-1}(\bar{h})$, and let $(g_{i})$ and $(h_{i})$ be
$\omega$–approximations to $g$ and $h$, respectively. Since
$[\bar{g},\bar{h}]=1$, we have $[g,h]\in Z^{\omega}(V)$, and so
$\omega$–almost surely $[g_{i},h_{i}]\in Z(\Gamma_{v_{i}})$. Since
$Z(\Gamma_{v_{i}})$ is the kernel of
$p_{i}:\Gamma_{v_{i}}\twoheadrightarrow\pi_{1}(O_{i})$, where $O_{i}$ is the
base orbifold of the LSF manifold corresponding to $\Gamma_{v_{i}}$, by Lemma
3.17, we have $[g_{i},h_{i}]=1$ $\omega$–almost surely. By Lemma 4.3
$[g,h]=1$, as required. ∎

Let $V$, $\overline{V}$ be as above, let $\mathcal{E}$ be the edge groups of
${{\mathbb{G}}}$ adjacent to $V$, and $\overline{\mathcal{E}}$ their image in
$\overline{V}$. By Corollary B.4 $\overline{V}$ admits a graph of groups
decomposition $\overline{\mathbb{L}}$ rel $\overline{\mathcal{E}}$ so that all
edge groups of $\overline{\mathbb{L}}$ have cardinality at most $4$, and no
vertex group of $\overline{\mathbb{L}}$ splits rel $\overline{\mathcal{E}}$
over a subgroup of size at most $4$. We call $\overline{\mathbb{L}}$ the
_relative $4$–Linnell decomposition of $\overline{V}$ rel
$\overline{\mathcal{E}}$_. Notice that by Corollary 3.19 the preimages in $V$
of edges in $\overline{\mathbb{L}}$ are abelian.

######  Definition 4.13.

Let $V$ be an LSF–type vertex group of ${{\mathbb{G}}}$. Let
$\overline{V}=V/Z^{\omega}(V)$. Let $\overline{\mathbb{L}}$ be the relative
$4$–Linnell decomposition of $\overline{V}$ relative to (images of) edge
groups of ${{\mathbb{G}}}$ adjacent to $V$, and $\mathbb{L}$ the induced
abelian splitting of $V$ relative to adjacent edge groups. The canonical
refinement of ${{\mathbb{G}}}$, denoted $\mathbb{G}^{r}$, is the refinement of
${{\mathbb{G}}}$ obtained by replacing each LSF-type vertex group $V$ by
$\mathbb{L}$.

######  Definition 4.14.

Suppose $V$ is a vertex group of $\mathbb{G}^{r}$. A good relative generating
set for $V$ is a finite set $A$ so that

  1. (1)

$A$ together with the adjacent edge groups generates $V$; and

  2. (2)

For each edge group $E$ adjacent to $V$, there exists $a\in A\cap E\setminus
Z^{\omega}(V)$.

###  4.1. The geometric decomposition

Let $(\phi_{i}\colon\thinspace G\to\Gamma_{i})$ be a sequence from
${\operatorname{Hom}}(G,\mathcal{M}_{\operatorname{Gen}}^{\pi})$ which is not
$\mathcal{T}$–divergent and let $T_{i}$ be the minimal $\phi_{i}(G)$–invariant
subtree of the tree dual to the geometric splitting of $\Gamma_{i}$. Let
$o_{i}$ be a point in $T_{i}$ so $\max\limits_{s\in
S}d_{T_{i}}(o_{i},\phi_{i}(s)o_{i})=\|\phi_{i}\|_{T_{i}}$.

Let $T_{\infty}={\lim}^{\omega}(T_{i},o_{i})$. Since each $T_{i}$ is a tree in
which edges have length one, $T_{\infty}$ is a simplicial tree. Moreover, the
actions of $G$ on $T_{i}$ induce a $2$–acylindrical action of $L$ on
$T_{\infty}$; see [12, Proposition 6.1].

######  Definition 4.15.

Let $G$, $S$, $\left(\phi_{i}\right)$, $T_{i}$, $L$ and $T_{\infty}$ be as
above. The geometric tree of $L$, denoted $T_{\operatorname{geom}}$, is the
minimal $L$–invariant subtree of $T_{\infty}$.

The _geometric decomposition_ of $L$ is the splitting dual to
$T_{\operatorname{geom}}$.

It is straightforward to check that the geometric decomposition of $L$ is in
fact a GGD in the sense of Definition 4.10, which we record in the following
lemma.

######  Lemma 4.16.

Let $L$ be an $\mathcal{M}_{\operatorname{Gen}}^{\pi}$–limit group defined by
a sequence $(\phi_{i}\colon\thinspace G\to\Gamma_{i})$ (so
$L=G/\operatorname{Ker}^{\omega}(\phi_{i})$). If $(\phi_{i})$ is not
$\mathcal{T}$–divergent then the geometric splitting of $L$ is a GGD with
respect to $(\phi_{i})$.

We now prove the following result from Section 2.

See 2.6

###### Proof.

Let ${{\mathbb{G}}}$ be the geometric decomposition of $L$. The edge groups of
${{\mathbb{G}}}$ are finitely generated, so the vertex groups are also and
hence each vertex group of ${{\mathbb{G}}}$ is an
$\mathcal{M}_{\operatorname{Geo}}^{\pi}$–limit group. By Theorem 3.21, the
defining sequence for each vertex group $\omega$–factors through the limit.
The edge groups of ${{\mathbb{G}}}$ are finitely generated and abelian, so $L$
is finitely presented relative to the vertex groups of ${{\mathbb{G}}}$, and
by Lemma 3.2 the defining sequence for $L$ $\omega$–factors through the limit,
as required. ∎

Given Theorem 2.6, our goal is to prove that the edge groups of
${{\mathbb{G}}}$ are finitely generated. It is tempting to think that this
follows from the fact that they are abelian edge groups in an acylindrical
splitting of a finitely generated group. Moreover, we also know that some (and
we believe all) vertex groups of this splitting have the property that their
finitely generated subgroups are $\mathcal{A}$–slender. The next example shows
that this is not enough.

######  Example 4.17.

Let $A=\langle\\{a_{i}\\}_{i\in\mathbb{N}}\rangle$ and
$B=\langle\\{b_{i}\\}_{i\in\mathbb{N}}\rangle$ be free abelian groups of
countably infinite rank, and let $H=A\ast B$. Define:

  * •

$A_{e}=\langle\\{a_{2k}\\}_{k\in\mathbb{N}}\rangle$
$A_{o}=\langle\\{a_{2k-1}\\}_{k\in\mathbb{N}}\rangle$;

  * •

$B_{e}=\langle\\{b_{2k}\\}_{k\in\mathbb{N}}\rangle$
$B_{o}=\langle\\{b_{2k+1}\\}_{k\in\mathbb{N}}\rangle$.

Define $\phi:A_{o}\to A_{e}$ by $\phi(a_{2k-1})=a_{2k}$ and $\psi:B_{e}\to
B_{o}$ by $\psi(b_{2k})=b_{2k+1}$. Then $\phi$ and $\psi$ are both
isomorphisms. Considering $\phi$ and $\psi$ to be isomorphisms between
subgroups of $H$, let $K$ be the the double HNN extension of $H$ via maps
$\phi$ and $\psi$, and observe $K$ is not finitely generated.

The group $K$ naturally contains $A$ and $B$ as subgroups. Let $\theta:A\to B$
be the isomorphism $\theta(a_{i})=b_{i}$, and $G$ the HNN extension of $K$
with map $\theta$. Then $G$ is generated by the stable letters of the HNN
extensions and $a_{1}$.

$G$ is finitely generated and the defining HNN extension of $G$ is a
$2$–acylindrical splitting with abelian edge groups, but the vertex group and
the edge groups of this splitting are not finitely generated. The vertex group
also has the property that finitely generated subgroups are
$\mathcal{A}$–slender (recall Definition 3.14).

###  4.2. Assumptions

It remains to prove Theorem 2.7. If $G$ is the group used to defined the limit
group $L$, then we can replace $G$ by a finitely presented group. For the
remainder of the paper we make the following assumptions, which are key for
the proof of Theorem 2.7.

######  Standing Assumption 4.18.

Let $G$ be a finitely presented group with finite generating set $S$. Let
$(\phi_{i}\colon\thinspace G\to\Gamma_{i}=\pi_{1}(M_{i}))$ be a sequence from
${\operatorname{Hom}}(G,\mathcal{M}_{\operatorname{Gen}}^{\pi})$ and let
$T_{i}$ be the minimal $\phi_{i}(G)$–invariant subtree of the geometric tree
associated to $M_{i}$. Suppose $(\phi_{i})$ is not $\mathcal{T}$–divergent.
Let $L=G/\operatorname{Ker}^{\omega}(\phi_{i})$ be the
$\mathcal{M}_{\operatorname{Gen}}^{\pi}$–limit group, and
$\phi_{\infty}\colon\thinspace G\twoheadrightarrow L$ the limit map. Let
${{\mathbb{G}}}$ be a GGD of $L$ wrt $(\phi_{i})$ and $T_{{\mathbb{G}}}$ the
Bass–Serre tree. Let $\mathbb{G}^{r}$ be the canonical refinement of
${{\mathbb{G}}}$ (Definition 4.13). Fix a vertex group $V$ of $\mathbb{G}^{r}$
associated to the vertex $v$, and a good relative generating set $A$ of $V$
(Definition 4.14). Let $(A_{i})$ be an $\omega$–approximation to $A$. Denote
by $\mathcal{H}_{V}$ the set of stably parabolic subgroups of $V$.

##  5\. The Collapsed Space

Suppose $\Delta\in\mathcal{M}_{\operatorname{Gen}}^{\pi}$, and consider the
geometric decomposition of $\Delta$. There are two kinds of vertex groups: (i)
Hyperbolic vertex groups which act on $\mathbb{H}^{3}$; and (ii) Large
Seifert-fibered vertex groups which (through projection to the base orbifold
and a choice of hyperbolic structure on this base orbifold) act on
$\mathbb{H}^{2}$. Each of $\mathbb{H}^{3}$ and $\mathbb{H}^{2}$ are
$\delta$–hyperbolic. However, these actions do not have properties that are
well suited for the arguments in this paper.

In this section we build the collapsed space, a variation on Farb’s
construction of the “Electric” space from [11]. The collapsed space is
geodesic and $\delta$–hyperbolic for a uniform $\delta$ (see Proposition 5.7
and Theorem 5.8). The key advantage of the collapsed space over
$\mathbb{H}^{2}$ and $\mathbb{H}^{3}$ is Lemma 5.15. As well as these results,
we prove a collection of basic structural results which are used in later
sections.

###  5.1. The collapsed space for Kleinian and Fuchsian groups

Let $\Gamma$ be a finitely generated Fuchsian or torsion-free Kleinian group.
In case $\Gamma$ is Kleinian, let $\mathbb{H}^{\ast}=\mathbb{H}^{3}$, and in
case $\Gamma$ is Fuchsian let $\mathbb{H}^{\ast}=\mathbb{H}^{2}$. Fix
$\delta_{\mathbb{H}}$ so that all geodesic triangles in $\mathbb{H}^{2}$ and
$\mathbb{H}^{3}$ are $\delta_{\mathbb{H}}$–thin. Let $\epsilon$ be the
$3$–dimensional (or $2$–dimensional) Margulis constant. (see [7] for more
information). Define the _$\epsilon$ –thin part_ of $\mathbb{H}^{\ast}$ with
respect to $\Gamma$:

$\mathcal{U}_{\epsilon}(\Gamma)=\\{x\in\mathbb{H}^{\ast}\mid d(x,\gamma
x)\leq\epsilon\text{ for some }\gamma\neq 1\in\Gamma\\}.$

The set $\mathcal{U}_{\epsilon}(\Gamma)$ is a disjoint union of a collection
of horoballs, balls and neighborhoods of geodesics (see, for example, [7,
Theorem D.3.3], though we work in the universal cover). We refer to the
neighborhoods of geodesics in the thin part of $\mathbb{H}^{\ast}$ as _tubes_.
The set $\mathcal{U}^{K}(\Gamma)$ below can be found by shrinking the
components of $\mathcal{U}_{\epsilon}(\Gamma)$ until they are distance $K$
apart, and then keeping only the tubes and balls which remain that are of
radius at least $K$.

######  Proposition 5.1.

For any $K>0$ there exists $D>0$ and a $\Gamma$–invariant open set
$\mathcal{U}^{K}(\Gamma)\subseteq\mathcal{U}_{\epsilon}(\Gamma)$ so that
$\mathcal{U}^{K}(\Gamma)$ satisfies the following:

  1. (1)

The components of $\mathcal{U}^{K}(\Gamma)$ are tubes, horoballs and balls;

  2. (2)

Distinct components of $\mathcal{U}^{K}(\Gamma)$ are at least $K$ apart;

  3. (3)

Suppose $x\in\partial U$ for some connected component of $U$ of
$\mathcal{U}^{K}(\Gamma)$. Then every $g\in\Gamma$ moves $x$ by at least $D$;

  4. (4)

Tubes and balls in $\mathcal{U}^{K}(\Gamma)$ have radius at least $K$; and

  5. (5)

Any horoball in $\mathcal{U}_{\epsilon}(\Gamma)$, and every tube and ball of
radius at least $2K$ is contained in the $K$–neighborhood of
$\mathcal{U}^{K}(\Gamma)$.

Given points $x,y$ in a tube $U$ around a geodesic $\gamma$ in
$\mathbb{H}^{*}$, let $\pi_{\gamma}\colon\thinspace U\to\gamma$ be the closest
point projection and let
$d_{\gamma}\left(x,y\right)=d_{\mathbb{H}^{*}}(\pi_{\gamma}(x),\pi_{\gamma}(y))$.
Suppose $K>1$ (a specific value for $K$ is fixed later). Define an “electric”
distance $d$ on $\mathbb{H}^{*}$ by

$d_{el}(x,y)=\left\\{\begin{array}[]{ll}0&\mbox{if $x,y$ lie in the same
horoball or ball of $\mathcal{U}^{K}(\Gamma)$;}\\\ d_{\gamma}(x,y)&\mbox{if
$x,y$ both lie in a tube in $\mathcal{U}^{K}(\Gamma)$; and}\\\
d_{\mathbb{H}^{\ast}}(x,y)&\mbox{otherwise.}\end{array}\right.$

Define a pseudo-metric $\widehat{d}$ on $\mathbb{H}^{*}$ by setting

$\widehat{d}(x,y)=\inf_{\\{x_{i}\\}_{i=0}^{n}}\left\\{\sum_{i=1}^{n}d_{el}(x_{i},x_{i-1})\right\\},$

where the infimum is taken over all finite sets
$x_{0}=x,\ldots,x_{n-1},x_{n}=y$.

