# Thurston’s broken windows only theorem revisited

Ken’ichi Ohshika  Department of Mathematics, Gakushuin University, Mejiro,
Toshima-ku, Tokyo, Japan<EMAIL_ADDRESS>
###### Abstract.

The ‘broken windows only theorem’ is the main theorem of the third paper among
a series of the paper in which Thurston proved his uniformisation theorem for
Haken manifolds. In this chapter, we show that the second statement of this
theorem is not valid, giving a counter-example. We also give a weaker version
of this statement with a proof. In the last section, we speculate on how this
second statement was intended to constitute a proof of the bounded image
theorem, which constituted a key of the uniformisation theorem. The proof of
the bounded image theorem was obtained only quite recently, although its
weaker version, which is sufficient for the proof of the uniformisation
theorem, had already been proved.

##  0\. Introduction

One of the most celebrated achievements of Thurston is the uniformisation of
Haken manifolds, which motivated the general geometrisation theorem, proved
later by Perelman. Thurston planned to publish the proof of the uniformisation
theorem in a series of seven papers. Only the first one was officially
published, and the second and the third papers exist as preprints, and are now
included in the collected works of Thurston [33]. In the first paper, the
compactness theorem for the deformation spaces of acylindrical hyperbolic
3-manifolds is proved. The third paper generalises this compactness to a
general atoroidal 3-manifold with incompressible boundary as a relative
convergence theorem, which is dubbed as the ‘broken windows only’ theorem
(Theorem 0.1 in [37]). This should have constituted the key part of the proof
of what is called the bounded image theorem, one of the main ingredients of
the entire proof. The theorem consists of two statements and we shall show
that in fact the second statement is not valid, giving a counter-example.

Although Thurston did not write a proof of the bounded image theorem, a weaker
version of the bounded image theorem, the bounded orbit theorem, which first
appeared in [34], is sufficient for the proof of the uniformisation theorem as
was observed also by Morgan [22], and a detailed proof of this weaker version
was given in Kapovich [13]. A proof of the bounded image theorem by Thurston
had been unknown, except for the case of acylindrical case given by Kent [14]
and Brock–Bromberg-Canary–Minsky [7], but quite recently a proof relying on
modern techniques of Kleinian group theorem (in particular the work of
Brock–Bromberg–Canary–Lecuire [6] among other works) was given by
Lecuire–Ohshika [16]. It is still worthwhile to consider how Thurston himself
tried to prove the bounded image theorem using the tools available at that
time. We shall show how this should have been done provided that the broken
windows only theorem were true, in the last section.

We now state this broken windows only theorem and explain what is problematic.
For that, we first need to explain some terms used in the theorem, whose
precise definitions will be given in the following section. Let $(M,P)$ be a
pared manifold, that is, a compact irreducible 3-manifold $M$ with a union $P$
of incompressible tori and annuli on $\partial M$ such that $\partial
M\setminus P$ is incompressible, and every $\pi_{1}$-injective, immersed torus
or properly immersed annulus can be properly homotoped into $P$. We denote by
$AH(M,P)$ the set of complete hyperbolic metrics on $\mathrm{Int}M$ modulo
isotopy such that each component of $P$ corresponds to a parabolic end. The
set $AH(M,P)$ corresponds one-to-one to the set of faithful discrete
representations of $\pi_{1}(M)$ into ${\rm PSL}_{2}(\mathbb{C})$ sending
$\pi_{1}(P_{0})$ to parabolic a parabolic subgroup for each component $P_{0}$
of $P$, modulo conjugacy.

We consider the characteristic submanifold $\Phi$ of $(M,P)$ in the sense of
Jaco–Shalen–Johannson, consisting of $I$-bundles (over incompressible surfaces
which are neither disc nor annuli nor Möbius bands), solid tori intersecting
$\partial M$ along incompressible annuli on their boundaries, and thickened
torus intersecting $\partial M$ along a torus and annuli. The window $F$ of
$(M,P)$ is an $I$-bundle constructed from the characteristic submanifold as
follows. All $I$-bundle components of $\Phi$ are contained in $F$. For each
solid torus or thickened torus component $V$ of $\Phi$, we consider each
component of $\operatorname{Fr}V$, which is an annulus. If it is not properly
isotopic into the $I$-bundle component we consider its thin regular
neighbourhood, which is an $I$-bundle over an annulus, and make it be included
in $F$. Otherwise we abandon that component. If there are more than two
properly homotopic frontier components, we do this procedure only for one
among them. Thus obtained $F$, which is an $I$-bundles over possibly
disconnected surface, is called the window of $(M,P)$. The window of $(M,P)$
is denoted by $\mathrm{window}(M,P)$ and its base space by $\mathrm{wb}(M,P)$.

Now Thurston’s ‘broken windows only’ theorem reads as follows:

######  Theorem 0.1 (Thurston [37]).

If $\Gamma\in\pi_{1}(M)$ is any subgroup which is conjugate to the fundamental
group of a component of $M\setminus\mathrm{window}(M,P)$, then the set of
representations of $\Gamma$ in $\mathrm{Isom}(\mathbb{H}^{3})$ induced from
$AH(M,P)$ are bounded, up to conjugacy.

Given any sequence $N_{i}\in AH(M,P)$, there is a subsurface with
incompressible boundary $x\subset\mathrm{wb}(M,P)$ and a subsequence
$N_{i(j)}$ such that the restriction of the associated sequence of
representations $\rho_{i(j)}:\pi_{1}(M)\to\mathrm{Isom}(\mathbb{H}^{3})$ to a
subgroup $\Gamma\subset\pi_{1}(M)$ converges if and only if $\Gamma$ is
conjugate to the fundamental group of a component of $M\setminus X$, where $X$
is the total space of the interval bundle above $x$. Furthermore, no
subsequence of $\rho_{i(j)}$ converges on any larger subgroup.

Thurston gave a fairly detailed proof for the first sentence. There is also an
alternative approach for the first part using Morgan–Shalen theory [24, 25,
26], and also a stronger version of this first part has been proved by
Brock–Bromberg–Canary–Minsky [7]. We shall give a counter-example to the
second sentence of Theorem 0.1. The problem lies in the fact that there may be
a solid torus $V$ in the characteristic submanifold of $M$ which intersects
$\partial M$ at more than three annuli. Such a solid torus may bring about a
$3$-manifold homotopy equivalent to $M$ which is not homeomorphic to $M$, by
changing the order of attaching the components of $M\setminus V$ along
$\partial V$. We call this operation shuffling along $V$.

We shall show that even though Theorem 0.1 does not hold in the original form,
a weaker version as follows can be proved making use of the Morgan–Shalen
theory and the efficiency of pleated surfaces due to Thurston [36].

######  Theorem 0.2.

Let $(M,P)$ and $N_{i}$ be as given in Theorem 0.1. Then there is a pared
manifold $(M^{\prime},P^{\prime})$ obtained from $(M,P)$ by shuffling around
solid pairs and thickened torus pairs of the characteristic submanifold of
$(M,P)$ for which the following hold.

Let $W^{\prime}$ be the characteristic submanifold of
$(M^{\prime},P^{\prime})$. Let $\mathcal{W}$ be the union of its $I$-pairs,
and $\mathcal{V}$ the union of solid torus pairs and thickened torus pairs of
$W^{\prime}$. Let $p\colon\mathcal{W}\to w$ be the fibration as an $I$-bundle
for $\mathcal{W}$. Then there is an incompressible subsurface $x$ of $w$ and a
union of essential annuli (possibly with singular axes) $\mathcal{A}$ in
$\mathcal{V}$ such that the restriction of the associated sequence of
representations $\rho_{i(j)}:\pi_{1}(M)\to\mathrm{Isom}(\mathbb{H}^{3})$ to a
subgroup $\Gamma\subset\pi_{1}(M)$ converges if and only if $\Gamma$ is
conjugate to the fundamental group of a component of
$M\setminus(X\cup\mathcal{A})$ under the identification of $\pi_{1}(M)$ with
$\pi_{1}(M^{\prime})$, where $X=p^{-1}(x)$. (When we cut $M$ along an annulus
with singular axis, we regard a regular neighbourhood of the singular axis as
contained in both sides after the operation.) Furthermore, no subsequence of
$\rho_{i(j)}$ converges on any larger subgroup.

Unfortunately, this weaker version is not sufficient for the proof of the
bounded image theorem.

##  1\. Preliminaries

In this section, we explain notions and terminologies which are necessary for
Theorems 0.1 and 0.2 and their proofs.

###  1.1. JSJ decompositon

In this subsection, we shall review the theory of Jaco–Shalen and Johannson
concerning characteristic decomposition of compact irreducible 3-manifolds by
tori and annuli and homotopy equivalences between such 3-manifolds.

Unless otherwise mentioned, throughout this section $M$ will denote a compact
irreducible 3-manifold with (possibly empty) incompressible boundary. We
always assume 3-manifolds to be orientable. Jaco–Shalen and Johannson proved
that there exists a disjoint family of embedded incompressible tori
$T_{1},\dots,T_{p}$ and properly embedded incompressible annuli
$A_{1},\dots,A_{q}$ with the following properties.

  1. (1)

Cutting $M$ along $(T_{1}\cup\dots\cup T_{p}\cup A_{1}\cup\dots\cup A_{q})$,
we obtain compact 3-submanifolds of $M$ each of which is one of the following.

    1. (a)

A Seifert fibred manifold whose frontier in $M$ consists of fibred tori and
fibred annuli.

    2. (b)

An atoroidal manifold, i.e. a manifold such that every $\pi_{1}$-injective map
from a torus to the manifold is homotoped into a boundary component. We call
each of such manifolds a hyperbolic piece. (In particular the definition
implies that each of the tori $T_{1},\dots,T_{p}$ is disjoint from the
interior of any hyperbolic piece.)

