Cyril Lecuire

Ken ' Ichi

Ohshika Thurston's

Ken ' Ichi Ohshika

THURSTON'S BOUNDED IMAGE THEOREM

come

Introduction

From the late 1970s to the early 1980s, Thurston gave lectures on his uniformisation theorem for Haken manifolds ( [START_REF] Thurston | The Geometry and Topology of Three-manifolds: Lecture Notes from Princeton University[END_REF][START_REF] Thurston | Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds[END_REF]). The theorem states that every atoroidal Haken 3-manifold with its (possibly empty) boundary consisting only of incompressible tori admits a complete hyperbolic metric in its interior. His proof of this theorem is based on an induction making use of a hierarchy for Haken manifolds invented by Waldhausen [START_REF] Waldhausen | On irreducible 3-manifolds which are sufficiently large[END_REF], i.e., a system of incompressible surfaces cutting the manifold down to balls, together with Maskit's combination theorem (see for instance [START_REF] Maskit | Grundlehren der Mathematischen Wissenschaften[END_REF]§VII]).

For simplicity, we now focus on the case of closed atoroidal Haken manifolds. In the last step of the induction, we are in the situation where N is a closed atoroidal Haken manifold obtained from a 3-manifold M with non-empty boundary (without torus components) by pasting ∂M to itself by an orientation reversing involution. The induction hypothesis guarantees the existence of a convex cocompact hyperbolic structure on M . There, Thurston used the so-called bounded image theorem to find a convex compact hyperbolic structure on M , obtained by quasi-conformally deforming the given hyperbolic structure, which can be pasted up along ∂M to give a hyperbolic structure on N .

Let us explain the setting in more detail. Let M be an atoroidal Haken manifold with an even number of boundary components all of which are incompressible. In the same way as we assumed that N is closed, we assume that no boundary component of M is a torus, for simplicity. Suppose that there is an orientation-reversing involution ι : ∂M → ∂M taking each component of ∂M to another one. Let N be the closed manifold obtained from M by identifying the points on ∂M with their images under ι. Suppose moreover that N is also atoroidal.

We assume, as the hypothesis of induction, that M admits a convex compact hyperbolic structure; in other words, that the interior of M is homeomorphic to H 3 /Γ for a convex cocompact Kleinian group Γ. The space of convex compact hyperbolic structures on M , which is not empty by assumption, modulo isotopy is parameterised by T(∂M ), as can be seen in the works of Ahlfors, Bers, Kra, Maskit, Marden and Sullivan. From each convex compact hyperbolic structure on M , by taking the covering of M associated with each component S of ∂M , we get a quasi-Fuchsian group isomorphic to π 1 (S), and by considering the second coordinate of the parameterisation T(S) × T( S) of the quasi-Fuchsian space, we obtain a map from T(∂M ) to T( S), where T( S) denotes the Teichmüller space of S with orientation reversed. By considering this for every component of ∂M , we get a map σ : T(∂M ) → T( ∂M ) called the skinning map, where T( ∂M ) denotes the product of T( S) for the components S of ∂M . Since ι is orientationreversing, it induces a homeomorphism ι * : T( ∂M ) → T(∂M ).

Then the bounded image theorem can be stated as follows.

Theorem 1.1. Suppose that M is a compact (orientable) atoroidal Haken manifold having an even number of boundary components all of which are incompressible and none of which are tori, and assume that M is not homeomorphic to an I-bundle over a closed surface. Assume moreover that M admits a convex compact hyperbolic structure. Suppose that there is an orientation reversing involution ι : ∂M → ∂M taking each component of ∂M to another component, and that by pasting each component of ∂M to its image under ι, we get a closed atoroidal manifold N . Then there exists n ∈ N depending only on the topological type of M such that the image of (ι * • σ) n is bounded (i.e. precompact) in T(∂M ).

There are several expository papers and books on Thurston's uniformisation theorem ( [START_REF] Morgan | On Thurston's uniformization theorem for three-dimensional manifolds[END_REF][START_REF] Otal | s hyperbolization of Haken manifolds[END_REF][START_REF] Kapovich | Hyperbolic manifolds and discrete groups[END_REF] among others). In all of them, a weaker version of the bounded image theorem called the bounded orbit theorem, which is sufficient for the proof of the uniformisation theorem, was proved and used, instead of this original one.

Up to now, no complete proof of the bounded image theorem as stated above was known. Kent [START_REF] Kent | Skinning maps[END_REF] gave a proof of this theorem under the assumption that M is acylindrical, in which case the deformation space of hyperbolic structures on M is compact.

The purpose of this paper is to give a proof of the original bounded image theorem. Our argument relies on recent progress in Klenian group theory, in particular, the embedding of partial cores in the geometric limit from [START_REF] Brock | Windows, cores and skinning maps[END_REF], the relation between the presence of short curves and their relative positions and the behaviour of ends invariant from [START_REF] Brock | Convergence properties of end invariants[END_REF], and criteria of convergence/divergence given in [START_REF] Brock | Convergence and divergence of Kleinian surface groups[END_REF].

1.1. Outline. We are going to find n such that if the image of (ι * • σ m ) n is unbounded then N contains a non-peripheral incompressible torus, contradicting our assumption. For that purpose we shall use the invariant m introduced in [START_REF] Brock | Convergence and divergence of Kleinian surface groups[END_REF].

Given a simple closed curve d on a closed surface S equipped with a hyperbolic metric g, we define

m(g, d, µ) = max { sup Y : d⊂∂Y d Y (µ(g), µ), 1 length g (d) } ,
where µ(g) is a shortest marking for (S, g), µ is a full marking, and the supremum of the first term in the maximum is taken over all incompressible subsurfaces Y of S whose boundaries ∂Y contain d. See Definition 4.1 for more details. It is not difficult to see that in the setting of Theorem 1.1, for a given sequence {m i } in T(∂M ), if the sequence {σ(m i )} is unbounded, then there is a simple closed curve d such that m(σ(m i ), d, µ) is unbounded (see Lemma 4.3). The core of our argument consists in showing, with the help of arguments from [START_REF] Brock | Convergence properties of end invariants[END_REF] and [START_REF] Brock | Convergence and divergence of Kleinian surface groups[END_REF], that in this situation, there is a simple closed curve d ′ ⊂ ∂M such that {m(m i , d ′ , µ)} is unbounded and that d∪d ′ bounds an essential annulus in M . Using this argument repeatedly, we build (when {(ι * • σ) n (m i )} is unbounded) an annulus in N which goes through the interior of M (viewed as a subset of N ) n times. If n is large enough, this annulus must create an essential torus in N , and contradicts the assumption that N is atoroidal.

Although this is the overall logic of the proof, in the following sections, we shall present the main steps in a different order. After setting up some preliminary definitions in Section 2, we shall discuss the topological part of the proof in Section 3. First we show that we can add some assumptions on the topology of M which will simplify the arguments later on. Next, we study incompressible surfaces on ∂M which can be extended multiple times through the characteristic submanifold of M when it is viewed as a submanifold of N . This will give us an integer n which appears in Theorem 1.1. In Section 4 we shall discuss the relation between the behaviour of the invariant m defined above, and the convergence and divergence of Kleinian groups. In Section 5 we shall prove our key proposition, and obtain the curve d ′ mentioned above. Finally in Section 6 we shall put these pieces together to prove our main theorem.

We would like to express our gratitude to the referee for his/her valuable suggestions, which have made it possible to improve our exposition.

Preliminaries

2.1. Haken manifolds and characteristic submanifolds. An orientable irreducible compact 3-manifold which contains a non-peripheral incompressible surface is called a Haken manifold. We note that a compact irreducible 3-manifold with non-empty boundary is always Haken except for a 3-ball. We say that a Haken manifold is atoroidal when it does not contain a nonperipheral incompressible torus, and acylindrical when it does not contain a non-peripheral incompressible annulus. By the torus theorem for Haken manifolds ( [START_REF] Waldhausen | On the determination of some bounded 3-manifolds by their fundamental groups alone[END_REF][START_REF] Jaco | Seifert fibered spaces in 3-manifolds[END_REF][START_REF] Johannson | Homotopy equivalences of 3-manifolds with boundaries[END_REF]), the former condition of the atoroidality is equivalent to the one that every monomorphism from Z×Z into the fundamental group is peripheral, i.e. is conjugate to the image of the fundamental group of a boundary component.

The Jaco-Shalen-Johannson theory [START_REF] Jaco | Seifert fibered spaces in 3-manifolds[END_REF][START_REF] Johannson | Homotopy equivalences of 3-manifolds with boundaries[END_REF] tells us that in a Haken manifold, incompressible tori and incompressible annuli can stay only in a very restricted place. Let us state what the theory says in the case when a Haken manifold M is atoroidal and boundary-irreducible, the latter of which means that ∂M is incompressible.

For an orientable atoroidal Haken boundary-irreducible 3-manifold M , there exists a 3-submanifold X of M each of whose components is one of the following: (a) An I-bundle over a surface with negative Euler characteristic whose associated ∂I-bundle coincides with its intersection with ∂M . Such a component is called a characteristic I-pair. (b) A solid torus Ξ such that Ξ ∩ ∂M consists of annuli which are incompressible on both ∂Ξ and ∂M . When Ξ ∩ ∂M is connected, it winds around the core curve of Ξ more than once. (c) A thickened torus S 1 ×S 1 ×I at least one of whose boundary components lies on a component of ∂M .

and which satisfies the following condition: Every properly embedded essential annulus (i.e. an incompressible annulus which is not homotopic into the boundary) is properly isotopic into X, and no component of X is properly isotopic into another component. Such X is unique up to isotopy, and is called the characteristic submanifold of M . We note that in the case when M has no torus boundary component, which is the assumption of our main theorem, a component of the last type (c) does not appear.