Denote by $(\mathcal{C}^{K}(\Gamma),d)$ the induced metric space of
$(\mathbb{H}^{*},\widehat{d})$, and let
$q^{K}_{\Gamma}\colon\thinspace\mathbb{H}^{\ast}\to\mathcal{C}^{K}(\Gamma)$ be
the canonical quotient map. Observe $(\mathcal{C}^{K}(\Gamma),d)$ is a length
space. Denote the length of a path $p$ in $\mathcal{C}^{K}(\Gamma)$ by
$\widehat{\ell}(p)$.

######  Definition 5.2.

The collapsing locus of $\mathcal{C}^{K}(\Gamma)$ is the collection of points
$x\in\mathcal{C}^{K}(\Gamma)$ so that
$\left|(q^{K}_{\Gamma})^{-1}(x)\right|>1$.

Let $\gamma$ be a path from $x$ to $y$ in $\mathcal{C}^{K}(\Gamma)$ so that
the intersection of $\gamma$ with each component of the collapsing locus is
connected. Let the components of the intersection of $\gamma$ with the
collapsing locus of $\mathcal{C}^{K}(\Gamma)$ be
$\mathbf{C}=\\{C_{1},\dots,C_{k}\\}$. The collapsed set associated to $\gamma$
is $\left\\{U_{i}=(q^{K}_{\Gamma})^{-1}(C_{i})\right\\}$, a disjoint
collection of horoballs, balls and tubes. If $C_{i}$ does not contain $x$ or
$y$ then $\gamma$ penetrates $U_{i}$. A component of the collapsed set is a
horoball, a ball, or a tube around a geodesic segment. Let
$\\{\gamma_{0},\dots,\gamma_{k}\\}$ be the components of
$\gamma\smallsetminus\mathbf{C}$ ($\gamma_{0}$ and $\gamma_{k}$ may be empty).
Let $\widetilde{\gamma}_{i}$ be the closure of
$(q^{K}_{\Gamma})^{-1}(\gamma_{i})$ in $\mathbb{H}^{\ast}$. The _entry point_
of $\gamma$ into $U_{i}$ is $e(\gamma,U_{i})=\widetilde{\gamma}_{i-1}\cap
U_{i}$ and the _exit point_ out of $U_{i}$ is
$o(\gamma,U_{i})=\widetilde{\gamma}_{i}\cap U_{i}$. Let
$\widetilde{\gamma}_{i}^{p}$ be the geodesic between $e(\gamma,U_{i})$ and
$o(\gamma,U_{i})$. The _lift_ of $\gamma$ is
$\\{\widetilde{\gamma}_{0},\widetilde{\gamma}_{1}^{p},\widetilde{\gamma}_{1},\dots,\widetilde{\gamma}_{k}^{p},\widetilde{\gamma}_{k}\\}$.
If each $\widetilde{\gamma}_{i}$ is geodesic then $\gamma$ is a local
collapsed geodesic.

A local collapsed geodesic $\gamma$ between $x,y\in\mathcal{C}^{K}(\Gamma)$ is
an almost geodesic if $\widehat{\ell}(\gamma)\leq\widehat{d}(x,y)+1$. Any path
$[x,y]$ can be replaced by a local collapsed geodesic connecting $x,y$ without
increasing its length, so an almost geodesic always exists between any two
points in $\mathcal{C}^{K}(\Gamma)$.

######  Notation 5.3.

Suppose $(X,d_{X})$ is a metric space, and $A,B\subset X$ are closed. If $x\in
X$ then $\pi_{A}(x)=\left\\{y\in A\mid d_{X}(x,y)=d_{X}(x,A)\right\\}$ and
$\pi_{A}(B)=\bigcup\limits_{b\in B}\pi_{A}(b)$.

Here is a standard fact about hyperbolic spaces.

######  Lemma 5.4.

Let $(X,d_{X})$ be a $\nu$–hyperbolic metric space and suppose that
$W_{1},W_{2}\subset X$ are convex and that $d(W_{1},W_{2})>6\nu$. There exists
$x\in X$ with $d_{X}(x,W_{2})\leq 3\nu$ so that
$\pi_{W_{1}}(W_{2})\subseteq\pi_{W_{1}}(B(x,2\nu))$.

######  Lemma 5.5.

Suppose that $U_{1}$ and $U_{2}$ are convex subsets of $\mathbb{H}^{\ast}$
that are at distance $t\geq 6\delta_{\mathbb{H}}$ apart. Then
$\mathrm{Diam}\left(\pi_{U_{1}}(U_{2})\right)\leq
4\delta_{\mathbb{H}}e^{-t+5\delta_{\mathbb{H}}}$.

###### Proof.

This follows from Lemma 5.4 and the fact that for any $t^{\prime}$ the length
of paths outside of $N(U_{2},t^{\prime})$ decreases by a factor of at least
$e^{-t^{\prime}}$ under projection to $U_{2}$. ∎

The following lemma is similar to [11, Lemma 4.5].

######  Lemma 5.6.

There an absolute constant $l$ with the following property: Let
$\widetilde{\gamma}$ be the geodesic between
$\widetilde{x},\widetilde{y}\in\mathbb{H}^{\ast}$ and let
$\gamma=q^{K}_{\Gamma}(\widetilde{\gamma})$. Let $\beta$ be any almost
geodesic in $\mathcal{C}^{K}(\Gamma)$ between
$x=q^{K}_{\Gamma}(\widetilde{x})$ and $y=q^{K}_{\Gamma}(\widetilde{y})$. Any
subsegment of $\beta$ which lies outside of $N(\gamma,l)$ has length at most
$4l+2$. In particular, any almost geodesic from $x$ to $y$ stays completely
inside $N(\gamma,3l+1)$.

###### Proof.

We claim any $l$ satisfying the following conditions suffices:

  1. (1)

$l\geq 6\delta_{\mathbb{H}}$; and

  2. (2)

$[1-(4\delta_{\mathbb{H}}+1)e^{-l+5\delta_{\mathbb{H}}}]\geq 1/2>0$.

Let $\beta^{\prime}$ be a subsegment of $\beta$ lying completely outside
$N(\gamma,l)$, and let $z$ and $w$ be the first point and the last point of
$\beta^{\prime}$, respectively. Since $\beta^{\prime}$ lies outside
$N(\gamma,l)$, all collapsed sets penetrated by $\beta^{\prime}$ are at least
$l$ away from $\widetilde{\gamma}$. Let $\\{U_{1},\dots,U_{k}\\}$ be the
collapsed set for $\beta^{\prime}$, and let

$\widetilde{\beta}_{0},\widetilde{\beta}_{1}^{p},\widetilde{\beta}_{1},\dots,\widetilde{\beta}_{k}^{p},\widetilde{\beta}_{k}$

be the lift of $\beta^{\prime}$ to $\mathbb{H}^{\ast}$. Then
$q^{K}_{\Gamma}(\widetilde{\beta}_{i})$ is subsegment of $\beta^{\prime}$ and
hence $\widetilde{\beta}_{i}$ is at least $l$ away from $\widetilde{\gamma}$.
Since $U_{i}$ is at least $l$ away from $\widetilde{\gamma}$ and $l\geq
6\delta_{\mathbb{H}}$, by Lemma 5.5 the projection of $\widetilde{\beta}_{i}$
to $\widetilde{\gamma}$ has size at most
$4\delta_{\mathbb{H}}e^{-l+5\delta_{\mathbb{H}}}$. Let
$l^{\prime}=l-5\delta_{\mathbb{H}}$.

Let $\widetilde{z}$ and $\widetilde{w}$ be the $q^{K}_{\Gamma}$–pre-image of
$z$ and $w$, respectively, and let $\widetilde{z}^{\prime}$ and
$\widetilde{w}^{\prime}$ denote the images of $\widetilde{z}$ and
$\widetilde{w}$ under orthogonal projection onto $\widetilde{\gamma}$,
respectively. Let $z^{\prime}=q^{K}_{\Gamma}(\widetilde{z}^{\prime})$ and
$w^{\prime}=q^{K}_{\Gamma}(\widetilde{w}^{\prime})$. Then

$\displaystyle\widehat{\ell}(\beta^{\prime})$ $\displaystyle\leq$
$\displaystyle d(z,z^{\prime})+d(z^{\prime},w^{\prime})+d(w^{\prime},w)+1$
$\displaystyle\leq$ $\displaystyle
l+d(\widetilde{z}^{\prime},\widetilde{w}^{\prime})+l+1$ $\displaystyle\leq$
$\displaystyle l+\widehat{\ell}(\beta^{\prime})\cdot
e^{-l^{\prime}}+4\delta_{\mathbb{H}}\cdot k\cdot e^{-l^{\prime}}+l+1$

Components are at least $K$ apart (in $\mathbb{H}^{\ast}$–distance) with
$K>1$, so $k+1\leq\widehat{\ell}(\beta^{\prime})$, and

$\displaystyle\widehat{\ell}(\beta^{\prime})$ $\displaystyle\leq$
$\displaystyle 2l+\widehat{\ell}(\beta^{\prime})\cdot
e^{-l^{\prime}}+4\delta_{\mathbb{H}}\cdot(\widehat{\ell}(\beta^{\prime})-1)\cdot
e^{-l^{\prime}}+1$

Therefore

$\displaystyle\widehat{\ell}(\beta^{\prime})\cdot[1-(4\delta_{\mathbb{H}}+1)e^{-l^{\prime}}]$
$\displaystyle\leq$ $\displaystyle 2l-4\delta_{\mathbb{H}}\cdot
e^{-l^{\prime}}+1$

We chose $l$ so that $[1-(4\delta_{\mathbb{H}}+1)e^{-l^{\prime}}]\geq 1/2>0$,
so

$\widehat{\ell}(\beta^{\prime})\leq 2(2l-4\delta_{\mathbb{H}}\cdot
e^{-l^{\prime}}+1)\leq 4l+2,$

as required. ∎

######  Proposition 5.7.

If $K\geq 8l$ then $(\mathcal{C}(\Gamma),d)$ is a geodesic metric space.

###### Proof.

Since $(\mathcal{C}(\Gamma),d)$ is a length space, for any
$x,y\in\mathcal{C}(\Gamma)$ there is a sequence of almost geodesics $p_{i}$
from $x$ to $y$ such that $\widehat{\ell}(p_{i})\rightarrow d(x,y)$. Choose
lifts $\widetilde{x}$ and $\widetilde{y}$ of $x$ and $y$ to
$\mathbb{H}^{\ast}$, let $\widetilde{\gamma}$ denote the
$\mathbb{H}^{\ast}$–geodesic from $\widetilde{x}$ to $\widetilde{y}$ and let
$\gamma=q_{\Gamma}(\widetilde{\gamma})$. By Lemmas 5.6, each of the $p_{i}$
lies within $3l+1$ of $\gamma$. Therefore, there is a finite collection of
components of the collapsing locus which any of the $p_{i}$ intersect, and
passing to a subsequence we may assume that the $p_{i}$ intersect exactly the
same collection of components of the collapsing locus, say
$C_{1},\ldots,C_{n}$.

The space $\mathcal{C}(\Gamma)$ is locally compact except at points in the
collapsing locus corresponding to the $q_{\Gamma}$–image of horoballs in
$\mathcal{U}^{K}(\Gamma)$. Suppose that $C_{i}$ is a component corresponding
to a horoball $U_{i}$. Then an almost geodesic travelling through $C_{i}$ is
locally the concatenation of two paths, each of which lift to paths in
$\mathbb{H}^{\ast}$ which travel almost as quickly as possible towards
$U_{i}$. In particular, if $a_{i}$ is a point on $p_{i}$ at distance $4l$ from
$C_{i}$, then $d(a_{i},C_{i})>3l+1$. The path from $a_{i}$ to $\gamma$ of
length at most $3l+1$ cannot intersect $C_{i}$, and since $K\geq 8l$ it cannot
intersect any other component of the collapsing locus either. It follows that
such $a_{i}$ lift to a bounded region of $\mathbb{H}^{\ast}$, and hence all of
$p_{i}$ lifts to a bounded region of $\mathbb{H}^{\ast}$. Since
$\mathbb{H}^{\ast}$ is locally compact, $(p_{i})$ sub-converges to a geodesic
$p$ from $x$ to $y$. ∎

The hyperbolicity (along with calculation of constants) of
$(\mathcal{C}^{K}(\Gamma),d)$ follows from Lemma 5.6 in the same way that [11,
Proposition 4.6] follows from [11, Lemma 4.5]. In the following, $l$ is as in
Lemma 5.6.

######  Theorem 5.8.

$(\mathcal{C}^{K}(\Gamma),d)$ has $\left(6l+\delta_{\mathbb{H}}+2\right)$–slim
triangles. Therefore, $(\mathcal{C}^{K}(\Gamma),d)$, is $\delta$–hyperbolic,
where $\delta=4\left(6l+\delta_{\mathbb{H}}+2\right)$.

######  Notation 5.9.

Fix $K=40\delta$, and let $l$ be the constant from Lemma 5.6.

We write $\mathcal{C}(\Gamma)=\mathcal{C}^{K}(\Gamma)$, and drop the
superscript $K$ from other constructions also. Therefore, for example, we have
the map $q_{\Gamma}\colon\thinspace X\to\mathcal{C}(\Gamma)$, and when
$\Gamma$ is assumed we just write $q\colon\thinspace X\to\mathcal{C}$, etc.

######  Lemma 5.10.

Let $x,y\in\mathcal{C}(\Gamma)$ and let $\gamma_{2}=[x,y]$ be a
$\mathcal{C}(\Gamma)$–geodesic. Let
$\widetilde{x},\widetilde{y}\in\mathbb{H}^{\ast}$ be lifts of $x,y$,
respectively. Let $\widetilde{\gamma_{1}}=[\widetilde{x},\widetilde{y}]$ be
the $\mathbb{H}^{\ast}$–geodesic and
$\gamma_{1}=q_{\Gamma}(\widetilde{\gamma_{1}})$. The Hausdorff distance
between $\gamma_{1}$ and $\gamma_{2}$ is at most $9\delta$.