  2. (2)

If a hyperbolic piece $N$ contains some of $A_{1},\dots,A_{q}$, then the
3-manifold $N^{\prime}$ obtained by cutting $N$ along the annuli contained in
$N$ consists of three types of manifolds.

    1. (a)

An $I$-bundle over a surface such that its intersection with $N$ coincides
with the associated $\partial I$-bundle and is an incompressible subsurface of
$\partial N$. Such a component is called an $I$-pair, and we denote each of
them in the form of a pair $(\Phi,\Sigma)$, where $\Phi$ is an $I$-bundle and
$\Sigma=N\cap\partial M$, or simply by $\Phi$ if there is no fear of
confusion.

    2. (b)

A solid torus $V$ whose intersection with $\partial N$ consists of annuli
which are incompressible both on $\partial V$ and $\partial N$. Such a
component $(V,V\cap\partial N)$ is called a solid torus pair.

    3. (c)

A thickened torus $U\cong S^{1}\times S^{1}\times I$ such that $S^{1}\times
S^{1}\times\\{0\\}$ is contained in $\partial N$ and $S^{1}\times
S^{1}\times\\{1\\}$ intersects $\partial N$ along a union of incompressible
annuli no two of which are homotopic in $\partial N$. For such a component
$(U,U\cap\partial N)$ is called a thickened torus pair.

    4. (d)

An acylindrical manifold, i.e. a compact 3-manifold such that every properly
embedded annulus is relatively properly homotopic into a boundary component.

We note that every $\pi_{1}$-injective proper map from an annulus into a
hyperbolic piece is properly homotopic into either an $I$-pair or a solid
torus pair or a thickened torus pair. We take a family of incompressible tori
and annuli having the above properties which is minimal with respect to the
inclusion among the families with the same properties. We call a decomposition
of $M$ by such a minimal family a JSJ-decomposition. Jaco–Shalen [11] and
Johannson [12] proved that JSJ-decomposition is unique up to ambient isotopy.

We now assume that $M$ is atoroidal. In this case, the union of the $I$-pairs,
the solid torus pairs and the thickened torus pairs is called the
characteristic submanifold of $M$. The window $F$ of $M$ is constructed from
the characteristic submanifold as follows. We first consider the $I$-pairs in
the characteristic submanifold, and let them be included in $F$. For a solid
torus pair or a thickened torus pair $V$, we replace it with regular
neighbourhoods $V^{\prime}_{1},\dots,V^{\prime}_{k}$ of the components of
$\operatorname{Fr}V$, which are solid tori and can be regarded as $I$-bundles
over annuli. If $V_{j}^{\prime}$ is properly homotopic into an $I$-pair or
another solid torus already added to $F$, we discard it, and otherwise we let
$V_{j}^{\prime}$ be included in $F$.

###  1.2. Homotopy equivalences and books of $I$-bundles

Each $I$-pair, solid torus pair and thickened torus pair in the characteristic
submanifold of an atoroidal Haken manifold may give rise to a homotopy
equivalence which is not realised by a homeomorphism under some conditions as
will be explained below. Let $(\Phi,\Sigma)$ be an $I$-pair in a JSJ-
decomposition of an atoroidal Haken manifold $M$. By definition, $\Phi$ has an
$I$-bundle structure whose projection to its base surface we denote by
$p\colon\Phi\rightarrow B$. Suppose that $\partial B$ has more than one
component, and let $b$ be a component of $\partial B$. Its inverse image
$p^{-1}(b)$ is an annulus which is a union of fibres. We parametrise
$p^{-1}(b)$ as $\\{(z,t)\mid z=e^{i\theta},\theta\in[0,2\pi),t\in[0,1]\\}$ so
that the second coordinate corresponds to fibres. We consider an involution
$\iota$ on $p^{-1}(b)$ taking $(z,t)$ to $(z,1-t)$. Now we remove $N$ from $M$
and paste it back to $M\setminus N$ by reversing the direction of each fibre
in $p^{-1}(b)$ by the involution $\iota$ just defined. Then we get another
3-manifold $M^{\prime}$ homotopy equivalent to $M$, and this operation gives a
homotopy equivalence from $M$ to $M^{\prime}$. We call this operation flipping
on the $I$-pair $(\Phi,\Sigma)$, following Canary-McCullough [9], where
homotopy equivalences of pared manifolds are thoroughly studied. We note that
$M^{\prime}$ is non-orientable if the annulus $p^{-1}(b)$ is non-separating.
The same kind of operation can be defined on a frontier component of a solid
torus pair or a thickened torus pair in the characteristic submanifold, which
we also call flipping.

Next let $(V,C)$ be a solid torus pair in the JSJ-decomposition of $M$. We
call the number of components of $V\cap\partial M$ the valency of $V$. It is
evident that the valency is also equal to the number of the components of
$\operatorname{Fr}V$. Suppose that the valency of $V$ is greater than $2$.
Then we can remove $V$ from $M$ and paste it back after changing the cyclic
order of the annuli in $\operatorname{Fr}V$. In this way, we get a 3-manifold
homotopy equivalent to $M$. We call this operation shuffling around $V$. We
can define shuffling around a thickened torus pair $(U,U\cap\partial M)$ in
the same way, to be changing the order of pasting the components of
$M\setminus U$ to $\operatorname{Fr}U$.

Jaco–Shalen [11] and Johannson [12] proved that any homotopy equivalence
between two compact irreducible 3-manifolds with (non-empty) incompressible
boundaries is obtained by repeating flipping and shuffling.

We now consider a special case of 3-manifolds obtained by pasting more than
two product $I$-bundles along one solid torus. Let $V$ be a solid torus and
take a simple closed curve $\alpha$ on $\partial V$ homotopic to the $p$-time
iteration of a core curve with $p\geq 1$. Letting $n$ be an integer greater
than $2$, we consider $n$ compact surfaces $\Sigma_{1},\dots,\Sigma_{n}$ each
of which has only one boundary component, denoted by $c_{1}\dots c_{n}$. We
then consider product $I$-bundles $\Sigma_{1}\times I,\dots,\Sigma_{n}\times
I$, and paste them along $c_{1}\times I,\dots,c_{n}\times I$ to $\partial V$
so that all of these annuli are identified with disjoint annuli with core
curves isotopic to $\alpha$ which lie on $\partial V$ in the (cyclic) order of
the subscripts. We get a compact, irreducible, atoroidal 3-manifold $N$ in
this way, and it is called a book of (product) $I$-bundles with $n$ pages.
Such manifolds were studied in the context of deformation spaces of Kleinian
groups for the first time by Anderson-Canary [1]. In particular when $p=1$,
the book of $I$-bundles is said to be primitive. The characteristic
submanifold of $N$ consists of $n+1$ components, i.e. $I$-pairs corresponding
to $\Sigma_{1}\times I,\dots,\Sigma_{n}\times I$ and a solid torus pair
$(V,V\cap\partial N)$. The window consists of $n$ components: the
characteristic submanifold with the solid torus pair $(V,V\cap\partial N)$
removed. By the theory of Jaco-Shalen and Johannson mentioned above, any
homotopy equivalence from $N$ to another compact irreducible 3-manifold is
obtained by shuffling along $V$ since no flipping gives rise to a new
manifold.

###  1.3. Pared manifolds

A pared manifold is a pair $(M,P)$ of a compact irreducible 3-manifold $M$
(with possibly compressible boundary) and a disjoint union of incompressible
tori and annuli $P$ lying on $\partial M$ with the following properties.

  1. (i)

Any $\pi_{1}$-injective immersion from a torus is homotopic into $P$.

  2. (ii)

Any $\pi_{1}$-injective proper immersion of an annulus whose boundaried is
mapped into $P$ is properly homotopic into $P$.

We also assume the following:

  1. (iii)

Every component of $\partial M\setminus P$ is incompressible.

We define the characteristic submanifold of a pared manifold $(M,P)$ in the
same way as before, just imposing the condition that no component of $P$ is
contained in the associated $\partial I$-bundle of an $I$-pair.

###  1.4. Deformation spaces

Let $M$ be a compact, irreducible, atoroidal $3$-manifold. We say that $M$
admits a convex compact hyperbolic structure when there is a convex compact
hyperbolic 3-manifold $N$ and a homeomorphism from $M$ to $N$ is given. The
hyperbolic manifofold $N$ can be embedded in a complete hyperbolic manifold
homeomorphic to $\mathrm{Int}N$ as a convex core. Therefore a convex compact
hyperbolic structure on $M$ also induces a complete hyperbolic structure on
$\mathrm{Int}M$.

Each complete hyperbolic structure on $\mathrm{Int}M$ corresponds to a
faithful discrete representation of $\pi_{1}(M)$ into ${\rm
PSL}_{2}(\mathbb{C})$, which is uniquely determined up to conjugacy. The set
of faithful discrete representations of $\pi_{1}(M)$ into ${\rm
PSL}_{2}(\mathbb{C})$ modulo conjugacy is denoted by $AH(M)$ and is given a
topology induced from the topology of element-wise convergence of the
representation space. We consider the space of convex compact hyperbolic
structures on $M$ modulo isotopy, which we denote by $QH_{0}(M)$, and is
regarded as a subspace of $AH(M)$ by using extensions of such structures to
complete hyperbolic structures as mentioned above. By the theory of Ahlfors,
Bers, Kra, Maskit, Marden and Sullivan, there is a covering map
$q\colon\mathcal{T}(\partial M)\to QH_{0}(M)$. (The definition of the map in
this general setting is due to Bers [3], Kra [15], Maskit [18] and Marden
[17]. Sullivan [31] showed that the map is surjective.) Under the present
assumption that $\partial M$ is incompressible $q$ is known to be a
homeomorphism ([3, 15, 18, 17]). In the special case when $M$ is homeomorphic
to $S\times I$ for a closed surface, this parametrisation is what is called
the Bers simultaneous uniformisation [2], and is expressed as
$qf\colon\mathcal{T}(S)\times\mathcal{T}(\bar{S})\to QH_{0}(M)$, where
$\mathcal{T}(S)$ denotes the Teichmüller space of oriented $S$, whereas
$\mathcal{T}(\bar{S})$ denotes that of $S$ with orientation reversed.
Representations contained in $QH_{0}(M)$ in this case are called quasi-
Fuchsian representations.