We need to consider characteristic submanifolds in a slightly general setting, for pared manifolds. We shall first give a definition of pared manifold.

Definition 2.1.

A pared manifold is a pair (M, P ), where M is a boundaryirreducible Haken 3-manifold, and P is a union of incompressible tori and annuli on ∂M , with the following properties.

(1) Every π 1 -injective map from a torus f : T → M is homotopic to a map into a component of P . (2) Every π 1 -injective map from an annulus g : A → M with g(∂A) ⊂ P is properly homotopic to a map whose image is contained in P . The subsurface P above is called the paring locus.

Let (M, P ) be a pared manifold. There exists a submanifold X of M disjoint from P each of whose components is either (a) or (b) listed in the definition of characteristic manifolds above, and which satisfies the following conditions.

(i) No component of X is a solid torus properly homotopic into P .

(ii) No component of X is properly homotopic into another component of X. (iii) Every properly emessential annulus A in M that is not properly homotopic into P can be properly homotoped into X. Such a submanifold is unique up to proper isotopy, and is called the characteristic submanifold of (M, P ).

Thurston's celebrated uniformisation theorem for Haken manifolds says that every atoroidal Haken manifold whose boundary consists of incompressible tori admits a hyperbolic structure of finite volume. More generally, he proved that every atoroidal Haken manifold, including the case when it has non-torus boundary components, admits a (minimally parabolic) convex hyperbolic structure of finite volume. The term 'convex hyperbolic structure' will be explained in the following subsection.

Kleinian groups and their deformation spaces.

A Kleinian group is a discrete subgroup of PSL 2 (C). In this paper, we always assume Kleinian groups to be torsion free, and finitely generated except for the case when we talk about geometric limits. For a Kleinian group Γ, we can consider the complete hyperbolic 3-manifold H 3 /Γ. The convex core of H 3 /Γ is the smallest convex submanifold that is a deformation retract. The Kleinian group Γ and the corresponding hyperbolic 3-manifold H 3 /Γ are said to be geometrically finite when the convex core of H 3 /Γ has finite volume. In particular, H 3 /Γ is said to be convex compact, and Γ to be convex cocompact if the convex core is compact. We also say that Γ is minimally parabolic when every parabolic element in Γ is contained in a rank-2 parabolic subgroup. Any convex cocompact Kleinian group is automatically minimally parabolic since it does not have parabolic elements.

A 3-manifold M is said to have a hyperbolic structure when Int M is homeomorphic to H 3 /Γ for a Kleinian group Γ, and we regard the pullback of the hyperbolic metric to Int M as a hyperbolic structure on M . In particular if Γ is taken to be geomerically finite or convex cocompact, we say that M has a geometrically finite or convex compact hyperbolic structure. If M admits a hyperbolic structure, then M must be atoroidal.

The set of hyperbolic structures on M modulo isotopy, which we denote by AH(M ), can be identified with a subset of the set of faithful discrete representations of π 1 (M ) into PSL 2 (C) modulo conjugacy. We put on AH(M ) a topology induced from the weak topology on the representation space. We regard an element of AH(M ) both as a hyperbolic structure on M and as a representation of π 1 (M ) into PSL 2 (C) depending on the situation.

A Kleinian group G is said to be a quasi-conformal deformation of another Kleinian group Γ if there is a quasi-conformal homeomorphism f : Ĉ → Ĉ such that G = f Γf -1 as Möbius transformations on Ĉ. When G is a quasiconformal deformation of Γ, there is a diffeomorphism from H 3 /Γ to H 3 /G preserving the parabolicity in both directions, which induces an isomorphism between the fundamental groups coinciding with the isomorphism given by the conjugacy G = f Γf -1 . We note that a quasi-conformal deformation of geometrically finite (resp. convex cocompact, minimally parabolic geometrically finite) group is again geometrically finite (resp. convex cocompact, minimally parabolic geometrically finite).

Let M be a compact 3-manifold admitting a minimally parabolic geometrically finite hyperbolic structure m. Let QH(M ) denote the set of all minimally parabolic geometrically finite hyperbolic structures on M modulo isotopy, which is regarded as a subset of AH(M ). Marden [START_REF] Marden | The geometry of finitely generated kleinian groups[END_REF] showed that every minimally parabolic geometrically finite hyperbolic structures on M is obtained as a quasi-conformal deformation of m. Therefore we call QH(M ) the quasi-conformal deformation space. Furthermore, if ∂M is incompressible, combined with the work of Ahlfors, Bers, Kra, Maskit and Sullivan, there is a parameterisation q : T(∂M ) → QH(M ), where T(∂M ) denotes the Teichmüller space of ∂M , i.e. the direct product of the Teichmüller spaces of the components of ∂M . We shall refer to this map as the Ahlfors-Bers map.

In the case when M is homeomorphic to S ×[0, 1] for a closed oriented surface S, the deformation spaces AH(M ), QH(M ) are denoted by AH(S), QF(S) respectively. The quasi-conformal deformation space QF(S) consists of quasi-Fuchsian representations of π 1 (S), i.e. quasi-conformal deformations of a Fuchsian representation, and is therefore called the quasi-Fuchsian space.

The Ahlfors-Bers map can be expressed as qf : T(S) × T( S) → QF(S), where the second coordinate T( S) denotes the Teichmüller space of S with orientation reversed, which is a more natural way for parametrisation since the boundary component S × {1} has the opposite orientation from the one given on S × {0} if we identify them with S by dropping the second factor. Now, let M be an atoroidal Haken 3-manifold with non-empty incompressible boundary which does not contain a torus. Suppose that M has a convex compact hyperbolic metric m, and let S be a component of ∂M . Take a covering of M associated with π 1 (S) ⊂ π 1 (M ), and lift the hyperbolic structure m to the hyperbolic structure m on S × [0, 1]. It is known (see [START_REF] Morgan | On Thurston's uniformization theorem for three-dimensional manifolds[END_REF]Proposition 7.1]) that the lifted structure m is also convex cocompact, hence can be regarded as an element of QF(S). Therefore m in turn corresponds to a point (g S (m), h S (m)) in T(S)×T( S). Let S 1 , . . . , S k be the components of ∂M that are not tori, and we consider the point h S i (m) ∈ T( Si ) for each i = 1, . . . , k. We define T( ∂M ) to be T( S1 ) × • • • × T( Sk ). The map taking g ∈ T(∂M ) to (h S 1 (q(g)), . . . , h S k (q(g))) ∈ T( ∂M ) is called the skinning map, which we shall denote by σ.

Curve complexes and projections.

Let S be a connected compact orientable surface possibly with boundary, satisfying ξ(S) = 3g + n ≥ 4 where g denotes the genus and n denotes the number of the boundary components. The curve complex CC(S) of S with ξ(S) ≥ 5 is a simplicial complex whose vertices are isotopy classes of non-peripheral, non-contractible simple closed curves on S such that n + 1 vertices span an n-simplex when they are represented by pairwise disjoint simple closed curves. In the case when ξ(S) = 4, we define CC(S) to be a graph whose vertices are isotopy classes of simple closed curves such that two vertices have smallest possible intersection. In the case when S is an annulus, we define CC(S) to be a graph whose vertices are isotopy classes (relative to the endpoints) of nonperipheral simple arcs in S such that two vertices are connected when they can be represented by arcs which are disjoint at their interiors. Masur-Minsky [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF] proved that CC(S) is Gromov hyperbolic with respect to the path metric for any S.

A marking µ on S consists of a pants decomposition of S, which is denoted by base(µ) and whose components are called base curves, and a collection t(µ) of simple closed curves, called transversals of base(µ), such that each component of base(µ) intersects at most one among them essentially. For two markings µ, ν on S and a subsurface Y , we define d Y (µ, ν) to be the distance between π Y (base(µ) ∪ t(µ)) and π Y (base(ν) ∪ t(ν)), where the projection π Y : CC(S) → P(CC(Y )) is obtained by taking the intersection of curves on S with Y and connecting the endpoints by arcs on Fr Y when the intersection contains arcs. In [START_REF] Masur | Geometry of the complex of curves. II. Hierarchical structure[END_REF], a marking defined as such is called clean. In this paper, we only consider clean markings. A marking is called full when every base curve has a transversal. In general, for two sets of simple closed curves a, b and a subsurface Y of S, we define d Y (a, b) to be the distance in CC(Y ) between π Y (a) and π Y (b) provided that both of them are non-empty. If one of them is empty, the distance is not defined.

For a point m in T(S), its shortest marking, which is a full marking and is denoted by µ(m), has a shortest pants decomposition of (S, m) as base(µ(m)), and t(µ(m)) consisting of shortest transversals, one for each component of base(µ(m)). When we talk about the distance d Y between two points in T(S) or between a point in T(S) and a marking, we identify points m ∈ T(S) with µ(m).

2.4. Geometric limits and compact cores. Let M be an atoroidal boundary-irreducible Haken 3-manifold. Let {ρ i } be a sequence of faithful discrete representations of π 1 (M ) into PSL 2 (C). We define a geometric limit of {ρ i (π 1 (M ))} to be a Kleinian group Γ such that every element γ of Γ is a limit of some sequence {g i ∈ ρ i (π 1 (M ))}, and every convergent sequence {γ i j ∈ ρ i j (π 1 (M ))} has its limit in Γ.