###### Proof.

By Lemma 5.6 $\gamma_{2}\subseteq N(\gamma_{1},3l+1)$, and $3l+1\leq 9\delta$.
So it remains to prove that $\gamma_{1}\in N(\gamma_{2},9\delta)$.

First note that any segment of $\gamma_{1}$ which lies outside the
$4l$–neighborhood of the collapsing locus is a geodesic. For, if not the
geodesic must intersect the collapsing locus. But the geodesic cannot cannot
intersect the collapsing locus, since by Lemma 5.6 the geodesic remains within
distance $3l+1$ of $\gamma_{1}$.

Given a convex set $W\in\mathbb{H}^{\ast}$ and a geodesic $\gamma$, the
distance from $\gamma(t)$ to $W$ is a strictly convex function, so for any $r$
there are at most two points on $\gamma$ lying at distance exactly $r$ from
$W$.

Now, let $C_{1},\ldots,C_{k}$ denote the components of the collapsing locus
that lie within $6\delta$ of $\gamma_{2}$, and suppose that
$\gamma_{1}=p_{1}\cdot p_{2}\cdots p_{j}$ (where $j\in\\{2k-1,2k,2k+1\\}$) is
so that the endpoints of each $p_{i}$ lie at distance exactly $12\delta$ from
$C_{i}$. Thus, the $p_{i}$ alternate between paths which lie outside the
$12\delta$–neighborhood of the collapsing locus, and within $12\delta$ of some
$C_{l}$. Suppose that $p_{i}$ lies within $12\delta$ of $C_{l}$, and let $q$
be a $\mathcal{C}(\Gamma)$-geodesic with the same endpoints as $p_{i}$. By
lifting $p_{i}$ to $\mathbb{H}^{\ast}$ and considering projection to
$q_{\Gamma}^{-1}(C_{l})$, we see that $p_{i}$ is within
$2\delta_{\mathbb{H}}\leq\delta$ of the projection of the endpoints of $p_{i}$
to $C_{l}$ together with (possibly) a geodesic through $C_{l}$. By considering
the projection of the endpoints to $C_{l}$, and slim quadrilaterals, it is
straightforward to check that the Hausdorff distance between $q$ and this
union of projections to $C_{l}$ and a geodesic through $C_{l}$ is at most
$\delta$ (since $q$ intersect $C_{l}$ or $p_{i}=q$). Thus, the Hausdorff
distance between $p_{i}$ and $q$ is at most $2\delta$. Thus, if we replace
each of the $p_{i}$ which lies in the $12\delta$–neighborhood of some $C_{l}$
by a $\mathcal{C}(\Gamma)$–geodesic, then we obtain a concatenation of
$\mathcal{C}(\Gamma)$–geodesics which lies at Hausdorff distance at most
$3\delta$ from $\gamma_{2}$.

It is straightforward to check (using projections to the collapsing locus)
that the Gromov product at each point of concatenation is at most $2\delta$,
and that each of the paths in this concatenation, except possibly the first
and last, have length greater than $12\delta$ (since $K\geq
40\delta>24\delta$). Therefore, by [2, Lemma 4.9] (with $l=2\delta$), the
Hausdorff distance between this concatenation and $\gamma_{2}$ is at most
$7\delta$, and the result follows. ∎

The following is similar to [11, Lemma 4.8].

######  Lemma 5.11.

There exists $D_{1}$ satisfying the following: Let $x,y\in\mathcal{C}(\Gamma)$
and let $\gamma_{2}=[x,y]$ be $\mathcal{C}(\Gamma)$–geodesic and
$\widetilde{\gamma_{2}}$ be its lift. Let
$\widetilde{x},\widetilde{y}\in\mathbb{H}^{\ast}$ be lifts of $x,y$,
respectively. Let $\widetilde{\gamma_{1}}=[\widetilde{x},\widetilde{y}]$ and
$\gamma_{1}=q_{\Gamma}(\widetilde{\gamma_{1}})$. If precisely one of
$\\{\gamma_{1},\gamma_{2}\\}$ penetrates a component $U$ of the collapsed set
then $d_{\mathbb{H}^{\ast}}\left(e(\gamma_{i},U),o(\gamma_{i},U)\right)\leq
D_{1}$.

###### Proof.

Suppose $\gamma_{2}$ penetrates $U$ and $\gamma_{1}$ does not. Let
$\widetilde{z}$ be the point on $\widetilde{\gamma}_{2}$ between
$\widetilde{x}$ and $e(\gamma_{2},U)$ and $10l$ away from $U$, or
$\widetilde{z}=\widetilde{x}$ if no such point exists. Let $\widetilde{w}$ be
the point on $\widetilde{\gamma}_{2}$ between $\widetilde{y}$ and
$o(\gamma_{2},U)$ and $10l$ away from $U$, or $\widetilde{w}=\widetilde{y}$ if
no such point exists. Let $z=q_{\Gamma}(\widetilde{z})$ and
$w=q_{\Gamma}(\widetilde{w})$. By Lemma 5.6, there exist
$z^{\prime},w^{\prime}$ on $\gamma_{1}$ with
$d(z,z^{\prime}),d(w,w^{\prime})\leq 5l$. Let $\widetilde{z}^{\prime}$ and
$\widetilde{w}^{\prime}$ be the lifts of $z^{\prime}$ and $w^{\prime}$,
respectively. By construction $[z^{\prime},z]$ and $[w^{\prime},w]$ are
disjoint from the collapsing locus. Hence
$d_{\mathbb{H}^{\ast}}(\widetilde{z},\widetilde{z}^{\prime}),d_{\mathbb{H}^{\ast}}(\widetilde{w},\widetilde{w}^{\prime})\leq
5l$. Since these paths are in $\mathbb{H}^{\ast}$,
$[\widetilde{z}^{\prime},\widetilde{w}^{\prime}]\subseteq
N_{5l}\left([\widetilde{z},\widetilde{w}]\right)$. Thus,
$q_{\Gamma}([\widetilde{z}^{\prime},\widetilde{w}^{\prime}])\subseteq
N_{5l}\left([z,w]\right)$, since
$[z,w]=q_{\Gamma}([\widetilde{z},\widetilde{w}])$. By Lemma 5.6, the
subsegment $\beta$ of $\gamma_{1}$ between $z^{\prime}$ and $w^{\prime}$ lies
in
$N_{3l+1}\left(q_{\Gamma}([\widetilde{z}^{\prime},\widetilde{w}^{\prime}])\right)$,
so $\beta\subseteq N_{8l+1}\left([z,w]\right)\subseteq
N_{18l+1}\left(q_{\Gamma}(U)\right)$. Components of the collapsing locus are
$40l$–separated, so $\beta$ does not penetrate any component of the collapsed
set except $U$. On the other hand, since $\beta$ is a subsegment of
$\gamma_{1}$, it does not penetrate $U$. Therefore the lift of $\beta$ is
$[\widetilde{z}^{\prime},\widetilde{w}^{\prime}]$ and
$[\widetilde{z}^{\prime},\widetilde{w}^{\prime}]$ is disjoint from $U$, so the
projection of $[\widetilde{z}^{\prime},\widetilde{w}^{\prime}]$ onto $U$ is
bounded by a uniform constant $D$. Since $[z,z^{\prime}]$ and $[w,w^{\prime}]$
do not penetrate any component of the collapsing locus, the projections of
$[\widetilde{z},\widetilde{z}^{\prime}]$ and
$[\widetilde{w},\widetilde{w}^{\prime}]$ onto $U$ are also bounded by $D$. By
choice of $\widetilde{z}$ and $\widetilde{w}$,
$d_{\mathbb{H}^{\ast}}(\pi_{U}(\widetilde{z}),e(\gamma_{2},U))$ and
$d_{\mathbb{H}^{\ast}}(\pi_{U}(\widetilde{w}),o(\gamma_{2},U))$ are bounded by
a uniform constant $C_{2}$. The above distance bounds imply
$d_{{\mathbb{H}^{\ast}}}(o(\gamma_{2},U),e(\gamma_{2},U))\leq 3C_{1}+2C_{2}$.

The case where $\gamma_{1}$ penetrates $U$ and $\gamma_{2}$ does not is
similar, the only difference being that instead of using Lemma 5.6, we use
Lemma 5.10. As a result, the constants are different but they are again
uniform constants. ∎

The following is similar to [11, Lemma 4.9].

######  Lemma 5.12.

There exists $D_{2}$ satisfying: Let
$x,y,\widetilde{x},\widetilde{y},\gamma_{1}$ and $\gamma_{2}$ be as in Lemma
5.11. If both $\gamma_{1}$ and $\gamma_{2}$ penetrate a component $U$ of the
collapsed set then
$d_{\mathbb{H}^{\ast}}\left(e(\gamma_{1},U),e(\gamma_{2},U)\right)\leq D_{2}$.
The same is true for exit points.

###### Proof.

Let $\widetilde{z}$ be the point on $\widetilde{\gamma}_{2}$ between
$\widetilde{x}$ and $e(\gamma_{2},U)$ and $10l$ away from $U$, or
$\widetilde{z}=\widetilde{x}$ if no such point exists. The distance between
$\widetilde{z}$ and $e(\gamma_{2},U)$ is bounded by a constant depending only
on $l$. By Lemma 5.6, there exists $w$ on $\gamma_{1}$, so that the collapsed
distance between $w$ and $z=q_{\Gamma}(\widetilde{z})$ is at most $5l$. There
is no other connected component of the collapsing locus in the
$20l$–neighborhood of $q_{\Gamma}(U)$, so the geodesic $[z,w]$ does not
penetrate any component of the collapsing locus. Let
$\widetilde{w}=q_{\Gamma}^{-1}(w)$. The geodesic
$[\widetilde{z},\widetilde{w}]$ has length at most $5l$. The distance between
$\widetilde{w}$ and $e(\gamma_{2},U)$ is bounded by a constant depending only
on $l$. By the triangle inequality, the distance between $e(\gamma_{1},U)$ and
$e(\gamma_{2},U)$ is bounded by a constant depending only on $l$. ∎

######  Definition 5.13.

Let $S\subseteq{\mathbb{H}}^{*}$ , and let $\gamma$ be a geodesic in
${\mathbb{H}}^{*}$ so $\gamma\cap S=\emptyset$. Let $T$ be the set of points
$s\in S$ so that there exists some $t$ for which $[\gamma(t),s]\cap S={s}$.
The visual size of $S$ with respect to $\gamma$ is the diameter of $T$. The
visual size of the $S$ is the supremum of the visual size of $S$ with respect
to $\gamma$, where the supremum is taken over all geodesics $\gamma$ disjoint
from $S$.

The following is similar to [11, Lemma 4.4].

######  Lemma 5.14.

There exists $D_{3}$ such that for each connected component $C$ of the
collapsing locus, the visual size of $q_{\Gamma}^{-1}(C)$ is less than
$D_{3}$.

###### Proof.

The case where $U=q_{\Gamma}^{-1}(C)$ is a horoball is proved in [11, Lemma
4.4]. The other cases are similar. In the last step, use that fact that (by
Proposition 5.1.(4)) the radius of $U$ is at least $K$, which is much bigger
than the hyperbolicity constant of $\mathbb{H}^{*}$. ∎

######  Lemma 5.15.

There exist uniform constants $B_{1}$ and $B_{2}$ satisfying: Let
$g\in\Gamma\smallsetminus\\{1\\}$ and $x\in\mathcal{C}(\Gamma)$. If $g$ is
parabolic, let $\gamma$ be the component of the collapsing locus fixed by $g$.
Otherwise let $\gamma$ be the $q_{\Gamma}$-image of the minimal invariant set
of $g$ in $\mathbb{H}^{\ast}$. Then $d(x,gx)-2d(x,\gamma)\geq B_{1}$ and if
$g$ is elliptic or parabolic then $d(x,gx)-2d(x,\gamma)\leq B_{2}$.

###### Proof.

Let $\widetilde{x}\in q_{\Gamma}^{-1}(x)$. Let $U_{1}=q^{-1}_{\Gamma}(\gamma)$
and $\widetilde{y}=\pi_{U_{1}}(\widetilde{x})$. Let $U_{2}$ be a horoball (or
ball or tube) with the same center (or axis) as $U_{1}$ so $g$ moves points on
$\partial U_{2}$ by $5\delta+l_{g}$, where $l_{g}$ is the translation length
of $g$ on $\mathbb{H}^{\ast}$. Let $\widetilde{w}=\pi_{U_{2}}(\widetilde{x})$.
We claim that $d(\widetilde{w},U_{1})\leq D_{1}$ for some absolute constant
$D_{1}$ . The claim is trivial if $U_{2}\subset U_{1}$, so suppose
$U_{1}\subset U_{2}$.

First suppose that $\gamma$ is a component of the collapsing locus. In this
case by Proposition 5.1.(3) $d(g\widetilde{y},\widetilde{y})\geq{\rm
max}\\{D,l_{g}\\}$, so the length of $[\widetilde{w},\widetilde{y}]$ is
bounded above by an absolute constant.

Now suppose $\gamma$ is _not_ a component of the collapsing locus. In this
case, $g$ is either loxodromic and $l_{g}$ is bounded from below by an
absolute positive constant, or $g$ is elliptic and the angle of rotation is
bounded below by an absolute positive constant, so $U_{2}$ has uniformly
bounded size (tube radius).

The claim gives an absolute upper bound on $d(w,\gamma)$, where
$w=q_{\Gamma}(\widetilde{w})$. By hyperbolicity of $\mathbb{H}^{\ast}$, up to
an error of $4\delta_{\mathbb{H}}$, $[\widetilde{x},g\cdot\widetilde{x}]$
tracks
$[\widetilde{x},\widetilde{w}]\cup[\widetilde{w},g\cdot\widetilde{w}]\cup[g\widetilde{w},g\cdot\widetilde{x}]$.
By Lemma 5.6, $[x,gx]$, $[x,w]$ and $[gx,gw]$ are in the
$\delta$-neighborhoods of $q_{\Gamma}([\widetilde{x},g\cdot\widetilde{x}])$,
$q_{\Gamma}([\widetilde{x},\widetilde{w}])$ and
$q_{\Gamma}([g\cdot\widetilde{x},g\cdot\widetilde{w}])$, respectively. Hence,
up to an error depending on $\delta$, $[x,g\cdot x]$ tracks
$[x,w]\cup[w,g\cdot w]\cup[g\cdot w,g\cdot x]$. Thus, there is a constant
$L_{1}$ depending only on $\delta$ so that $d(x,g\cdot x)>2d(x,w)-L_{1}$. If
$g$ is elliptic or parabolic, then $d(w,g\cdot w)\leq 5\delta$ and hence
$\left|d(x,g\cdot x)=2d(x,w)\right|$ is bounded by a constant depending only
on $\delta$. Combining this with the fact that $d(w,\gamma)$ has an absolute
upper bound, the lemma follows. ∎

######  Lemma 5.16.

Suppose $g\in\Gamma$ acts loxodromically on $\mathbb{H}^{\ast}$ with axis
$\widetilde{\gamma}$, and let $\gamma=q(\widetilde{\gamma})$. If $\gamma$
comes within $5\delta$ of the collapsing locus but is not contained in the
collapsing locus then $g$ moves every point of $\gamma$ by at least
$30\delta$.