For a pared manifold $(M,P)$, we define $AH(M,P)$ to be the set of faithful
discrete representations of $\pi_{1}(M)$ into ${\rm PSL}_{2}(\mathbb{C})$
which send each non-trivial element of $\pi_{1}(P_{0})$ to a parabolic element
for each component $P_{0}$ of $P$. The set modulo isotopy of convex finite-
volume complete hyperbolic structures on $(M,P)$ for which $P$ corresponds to
parabolic ends forms a subspace of $AH(M,P)$, which we denote by
$QH_{0}(M,P)$. As in the case of convex compact hyperbolic structures, there
is a homeomorphism $q\colon\mathcal{T}(\partial M\setminus P)\to QH_{0}(M)$.

###  1.5. Morgan–Shalen compactification

In this subsection we summarise some results contained in Morgan-Shalen [24,
25, 26], which will be used in the proof of Theorem 0.2 in Section 3.

Let $M$ be an irreducible, atoroidal compact 3-manifold with incompressible
boundary. We may also consider a pared manifold $(M,P)$ as will be explained
later. What interests us is the case when $M$ has a boundary component which
is not a torus. Let $\\{\rho_{i}\\}$ be a sequence in $AH(M)$. Suppose that
$\\{\rho_{i}\\}$ does not have a convergent subsequence in $AH(M)$. Fix a
generator system $g_{1},\dots,g_{k}$ of $\pi_{1}(M)$ such that the trace
function $\mathrm{tr}_{g_{1}},\dots,\mathrm{tr}_{g_{k}}$ generates the trace
functions of elements of $\pi_{1}(M)$ as a ring. (See Culler–Shaen [10] for
the existence of such a system.) Let $\mu_{i}$ be
$\max_{j=1}^{k}\\{\mathrm{trlength}(\rho_{i}(g_{j}))\\}$, where
$\mathrm{trlength}$ denotes the translation length. By changing the ordering
of the generator system and passing to a subsequence, we can assume that
$\mu_{i}$ is attained by $\mathrm{trlength}(\rho_{i}(g_{1}))$. Put a basepoint
$x_{i}$ in $\mathbb{H}^{3}$ which is within bounded distance from the axis of
$\rho_{i}(g_{1})$. Consider the rescaled hyperbolic space
$\mathrm{trlength}(\rho_{i}(g_{1}))^{-1}\mathbb{H}^{3}$ with basepoint at
$x_{i}$, and take an equivariant Gromov limit. Then we get an isometric action
of $\pi_{1}(M)$ on an $\mathbb{R}$-tree. (Morgan–Shalen [24], Paulin [29],
Bestvina [4])

Here an $\mathbb{R}$-tree is a geodesic space in which for any two points,
there is only one simple arc connecting them. In other words, it is a
$0$-hyperbolic space in sense of Gromov. An isometric action of $G$ on an
$\mathbb{R}$-tree $T$ is said to have small edge stabilisers when for any non-
trivial arc $a$ in $T$, its stabiliser is a virtually abelian group. It is
said to be minimal when $T$ has no $\rho_{G}(G)$-invariant proper subtree.

In the case when we consider a pared manifold $(M,P)$, we assume that for any
component $P_{0}$ of $P$ and any element $\gamma\in\pi_{1}(P_{0})$, the
translation length of $\rho_{i}(\gamma)$ is bounded as $i\to\infty$. Then the
limit action on an $\mathbb{R}$-tree has a property that for any element
$\gamma$ in $\pi_{1}(P_{0})$ as above, the action of $\gamma$ on the tree is
trivial.

######  Theorem 1.1 (Morgan–Shalen, Paulin, Bestvina).

As a rescaled Gromov limit, we obtain a minimal isometric action of $\pi(M)$
on an $\mathbb{R}$-tree $T$ with small edge stabilisers.

A codimension-1 lamination in a compact $3$-manifold $M$ is a closed subset
$L$ of $M$ such that for any point $x\in M$ there is a coordinate
neighbourhood $U$ of $M$ in the form $D^{2}\times[0,1]$, where $U\cap
L=D^{2}\times J$ with a closed subset $J$ of $[0,1]$. We can define a leaf of
$L$ in the same way as for a foliation. A codimension-1 lamination is said to
be measured when there is a positive Borel measure on $[0,1]$ supported on $J$
which is invariant under coordinate changes.

We equip the set of codimension-1 measured laminations with a topology induced
from the weak topology on the set of measures on transverse arcs. A positively
weighted disjoint union of embedded surfaces is an example of a codimension-1
measured lamination. A codimension-1 measured lamination is said to be
incompressible when it is a limit of positively weighted unions of properly
embedded incompressible surfaces. An incompressible codimension-1 measured
lamination is said to be annular when it is a limit of weighted properly
embedded incompressible annuli.

######  Proposition 1.2 (Morgan–Shalen [25]).

Any codimension-1 annular lamination $L$ is properly isotopic into the
characteristic submanifold $W$ of $M$. For each $I$-pair $W_{0}$ of $W$, the
intersection $L\cap W_{0}$ can be isotoped so that it becomes vertical with
respect to the fibration.

Let $L$ be an incompressible codimension-1 measured lamination, and consider
the universal cover $\tilde{M}$ of $M$. (In our setting, $\tilde{M}$ is always
simply connected.) Then each leaf of $L$ can be lifted to a topological plane
in $\tilde{M}$, and the union of lifted leaves constitute a codimension-1
measured lamination in $\tilde{M}$.

For an incompressible measured lamination $L$, we can construct an
$\mathbb{R}$-tree dual to $L$ in the following way. In the universal cover
$\tilde{M}$, we collapse each component of $\tilde{M}\setminus\tilde{L}$ to a
point, replace isolated leaves with arcs, and put a metric coming from the
lift of the transverse measure of $L$. Then we get an $\mathbb{R}$-tree on
which $\pi_{1}(M)$ acts by isometries. We call this the $\mathbb{R}$-tree dual
to $L$, and denote it by $T_{L}$. We note that we can construct an
$\mathbb{R}$-tree dual to a measured geodesic lamination on a hyperbolic
surface in the same way, on which its fundamental group acts by isometries.

A continuous map $f$ from an $\mathbb{R}$-tree $T$ to another
$\mathbb{R}$-tree $T^{\prime}$ is said to be a morphism if for each point $x$
in $T$, there is an geodesic segment $I$ in $T$ containing $x$ such that $f|I$
is an isometry. An isomorphism from $T$ to $T^{\prime}$ is a surjective
isometry. A morphism may fail to be isomorphism, for $f$ may be ‘folded’ at
some points of $T$.

######  Theorem 1.3 (Morgan–Shalen [26], Morgan–Otal [23]).

Let $\rho\colon\pi_{1}(M)\to\mathrm{Isom}(T)$ be a minimal isometric action on
an $\mathbb{R}$-tree $T$ with small edge stabilisers. Then there is an annular
codimension-1 lamination in $M$ with an $\pi_{1}(M)$-equivariant morphism
$t\colon T_{L}\to T$, where $T_{L}$ is the $\mathbb{R}$-tree dual to $L$.

Furthermore, let $(M,P)$ be a pared manifold, and
$\rho\colon\pi_{1}(M)\to\mathrm{Isom}(T)$ as above has a property that for any
component $P_{0}$ of $P$, its fundamental group $\pi_{1}(P_{0})$ lies in the
kernel of $\rho$. Then the annular codimension-1 lamination $L$ above can be
taken to be disjoint from $P$.

We note that this theorem only guarantees the existence of morphism, not an
isomorphism. As we shall see in Section 2, there is an example for which we
cannot have an annular codimension-1 lamination giving an isomorphism.

We need to generalise slightly the notion of annular codimension-1 lamination
for Theorems 0.2 and 3.1. We define an annulus with a singular axis as
follows. Let $V$ be a solid torus pair in a pared manifold $(M,P)$. Then
$V\cap\partial M$ consists of annuli on $\partial V$ whose core curves are
homotopic in $V$ to the $p$-time iteration of a core curve of $V$ for some
positive integer $p$. This means that there is a Seifert fibration $\pi\colon
V\to D$ for a disc $D$ with a cone point $\mathfrak{c}$ of order $p$ with
respect to which each component of $V\cap\partial M$ is vertical, i.e. it is
expressed as $\pi^{-1}(\alpha)$ for a simple arc on $\partial D$. We denote
the union of the simple arcs corresponding to $V\cap\partial M$ by $A$. An
annulus with a singular axis is defined to be a singular annulus of the form
$p^{-1}(\beta_{1}\cup\dots\cup\beta_{k})$ for the union of simple arcs
$\beta_{1},\dots,\beta_{k}$ with $k\geq 2$ in $D$ connecting $A$ to
$\mathfrak{c}$ which intersect each other only at $\mathfrak{c}$. It has a
singular axis, which is equal to $\pi^{-1}(\mathfrak{c})$ around which any
non-singular fibre wraps $p$ times. We call such an annulus weighted if each
of $\beta_{1},\dots,\beta_{k}$, and correspondingly its preimage, has a
positive weight. In Theorem 3.1, we shall allow such weighted annuli with
singular axes in solid torus pairs to be included as components of an annular
codimension-1 measured lamination.

###  1.6. Skora’s theorem

In the previous subsection, we talked about an action on an $\mathbb{R}$-trees
dual to a codimension-1 measured lamination in a 3-manifold. In this
subsection, in the case of (possibly non-orientable) surfaces of hyperbolic
type, any minimal isometric action on an $\mathbb{R}$-tree with small edge
stabilisers is dual to a measured geodesic lamination on the surface.

