# Realisation of bending measured laminations by Kleinian surface groups

Shinpei Baba and Ken’ichi Ohshika  S.B.: Department of Mathematics, Graduate
School of Science, Osaka University, Toyonaka, Osaka, Japan
<EMAIL_ADDRESS>K.O.: Department of Mathematics, Faculty of
Science, Gakushuin University, Toshima-ku, Tokyo 171-8588, Japan
<EMAIL_ADDRESS>
###### ?abstractname? .

For geometrically finite Kleinian surface groups, Bonahon and Otal proved the
existence part, and partly the uniqueness part of the bending lamination
conjecture. In this paper, we generalise the existence part to general
Kleinian surface groups including geometrically infinite ones. Along the way,
we also prove the compactness of the set of Kleinian surface groups realising
an arbitrarily fixed data of bending laminations and ending laminations. Our
proof is independent of that of Bonahon and Otal.

##  1\. Introduction

The simultaneous uniformisation theorem by Bers [2] gives a parametrisation of
the quasi-Fuchsian space for a closed oriented surface $S$ by the product of
two Teichmüller spaces $\mathcal{T}(S)\times\mathcal{T}(\bar{S})$, where
$\bar{S}$ denotes $S$ with its orientation reversed. This was generalised by
the work of Kra, Maskit, Marden and Sullivan, which shows that for any
Kleinian surface group, or more generally for any freely indecomposable
Kleinian group $G$, its quasi-conformal deformation space is parametrised by
$\mathcal{T}(\Omega_{G}/G)$, where $\Omega_{G}$ is the region of discontinuity
of $G$ in the Riemann sphere.

In his lecture notes [24], Thurston considered the convex core of the quotient
hyperbolic 3-manifold $\mathbb{H}^{3}/G$ for a Kleinian (surface) group $G$.
He noticed that the boundary of the convex core has two pieces of information:
the hyperbolic structure and the bending lamination. In contrast to the
previous work of Bers et al, these are obtained by just considering the
quotient hyperbolic manifolds, without looking at structures at infinity. He
seems to have conjectured that both hyperbolic structures and bending
laminations on the boundaries serve as other kinds of parametrisation of the
quasi-Fuchsian space, or more generally, the quasi-conformal deformation space
of a freely indecomposable Kleinian group.

As for the first of these two, the hyperbolic structures on the boundaries of
convex cores, Sullivan’s lemma (see e.g. [9]) shows that they are within
universally bounded distance in the corresponding Teichmüller spaces from the
conformal structures at infinity on $\Omega_{G}/G$. Nevertheless, it is still
unknown if they really give a parametrisation of the deformation space.

As for bending laminations, Bonahon-Otal showed in [4] that every pair of
measured laminations on a closed orientable surface $S$, without homotopic
components and without compact leaves with weight larger than or equal to
$\pi$, can be realised as bending laminations of a quasi-Fuchsian group
corresponding to $S$. In particular, when both of the measured laminations are
weighted multi-curves, it was proved that the realising quasi-Fuchsian group
is unique (up to conjugation). They also showed the same result for quasi-
conformal deformation spaces of general freely indecomposable geometrically
finite groups. Their result was generalised to freely decomposable Kleinian
groups by Lecuire [12].

In this paper, we prove a generalisation of the existence part of this result
by Bonahon-Otal to general Kleinian surface groups including geometrically
infinite ones (Theorem 3.1-(1)), by adding ending laminations to the data. We
shall furthermore prove the compactness of the set of representations (up to
conjugacy) realising given bending laminations and ending laminations (Theorem
3.1-(2)). The proofs of both are different from and independent of the results
by Bonahon-Otal. On the other hand, we do not have the partial uniqueness
result as was given by Bonahon-Otal, for we cannot invoke the theory of cone
manifold deformation, whose generalisation to the case of geometrically
infinite groups does not exist for the moment.

We shall prove the existence part and the compactness by showing the
properness of the following composition of maps in Theorem 3.3. For a Kleinian
surface group $G$, the theory of Bers-Kra-Maskit-Marden-Sullivan gives a
parametrisation of the quasi-conformal deformation space
$q\colon\mathcal{T}(\Omega_{G}/G)\to\mathsf{QH}(G)$, where $\Omega_{G}$
denotes the region of discontinuity of $G$, $\mathcal{T}(\Omega_{G}/G)$ the
Teichmüller space of the Riemann surface $\Omega_{G}/G$, and $\mathsf{QH}(G)$
the space of quasi-conformal deformations of $G$ modulo conjugacy. Sending
each Kleinian group to its bending lamination, we get a map
$b\colon\mathsf{QH}(G)\to\mathcal{ML}(\Omega_{G}/G)$, where $\mathcal{ML}$
denotes the space of measured laminations. Let
$D\subset\mathcal{ML}(\Omega_{G}/G)$ be the set of measured laminations
evidently unrealisable, whose exact definition is given in Theorem 3.1. In
this setting, Theorem 3.3 states that $q\circ b$ is a proper, degree-1 map to
$\mathcal{ML}(\Omega_{G}/G)\setminus D$. This in particular says that $q\circ
b$ is surjective to $\mathcal{ML}(\Omega_{G}/G)\setminus D$, and hence we
obtain the existence part of the main result. The compactness part is derived
from the properness of the map.

Our proof of Theorem 3.3, is divided into two parts: we shall first show the
properness of the map $b\circ q$ in Section 5, and then that $b\circ q$ has
degree 1 in Section 6. In the first part, relying on the analysis of geometric
limits, as was given in Ohshika-Soma [20] and Ohshika [19], we shall show that
any sequence going to infinity in $\mathcal{T}(\Omega_{G}/G)$ has image under
$b\circ q$ which cannot stay in a compact set disjoint from $D$. In the second
part, we shall show that $q\circ b$ can be properly homotoped to a local
degree-1 map which is constructed using the earthquake map. Since the degree
is invariant under a proper homotopy, this implies that $b\circ q$ has also
degree 1.

The authors would like to express their hearty gratitude to the referee for
his/her careful reading and suggestions, due to which we could in particular
remove some mistakes in the first version.

##  2\. Preliminaries

###  2.1. Basics of Kleinian groups

A Kleinian group is a discrete subgroup of ${\rm PSL}_{2}(\mathbb{C})$. In
this paper, we only consider Kleinian groups isomorphic to the fundamental
groups of closed orientable surfaces of genus greater than $1$, which we call
Kleinian surface groups. A Kleinian group acts on the Riemann sphere
$\hat{\mathbb{C}}$ by linear fractional transformations and on the hyperbolic
space $\mathbb{H}^{3}$ by orientation-preserving isometries. By considering
the Poincaré model of $\mathbb{H}^{3}$, the Riemann sphere $\hat{\mathbb{C}}$
is regarded as the sphere at infinity of $\mathbb{H}^{3}$. The action on
$\hat{\mathbb{C}}$ is a continuous extension of the action on $\mathbb{H}^{3}$
if we regard $\hat{\mathbb{C}}$ as the points at infinity in this way. For a
Kleinian group $G$, its limit set $\Lambda_{G}$ is the closure of the set of
fixed points of non-trivial elements of $G$. The complement of $\Lambda_{G}$
in $\hat{\mathbb{C}}$ is called the region of discontinuity of $G$, and is
denoted by $\Omega_{G}$.

The smallest convex subset of $\mathbb{H}^{3}$ containing all geodesics both
of whose endpoints at infinity lie on $\Lambda_{G}$ is called the Nielsen
convex hull and is denoted by $H_{G}$. Since $H_{G}$ is a closed convex subset
invariant under $G$, its quotient $H_{G}/G$ is a closed convex subset of
$\mathbb{H}^{3}/G$, which is a 3-submanifold except for the case when $G$ is
Fuchsian. The quotient $H_{G}/G$ is called the convex core of
$\mathbb{H}^{3}/G$ and is denoted by $C(\mathbb{H}^{3}/G)$. The Kleinian group
$G$ is said to be geometrically finite if $C(\mathbb{H}^{3}/G)$ has finite
volume.

###  2.2. Geodesic and measured laminations

In this section, we consider an orientable surface $S$ which may have
punctures but has no boundary. We fix a complete hyperbolic metric on $S$
which makes punctures cusps. A geodesic lamination $\lambda$ on $S$ is a
closed subset consisting of disjoint simple geodesics which do not tend to
cusps. A geodesic constituting $\lambda$ is called a leaf. A geodesic
lamination is said to be minimal when it does not have a non-empty proper
sublamination. Any geodesic lamination is decomposed into disjoint finitely
many minimal sublaminations, which we call minimal components, and isolated
leaves spiralling around minimal components. We say that a geodesic lamination
is arational when every component of its complement is either simply connected
or an annulus containing a cusp.

A measured lamination is a geodesic lamination equipped with a transverse
invariant measure. The support of a measured lamination is a geodesic
lamination having the property that the entire lamination coincides with the
union of its minimal components. We denote the support of a measured
lamination $\lambda$ by $|\lambda|$. Conversely, a geodesic lamination with
this property always supports a transverse invariant measure. We always assume
that the support of a measured lamination is the entire lamination.

For two measured laminations $\lambda$ and $\mu$, their intersection number
$\iota(\lambda,\mu)$ is defined to be the integral of the product of the
transverse measures of $\lambda$ and $\mu$ over the surface $S$. In
particular, when $c$ is a simple closed curve, we regard $c$ as having the
unit Dirac transverse measure and define $\iota(\lambda,c)$ as such. For two
geodesic laminations $\lambda$ and $\lambda^{\prime}$ on a hyperbolic surface
$S$, and a point $p\in\lambda\cap\lambda^{\prime}$, we can consider the angle
formed by $\lambda$ and $\lambda^{\prime}$ at $p$ taking the value in
$[0,\pi/2]$, which we denote by $\angle_{p}(\lambda,\lambda^{\prime})$. We
define the angle between $\lambda$ and $\lambda^{\prime}$ to be
$\sup_{p\in\lambda\cap\lambda^{\prime}}\angle_{p}(\lambda,\lambda^{\prime})$
and denote it by $\angle_{S}(\lambda,\lambda^{\prime})$.

A measured lamination $\lambda$ is said to be uniquely ergodic if the
transverse measure of $\lambda$ is a unique transverse measure on its support
up to scaling. A pair of geodesic (or measured) laminations $\lambda_{1}$ and
$\lambda_{2}$ is said to fill up $S$ if every geodesic lamination $\mu$ on $S$
intersects $\lambda_{1}$ or $\lambda_{2}$ transversely.

For a measured lamination $\lambda$ or a geodesic lamination supporting a
measured lamination on $S$, its minimal supporting surface $S(\lambda)$ is an
incompressible compact subsurface containing $\lambda$ and is minimal with
respect to the inclusion, which is unique up to isotopy.

Geodesic laminations and measured laminations defined above depend on the
hyperbolic metric given on $S$. Still for two complete hyperbolic metrics
$m,n$ on $S$, and a geodesic lamination $\lambda$ on $(S,m)$, there is a
unique geodesic lamination $\lambda^{\prime}$ on $(S,n)$ which is isotopic to
$\lambda$. By identifying $\lambda$ and $\lambda^{\prime}$ as above, we can
talk about geodesic laminations and measured laminations without specifying a
hyperbolic metric. We note the intersection number does not depend on the
choice of a hyperbolic metric whereas the angle does depend on it.

Thurston proved that the space of measured laminations with the weak topology
with respect to the transverse measures is homeomorphic to the Euclidean space
of dimension $6g-6+2b$, where $g$ is the genus and $b$ is the number of
punctures of $S$. We denote this space by $\mathcal{ML}(S)$ and call it the
measured lamination space of $S$. There is a PL local chart of
$\mathcal{ML}(S)$, which can be constructed using train tracks as in the next
section.

###  2.3. Train tracks

We shall define basic terms on train tracks in this subsection. We refer the
reader to Penner-Harer [22] for a more detailed account.

A train track $\tau$ on $S$ is a $C^{1}$-graph (i.e. a graph whose edges are
$C^{1}$-arcs and tangent to each other at vertices) embedded in $S$ whose
edges are called branches and whose vertices are called switches, such that no
component of $S\setminus\tau$ is a disc with one corner or an annulus with
$C^{1}$-smooth boundary. A weight system $\omega$ on a train track $\tau$ is a
system of non-negative numbers, called weights, given on branches of $\tau$
such that at each switch the sum of the weights on the incoming branches
coincides with the sum of the weights on the outgoing branches.

A geodesic lamination $\lambda$ is said to be carried by a train track $\tau$,
when it can be regularly homotoped to an immersion in $\tau$. Any geodesic
lamination has a train track carrying it. In particular, if a measured
lamination $\lambda$ is carried by a train track $\tau$, it induces a weight
system on $\tau$, by defining the weight of a branch to be the total
transverse measure of the leaves lying there (after a regular homotopy). We
denote this weight system induced from $\lambda$ by $w(\lambda)$. Conversely,
for any weight system $\omega$ on a train track, we can construct a measured
lamination $\lambda$ such that $w(\lambda)=\omega$.

A train track $\tau$ is said to be recurrent if it has a weight system which
takes only positive values, and transversely recurrent if for each branch $b$
of $\tau$, there is a simple closed curve intersecting $\tau$ essentially
(i.e. without cobounding a bigon) and transversely with non-empty intersection
with $b$. Train tracks which are both recurrent and transversely recurrent are
called bi-recurrent. Every measured lamination is carried by a bi-recurrent
train track. For a bi-recurrent train track $\tau$, the set of measured
laminations inducing weight systems with positive values on $\tau$ forms an
open set in $\mathcal{ML}(S)$, which we denote by $U(\tau)$. For an arational
measured lamination $\lambda$, the open sets $U(\tau)$ for all bi-recurrent
train tracks $\tau$ carrying $\lambda$ form a base of neighbourhoods of
$\lambda$ in $\mathcal{ML}(S)$.