Fixing a point x ∈ H 3 , and considering its projections

x i in H 3 /ρ i (π 1 (M )) and x ∞ in H 3 /Γ, the geometric convergence implies the existence of pointed Gromov-Hausdorff convergence of ((H 3 /ρ i (π 1 (M )), x i )) to (H 3 /Γ, x ∞ )
. This latter convergence means that there exist real numbers r i going to ∞, K i converging to 1, and

K i -bi-Lipschitz diffeomorphisms f i (called approximate isometries) between r i -balls B r i (H 3 /ρ i (π 1 (M )), x i ) and B K i r i (H 3 /Γ, x ∞ ). Suppose that {ρ i } converges to ρ ∞ : π 1 (M ) → PSL 2 (C) as representations and that {ρ i (π 1 (M ))} converges to Γ geometrically. Then, ρ ∞ (π 1 (M )) is a subgroup of the geometric limit Γ.
For an open irreducible 3-manifold V with finitely generated fundamental group, a compact 3-dimensional submanifold C in V is called a compact core when the inclusion induces an isomorphism between their fundamental groups. The existence of compact cores was proved by Scott [START_REF] Scott | Compact submanifolds of 3-manifolds[END_REF]. The case which interests us is when V is a hyperbolic 3-manifold.

Let H 3 /G be a hyperbolic 3-manifold associated with a finitely generated, torsion free Kleinian group G. By Margulis's lemma, there is a positive constant ε 0 such that the set of points of H 3 /G where the injectivity radii are less than ε 0 consists of a finite disjoint union of tubular neighbourhoods of closed geodesics of length less than ε 0 , called Margulis tubes, and cusp neighbourhoods each of which is stabilised by a maximal parabolic subgroup of G, and whose quotient by its stabiliser is homeomorphic to S 1 × R 2 when the stabiliser has rank 1, and to S 1 × S 1 × R when the stabiliser has rank 2. The former cusp neighbourhood is called a Z-cusp neighbourhood, and the latter a torus cusp neighbourhood. The union of the cusp neighbourhoods is called the cuspidal part of H 3 /G. The complement of the cuspidal part is called the non-cuspidal part and is denoted by (H 3 /G) 0 . Each boundary component of (H 3 /G) 0 is either an open annulus or a torus. By the relative compact core theorem by McCullough [START_REF] Mccullough | Compact submanifolds of 3-manifolds with boundary[END_REF], there is a compact core C G ⊂ (H 3 /G) 0 such that for each boundary component B of (H 3 /G) 0 , the intersection C G ∩ B is a core annulus when B is an open annulus, and is the entire B when B is a torus. We call such a compact core a relative compact core of (H 3 /G) 0 .

Let p : H 3 /ρ ∞ (π 1 (M )) → H 3 /Γ be the covering map associated with the inclusion of ρ ∞ (π 1 (M )) into the geometric limit Γ. Let C be a relative compact core of (H 3 /ρ ∞ (π 1 (M ))) 0 . Suppose that H 3 /Γ has a torus cusp neighbourhood T . We say that H 3 /ρ ∞ (π 1 (M )) wraps around T when p|C is homotoped to an immersion which goes around T non-trivially, and hence cannot be homotoped to an embedding.

Topological features

3.1. Coverings. In this subsection, we shall show that to prove Theorem 1.1, we can put an extra assumption that all the characteristic I-pairs of M are product bundles.

We consider an atoroidal Haken manifold as is given in Theorem 1.

* ∂ •(ι•σ) = (ι• σ)•p * ∂ , and hence p * ∂ •(ι•σ) n = (ι • σ) n • p * ∂ .
Therefore, if the image of (ι • σ) n is bounded, the properness of p * ∂ implies that the image of (ι • σ) n must also be bounded. □

This result allows us to work on manifolds with topological features that will make the arguments simpler.

Lemma 3.2. Let M be an orientable atoroidal Haken manifold with incompressible boundary. Then there is a double covering of M all of whose characteristic I-pairs are product I-bundles.

Proof. Let Ξ 1 , . . . , Ξ p be the characteristic I-pairs of M that are twisted Ibundles. Take their double coverings Ξ1 , . . . , Ξp corresponding to the orientation double coverings of the base surfaces. For each Ξ j among Ξ 1 , . . . , Ξ p , its frontier components (i.e. the closures of the components of ∂Ξ j \ ∂M ) are annuli. Each of such annuli has two pre-images in Ξj which are taken to each other by the unique non-trivial covering translation.

Let C be the closure of a component of

M \ (Ξ 1 ∪ • • • ∪ Ξ p ). Let A 1 , . . . , A k be the components of C ∩ (Ξ 1 ∪ • • • ∪ Ξ p ), which are annuli on ∪ p l=1 ∂Ξ l \ ∂M . We prepare two copies C + and C -of C. Each A j among A 1 , . . . A k , which is contained some Ξ i among Ξ 1 , . . . , Ξ p , has two lifts A + j and A - j in Ξi . Now
we identify the copy of A j in C + to A + j in Ξi and the one in C -to A - j in Ξi for each annulus among A 1 , . . . A k . We repeat the same procedure for every component

C of M \ (Ξ 1 ∪ • • • ∪ Ξ p )
, and get a manifold M , which will turn out to be a double cover of M as shown below.

Define a homeomorphism t : M → M to be the covering translation on each Ξi and the map taking C ± to C ∓ preserving the identification with C for each of C ± . It is clear from the definition that this homeomorphism t is a free involution. By taking the quotient of M under ⟨t⟩ ∼ = Z 2 , we get a manifold naturally identified with M . Thus we see that M is a double cover of M . Since Ξi is a product I-bundle and all of the characteristic I-pairs contained in C ± are product bundles by our definition of Ξ 1 , . . . , Ξ p , we see that M is a double cover as desired.

□

For some of our arguments we shall need a stronger assumption than having only product bundles: Definition 3.3. Let M be a compact orientable Haken 3-manifold with incompressible boundary. We say that M is strongly untwisted if and only if: (A) Every characteristic I-pair is a product bundle. (B) For any characteristic I-pair Ξ and any simple closed curve d ⊂ ∂M , the simple closed curve d can be homotoped on ∂M into at most one component of Ξ ∩ ∂M .

We are going to construct a cover with the properties (A) and (B) above. In order to do that, we need to examine how characteristic I-pairs are attached to other components of the characteristic submanifold. In the following proof of Lemma 3.4, it will turn out that there are two situations ((a) and (b) below) where the second condition of 'strong untwistedness' breaks down. Lemma 3.4. Let M be a compact orientable atoroidal Haken manifold with incompressible boundary. Then there is a finite-sheeted regular covering of M which is strongly untwisted. Proof. By Lemma 3.2, we have a double covering all of whose characteristic I-pairs are product I-bundles. Therefore, we may assume that M satisfies the first condition (A) of 'strong untiwistedness', and we shall construct a covering satisfying the second condition.

To construct such a covering, let us analyse how this second condition (B) can fail to hold. Let d ⊂ ∂M be a simple closed curve, and let W a characteristic I-pair. Since no components of W ∩ ∂M are annuli, d can be homotoped on ∂M into at most two components of W ∩ ∂M . Furthermore, if d can be homotoped into two such components, then d lies (up to isotopy on ∂M ) on a component T j (characteristic solid torus or a characteristic thickened torus) of T , and there are two possibilities: (a) T j ∩ ∂M is an annulus when T j is a solid torus, and is the union of an annulus and a torus when T j is a thickened torus; or (b) d separates two consecutive components of T j ∩ M \ T both lying in the same characteristic I-pair. We shall show that we can take a finite-sheeted covering of M so that neither (a) nor (b) can happen.

First, we consider the condition (a). Let T j be a component of T such that T j ∩ ∂M is an annulus (and T j is a solid torus) or the union of an annulus and a torus (when T j is a thickened torus). This implies that T j ∩ M \ T j is connected, hence is an annulus, which we denote by A. Since T j is a characteristic solid torus or characteristic thickened torus, the annulus A is essential, and hence is not homotopic to T j ∩∂M fixing the boundary. Then, we can choose a simple closed curve α, which is not contractible in M , on the component of ∂T j on which d lies so that both α ∩ A and α ∩ ∂M are connected, i.e. arcs. Since π 1 (T j ) is either Z or Z×Z, we can take a k-sheeted cyclic covering Tj of T j so that α cannot be lifted homeomorphically, whereas the annulus A is homeomorphically lifted. (For instance, in the case when π 1 (T j ) ∼ = Z, we choose k which is coprime with the element represented by α.) Then the preimage of the annulus A is k copies of A, which we denote by A 1 , . . . , A k . Let C be M \ T j . We prepare k copies of C, which we denote by C 1 , . . . , C k . By pasting C j along A j to Tj , we can make a k-sheeted cyclic covering of M in which Tj does not satisfy the condition (a). If there is another component T j ′ of T with the condition (a), we repeat the same process for all the k lifts of T j ′ at the same time. Repeating the process, we get a finite-sheeted covering of M in which there is no characteristic solid torus or a characteristic thickened torus with the condition (a). We use the same symbol M and T for this finite-sheeted covering, abusing the notation. Now we turn to the condition (b). We consider three colours named the colour 0, 1 and 2 and we choose a colour for each annulus of T ∩ M \ T so that, on ∂T , no two consecutive annuli have the same colour. We take three copies of each component of T and of M \ T which we name the lift 0, 1 and 2. Consider a component U of T , a component V of M \ T and an annulus E ⊂ U ∩ V with the colour k ∈ {0, 1, 2}. For every j ∈ {0, 1, 2}, we glue the lift j of V to the lift (j + k) mod 3 of U along the appropriate lifts of E. Using the same construction for each component of T ∩ M \ T , we get a triple cover M of M in which any two consecutive components of T ∩ M \ T lie in different components of M \ T . In particular there is no characteristic solid torus or characteristic thickened torus in M for which the condition (b) holds. Thus, we have shown that by taking a finite-sheeted covering, we can make both of the situations (a) and (b) disappear, which means, as we saw above, that the covering is strongly untwisted. □ Lemmas 3.1 and 3.4 show that to prove Theorem 1.1, we have only to consider the case when M is strongly untwisted.