The following theorem was proved by Skora [30].

######  Theorem 1.4 (Skora).

Let $S$ be a closed hyperbolic surface, and suppose that
$\rho\colon\pi_{1}(S)\to\mathrm{Isom}(T)$ is a minimal action with small edge
stabilisers. Then there is a measured geodesic lamination $\lambda$ on $S$
with a $\pi_{1}(S)$-equivariant isomorphism between the $\mathbb{R}$-tree
$T_{L}$ dual to $L$ and $T$.

We note that even in the case when $S$ has punctures or boundaries, Theorem
1.4 holds under the assumption that elements of $\pi_{1}(S)$ corresponding to
loops around single punctures or boundaries act on $T$ trivially.

##  2\. Counter-example

In this section, we shall construct a counter-example to Theorem 0.1. We
consider a primitive book of product $I$-bundles explained in the previous
section consisting of more than three pages. It is a special case of
Thurston’s theorem that such a 3-manifold admits convex compact hyperbolic
structure in its interior, but even without using Thurston’s theorem, we can
construct such a structure just by the Klein–Maskit combination theorem (see
[19] for various versions of the combination theorem). To fix symbols, let $M$
be a primitive book of $I$-bundles which is a union of product $I$-bundles
$\Sigma_{1}\times[0,1],\dots,\Sigma_{n}\times[0,1]$ and a solid torus $V$,
where each $\Sigma_{j}$ has only one boundary component, and
$\partial\Sigma_{i}\times[0,1]$ is attached to $\partial V$ along an essential
annulus $A_{i}$ whose core curve is homotopic to a core curve of $V$.

We order the $I$-bundles so that the attaching annuli $A_{1},\dots,A_{n}$ lie
on $\partial V$ in this (cyclic) order. We assume that the ordering is
compatible with the parametrisation of $[0,1]$, that is,
$\partial\Sigma_{i}\times\\{1\\}$ and $\Sigma_{i+1}\times\\{0\\}$ are
juxtaposed. We let $\Sigma_{i}^{\prime}$ be the complement of a thin annular
neighbourhood of $\partial\Sigma_{i}$ in $\Sigma_{i}$. The characteristic
submanifold $W$ of $M$ consists of a solid torus $V$ and $I$-bundles
$\Sigma_{1}^{\prime}\times[0,1],\dots,\Sigma^{\prime}_{n}\times[0,1]$, and the
window consists of
$\Sigma_{1}^{\prime}\times[0,1],\dots,\Sigma^{\prime}_{n}\times[0,1]$. We call
the boundary component of $A_{i}$ nearer to $A_{i+1}$, i.e.
$\partial\Sigma_{i}\times\\{1\\}$ (we regard $A_{n+1}$ as $A_{1}$) the upper
boundary. For $1\leq i<j\leq n$, we denote by $B_{i,j}$ the closure of a
component of the complement of $A_{i}\cup A_{j}$ in $\partial V$ which
contains the upper boundary of $A_{i}$. We let $S_{i,j}$ be
$\Sigma_{i}\times\\{1\\}\cup B_{i,j}\cup\Sigma_{j}\times\\{0\\}$ which is a
closed surface embedded in $M$.

We now define the (right) Dehn twist of $M$ along $A_{j}$. The right Dehn
twist of $M$ along $A_{j}$ is obtained by cutting $M$ along $A_{i}$ into
$\Sigma_{j}\times I$ and the rest, and glue them back after twisting $A_{i}$
on $\Sigma_{j}\times I$ by $2\pi$ in the direction of $S^{1}$ keeping
$I$-direction unchanged to the right when we look at it from the inside of
$\Sigma_{j}\times I$. In other words, the right Dehn twist along $A_{j}$ is a
product of the right Dehn twist around $\operatorname{Fr}\Sigma_{j}$ and the
identify on the fibre direction. We denote this homemorphism by
$\tau_{A_{j}}\colon M\to M$. The left Dehn twist along $A_{j}$ is defined to
be $\tau_{A_{j}}^{-1}$.

######  Proposition 2.1.

Let $M$ be a primitive book of product $I$-bundles of $n$ pages. Fix a convex
cocompact hyperbolic structure $m_{0}$ on $M$, which is also regarded as a
point in $AH(M)$. Let $\\{m_{i}\in AH(M)\\}$ be a sequence obtained from
$m_{0}$ by pushing forward $m_{0}$ by the composition of the $i$-time
iteration of the right Dehn twist $\tau_{A_{1}}$ along $A_{1}$ and the
$i$-time iteration of the left Dehn twist $\tau_{A_{3}}^{-1}$ along $A_{3}$,
i.e. we define $m_{i}=(\tau_{A_{1}}\circ\tau_{A_{3}}^{-1})_{*}^{i}(m_{0})$.
Then the following hold:

  1. (a)

The sequence $\\{m_{i}\\}$ does not have a convergent subsequence in $AH(M)$.

  2. (b)

The sequence of restrictions $\\{m_{i}|\pi_{1}(S_{1,2})\\}$ does not have a
convergent subsequence in $AH(S_{1,2}\times I)$.

  3. (c)

The restrictions $m_{i}|\pi_{1}(S_{1,3})$ converge up to conjugation.

  4. (d)

The restrictions $m_{i}|\pi_{1}(S_{2,4})$ converge up to conjugation.

###### Proof.

Since (a) follows from (b), we shall first prove (b). Take a covering
$\tilde{M}_{1,2}$ of $M$ associated with $\pi_{1}(S_{1,2})$. The open
3-manifold $\tilde{M}_{1,2}$ has a compact core $C_{1,2}$ homeomorphic to
$S_{1,2}\times I$ containing homeomorphic lifts of $\Sigma_{1}\times I$ and
$\Sigma_{2}\times I$. Since convex compact hyperbolic structures are lifted to
convex compact hyperbolic structures (see Proposition 7.1 of Morgan [22]), the
sequence $\\{m_{i}\\}$ is lifted to a sequence $\\{\tilde{m}_{i}\\}$ in
$QH_{0}(S_{1,2}\times I)\subset AH(\tilde{M}_{1,2})$. We shall show that
$\\{\tilde{m}_{i}\\}$ does not have a convergent subsequence.

Recall that $m_{i}$ is obtained from $m_{0}$ by the composition of the
$i$-time iterations of the right Dehn twists along $A_{1}$ and $A_{3}$.
Putting a basepoint $x_{0}$ on $\Sigma_{1}\times\\{0\\}$, we see that every
closed loop representing an element in $\pi_{1}(S_{1,2},x_{0})$ can be
homotoped off $A_{3}$ (fixing the basepoint). On the other hand, the annulus
$A_{1}$ is lifted to an annulus $\tilde{A}_{1}$ properly embedded in
$C_{1,2}$, and $\tilde{m}_{i}$ is obtained from $\tilde{m}_{0}$ by the
$i$-time iteration of the right Dehn twist along $\tilde{A}_{1}$. Since
$\tilde{m}_{0}$ is convex compact, $\tilde{m}_{0}$ corresponds to a quasi-
Fuchsian representation. By the Bers simultaneous uniformisation, there
$\tilde{m}_{0}$ is expressed as $qf(g_{0},h_{0})$, with
$(g_{0},h_{0})\in\mathcal{T}(S_{1,2})\times\mathcal{T}(\bar{S}_{1,2})$. Recall
that $\mathbb{H}^{3}/qf(g_{0},h_{0})$ is homeomorphic to $S_{1,2}\times(0,1)$.
The compact core $C_{1,2}$ is homeomorphic to $S_{1,2}\times[0,1]$, and we can
identify $S_{1,2}$ and $\bar{S}_{1,2}$ with $S_{1,2}\times\\{0\\}$ and
$S_{1,2}\times\\{1\\}$ respectively by homeomorphisms preserving markings,
where for the latter the orientation of $\bar{S}_{1,2}$ is the reverse of the
natural one of $S_{1,2}\times\\{1\\}$. Let $\gamma,\bar{\gamma}$ be simple
closed curve on $S_{1,2}$ and $\bar{S}_{1,2}$ isotopic to
$\partial\tilde{A}_{1}\cap S_{1,2}\times\\{0\\}$ and $\partial A_{1}\cap
S_{1,2}\times\\{1\\}$ respectively. The simple closed curve $\bar{\gamma}$
corresponds to $\gamma$ on $\bar{S}$ by the identification above. By this,
$\tilde{m}_{i}$ is expressed as
$qf(\tau_{\gamma}^{i}(g_{0}),\tau_{\gamma}^{i}(h_{0}))$, where $\tau_{\gamma}$
denotes the right Dehn twist around $\gamma$. Then either by considering the
geometric limit of the conformal structures at infinity, or just applying [27]
or [6], we see that $\\{\tilde{m}_{i}\\}$ does not have a convergent sequence
in $AH(S\times I)$, and hence $\\{m_{i}\\}$ does not have a convergent
sequence in $AH(M)$. This completes the proof of (b), and we are also done
with (a).

The parts (c) and (d) can be proved by the exact same argument. Therefore, we
shall only prove (c). As in the case (a), take a covering $\tilde{M}_{1,3}$
associated with $\pi_{1}(S_{1,3})$. Let $\tilde{m}_{i}$ be the convex compact
hyperbolic structure on $\mathrm{Int}\tilde{M}_{1,3}$ which is the lift of
$m_{i}$. In the present case, both of the annuli $A_{1}$ and $A_{3}$ are
lifted to properly embedded annuli $\tilde{A}_{1}$ and $\tilde{A}_{3}$ in
$\tilde{M}_{1,3}$. We can also take a compact core $C_{1,3}$ containing
homeomorphic lifts of $\Sigma_{1}\times[0,1]$ and $\Sigma_{3}\times[0,1]$. The
right Dehn twist along $A_{1}$ is lifted to the one along $\tilde{A}_{1}$, and
the left Dehn twist along $A_{3}$ is lifted to the one along $\tilde{A}_{3}$.
Now the annuli $\tilde{A}_{1}$ and $\tilde{A}_{3}$ are parallel in $C_{1,3}$,
and hence the right Dehn twist along $\tilde{A}_{1}$ and the left Dehn twist
along $\tilde{A}_{3}$ cancel out each other. Therefore we see that
$\tilde{m}_{i}=\tilde{m}_{0}$ for all $i$, which evidently implies that
$\\{m_{i}|\pi_{1}(S_{1,3})\\}$ converges up to conjugation. ∎

Now we show that the sequence in Proposition 2.1 is a counter-example of the
second claim of Theorem 0.1.