###  2.4. Pleated surfaces

Let $M$ be a hyperbolic 3-manifold, and $F$ an orientable surface, which we
assume to be either closed or the interior of a compact surface. A pleated
surface $f\colon(F,m)\to M$, where $m$ is a complete hyperbolic metric on $F$,
is a continuous map taking each cusp of $(F,m)$ to a cusp of $M$ such that for
every point $x\in F$, there is at least one geodesic segment containing $x$ in
its interior which is mapped isometrically to a geodesic segment in $M$ by
$f$. The set of points on $F$ at which only one direction is mapped
geodesically constitutes a geodesic lamination $\mu$ on $(F,m)$. We call $\mu$
the pleating locus of the pleated surface $f$. More generally, if a geodesic
(or a measured) lamination $\lambda$ is mapped geodesically by a pleated
surface $f$, we say that $f$ realises $\lambda$.

The boundary component of the convex core of a hyperbolic 3-manifold is an
example of pleated surface. It has moreover a special property that the
surface is bent only in one direction. The pleating locus of such a surface
has a transverse measure coming from bending angles, and is called the bending
lamination when it is regarded as a measured lamination.

###  2.5. Ending laminations

By Bonahon’s tameness theorem [3], it is known that for any faithful discrete
representation $\phi\colon\pi_{1}(S)\to{\rm PSL}_{2}(\mathbb{C})$, there is an
orientation-preserving homeomorphism $\Phi\colon
S\times(0,1)\to\mathbb{H}^{3}/\phi(\pi_{1}(S))$ which induces $\phi$ between
their fundamental groups.

For a hyperbolic 3-manifold $M=\mathbb{H}^{3}/\phi(\pi_{1}(S))$, its non-
cuspidal part, denoted by $M_{0}$, is the complement of $\epsilon$-thin cusp
neighbourhoods for some fixed positive number $\epsilon$ smaller than the
three-dimensional Margulis constant. The boundary of $M_{0}$ consists of
incompressible open annuli. (In the case of general Kleinian groups,
incompressible tori may appear. We do not have such components since we only
deal with Kleinian surface groups.) By the relative core theorem ([23, 13]),
there is a compact submanifold $C$ of $M_{0}$, called a relative compact core
of $M_{0}$, such that the inclusion is a homotopy equivalence and
$C\cap\partial M_{0}$ is the union of core annuli of the components of
$\partial M_{0}$. A core curve of each component of $C\cap\partial M_{0}$
represents a generator of a maximal parabolic subgroup of $\phi(\pi_{1}(S))$.
We call these curves parabolic curves. A relative compact core $C$ of $M_{0}$
is always homeomorphic to $S\times[0,1]$ in our setting. We fix orientations
on $S$ and $\mathbb{H}^{3}$, and assume the identification of $S\times[0,1]$
with $C$ to preserve the orientations.

An end of $M_{0}$ is an inverse limit (with respect to the inclusion) of
complementary components of compact sets in $M_{0}$. Each component $U$ of
$M_{0}\setminus C$ contains a unique end $e$ of $M_{0}$, and also its closure
contains a unique component $\Sigma$ of $\operatorname{Fr}_{M_{0}}C$, where
$\operatorname{Fr}_{M_{0}}$ denotes the frontier as a subspace of $M_{0}$. In
this situation, we say that $\Sigma$ faces the end $e$. The end $e$ is said to
be geometrically finite when it has a neighbourhood disjoint from any closed
geodesic, and otherwise geometrically infinite. If $e$ is geometrically
finite, there is a boundary component $F$ of the convex core $C(M)$ such that
$F\cap M_{0}$ is isotopic to $\Sigma$. This component $F$ in turn corresponds
to a component of $\Omega_{G}/G$ which is regarded as lying at infinity.

When $e$ is geometrically infinite, it was proved in [3] that there is a
sequence of simple closed curves $c_{i}$ on $\Sigma$ which are homotopic in
$U\cup\Sigma$ to closed geodesics $c_{i}^{*}$ tending to the end. Such an end
is called simply degenerate. Regarding $c_{i}$ as a geodesic lamination on
$\Sigma$, after fixing any hyperbolic metric on $\Sigma$, we consider the
Hausdorff limit $c_{\infty}$ of $c_{i}$, which is a geodesic lamination. It
was shown by Thurston [24] and Bonahon [3] that $c_{\infty}$ has only one
minimal component $\lambda$, which is called the ending lamination of $e$, and
that $S(\lambda)=\Sigma$. The geodesic lamination $\lambda$ is the support of
a measured lamination which is a limit of $\\{r_{i}c_{i}\\}$ in the space of
measured laminations, where $r_{i}$ is a positive scalar.

The notion of ending lamination was first introduced by Thurston using pleated
surfaces as follows. Let $\Sigma$ be a subsurface of $S$ as above. If the end
$e$ facing $\Sigma$ is geometrically infinite, there is a sequence of pleated
surfaces $\\{f_{i}\\}$ homotopic to the inclusion of $\Sigma$ which tends to
$e$. For instance, in the setting of the preceding paragraph, pleated surfaces
realising the simple closed curves $c_{i}$ are such pleated surfaces. Thurston
considered the Hausdorff limit of the geodesic laminations realised by such
pleated surfaces, and proved that the limit has only one minimal component,
which is defined to be the ending lamination of $e$. He also showed that the
ending lamination thus defined does not depend on the choice of pleated
surfaces.

Recall that the relative compact core $C$ is identified with $S\times[0,1]$.
When a parabolic curve lies on $S_{+}=S\times\\{1\\}$ (resp.
$S_{-}=S\times\\{0\\}$), we call it an upper (resp. a lower) parabolic curve.
When the end $e$ is above $C$, i.e., when $\Sigma$ lies on $S\times\\{1\\}$
(resp. $S\times\\{0\\}$), we say that the ending lamination $\lambda$ is an
upper (resp. a lower) ending lamination. It was also proved in [24] and [3]
that for each upper (resp. lower) ending lamination $\lambda$ of
$\mathbb{H}^{3}/\phi(\pi_{1}(S))$, each boundary component of $S(\lambda)$ is
an upper (resp. lower) parabolic curve. We call the union of the parabolic
curve and the ending laminations regarded as lying on $S_{-}\sqcup S_{+}$ the
qi(quasi-isometric)-end invariant of $\phi$ (or
$\mathbb{H}^{3}/\phi(\pi_{1}(S))$). In particular, the union of those lying on
$S_{+}$ (resp. $S_{-}$) is called the upper (resp. lower) qi-end invariant.

###  2.6. Deformation spaces

The space of faithful discrete representations of $\pi_{1}(S)$ into ${\rm
PSL}_{2}(\mathbb{C})$ modulo conjugacy is denoted by $\mathsf{AH}(S)$. We
endow $\mathsf{AH}(S)$ with the topology induced from the representation
space. Although each element of $\mathsf{AH}(S)$ is a conjugacy class of
representations, by abusing notation, we denote it by its representative.

The interior of $\mathsf{AH}(S)$ is known to be the quasi-Fuchsian space
$\mathsf{QF}(S)$. When $\phi\in\mathsf{AH}(S)$ is quasi-Fuchsian, letting $G$
be $\phi(\pi_{1}(S))$, the conformal structure on $\Omega_{G}/G$ induces a
marked conformal structures at infinity, on $S\times\\{0\\}$ and
$S\times\\{1\\}$. We note that the identification of $S\times\\{0\\}$ to a
component of $\Omega_{G}/G$ is orientation-preserving, but that of
$S\times\\{1\\}$ is orientation-reversing. We use the symbol
$\mathcal{T}(\bar{S})$ to denote the Teichmüller space of $S$ with its
orientation reversed. Then the conformal structure on $\Omega_{G}/G$
determines a point in $\mathcal{T}(S)\times\mathcal{T}(\bar{S})$. Bers showed
that this identification of the conformal structure on $\Omega_{G}/G$ and a
point in $\mathcal{T}(S)\times\mathcal{T}(\bar{S})$ gives a parametrisation of
$\mathsf{QF}(S)$, which we denote by
$q\colon\mathcal{T}(S)\times\mathcal{T}(\bar{S})\to\mathsf{QF}(S)$.

For a general point $\phi\in\mathsf{AH}(S)$ and $G=\phi(\pi_{1}(S))$, let
$\lambda_{-}$ and $\lambda_{+}$ be the lower and upper qi-end invariants of
$\mathbb{H}^{3}/\phi(\pi_{1}(S))$. We recall that $\lambda_{-}$ (resp.
$\lambda_{+}$) has the property that for any component $\lambda$ of
$\lambda_{-}$ (resp. $\lambda_{+}$), every boundary component of $S(\lambda)$
is contained in $\lambda_{-}$ (resp. $\lambda_{+}$). The quotient of the
region of discontinuity $\Omega_{G}/G$ is identified with the disjoint union
of $S_{-}\setminus S(\lambda_{-})$ and $S_{+}\setminus S(\lambda_{+})$, which
we denote by $\Sigma_{-}$ and $\Sigma_{+}$. The ending lamination theorem,
proved by Minsky and Brock-Canary-Minsky [14, 7], shows that any Kleinian
group in $\mathsf{AH}(S)$ having $\lambda_{-}$ and $\lambda_{+}$ as lower and
upper qi-end invariants is a quasi-conformal deformation of $G$. Therefore, we
denote the quasi-conformal deformation space of $G$ by
$\mathsf{QH}_{\lambda_{-},\lambda_{+}}$. The theory of Bers-Kra-Maskit-Marden-
Sullivan shows that the conformal structures at infinity give a
parametrisation
$q\colon\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})\to\mathsf{QH}_{\lambda_{-},\lambda_{+}}$.

###  2.7. Geometric limits

For a sequence of Kleinian groups $\\{G_{i}\\}$, we say that $\\{G_{i}\\}$
converges to a Kleinian group $\Gamma$ geometrically if (i) every element
$\gamma\in\Gamma$ is a limit of some sequence $\\{g_{i}\in G_{i}\\}$, and (ii)
for every convergent subsequence $\\{g_{i_{j}}\in G_{i_{j}}\\}$, its limit
lies in $\Gamma$. The geometric convergence is equivalent to the pointed
Gromov-Hausdorff convergence of the corresponding hyperbolic 3-manifolds:
fixing a basepoint $x\in\mathbb{H}^{3}$ and letting $x_{i}$ and $x_{\infty}$
be the projections of $x$ to $\mathbb{H}^{3}/G_{i}$ and
$\mathbb{H}^{3}/\Gamma$ respectively, the sequence of pointed hyperbolic
3-manifolds $\\{(\mathbb{H}^{3}/G_{i},x_{i})\\}$ converges to
$(\mathbb{H}^{3}/\Gamma,x_{\infty})$ in the sense of Gromov-Hausdorff if and
only if $\\{G_{i}\\}$ converges to $\Gamma$ geometrically. Due to this fact,
we also refer to Gromov-Hausdorff limits as geometric limits. The compactness
of Gromov-Hausdorff topology shows that every sequence of non-elementary
Kleinian groups has a geometric limit after passing to a subsequence. We
recall that, by definition, if $\\{(\mathbb{H}^{3}/G_{i},x_{i})\\}$ converges
to $(\mathbb{H}^{3}/\Gamma,x_{\infty})$ in the sense of Gromov-Hausdorff, then
there is a $K_{i}$-bi-Lipschitz diffeomorphism, which is called an approximate
isometry between the $R_{i}$-ball around $x_{i}$ and the $K_{i}R_{i}$-ball
around $x_{\infty}$, with $K_{i}\longrightarrow 1$ and
$R_{i}\longrightarrow\infty$.

Suppose that a sequence $\\{\phi_{i}\\}$ in $\mathsf{AH}(S)$ converges to
$\psi\in\mathsf{AH}(S)$. In this situation, we always assume that we take
representatives so that $\\{\phi_{i}\\}$ converges to $\psi$ as genuine
representations from $\pi_{1}(S)$ into ${\rm PSL}_{2}(\mathbb{C})$. Then,
passing to a subsequence, $\\{\phi_{i}(\pi_{1}(S))\\}$ converges to some
Kleinian group $\Gamma$. From the definition of geometric limits, it is easy
to see that $\Gamma$ contains $\psi(\pi_{1}(S))$. If
$\Gamma=\psi(\pi_{1}(S))$, we say that $\\{\phi_{i}\\}$ converges to $\psi$
strongly.

In Ohshika-Soma [20] a classification of geometric limits of Kleinian surface
groups was given. An alternative description was also given in [19]. We shall
now review some of the results there which will be used in the proof of the
main theorem, in particular, in Section 5.2.

Let $\\{\phi_{i}\\}$ be a sequence in $\mathsf{AH}(S)$, and suppose that
$\\{\phi_{i}(\pi_{1}(S))\\}$ converges geometrically to a Kleinian group
$\Gamma$. The following is a paraphrase of a part of Theorem A in [20].

######  Theorem 2.1.

The non-cuspidal part $(\mathbb{H}^{3}/\Gamma)_{0}$ is topologically embedded
in $S\times(0,1)$ in such a way that the following hold.

  1. (a)

Every end of $(\mathbb{H}^{3}/\Gamma)_{0}$ is mapped to a horizontal surface
$\Sigma\times\\{t\\}$, where $\Sigma$ is an incompressible subsurface of $S$
and $t$ lies in $[0,1]$.

  2. (b)

Every geometrically finite end of $(\mathbb{H}^{3}/\Gamma)_{0}$ lies on either
$S\times\\{0\\}$ or $S\times\\{1\\}$.

  3. (c)

Each boundary component of the convex core of $(\mathbb{H}^{3}/\Gamma)_{0}$ is
a Gromov-Hausdorff limit of boundary components of convex cores of
$(\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))_{0}$ with some base points.

  4. (d)

Every geometrically infinite end is either simply degenerate or an
accumulation set of countably many torus cusps or simply degenerate ends or
both.

Identifying $(\mathbb{H}^{3}/\Gamma)_{0}$ with the image of its embedding in
$S\times(0,1)$ as above, we can talk about the horizontal direction and the
vertical direction in $(\mathbb{H}^{3}/\Gamma)_{0}$.

The following lemma, which is [19, Lemma 4.13] and also can be found in [6,
§3], will be used in Section 5.2.