###### Proof.

By assumption, there exists $x$ on $\gamma$ within $5\delta$ of a component
$C$ of the collapsing locus and $\gamma$ is not contained in $C$. Hence $C$
and $g\cdot C$ are distinct components of the collapsing locus, which are
separated by $K=40\delta$. Hence $x$ and $gx$ are separated by a distance at
least $40\delta-2\left(5\delta\right)=30\delta$. So the translation length of
$g$ on $\widetilde{\gamma}$ is at least $30\delta$. Let $y$ be a point on
$\gamma$. Suppose $d(y,gy)<30\delta$. Since the translation length of $g$ on
$\widetilde{\gamma}$ is at least $30\delta$, $[y,gy]$ intersects some
connected component $C^{\prime}$ of the collapsing locus and we have
$d(y,C^{\prime})+d(C^{\prime},gy)\leq 30\delta$. Hence
$d(gy,gC^{\prime})+d(C^{\prime},gy)\leq 30\delta$, so
$d(C^{\prime},gC^{\prime})\leq 30\delta$. Thus $gC^{\prime}=C^{\prime}$, so
$C^{\prime}$ is a line and $\gamma=C^{\prime}$. This is a contradiction. ∎

##  6\. Sequences which are not $\mathcal{C}$–divergent

Throughout this section make Standing Assumption 4.18. We apply the results in
the previous section to the vertex groups of the geometric decomposition of a
group in $\mathcal{M}_{\operatorname{Gen}}^{\pi}$. If $\Gamma_{v}$ is such a
vertex group, and it is hyperbolic, then it admits a hyperbolic structure,
unique by Mostow–Prasad Rigidity, and this exhibits $\Gamma_{v}$ as a Kleinian
group. If $\Gamma_{v}$ is an LSF-type vertex group, let
$\overline{\Gamma_{v}}$ be the quotient (hyperbolic) $2$–orbifold group, and
fix a hyperbolic structure on the orbifold, witnessing $\overline{\Gamma_{v}}$
as a Fuchsian group. There are many such hyperbolic structures, but any
suffices for our purposes.

The purpose of this section is to prove the following result, required for the
proof of Theorem 2.7. The proof of Theorem 6.2 is contingent on the technical
result Theorem 6.7, proved in Appendix A.

Recall from Assumption 4.18 that $A_{i}$ is the $\omega$–approximation to the
good relative generating set $A$ of the vertex group $V$ of the refined GGD
$\mathbb{G}^{r}$ of $L$. By Definition 4.10 $\omega$–almost surely there are
vertices $v_{i}$ of $T_{i}$ so $A_{i}$ fixes $v_{i}$.

######  Definition 6.1 ($\mathcal{C}$–divergent).

Let $\Gamma_{v_{i}}$ be the vertex group of $\Gamma_{i}$ associated to $v_{i}$
and let $\mathcal{C}(\Gamma_{v_{i}})$ be the associated collapsed space. We
set

$\|\phi_{i}\|_{\mathcal{C},v}=\inf_{x\in\mathcal{C}(\Gamma_{v_{i}})}\max_{s\in
A_{i}}d_{i}\left(s\cdot x,x\right),$

which is defined $\omega$–almost surely. The sequence $\left(\phi_{i}\right)$
is $\mathcal{C}$–divergent with respect to ${{\mathbb{G}}}$ if
${\lim}^{\omega}\|\phi_{i}\|_{\mathcal{C},v}=\infty$ for some vertex $v$ of
$\mathbb{G}^{r}$.

######  Theorem 6.2.

Let $L$ be an $\mathcal{M}_{\operatorname{Gen}}^{\pi}$–limit group defined by
a non-$\mathcal{T}$–divergent sequence $(\phi_{i})$. If there is a GGD
${{\mathbb{G}}}$ of $L$ with respect to which $(\phi_{i})$ is not
$\mathcal{C}$–divergent then all stably parabolic subgroups of $L$ are
finitely generated.

When $(\phi_{i})$ is not $\mathcal{C}$–divergent with respect to
${{\mathbb{G}}}$, we obtain the following crucial information about the vertex
group $V$ of $\mathbb{G}^{r}$:

######  Proposition 6.3.

Suppose $(\phi_{i})$ is not $\mathcal{C}$–divergent with respect to
${{\mathbb{G}}}$. Then $V$ admits a splitting $\mathbb{D}$ so:

  1. (1)

The underlying graph of $\mathbb{D}$ is a star, i.e. a bipartite graph where
one type of vertices has a single vertex.

  2. (2)

$Z^{\omega}(V)$ is contained in each edge group of $\mathbb{D}$.

  3. (3)

For each the edge group $H$ of $\mathbb{D}$, $H/Z^{\omega}(V)$ has rank at
most two.

  4. (4)

The non-central vertices of $\mathbb{D}$ are in bijection with the edges in
$\mathbb{G}^{r}$ adjacent to $V$. This bijection induces isomorphisms of
associated groups.

Furthermore, if $V$ is of hyperbolic type, and $P\in\mathcal{H}_{V}$ is not
conjugate into an edge group of $\mathbb{G}^{r}$ adjacent to $V$, then $P$ is
finitely generated.

We prove Proposition 6.3 after we use it to prove Theorem 6.2.

###  6.1. The proof of Theorem 6.2

As a consequence of Proposition 6.3, each vertex group of $\mathbb{G}$ also
enjoys the properties (1)–(4) from the conclusion of Proposition 6.3 enjoyed
by $V$:

######  Corollary 6.4.

Suppose $(\phi_{i})$ is not $\mathcal{C}$–divergent with respect to
${{\mathbb{G}}}$. Then any vertex group ${{\mathbb{G}}}_{v}$ of
${{\mathbb{G}}}$ admits a splitting $\mathbb{D}(v)$ so that properties of
Proposition 6.3 hold when $\mathbb{D}$ and $\mathbb{G}^{r}$ are replaced by
$\mathbb{D}(v)$ and ${{\mathbb{G}}}$, respectively.

###### Proof.

Let $\mathbb{R}_{v}$ be the sub-splitting of $\mathbb{G}^{r}$ corresponding to
${{\mathbb{G}}}_{v}$. Let $V_{1},\dots,V_{t}$ be the vertex groups of
$\mathbb{R}_{v}$ and let $\mathbb{D}_{1},\dots,\mathbb{D}_{t}$ be the
corresponding splittings given by Proposition 6.3, respectively. Refined
$\mathbb{R}_{v}$ by $\mathbb{D}_{1},\dots,\mathbb{D}_{t}$ and collapse the
sub-graph of group of this refinement spanned by the central vertex groups of
$\mathbb{D}_{1},\dots,\mathbb{D}_{t}$. The resulting splitting is the required
splitting $\mathbb{D}(v)$. ∎

######  Definition 6.5.

A graph of groups $\mathbb{E}$ is edge-twisted if:

The underlying graph of $\mathbb{E}$ is bipartite with colors $A$ and $B$.
Type $A$ vertices have valence $2$, and abelian vertex groups (thus the edge
groups of $\mathbb{E}$ are also abelian). Let $W$ be a Type $A$ vertex group
of $\mathbb{E}$ and let $E_{1}$ and $E_{2}$ be the images in $W$ of the
adjacent edge groups. There are subgroups $K_{j}\leq E_{j}$ (for $j=1,2$) so
that

  1. (1)

$K_{1}\cap K_{2}=\\{1\\}$; and

  2. (2)

For $j=1,2$, the group $E_{j}/K_{j}$ is finitely generated.

Refine ${{\mathbb{G}}}$ by replacing each vertex group ${{\mathbb{G}}}_{v}$ by
the splitting given by Corollary 6.4. Collapse the edges in the resulting
splitting corresponding to the edges in ${{\mathbb{G}}}$, and denote the
resulting splitting of $L$ by $\mathbb{K}$.

######  Proposition 6.6.

$\mathbb{K}$ is an edge-twisted splitting.

###### Proof.

The vertices of $\mathbb{K}$ corresponding to the central vertices of the
splittings given by Corollary 6.4 are of Type B and the vertices corresponding
to the non-central vertices are of Type $A$. That Type $A$ vertex groups are
abelian follows from the last property of Corollary 6.4 and the fact that edge
groups of $\mathbb{G}^{r}$ are abelian. Type $A$ vertex groups clearly have
valence $2$. By Lemma 3.20 and Definition 4.10(2) $Z^{\omega}(W_{1})\cap
Z^{\omega}(W_{2})=\\{1\\}$ for adjacent vertex groups $W_{1},W_{2}$ of
${{\mathbb{G}}}$. For the $K_{i}$ from Definition 6.5 we choose the stable
center of the vertex groups of ${{\mathbb{G}}}$. With this choice, the first
condition of Definition 6.5 is satisfied. The last condition of Definition 6.5
is satisfied because of the third property of Corollary 6.4. ∎

The following Theorem 6.7 is proved in Appendix A.

######  Theorem 6.7.

Let $\mathbb{E}$ be a finite edge-twisted graph of groups so that
$\pi_{1}(\mathbb{E})$ is finitely generated. The Type B vertex groups of
$\mathbb{E}$ are finitely generated.

###### Proof of Theorem 6.2.

Let $\mathbb{K}$ be as in Proposition 6.6. By Proposition 6.6 and Theorem 6.7,
Type B vertex groups of $\mathbb{K}$ are finitely generated. Thus, each Type B
vertex group which is a subgroup of an LSF-type vertex groups of
${{\mathbb{G}}}$ is an $\mathcal{M}_{\operatorname{LSF}}^{\pi}$–limit group.
By Corollary 3.16, all abelian subgroups of these vertex groups are finitely
generated, so edge groups adjacent to these vertex groups are finitely
generated. The other edge groups are adjacent to a hyperbolic type vertex
group of $\mathbb{K}$ and are finitely generated by the third property of
Corollary 6.4, so all vertex groups of $\mathbb{K}$ are finitely generated
also. Hence all vertex groups of ${{\mathbb{G}}}$ are finitely generated. A
stably parabolic subgroup is finitely generated by Corollary 3.16 if it is in
an LSF-type vertex group or by Corollary 6.4 if it is in a hyperbolic type
vertex group. ∎

The rest of the section is devoted to the proof of Proposition 6.3.

###  6.2. The cut point tree

The defining sequence for $L$ is not $\mathcal{C}$–divergent with respect to
${{\mathbb{G}}}$. We consider how $V$ maps into collapsed spaces of the vertex
groups of the geometric decompositions of the $\Gamma_{i}$.

Now, $\omega$–almost surely $A_{i}\subset\Gamma_{v_{i}}$. For $\omega$–almost
every $i$, fix $\mathfrak{o}^{v}_{i}\in\mathcal{C}(\Gamma_{v_{i}})$ satisfying

$\max_{g\in
A_{i}}\left\\{d_{i}(g.\mathfrak{o}^{v}_{i},\mathfrak{o}^{v}_{i})\right\\}\leq\|\phi_{i}\|_{\mathcal{C},v}+\frac{1}{i}.$

By Definition 4.14 $V$ is generated by $A$ and the adjacent edge groups. Let
$S_{v}$ be a (possibly infinite) generating set for $V$ consisting of $A$ and
elements from the adjacent edges groups. Choose a lift
$\widetilde{S}_{v}\subset G$ of $S_{v}$.

Fix $s\in S_{v}$, and denote by $\widetilde{s}$ the corresponding element in
$\widetilde{S}_{v}$. For each $i$ consider a geodesic $\gamma_{i,s}$ in
$\mathcal{C}(\Gamma_{v_{i}})$ between $\mathfrak{o}^{v}_{i}$ and
$\phi_{i}(\widetilde{s}).\mathfrak{o}^{v}_{i}$. Since $(\phi_{i})$ is not
$\mathcal{C}$–divergent, by Lemma 5.15 the $\omega$-limit of the length of
$\gamma_{i,s}$ is finite; denote this limiting length by $l_{s}$. Let
$P_{i,s}$ denote the set of points on $\gamma_{i,s}$ which map to parabolic
collapsed points in $\mathcal{C}(\Gamma_{v_{i}})$. Since $K\geq 1$, distinct
parabolic collapsed points lie at least distance $1$ from each other. Hence
the size of $P_{i,s}$ is $\omega$–almost surely bounded independent of $i$.
Let $n_{s}$ be the $\omega$–limit of the size of $P_{i,s}$. The positions of
these points on $\gamma_{i,s}$ $\omega$–converge to $n_{s}$ distinct points on
a segment of length $l_{s}$. Let $\gamma_{s}$ be a segment of length $s$
marked with $n_{s}$ points whose positions correspond to the $\omega$–limit of
the $n_{s}$ marked points on the $\gamma_{i,s}$.

Let $\operatorname{Cay}(V,S_{v})$ be the (possibly locally infinite) Cayley
graph of $V$ with respect to $S_{v}$. For $g\in V$ and $s\in S_{v}$ let the
length of the edge $[g,gs]$ be $l_{s}$ and mark the edge with $n_{s}$ _marked_
points. The locations of these marked points are chosen so there is an
isometry between $[g,gs]$ and $\gamma_{s}$ taking marked points to marked
points.

Now define a partial map $f_{i}$ from $\operatorname{Cay}(V,S_{v})$ to
$\mathcal{C}(\Gamma_{v_{i}})$ as follows:

For each $s\in S_{v}$, let $f_{i}$ be a continuous map from $[1,s]$ to
$\gamma_{i,s}$. By our choice of length for the edges and the number of the
marked points on them, for $\omega$–almost all $i$ we can choose $f_{i}$ so
that:

  1. (1)

$f_{i}$ maps the marked points of $[1,s]$ bijectively to the parabolic
collapsed points on $\gamma_{i,s}$; and

  2. (2)

$f_{i}$ is bi-Lipschitz on $[1,s]$ with constant $2$.