Vertically extendible surfaces.

Let M be an atoroidal Haken manifold as in Theorem 1.1. Let X be the characteristic submanifold of M . Assume that every I-bundle in X is a product I-bundle. Definition 3.5. Given an incompressible subsurface F ⊂ ∂M , we say that F is one-time vertically extendible if there is an incompressible surface F 1 ⊂ ∂M and an essential I-bundle V F ⊂ M with V F ∩∂M = F ∪F 1 and F 1 ⊂ ∂X up to isotopy. We call F 1 a first elevation of F .

It follows from the definition of characteristic submanifold that there is an isotopy which takes V F into the characteristic submanifold X. From now on, we assume that if F is one-time vertically extendible then F ⊂ X and V F ⊂ X.

We note solid torus components in X may add some complications in the case when F is an annulus. If F is contained in such a component of X, there may be more than one possible first elevation (even up to isotopy) and the I-bundles corresponding to two disjoint annuli may intersect (even up to isotopy).

We now define multiple elevations by induction. Definition 3.6. Given an incompressible subsurface F in ∂M and n ≥ 2, we say that F is n-time vertically extendible if there is an essential surface 1 ) is (n -1)-time vertically extendible. An (n -1)-th elevation F n of ι(F 1 ) is defined to be an n-th elevation of F .

F 1 ⊂ ∂M and an essential I-bundle V F ⊂ M with V F ∩ ∂M = F ∪ F 1 and ι(F
We say that two multi-curves c, d ⊂ ∂M intersect minimally if for every multicurves c ′ , d ′ homotopic to c and d respectively, {c ∩ d} ≤ {c ′ ∩ d ′ }. Let F, G ⊂ X ∩ ∂M be two incompressible surfaces. We say that F and G intersect minimally if ∂F intersects ∂G minimally. Lemma 3.7. Let F, G ⊂ ∂M be connected incompressible subsurfaces which intersect minimally and are not disjoint. If F and G are n-time vertically extendible, then so is F ∪ G.

Proof. If F and G are one-time vertically extendible, as was remarked before, we may assume that F, G ⊂ X ∩ ∂M . Since they intersect minimally and are not disjoint, they must lie in the same component H of X ∩ ∂M which is not an annulus. Then the component V of X containing H is an I-bundle, which is a product I-bundle by assumption, and can be parametrised as

H × [0, 1].
Then, by moving F and G by isotopies, we have

V F = F ×[0, 1] ⊂ H ×[0, 1] and V G = G×[0, 1] ⊂ H ×[0, 1], and F 1 = F ×{0, 1}\F, G 1 = G×{0, 1}\G. Since F 1 , resp. G 1 ,
lies in the component of X ∩ ∂M which does not contain F and G, F 1 and G 1 lie in the same component of X ∩ ∂M . Therefore F 1 ∪ G 1 lies in X ∩ ∂M . Thus we have proved that if F and G are one-time vertically extendible then F ∪ G is also one-time vertically extendible and Otherwise, there is an n-time vertically extendible surface F which cannot be isotoped into Σ. Moving F by an isotopy we can assume that F intersects Σ minimally. By Lemma 3.7, each connected component of F ∪ Σ is n-time vertically extendible, and we replace Σ with Σ ∪ F , and call this enlarged surface Σ. We repeat this operation as long as there is an n-time vertically extendible surface which cannot be isotoped into Σ. Every time we add a surface, either we decrease the Euler characteristic of Σ or we add a disjoint annulus which cannot be isotoped into Σ. Hence this process must terminate after finitely many steps. The final resulting surface is Σ n . □

F 1 ∪ G 1 is its first elevation.
Since an n-time vertically extendible surface is m-vertically extendible for any m ≤ n, we have Σ n ⊂ Σ m up to isotopy.

In the next lemma we show that, when N is atoroidal, M cannot contain an n-time extendible surface for sufficiently large n. In the last section, this result will lead us to the constant n of Theorem 1.1.

Lemma 3.9.

There is L depending only on the topological type of ∂M such that if there is an L-time vertically extendible surface, then N is not atoroidal.

Proof. Letting g denote the genus of ∂M , we set K = 3g -3, which is the number of curves in a pants decomposition of ∂M . Since no components of Σ n can be isotoped into another component, ∂Σ n has at most 2K boundary components. Using this observation, we show in the following claim that Σ n+2K must be a proper subsurface of Σ n even up to isotopy. Claim 3.10. For any n ∈ N, if Σ n is non-empty and any component of Σ n can be isotoped into Σ n+2K , then N cannot be atoroidal.

Proof. Suppose that Σ n ̸ = ∅, and that any component of Σ n can be isotoped into Σ n+2K . Since Σ n+j is contained in Σ n for any j ≥ 0 up to isotopy as observed above, and no component of Σ n+j can be isotoped into another component, we have then Σ n+j = Σ n for any j ≤ 2K up to isotopy. Let F be a component of Σ n+2K with minimal Euler characteristic, and F j its j-th elevation. By definition, ι(F j ) is (n + 2K -j)-time vertically extendible for any j ≤ 2K. Therefore ι(F j ) can be isotoped into Σ n . Since Σ n = Σ n+2K and F has minimal Euler characteristic, ι(F j ) is a component of Σ n , up to isotopy. In particular ∂(ι(F j )) ⊂ ∂Σ n up to isotopy. Let V j be the I-bundle cobounded by ι(F j-1 ) and F j . We note that by definition, F j and ι(F j ) are identified in N and that the interior of V j is embedded in N . Taking the union of the I-bundles V j in N for j ≤ 2K, we get a map F × [0, 2K] → N such that F × {j} is sent to F j . Let c be a component of ∂F . The image of the annulus c × [0, 2K] goes 2K + 1 times through ∂Σ n . Since ∂Σ n has at most 2K components, there is a component c ′ of ∂Σ n through which c×[0, 2K] goes at least twice. The image of the part of this annulus between two such instances forms a torus T embedded in N . Considering the component of ∂M through which T goes, we can construct an infinite cyclic covering of N in which T lifts to an infinite incompressible annulus. It follows that T is incompressible and non-peripheral. Hence N is not atoroidal. □

As mentioned before, we have Σ n ⊂ Σ m for any m ≤ n. Consider monotone increasing indices n j such that Σ n j +1 is smaller than Σ n j in the sense that at least one component of Σ n j cannot be isotoped into Σ n j +1 . Since no component of Σ n can be isotoped into another component, we have then either χ(Σ n j +1 ) > χ(Σ n j ) or Σ n j +1 has fewer connected components than Σ n j . It follows that there are at most K such n j , namely, there is J ≤ K such that for any n ≥ n J + 1, we have Σ n = Σ n+1 . By Claim 3.10, if n j -n j-1 ≥ 2K for some j ≤ J or if Σ n J ̸ = ∅, then N is not atoroidal. Since J ≤ K, we can now conclude the proof just by setting L = 2K 2 . □

Convergence, divergence and subsurface projections

In this section, we shall review the relations between the invariant m mentioned in the introduction and the convergence and divergence of Fuchsian and Kleinian groups.

4.1. Subsurface projections and Fuchsian groups. We first recall the definition of the invariant m from [START_REF] Brock | Convergence and divergence of Kleinian surface groups[END_REF], and see how it controls the behaviour of sequences of Fuchsian groups. Definition 4.1. Let S be a (possibly disconnected) closed surface of genus at least 2 and g a point in its Teichmüller space. Regarding g as a hyperbolic structure on S, we let µ(g) be a shortest marking for (S, g) (See Section 2.3). Although there might be more than one shortest markings, its choice does not matter for our definition and arguments. We fix a full and clean marking µ consisting of a pants decomposition and transversals on S independent of g. For any essential simple closed curve d on S, we define

m(g, d, µ) = max { sup Y : d⊂∂Y d Y (µ(g), µ), 1 length g (d) } ,
where the supremum of the first term in the maximum is taken over all incompressible subsurfaces Y of S whose boundaries ∂Y contain d.

It follows from [2, Lemma 5.2] that two curves with unbounded m cannot intersect: Lemma 4.2. Let {m j } be a sequence in T(S) and let c 1 , c 2 be simple closed curve on S. If m(m j , c i , µ) -→ ∞ as j → ∞ for both i = 1, 2, then i(c 1 , c 2 ) = 0.

Proof. This is just a special case of [2, Lemma 5.2] for Fuchisan groups. We note that the assumption of bounded projections of end invariants is unnecessary in this special case for which end invariants are empty. □

The invariant m is related to the divergence and convergence of a sequence by the following lemma: Lemma 4.3. Let µ be a full marking on S, and let {m i } be a sequence in T(S). Then every subsequence of {m i } contains a convergent subsequence if and only if {d S (µ(m i ), µ)} is bounded for a shortest marking µ(m i ) of m i and {m(m i , c, µ)} is bounded for every essential simple closed curve c on S.

Proof. It follows from classical results on Fenchel-Nielsen coordinates that any subsequence of {m i } contains a converging subsequence if and only if the sequence {µ(m i )} is a finite set and {length m i (µ(m i ))} is bounded. By [2, Lemma 2.3], the sequence {µ(m i )} is infinite if and only if passing to a subsequence, either {d S (µ(m i ), µ)} is unbounded or there is an incompressible subsurface Y such that d Y (µ(m i ), µ) -→ ∞ (and hence m(m i , c, µ) -→ ∞ for any component c of ∂Y ).