######  Lemma 2.2.

In the setting as above, suppose moreover that $\\{\phi_{i}\\}$ converges to
$\psi\in\mathsf{AH}(S)$. Then the image of the inclusion of $\psi(\pi_{1}(S))$
into $(\mathbb{H}^{3}/\Gamma)_{0}$ is represented by an immersion
$f_{\infty}\colon S\to(\mathbb{H}^{3}/\Gamma)_{0}$ which is horizontal except
for disjoint annuli in $S$ whose images wrap around torus cusps.

An immersion $f_{\infty}$ as above is called an algebraic locus. We note that
for an approximate isometry $\rho_{i}$ from
$\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ to $\mathbb{H}^{3}/\Gamma$, the
composition $\rho_{i}^{-1}\circ f_{\infty}$ induces the same isomorphism as
$\phi_{i}$ between the fundamental groups for sufficiently large $i$.

By Theorems 2.1 and 2.2, we can show the following.

######  Corollary 2.3.

Suppose that $\\{\phi_{i}\\}$ in $\mathsf{AH}(S)$ converges to
$\psi\in\mathsf{AH}(S)$. If $(\mathbb{H}^{3}/\Gamma)_{0}$ does not have a
torus boundary component and every geometrically infinite end is homotopic
into an algebraic locus, then $\\{\phi_{i}\\}$ converges to $\psi$ strongly.

###### ?proofname? .

Since we assumed that $(\mathbb{H}^{3}/\Gamma)_{0}$ does not have a torus
boundary component, the algebraic locus $f_{\infty}$ in Lemma 2.2 cannot wrap
around a boundary component, and hence is a horizontal surface. By assumption,
every geometrically infinite end can be lifted to the algebraic limit, and
hence simply degenerate. This implies that in the embedded image of
$(\mathbb{H}^{3}/\Gamma)_{0}$ in $S\times(0,1)$, there is no other end on the
side farther from $f_{\infty}(S)$ of each geometrically infinite end.
Therefore, we can isotope the embedding so that every geometrically infinite
end lies on $S\times\\{0\\}\cup S\times\\{1\\}$.

By Theorem 2.1-(a), every geometrically finite also lies on
$\Sigma\times\\{0,1\\}$. Each boundary component of
$(\mathbb{H}^{3}/\Gamma)_{0}$, which is an open annulus, has both ends on the
same level, either on $S\times\\{0\\}$ or $S\times\\{1\\}$, for it cannot pass
through the algebraic locus. This means that $(\mathbb{H}^{3}/\Gamma)_{0}$
coincides with the complement of finitely many half solid tori lying on a
neighbourhood of $S\times\\{0\\}$ or $S\times\\{1\\}$. Therefore it is
homeomorphic to $S\times(0,1)$, and hence $\psi(\pi_{1}(S))=\Gamma$. ∎

As explained in [20, 19], if a sequence $\\{\phi_{i}\\}$ in $\mathsf{AH}(S)$
converges geometrically to a Kleinian group $\Gamma$, a geometric limit of
(uniform) bi-Lipschitz model manifolds of $(\mathbb{H}^{3}/\phi_{i}(S))_{0}$
(due to Minsky [14]) serves as a bi-Lipschitz model manifold of
$(\mathbb{H}^{3}/\Gamma)_{0}$. Suppose that
$\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ has a lower qi-end invariant
$\lambda_{-}$ and an upper qi-end invariant $\lambda_{+}$. Let $\Sigma_{-}$
and $\Sigma_{+}$ be $S\setminus S(\lambda_{-})$ and $S\setminus
S(\lambda_{+})$ respectively, and $\mathbf{m}_{i}^{-}$ and
$\mathbf{m}_{i}^{+}$ the structures at infinity of
$\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ on $\Sigma_{-}$ and $\Sigma_{+}$
respectively. Let $P_{i}^{-}$ and $P_{i}^{+}$ be shortest pants decompositions
(with respect to the hyperbolic length) of $(\Sigma_{-},\mathbf{m}_{i}^{-})$
and $(\Sigma_{+},\mathbf{m}_{i}^{+})$ respectively. By adding a shortest
transversal simple closed curve to each component of $P_{i}^{-}$
(resp.$P_{i}^{+}$) disjoint from all the other components, we get shortest
markings $M_{i}^{-}$ (resp. $M_{i}^{+}$). Minsky’s model manifold is
constructed from the hierarchy of tight geodesics connecting the generalised
markings $M_{i}^{-}\cup\lambda_{-}$ and $M_{i}^{+}\cup\lambda_{+}$.

In this setting, we have the following lemmas.

######  Lemma 2.4.

Suppose that $\\{\phi_{i}(\pi_{1}(S))\\}$ converges geometrically to $\Gamma$.
Let $l$ be a non-contractible simple closed curve on $S$. We assume that the
translation length of $\phi_{i}(l)$ is bounded from above as
$i\longrightarrow\infty$. Let $\epsilon$ be a positive constant less than the
three-dimensional Margulis constant.

  1. (1)

If the $\epsilon$-Margulis tube around the closed geodesic representing
$\phi_{i}(l)$ converges geometrically to a torus cusp neighbourhood in
$\mathbb{H}^{3}/\Gamma$ as $i\longrightarrow\infty$, then both
$\mathrm{length}_{\mathbf{m}_{i}^{+}}(l)$ and
$\mathrm{length}_{\mathbf{m}_{i}^{-}}(l)$ (if one or both of them are defined)
are bounded from below by a positive constant, and
$d_{A(l)}(M_{i}^{-}\cup\lambda_{-},M_{i}^{+}\cup\lambda_{+})$ goes to
$\infty$. Here $A(l)$ denotes an annular neighbourhood of $l$ and $d_{A(l)}$
the distance in the curve complex of $A(l)$. Conversely if both
$\mathrm{length}_{\mathbf{m}_{i}^{+}}(l)$ and
$\mathrm{length}_{\mathbf{m}_{i}^{-}}(l)$ are bounded from below by a positive
constant and $d_{A(l)}(M_{i}^{-}\cup\lambda_{-},M_{i}^{+}\cup\lambda_{+})$
goes to $\infty$, then the $\epsilon$-Margulis tube around the closed geodesic
representing $\phi_{i}(l)$ converges geometrically to either a torus cusp
neighbourhood or a $\mathbb{Z}$-cusp attached to a geometrically infinite end.

  2. (2)

Suppose moreover that $\\{\phi_{i}\\}$ converges in $\mathsf{AH}(S)$, and let
$k$ be a simple closed curve intersecting $l$ essentially. If the
$\epsilon$-Margulis tube around the closed geodesic representing $\phi_{i}(l)$
converges geometrically to a torus cusp lying above (resp. below) an algebraic
locus as $i\longrightarrow\infty$, then $d_{A(l)}(k,M_{i}^{+}\cup\lambda_{+})$
(resp. $d_{A(l)}(k,M_{i}^{-}\cup\lambda_{-})$) goes to $\infty$. In the case
when an algebraic locus wraps around a torus cusp, we regard the cusp as lying
both above and below the locus. Conversely, if
$d_{A(l)}(k,M_{i}^{+}\cup\lambda_{+})$ (resp.
$d_{A(l)}(k,M_{i}^{-}\cup\lambda_{-})$) goes to $\infty$, then the
$\epsilon$-Margulis tube around the closed geodesic representing $\phi_{i}(l)$
converges geometrically to either a torus cusp neighbourhood or a
$\mathbb{Z}$-cusp neighbourhood attached to a geometrically infinite end.

###### ?proofname?.

Recall that the model manifold of $(\mathbb{H}^{3}/\Gamma)_{0}$ can be taken
to be a geometric limit of models constructed from hierarchies of tight
geodesics. If $\mathrm{length}_{\mathbf{m}_{i}^{-}}(l)$ or
$\mathrm{length}_{\mathbf{m}_{i}^{+}}(l)$ goes to $0$ after passing to a
subsequence, then the corresponding geometrically finite block of the model
manifold of $(\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S)))_{0}$ splits along a simple
closed curve representing $l$ as $i\longrightarrow\infty$. This shows that the
geometric limit of the Margulis tubes around the closed geodesic representing
$\phi_{i}(l)$ must be a $\mathbb{Z}$-cusp in $\mathbb{H}^{3}/\Gamma$ in this
case.

Now suppose that $\mathrm{length}_{{\mathbf{m}}_{i}^{-}}(l)$ and
$\mathrm{length}_{\mathbf{m}_{i}^{+}}(l)$ are bounded from below by a positive
constant if one or both of them are defined. Then by [14, Lemma 9.4], we see
that the Margulis tubes around the closed geodesic corresponding to
$\phi_{i}(l)$ converge geometrically to a torus cusp neighbourhood if and only
if $d_{A(l)}(M_{i}^{-}\cup\lambda_{-},M_{i}^{+}\cup\lambda_{+})$ goes to
$\infty$. Thus we are done for the part (1).

A proof of the part (2) can be found in [19, Proof of Theorem 5.2]. We
summarise its argument here. Consider the situation where the Margulis tube
$V_{i}$ around the closed geodesic corresponding to $\phi_{i}(l)$ converges to
a torus cusp neighbourhood $V_{\infty}$. First suppose that the algebraic
locus does not wrap around $V_{\infty}$ and that $V_{\infty}$ lies above an
algebraic locus. Recall that Minsky’s model manifold is constructed from a
hierarchy $H_{i}$ of tight geodesics in the curve complex of (subsurfaces of)
$S$. If the Margulis tube $V_{i}$ converges to a torus cusp neighbourhood,
then the annular neighbourhood $A(l)$ supports a geodesic $g_{i}$ in $H_{i}$
whose length goes to $\infty$. The last vertex of $g_{i}$ is within bounded
distance from $\pi_{A(l)}(M_{i}^{+}\cup\lambda_{+})$ as
$i\longrightarrow\infty$, where $\pi_{A(l)}$ denotes the projection between
the curve complexes $\mathcal{CC}(S)$ and $\mathcal{CC}(A(l))$ induced by
restricting curves to $A(l)$. On the other hand, since the torus cusp lies
above an algebraic locus, the initial vertex of $g_{i}$ is within bounded
distance from $\pi_{A(l)}(k)$ as $i\longrightarrow\infty$. Thus, we have
$d_{A(l)}(k,M_{i}^{+}\cup\lambda_{+})\longrightarrow\infty$. The same argument
works also in the case when $V_{\infty}$ lies below an algebraic locus. (See
[19, Claim 5.3] for more details.)

Next suppose that an algebraic locus wraps $n$-times around $V_{\infty}$ with
$n\neq 0$. Then there is a sequence of integers $r(i)$ whose absolute values
go to $\infty$ such that the initial vertex of $h_{i}$ is within bounded
distance from $\pi_{A(l)}(\tau_{l}^{nr(i)}(k))$ and the last vertex is within
bounded distance from $\pi_{A(l)}(\tau_{l}^{(n+1)r(i)}(k))$, where $\tau_{l}$
denotes the Dehn twist around $l$. The initial vertex of $h_{i}$ is also
within bounded distance from $\pi_{A(l)}(M_{i}^{-}\cup\lambda_{-})$, and the
last vertex of $h_{i}$ is also within bounded distance from
$\pi_{A(l)}(M_{i}^{+}\cup\lambda_{+})$. Therefore both
$d_{A(l)}(k,M_{i}^{-}\cup\lambda_{-})$ and
$d_{A(l)}(k,M_{i}^{+}\cup\lambda_{+})$ go to $\infty$ in this case.

Finally, suppose that $d_{A(l)}(k,M_{i}^{-}\cup\lambda_{-})$ goes to $\infty$.
Then by [14, Lemma 9.4] again, we see that the boundary of the Margulis tube
$\partial V_{i}$ either converges to the boundary of a torus cusp
neighbourhood or diverges to give rise to an open annulus whose end is
attached to a lower end of $(\mathbb{H}^{3}/\Gamma)_{0}$. Since
$d_{A(l)}(k,M_{i}^{-}\cup\lambda_{-})\longrightarrow\infty$, the curve $l$
cannot be contained in the shortest pants decomposition $M_{i}$. Therefore,
the upper geometrically finite block of the model manifold of
$(\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S)))_{0}$ cannot split along $l$, and hence
neither end of the geometric limit of $\partial V_{i}$ cannot tend to
$S\times\\{0\\}$. It follows that the end of $(\mathbb{H}^{3}/\Gamma)_{0}$ to
which an end of the geometric limit of $\partial V_{i}$ tends must be
geometrically infinite by Theorem 2.1-(b). Thus we have shown that the second
part of (2) holds. ∎

######  Lemma 2.5.

Suppose that $\\{\phi_{i}\\}$ converges in $\mathsf{AH}(S)$ to $\psi$. Let
$\Sigma$ be an open incompressible subsurface of $S$, with negative Euler
characteristic. Then, $\\{P_{i}^{+}|\Sigma\\}$ (resp.
$\\{P_{i}^{-}|\Sigma\\}$) converges (in the Hausdorff topology) to a geodesic
lamination containing a minimal component $\lambda$ whose minimal supporting
surface is $\Sigma$ if and only if there is an upper (resp. a lower) simply
degenerate end of $(\mathbb{H}^{3}/\psi(\pi_{1}(S)))_{0}$ whose ending
lamination is $\lambda$.

###### ?proofname?.

This is contained in [19, Proposition 4.18, Theorem 5.2] and their
generalisations explained in [19, §10], or alternatively in Brock-Bromberg-
Canary-Lecuire [6, Theorem 1.2]. ∎

##  3\. Main theorems

The purpose of this paper is to prove the following theorem, which is a
generalisation of the existence part of the theorem of Bonahon-Otal [4] to
general, possibly geometrically infinite Kleinian surface groups. We do not
have the uniqueness part of Bohaon-Otal’s theorem (in the case when the
bending laminations are multi-curves), but instead have compactness for the
set of Kleinian groups realising given data of qi-end invariants and bending
laminations.

######  Theorem 3.1.

Let $S$ be a closed oriented surface of genus greater than $1$, and
$S_{-},S_{+}$ two copies of $S$, where $S_{+}$ has the same orientation as $S$
whereas $S_{-}$ has the opposite orientation.