######  Corollary 2.2.

The sequence $\\{m_{i}\\}$ in Proposition 2.1 disproves the second sentence of
Theorem 0.1.

###### Proof.

Recall that $M$ is a primitive book of product $I$-bundles, and its window,
which we denote by $F$, is the union of $\Sigma_{1}^{\prime}\times
I,\dots,\Sigma^{\prime}_{n}\times I$. We denote the fibring of $F$ as an
$I$-bundle by $p\colon F\to B$. Suppose, seeking a contradiction, that the
second sentence of Theorem 0.1 holds. Since $\\{m_{i}\\}$ in Proposition 2.1
diverges, there must be an incompressible subsurface $x$ of the base surface
$B$ with the property of Theorem 0.1 that the restriction of $m_{i}$ to a
subgroup of $\pi_{1}(M)$ converges if and only if $\Gamma$ corresponds to the
fundamental group of a component of $M\setminus X$ for $X=p^{-1}(x)$.

Recall that for each component $\Sigma_{j}^{\prime}\times I$ of $F$, the
restrictions $m_{i}|\pi_{1}(\Sigma_{j}^{\prime}\times I)$ converge. Therefore,
$\Sigma^{\prime}_{j}\times I$ must lie outside $X=p^{-1}(x)$ for every $j$.
This forces $x$ to be a union of annular neighbourhoods of some of the
boundaries $\partial\Sigma^{\prime}_{j}\ (j=1,\dots,n)$. If $x$ contains an
annular neighbourhood of $\partial\Sigma^{\prime}_{1}$, then there is no
component of $M\setminus X$ corresponding the subgroup $\pi_{1}(S_{1,3})$,
contradicting (c) of Proposition 2.1. Therefore, $x$ cannot contain such an
annulus. In the same way, by (c) of Proposition 2.1, $x$ cannot contain an
annular neighbourhood of $\partial\Sigma^{\prime}_{2}$. Now, these show that
$S_{1,2}\times I$ is isotoped into a component of $M\setminus X$. This
contradicts (b) of Proposition 2.1. Thus we are lead to a contradiction. ∎

We next show that Proposition 2.1 gives an example of $\mathbb{R}$-tree action
which is not dual to any codimension-1 annular measured lamination. We note
that this also shows that the second sentence of Theorem 0.1 cannot be
remedied just by taking into account annuli embedded in solid torus pairs,
replacing the window in the statement to the entire characteristic
submanifold. We recall that in two-dimensional setting, by Skora’s theorem,
every isometric action of $\pi_{1}(S)$ on an $\mathbb{R}$-tree with small
stabilisers is dual to a measured lamination.

######  Corollary 2.3.

Let $\\{m_{i}\\}$ be a sequence in $AH(M)$ given in Proposition 2.1. Let
$\rho\colon\pi_{1}(M)\to\mathrm{Isom}(T)$ be a rescaled limit of
$\\{m_{i}\\}$, which is a minimal action on an $\mathbb{R}$-tree $T$ with
small edge-stabilisers as in Theorem 1.1. Then there is no codimension-1
annular measured lamination $L$ in $M$ with an equivariant isomorphism from
the dual tree $T_{L}$ to $T$.

###### Proof.

Suppose, seeking a contradiction, that there is such a codimension-1 measured
lamination $L$. By Proposition 1.2, $L$ can be assumed to lie in the
characteristic submanifold $W$ of $M$. Since the restrictions of $m_{i}$ to
$\pi_{1}(\Sigma_{j})$ converges for every $j=1,\dots,n$, we see that $L$ must
be disjoint from these pages of the book, hence must be contained in the solid
torus $V$. By (c) of Proposition 2.1, $S_{1,3}\times I$ can be homotoped so
that it becomes disjoint from $L$. This implies that core curves of the annuli
$A_{1}$ and $A_{3}$ are homotopic in $M\setminus L$. In the same way, by (d)
of Proposition 2.1, core curves of $A_{2}$ and $A_{4}$ are homotopic in
$M\setminus L$. In the proof of (d), we only used the fact that $m_{i}$ is
defined by the $i$-time iterations of Dehn twists along $A_{1}$ and $A_{3}$,
and that $S_{2,4}$ can be homotoped off from them. Therefore, the same
argument works also for $S_{2,j}$ with $j=5,\dots,n$. This implies that $L\cap
V$ must be empty. Since $\\{m_{i}\\}$ diverges, $L$ itself cannot be empty.
This is a contradiction. ∎

##  3\. A weaker version of Thurston’s theorem

As can be seen in the proof of Corollary 2.2, the existence of a solid torus
pair in $M$ is the cause of the problem. On the other hand, such a solid torus
gives rise to a homotopy equivalence by shuffling. In this section, we shall
show by moving from $M$ to a homotopy equivalent manifold, and taking into
account not only the window but also solid torus pairs and thickened torus
pairs of the characteristic submanifold, we can prove a weak version of
Thurston’s original theorem, stated as Theorem 0.2. Before starting to prove
this theorem, we prove the following theorem showing that a minimal isometric
action of $\pi_{1}(M)$ with small edge stabilisers on an $\mathbb{R}$-tree is
isomorphic to an action on an $\mathbb{R}$-tree dual to an annular measured
lamination in a 3-manifold homotopy equivalent to $M$. This should be
contrasted with Corollary 2.3.

######  Theorem 3.1.

Let $(M,P)$ be a pared manifold as given in Theorem 0.1. Let
$\rho\colon\pi_{1}(M)\to\mathrm{Isom}(T)$ be a minimal isometric action on an
$\mathbb{R}$-tree $T$ with small edge stabilisers such that every element in
$\pi_{1}(M)$ conjugate into the fundamental group of a component of $P$ is
mapped to the identity by $\rho$. Then there is a codimension-1 annular
measured lamination $L^{\prime}$, with weighted annuli with singular axes
allowed, in a pared manifold $(M^{\prime},P^{\prime})$ obtained from $(M,P)$
by shuffling, and an equivariant isomorphism from the dual tree
$T_{L^{\prime}}$ to $T$.

###### Proof.

By Theorem 1.3, there is a codimension-1 annular measured lamination $L$ in
$M$ such that there is a $\pi_{1}(M)$-equivariant morphism $f$ from the dual
tree $T_{L}$ to $T$. For any component $P_{0}$ of $P$, an element in
$\pi_{1}(P_{0})$ acts trivially on $T$. Therefore we can choose $L$ which is
disjoint from $P$ as stated in Theorem 1.3.

Let $W_{0}$ be an $I$-pair in the characteristic submanifold $W$ which
contains a component of $L$. Then $W_{0}$ is an $I$-bundle over a surface
$\Sigma$. We denote this fibration by $p\colon W_{0}\to\Sigma$. By Proposition
1.2, $L_{0}:=L\cap W_{0}$ can be assumed to be vertical with respect to $p$.
Let $\tilde{W}_{0}$ be a component of the preimage of $W_{0}$ in $\tilde{M}$
which is invariant under $\pi_{1}(W_{0})\subset\pi_{1}(M)$. Let $T^{W_{0}}$ be
the sub-tree of $T$ corresponding to the rescaled limit of $\tilde{W}_{0}$,
and $T_{L}^{W_{0}}$ the subtree of $T_{L}$ corresponding to $\tilde{W}_{0}$.
Then the restriction $f|T_{L}^{W_{0}}$ is a $\pi_{1}(W_{0})$-equivariant
morphism. Since $\pi_{1}(W_{0})$ is isomorphic to $\pi_{1}(\Sigma)$, and an
element of $\pi_{1}(\Sigma)$ corresponding to a boundary component of $\Sigma$
acts on $T$ trivially, we can apply Theorem 1.4, and see that $T^{W_{0}}$ is
$\pi_{1}(W_{0})$-equivariantly isometric to an $\mathbb{R}$-tree
$T^{{}^{\prime}W_{0}}$ which is dual to an annular codimension-1 measured
lamination $L^{W_{0}}$ in $W_{0}$. We replace $L\cap W_{0}$ with $L^{W_{0}}$
and hence $T_{L}^{W_{0}}$ with $T^{{}^{\prime}W_{0}}$. Then $f|T^{W_{0}}_{L}$
can be replaced with the equivariant isometry between $T^{{}^{\prime}W_{0}}$
and $T^{W_{0}}$. At the same time, every image of $T_{L}^{W_{0}}$ under the
action of $\pi_{1}(M)$ and the restriction of $f$ there are modified. Thus we
get an $\mathbb{R}$-tree $T^{\prime}$ and a morphism $f^{\prime}\colon
T^{\prime}\to T$ which is isometry on the orbit of the subtree corresponding
to $\tilde{W}_{0}$ which is dual to $L^{W_{0}}$. Repeating this modification
for each $I$-pair of $W$ intersecting $L$, we get an $\mathbb{R}$-tree
$\hat{T}$ and a morphism $\hat{f}\colon\hat{T}\to T$ with an annular
codimension-1 measured lamination $\hat{L}$ to which $\hat{f}$ is dual. The
morphism $\hat{f}$ is isometric outside the part corresponding to the preimage
of the solid torus pairs and thickened torus pairs in $W$.