On the other hand, if the sequence {µ(m i )} consists of finite elements, then length m i (µ(m i )) is unbounded if and only if passing to a subsequence, there is a curve c with length m i (c) -→ 0 (and hence m(m i , c, µ) -→ ∞). □ 4.2. Relative convergence of Kleinian groups. We shall next establish a necessary condition on the invariant m for algebraic convergence on a submanifold. We start with a fundamental result. Thurston proved in [START_REF] Thurston | Hyperbolic Structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary[END_REF] the following which is the first half of the theorem often referred to as the 'broken window only' theorem. We note that the latter half of the broken window only theorem should need some rectification (see [START_REF] Ohshika | Degeneration of marked hyperbolic structures in dimensions 2 and 3[END_REF]) but is irrelevant to the present paper.

Theorem 4.4. Let M be an atoroidal Haken 3-manifold and X its characteristic submanifold. Then for any curve γ in M \ X and any sequence {ρ i ∈ AH(M )}, the length of the closed geodesic in

H 3 /ρ i (π 1 (M )) represent- ing the free homotopy class of ρ i (γ) is bounded as i -→ ∞.
Using arguments from [START_REF] Brock | Convergence and divergence of Kleinian surface groups[END_REF], we establish the following necessary condition for algebraic convergence on a submanifold. Then the sequence of the restrictions {ρ i |π 1 (W )} has a convergent subsequence up to conjugation.

Proof. We follow the argument of [2, Proposition 6.1] with some modifications as below. We note that the condition (c) above will replace the assumption of 'bounding projection without combinatorial wrapped parabolics' imposed there. The condition (a) will allow us to work on the submanifold W rather than the whole manifold. Following [2, Lemma 6.2], we start with constructing a pants decomposition r of ∂W with uniformly bounded length. In the first paragraphs of the proof of [2, Lemma 6.2], the assumption of bounded projection is used to find the first curves in r. In our relative setting, we do not have an equivalent assumption. Instead, we use the conditions (b) and (d) above and Theorem 4.4 to find the first curves as below.

Denote by c i a shortest pants decomposition of ∂M with respect to m i . Note that {d Y (m i , µ) = d Y (µ(m i ), µ)} is bounded for any essential subsurface Y that is not an annulus with its core curve in c i if and only if {d Y (c i , µ)} is bounded. Let X be the characteristic submanifold of the pared manifold (W, P ). Consider a multicurve r on ((W \ X) ∩ ∂W ) ∪ P which is maximal in the sense that any simple closed curve in ((W \ X)∩∂W )∪P either intersects r or is homotopic on ∂W to a component of r. We note that it contains a curve isotopic to each boundary component of X ∩ ∂W by the maximality, and is not empty by (d). By our assumption (b) and Theorem 4.4, there is L such that length ρ i (r) ≤ L.

Next, following the proof of [2, Proposition 6.1], we add curves to r until we get a pants decomposition. Since r is already maximal in ((W \ X) ∩ ∂W ) ∪ P we only need to extend it to the union Z of the characteristic I-pairs in X. By assumption, Z is a product I-bundle in the form Σ × I (Σ may be disconnected). We denote by f : Z → Σ the projection along the fibres, and for a subsurface F ⊂ Σ and for j = 0, 1, we use the symbol F j to denote f -1 (F ) ∩ Σ × {j}. For each component F of Σ that is not a pair of pants, by the assumption (c), there is j ∈ {0, 1} such that {d F j (c i , µ)} is bounded. Let S j be the component of ∂M containing F j , and denote by θ i = ρ i • I * : π 1 (S j ) → PSL 2 (C) the representation induced by the inclusion I : S j → M . The quotient manifold H 3 /θ i (π 1 (S j )) covers H 3 /ρ i (π 1 (M )), and has end invariant m i | S j on one side. Now, replacing ρ i with θ i , we can follow the proof of [2, Lemma 6.2] starting at the penultimate paragraph (i.e. the third paragraph on p. 836) to find a simple closed curve a i contained in F as follows. By Theorem 4.4 and the assumptions (a,b), each boundary component of F is homotopic to a closed geodesic in H 3 /θ i (π 1 (S j )) with length bounded as i -→ ∞. We note that the assumption (d) implies that F is not the entire S, and hence has non-empty boundary. Then the argument of the proof of Lemma 6.2, which makes use of [START_REF] Brock | Convergence and divergence of Kleinian surface groups[END_REF]Lemma 2.11], gives us a simple closed curve a i in F homotopic to a closed geodesic in H 3 /θ i (π 1 (S j )) with bounded length such that the distance from f (c i ∩ F j ) in CC(F ) is also bounded as i -→ ∞. Thus we have a constant L ′ and a sequence of curves

{a i } on F such that ρ i (a i ) ≤ L ′ and {d F j (f -1 (a i ) ∩ F j , c i )} is bounded as i -→ ∞.
Up to isotopy, f (r ∩ Z) consists of boundary components of Σ since Z is contained in X and r lies outside X. We denote f (r ∩ Z) by s. If {a i } has a constant subsequence, then we pass to an appropriate subsequence of {ρ i }, and add a i (independent of i) to s.

If not, by [2, Lemma 2.3], there is a subsurface Y ⊂ F with d Y (a i , µ) -→ ∞, passing to a subsequence. Since {d F j (c i , µ)} and {d F j (f -1 (a i ) ∩ F j , c i )} are bounded, Y must be a proper subsurface of F (even up to isotopy). If, passing to a subsequence, there is k ∈ {0, 1} such that Y k = f -1 (Y ) ∩ S k
is an annulus containing a component of c i for all i, we add the projection by f of this component of c i to s. Otherwise, by the assumption (c), there exists k ∈ {0, 1} with bounded {d Y k (c i , µ)}. Hence, passing to a subsequence,

d Y k (c i , f -1 (a i ) ∩ S k ) -→ ∞,
and by [START_REF] Minsky | Kleinian groups and the complex of curves[END_REF]Theorem B], ρn (∂Y ) → 0. In this case, we add ∂Y to s. We repeat the above construction letting F be a component of Σ \ s until Σ \ s becomes a union of annuli and pair of pants. Adding f -1 (s) to r, we obtain a pants decomposition of ∂W , which we shall still denote by r, such that { ρn (r)} is bounded.

Next we attach a transversal with bounded length to each component d of r such that there is an essential annulus E with d ⊂ ∂E and that {m(m i |S, d, µ)} is bounded (where S is the component of ∂M on which d lies). Let d be such a curve. If d is contained in c i , we replace c i with a shortest pants decomposition not containing d. Since {m(m i |S, d, µ)} is bounded, there is a positive lower bound on { m i (d)}, and there is an upper bound on { m i (c i )} by our definition of c i . Considering the covering associated with the inclusion S → M we can use the arguments of [START_REF] Brock | Convergence and divergence of Kleinian surface groups[END_REF] (proof of Proposition 6.1, the part after the proof of Lemma 6.2, starting from the fifth paragraph on p. 836) to obtain a transversal t d to d with bounded length ρ i (t d ).

By the assumption (c), the union of r and all the transversals defined above is doubly incompressible in Thurston's sense [START_REF] Thurston | Hyperbolic Structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary[END_REF]Section 2]. Then we can deduce from Thurston's relative boundedness theorem [23, Theorem 3.1] that the restriction of {ρ i | π 1 (W ) } has a convergent subsequence. □

Unbounded skinning and annuli

The following proposition is the main step of our proof of Theorem 1.1.

Proposition 5.1. Let M be an orientable atoroidal boundary-irreducible Haken 3-manifold that is strongly untwisted. Let {m i } be a sequence in T(∂M ), let σ be the skinning map, and assume that there is a simple closed curve d on ∂M such that m(σ(m i ), d, µ) -→ ∞ for a full clean marking µ. Then, passing to a subsequence, there is a properly embedded essential annulus

A ⊂ M with ∂A = d ∪ d ′ such that m(m i , d ′ , µ) -→ ∞.
We are going to show that any subsequence of {m i } contains a further subsequence for which the conclusion holds. To simplify the notations we shall use the same subscript i for all subsequences.

5.1. Re-marking. Our manifold M is either connected or has two components. In the case when M has two components, by considering the component on which d lies, and abusing the symbol M to denote this component, we can assume that M connected. Recall that, by the assumption throughout this section, M is strongly untwisted. Let ρ i : π 1 (M ) → PSL 2 (C) be a representation corresponding to q(m i ).

As a first step for the proof of Proposition 5.1, we change the markings of M so that the behaviour of the ρ i can be read more easily from the behaviour of their end invariants. Furthermore, we assume that (*) {m(m i , d j , µ)} is bounded for every j = 2, . . . , p. Then there is a sequence of orientation-preserving homeomorphisms {ψ i : M → M } such that, passing to a subsequence, the following hold: [START_REF] Anderson | Cores of hyperbolic 3-manifolds and limits of Kleinian groups[END_REF] For any essential simple closed curve c ⊂ ∂M , either {m(

ψ i * (m i ), c, µ)} is bounded or m(ψ i * (m i ), c, µ) -→ ∞,. (2) If A ⊂ M is an essential annulus disjoint from all the d j such that m(ψ i * (m i ), ∂ k A, µ) -→ ∞ for both boundary components ∂ 1 A and ∂ 2 A of A, then ρ i •ψ -1 i * (∂A) -→ 0. (
(iii) {m(σ • ψ i * (m i ), d j , µ)} is bounded if and only if {m(σ(m i ), d j , µ)} is bounded.
Proof. We shall first define the homeomorphisms ψ i , and then verify the desired properties. Let Ξ be a component of the characteristic submanifold X of M \ d. Suppose first that Ξ is a solid torus. The components of ∂Ξ \ ∂M are incompressible annuli. We define ψ i on soid-torus components Ξ of X to be a composition of Dehn twists along these frontier annuli with the following properties:

(a) If Ξ is a solid torus, then π F (µ(ψ i * (m i ))) is constant with respect to i for every component F of Ξ ∩ ∂M except for at most one.