  * •

Let $\lambda_{-}$ and $\lambda_{+}$ be (possibly empty) geodesic laminations
without non-compact isolated leaves on $S_{-}$ and $S_{+}$ respectively, such
that for every component $\lambda$ of $\lambda_{-}$ (resp. $\lambda_{+}$),
each boundary component of the minimal supporting surface $S(\lambda)$ is
isotopic to a closed geodesic contained in $\lambda_{-}$ (resp.
$\lambda_{+}$).

  * •

Let $\mu_{-}$ and $\mu_{+}$ be measured laminations on $S_{-}$ and $S_{+}$
such that

    1. (a)

$\lambda_{-}\cap\mu_{-}=\emptyset$ and $\lambda_{+}\cap\mu_{+}=\emptyset$;

    2. (b)

neither $\mu_{-}$ nor $\mu_{+}$ contains a compact leaf of weight larger than
or equal to $\pi$; and

    3. (c)

$\lambda_{-}\sqcup\mu_{-}$ and $\lambda_{+}\sqcup\mu_{+}$ fill up $S$ if we
identify both $S_{-}$ and $S_{+}$ with $S$.

Then the following hold.

  1. (1)

There is $\phi\in\mathsf{AH}(S)$ such that

    1. (i)

the hyperbolic 3-manifold $\mathbb{H}^{3}/\phi(\pi_{1}(S))$ has $\lambda_{-}$
as its lower qi-end invariant and $\lambda_{+}$ as its upper qi-end invariant,
and

    2. (ii)

the hyperbolic 3-manifold $\mathbb{H}^{3}/\phi(\pi_{1}(S))$ realises $\mu_{-}$
and $\mu_{+}$ as the bending laminations on the lower and the upper boundaries
respectively of its convex core $C(\mathbb{H}^{3}/\phi(\pi_{1}(S))$.

  2. (2)

The set of all $\rho\in\mathsf{AH}(S)$ satisfying the condition (1) is a
compact subset of $\mathsf{QH}_{\lambda_{-},\lambda_{+}}$.

######  Remark 3.2.

In the theorem above, the existence of $\mu_{-},\mu_{+}$ satisfying (c)
imposes on $\lambda_{-},\lambda_{+}$ the condition that they share no minimal
component.

This theorem is an immediate consequence of following Theorem 3.3, which
states something a bit stronger.

Let $\lambda_{-}$ and $\lambda_{+}$ be geodesic laminations on $S_{-}$ and
$S_{+}$ as in Theorem 3.1, which share no minimal component. Let $\Sigma_{-}$
and $\Sigma_{+}$ be the complements $S_{-}\setminus S(\lambda_{-})$ and
$S_{+}\setminus S(\lambda_{+})$ respectively. As we explained in Section 2.6,
we have a parametrisation
$q\colon\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})\to\mathsf{QH}_{\lambda_{-},\lambda_{+}}$.
Let
$b\colon\mathsf{QH}_{\lambda_{-},\lambda_{+}}\to\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$
be the map taking $\rho\in\mathsf{QH}_{\lambda_{-},\lambda_{+}}$ to the
bending lamination of $C(\mathbb{H}^{3}/\rho(\pi_{1}(S)))$.

######  Theorem 3.3.

Let $D$ be the subset of
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$ consisting of
measured laminations which do not satisfy at least one of the conditions (b)
and (c) in Theorem 3.1. Then the map $b\circ
q\colon\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})\to\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus
D$ is a proper, degree-1 map.

##  4\. Bending laminations

We present some lemmas which will be useful for the proofs of our main
theorems.

######  Lemma 4.1.

For every $K>0$, there is a positive constant $L(K)$ depending only on $K$
which goes to $0$ as $K\longrightarrow 0$ with the following property. For
every $\phi\in\mathsf{AH}(S)$, letting $M=\mathbb{H}^{3}/\phi(\pi_{1}(S))$, if
$\Sigma$ is a boundary component of $C(M)$ with bending lamination $\lambda$
and c is a simple closed curve on $\Sigma$ with length less than $K$, then
$\iota(c,\lambda)<L(K)$.

###### ?proofname?.

This is a direct consequence of the boundedness of the average bending measure
proved by Bridgeman [5]. ∎

As a consequence of this lemma, we have the following.

######  Corollary 4.2.

Consider the situation in Theorem 3.1, and let $\Sigma$ be a component of
$\Sigma_{-}\sqcup\Sigma_{+}$. Let $\\{\mathbf{g}_{i}\\}$ be a sequence in
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$. Let $\gamma_{i}$ be a
shortest pants decomposition of $\mathbf{g}_{i}|\Sigma$. Then the sequence
$\\{\iota(b\circ q(\mathbf{g}_{i}),\gamma_{i})\\}$ is bounded.

###### ?proofname?.

By Bers’s lemma, there is a constant $C$ depending only on $S$ such that each
component of the shortest pants decomposition of $(\Sigma,\mathbf{g}_{i})$ has
length less than $C$. Therefore, by Lemma 4.1, $\iota(b\circ
q(\mathbf{g}_{i}),\gamma_{i})$ is bounded as $i\longrightarrow\infty$. ∎

######  Lemma 4.3.

For any $\epsilon>0$, there is a positive constant $\delta>0$ depending only
on $\epsilon$ satisfying the following.

Let $\\{\phi_{i}\\}$ be a sequence in $\mathsf{QH}_{\lambda_{-},\lambda_{+}}$,
and $\Sigma$ a component of $\Sigma_{-}\sqcup\Sigma_{+}$. Let $\mu_{i}$ be the
restriction of the bending lamination of the convex core of
$\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ to (the component corresponding to)
$\Sigma$. We denote the hyperbolic metric on $\Sigma$ as a boundary component
of the convex core $C(\mathbb{H}^{3}/\phi_{i}(\pi_{1}(M)))$ by
$m_{i}^{\prime}$.

Suppose

  1. (i)

that $\\{\mu_{i}\\}$ converges to a measured lamination $\mu_{\infty}$,

  2. (ii)

that $\mu_{\infty}$ is decomposed into disjoint (possibly empty)
sublaminations $\mu_{\infty}^{1}$ and $\mu_{\infty}^{2}$,

  3. (iii)

and that $c$ is a simple closed curve on $\Sigma$ such that
$\iota(c,\mu_{\infty}^{1})<\delta$ and
$\angle_{(\Sigma,m_{i}^{\prime})}(c,\mu_{\infty}^{2})<\delta$ for large $i$.

Let $c_{i}$ be the closed geodesic on $\Sigma$ with respect to
$m_{i}^{\prime}$ freely homotopic to $c$, and $c_{i}^{*}$ the closed geodesic
in $\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ freely homotopic to $\phi_{i}(c)$.

Then we have

$1\leq\frac{\mathrm{length}_{(\Sigma,m_{i}^{\prime})}(c_{i})}{\mathrm{length}_{\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))}(c_{i}^{*})}\leq
1+\epsilon.$

###### ?proofname?.

Since the length of $c_{i}^{*}$ is less than or equal to that of $c_{i}$, the
first inequality obviously holds.

Since the Hausdorff limit of the support $|\mu_{i}|$ is a geodesic lamination
containing $\mu_{\infty}=\mu_{\infty}^{1}\sqcup\mu_{\infty}^{2}$, for
sufficiently large $i$, we can decompose $c_{i}$ into two parts $c_{i}^{2}$
and $c_{i}^{1}$ such that
$\angle_{(\Sigma,m_{i}^{\prime})}(c_{i}^{2},\mu_{i})<2\delta$ and
$\iota(c_{i}^{1},\mu_{i})<2\delta$. By applying [1, Proposition 4.1] for
$c_{i}^{2}$ and [8, Corollary 4.6] for $c_{i}^{1}$, we obtain the second
inequality. ∎

##  5\. properness

In this section, we shall prove the properness of $b\circ q$ in Theorem 3.3.
The argument is by contradiction. Suppose that $b\circ q$ is not proper. Then,
there is a sequence
$\\{(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}\subset\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$
without a convergent subsequence such that $\\{b\circ
q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$ converges in
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus D$ to a
measured lamination $\nu$ on $\Sigma_{-}\sqcup\Sigma_{+}$.

We are going to analyse a geometric limit of
$\\{\mathbb{H}^{3}/q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$, making use of
results in Ohshika-Soma [20] and Ohshika [19]. Let
$\phi_{i}\colon\pi_{1}(S)\to{\rm PSL}_{2}(\mathbb{C})$ be a representative of
$q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})$ for each $i$. We take the
$\phi_{i}$ to converge to some $\psi\in\mathsf{AH}(S)$ if
$\\{q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$ converges in
$\mathsf{AH}(S)$.

We divide our argument into the following three cases:

  1. (a)

The case when $\\{q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$ converges in
$\mathsf{AH}(S)$ strongly after passing to a subsequence.

  2. (b)

The case when $\\{q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$ converges in
$\mathsf{AH}(S)$ after passing to a subsequence, but not strongly.

  3. (c)

The case when $\\{q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$ diverges in
$\mathsf{AH}(S)$ (even after passing to a subsequence).

In the cases (a) and (b), we put a basepoint $\tilde{x}$ in $\mathbb{H}^{3}$,
and by projecting it to $\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$, we get a
basepoint $x_{i}$. Taking the Gromov-Hausdorff limit of
$(\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S)),x_{i})$, we obtain a geometric limit
$(M_{\infty},x_{\infty})$ which is covered by the algebraic limit
$\mathbb{H}^{3}/\psi(\pi_{1}(S))$.

Now we start to show that in each of the three cases, we get a contradiction.

###  5.1. Case (a)

In this case, $M_{\infty}=\mathbb{H}^{3}/\psi(\pi_{1}(S))$. There are three
possibilities for $\psi$:

  1. (a)-(1)

The case when $\psi$ is also contained in
$\mathsf{QH}_{\lambda_{-},\lambda_{+}}$.

  2. (a)-(2)

The case when $(\mathbb{H}^{3}/\psi(\pi_{1}(S))_{0}$ has a ‘new simply
degenerate end’ not corresponding to a simply degenerate end of
$(\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S)))_{0}$.

  3. (a)-(3)

The case when $\mathbb{H}^{3}/\psi(\pi_{1}(S))$ does not have a new simply
degenerate end, but has a ‘new $\mathbb{Z}$-cusp’ not corresponding to a
$\mathbb{Z}$-cusp of $(\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S)))_{0}$.

Since we are assuming that $\\{(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$ has
no convergent subsequence, we can exclude the case (a)-(1).

We first assume that the condition (a)-(2) holds. Let $\Psi\colon
S\times(0,1)\to\mathbb{H}^{3}/\psi(\pi_{1}(S))$ be an orientation-preserving
homeomorphism inducing $\psi$ between the fundamental groups. For brevity of
description, we now assume that a new simply degenerate end is lower. The case
when the new simply degenerate end is upper can be dealt with in the same way.

Let $\gamma_{i}$ be a shortest pants decomposition of
$(\Sigma_{-},\mathbf{m}_{i}^{-})$. Let $\gamma_{\infty}$ be the Hausdorff
limit of the $\gamma_{i}$ regarded as geodesic laminations. Since we assumed
that there is a new lower simply degenerate end, by Lemma 2.5, there is a
minimal component $\ell$ of $\gamma_{\infty}$ which is the ending lamination
of such an end. Let $\Sigma$ be the minimal supporting surface of $\ell$. Let
$\lambda$ be a measured lamination supported on $\ell$. Let $\delta>0$ be the
positive constant given in Lemma 4.3 for $\epsilon=1$. Now, take an essential
simple closed curve $d$ on $\Sigma$ such that $\iota(\nu,d)<\delta$. Since we
have $\iota(\lambda,d)>0$ from the fact that $\Sigma=S(\lambda)$, and
$\gamma_{i}$ converges to $\gamma_{\infty}$ containing $\ell=|\lambda|$ in the
Hausdorff topology, the intersection number $\iota(\gamma_{i},d)$ goes to
$\infty$, which implies the length of $d$ with respect to $\mathbf{m}_{i}^{-}$
also goes to $\infty$, for $\gamma_{i}$ is a shortest pants decomposition. On
the other hand, $\\{b\circ q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$
converges to $\nu$ by our assumption. Then, by setting $\mu_{\infty}^{1}$ in
the statement of Lemma 4.3 to be $\nu$ and $\mu^{2}_{\infty}$ to be empty, it
follows that the translation length of $\phi_{i}(d)$ goes to $\infty$ as
$i\longrightarrow\infty$, contradicting the assumption that $\\{\phi_{i}\\}$
converges.

We now assume the condition (a)-(3) that $\mathbb{H}^{3}/\psi(\pi_{1}(S))$ has
a new $\mathbb{Z}$-cusp, represented by a non-contractible, non-peripheral
simple closed curve $c$ on $\Sigma_{-}\sqcup\Sigma_{+}$, but does not have a
new simply degenerate end. In the same way as in the preceding paragraph, we
can assume that $c$ lies on $\Sigma_{-}$. Since we assumed that there is no
new simply degenerate end, if the $\mathbb{Z}$-cusp represented by $\psi(c)$
touches a simply degenerate end, it corresponds to a simply degenerate end of
$(\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))_{0}$, which implies that $\phi_{i}(c)$
is parabolic, contradicting our assumption that $c$ is a new parabolic curve.
Therefore, the lower $\mathbb{Z}$-cusp represented by $c$ has geometrically
finite ends on its both sides, which may coincide. Therefore, the lower
boundary of the convex core of $\mathbb{H}^{3}/\psi(\pi_{1}(S))$ has bending
angle $\pi$ along $c$. Since we assumed that $\\{\phi_{i}\\}$ converges to
$\psi$ strongly, the convex core of $\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$
converges to that of $\mathbb{H}^{3}/\psi(\pi_{1}(S))$ (see [17, Proof of
Theorem 5]), and hence the bending angle along $c\subset\Sigma_{-}$ of the
convex core of $\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ converges to $\pi$. This
means that $\\{b\circ q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$ tends to a
point in $D$, contradicting our assumption. Thus, in every sub-case of the
case (a), we have obtained a contradiction.

We state what we have proved in the last paragraph as a lemma to refer to it
in the following section.