Now let $V$ be either a solid torus pair or a thickened torus pair in $W$.
Since $\hat{L}$ is annular, by isotopying it, we can assume that
$V\cap\hat{L}$ consists of properly embedded incompressible annuli whose
boundaries lie in $V\cap(\partial M\setminus P)$, and that no two components
of $V\cap\hat{L}$ are parallel as embeddings in $(V,V\cap(\partial M\setminus
P))$. In the case when $V$ is a solid torus, we can also assume that it is
disjoint from a core curve of $V$. By isotoping $V\cap\hat{L}$, we can regard
$V$ as a Seifert fibred manifold with base surface $D$, which is a disc with
singular point $\mathfrak{c}$ of order $p$ when $V$ is a solid torus and
annulus when $V$ is a thickened torus in such a way that these annuli
$V\cap\hat{L}$ are all vertical with respect to the fibration $\pi\colon V\to
D$. We denote the union of arcs on $\partial D$ which is the image of
$p(V\cap(\partial M\setminus P))$ by $A$

The subtree $T_{V}$ of $\hat{T}$ corresponding to a component of the preimage
of $V$ in $\tilde{M}$ which is invariant under $\pi_{1}(V)$ is a simplicial
tree each of whose edges is dual to a component of $V\cap\hat{L}$ and has
length equal to the weight given to that component. We note that the action of
the group $\mathbb{Z}$ corresponding to a regular fibre is trivial. In the
case when $V$ is a solid torus pair, and $p>1$, the cyclic group
$\mathbb{Z}_{p}$ acts on $T_{V}$ by isometries, whereas in the case when $V$
is a thickened torus, the infinite cyclic subgroup $\mathbb{Z}$ acts on
$T_{V}$ by isometries. Both of them are quotients of $\pi_{1}(V)$ by a normal
subgroup generated by a regular fibre. In the latter case, the stabiliser of
the group $\mathbb{Z}$ consists of one point since there is no leaf of
$\hat{L}$ which is a torus. We modify $T_{V}$ and $\hat{f}|T_{V}$, by removing
folds as in the proof of Skora’s theorem [30], and make $\hat{f}|T_{V}$ an
embedding to $T$, whose image we denote by $T^{\prime}_{V}$. We can do this
equivariantly with respect to the action of either $\mathbb{Z}_{p}$ or
$\mathbb{Z}$. Then $T^{\prime}_{V}$ is a subtree of $T$ on which either
$\mathbb{Z}_{p}$ or $\mathbb{Z}$ acts by isometries. We consider the quotient
$\bar{T}_{V}$ of $T^{\prime}_{V}$ by this isometric action of $\mathbb{Z}_{p}$
or $\mathbb{Z}$, which is a finite tree.

Each of the endpoints of $\bar{T}_{V}$, i.e. each vertex of valency $1$,
corresponds to a component (or components) of $\operatorname{Fr}V$.
Conversely, each component of $\operatorname{Fr}V$ corresponds to a vertex of
$\bar{T}_{V}$, but not necessarily an endpoint. We give numbers to the
components of $\operatorname{Fr}V$ as $A_{1},\dots,A_{n}$ so that the
subscripts coincide with the cyclic order of these components on $\partial V$.
Correspondingly, we number the associated vertices of $\bar{T}_{V}$. Note that
it is possible that more than one component of $\partial V$ correspond to the
same vertex, and hence the vertex has more than one number. We also note that
in the case when $V$ is a thickened torus pair, there is a vertex $v_{P}$ in
$\bar{T}_{V}$ corresponding to the torus component of $V\cap\partial M$ which
we denote by $P_{0}$. In the case when $V$ is a solid torus and $p>1$, there
is a subtree $t_{c}$ of $\bar{T}_{V}$ which is the quotient of the stabiliser
of the $\mathbb{Z}_{p}$ action.

We realise $\bar{T}_{V}$ in the base surface $D$ so that the vertices numbered
lie on $\partial D$. In the case when $V$ is a solid torus pair with $p>1$, we
collapse $t_{c}$ into $\mathfrak{c}$. In the case when $V$ is a thickened
torus pair, we assume that $v_{P}$ lies on $P_{0}$. Then it can be regarded as
a tree dual to a partition of a disc (possibly with a cone point) or an
annululs $D$ by disjoint weighted simple arcs $\alpha$ with endpoints at
$A\cup\\{\mathfrak{c}\\}$, that is, each region of the complement corresponds
to a vertex, and each arc corresponds to an edge and has weight equal to the
length of the edge. If an edge has a numbered vertex, the dual arc should cut
off components of $A$ which are the images of components of $V\cap(\partial
M\setminus P)$ having the numbers given to the vertices. (See Figure 1.) In
the case when $t_{c}$ contains an edge, each edge is dual to a weighted simple
proper arc passing through $\mathfrak{c}$, and hence corresponds to a vertical
annulus containing the singular axis at the centre. Then the union of such
weighted arcs is realised as a union of weighted arcs connecting $\partial D$
to $\mathfrak{c}$ intersecting only at $\mathfrak{c}$.

Figure 1. The tree $T^{\prime}_{V}$ and its dual disc when $V$ is a thickened
torus.

We perform shuffling around $V$ so that the order of the components of
$\operatorname{Fr}V$ coincides with the order of the corresponding vertices of
$\bar{T}_{V}$ on $D$. Then we define a new annular codimension-1 lamination in
$V$ to be the union of vertical annuli (possibly with singular axes) over the
weighted union of arcs $\alpha$. We note that if one vertex of $\bar{T}_{V}$
has more than one number, it does not matter how to perform shuffling among
them. Repeating this operation for every torus pair and thickened torus pair
intersecting $\hat{L}$, we have a pared manifold $(M^{\prime},P^{\prime})$
obtained from $M$ by shuffling without modifying the paring locus $P$, and an
annular codimension-1 measured lamination $L^{\prime}$ (possibly containing
annuli with singular axes) in $M^{\prime}$ such that there is a
$\pi_{1}(M^{\prime})=\pi_{1}(M)$-equivariant isometry from the dual tree
$T_{L^{\prime}}$ of $L^{\prime}$ to $T$. ∎

To derive Theorem 0.2 from Theorem 3.1, we need some facts relating three-
dimensional hyperbolic geometry and two-dimensional one. We first prove the
following rather elementary fact about degeneration of hyperbolic structures
on surfaces. This can be regarded as a two-dimensional analogue of the
problematic second sentence of Theorem 0.1.

######  Lemma 3.2.

Let $S$ be an orientable surface of hyperbolic type, and let $\\{m_{i}\\}$ be
a sequence in $\mathcal{T}(S)$. Then, passing to a subsequence, there is a
possibly disconnected subsurface $S^{\prime}$ of $S$ such that
$\\{m_{i}|S^{\prime}\\}$ converges in $\mathcal{T}(S^{\prime})$ and such that
$S^{\prime}$ is maximal up to isotopy among the subsurfaces with the same
property.

###### Proof.

Let $\alpha_{i}$ be a sequence of shortest pants decomposition of $(S,m_{i})$,
each of which is regarded as a collection of simple closed geodesics. Passing
to a subsequence, we can assume that there is a sub-collection $\beta_{i}$ of
$\alpha_{i}$ which is constant, and no component of
$\alpha_{i}\setminus\beta_{i}$ has a constant subsequence. Since $\beta_{i}$
is constant, we denote it by $\beta$. We take a union of $\beta$ and all pairs
of pants in $S\setminus\alpha_{i}$ all of whose boundary components belong to
$\beta$, and denote it by $\Sigma$.

Let $\Sigma_{0}$ be a component of $\Sigma$, and denote $\beta\cap\Sigma$ by
$\beta_{0}$. For a component $b$ of $\beta_{0}$, by passing to a subsequence,
we can assume that either $\mathrm{length}_{m_{i}}(b)\to 0$ or
$\mathrm{length}_{m_{i}}(b)$ converges to a positive constant. In the first
case, we cut $\Sigma_{0}$ along $b$. In the second case, passing to a
subsequence again, the twisting parameter (with respect to the Fenchel-Nielsen
coordinate) around $b$ can be assumed to converge or goes to $\pm\infty$. In
the latter case, we cut $\Sigma_{0}$ along $b$. Repeating this for all
components of $\beta_{0}$, we get a subsurface $\Sigma_{1}$ of $\Sigma_{0}$,
and $m_{i}|\Sigma_{1}$ converges in $\mathcal{T}(\Sigma_{1})$. Taking the
union of such $\Sigma_{1}$ for all components $\Sigma_{0}$ of $\Sigma$, we
obtain $S^{\prime}$. The maximality follows from our construction easily. ∎

Let $(S,m)$ be a hyperbolic surface of finite area and $M$ be a complete
hyperbolic 3-manifold. A pleated surface $f\colon S\to M$ is a continuous map
such that for each point $x$ on $S$, there is at least one segment $s$
containing $x$ at its interior such that $f|s$ is totally geodesic. The set
consisting of points where there is exactly one germ for such segments is
called the pleating locus of $f$, and it constitutes a geodesic lamination on
$(S,m)$. The following is a consequence of ‘efficiency of pleated surfaces’
proved in Theorems 3.3 and 3.4 in [36] combined with Lemma 3.2.

######  Proposition 3.3.

Let $(M,P)$ be a pared manifold, and let $\rho_{i}\in AH(M,P)$ be a sequence.
Then for every component $S$ of $\partial M\setminus P$, there is a possibly
disconnected subsurface $S^{\prime}$ such that
$\\{\rho_{i}|\pi_{1}(S^{\prime})\\}$ converges up to conjugacy, and
$S^{\prime}$ is maximal up to isotopy among the subsurfaces of $S$ with this
property. If $S^{\prime}$ intersects essentially a component $T$ of $\partial
M\setminus W$ for the characteristic submanifold $W$, then $S^{\prime}$
contains the entire $T$ (up to isotopy).