By the assumption (*), passing to a subsequence, we need not compose Dehn twists along annuli of the frontier components of Ξ to achieve the condition (a) when Ξ ∩ ∂M contains an annular neighbourhood of d (up to isotopy), and hence ψ i , as defined for the moment, also satisfies the following:

(b) For every j = 1, . . . , p, we have

ψ i (d j ) = d j and π A j (µ(ψ i * (m i ))) = π A j (µ(m i ))
for an annulus A j on ∂M whose core curve is d j .

If Ξ is not a solid torus, Ξ is a product Ξ = F ×I. (Recall that we have an assumption that every characteristic I-pair of M is a product bundle. This implies that an I-pair in the characteristic submanifold X of M \ d is also a product I-bundle.) Let F 0 be a component of Ξ∩∂M which does not contain a curve homotopic on ∂M to d 1 (there is always such a component since M is strongly untwisted). Since the curve complex of F 0 has finitely many orbits under the action of the mapping class group of F 0 (relative to ∂F 0 ), there is a sequence of orientation-preserving homeomorphisms g i : F 0 → F 0 fixing ∂F 0 such that, passing to a subsequence, π F 0 (µ(g i * (m i ))) is constant. We then define ψ i on Ξ by extending g i along the fibres, i.e. ψ i (x, t) = (g i (x), t) for any (x, t) ∈ Ξ = F 0 × I.

Thus we have the following.

(c) there are R > 0 and a component F 0 of Ξ ∩ ∂M not containing any curve homotopic on ∂M to

d 1 such that d Y (µ(ψ i * (m i )), µ) ≤ R for any incompressible subsurface Y ⊂ F 0 .
We note that since Ξ is a component of the characteristic submanifold of M \ d, if ∂Ξ contains a curve d j , then it must be peripheral, and hence the action of ψ i on Ξ does not affect the property (b).

We repeat the construction above for all the components of the characteristic submanifold X, and we extend the resulting homeomorphisms to a homeomorphism of M which is isotopic to the identity on the complement of the characteristic submanifold.

We now verify the properties (1, 2, 3) for ψ i thus constructed. The first property (1) can be obtained by passing to a subsequence for any sequence of homeomorphisms. Therefore, we are done with [START_REF] Anderson | Cores of hyperbolic 3-manifolds and limits of Kleinian groups[END_REF].

We next turn to proving the property [START_REF] Brock | Convergence properties of end invariants[END_REF]. By the assumption (*), taking a subsequence, we may assume that π F (µ(m i )) is constant whenever F is an annulus containing a curve d j for j ̸ = 1. Wet first show the following claim. Claim 5.3. For every j = 1, . . . , p and for any sequence of incompressible subsurfaces Y i ⊂ ∂M with its boundary containing d j which are not a pair of pants, {d Y i (µ, ψ i (µ))} is bounded.

Proof. Fix j = 1, . . . , p, and consider a sequence of incompressible subsurfaces Y i ⊂ ∂M each of which contains d j in its boundary. If all of the Y i are annulli after passing to a subsequence, the conclusion follows from the property (b). From now on, taking a subsequence, we assume that none of the Y i are annuli.

Assume first that there is a simple closed curve c ⊂ ∂M intersecting Y i which lies outside the characteristic submanifold X. Then by our construction of ψ i , we have ψ i (c) = c, and hence

d Y i (µ, ψ i (µ)) ≤ d Y i (µ, c) + d Y i (c, ψ i (µ)) ≤ d Y i (µ, c) + d Y i (ψ i (c), ψ i (µ)) ≤ d Y i (µ, c) + d ψ -1 i (Y i ) (µ, c) ≤ 4i(c, µ) + 2,
where the last inequality is due to Masur-Minsky [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF]Lemma 2.1]. Thus we are done in this case.

Otherwise, taking a subsequence, we may assume that Y i is contained in Ξ i ∩ ∂M for a component Ξ i of the characteristic submanifold X. Taking a further subsequence, we may assume that Ξ i = Ξ does not depend on i. Since Y i is not an annulus, Ξ is a product I-pair F × I. Let F 0 be the component of Ξ ∩ ∂M given by the property (c). Let us denote by Y ′ i the projection of Y i to F 0 along the fibres, (setting

Y ′ i = Y i if Y i ⊂ F 0 ).
{d Y ′ i (µ(ψ i (m i )), ψ i (µ)) = d ψ -1 i (Y ′ i ) (µ(m i ), µ)} is bounded. On the other hand, by the property (c), {d Y ′ i (µ(ψ i * (m i )), µ)} is bounded. Thus we see that {d Y ′ i (µ, ψ i (µ)) ≤ d Y ′ i (µ, µ(ψ i * (m i ))+d Y ′ i (µ(ψ i * (m i ), ψ i (µ))} is bounded. It follows from the construction of ψ i that d Y i (µ, ψ i (µ)) = d Y ′ i (µ, ψ i (µ)), and hence {d Y i (µ, ψ i (µ))} is also bounded. □
Now we can show that the sequence {ψ i } satisfies the property (3) by the condition (*) and the following claim. Claim 5.4. For any j = 1, . . . , p, the sequence {m(

ψ i * (m i ), d j , µ)} is bounded if and only if {m(m i , d j , µ)} is bounded, and {m(σ • ψ i * (m i ), d j , µ)} is bounded if and only if {m(σ(m i ), d j , µ)} is bounded. Proof. Let {Y i ⊂ ∂M } be a sequence of incompressible subsurfaces with d j ⊂ ∂Y i which are not pairs of pants. Since d Y i (m i , µ) = d ψ i (Y i ) (µ(ψ i * (m i )), ψ i (µ)), the triangle inequalities d Y i (µ(m i ), µ) ≤ d ψ i (Y i ) (µ(ψ i * (m i )), µ) + d ψ i (Y i ) (µ, ψ i (µ)), and
d ψ i (Y i ) (µ(ψ i * (m i )), µ) ≤ d ψ i (Y i ) (µ(ψ i * (m i )), ψ i (µ)) + d ψ i (Y i ) (ψ i (µ), µ) lead to d ψ i (Y i ) (µ(ψ i * (m i )), µ) -d ψ i (Y i ) (µ, ψ i (µ)) ≤ d Y i (µ(m i ), µ) ≤ d ψ i (Y i ) (µ(ψ i * (m i )), µ) + d ψ i (Y i ) (µ, ψ i (µ)).
Thus by applying Claim 5.3, we see that

{d Y i (µ(m i ), µ)} is bounded if and only if {d ψ i (Y i ) (µ(ψ i * (m i )), µ)} is bounded. Since ψ i (d j ) = d j by the property (b), we also have length m i (d j ) = length ψ i * (m i ) (d j ), and we conclude that {m(ψ i * (m i ), d j , µ)} is bounded if and only if {m(m i , d j , µ)} is bounded.
Since σ commutes with ψ i * , the same argument shows that that {m(σ •

ψ i * (m i ), d, µ)} is bounded if and only if {(σ(m i ), d, µ)} is bounded. □
To conclude the proof of Lemma 5.2, it remains to establish the property (2). We restate the property as a claim.

Claim 5.5. Let A ⊂ M be an essential annulus with its boundary components denoted by

∂ 1 A and ∂ 2 A. Suppose that m(ψ i * (m i ), ∂ k A, µ) -→ ∞ for both k = 1 and k = 2. Then length ρ i •ψ -1 i * (∂ 1 A) -→ 0.
Proof. Let a 1 , . . . , a q be homotopically distinct simple closed curves on ∂M representing all the homotopy classes (in ∂M ) homotopic to ∂ 1 A in M . By renumbering them, we can assume

a k = ∂ k A for k = 1, 2. If length ψ i * (m i ) (a k ) -→
0 for some k = 1, . . . , q, we are done.

To deal with the remaining case, we now assume that there is a positive constant such that length ψ i * (m i ) (a k ) ≥ for every i ∈ N and k = 1, . . . , q. Then, there are a constant L and simple closed curves c k,i for every i ∈ N and k = 1, . . . , q such that c k,i intersects a k essentially and length ψ i * (m i ) (c k,i ) ≤ L. There is also K 1 such that d Y (c k,i , µ(ψ i * (m i ))) ≤ K 1 for any j, i and any incompressible subsurface Y ⊂ ∂M intersecting c k,i that is neither an annulus nor a pair of pants, since by definition, the length of µ(ψ i * (m i )) is also bounded from above by a constant.

Since

m(ψ i * (m i ), a k , µ) -→ ∞ and ψ i * (m i ) (a k ) ≥ for k = 1, 2, there are incompressible subsurfaces Y k,i such that a k ⊂ ∂Y k,i and d Y k,i (µ(ψ i * (m i )), µ) -→ ∞ for k = 1, 2.
If, passing to a subsequence, Y 1,i and Y 2,i are both annuli, then, up to homotopy, they lie on the boundary of the same component Ξ of the characteristic submanifold (which is, up to passing to a further subsequence independent of i). However, the assumption that m(ψ i * (m i ), a k , µ) -→ ∞ contradicts (a) when Ξ is a solid torus, and (c) when Ξ is an I-pair. Therefore, we can assume that one of the Y k,i (k = 1, 2), say Y 1,i is not an annulus.