######  Lemma 5.1.

Suppose that $\\{\phi_{i}\\}$ converges to $\psi$ strongly, and that
$\mathbb{H}^{3}/\psi(\pi_{1}(S))$ has a $\mathbb{Z}$-cusp whose core curve is
a simple closed curve $c$ on a component $\Sigma$ of
$\Sigma_{-}\sqcup\Sigma_{+}$. Suppose furthermore that the $\mathbb{Z}$-cusp
has geometrically finite ends on its both sides. Then the bending angle on the
boundary of the convex core of $\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ along $c$
converges to $\pi$.

###  5.2. Case (b)

By Corollary 2.3, if $\\{q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$
converges algebraically but not strongly, then the geometric limit either has
a $\mathbb{Z}\times\mathbb{Z}$-cusp or a new simply degenerate end. We note
that the following arguments only use the assumption that $\nu$ satisfies the
condition (c) of Theorem 3.1. Later in §6, we shall use the arguments again in
the case when $\nu$ satisfies (c) but not (b) of Theorem 3.1.

By Theorem 2.1, the geometric limit $M_{\infty}$ is topologically embedded in
$S\times(0,1)$. We regard $M_{\infty}$ as embedded in $S\times(0,1)$ from now
on, and we call the direction of $S\times\\{t\\}$ horizontal. By Lemma 2.2,
there is an algebraic locus $f_{\infty}\colon S\to M_{\infty}$ which can be
lifted to an immersion into $\mathbb{H}^{3}/\psi(\pi_{1}(S))$ inducing $\psi$
between the fundamental groups such that $f_{\infty}(S)$ is horizontal except
for the part where it goes around a torus cusp. We recall that the pull-back
of $f_{\infty}$ to $\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ by an approximate
isometry between $\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ and $M_{\infty}$
induces the isomorphism $\phi_{i}$ between the fundamental groups.

A geometrically infinite end or a torus cusp is situated either above
$f_{\infty}(S)$ or below $f_{\infty}(S)$, except for the case of a torus cusp
around which $f_{\infty}(S)$ goes. When $f_{\infty}(S)$ goes around a torus
cusp $T$, we regard $T$ as being situated both above and below
$f_{\infty}(S)$. We say that a simply degenerate end or a torus cusp is
nearest to $f_{\infty}(S)$ when pleated surfaces tending to the end can be
homotoped into $f_{\infty}(S)$ in $M_{\infty}$ for a simply degenerate end,
and when a longitude can be homotoped into $f_{\infty}(S)$ for a torus end.
Since a wild geometrically infinite end is an accumulation of simply
degenerate ends, it cannot be nearest to $f_{\infty}(S)$.

######  Claim 5.2.

Unless $\\{\phi_{i}\\}$ converges to $\psi$ which has no new simply degenerate
ends, there is either a new simply degenerate end or a torus cusp, which is
nearest to $f_{\infty}(S)$. If a simply degenerate end is nearest to
$f_{\infty}(S)$, then it can be lifted to the algebraic limit.

###### ?proofname?.

Since $M_{\infty}$ is embedded in $S\times(0,1)$, a simply degenerate end or a
longitude of a torus end can be vertically homotoped into $f_{\infty}(S)$
unless there is another simply degenerate end or a torus end which impedes
this. Since ends or torus cusps can accumulate only into a horizontal surface
containing an end, there cannot be ends which accumulate into $f_{\infty}(S)$.
Therefore, there must be a simply degenerate end or a torus cusp which can be
homotoped into $f_{\infty}(S)$ without being obstructed by other ends. This
shows the first statement.

If a simply degenerate end is nearest to $f_{\infty}(S)$, it is homotopic into
$f_{\infty}(S)$, and hence can be lifted to the algebraic limit. ∎

####  5.2.1. Nearest new simply degenerate end.

Suppose that there is a new simply degenerate end $E$ which is nearest to
$f_{\infty}(S)$. We assume that $E$ is situated above $f_{\infty}(S)$. The
case when $E$ is situated below $f_{\infty}(S)$ can be dealt with in the same
way just by turning everything upside down. The end $E$ has a neighbourhood of
the form $\Sigma\times(s,t)$ for an incompressible subsurface $\Sigma$ of $S$,
where $\Sigma\times\\{t\\}$ corresponds to the end $E$ (Figure 1). Let
$\delta$ be the constant given in Lemma 4.3 for $\epsilon=1$. Take a simple
closed curve $d$ contained in $\Sigma$ satisfying $\iota(d,\nu)<\delta$. Let
$\lambda$ be a measured lamination on $\Sigma$ whose support is the ending
lamination of $E$. Take a shortest pants decomposition $C_{i}$ of
$(\Sigma_{+},\mathbf{m}_{i}^{+})$. Then by Lemma 2.5, we see that
$C_{i}|\Sigma$ converges in the Hausdorff topology to a geodesic lamination
whose only minimal component is $|\lambda|$. Since $\iota(d,\lambda)>0$, we
have $\iota(C_{i},d)\longrightarrow\infty$ and see that the length of $d$ with
respect to $\mathbf{m}_{i}^{+}$ goes to $\infty$. On the other hand,
$\\{\iota(d,b\circ q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+}))\\}$ is bounded
since $b\circ q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})$ converges to $\nu$. We
set $\mu_{\infty}^{1}$ in the statement of Lemma 4.3 to be this $\nu$ and
$\mu_{\infty}^{2}$ to be empty. Then we see that the translation length of
$\phi_{i}(d)$ goes to $\infty$, contradicting our assumption that
$\\{\phi_{i}\\}$ converges.

\begin{overpic}[scale={.15}]{NearestSimplyDegeneratedEnd} \put(61.0,24.0){$E$}
\put(0.0,10.0){$f_{\infty}(S)$} \end{overpic} ?figurename? 1\. A nearest end
above is a simply degenerate end

####  5.2.2. Nearest torus cusp.

Let $T$ be such a nearest torus cusp. Again, the case when there is a nearest
torus cusp $T$ below $f_{\infty}(S)$ can also be dealt with in the same way,
by turning everything upside down. Let $l$ be a simple closed curve on
$\Sigma_{+}$ whose image under $f_{\infty}$ is homotopic to a longitude of
$T$.

Let $\delta>0$ be the constant given in Lemma 4.3 for $\epsilon=1$, and take a
simple closed curve $d$ on $\Sigma_{+}$ with $\iota(d,l)>0$ and
$\iota(d,\nu\setminus l)<\delta$. Since $l$ is homotopic to a longitude of a
torus cusp in $M_{\infty}$, by Lemma 2.4-(1), the length of $l$ with respect
to $\mathbf{m}_{i}^{+}$ is bounded from below by a positive constant. Consider
the shortest pants decomposition $P_{i}$ of $(\Sigma_{+},\mathbf{m}_{i}^{+})$,
and extend it to a shortest marking $M_{i}$ of $\Sigma_{+}$. Then by Lemma
2.4-(2), if we choose a simple closed curve $k$ on $S$ intersecting $l$
essentially, we have $d_{A(l)}(M_{i},k)\longrightarrow\infty$. This means that
unless $P_{i}$ contains $l$ for all $i$ after passing to a subsequence, there
is a component $a_{i}$ of $P_{i}$ which spirals around $l$ more and more as
$i\longrightarrow\infty$. Then, since $\iota(P_{i},d)\geq\iota(a_{i},d)$ and
the right hand side goes to $\infty$, the length of $d$ with respect to
$\mathbf{m}_{i}^{+}$ goes to $\infty$.

In the case when $P_{i}$ contains $l$ for all $i$, there is a curve $t_{i}$ in
$M_{i}$ with $\iota(l,t_{i})>0$ which is shortest among such curves. Since we
are assuming that the length of $l$ with respect to $\mathbf{m}_{i}^{+}$ is
bounded away from $0$, the length of $t_{i}$ with respect to
$\mathbf{m}_{i}^{+}$ is bounded above. Then by Lemma 2.4-(2), we see that
$t_{i}$ spirals around $l$ more and more as $i\longrightarrow\infty$. It
follows that $d_{A(l)}(t_{i},d)$ goes to $\infty$. Since $t_{i}$ has bounded
length and the length of $l$ with respect $\mathbf{m}_{i}^{+}$ does not go to
$0$, we see that the length of $d$ with respect to $\mathbf{m}_{i}^{+}$ goes
to $\infty$ also in this case.

By the observation above, the closed geodesic representing $d$ in
$(\Sigma_{+},\mathbf{m}_{i}^{+})$ spirals around $l$ more and more as
$i\longrightarrow\infty$. This implies that
$\angle_{(\Sigma_{+},\mathbf{m}_{i}^{+})}(d,l)$ goes to $0$. We set
$\mu_{\infty}^{1}$ to be $\nu\setminus l$ and $\mu_{\infty}^{2}$ to be $l$
with a positive weight given by $\nu$ if $l$ is contained in $|\nu|$, and
$\mu_{\infty}^{1}$ to be $\nu$ and $\mu_{\infty}^{2}$ to be empty otherwise.
Then we apply Lemma 4.3, and see that the translation length of $\phi_{i}(d)$
goes to $\infty$, contradicting our assumption that $\\{\phi_{i}\\}$
converges.

###  5.3. Case (c)

Suppose that $\\{q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$ does not
converge algebraically. By considering efficient pleated surfaces as in
Thurston [26], the following can be proved. See Ohshika [18, Theorem 3.1] for
a complete proof.

######  Lemma 5.3.

There is a vertical codimension-1 lamination $L$ properly embedded in
$S\times[0,1]$, which is disjoint from both $\lambda_{-}$ and $\lambda_{+}$,
such that for any sequence of weighted simple closed curves $s_{i}\gamma_{i}$
on $S$ converging to a measured lamination $\mu$, if $\iota(L,\mu)>0$, then
$s_{i}\mathrm{length}(\phi_{i}(\gamma_{i}))\longrightarrow\infty$, where
$\mathrm{length}(\phi_{i}(\gamma_{i}))$ denotes the length of the closed
geodesic representing $\phi_{i}(\gamma_{i})$ in
$\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$. Indeed, there is a pleated surface
called an ‘efficient’ pleated surface in $\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$
on which the growth of the length of every measured lamination is comparable
to that in $\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$.

By the condition (c) of Theorem 3.1, which $\nu$ must satisfy, there is a
component $\nu_{0}$ of $\nu$ intersecting a component $L_{0}$ of $L$
essentially. Let $\Sigma$ be a component of $\Sigma_{-}\sqcup\Sigma_{+}$
containing $\nu_{0}$, and take a shortest pants decomposition $P_{i}$ of
$(\Sigma,(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})|\Sigma)$. Since $P_{i}$ is a
shortest pants decomposition, by Sullivan’s lemma or [15, Proposition 2.1],
$\\{\mathrm{length}(\phi_{i}(P_{i}))\\}$ is bounded. By Lemma 5.3 we can apply
the proof of [19, Lemma 5.7] using the length in
$\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ instead of
$(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})|\Sigma$, we see that the Hausdorff
limit of $P_{i}$ contains $L_{0}\cap\Sigma$. This implies in turn that there
is a sequence of positive numbers $r_{i}$ tending to $0$ and a component
$P_{i}^{\prime}$ of $P_{i}$ such that $r_{i}P^{\prime}_{i}$ converges to a
measured lamination containing $L_{0}\cap\Sigma$ on
$(\Sigma,(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})|\Sigma)$ unless
$L_{0}\cap\Sigma$ coincides with $P_{i}^{\prime}$ for every sufficiently large
$i$. Even in the latter case, unless the length of $L_{0}\cap\Sigma$ with
respect to $(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})|\Sigma$ goes to $0$, by
letting $P_{i}^{\prime\prime}$ be the shortest simple closed curve transverse
to $P_{i}^{\prime}$, we can find $s_{i}$ going to $0$ such that
$s_{i}P_{i}^{\prime\prime}\longrightarrow L_{0}\cap\Sigma$. Since the length
of $P_{i}^{\prime\prime}$ with respect to
$(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})|\Sigma$ is bounded, abusing the
symbols, we denote $s_{i}P_{i}^{\prime\prime}$ also by $r_{i}P_{i}^{\prime}$
in this case. The properties which we shall use below are that
$r_{i}\longrightarrow 0$ and that
$\\{\mathrm{length}_{(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})|\Sigma}(P_{i})\\}$
is bounded.

If the length of $L_{0}\cap\Sigma$ with respect to
$(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})|\Sigma$ goes to $0$, the efficient
pleated surface is pinched along $L_{0}\cap\Sigma$. This implies that
$\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$ is also pinched along $L_{0}$, and hence
$(\Sigma,(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})|\Sigma)$ is also pinched
along $L_{0}\cap\Sigma$. Since $\nu_{0}$ intersects $L_{0}$, this contradicts
Corollary 4.2.

It remains to consider the case when $\\{r_{i}P_{i}\\}$ converges to
$L_{0}\cap\Sigma$ with $r_{i}\longrightarrow 0$. Since
$\iota(\nu_{0},L_{0})>0$ and $r_{i}$ goes to $0$, we see that
$\iota(P_{i}^{\prime},\nu_{0})\longrightarrow\infty$. This contradicts
Corollary 4.2, for
$\\{\mathrm{length}_{(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})|\Sigma}(P_{i})\\}$
is bounded. Thus we have completed the proof of the case (c).

We note that we did not use the assumption (b) of Theorem 3.1. We state what
we have proved in the case (c) as a lemme for later use.

######  Lemma 5.4.

Suppose that $\\{b\circ q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$ converges
to a measured lamination $\nu$ satisfying the assumption (c) of Theorem 3.1.
Then $\\{q(\mathbf{m}_{i}^{-},\mathbf{m}_{i}^{+})\\}$ converges in
$\mathsf{AH}(S)$.

##  6\. Homotopy to a degree-1 map

In this section, we shall prove that $b\circ q$ is a degree-1 map to
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus D$ by
constructing a homotopy in the one-point compactification of
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus D$ between
the map induced from $b\circ q$ and a degree-1 map. First, we shall define
compactification where a homotopy will take place.