###### Proof.

Let $f_{i}\colon(S,m_{i})\to\mathbb{H}^{3}/\rho_{i}(\pi_{1}(M))$ be a pleated
surface taking each frontier of $S$ to a cusp, which is efficient in the
following sense:

  * (*)

For any simple closed curve $\gamma$ on $S$, the translation length of
$\rho_{i}(\gamma)$ is bounded if and only
$\\{\mathrm{length}_{m_{i}}(\gamma)\\}$ is bounded, where
$\mathrm{length}_{m_{i}}$ denotes the geodesic length.

The existence of such a sequence of pleated surfaces was proved in Theorem 3.3
of Thurston [36]. (We need to take a covering associated with $\pi_{1}(S)$ to
apply this result of Thurston.) Now the first statement of our proposition
follows from Lemma 3.2. Combining this with the first part of Theorem 0.1, we
obtain the second statement. ∎

Although this surface $S^{\prime}$ is determined only up to isotopy, we always
assume that no frontier component of $S^{\prime}$ is isotopic into $P$, and
any component $T$ of $\partial M\setminus W$ intersecting $S^{\prime}$ is
contained in $S^{\prime}$.

We need to refine Proposition 3.3 for twisted $I$-pairs in the characteristic
submanifold as follows.

######  Corollary 3.4.

In the setting of Proposition 3.3, suppose that there is a twisted $I$-pair
$(W_{0},W_{0}\cap\partial M)$ in the characteristic submanifold of $(M,P)$,
whose base surface we denote by $\Sigma_{0}$. Let $S$ be a component of
$\partial M\setminus P$ containing $W_{0}\cap\partial M$. Then the subsurface
$S^{\prime}$ as obtained in Proposition 3.3 doubly covers a subsurface of
$\Sigma_{0}$.

###### Proof.

We can consider a non-orientable efficient pleated surface
$g_{i}\colon\Sigma_{0}\to\mathbb{H}^{3}/\rho_{i}(\pi_{1}(M))$. Consider a
double cover $\tilde{M}$ of $M$ into which $W_{0}$ is lifted to a product
$I$-bundle. Then $g_{i}$ is lifted to an orientable pleated surface
$\tilde{g}_{i}\colon(\tilde{\Sigma}_{0},n_{i})\to\mathbb{H}^{3}/\rho_{i}(\pi_{1}(\tilde{M}))$,
and both the length with respect to $n_{i}$ and the translation length with
respect to $\rho_{i}$ are invariant under the covering involution. This
implies Corollary 3.4. ∎

###### Proof of Theorem 0.2.

We have only to consider the case when $\\{\rho_{i}\\}$ diverges in $AH(M,P)$.
(Otherwise we can just take $X$ to be empty.) Let $W$ be the characteristic
submanifold of $(M,P)$. For every component $S$ of $\partial M\setminus P$,
there is a subsurface $S^{\prime}$ as was given in Proposition 3.3. Since any
component of $\partial M\setminus W$ that intersects $S^{\prime}$ must be
entirely contained in $S^{\prime}$, every frontier component of $S^{\prime}$
is contained in either an $I$-pair or a solid torus pair or a thickened torus
pair.

We only consider the frontier components contained in $I$-pairs. Let $c$ be
such a frontier component of $S^{\prime}$ contained an $I$-pair $W_{0}$. If
$W_{0}$ is a product $I$-pair, then there is a simple closed curve
$c^{\prime}$ on $W_{0}\cap\partial M$ such that $c\cup c^{\prime}$ bounds an
essential (vertical) annulus $A$ in $W_{0}$. By our definition of
$S^{\prime}$, if $c^{\prime}$ is contained in $S$, then it must also be a
frontier component of $S^{\prime}$. Otherwise, let $\bar{S}$ be a component of
$\partial M\setminus P$ containing $c^{\prime}$. Then we have a subsurface
$\bar{S}^{\prime}$ of $\bar{S}$ corresponding to $S^{\prime}$ in Proposition
3.3, and $c^{\prime}$ is a frontier component of $\bar{S}^{\prime}$.

If $W_{0}$ is a twisted $I$-pair, then by Corollary 3.4, either $c$ is
homotopic to $c^{\prime}$ as above and $c\cup c^{\prime}$ bounds an annulus
$A$, or $c$ doubly covers an orientation-reversing curve in the base surface.
In the latter case, $c$ bounds an essential one-sided Möbius band, which we
also denote by $A$.

We remove from $M$ thin regular neighbourhoods of all annuli and Möbius bands
$A$ as above for all frontier components of $S^{\prime}$ contained in
$I$-pairs, and obtain a 3-submanifold $M_{0}$ of $M$. We let the union of $P$
and the frontier components of $M_{0}$ in $M$, the latter of which are all
annuli, be a paring locus $P_{0}$ of $M_{0}$. By our construction, it is easy
to check that $(M_{0},P_{0})$ is a pared manifold, and that $M\setminus M_{0}$
is contained in the union of $I$-pairs in $W$, and can be assumed to be
vertical with respect to the fibration. We let the projection of $M\setminus
M_{0}$ to the base surface be $x$ in our statement, and hence $X$ coincides
with $M\setminus M_{0}$.

Now restricting $\rho_{i}$ to $(M_{0},P_{0})$, we get a sequence
$\\{\rho_{i}^{\prime}\\}$ of $\pi_{1}(M_{0})$. If $\\{\rho_{i}\\}$ converges
after passing to a subsequence, then by letting $\mathcal{A}$ be empty and
setting $M^{\prime}=M$, we are done. Suppose that $\\{\rho_{i}^{\prime}\\}$
does not have a convergent subsequence. We note that the translation length of
every curve on a component of $P_{0}$ with respect to $\rho_{i}$ is bounded,
by our definition of $S^{\prime}$ in Proposition 3.3. Therefore, we can apply
Theorem 3.1, and there are a pared manifold $(M_{0}^{\prime},P_{0})$ obtained
from $(M_{0},P_{0})$ by shuffling and a codimension-1 annular measured
lamination $L_{0}$ properly embedded in $(M_{0}^{\prime},P_{0})$ whose dual
tree is equivariantly isometric to the Gromov limit of the action of
$\pi_{1}(M_{0})$ on rescaled $\mathbb{H}^{3}$. By performing the same
shuffling that gives $M_{0}^{\prime}$ from $M_{0}$, we obtain a pared manifold
$(\hat{M}_{0},P)$ homotopy equivalent of $(M,P)$.

By our definition of $M_{0}$, and by Proposition 3.3, the lamination $L_{0}$
is disjoint from the $I$-pairs in the characteristic submanifold of
$(M_{0}^{\prime},P_{0})$, and hence contained in the union of solid torus
pairs and the thickened torus pairs. This means that $L_{0}$ is a union of
weighted incompressible annuli, possibly with singular axes. We let
$\mathcal{A}_{0}$ be the support of $L_{0}$. Now, we cut $M_{0}^{\prime}$
along $\mathcal{A}_{0}$, and obtain a 3-manifold $M_{1}$. In the case when
$\mathcal{A}_{0}$ contains an annulus with singular axis, we regard
$M_{0}^{\prime}$ as containing a regular neighbourhood of the singular axis on
both sides. By letting $P_{1}$ be the union of $\mathcal{A}_{0}\cup(P_{0}\cap
M_{1})$ when there are no singular axes, and when a component of
$\mathcal{A}_{0}$ has a singular axis, we let the intersection of the regular
neighbourhood of the singular axis with the boundary be contained in $P_{1}$,
we get a pared manifold $(M_{1},P_{1})$. We consider restrictions of
$\rho_{i}$ to $\pi_{1}(M_{1})$, and if they converge passing to a subsequence,
we let $\mathcal{A}$ be $\mathcal{A}_{0}$, setting $(M^{\prime},P^{\prime})$
to be $(\hat{M}_{0},P)$. Otherwise, we repeat the same procedure for
$(M_{1},P_{1})$ as for $(M_{0},P_{0})$, and obtain the next pared manifold
$(M_{2},P_{2})$ contained in a pared manifold $(\hat{M}_{1},P)$ obtained by
shuffling from $(\hat{M}_{0},P)$. Since there are only finitely many disjoint
incompressible annuli in $M$, this process terminates in finite steps, and
there is $n_{0}$ such that $\rho_{i}|\pi_{1}(M_{n_{0}})$ converges. Then we
set $(\hat{M}_{n_{0}-1},P)$ to be $(M^{\prime},P^{\prime})$, and
$\mathcal{A}=\cup_{j=0}^{n_{1}}\mathcal{A}_{j}$ regarded as contained in
$(M^{\prime},P^{\prime})$. Then together with $x$ which we defined above, also
regarded as contained in $(M^{\prime},P^{\prime})$, we are done. ∎

##  4\. Implication of the bounded image theorem

To explain the significance of Theorem 0.1 in the proof of Thurston’s
uniformisation theorem for Haken manifolds, we first review the overall
structure of the proof. Let $M$ be a closed atoroidal Haken manifold. In
general, we should consider a pared manifold $(M,P)$ and show the existence of
a geometrically finite hyperbolic structure, but for simplicity, here we
assume that $M$ is closed, and hence $P=\emptyset$.

By Waldhausen’s result [38], a Haken manifold admits a ‘hierarchy’, that is, a
finite sequence of compact 3-manifolds $M=M_{0},M_{1},\dots,M_{n}$ such that
$M_{j+1}$ is obtained by cutting $M_{j}$ along a non-peripheral incompressible
surface, and $M_{n}$ is a disjoint union of 3-balls. The basic strategy of the
proof of the uniformisation theorem is to construct a convex hyperbolic
structure of $M_{j}$ from that of $M_{j+1}$. Topologically $M_{j}$ is obtained
from $M_{j+1}$ by pasting an incompressible subsurface on the boundary of
$M_{j+1}$ to another incompressible subsurface disjoint from the first one.
For a general step, we need to separate the boundary of $M_{j+1}$ into two
parts: the part which is glued each other to get $M_{j}$, and the part which
remains to be the boundary of $M_{j}$. This makes the argument complicated.
Therefore, we here consider only the last step in which each boundary
component $M_{1}$ is glued to another component to get $M_{0}$.