Suppose now that Y 1,i is not eventually contained in the characteristic submanifold X (up to homotopy), even after passing to a subsequence. By taking a subsequence, we can assume that none of the Y 1,i are contained in X. Then, there is a simple closed curve c ⊂ ∂M disjoint from X which intersects Y 1,i for all i, by passing to a further subsequence. By Theorem 4.4 there is a constant L such that length

ρ i •ψ -1 i * (c) ≤ L. Since d Y 1,i (µ(ψ i * (m i )), µ) -→ ∞ by our assumption, we have d Y 1,i (c 1,i , c) -→ ∞. Then it follows from [15, Theorem B] that length ρ i •ψ -1 i *
(∂Y 1,i ) -→ 0, and hence in particular, we have length

ρ i •ψ -1 i * (∂ 1 A) -→ 0.
Next suppose that Y 1,i eventually lies in X. Taking a subsequence, we can assume that all the surfaces Y 1,i lie in the same component Ξ of X. Since Y 1,i is not an annulus, Ξ must be an I-bundle, which has a form of Ξ = F × I. By (c), there is another surface Y 3,i ⊂ ∂Ξ such that Y 1,i and Y 3,i bound an I-bundle compatible with the I-bundle structure of Ξ, and are projected along the fibres of Ξ = F × I to the same surface Z i in F and d Y 3,i (µ(ψ i * (m i )), µ) ≤ R. We note that by our definition of a 1 , . . . , a q , there is k 0 ≥ 2 such that a k 0 lies on ∂Y 3,i . Then since

d Y 3,i (µ(ψ i * (m i )), c k 0 ,i ) ≤ K 1 , we have d Y 3,i (c k 0 ,i , µ) ≤ R + K 1 .
We shall make use of {c 1,i } and {c k 0 ,i } to apply [START_REF] Minsky | Kleinian groups and the complex of curves[END_REF]Theorem B] as before. Since they do not lie on the same surface, we first need to project them to F . This leads to the following claim: Claim 5.6. There are K > 0 and two sequences of simple closed curves {d 1,i } and {d k 0 ,i } on F such that length

ρ i •ψ -1 i * (d k,i ) ≤ K for all i and k = 1, k 0 , and d Z i (d 1,i , d k 0 ,i ) -→ ∞.
Proof. Let k be either 1 or k 0 . If c k,i is contained in Ξ for sufficiently large i, then we let d k,i be the projection of c k,i to F . We also note that length

ρ i •ψ -1 i * (d k,i ) ≤ L then.
Suppose that this is not the case. We let S be the component of ∂M containing c k,i . Following [15, page 138] we extend the multicurve B := Fr(Ξ ∩ S) to a complete geodesic lamination λ by performing Dehn twists around B infinitely many times to c k,i and adding finitely many isolated leaves spiralling around B. There is a unique pleated surface h k,i : S → H 3 /ρ i (π 1 (S)) realising λ which induces ρ i • ψ -1 i * between the fundamental groups. Let R λ be the -thick part of S with respect to the hyperbolic metric induced by h k,i . By the efficiency of pleated surfaces ( [START_REF] Thurston | Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle[END_REF]Theorem 3.3], [START_REF] Minsky | Kleinian groups and the complex of curves[END_REF]Theorem 3.5]), there is a constant K 2 such that length h k,i (c k,i ∩ R λ ) ≤ L+K 2 i(c k,i , B) (the relation between the alternation and intersection numbers comes from (4.3) in [START_REF] Minsky | Kleinian groups and the complex of curves[END_REF]). In particular, there is an arc κ k,i in c k,i ∩ (Ξ ∩ S) ∩ R λ intersecting Y k,i and having length at most L + K 2 . By Theorem 4.4, the length of each component of B on h k,i is bounded by a constant L ′ independent of i. By joining one or two copies of κ k,i (depending on whether κ k,i intersects one or two components of B ∪ Fr R λ ) with arcs on B ∪ Fr R λ , we can construct in S ∩ Ξ a simple closed curve d k,i such that length h k,i (d k,i ) ≤ 2(L + K 2 + L ′ + ). Furthermore, this construction implies that there is a constant K 3 such that d Y (d k,i , c k,i ) ≤ K 3 for any incompressible subsurface Y ⊂ S ∩ Ξ intersecting both d k,i and c k,i , and in particular for Y = Y k,i . We use the same symbol d k,i to denote the projection of d k,i on F along the fibres of Ξ = F × I.

Thus we have length

ρ i •ψ -1 i * (d k,i ) ≤ 2(L+K 2 +L ′ + ), and d Z i (d 1,i , d k 0 ,i ) ≥ d Y 1,j (c 1,i , µ)-d Y k 0 ,i (c k 0 ,i , µ)-2K 3 ≥ d Y 1,j (c 1,i , µ)-R-K 1 -2K 3 -→ ∞. □ case, we let T be A × [0, 1] such that A × {0} is an annular neighbourhood of d whereas A × {1} is that of d 2 .
Since Fr V d consists of annuli, by the condition (i) above, T can be assumed to be contained in V d by moving it by an isotopy in both cases. If p = 1, we set T = ∅.

Given j = 1, . . . , p, we denote by F j the component of

V d ∩ ∂M \ ∪ k̸ =j d k containing d j .
The sequence of groups {ρ i (π 1 (V d ))} converges geometrically to a Kleinian group Γ containing ρ ∞ (π 1 (V d )), passing to a subsequence.

In the next section, we shall construct V d having the properties above, which shows that our argument in the present section really works.

Assuming the existence of V d for the moment, we now prove that every component of V d \ T has a compact core which is embedded in the geometric limit H 3 /Γ making use of the work of [START_REF] Brock | Windows, cores and skinning maps[END_REF]. Lemma 5.8. In Setting 5.7, let W be a submanifold of V d which is the closure of a component of V d \T . Then there is a relative compact core C W ⊂ H 3 /ρ ∞ (π 1 (W )) which is homeomorphic to W and on which the restriction of the covering projection H 3 /ρ ∞ (π 1 (W )) → H 3 /Γ induced by the inclusion is injective. Furthermore, for the closures of two components W 1 , W 2 of V d \ T (in the case when T is non-empty and separates W ), the compact cores C W 1 and C W 2 can be taken so that their images in H 3 /Γ are disjoint.

Proof. Our conditions in Setting 5.7 imply the assumptions of [4, Proposition 4.4], and applying this proposition, we see that there is a compact submanifold of H 3 /Γ which lifts to a compact core C W of H 3 /ρ ∞ (π 1 (W )) such that the restriction of the covering projection H 3 /ρ ∞ (π 1 (W )) → H 3 /Γ to C W is injective. Let Γ W ⊂ Γ be the geometric limit of {ρ i (π 1 (W ))}. Then the restriction of the covering projection H 3 /ρ ∞ (π 1 (W )) → H 3 /Γ W to C W must also be injective.

By [START_REF] Brock | Windows, cores and skinning maps[END_REF]Lemma 4.6], ρ ∞ (π 1 (W )) is either a generalised web group or a degenerate group without accidental parabolic elements. It follows then from [1, Corollary C and Theorem E] that C W is homeomorphic to W . The last sentence of our lemma also follows from [START_REF] Brock | Windows, cores and skinning maps[END_REF]Proposition 4.4]. □

We next show that by performing Dehn twists along embedded annuli bounded by d and d j (j = 2, . . . , p), we can make each F j embedded in the algebraic limit and mapped injectively in the geometric limit by the covering projection.

In the next lemma and the following, we shall use the expression 'the outward side of a cusp'. We say that an embedding of the surface F j ⊂ ∂V d into the geometric limit H 3 /Γ lies on the outward side of a cusp if the cusp lies on the same side of the embedding of F j as the embeddings of the components of V d \ T intersecting F j . Otherwise we say that the embedding of F j lies on the inward side of the cusp. Lemma 5.9. In Setting 5.7, we denote by D j the right-hand Dehn twist along an embedded annulus bounded by d = d 1 and d j (j = 2, . . . , p). Then for each j, there is a sequence {a i (j)} of integers with the following properties:

-The sequence

{θ i = ρ i • D a i (j) j * | π 1 (F j ) } converges algebraically to a repre- sentation θ ∞ : π 1 (F j ) → PSL 2 (C).
-There is an embedding h j :

F j → H 3 /θ ∞ (π 1 (F j )) inducing θ ∞ such that the restriction of the covering projection Π F j : H 3 /θ ∞ (π 1 (F j )) → H 3 /Γ to h j (F j )
′ j = F j ∩ W ′ and F ′′ j = F j ∩ W ′′ . By Lemma 5.8, there are compact cores C W ′ ⊂ H 3 /ρ ∞ (π 1 (W ′ )) and C W ′′ ⊂ H 3 /ρ ∞ (π 1 (W ′′ ))
, homeomorphic to W ′ and W ′′ respectively, on which the restrictions of the covering projections to H 3 /Γ are injective. The inclusions induce embeddings f ′ : F

′ j → ∂C W ′ and f ′′ : F ′′ j → ∂C W ′′ which lift to embeddings g ′ : F ′ j → H 3 /ρ ∞ (π 1 (F j )) and g ′′ : F ′′ j → H 3 /ρ ∞ (π 1 (F j )). The restrictions of the covering projection Π F j : H 3 /ρ ∞ (π 1 (F j )) → H 3 /Γ to g ′ (F j ∩ W ′ ) and to g ′′ (F j ∩ W ′′ ) are embeddings.
If T does not separate F j , we set ǧ = g ′ = g ′′ , otherwise, we put g ′ and g ′′ together to get an embedding ǧ :

F j \ T → H 3 /ρ ∞ (π 1 (F j )). Mov- ing C W ′ , C W ′′ , f ′
and f ′′ by isotopies, we may assume that they send the boundary of F j \ T into the -thin part. Then for an appropriate choice of , the map ǧ sends the boundary of F j \ T to the boundary of the 1 -thin part of H 3 /ρ ∞ (π 1 (F j )), where 1 is smaller than the threedimensional Margulis constant. It is then easy to extend ǧ to an embedding g : F j → H 3 /ρ ∞ (π 1 (F j )) such that g(T ∩ F j ) lies on the boundary of the 2 -thin part with 2 ≤ 1 . By Lemma 5.8 and by our construction, the restriction of Π F j • g to F j \ T , which is Π F j • ĝ, is an embedding and with an appropriate choice of , the composition Π F j •g maps F j ∩T to the boundary of the 0 -thin part of H 3 /Γ.