######  Definition 6.1.

We let $\widehat{\mathrm{T}}$ be the one-point compactification of
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$, and
$\widehat{\mathrm{ML}}$ the one-point compactification of
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus D$. Since
$b\circ q$ is proper, it induces a continuous map $\widehat{b\circ
q}\colon\widehat{\mathrm{T}}\to\widehat{\mathrm{ML}}$.

The following is immediate from the definition of one-point compactification.

######  Lemma 6.2.

The map $b\circ q$ has degree 1 if and only if $\widehat{b\circ q}$ has degree
1.

Therefore, we have only to show that $\widehat{b\circ q}$ has degree $1$. For
that, we shall construct a homotopy in an open set from $\widehat{b\circ q}$
to a locally degree-1 map. To construct the latter map, we shall make use of
the following homeomorphism derived from the earthquake introduced by Thurston
(see Thurston [25] and Kerckhoff [11]).

######  Definition 6.3.

Fix $g_{-}\in\mathcal{T}(\Sigma_{-})$ and $g_{+}\in\mathcal{T}(\Sigma_{+})$.
For $j=-,+$, we let
$E_{j}\colon\mathcal{ML}(\Sigma_{j})\to\mathcal{T}(\Sigma_{j})$ be the left
earthquake map, that is, a homeomorphism sending a measured lamination
$\lambda\in\mathcal{ML}(\Sigma_{j})$ to the marked hyperbolic structure
obtained by the left earthquake along $\lambda$ on $(\Sigma_{j},g_{j})$. Then,
we have a homeomorphism $E_{-}\times
E_{+}\colon\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\to\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$.
We define
$\mathcal{E}\colon\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})\to\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$
to be the inverse of $E_{-}\times E_{+}$.

To construct a homotopy, we shall first define its support, which will be done
by using an open neighbourhood of a point contained in the ‘corner’ of the
product of the Thurston compactifications of the components of
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$. We now describe it
more concretely. Let $\Sigma_{1},\dots,\Sigma_{n}$ be the components of
$\Sigma_{-}\sqcup\Sigma_{+}$ that are not thrice-punctured spheres. We
compactify each $\mathcal{T}(\Sigma_{j})$ by attaching
$\mathcal{PML}(\Sigma_{j})$ as its boundary. We call their product
$\prod_{j=1}^{n}(\mathcal{T}(\Sigma_{j})\cup\mathcal{PML}(\Sigma_{j}))$ the
Thurston compactification product of
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ and denote its boundary
by $\mathcal{PML}$. A point of $\mathcal{PML}$ has a form $(x_{j})_{j=1}^{n}$,
where $x_{j}$ is either $\mathcal{T}(\Sigma_{j})$ or
$\mathcal{PML}(\Sigma_{j})$ and at least one of the $x_{j}$ is contained in
the boundary $\mathcal{PML}(\Sigma_{j})$. We put the product topology on the
compactification. We call the subset
$\prod_{j=n}^{n}\mathcal{PML}(\Sigma_{j})$ of the boundary the corner and
denote it by $\mathcal{PML}_{c}$. A sequence $\\{\mathbf{m}_{i}\\}$ of
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ converges to a point in
$\mathcal{PML}_{c}$ after passing to a subsequence if
$\\{\mathbf{m}_{i}|\Sigma_{j}\\}$ diverges for every component $\Sigma_{j}$ of
$\Sigma_{-}\sqcup\Sigma_{+}$ that is not a thrice-punctured sphere.

######  Definition 6.4.

Let $\Lambda$ be a point in $\mathcal{ML}(\Sigma_{-}\sqcup\Sigma_{+})$ not
contained in $D$ such that, for each component $\Sigma_{j}$ of
$\Sigma_{-}\sqcup\Sigma_{+}$ that is not thrice-punctured sphere, the
restriction $\Lambda\cap\Sigma_{j}$ is an arational uniquely ergodic measured
lamination. By setting its $j$-th coordinate to be $[\Lambda|\Sigma_{j}]$, we
define $[\Lambda]\in\mathcal{PML}_{c}$.

We note that by the arationality, the condition that $\Lambda$ is not
contained in $D$ is equivalent to, when $\lambda_{-}=\lambda_{+}=\emptyset$,
the condition that no two components of $\Lambda$ are homotopic, and
otherwise, the condition that no component of the support of $\Lambda$ is
homotopic to an ending lamination.

######  Definition 6.5.

Let $\tau$ be a bi-recurrent train track on $\Sigma_{-}\sqcup\Sigma_{+}$
carrying $\Lambda$ by a weight system $\omega$ in such a way that $\omega$
takes a positive value on every branch of $\tau$. We call an arc connecting
two measured laminations $\mu_{1},\mu_{2}$ carried by $\tau$ a segment when it
is a linear path with regard to the weight system. We note that this notion is
independent of the choice of $\tau$ since the transition function between two
weight systems is linear.

For two measured laminations $\lambda_{1},\lambda_{2}$ carried by $\tau$ with
weight systems $\omega_{1},\omega_{2}$ respectively, we define
$d_{\tau}(\lambda_{1},\lambda_{2})$ to be the sum of the differences of the
weights of $\omega_{1}$ and $\omega_{2}$ on the branches of $\tau$.

From now on until the end of this section, we fix a train track $\tau$ as
above.

In the same way as we did for
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$, for each component
$\Sigma_{j}\ (j=1,\dots,n)$ of $\Sigma_{-}\sqcup\Sigma_{+}$, we consider the
ray compactification of $\mathcal{ML}(\Sigma_{j})$ and regard
$\mathcal{PML}(\Sigma_{j})$ as its boundary at infinity, and define the ray
compactification of $\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$
with boundary at infinity $\mathcal{PML}$ to be the product of the ray
compactification of the components
$\mathcal{ML}(\Sigma_{1}),\dots,\mathcal{ML}(\Sigma_{n})$. As before, we call
$\mathcal{PML}_{c}$ the corner also in this ray compactification, and we see
that a sequence $\\{\lambda_{i}\\}$ in
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$ converges to a point
in $\mathcal{PML}$ after passing to a subsequence if
$\\{\lambda_{i}|\Sigma_{j}\\}$ diverges for every $j=1,\dots,n$.

######  Definition 6.6.

We call a subset $U$ of
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$ a truncated cone if
$U$ consists of all measured laminations $\lambda$ that satisfy the following
conditions.

  1. (a)

The train track $\tau$ carries $\lambda$.

  2. (b)

Each weight of $w(\lambda)$ is greater than a fixed positive constant $K$.
(Recall from Definition 6.5 that we denote the weight system on $\tau$
corresponding to $\lambda$ by $w(\lambda)$.)

We say that a truncated cone $U$ is a truncated cone neighbourhood of
$[\Lambda]$ (supported on $\tau$) when there is a neighbourhood $V$ of
$[\Lambda]$ in the boundary at infinity $\mathcal{PML}$ such that the closure
of $U$ in the ray compactification
$(\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+}))$ coincides with $V$.

The positive weight systems on $\tau$ form an open set in
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$ since $\tau$ is bi-
recurrent, and its ray compactification contains $[\Lambda]$ since each
$\Lambda|\Sigma_{j}$ is arational and uniquely ergodic. Therefore we see that
(the ray compactifications of) the truncated cone neighbourhoods form a basis
of neighbourhoods of $[\Lambda]$ in the ray compactification
$(\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+}))\cup\mathcal{PML}$.

The following is a well-known property of the earthquake map. (See
Papadopoulos [21] for instance.)

######  Lemma 6.7.

The map
$\mathcal{E}\colon\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})\to\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$
is a homeomorphism extending continuously to the identity on the boundary
$\mathcal{PML}$.

We now show some lemmas and their corollaries which will be used in the main
step of the proof of Theorem 3.3.

######  Lemma 6.8.

Let $[\Lambda]$ is a projective lamination in $\mathcal{PML}_{c}$ as in
Definition 6.4. Let $\\{\mathbf{m}_{i}\\}$ be a sequence in
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ converging to
$[\Lambda]$ in the Thurston compactification product. Then $\\{b\circ
q(\mathbf{m}_{i})\\}$ also converges to $[\Lambda]$ in the ray
compactification
$(\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+}))\cup\mathcal{PML}$.

###### ?proofname?.

Let $\Sigma_{j}$ be a component of $\Sigma_{-}\sqcup\Sigma_{+}$. Since
$\\{\mathbf{m}_{i}\\}$ converges to $[\Lambda]$ contained in the corner, the
restriction $\\{\mathbf{m}_{i}|\Sigma_{j}\\}$ converges in the Thurston
compactification to $[\Lambda|\Sigma_{j}]\in\mathcal{PML}(\Sigma_{j})$. Let
$k_{i}$ be a shortest simple closed geodesic in
$(\Sigma_{j},\mathbf{m}_{i}|\Sigma_{j})$. Since $\Lambda|\Sigma_{j}$ is
arational and uniquely ergodic, there is a sequence of positive numbers
$\\{s_{i}\\}$ going to $0$ such that $\\{s_{i}k_{i}\\}$ converges to the
measured lamination $\Lambda|\Sigma_{j}$. By Corollary 4.2, $\\{\iota(b\circ
q(\mathbf{m}_{i}),k_{i})\\}$ is bounded. By the continuity of intersection
numbers combined with the arationality and the unique ergodicity of
$\Lambda|\Sigma_{j}$, we see that either $\\{b\circ q(\mathbf{m}_{i})\\}$
converges to $s(\Lambda|\Sigma_{j})$ for some positive scalar $s$, or
$\\{b\circ q(\mathbf{m}_{i})\\}$ converges in the ray compactification to the
point at infinity $[\Lambda|\Sigma_{j}]\in\mathcal{PML}(\Sigma_{j})$.

It remains to show that the former case cannot happen. Suppose, seeking a
contradiction, that $\\{b\circ q(\mathbf{m}_{i})\\}$ converges to
$s(\Lambda|\Sigma_{j})$. Then, by Lemma 5.4, we see that
$\\{q(\mathbf{m}_{i})|\pi_{1}(\Sigma_{j})\\}$ converges algebraically. Let the
constant $\delta>0$ given in Lemma 4.3 for $\epsilon=1$. We then take a simple
closed curve $c$ on $\Sigma_{j}$ approximating $|\Lambda|\Sigma_{j}|$ such
that $\iota(c,s\Lambda)<\delta$. Since $\\{\mathbf{m}_{i}|\Sigma_{j}\\}$
converges to $[\Lambda|\Sigma_{j}]$ in the Thurston compactification, we see
that $\mathrm{length}_{\mathbf{m}_{i}}(c)$ goes to $\infty$ as
$i\longrightarrow\infty$. By Lemma 4.3, this implies that
$\mathrm{length}_{q(\mathbf{m}_{i})}(c)$ also goes to $\infty$, contradicting
the fact that $\\{q(\mathbf{m}_{i})|\pi_{1}(\Sigma_{j})\\}$ converges.

Thus we have shown that $\\{b\circ q(\mathbf{m}_{i})|\Sigma_{j}\\}$ converges
to $[\Lambda|\Sigma_{j}]$ for every $j=1,\dots,n$, and hence $\\{b\circ
q(\mathbf{m}_{i})\\}$ converges to $[\Lambda]$ in the ray compactification. ∎

The lemma implies the following corollary.

######  Corollary 6.9.

Let $[\Lambda]$ be a projective measured lamination in $\mathcal{PML}_{c}$ as
given in Definition 6.4. For any truncated cone neighbourhood $U$ of
$[\Lambda]$, there is a neighbourhood $V$ of $[\Lambda]$ in the Thurston
compactification product of
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ such that $b\circ
q(V\cap(\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})))$ is contained
in $U$.

###### ?proofname?.

Consider a sequence $\\{\mathbf{m}_{i}\\}$ in
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ converging to
$[\Lambda]$ in the Thurston compactification product. Then by Lemma 6.8, for
any given truncated cone neighbourhood $U$, the sequence $\\{b\circ
q(\mathbf{m}_{i})\\}$ is contained in $U$ for sufficiently large $i$. This
implies the existence of a neighbourhood of $[\Lambda]$ as desired. ∎

In a more general case where we do not assume that the limit of
$\\{\mathbf{m}_{i}\\}$ is $[\Lambda]$, we have the following.

######  Lemma 6.10.

Let $\\{\mathbf{m}_{i}\\}$ be a sequence in
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ which converges to a
point $(x_{j})\in\mathcal{PML}$ in the Thurston compactification product.
Suppose moreover that $\\{b\circ q(\mathbf{m}_{i})\\}$ converges to a point
$(y_{j})$ in the ray compactification. For each $x_{j}$ that lies on the
boundary at infinity $\mathcal{PML}(\Sigma_{j})$, let $\mu_{j}$ be a measured
lamination with $[\mu_{j}]=x_{j}$. Then $y_{j}$ lies either in
$\mathcal{ML}(\Sigma_{j})$ and $\iota(y_{j},\mu_{j})=0$ or on the boundary at
infinity $\mathcal{PML}(\Sigma_{j})$ and is represented by a measured
lamination $\nu_{j}$ with $\iota(\mu_{j},\nu_{j})=0$. Moreover, for every $j$
such that $y_{j}$ lies in $\mathcal{PML}(\Sigma_{j})$, the point $x_{j}$ also
lies in $\mathcal{PML}(\Sigma_{j})$.

###### ?proofname?.

Suppose that $x_{j}=[\mu_{j}]$ lies in $\mathcal{PML}(\Sigma_{j})$. Let
$\mu^{0}_{j}$ be a connected component of $\mu_{j}$. If $\mu_{j}^{0}$ is a
simple closed curve, either (a)
$\mathrm{length}_{\mathbf{m}_{i}}(\mu_{j}^{0})\rightarrow 0$ or (b) there are
simple closed curves $d_{i}$ on $\Sigma_{j}$ with bounded
$\mathrm{length}_{\mathbf{m}_{i}}(d_{i})$ and positive numbers
$r_{i}\longrightarrow 0$ such that $\\{r_{i}d_{i}\\}$ converges to a measured
lamination $\hat{\mu}_{j}$ containing $\mu_{j}^{0}$. The latter curve $d_{i}$
can be chosen to be a shortest simple closed curve on $\Sigma_{j}$ with
respect to $\mathbf{m}_{i}$ intersecting $\mu_{j}^{0}$. Moreover, if
$\mu_{j}^{0}$ is not a simple closed curve, the condition (b) always holds.