Extend $f_{i}$ to other edges of $\operatorname{Cay}(V,S_{v})$ equivariantly
whenever possible: For each $g\in V$, choose an $\omega$–approximation
$(g_{i})$ of $g$. For each $g\in V$ so $g_{i}\in\Gamma_{v_{i}}$, extend
$f_{i}$ to $[g,gs]$ equivariantly.

Since $f_{i}$ depends on the choice of $(g_{i})$ of $g\in V$ and there is no
canonical choice of $(g_{i})$, $f_{i}$ is not well defined on the vertices of
$\operatorname{Cay}(V,S_{v})$. So formally, each $f_{i}$ is a map defined on
the interior of some edges of $\operatorname{Cay}(V,S_{v})$. However, we have
the following:

######  Lemma 6.8.

Let $B$ be a finite set of edges in $\operatorname{Cay}(V,S_{v})$. For
$\omega$–almost all $i$, $f_{i}$ is defined on $B$ and can be continuously
extended to the closure of $B$.

Define an equivalence relation on the set of marked points of
$\operatorname{Cay}(V,S_{v})$ by setting marked points $x$ and $y$ to be
equivalent if $\omega$–almost surely $f_{i}(x)=f_{i}(y)$. Let $Q$ be the
quotient of $\operatorname{Cay}(V,S_{v})$ obtained by identifying equivalent
marked points.

This construction is clearly equivariant, so the $V$–action on
$\operatorname{Cay}(V,S_{v})$ descends to an $V$–action on $Q$ by isometries.
We continue to refer to images in $Q$ of the marked points in
$\operatorname{Cay}(V,S_{v})$ as marked points of $Q$. The following lemma is
straightforward.

######  Lemma 6.9.

The equivalence relation on the marked points does not depend on the choices
of $(g_{i})$.

Since the marked points in $\operatorname{Cay}(V,S_{v})$ are discrete in each
edge, we may consider $Q$ to be a metric graph in the obvious way.

By construction, each point in the complement of the marked points in $Q$ has
a unique pre-image in $\operatorname{Cay}(V,S_{v})$. Hence $f_{i}$ induces a
partial map from the interiors of the edges of $Q$ to
$\mathcal{C}(\Gamma_{v_{i}})$, which we denote by $f^{Q}_{i}$. Although
$f^{Q}_{i}$ is only partially defined, we have the following:

######  Lemma 6.10.

Suppose $B$ is a finite collection of edges in $Q$. For $\omega$–almost all
$i$ the map $f^{Q}_{i}$ well-defined on $B$. For any finite path $\gamma$ in
$Q$, the map $f^{Q}_{i}$ is $\omega$–almost surely well-defined on $\gamma$.
Moreover $f^{Q}_{i}$ is $\omega$–equivariant in the sense that for any $x\in
B$ and any $g\in V$ $\omega$–almost surely
$f^{Q}_{i}(g.x)=g_{i}.f^{Q}_{i}(x)$.

Let $\mathcal{C}_{\infty,v}$ be the $\omega$–limit of the collapsed spaces
$\mathcal{C}(\Gamma_{v_{i}})$. The sequence $\\{f_{i}^{Q}\\}$ of partially
defined maps induces a map $f_{\infty}\colon\thinspace
Q\to\mathcal{C}_{\infty,v}$.

######  Lemma 6.11.

The map $f_{\infty}\colon\thinspace Q\to\mathcal{C}_{\infty,v}$ is well-
defined, continuous and $V$–equivariant. The map $f_{\infty}$ restricts to an
injective map from the marked points of $Q$ to the set of limits of parabolic
collapsed points in $\mathcal{C}_{\infty,v}$. Moreover, for any
$p\in\mathcal{C}_{\infty,v}$ which is a limit of parabolic collapsed points,
$f_{\infty}^{-1}(p)$ is either a single marked point in $Q$ or is empty.

######  Definition 6.12.

Suppose $H\in\mathcal{H}_{V}$. If $g\in H$ and $(g_{i})$ is an
$\omega$–approximation of $g$, then $\omega$–almost surely $g_{i}$ is
contained in $\Gamma_{v_{i}}$ and fixes a parabolic collapsed point
$\xi_{i}(g)$ in $\mathcal{C}(\Gamma_{v_{i}})$.

An element $g\in H$ has _Property $\mathcal{B}$_ if for some $x\in Q$ there is
a path $\tau$ in $Q$ between $x$ and $g\cdot x$ so $\omega$–almost surely
$\xi_{i}(g)\not\in f_{i}^{Q}(\tau)$.

The point $\xi_{i}(g)$ $\omega$–almost surely depends only on $g$ and not on
the choices of $(g_{i})$.

######  Lemma 6.13.

Let $H\in\mathcal{H}_{V}$ be such that every element of $H$ has Property
$\mathcal{B}$. Then $H/Z^{\omega}(V)$ is a free abelian group of rank at most
two.

###### Proof.

Let $g\in H$ and fix an $\omega$–approximation $(g_{i})$ of $g$. By the
definition of Property $\mathcal{B}$, there is a path $\tau$ in $Q$ from $x$
to $g.x$ such that $\omega$–almost surely $f_{i}^{Q}(\tau)$ does not intersect
the parabolic collapsed point $\xi_{i}(g)$ in $\mathcal{C}(\Gamma_{v_{i}})$.
Let $x_{i}=f^{Q}_{i}(x)$. By Lemma 6.10 $\omega$–almost surely
$f_{i}^{Q}(\tau)$ is a path $\tau_{i}$ in $\mathcal{C}(\Gamma_{v_{i}})$
connecting $x_{i}$ and $g_{i}.x_{i}$. This path $\tau_{i}$ can be lifted to a
path $\widetilde{\tau_{i}}$ in $\mathbb{H}^{\ast}$ connecting
$\widetilde{x}_{i}$ and $g_{i}.\widetilde{x}_{i}$, where is
$\widetilde{x}_{i}$ is the pre-image of $x_{i}$ under the natural projection
from $\mathbb{H}^{\ast}$ to $\mathcal{C}(\Gamma_{v_{i}})$. Let
$U_{i}\subset\mathbb{H}^{\ast}$ be the maximal horoball in $\mathbb{H}^{\ast}$
that is collapsed to make the collapsed point $\xi_{i}$. Let
$\widetilde{y}_{i}$ be the closest point projection of $\widetilde{x}_{i}$ to
$U_{i}$. Then $g_{i}.\widetilde{y}_{i}$ is the closest point projection of
$g_{i}.\widetilde{x}_{i}$ to $A_{i}$. Moreover, the path
$\widetilde{\tau_{i}}$ projects to a path in the boundary of $U_{i}$
connecting $\widetilde{y}_{i}$ and $g_{i}.\widetilde{y}_{i}$. The length of
$\tau_{i}$ is $\omega$–almost surely bounded. Therefore, $\omega$–almost
surely there is a bound on the number of components of the pre-image of the
collapsing locus which $\widetilde{\tau_{i}}$ intersects. Then $\omega$–almost
surely $\widetilde{\tau_{i}}$ does not intersect $U_{i}$. If $C$ is a
component of the pre-image of the collapsing locus other than $U_{i}$ then by
Lemma 5.5 the projection of $C$ to $U_{i}$ has uniformly bounded diameter.
Thus there exists $N$ so that $\omega$–almost surely the projection of
$\widetilde{\tau_{i}}$ to $U_{i}$ has length at most $N$. Hence, the
translation length of $g_{i}$ on $U_{i}$ is $\omega$–almost surely bounded.

Let $P_{i}$ be the subgroup of $\Gamma_{v_{i}}$ fixing $\xi_{i}(g)$. We may
identify each $P_{i}$ with $\mathbb{Z}$ or $\mathbb{Z}^{2}$ via a choice of
short generators. Suppose that $P_{i}\cong\mathbb{Z}^{2}$. Using this
identification, we have that $\omega$–almost surely
$\phi_{i}(g_{i})\in\mathbb{Z}^{2}$. The upper bound on the translation length
of $g_{i}$ on $U_{i}$ imply that there exist $a,b\in\mathbb{Z}$, independent
of $i$, so that $\phi_{i}(g_{i})=(a,b)$ $\omega$–almost surely. Note that
$(a,b)$ does not depend on the choice of $(g_{i})$.

By Proposition 5.1.(3), as long as $g\notin Z^{\omega}(V)$ there is a lower
bound, independent of $i$, on the translation length of $g_{i}$ on $U_{i}$.
Defining $\phi^{\prime}(g)=(a,b)$ for each $g\in H$ induces a homomorphism
$H\to\mathbb{Z}^{2}$, and the lower bound on translation length for $g\notin
Z^{\omega}(V)$ implies that this homomorphism has kernel $Z^{\omega}(V)$. The
case where $P_{i}\cong\mathbb{Z}$ is entirely similar. The proof of the lemma
is complete. ∎

######  Lemma 6.14.

Let $H\in\mathcal{H}_{V}$. Then either $H/Z^{\omega}(V)$ has rank at most two
or $H$ is the stabilizer of a marked point in $Q$.

###### Proof.

Let $P_{i}$ and $\xi_{i}$ be as above. By Lemma 5.15 and the fact the
translation length of $\phi_{i}$ with respect to $S_{v}$ and
$\mathfrak{o}_{i}^{v}$ is bounded, the distance between $\xi_{i}$ and
$\mathfrak{o}_{i}^{v}$ is $\omega$–almost surely bounded independent of $i$.
As a result, $\\{\xi_{i}\\}$ defines a point $\xi$ in
$\mathcal{C}_{\infty,v}$. The stabilizer of $\xi$ is $H$. If $\xi$ is the
image of a marked point under the map $f_{\infty}$, then by Lemma 6.11 $H$ is
the stabilizer of a marked point of $Q$. Now suppose $\xi$ is _not_ the image
of any marked point under the map $f_{\infty}$. Then $\omega$–almost surely
the $f^{Q}_{i}$–image of any segment in $Q$ does not contain $\xi_{i}$, so for
any $h\in H$, $\omega$–almost surely $f_{i}^{Q}(1,h.1)$ does not contain
$\xi_{i}$. Hence every element of $H$ has Property $\mathcal{B}$. Therefore,
by Lemma 6.13, $H/Z^{\omega}(V)$ has rank at most two. ∎

By Lemma 6.14, to further understand the elements of $\mathcal{H}_{V}$, we
study the case when they are stabilizers of marked points in $Q$.

######  Lemma 6.15.

Let $H\in\mathcal{H}_{V}$. Suppose $H$ is the stabilizer of a marked point
$z\in Q$ which is not a cut point. Then $H/Z^{\omega}(V)$ has rank at most two
.

###### Proof.

By Lemma 6.13, it suffices to show that every element of $H$ has Property
$\mathcal{B}$. Let $g\in H$ and $x\in Q\smallsetminus\\{z\\}$. Since $z$ is
not a cut point, there is a path $\tau$ in $Q\smallsetminus\\{z\\}$ from $x$
to $g\cdot x$. By Lemma 6.11, $f_{\infty}(z)\not\in f_{\infty}(\tau)$. Note
that $\xi=f_{\infty}(z)$ is the fixed point of $H$ in
$\mathcal{C}_{\infty,v}$. Hence, $\omega$–almost surely $f_{i}^{Q}(\tau)$ does
not intersect the parabolic collapsed point $\xi_{i}$ associated to $H$, so
each $g\in H$ has Property $\mathcal{B}$ as required. ∎

By Lemma 6.15, to understand stabilizers of marked points we may assume that
$H$ is the stabilizer of a marked point $z$ which is a cut point of $Q$. Let
$M\subset Q$ be the collection of all marked points that are cut points.
Consider the cut-point tree $T_{\operatorname{cut}}$ associated to $(Q,M)$.
The vertices of $T_{\operatorname{cut}}$ are:

  1. (1)

Points in $M$; and

  2. (2)

The maximum connected subsets of $Q$ not separated by an element of $M$,

Vertices of the first type are cut points, and vertices of the second type are
_blocks_. Edges of $T_{\operatorname{cut}}$ correspond to the inclusion of a
cut point in a block. Clearly, $V$ acts on $T_{\operatorname{cut}}$.

######  Lemma 6.16.

Let $z\in M$ and let $B$ be a block containing $z$. Let $e=[z\subset B]$ be
the corresponding edge of $T_{\operatorname{cut}}$, and let $H$ be the
stabilizer in $V$ of $e$. Then $H/Z^{\omega}(V)$ has rank at most two.

###### Proof.

Denote the stabilizer of $z$ by $H^{\prime}$. By Lemma 6.11, $H^{\prime}$
fixes $\xi=f_{\infty}(z)\in\mathcal{C}_{\infty,v}$, a limit point of parabolic
points $\xi_{i}\in\mathcal{C}(\Gamma_{v_{i}})$. In particular,
$H^{\prime}\in\mathcal{H}_{V}$, hence $H\in\mathcal{H}_{V}$. By Lemma 6.13, it
suffices to show that $H$ consists entirely of elements satisfying Property
$\mathcal{B}$. Let $g\in H$ and $x\in B$. Since $g$ fixes $B$ as a set, we
know $g\cdot x\in B$. By definition of blocks, $B\smallsetminus\\{z\\}$ is
connected. Hence there is a path $\tau$ connecting $x$ and $g\cdot x$ and
$\tau$ does not intersect $z$. Thus $\omega$–almost surely $f^{Q}_{i}(\tau)$
does not intersect $\xi_{i}$. Therefore, $g$ has Property $\mathcal{B}$. ∎

We are now ready to prove Proposition 6.3. As we showed above, this is enough
to finally complete the proof of Theorem 6.2

###### Proof of Proposition 6.3.

Let $\mathbb{D}_{0}$ be the splitting of $V$ dual to the action of $V$ on
$T_{\operatorname{cut}}$. Note that $Z^{\omega}(V)$ acts on
$T_{\operatorname{cut}}$ trivially, hence $Z^{\omega}(V)$ is contained in all
the edge groups of $\mathbb{D}_{0}$. The underlying graph of $\mathbb{D}_{0}$
is bipartite, with one type of vertices corresponding to cut point vertices in
$T_{\operatorname{cut}}$ and the other type corresponding to block vertices.
For each cut point vertex of $\mathbb{D}_{0}$, fold all the adjacent edges
together to one edge, and denote the resulting splitting by
$\mathbb{D}^{\prime}$. Now, each cut point vertex of $\mathbb{D}^{\prime}$ has
one adjacent edge and $\mathbb{D}^{\prime}$ is still connected, so the
underlying graph of $\mathbb{D}^{\prime}$ is a star.