In the logical structure of the proof given by Thurston in the form of a chart
on the page 206 of [35], Theorem 0.1 is meant to be used to prove the ‘bounded
image theorem’, which is a key step for showing that the ‘skinning map’
defined below has a fixed point in the case when $M_{1}$ is not an $I$-bundle
over a surface. The bounded image theorem in the original version is as below,
restricted to the case which we are considering, i.e. the case when the paring
locus is empty.

To state the theorem, we need to define the skinning map. Let $M$ be a
3-manifold admitting a convex compact hyperbolic metric. Let
$q\colon\mathcal{T}(\partial M)\to QH_{0}(M)$ be the parametrisation of
Marden. For each element of $QH_{0}(M)$, by taking the covering associated
with a component $S$ of $\partial M$, we get a quasi-Fuchsian representation,
and by the Bers simultaneous uniformisation, we have a point in
$\mathcal{T}(S)\times\mathcal{T}(\bar{S})$. We consider the projection to the
second coordinate $\mathcal{T}(\bar{S})$, and taking the product by
considering all the components of $\partial M$, we have a map
$\sigma\colon\mathcal{T}(\partial M)\to\mathcal{T}(\bar{\partial}M)$, where
the latter denotes the Teichmüller space of $\partial M$ with its orientation
reversed.

######  Theorem 4.1 (The bounded image theorem).

Let $M$ be an atoroidal Haken manifold with incompressible boundary admitting
a convex compact hyperbolic metric, and $\tau\colon\partial M\to\partial M$ be
an orientation-reversing involution taking each component of $\partial M$ to
another component. Suppose that $N=M/\tau$ is also atoroidal. Let
$\sigma\colon\mathcal{T}(\partial M)\to\mathcal{T}(\bar{\partial}M)$ be the
skinning map. Then there is $n_{0}\in\mathbb{N}$ depending only on the
topological type of $M$ such that
$(\tau_{*}\circ\sigma)^{n}\mathcal{T}(\partial M)$ is precompact for every
$n\geq n_{0}$.

There are three important books which explain the proof of the uniformisation
theorem in detail: Morgan [22], Kapovich [13] and Otal [28]. In the first two
of them, the bounded image theorem is mentioned, but it is noted that its
proof is unknown to the authors. Instead, they gave a proof of a weaker
version of the theorem, which should be called the ‘bounded orbit theorem’ as
follows. This theorem was first stated in Thurston’s preprint [34]. To prove
this, only the first part of Theorem 0.1, which is valid, and some argument
involving the ‘covering theorem’ are necessary. In the third book [28], the
author invokes McMullen’s results [20, 21], which implies Theorem 4.2 without
even using the first part of Theorem 0.1.

######  Theorem 4.2 (The bounded orbit theorem).

In the setting of Theorem 4.1, let $g$ be any point in $\mathcal{T}(\partial
M)$. Then the sequence $\\{(\tau\circ\sigma)^{n}_{*}(g)\\}$ has a convergent
subsequence in $\mathcal{T}(\partial M)$.

Quite recently, Lecuire and the present author [16] succeeded in proving the
bounded image theorem in the original general form, in particular Theorem 4.1,
making use of modern technique of Kleinian group theory, but not using the
second part of Theorem 0.1. In the process of proving this, it has become
possible to imagine what was Thurston’s plan to prove the bounded image
theorem. As far as we understand it, his proof essentially relies on the
second part of Theorem 0.1. We here try to recover the ‘proof’ which we
speculate that he should have had in mind.

The ‘proof’ is by contradiction. By the continuity of $\tau_{*}\circ\sigma$,
we see that if $(\tau_{*}\circ\sigma)^{n}\mathcal{T}(S)$ is precompact, then
so is $(\sigma\circ\tau)^{m}$ for every $m\geq n$. Therefore, what we have to
exclude is the case when $(\tau_{*}\circ\sigma)^{n}_{*}\mathcal{T}(S)$ is
unbounded for every $n$. If the image
$(\tau_{*}\circ\sigma)^{n}\mathcal{T}(S)$ is unbounded for every $n$, there
must exist a sequence $\\{m_{i}\\}$ in $\mathcal{T}(S)$ such that
$\\{(\tau_{*}\circ\sigma)^{n}m_{i}\\}_{i}$ diverges in $\mathcal{T}(S)$ for
every $n$. Then, passing to a subsequence, there are two possibilities to
consider: the first is when $\\{(q(\tau_{*}\circ\sigma)^{n}(m_{i})\\}_{i}$
diverges in $AH(M)$; the second is when
$\\{q((\tau_{*}\circ\sigma)^{n}(m_{i})\\}_{i}$ converges in $AH(M)$.

For the first possibility, the second part of Theorem 0.1 is essential to
reach a contradiction. We assumed that
$\\{(q(\tau_{*}\circ\sigma)^{n}(m_{i})\\}_{i}$ diverges in $AH(M)$ for every
$n$. Then, for a fixed $n$, provided that the second part of Theorem 0.1 were
valid, it would give us a collection $\mathcal{A}_{n}$ of incompressible
annuli which separates $M$ into ‘converging part’ and ‘diverging part’ for
every $n$. It can be proved that this division into two parts is preserved
under $\tau$, by the argument similar to the one employed in §16, in
particular Proposition 16.5 of [13]. Therefore for each annulus in
$\mathcal{A}_{n}$, its boundary is mapped by $\tau$ to a simple closed curve
which lies on the boundary of an annulus in $\mathcal{A}_{n+1}$. We see that
by continuing to connect an annulus in $\mathcal{A}_{n}$ to that of
$\mathcal{A}_{n+1}$ along a boundary starting from $n=1$, we have either an
incompressible torus or an incompressible Klein bottle in $N$, contradicting
the assumption that $N$ is atoroidal.

Now, suppose that $\\{q(\tau_{*}\circ\sigma)^{n}(m_{i})\\}$ converges in
$AH(M)$ as $i\to\infty$ for every $n$. Then the argument is quite similar to
that of the bounded orbit theorem explained in [13], or the proof of the
bounded image theorem for the special case when $M$ is acylindrical explained
in [14]. We can consider the limit representation
$\phi_{\infty}\colon\pi_{1}(M)\to{\rm PSL}_{2}(\mathbb{C})$ of
$\\{q(\sigma\circ\tau)^{n}(m_{i})\\}$ as $i\to\infty$ for each $n$. Then
either $\phi_{\infty}(\pi_{1}(M))$ has a parabolic element or
$\mathbb{H}^{3}/\phi_{\infty}(\pi_{1}(M))$ has a totally degenerate end, i.e.
, an end with a neighbourhood of a form $S\times(0,\infty)$ for a closed
surface $S$ which is entirely contained in the convex core. In the latter
case, there is a sequence of pleated surfaces $f_{k}\colon
S\to\mathbb{H}^{3}/\phi_{\infty}(\pi_{1}(M))$ tending to the end as
$k\to\infty$. These facts are proved by Bonahon [5] in more general setting,
but for the present case, there is an explanation by Thurston himself in §9 of
his lecture notes [32].

In the first case when $\phi_{\infty}(\pi_{1}(M))$ has a parabolic element,
there must be a simple closed curve $c$ on $\partial M$ representing a
parabolic element. (This follows from the fact that otherwise the limit
$\rho_{\infty}$ is a strong limit.) This will force $\tau(c)$ to be homotopic
in $M$ to a another simple closed curve $c^{\prime}$ on $\partial M$
homotopically distinct from $\tau(c)$, which implies that $\tau(c)$ and
$c^{\prime}$ bound an incompressible annulus in $M$. We note here that we
cannot prove this by considering the limit group $\phi_{\infty}(\pi_{1}(M))$
since $\mathbb{H}^{3}/\phi_{\infty}(\pi_{1}(M))$ may not be homeomorphic to
the interior of $M$ although they are homotopy equivalent. Therefore we need
to work in $\mathbb{H}^{3}/q(\tau_{*}\circ\sigma)^{n}(m_{i})$ for very large
$i$. The same argument can be found in §17.2.3 in [13]. Once we have such an
annulus, we repeat the same argument as the one which we used for the case
when $\\{q(\tau_{*}\circ\sigma)^{n}(m_{i})\\}$ diverges, and reach a
contradiction to the assumption that $N$ is atoroidal.

We next consider the latter case when
$\mathbb{H}^{3}/\phi_{\infty}(\pi_{1}(M))$ has a totally degenerate end. Then
we see that there must be one corresponding to a boundary component $S$ of
$M$, for under the assumption that there is no parabolic element in
$\phi_{\infty}(\pi_{1}(M))$, the convergence must be strong. By the covering
theorem, which first appeared in §9 of [32] (a generalised version can be
found in Canary [8]), this implies that $M$ must have a boundary component
which is homotopic to $\tau(S)$, but is distinct from $\tau(S)$. This is
impossible since $M$ is not a product $I$-bundle over a closed surface.

Thus we have reached a contradiction in either case, and this would give a
proof of Theorem 4.1 if the second part of Theorem 0.1 were valid. On the
other hand, a weaker version as we showed in Theorem 0.2 is not sufficient for
the proof of Theorem 4.1 since to construct a torus or a Klein bottle to reach
a contradiction, we need annuli embedded in $M$, not those embedded in a
homotopy equivalent manifold $M^{\prime}$. This observation also highlights
why a proof of the bounded image theorem given in [16] without relying on the
second part of Theorem 0.1 needed much sophisticated technique of Kleinian
group theory.

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