In the case (a), Lemma 4.1 implies that $\iota(\mu_{j}^{0},b\circ
q(\mathbf{m}_{i})|\Sigma_{j})\longrightarrow 0$, which implies that
$\iota(\mu_{j}^{0},y_{j})=0$ when $y_{j}$ lies in $\mathcal{ML}(\Sigma_{j})$
and $\iota(\mu_{j}^{0},\nu_{j})=0$ when $x_{j}=[\nu_{j}]$ lies on the boundary
at infinity, by the continuity of the intersection number. In the case (b),
Lemma 4.1 implies that $\\{\iota(d_{i},b\circ q(\mathbf{m}_{i}))\\}$ is
bounded. Since $r_{i}$ tends to $0$, by the continuity of the intersection
number, we have $\iota(\hat{\mu}_{j},y_{j})=0$, and hence
$\iota(\mu^{0}_{j},y_{j})=0$ when $y_{j}$ lies in $\mathcal{ML}(\Sigma_{j})$,
and in the same way, $\iota(\mu^{0}_{j},\nu_{j})=0$ when $x_{j}=[\nu_{j}]$
lies on the boundary at infinity. Thus, we have $\iota(\mu_{j}^{0},y_{j})=0$
or $\iota(\mu^{0}_{j},\nu_{j})=0$ for every connected component $\mu_{j}^{0}$
of $\mu_{j}$, which implies that $\iota(\mu_{j},y_{j})=0$ or
$\iota(\mu_{j},\nu_{j})=0$.

To show the last statement, suppose that $x_{j}$ lies in
$\mathcal{T}(\Sigma_{j})$. Then, by Lemma 5.3, we see that
$\\{q(\mathbf{m}_{i})|\pi_{1}(\Sigma_{j})\\}$ converges. Since
$\\{\mathbf{m}_{i}|\Sigma_{j}\\}$ converges by assumption, the boundary
component $\Sigma_{j}^{i}$ of the convex core
$C(\mathbb{H}^{3}/q(\mathbf{m}_{i}))$ corresponding to $\Sigma_{j}$ converges
geometrically to a boundary component $\Sigma_{j}^{\infty}$ of the convex core
of the geometric limit, which is homotopic to an algebraic locus of
$\Sigma_{j}$ (see the argument of [16, p.103]). Then the $j$-th component of
$b\circ q(\mathbf{m}_{j})$ converges to the bending lamination of
$\Sigma_{j}^{\infty}$, and hence $y_{j}$ must be inside
$\mathcal{ML}(\Sigma_{j})$. This shows that if $y_{j}$ lies on the boundary at
infinity, then so does $x_{j}$. ∎

By a similar argument, we can also show the following proposition.

######  Proposition 6.11.

Let $[\Lambda]$ be a projective lamination as in Definition 6.4. Let
$\\{\mathbf{m}_{i}\\}$ be a sequence in
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ such that $\\{b\circ
q(\mathbf{m}_{i})\\}$ converges to a point at infinity represented by
$[\Lambda]$ in the ray compactification. Then $\\{\mathbf{m}_{i}\\}$ converges
to $[\Lambda]$ in the Thurston compactification product of
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$.

###### ?proofname?.

We have only to show that any subsequence of $\\{\mathbf{m}_{i}\\}$ has a
subsequence converging to $[\Lambda]$ in the Thurston compactification
product. Passing to a subsequence, we can assume that $\\{\mathbf{m}_{i}\\}$
converges to either a point $\mathbf{n}$ in
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ or a projective
lamination $(y_{j})\in\mathcal{PML}$ in the Thurston compactification product.
In the former case, by the continuity of the function $b$ due to [10],
$\\{b\circ q(\mathbf{m}_{i})\\}$ converges to $b\circ q(\mathbf{n})$,
contradicting our assumption.

Suppose that $\\{\mathbf{m}_{i}\\}$ converges to $(y_{j})$ in the Thurston
compactification product. Since $[\Lambda]$ lies in the corner, and
$\Lambda\cap\Sigma_{j}$ is arational and uniquely ergodic for every $j$, by
Lemma 6.10, we have we have $y_{j}=\Lambda\cap\Sigma_{j}$, and hence
$(y_{j})=[\Lambda]$. This completes the proof. ∎

Since truncated cone neighbourhoods form a basis of neighbourhoods of
$[\Lambda]$ as remarked before, we have the following corollary.

######  Corollary 6.12.

Let $V$ be a neighbourhood of $[\Lambda]$ in the Thurston compactification
product of $\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ for
$[\Lambda]$ as in Definition 6.4. Then, there is a truncated cone
neighbourhood $U$ of $[\Lambda]$ in the ray compactification such that
$(b\circ q)^{-1}(U)$ is contained in
$V\cap(\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+}))$.

By combining these results, we obtain the following technical proposition,
which constitutes an essential step for our construction of a homotopy.

######  Proposition 6.13.

Let $\Lambda$ be a measured lamination given in Definition 6.4. Then there are
three nested truncated cone neighbourhoods $U_{0}\subset U_{1}\subset U_{2}$
which satisfy the following.

  1. (1)

Every measured lamination in $U_{2}$ satisfies the condition (c) of Theorem
3.1 (with $\lambda_{-}$ and $\lambda_{+}$).

  2. (2)

Let $V_{1}$ be $(b\circ q)^{-1}(U_{1})$. Then there is an open set $V_{2}$
containing the closure $\bar{V}_{1}$ such that both $b\circ q(V_{2})$ and
$\mathcal{E}(V_{2})$ are contained in $U_{2}$.

  3. (3)

Neither $\mathcal{E}$ nor $b\circ q$ maps a point outside $V_{1}$into $U_{0}$.

  4. (4)

For any point $m\in V_{2}\setminus V_{1}$, the segment connecting $b\circ
q(m)$ and $\mathcal{E}(m)$ is disjoint from $U_{0}$.

###### ?proofname?.

If a measured lamination on $\Sigma_{-}\sqcup\Sigma_{+}$ does not satisfy the
condition (c) of Theorem 3.1, then either it has a component homotopic to a
component of $\lambda_{-}$ or $\lambda_{+}$, or it has two components which
are homotopic in $S\times[0,1]$. Since $\Lambda$ is arational and is not
contained in $D$, every sufficiently small truncated cone neighbourhood of
$[\Lambda]$ contains no such measured laminations. Therefore, by choosing a
truncated cone neighbourhood $U_{2}$ to be sufficiently small, the condition
(1) is satisfied.

By Lemmas 6.7 and 6.9, there is a neighbourhood $V_{2}$ of $[\Lambda]$ in the
Thurston compactification product such that both $\mathcal{E}(V_{2})$ and
$b\circ q(V_{2})$ are contained in $U_{2}$. By Lemmas 6.7 and 6.12, we can
take a neighbourhood $U_{1}^{\prime}$ of $[\Lambda]$ such that both
$\mathcal{E}^{-1}(U_{1}^{\prime})$ and $(b\circ q)^{-1}(U_{1}^{\prime})$ are
contained in $V_{2}$. Again by Lemmas 6.7, 6.9 and 6.12, we can take a
neighbourhood $V_{1}^{\prime}$ of $[\Lambda]$ in the Thurston compactification
product such that $\overline{V_{1}^{\prime}}$ is contained in
$\mathcal{E}^{-1}(U_{1}^{\prime})\cap(b\circ q)^{-1}(U_{1}^{\prime})$, and a
truncated cone neighbourhood $U_{1}$ of $[\Lambda]$ in the ray
compactification such that both $\mathcal{E}^{-1}(U_{1})$ and $(b\circ
q)^{-1}(U_{1})$ are contained in $V_{1}^{\prime}$. These $U_{1},U_{2},V_{2}$
and $V_{1}=(b\circ q)^{-1}(U_{1})$ satisfy the condition (2).

Now, we shall show that we can take a truncated cone neighbourhood $U_{0}$ of
$[\Lambda]$ in the ray compactification satisfying (3) and (4). Let
$\\{\mathbf{m}_{i}\\}$ be an arbitrary sequence in
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ lying outside $V_{1}$.
Since $V_{1}$ is a neighbourhood of $[\Lambda]$ in the Thurston
compactification product, Lemma 6.7 implies that
$\\{\mathcal{E}(\mathbf{m}_{i})\\}$ cannot converge to $[\Lambda]$ in the ray
compactification. Similarly, by Proposition 6.11, we see that $\\{b\circ
q(\mathbf{m}_{i})\\}$ cannot converge to $[\Lambda]$ in the ray
compactification either. Therefore, if we take $U_{0}$ to be a sufficiently
small truncated cone neighbourhood of $[\Lambda]$, then the condition (3) is
satisfied.

We next show that $U_{0}$ can be taken to satisfy the condition (4). Let
$\\{\mathbf{m}_{i}\\}$ be a sequence in $V_{2}\setminus V_{1}$, and
$\lambda_{i}$ a point on the segment connecting $b\circ q(\mathbf{m}_{i})$ and
$\mathcal{E}(\mathbf{m}_{i})$ such that $\\{\lambda_{i}\\}$ converges to
$[\Lambda]$ in the ray compactification. Taking a subsequence, we can assume
that either $\\{\mathbf{m}_{i}\\}$ converges to a point
$\mathbf{n}\in\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ or does
not have a convergent sequence inside
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$, and converges to a
point $(w_{j})\in\mathcal{PML}$ in the Thurston boundary. The boundary point
$(w_{j})\in\mathcal{PML}$ in the latter case is distinct from $[\Lambda]$
since $\\{\mathbf{m}_{i}\\}$ lies outside $V_{1}$. In the former case, passing
to a subsequence $\\{\lambda_{i}\\}$ converges to a point inside
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$ which lies on a
segment connecting $\mathcal{E}(\mathbf{n})$ and $b\circ q(\mathbf{n})$,
contradicting our assumption. In the latter case, by Lemma 6.7, we see that
$\\{\mathcal{E}(\mathbf{m}_{i})\\}$ converges to
$(w^{\prime}_{j})\in\mathcal{PML}$ such that
$w_{j}^{\prime}=\mathcal{(}w_{j})$ if $w_{j}\in\mathcal{T}(\Sigma_{j})$ and
$w_{j}^{\prime}=w_{j}$ otherwise. The sequence $\\{b\circ
q(\mathbf{m}_{i})\\}$ converges to a point $(y_{j})$ in the ray
compactification passing to a subsequence. If there is $j$ such that
$w_{j}^{\prime}$ lies in $\mathcal{ML}(\Sigma_{j})$, then $y_{j}$ must lie in
$\mathcal{ML}(\Sigma_{j})$. Then the limit of $\lambda_{j}$ also lies in
$\mathcal{ML}(\Sigma_{j})$ and contradicts the assumption that $[\Lambda]$ is
a corner point. Therefore, we have $(w_{j}^{\prime})=(w_{j})$ and
$w_{j}\in\mathcal{PML}(\Sigma_{j})$. Let $\mu$ be a measured lamination on
$\Sigma_{-}\sqcup\Sigma_{+}$ representing $(w_{j})$. By Lemma 6.10 we have
$\iota(y_{j},\mu|\Sigma_{j})=0$ if $y_{j}$ lies inside
$\mathcal{ML}(\Sigma_{j})$ and is represented by
$\mu_{j}^{\prime}\in\mathcal{ML}(\Sigma_{j})$ with $y_{j}=[\mu_{j}^{\prime}]$
and $\iota(\mu|\Sigma_{j},\mu_{j}^{\prime})=0$ if $y_{j}$ lies in
$\mathcal{PML}(\Sigma_{j})$. Recall that $\lambda_{i}$ lies on the segment
between $\mathcal{E}(\mathbf{m}_{i})$ and $b\circ q(\mathbf{m}_{i})$.
Therefore $\lambda_{j}$ converges to a point $(z_{j})$ in the ray
compactification such that $z_{j}$ is a weighted union of $\mu|\Sigma_{j}$ and
either $y_{j}$ or $\mu_{j}^{\prime}$. Therefore, we have
$\iota(\mu|\Sigma_{j},z_{j})=0$ if $z_{j}$ lies inside
$\mathcal{ML}(\Sigma_{j})$ and is represented by
$\nu_{j}\in\mathcal{ML}(\Sigma_{j})$ with $z_{j}=[\nu_{j}]$ and
$\iota(\nu_{j},\mu|\Sigma_{j})=0$ if $z_{j}$ lies in
$\mathcal{PML}(\Sigma_{j})$. Since $[\mu]\neq[\Lambda]$ and
$\Lambda|\Sigma_{j}$ is arational, we have $(z_{j})\neq[\Lambda]$, and we are
led to a contradiction. Thus we have shown that by taking $U_{0}$ to be
sufficiently small, the condition (4) also holds. This completes the proof. ∎

To define a locally degree-1 map
$\widehat{F}\colon\widehat{\mathrm{T}}\to\widehat{\mathrm{ML}}$, and a
homotopy from $\widehat{b\circ q}$ to $\widehat{F}$, we shall first define a
map
$F\colon\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})\to\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$
which induces $\widehat{F}$.

######  Definition 6.14.

Let
$U_{0},U_{1},U_{2}\subset\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$
and $V_{1},V_{2}\subset\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$
be open sets given in Proposition 6.13, and let $\tau$ be the train track
given in Definition 6.5. Let
$F\colon\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})\to\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$
be a continuous map defined as follows:

  1. (i)

For $x$ outside $V_{2}$, we define $F(x)=b\circ q(x)$.

  2. (ii)

For $x\in V_{1}$, we define $F(x)=\mathcal{E}(x)$.

  3. (iii)

For $x\in V_{2}\setminus V_{1}$, letting $t(x)$ be
$\displaystyle\frac{d_{\tau}(x,\overline{V_{1}})}{d_{\tau}(x,\overline{V_{1}})+d_{\tau}(x,\overline{V_{2}^{c}})}$,
we define $F(x)$ to be the point dividing the segment connecting
$\mathcal{E}(x)$ and $b\circ q(x)$ internally by $t(x):1-t(x)$ in the weight
system coordinates of $\tau$.