Collapse the cut point vertices of $\mathbb{D}^{\prime}$ whose stabilizers are
not conjugate into edge groups of $\mathbb{G}^{r}$ adjacent to $V$. Add
vertices to the resulting graph of groups corresponding to edge groups
adjacent to $V$ not corresponding to vertices of $\mathbb{D}^{\prime}$ (with
identical edge and vertex groups). It follows that the resulting graph of
groups $\mathbb{D}$ satisfies (4). Note that since $\mathbb{D}^{\prime}$
satisfies (1) and (2), so does $\mathbb{D}$. Statement (3) follows from Lemmas
6.14, 6.15 and 6.16.

Suppose $V$ is of hyperbolic type and that $P\in\mathcal{H}_{V}$ is not
conjugate into an adjacent edge group. Then $Z^{\omega}(V)=\\{1\\}$. If $P$
does not stabilize a cut point vertex of $T_{\operatorname{cut}}$ then $P$ is
finitely generated by Lemma 6.14 and Lemma 6.15. If $P$ does stabilize a cut
point vertex of $T_{\operatorname{cut}}$, then $P$ is in a valence-one vertex
group of $\mathbb{D}^{\prime}$ whose adjacent edge group is finitely generated
by Lemma 6.16. Refining $\mathbb{G}^{r}$ by $\mathbb{D}^{\prime}$ yields a
splitting $\mathbb{K}^{\prime}$ of $L$. Since $P$ is not contained in the
conjugate of any edge group of $\mathbb{G}^{r}$, the corresponding vertex
group in $\mathbb{K}^{\prime}$ containing $P$ is still valence-one with the
same adjacent edge group, so it is finitely generated since $L$ is. In all
cases, $P$ is finitely generated and the proof of Proposition 6.3 is complete.
∎

##  7\. Limits and $\mathbb{R}$–trees

Throughout this section make Standing Assumption 4.18. Suppose further that
$(\phi_{i})$ is $\mathcal{C}$–divergent with respect to ${{\mathbb{G}}}$ (see
Definition 6.1) and that the vertex $v$ of $\mathbb{G}^{r}$ associated to $V$
is one for which ${\lim}^{\omega}\|\phi_{i}\|_{\mathcal{C},v}=\infty$. The
space $\mathcal{C}_{\infty,v}$ is defined in Definition 7.4 below.

The goal of this section is to prove the following theorem, which is the only
result from this section needed in future sections.

######  Theorem 7.1.

  1. (1)

$\mathcal{C}_{\infty,v}$ is an $\mathbb{R}$–tree equipped with a nontrivial
isometric $V$–action.

  2. (2)

Let $H\in\mathcal{H}_{V}$. There exists $x_{H}\in\mathcal{C}_{\infty,v}$,
fixed by $H$. Each nontrivial element of $H\smallsetminus Z^{\omega}(V)$ fixes
only the point $x_{H}$.

  3. (3)

If $g\in V$ and $(g_{i})$ is an $\omega$–approximation to $g$ so
$\omega$–almost surely $g_{i}$ is (nontrivial and) elliptic then $g$ fixes a
unique point in $\mathcal{C}_{\infty,v}$.

  4. (4)

The $\overline{V}$–action on $\mathcal{C}_{\infty,v}$ has trivial tripod
stabilizers.

  5. (5)

The $\overline{V}$–action on $\mathcal{C}_{\infty,v}$ has abelian segment
stabilizers.

The above theorem, together with the Rips machine, implies that the
$\overline{V}$–action on $\mathcal{C}_{\infty,v}$ admits a graph of actions
decomposition with simplicial, axial and Seifert type vertex actions, which
induces an abelian splitting of $\overline{V}$. The splitting of $V$ (rel
$\mathcal{H}_{V}$) induced by this splitting also has abelian edge groups.
222See [14] for the definitions of graphs of actions, simplicial vertex
actions, axial vertex actions and vertex actions of Seifert type, as well as a
statement of the Rips machine (Theorem 5.1 in that paper).

The remainder of this section is dedicated to the proof of Theorem 7.1. Recall
the definitions from Section 4, particularly Definitions 4.10 and 4.14.

Let $A$ be the good relative generating set for $V$ from Assumption 4.18 and
let $(A_{i})$ be the $\omega$–approximation of $A$. Let $v_{i}\in T_{i}$ be
the sequence of vertices associated to $v$. Let
$\Gamma_{v_{i}}\subset\Gamma_{i}$ be the stabilizer of $v_{i}$. Then since
${{\mathbb{G}}}$ is a GGD of $L$ with respect to $(\phi_{i})$ $\omega$–almost
surely $A_{i}\subset\Gamma_{v_{i}}$. Let
$\mathcal{C}_{i}=\mathcal{C}(\Gamma_{v_{i}})$. For $\omega$–almost every $i$,
fix $\mathfrak{o}_{i}\in\mathcal{C}_{i}$ so that

$\max_{g\in
A_{i}}\left\\{d_{i}(g.\mathfrak{o}_{i},\mathfrak{o}_{i})\right\\}\leq\|\phi_{i}\|_{\mathcal{C},v}+\frac{1}{i}.$

######  Definition 7.2.

Let $\mathcal{C}_{\infty,v}^{0}$ be the ultra-limit (with respect to $\omega$)
of the sequence
$\left(\mathcal{C}_{i},\frac{1}{\|\phi_{i}\|_{\mathcal{C},v}}d_{i},\mathfrak{o}_{i}\right)$.

######  Proposition 7.3 (Limiting $\mathbb{R}$–tree).

The space $\mathcal{C}_{\infty,v}^{0}$ is a (pointed) $\mathbb{R}$–tree
equipped with a non-trivial isometric $V$–action.

###### Proof.

Most parts of this proposition are standard facts in the theory of ultra-
limits and $\mathbb{R}$–trees (see, for example, [12, $\S 4$] and the
references therein). The only new thing here is to show that $\\{\phi_{i}\\}$
induces an isometric action of $V$ on $\mathcal{C}_{\infty,v}^{0}$. This does
not follow from the standard theory since $A$ may not generate $V$. However,
$V$ is generated by $A\cup\mathbb{E}(V)$, where $\mathbb{E}(V)$ are the edge
groups of $\mathbb{G}^{r}$ adjacent to $V$.

Let $\mathfrak{o}=[\\{\mathfrak{o}_{i}\\}]\in\mathcal{C}_{\infty,v}^{0}$ be
the basepoint of $\mathcal{C}_{\infty,v}^{0}$. By the definition of
$\|\phi_{i}\|_{\mathcal{C},v}$, for all $g\in A_{v,i}$

($\dagger$)
${\lim}^{\omega}\frac{1}{\|\phi_{i}\|_{\mathcal{C},v}}d_{i}(\mathfrak{o}_{i},g\cdot\mathfrak{o}_{i})<\infty.$

Let $H\in\mathbb{E}(V)$, and suppose $h\in H$ and $g\in A\cap H$ (note that
such a $g$ exists by Definition 4.14). Let $(h_{i})$ and $(g_{i})$ be
$\omega$–approximations of $h$ and $g$, respectively. Then $\omega$–almost
surely $g_{i}$ and $h_{i}$ are parabolic fixing the same point in
$\mathcal{C}_{i}$. By Lemma 5.15, there exists some constant $D$ so that
$\omega$–almost surely
$\left|d_{i}(\mathfrak{o}_{i},h_{i}\mathfrak{o}_{i})-d_{i}(\mathfrak{o}_{i},g_{i}\mathfrak{o}_{i})\right|<D$.
Therefore, by ($\dagger$ ‣ 7)

${\lim}^{\omega}\frac{1}{\|\phi_{i}\|_{\mathcal{C},v}}d_{i}(\mathfrak{o}_{i},h_{i}\cdot\mathfrak{o}_{i})<\infty,$

as required. ∎

######  Definition 7.4.

Let $\mathcal{C}_{\infty,v}$ be the minimal $V$–invariant subtree of
$\mathcal{C}^{0}_{\infty,v}$.

Theorem 7.1.(1) follows immediately from Proposition 7.3.

Recall that $\overline{V}=V/Z^{\omega}(V)$, where $Z^{\omega}(V)$ is given by
Definition 4.9. Each element of $Z^{\omega}(V)$ $\omega$–almost surely acts
trivially on $\mathcal{C}_{i}$. Thus we have

######  Corollary 7.5.

The $V$–action descends to a non-trivial isometric $\overline{V}$–action on
$\mathcal{C}^{0}_{\infty,v}$, with minimal $\overline{V}$–invariant subtree
$\mathcal{C}_{\infty,v}$.

Theorem 7.1.(2) and (3) follow from the next lemma.

######  Lemma 7.6.

Let $g\in V$ and let $(g_{i})$ be an $\omega$–approximation of $g$. Suppose
that $\omega$–almost surely $g_{i}$ is parabolic (or nontrivial elliptic).
Then $g$ fixes a unique point in $\mathcal{C}_{\infty,v}$. If $g^{1},g^{2}\in
V$, $(g^{1}_{i})$ and $(g^{2}_{i})$ are $\omega$–approximations of $g^{1}$ and
$g^{2}$, respectively, and $\omega$–almost surely $g^{1}_{i}$ and $g^{2}_{i}$
are parabolic with the same fixed point then $g^{1}$ and $g^{2}$ fix the same
point in $\mathcal{C}_{\infty,v}$.

###### Proof.

Let $C_{i}$ be the component of the collapsing locus fixed by $g_{i}$ (or the
projection in $\mathcal{C}_{i}$ of the fixed point of the elliptic $g_{i}$ in
$\mathbb{H}^{\ast}$). Consider the geodesic triangle with vertices
$\mathfrak{o}_{i},g_{i}\cdot\mathfrak{o}_{i}$ and $C_{i}$. Let $x_{i}$ be the
point on $[\mathfrak{o}_{i},C_{i}]$ with
$d_{i}(x_{i},C_{i})=(\mathfrak{o}_{i}\mid g\cdot\mathfrak{o}_{i})_{C_{i}}$.
Note that $2d_{i}(x_{i},\mathfrak{o}_{i})\leq
d_{i}(\mathfrak{o}_{i},g_{i}\cdot\mathfrak{o}_{i})+4\delta$. Hence

${\lim}^{\omega}\frac{1}{\|\phi_{i}\|_{\mathcal{C},v}}d_{i}(\mathfrak{o}_{i},x_{i})<\infty$

Hence $\\{x_{i}\\}$ defines a point in $\mathcal{C}^{0}_{\infty,v}$. By
hyperbolicity of $\mathcal{C}_{i}$, $d_{i}(x_{i},g_{i}\cdot x_{i})\leq\delta$.
Hence $g$ fixes $x$. Therefore $g$ fixes a point in $\mathcal{C}_{\infty,v}$.
We now show that $g$ fixes exactly one point in $\mathcal{C}_{\infty,v}$.
Suppose $g$ fixes $x\in\mathcal{C}_{\infty,v}$ and
$\left(x_{i}\in\mathcal{C}_{i}\right)$ represents $x$. By Lemma 5.15,
$\left|2d_{i}(x_{i},C_{i})-d_{i}(x_{i},g_{i}x_{i})\right|$ is $\omega$–almost
surely bounded independent of $i$. Since $g$ fixes $x$,

${\lim}^{\omega}d_{i}(x_{i},g_{i}x_{i})/\|\phi_{i}\|_{\mathcal{C},v}=0$

so ${\lim}^{\omega}d_{i}(x_{i},C_{i})/\|\phi_{i}\|_{\mathcal{C},v}=0$, and
$x=[\\{C_{i}\\}]\in\mathcal{C}_{\infty,v}$.

The second assertion follows immediately from the above argument and the
assumption that $\omega$–almost surely $g^{1}_{i}$ and $g^{2}_{i}$ fix the
same point. ∎

######  Corollary 7.7.

Suppose $g\in V\smallsetminus Z^{\omega}(V)$ fixes a non-degenerate arc in
$\mathcal{C}_{\infty,v}$. If $(g_{i})$ is an $\omega$–approximation of $g$
then $\omega$–almost surely $g_{i}$ is loxodromic.

Theorem 7.1.(4) follows from Lemma 7.6 and the following lemma.

######  Lemma 7.8.

Let $g\in V$ and $(g_{i})$ be an $\omega$–approximation of $g$. Suppose that
$\omega$–almost surely $g_{i}$ is loxodromic. The fixed point set of $g$ in
$\mathcal{C}_{\infty,v}$ is either empty or a single geodesic. In case it is a
single geodesic, it is a limit of images in the collapsed space of geodesic
axes in $\mathbb{H}^{\ast}$.

###### Proof.

Suppose the fixed point set of $g$ is not empty. Let
$x=[\\{x_{i}\\}]\in\mathcal{C}_{\infty,v}$ be a point fixed by $g$. Let
$\mathbb{H}^{\ast}$ be the domain of the map $q_{i}=q_{\Gamma_{v_{i}}}$.
Denote the axis of $g_{i}$ in $\mathbb{H}^{\ast}$ by $\widetilde{\gamma}_{i}$.
Let $\gamma_{i}=q_{i}(\widetilde{\gamma}_{i})$. Let
$\widetilde{x_{i}}\in\mathbb{H}^{\ast}$ be a point in $q_{i}^{-1}(x_{i})$. By
Lemma 5.15, there is a constant $D$ so that $\omega$–almost surely
$d_{i}(x_{i},g_{i}x_{i})\geq 2d_{i}(x_{i},\gamma_{i})-D$. Since $x$ is fixed
by $g$, we have
${\lim}^{\omega}\frac{d_{{}_{i}}(x_{i},g_{i}x_{i})}{\|\phi_{i}\|_{\mathcal{C},v}}=0$,
and so
${\lim}^{\omega}\frac{d_{{}_{i}}(x_{i},\gamma_{i})}{\|\phi_{i}\|_{\mathcal{C},v}}=0$.
Therefore $x$ is on the limit of $\\{\gamma_{i}\\}$, which is a geodesic in
$\mathcal{C}_{\infty,v}$. ∎

The goal of the rest of this section is to prove Theorem 7.1.(5). To that end,
fix a non-trivial segment $I=[a,b]$ in $\mathcal{C}_{\infty,v}$, and suppose
that $\overline{g},\overline{h}\in\overline{V}$ stabilize $I$. Fix lifts
$g,h\in V$ of $\overline{g},\overline{h}$ and let $(g_{i})$ and $(h_{i})$ be
$\omega$–approximations of $g$ and $h$, respectively. We know that
$\omega$–almost surely $g_{i}$ and $h_{i}$ both lie in the same
$\Gamma_{v,i}$, and our goal is to show that $\omega$–almost surely
$[g_{i},h_{i}]\in Z(\Gamma_{v,i})$. Recall that if $\Gamma_{v,i}$ is of
hyperbolic type then $Z(\Gamma_{v,i})=\\{1\\}$, and that in either case
$Z(\Gamma_{v,i})$ is the kernel of the action of $\Gamma_{v,i}$ on the
associated hyperbolic space $\mathbb{H}^{\ast}$.