######  Lemma 6.15.

For the map $F$ defined above, there is no sequence in
$\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$ diverging to infinity,
whose image under $F$ has a subsequence converging in
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus D$.

######  ?proofname?.

Since both $\mathcal{E}$ and $b\circ q$ are proper, $\mathcal{E}$ as a map to
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$ and $b\circ q$ as a
map to $\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus D$, we
have only to consider the case when $\\{\mathbf{m}_{i}\\}$ lies in
$V_{2}\setminus V_{1}$. Let $\\{\mathbf{m}_{i}\\}$ be a sequence in
$V_{2}\setminus V_{1}$ which does not have a convergent subsequence. We can
assume that it converges in the Thurston compactification product to a point
$(y_{j})$ in $\mathcal{PML}$, passing to a subsequence. We need to show that
$\\{F(\mathbf{m}_{i})\\}$ does not have a convergent subsequence in
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus D$.

Recall from Proposition 6.13 that both $\\{\mathcal{E}(\mathbf{m}_{i})\\}$ and
$\\{b\circ q(\mathbf{m}_{i})\\}$ lie in $U_{2}$. Since $\mathcal{E}$ is a
proper map to $\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$, the
sequence $\\{\mathcal{E}(\mathbf{m}_{i})\\}$ diverges to infinity, necessarily
within $U_{2}$. If $\\{b\circ q(\mathbf{m}_{i})\\}$ also diverges to infinity
(within $U_{2}$), the segment connecting $\mathcal{E}(\mathbf{m}_{i})$ and
$b\circ q(\mathbf{m}_{i})$ also diverges to infinity within $U_{2}$ as
$i\longrightarrow\infty$, and we are done. It remains to deal with the case
when $\\{b\circ q(\mathbf{m}_{i})\\}$ converges to a point $\nu$ in $D$ after
passing to a subsequence. By the part (1) of Proposition 6.13, $\nu$ must
satisfy the condition (c) of Theorem 3.1, and hence contains a compact leaf
with weight larger than or equal to $\pi$. We denote the union of all such
components of $\nu$ by $\nu_{0}$. Let $\Sigma_{k}$ be a component of
$\Sigma_{-}\sqcup\Sigma_{+}$ containing a component of $\nu_{0}$. Then $y_{k}$
lies in $\mathcal{PML}(\Sigma_{k})$ since otherwise the component of the
convex core boundary corresponding to $\Sigma_{k}$ converges geometrically
without giving rise to a new parabolic curve, as argued in the previous
section.

We first remark the following, which was just a restatement of Lemma 5.4.

######  Claim 6.16.

Let $\phi_{i}\colon\pi_{1}(S)\to{\rm PSL}_{2}(\mathbb{C})$ be a representation
corresponding to $q(\mathbf{m}_{i})$. Then, $\\{\phi_{i}\\}$ converges to some
$\psi\in\mathsf{AH}(S)$ as $i\longrightarrow\infty$.

Let $P_{i}$ be a shortest pants decomposition of
$(\Sigma_{-}\sqcup\Sigma_{+},\mathbf{m}_{i})$. Let $P_{\infty}$ be the
Hausdorff limit of $\\{P_{i}\\}$ (after passing to a subsequence), which is a
geodesic lamination. Invoking an argument which we used in the preceding
section, we can show the following.

######  Claim 6.17.

Every minimal component of $P_{\infty}$ is a simple closed curve. The
geometric limit $M_{\infty}$ has neither a new geometrically infinite end
(i.e. one not corresponding to that of
$(\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))_{0}$) nor a torus cusp.

###### ?proofname?.

Suppose, seeking a contradiction, that $P_{\infty}$ has a minimal component
$\varrho$ which is not a simple closed curve. Let $S(\varrho)\subset S$ be the
minimal supporting surface of $\varrho$. Then $M_{\infty}$ must have a new
geometrically infinite end which lifts to the algebraic limit. (This follows
from the fact in Minsky’s model of $\mathbb{H}^{3}/\phi_{i}(\pi_{1}(S))$, the
subsurface$S(\varrho)$ must support a tight geodesic whose length goes to
$\infty$ in this case.)

Now, we put a basepoint on the algebraic locus $f_{\infty}(S)$, and consider a
geometric limit $M_{\infty}$ containing the algebraic limit of
$\\{\phi_{i}\\}$. As in Section 5.2.1, by taking a non-peripheral simple
closed curve $d$ on $S(\varrho)$ with $\iota(d,\varrho)<\delta$ for the
constant $\delta$ given in Lemma 4.3 with $\epsilon=1$, we are led to a
contradiction. Thus we have shown that every minimal component of $P_{\infty}$
is a simple closed curve. This argument also shows that $M_{\infty}$ cannot
have a nearest simply degenerate end, for such an end can be lifted to the
algebraic limit by 5.2.

Next suppose that $M_{\infty}$ has a torus cusp. Since there is no nearest
simply degenerate end for $M_{\infty}$, if there are torus cusps, we can take
a nearest one by 5.2. By repeating the arguments of Section 5.2.2, which can
be applied also to our situation as remarked there, we get a contradiction.

Since $M_{\infty}$ has neither a nearest simply degenerate end nor a nearest
torus cusp, by 5.2, we see that $M_{\infty}$ does not have a new geometrically
infinite end. ∎

We define a subset $P^{0}_{\infty}$ of $P_{\infty}$ to be the subset
consisting of simple closed curves whose lengths with respect to
$\mathbf{m}_{i}$ go to $0$ as $i\longrightarrow\infty$.

Since $\\{\phi_{i}\\}$ converges to $\psi\in\mathsf{AH}(S)$ as mentioned
above, by Corollary 2.3, the claim above implies that the convergence is
strong.

######  Claim 6.18.

The multi-curve $P_{\infty}^{0}$ is contained in $\nu_{0}$.

###### ?proofname?.

Let $c$ be a component of $P_{\infty}^{0}$. We regard $c$ as lying on a
compact core of $\mathbb{H}^{3}/\psi(\pi_{1}(S)))$. Then
$\mathrm{length}_{\mathbf{m}_{i}}(c)$ goes to $0$ by the definition of
$P_{\infty}^{0}$, and hence it represents a core curve of a $\mathbb{Z}$-cusp
neighbourhood lying outside the compact core. Since $\\{\phi_{i}\\}$ converges
to $\psi$ strongly and $M_{\infty}$ does not have a new geometrically infinite
end by 6.17, the $\mathbb{Z}$-cusp has geometrically finite ends on its both
sides. We can apply Lemma 5.1 to see that the bending angle along $c$
converges to $\pi$, and hence $c$ is contained in $\nu_{0}$. ∎

We can further see the following.

######  Claim 6.19.

Let $C$ be a minimal component of $P_{\infty}$ not contained in
$P^{0}_{\infty}$. If $P_{\infty}$ does not contain any other minimal component
homotopic to $C$ in $S\times[0,1]$, then the twisting parameter along $C$ of
$\mathbf{m}_{i}$ is bounded as $i\longrightarrow\infty$. If $P_{\infty}$ has
two distinct minimal components $C$ and $C^{\prime}$ homotopic to each other
in $S\times[0,1]$, then the difference of the twisting parameters of
$\mathbf{m}_{i}$ along $C$ and $C^{\prime}$ is bounded as
$i\longrightarrow\infty$.

###### ?proofname?.

Suppose first that there is no other component in $P_{\infty}$ homotopic to
$C$ in $S\times I$. If the twisting parameter of $\mathbf{m}_{i}$ along $C$
goes to $\infty$ after passing to a subsequence, then by the part (1) of Lemma
2.4, the geometric limit $M_{\infty}$ has a corresponding torus cusp. This
contradicts 6.17. Also in the latter case when $C$ is homotopic to
$C^{\prime}$ in $S\times I$, by the same argument as above involving Lemma
2.4, we see that if the difference of twisting parameters of $\mathbf{m}_{i}$
goes to $\infty$, then $M_{\infty}$ must have a torus cusp, contradicting
6.17.∎

This claim implies the following.

######  Claim 6.20.

The projective lamination $y_{k}$ is supported in $P^{0}_{\infty}$.

###### ?proofname?.

By the definition of the topology of the Thurston compactification product, we
see that $|y_{k}|$ is contained in $P_{\infty}$. By 6.17, every minimal
component of $P_{\infty}$ is a simple closed curve. If $|y_{j}|$ has a
component $C$ contained in $P_{\infty}$ but not in $P^{0}_{\infty}$, then the
twist parameter of $\mathbf{m}_{i}$ around $C$ must go to either $\infty$ or
$-\infty$. This contradicts 6.19 unless there is another component of
$P_{\infty}$ homotopic to $C$ in $S\times I$. If there is such a component
$C^{\prime}$, then the support of some coordinate $y_{l}$ of $(y_{j})$ must
contain $C^{\prime}$ as a component by 6.19 again. On the other hand,
$(y_{j})$ is contained in $U_{2}$, and hence cannot have two components of
projective laminations homotopic to each other in $S\times I$. This is a
contradiction. ∎

Having proved these claims, we can now complete the proof of Lemma 6.15. Since
$|y_{k}|$ is contained in $P^{0}_{\infty}$ by 6.20, it lies in $\nu_{0}$ by
6.18. Therefore the segment connecting $\mathcal{E}(\mathbf{m}_{i})$ and
$b\circ q(\mathbf{m}_{i})$ converges uniformly on any compact set to a ray
entirely lying in $D$. This completes the proof. ∎

Now we can define $\widehat{F}$ which we mentioned before. By Lemma 6.15,
$F\colon\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})\to\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$
induces a unique continuous map
$\widehat{F}\colon\widehat{\mathrm{T}}\to\widehat{\mathrm{ML}}$ similarly to
Definition 6.1. We shall show that $\widehat{F}$ has degree $1$. We use the
symbols
$\widehat{\mathcal{E}}\colon\widehat{\mathrm{T}}\to\widehat{\mathrm{ML}}$ to
denote the map induced by $\mathcal{E}$.

######  Proposition 6.21.

The map $\widehat{F}\colon\widehat{\mathrm{T}}\to\widehat{\mathrm{ML}}$ has
degree $1$.

###### ?proofname?.

Let $\widehat{U}_{0}$ be the open set in $\widehat{\mathrm{ML}}$ corresponding
to $U_{0}\setminus D$, where $U_{0}$ is the truncated cone neighbourhood in
Proposition 6.13. We shall show that the restriction of $\widehat{F}$ to
$\widehat{F}^{-1}(\widehat{U}_{0})$ has degree $1$, which immediately implies
that $\widehat{F}$ has degree $1$.

Since $F|\mathcal{E}^{-1}(U_{0})$ coincides with
$\mathcal{E}|\mathcal{E}^{-1}(U_{0})$, it is a homeomorphism to its image. We
have only to show that there is no point outside $\mathcal{E}^{-1}(U_{0})$
which is mapped into $U_{0}$. Since $F|V_{1}$ coincides with
$\mathcal{E}|V_{1}$, there are no points in
$V_{1}\setminus\mathcal{E}^{-1}(U_{0})$ mapped into $U_{0}$ by $F$. On the
other hand, the conditions (3) and (4) in Proposition 6.13 guarantee that no
points outside $V_{1}$ are mapped into $U_{0}$ by $F$. This completes the
proof. ∎

The next step is to show that there is a homotopy between $\widehat{b\circ q}$
and $\widehat{F}$. We shall first construct a homotopy between $b\circ q$ and
$F$.

We define a homotopy
$H\colon\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})\times[0,1]\to\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})$
from $b\circ q$ to $F$ as follows.

  1. (1)

For any $x$ outside $V_{2}$ and any $s\in[0,1]$, $H(x,s)$ is defined to be
$b\circ q(x)$.

  2. (2)

For any $x$ in $V_{2}$ and any $s\in[0,1]$, $H(x,s)$ is defined to be the
point dividing the segment connecting $b\circ q(x)$ and $F(x)$ internally by
$s:1-s$.

######  Lemma 6.22.

For any sequence $\\{t_{i}\in[0,1]\\}$ and any
$\\{\mathbf{m}_{i}\\}\subset\mathcal{T}(\Sigma_{-})\times\mathcal{T}(\Sigma_{+})$
diverging to infinity, $\\{H(\mathbf{m}_{i},t_{i})\\}$ cannot have a
subsequence converging in
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus D$ as
$i\longrightarrow\infty$.

###### ?proofname?.

The proof is quite similar to that of Lemma 6.15. We can assume that
$\\{\mathbf{m}_{i}\\}$ either lies outside $V_{2}$, in $V_{2}\setminus V_{1}$,
or in $V_{1}$, passing to a subsequence. In the case when
$\\{\mathbf{m}_{i}\\}$ lies outside $V_{2}$, the statement follows immediately
from the properness of $b\circ q$ as a map to
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus D$. In the
case when $\\{\mathbf{m}_{i}\\}$ lies in either $V_{2}\setminus V_{1}$ or
$V_{1}$, the point $H(\mathbf{m}_{i},t_{i})$ lies in the segment connecting
$b\circ q(\mathbf{m}_{i})$ and $\mathcal{E}(\mathbf{m}_{i})$. By the same
argument as in the proof of Lemma 6.15, such a segment converges to a ray on
$D$ as $i\longrightarrow\infty$. This shows that
$\\{H(\mathbf{m}_{i},t_{i})\\}$ cannot have a subsequence converging in
$\mathcal{ML}(\Sigma_{-})\times\mathcal{ML}(\Sigma_{+})\setminus D$. ∎

This lemma immediately implies the following.

######  Corollary 6.23.

The homotopy $H$ induces a homotopy
$\widehat{H}\colon\widehat{\mathrm{T}}\to\widehat{\mathrm{ML}}$ between
$\widehat{b\circ q}$ to $\widehat{\mathcal{E}}$.

Combining this corollary with Proposition 6.21, we conclude that
$\widehat{b\circ q}$ has degree $1$, which in turn implies that $b\circ q$ has
degree $1$. This completes the proof of the latter half of Theorem 3.3.

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