Observe that $[g_{i},h_{i}]$ is an $\omega$–approximation of $[g,h]$. We may
suppose $[g_{i},h_{i}]$ is $\omega$–almost surely non-trivial, or else there
is nothing to prove. Because $[g,h]$ stabilizes the non-trivial segment $I$ in
$\mathcal{C}_{\infty,v}$, by Corollary 7.7 $\omega$–almost surely
$[g_{i},h_{i}]$ corresponds to a loxodromic isometry of $\mathbb{H}^{\ast}$.
Let $\widetilde{\gamma}_{i}$ be the invariant geodesic for $[g_{i},h_{i}]$ in
$\mathbb{H}^{\ast}$ and let $\gamma_{i}=q_{i}(\widetilde{\gamma}_{i})$. By
Lemma 7.8, $I$ is a subsegment of the limit of $\gamma_{i}$. As a result,
there are $a_{i},b_{i}$ on $\gamma_{i}$ such that $a=[a_{i}]$ and $b=[b_{i}]$.
Let $I_{i}$ be a geodesic segment between $a_{i}$ and $b_{i}$.

That segment stabilizers are abelian is proved in the following three lemmas.

######  Lemma 7.9.

Suppose that $\omega$–almost surely $\gamma_{i}$ is not entirely contained in
the collapsing locus but that it comes within $5\delta$ of the collapsing
locus. Then $\omega$–almost surely $[g_{i},h_{i}]\in Z(\Gamma_{v,i})$.

###### Proof.

Since $a_{i}$ and $b_{i}$ are on $\gamma_{i}$, by Lemma 5.6 and the choice of
$\delta$, $I_{i}\subset N_{\delta}(\gamma_{i})$. Thus for $w_{i}\in I_{i}$
satisfying
$d_{i}(a_{i},w_{i}),d_{i}(b_{i},w_{i})\geq\frac{1}{10}d_{i}(a_{i},b_{i})$, we
have $d_{i}\left([g_{i},h_{i}]\cdot w_{i},w_{i}\right)\leq 16\delta$ (See [23,
Lemma 5.7]). Hence $[g_{i},h_{i}]$ moves points in the middle part of
$\gamma_{i}$ by less than $30\delta$, contradicting Lemma 5.16. Hence
$[g_{i},h_{i}]$ acts trivially on $\mathbb{H}^{\ast}$ $\omega$–almost surely,
so $\omega$–almost surely $[g_{i},h_{i}]\in Z(\gamma_{v,i})$, as required. ∎

######  Lemma 7.10.

Suppose that $\omega$–almost surely $\gamma_{i}$ does not come within
$5\delta$ of the collapsing locus. Then $\omega$–almost surely
$[g_{i},h_{i}]\in Z(\Gamma_{v,i})$.

###### Proof.

We consider an $\omega$–large set of indices $i$ so that $g_{i}$ and $h_{i}$
correspond to loxodromic isometries in $\mathbb{H}^{*}$, as above, and
implicitly concentrate only on such indices.

Suppose $[a_{i},g_{i}\cdot a_{i}]$ penetrates a component $W$ of the
collapsing locus. We claim the distance between the entry and exit points of
$[a_{i},g_{i}\cdot a_{i}]$ in $W$ is bounded independent of $i$. To that end,
let $\widetilde{a}_{i}=q_{i}^{-1}(a_{i})$ and
$\widetilde{b}_{i}=q_{i}^{-1}(b_{i})$. If
$q_{i}\left([\widetilde{a}_{i},g_{i}\cdot\widetilde{a}_{i}]\right)$ does not
penetrate $W$, the claim follows from Lemma 5.11. Suppose then that
$q_{i}([\widetilde{a}_{i},g_{i}\cdot\widetilde{a}_{i}])$ does penetrate $W$
and let $a^{\prime}_{i}$ and $u_{i}^{\prime}$ be the points of entry and exit
of $q_{i}([\widetilde{a}_{i},g_{i}\cdot\widetilde{a}_{i}])$ into $W$,
respectively. By Lemma 5.12, it suffices to show
$d_{\mathbb{H}^{\ast}}(a^{\prime}_{i},u_{i}^{\prime})$ is bounded independent
of $i$. Let $\pi_{\widetilde{W}}$ be the closest-point projection onto
$\widetilde{W}=q_{i}^{-1}(W)$. By Lemma 5.14, both
$d_{\mathbb{H}^{\ast}}(a^{\prime}_{i},\pi_{\widetilde{W}}(\widetilde{a}_{i}))$
and
$d_{\mathbb{H}}(u_{i}^{\prime},\pi_{\widetilde{W}}(g_{i}\cdot\widetilde{a}_{i}))$
are bounded independent of $i$. Suppose
$q_{i}([\widetilde{b}_{i},g_{i}\cdot\widetilde{b}_{i}])$ penetrates $W$. Then
by Lemma 5.6 $[b_{i},g_{i}\cdot b_{i}]\cap N_{\delta}(W)\neq\emptyset$. Since
$d(a_{i},b_{i})$ is much bigger than both $d(b_{i},g_{i}\cdot b_{i})$ and
$d(a_{i},g_{i}\cdot a_{i})$ for large $i$, by $\delta$–hyperbolicity,
$[a_{i},b_{i}]\cap N_{2\delta}(W)\neq\emptyset$. By Lemma 5.6, the subsegment
of $\gamma_{i}$ between $a_{i}$ and $b_{i}$ lies entirely within
$N_{\delta}\left([a_{i},b_{i}]\right)$. As a result, $\gamma_{i}\cap
N_{3\delta}(W)\neq\emptyset$, contradicting the assumptions of the lemma.
Hence $q_{i}([\widetilde{b}_{i},g_{i}\cdot\widetilde{b}_{i}])$ does not
penetrate $W$. It follows that the projections of
$[\widetilde{a}_{i},\widetilde{b}_{i}]$,
$[\widetilde{b}_{i},g_{i}\cdot\widetilde{b}_{i}]$ and
$g_{i}\cdot[\widetilde{a}_{i},\widetilde{b}_{i}]$ to $W$ have bounded length
as none of them intersect $W$. The above bounds together show that
$d_{\mathbb{H}^{\ast}}(a^{\prime}_{i},u_{i}^{\prime})$ is $\omega$–almost
surely bounded independent of $i$.

The number of components penetrated by $[a_{i},g_{i}\cdot a_{i}]$ is at most
$d(a_{i},g_{i}\cdot a_{i})$. Hence
$d_{\mathbb{H}^{\ast}}(\widetilde{a}_{i},g_{i}\cdot\widetilde{a}_{i})$ is
bounded by $d(a_{i},g_{i}\cdot a_{i})D$, where $D$ is independent of $i$. As a
result, the geodesic rectangle
$[\widetilde{a}_{i},\widetilde{b}_{i},g_{i}\cdot\widetilde{a}_{i},g_{i}\cdot\widetilde{b}_{i}]$
is arbitrarily thin in the middle of $[\widetilde{a}_{i},\widetilde{b}_{i}]$.
Similarly, the same is true for the geodesic rectangle
$[\widetilde{a}_{i},\widetilde{b}_{i},h_{i}\cdot\widetilde{a}_{i},h_{i}\cdot\widetilde{b}_{i}]$.
Hence $\omega$–almost surely, $[g_{i},h_{i}]$ moves a point in the middle of
$[\widetilde{a}_{i},\widetilde{b}_{i}]$ by arbitrarily small amount. If
$[g_{i},h_{i}]\notin Z(\Gamma_{v_{i}})$, then by the definition of collapsed
spaces $\gamma_{i}$ is in the collapsing locus $\omega$–almost surely. This
contradicts the assumption of the lemma. Hence $[g_{i},h_{i}]\in
Z(\Gamma_{v_{i}})$, as required. ∎

######  Lemma 7.11.

If $\omega$–almost surely $\gamma_{i}$ is contained in the
$5\delta$–neighborhood of the collapsing locus then $\omega$–almost surely
$[g_{i},h_{i}]\in Z(\Gamma_{v,i})$.

###### Proof.

We first show that $\gamma_{i}$ is entirely contained in the collapsing locus
of $\mathcal{C}_{i}$. By the choice of $K$, if $\gamma_{i}$ is entirely
contained in the $5\delta$–neighborhood of the collapsing locus, then it is
entirely contained in the $5\delta$–neighborhood of some connected component
$C$ of the collapsing locus. So $q_{i}^{-1}(C)$ is a tube around some geodesic
$\beta$ in $\mathbb{H}^{*}$. The $q_{i}$–pre-image of the $5\delta$
neighborhood of $C$ is the $D^{\prime}$ neighborhood of $\beta$ for some
constant $D^{\prime}$. The only bi-infinite geodesic contained in this
neighborhood is $\beta$, so $\widetilde{\gamma_{i}}=\beta$. In particular,
$\widetilde{\gamma_{i}}$ is contained in $q_{i}^{-1}(C)$, so $\gamma_{i}$ is
contained in $C$.

As above, choose subsegments of $\gamma_{i}$ denoted $[a_{i},b_{i}]$. Let
$C_{i}$ be the component of the collapsing locus containing $\gamma_{i}$.
Suppose $g_{i}\cdot C_{i}\neq C_{i}$. Then
$d_{\mathcal{C}_{i}}(C_{i},g_{i}\cdot C_{i})\geq K$. For $\omega$–almost all
$i$, $[a_{i},b_{i}]$ is much longer than both $[a_{i},g_{i}\cdot a_{i}]$ and
$[b_{i},g_{i}\cdot b_{i}]$. By $\delta$–hyperbolicity of $\mathcal{C}_{i}$,
the middle part of $[a_{i},b_{i}]$ is in a $2\delta$–neighborhood of
$g[a_{i},b_{i}]$. But $K\geq 2\delta$, so $C_{i}$ intersects the
$K$–neighborhood of $g_{i}\cdot C_{i}$, a contradiction. Thus, $g_{i}\cdot
C_{i}=C_{i}$, and similarly $h_{i}\cdot C_{i}=C_{i}$. Hence $g_{i}$ and
$h_{i}$ act as translations along $\widetilde{\gamma}_{i}$, and
$\omega$–almost surely $[g_{i},h_{i}]=1$, proving the lemma. ∎

The previous three lemmas finish the proof of Theorem 7.1.(5).

##  8\. JSJ-decompositions and modular automorphisms

We continue to make Standing Assumption 4.18 and use the notation from there.

###  8.1. Summary

In the last section we showed that if $v$ is the vertex associated to $V$ and
${\lim}^{\omega}\|\phi_{i}\|_{\mathcal{C},v}=\infty$ then $V$ admits a
nontrivial action on an $\mathbb{R}$-tree. By the Rips machine (and the
results in the last section) this induces a graph of groups decomposition of
$V$. Associated to any such decomposition is a group of _modular
automorphisms_ of $V$ (see Definition 8.11). For a group $H$ equipped with a
splitting $\mathbb{A}$ and a family of subgroups $\mathcal{H}$, the group of
modular automorphisms of $H$ rel $(\mathbb{A},\mathcal{H})$ is denoted
$\operatorname{Mod}^{\mathcal{H}}_{\mathbb{A}}(H)$.

To apply the shortening argument in the next section, we need to use modular
automorphisms of $V$ without a priori knowing the particular splitting of $V$.
For this we need a _JSJ–decomposition_ of $V$, which is a graph of groups
decomposition which “sees” all possible modular automorphisms of $V$. The
JSJ–decomposition we use was essentially constructed by Guirardel-Levitt in
[16]. Our situation does not quite satisfy the assumptions used in the
construction in [16], but the same arguments work with only minor changes.
These changes are outlined in Appendix B.

Recall $Z^{\omega}(V)$ is the stable center of $V$ and
$\overline{V}=V/Z^{\omega}(V)$. Let $\pi_{v}\colon V\to\overline{V}$ denote
the quotient map. If $V$ is a hyperbolic-type vertex group,
$Z^{\omega}(V)=\\{1\\}$ and $\overline{V}=V$ in this case. Let $\mathcal{A}$
be the family of all virtually abelian subgroups of $\overline{V}$, and
$\mathcal{H}=\mathcal{H}_{\overline{V}}$ the family of all stably parabolic
subgroups of $\overline{V}$ (recall Definition 4.5). When working with a
subgroup $H\leq\overline{V}$ or $H\leq V$, we abuse notation and write
$\mathcal{A}$ and $\mathcal{H}$ for the families of virtually abelian and
stably parabolic subgroups of $H$, respectively.

We construct a JSJ–decomposition for $\overline{V}$ in Appendix B. The next
theorem, which is the main result of this sections, shows that this
decomposition has the properties we need in Section 9.

######  Theorem 8.1.

  1. (1)

$\overline{V}$ has an $(\mathcal{A},\mathcal{H})$–splitting
$\overline{\mathbb{J}}(v)$ so that for any
$(\mathcal{A},\mathcal{H})$–splitting $\mathbb{A}$ of $\overline{V}$ we have
$\operatorname{Mod}^{\mathcal{H}}_{\mathbb{A}}(\overline{V})\leq\operatorname{Mod}^{\mathcal{H}}_{\overline{\mathbb{J}}(v)}(\overline{V})$.

  2. (2)

Any element of
$\operatorname{Mod}^{\mathcal{H}}_{\overline{\mathbb{J}}(v)}(\overline{V})$
lifts to an automorphism of $V$ which acts as the identity on $Z^{\omega}(V)$
and by conjugation on each stably parabolic subgroup.

######  Definition 8.2.

Let $\mathbb{J}(v)$ be the lift of $\overline{\mathbb{J}}(v)$ to $V$.

If $\overline{V}$ is finitely generated and $K$–CSA, the existence of
$\overline{\mathbb{J}}(v)$ follows from [16, Theorem 9.14]. In our case
$\overline{V}$ is not necessarily $K$–CSA since $\overline{V}$ can have
arbitrarily large finite subgroups. In the next section we show that
$\overline{V}$ is _weakly $K$–CSA_ (see Definition 8.4 below), which means
that it shares enough of the properties of $K$–CSA groups that the proof of
[16, Theorem 9.14] holds with only minor changes. We explain these changes in
Appendix B, see Theorem B.5.

###  8.2. Weakly $K$–CSA groups

######  Definition 8.3.

A group is _$K$ –virtually abelian_ if it has an abelian subgroup of index at
most $K$.

######  Definition 8.4.

Fix $K\geq 1$. A group is _weakly $K$–CSA_ if (i) any element $g$ of order
greater than $K$ is contained in a unique maximal virtually abelian subgroup
$M(g)$, so that $M(g)$ is $K$–virtually abelian and equal to its normalizer,
and (ii) every two infinite, virtually abelian subgroups $A$ and $B$ with
$\langle A,B\rangle$ not virtually abelian satisfy $|A\cap B|\leq K$.

Note that if a group $H$ is weakly $K$–CSA and all finite subgroups of $H$
have order $\leq K$, then $H$ is $K$–CSA.

######  Lemma 8.5.

Let $A$ and $B$ be virtually abelian subgroups of $\overline{V}$. Either
$|A\cap B|\leq 2$ or $\langle A,B\rangle$ is $2$–virtually abelian.

###### Proof.

If all finitely generated subgroups of a countable group $C$ are $2$–virtually
abelian, then $C$ is $2$–virtually abelian by [16, Lemma 9.6]. Thus we may
assume that $A$ and $B$ are finitely generated. Let $A_{i}$ and $B_{i}$ be
subgroups of $\Gamma_{i}$ generated by $\omega$–approximations of fixed finite
generating sets of $A$ and $B$, respectively. There exist vertex groups
$V_{i}$ in the geometric decomposition of $\Gamma_{i}$ so that $\omega$–almost
surely $A_{i},B_{i}\leq V_{i}$. Let $\overline{A}_{i}$ and $\overline{B}_{i}$
be the images of $A_{i}$ and $B_{i}$ in $V_{i}/Z(V_{i})$, respectively. Then
$\overline{A}_{i}$ and $\overline{B_{i}}$ are either both subgroups of
fundamental groups of orientable finite volume hyperbolic $3$–manifolds or
both subgroups of fundamental groups of orientable hyperbolic $2$–orbifolds.
Also, $A$ and $B$ are limit groups over the families $\\{\overline{A}_{i}\\}$
and $\\{\overline{B}_{i}\\}$, respectively.

Assume $|A|\geq 3$ and $|B|\geq 3$, otherwise the result is trivial. Since $A$
and $B$ are finitely generated and virtually abelian, $\overline{A}_{i}$ and
$\overline{B}_{i}$ are $\omega$–almost surely virtually abelian. Hence each
$\overline{A}_{i}$ is contained in a unique maximal virtually abelian subgroup
$M_{i}$ which is either cyclic or infinite dihedral. Similarly, each
$\overline{B}_{i}$ is contained in a unique maximal virtually abelian subgroup
$N_{i}$. Either $|M_{i}\cap N_{i}|\leq 2$, or $M_{i}=N_{i}$. If the first case
happens $\omega$–almost surely, then $|A\cap B|\leq 2$, and if the second
happens $\omega$–almost surely then $\langle A,B\rangle$ is $2$–virtually
abelian. ∎

The following lemma follows easily from the proof of Lemma 8.5.

######  Lemma 8.6.

Let $H\leq\overline{V}$ be virtually abelian but not abelian. Then $H$ has an
abelian subgroup $H^{+}$ of index two and an element $\tau\in H\setminus
H^{+}$ so that for all $g\in H^{+}$, $\tau^{-1}g\tau=g^{-1}$.

######  Lemma 8.7.

Let $g\in\overline{V}$ be an element of order at least 3. Let

$M(g)=\left\langle\\{h\in\overline{V}\;|\;\langle g,h\rangle\text{ is
virtually abelian}\\}\right\rangle$

Then $M(g)$ is $2$–virtually abelian and $M(g)$ is the unique, maximal
virtually abelian subgroup of $\overline{V}$ containing $g$.

###### Proof.

By induction and Lemma 8.5, all finitely generated subgroups of $M(g)$ are
$2$–virtually abelian so $M(g)$ is $2$–virtually abelian by [16, Lemma 9.6].
The maximality and uniqueness of $M(g)$ follow. ∎

######  Lemma 8.8.

Let $g\in\overline{V}$ have order at least 3. Then $M(g)$ is equal to its
normalizer.

###### Proof.

Suppose $x\in\overline{V}$ and $x^{-1}M(g)x=M(g)$. Let $V_{i}$ be as in the
proof of Lemma 8.5. Let $g_{i}$ and $x_{i}$ be $\omega$–approximations of $g$
and $x$ respectively, and let $\overline{g}_{i}$ and $\overline{x}_{i}$ be the
images of $g_{i}$ and $x_{i}$ in $V_{i}/Z(V_{i})$ respectively.

Then $\omega$–almost surely,
$\langle\overline{g}_{i},\overline{x}_{i}^{-1}\overline{g}_{i}\overline{x}_{i}\rangle$
is virtually abelian, so it is contained in a maximal virtually abelian
subgroup $M_{i}$, which is either cyclic or infinite dihedral. Since
$\overline{g}_{i}$ and $\overline{x}_{i}^{-1}\overline{g}_{i}\overline{x}_{i}$
both have order at least 3, they commute. Since $M_{i}$ is a maximal virtually
abelian subgroup of a hyperbolic 3–manifold group or a hyperbolic $2$–orbifold
group, $\omega$–almost surely $\overline{x}_{i}\in M_{i}$, so $x\in M(g)$. ∎

Lemmas 8.5, 8.7, and 8.8 together imply the following.

######  Proposition 8.9.

$\overline{V}$ is weakly $2$–CSA.

Using Proposition 8.9, Theorem B.5 applies to $\overline{V}$. Hence
$\overline{V}$ has an $(\mathcal{A},\mathcal{H})$–JSJ decomposition, and we
let $\overline{\mathbb{J}}(v)$ be the tree of cylinders of this splitting. By
Theorem B.5, $\overline{\mathbb{J}}(v)$ is compatible with every
$(\mathcal{A},\mathcal{H})$–splitting of $\overline{V}$.

###  8.3. Modular automorphisms

We define the group of modular automorphisms relative to a particular graph of
groups decomposition and a fixed family of subgroups. These definitions are
standard except that some care has to be taken to ensure we are able to lift
the automorphisms from $\overline{V}$ to $V$.

Suppose $H=A\ast_{C}B$ and $c\in Z_{H}(C)$. The _Dehn twist by $c$_ is the
automorphism of $H$ defined by fixing each $a\in A$ and mapping each $b\in B$
to $cbc^{-1}$. If $H=A\ast_{C}$ and $c\in Z_{H}(C)$ then the _Dehn twist by
$c$_ is the automorphism defined by fixing each $a\in A$ and mapping the
stable letter $t$ to $tc$. If $\mathbb{A}$ is a graph of groups decomposition
and $e$ is an edge of $\mathbb{A}$, a _Dehn twist over $e$_ is a Dehn twist in
the one edge splitting corresponding to collapsing all edges of $\mathbb{A}$
other than $e$.

Now suppose $H$ is finitely generated relative to a family of subgroups
$\mathcal{H}_{0}$. Let $\mathbb{A}$ be a splitting of $H$ rel
$\mathcal{H}_{0}$ and let $A$ be an abelian vertex group of $\mathbb{A}$. Let
$P_{A}$ be the minimal direct factor of $A$ which contains the edge groups of
$\mathbb{A}$ adjacent to $A$, the elements of $\mathcal{H}_{0}$ contained in
$A$, and the torsion of $A$. The subgroup $P_{A}$ can be characterized as the
intersection of the kernels of all homomorphisms $A\to\mathbb{Z}$ which are
identically 0 on the adjacent edge groups and on elements of
$\mathcal{H}_{0}$. We call $P_{A}$ the
_$(\mathbb{A},\mathcal{H}_{0}$)–envelope of $A$_. Write $A=A_{0}\oplus P_{A}$,
and note that since $A$ is finitely generated relative to adjacent edge groups
and $\mathcal{H}_{0}$, $A_{0}$ is a finitely generated free abelian group.
Thus the group of automorphisms of $A$ fixing $P_{A}$ can be identified with
$GL_{n}(\mathbb{Z})$ for some $n\geq 1$. In particular, this group is
generated by transvections: automorphisms sending $g$ to $gh$ for some basis
elements $g$ and $h$ of $A_{0}$ and fixing all other elements of the basis of
$A_{0}$, and all elements of $P_{A}$.

A _dihedral type_ vertex group of $\mathbb{A}$ is a vertex group $D$ which has
an abelian subgroup $D^{+}$ of index two and $\tau\in D\setminus D^{+}$ so
that for all $g\in D^{+}$, $\tau^{-1}g\tau=g^{-1}$. The
$(\mathbb{A},\mathcal{H}_{0}$)–envelope $P_{D^{+}}$ is defined exactly as for
abelian vertex groups. Write $D^{+}=D_{0}\oplus P_{D^{+}}$, and suppose $g$
and $h$ are distinct elements of a basis for $D_{0}$. A _squared Nielsen
transformation rel $(\mathbb{A},\mathcal{H}_{0}$)_ is an automorphism of $D$
fixing $\tau$ and extending the automorphism of $D^{+}$ sending $g$ to
$gh^{2}$ and fixing $P_{D^{+}}$ and all other basis elements of $D_{0}$.

Let ${\rm Aut}_{s}(D)$ be the subgroup of the automorphism group of $D$
generated by squared Nielsen transformations rel
$(\mathbb{A},\mathcal{H}_{0}$). In previous applications of the shortening
argument, one is allowed to pick $\alpha$ from the whole automorphism group of
$D$. In that case Lemma 8.10 below follows from the Euclidean algorithm. Our
situation, where fewer shortening automorphisms are allowed, requires only
slightly more care.

######  Lemma 8.10.

Suppose $D$ acts on the real line $T$ with indiscrete orbit and
$g_{1},\dots,g_{k}\in D$ project to form a basis of $D^{+}/P_{D^{+}}$. There
exists a sequence $\alpha_{i}\in{\rm Aut}_{s}(D)$ so the translation length of
$\alpha_{i}(g_{j})$ goes to zero as $i\rightarrow\infty$.

Suppose $\mathbb{A}$ is a graph of groups decomposition of a group $H$ and
$v_{0}$ is a vertex of $\mathbb{A}$. Any element of $g$ can be represented
(non-uniquely) by $[a_{0},e_{1},a_{1},...,e_{n},a_{n}]$ where
$e_{1},...,e_{n}$ is an edge path in $\mathbb{A}$ from $v_{0}$ to $v_{0}$,
$a_{0}\in\mathbb{A}_{v_{0}}$, and each $a_{i}\in\mathbb{A}_{v_{i}}$ where
$v_{i}$ is the terminal vertex of $e_{i}$ for $1\leq i\leq n$. Suppose $v$ is
a vertex of $\mathbb{A}$ and $\sigma\in\operatorname{Aut}(\mathbb{A}_{v})$
acts by conjugation on each adjacent edge group. Then $\sigma$ can be extended
to an automorphism of $G$ as in [36, Definition 4.13] as follows: for each
adjacent edge $e$ there is $\gamma_{e}\in\mathbb{A}_{v}$ so that
$\sigma(h)=\gamma_{e}h\gamma_{e}^{-1}$ for all $h$ in the image of
$\mathbb{A}_{e}$ in $\mathbb{A}_{v}$. Then $\sigma$ extends to
$\overline{\sigma}\in\operatorname{Aut}(H)$ by defining

$\overline{\sigma}([a_{0},e_{1},a_{1},...,e_{n},a_{n}])=[\overline{a}_{0},e_{1},\overline{a}_{1},...,e_{n},\overline{a}_{n}]$

where

$\overline{a_{i}}=\begin{cases}a_{i}\;&\text{if}\;a_{i}\notin\mathbb{A}_{v}\\\
\gamma_{e_{i}}^{-1}\sigma(a_{i})\gamma_{e_{i+1}}\;&\text{if}\;a_{i}\in\mathbb{A}_{v}.\end{cases}$

We call $\overline{\sigma}$ the natural extension of $\sigma$. This natural
extension is not unique: it depends on the choice of $\gamma_{e}$.

In the splittings we are interested in, some vertex groups are identified with
finite extensions of fundamental groups of hyperbolic orbifolds. These are
called _QH–vertex groups_.

######  Definition 8.11.

Let $\mathbb{A}$ be a reduced $(\mathcal{A},\mathcal{H})$ splitting of a group
$H$. The _modular automorphism group of $H$ rel $(\mathbb{A},\mathcal{H}$_),
denoted $\operatorname{Mod}^{\mathcal{H}}_{\mathbb{A}}(H)$, is the subgroup of
$\operatorname{Aut}(H)$ generated by:

  1. (1)

Inner automorphisms.

  2. (2)

Dehn twists over edges of $\mathbb{A}$ with abelian edge groups.

  3. (3)

Natural extensions of automorphisms of QH-vertex groups of $\mathbb{A}$
induced by homeomorphisms of the underlying orbifold fixing the boundary and
cone points which act by conjugation on elements of $\mathcal{H}$.

  4. (4)

Natural extensions of automorphisms of maximal abelian vertex groups of
$\mathbb{A}$ fixing the $(\mathbb{A},\mathcal{H}_{0}$)–envelope.

  5. (5)

Natural extensions of squared Nielsen transformations rel
$(\mathbb{A},\mathcal{H}_{0}$) of maximal dihedral-type vertex groups.

  6. (6)

Natural extensions of automorphisms of vertex groups $U$ where $U$ is a
central extension $1\to Z\to U\to B\to 1$, $Z$ is contained in all edge groups
adjacent to $U$, $B$ is a QH or maximal dihedral-type group and the
automorphism fixes $Z$ and projects to a type 3 or type 5 automorphism of $B$.

We only need automorphisms of type 1-5 when working with the quotient