If ρ ∞ (d) belongs to a rank-1 maximal parabolic subgroup of Γ, then it is easy to change g on F j ∩ T so that Π F j • g is an embedding. In this case, we simply take a i to be 0.

Otherwise, ρ ∞ (d) belongs to a rank-2 maximal parabolic subgroup of Γ. We denote by T 0 the boundary of the corresponding torus cuspneighbourhood in H 3 /Γ, i.e. the boundary of the corresponding component of the 2 -thin part. Let Z be the union of Π F j • g(F j \ T ) and T 0 . Then Π F j • g(F j ) is contained in Z by our way of extending ǧ to g as described above. As is explained in [2, Lemma 3.1], Π F j • g is homotopic to a standard map f k wrapping k times around T 0 for some k ∈ Z, and there are two standard embeddings f 0 , f 1 : F j → Z such that f 0 (F j ) lies on the outward side of the cusp associated with d and f 1 (F j ) lies on its inward side, both without wrapping around T 0 .

Let

{q i : B r i (H 3 /ρ i (π 1 (M )), x i ) → B K i r i (H 3 /Γ, x ∞
)} be a sequence of K ibi-Lipschitz approximate isometry on the r i -ball with r i -→ ∞, K i -→ 1 given by the geometric convergence as explained in Section 2.4. By [2, Lemma 3.1], there is

s i ∈ Z such that q -1 i •f 0 is homotopic to q -1 i •Π F i •g•D s i j .
The conclusion follows, taking a i (j) = s i and setting h j to be the lift of f 0 to H 3 /θ ∞ (π 1 (F j )).

□

Next we study how the embedding of a compact core in the geometric limit as above affects the end invariants. Lemma 5.10. In Setting 5.7, for each j = 1, . . . , p, suppose that there is an embedding h j :

F j → H 3 /ρ ∞ (π 1 (F j )) inducing ρ ∞ | π 1 (F j ) such that the restriction of the covering projection Π F j : H 3 /ρ ∞ (π 1 (F j )) → H 3 /Γ to h j (F j ) is an embedding. If Π F j (h j (F j )) lies on the outward side of the cusp associated with ρ ∞ (d) ∈ ρ ∞ (π 1 (M )) ⊂ Γ, then {m(m i , d j , µ)} is bounded whereas m(σ(m i ), d j , µ) -→ ∞. If Π F j (h j (F j )) lies on the inward side of the cusp associated with ρ ∞ (d) then {m(σ(m i ), d j , µ)} is bounded whereas m(m i , d j , µ) -→ ∞.
Proof. Suppose that Π F j (h j (F j )) lies on the outward side of the cusp associated with ρ ∞ (d) ∈ ρ ∞ (π 1 (M )) ⊂ Γ. Let c ⊂ F j be a simple closed curve intersecting d j essentially, c * the closed geodesic homotopic to Π F j (h j (c)), and denote by ψ i : B K i r i (H 3 /ρ i (π 1 (M )), x i ) → B r i (H 3 /Γ, x ∞ ) an approximate isometry associated with the geometric convergence of {ρ i (π 1 (M ))} to Γ as explained in Section 2.4. For i large enough, ψ -1 i (c * ) is a quasi-geodesic lying outside the thin part. on the same side as F j of the Margulis tube associated with ρ i (d). Let S j be the component of ∂M containing F j . In the covering H 3 /ρ i (π 1 (S j )) of H 3 /ρ i (π 1 (M )), the closed geodesic homotopic to ρ i (c) lies above the Margulis tube associated with ρ i (d). Therefore, by [START_REF] Brock | Convergence properties of end invariants[END_REF]Theorem 1.3] there is a constant D such that d Y (c, µ(m i )) ≤ D for any surface Y ⊂ S j with d j ⊂ Fr Y . Thus for any full marking µ, there is D ′ such that d Y (µ, µ(m i )) ≤ D ′ for any surface Y ⊂ S with d j ⊂ Fr Y .

To conclude that {m(m i , d j , µ)} is bounded, it remains to show that length m i (d j ) is bounded away from 0.

Assume the contrary, that length m i (d j ) -→ 0 after passing to a subsequence. Then, there is an annulus joining the closed geodesic d * j ⊂ H 3 /ρ i (π 1 (S)) representing ρ i (d) with d + j ⊂ ∂C(H 3 /ρ i (π 1 (S))) corresponding to d j , which lies entirely in the ithin part with i -→ 0. Since ψ -1 i (c * ) has bounded length, it cannot intersect such an annulus, whereas ψ -1 i (c * ) lies in a uniformly bounded neighbourhood of the convex core for large i. Since c * and Π F j (h j (F j )) lie on the same side of the cusp associated with ρ ∞ (d), this contradicts the assumption that Π F j (h j (F j )) lies on the outward side of the cusp associated with ρ ∞ (d) ∈ Γ.

Since length ρ i (d) -→ 0 and {m(m i , d j , µ)} is bounded, it follows from [14, Short Curve Theorem] that m(σ(m i ), d j , µ) -→ ∞.

A quite similar argument works also when Π F j (h j (F j )) lies on the inward side of the cusp associated with ρ ∞ (d) ∈ Γ. □ Corollary 5.11. In Setting 5.7, assume that p ≥ 2, and consider j ≤ p such that {m(m i , d j , µ)} is bounded. Then there is an embedding h :

F j → H 3 /ρ ∞ (π 1 (F j ))
inducing ρ ∞ such that the restriction of the covering projection Π F j : H 3 /ρ ∞ (π 1 (F j )) → H 3 /Γ to h(F j ) is an embedding whose image lies on the outward side of the cusp corresponding to ρ ∞ (d).

Proof. As can be seen in the proof of Lemma 5.9, if ρ ∞ (d) belongs to a rank-1 maximal parabolic subgroup of Γ, then a i (j) = 0 for any i and θ ∞ = ρ ∞ . Therefore, our claim of this corollary follows immediately from Lemmas 5.9 and 5.10. Otherwise, ρ ∞ (d) belongs to a rank-2 maximal parabolic subgroup of Γ. By Lemma 5.9, there is a sequence of integers {a i (j)} and an embedding h j : F j → H 3 /θ ∞ (π 1 (F j )) inducing θ ∞ between the fundamental groups such that the restriction of the covering projection Π F j : H 3 /θ ∞ (π 1 (F j )) → H 3 /Γ to h j (F j ) is an embedding and its image Π F j • h j (F j ) lies on the outward side of the cusp corresponding to θ ∞ (d) = ρ ∞ (d). By Lemma 5.10, {m(D a i j * m i , d j , µ)} is bounded. Since {m(m i , d j , µ)} is bounded by assumption, this is possible only when {a i (j)} is bounded. Then we may take a i (j) = 0 for any i in Lemma 5.9 so that θ ∞ = ρ ∞ , and the conclusion follows.

□

We now put these results together to get the result which we shall use to prove Proposition 5.1. Lemma 5.12. In Setting 5.7, suppose that {m(m i , d j , µ)} is bounded for j ̸ = 1. Then, there is a relative compact core for H 3 /ρ ∞ (π 1 (V d )) homeomorphic to V d on which the restriction of the covering projection Π d : H 3 /ρ ∞ (π 1 (V d )) → H 3 /Γ is injective. Furthermore, a cusp neighbourhood corresponding to ρ ∞ (d) intersects the compact core in an annular neighbourhood of d 1 .

Proof. By Lemma 5.8, for the components W of V d \ T , we have embeddings Suppose that p ≥ 2, and assume that {m(m i , d j , µ)} is bounded for every j ̸ = 1. Then by Corollary 5.11, for every j ̸ = 1, there is an embedding g j : F j → H 3 /ρ ∞ (π 1 (V d )) inducing ρ ∞ | π 1 (F j ) on which the restriction of Π d is injective. Furthermore, it follows from the construction that g j and g W agree on F j ∩W . Putting together the maps g W for all the components W of V d \T and the g j for all j ̸ = 1, we get an embedding g :

g W : W → H 3 /ρ ∞ (π 1 (V d )) inducing ρ ∞ | π 1 (W ) ,
V d → H 3 /ρ ∞ (π 1 (V d )) inducing ρ ∞ | π 1 (V d )
on which the restriction of Π d is injective.

Changing g by an isotopy, we may assume that g(V d ) intersects a cusp neighbourhood C associated with ρ ∞ (d) along an annulus A ⊂ g(∂V d ) which is a regular neighbourhood of g(d k ) for some k = 1, . . . , p. Then g(F k ) lies on the inward side of C. This is possible only if Π d • g(F k ) lies on the inward side of C; for the restriction of Π d is injective on g(V d ), and hence it cannot wrap around C.

By assumption, for every j ̸ = 1, {m(m i , d j , µ)} is bounded. It follows then from Corollary 5.11 that Π d (g(F j )) lies on the outward side of C for j ̸ = 1. Hence the only possibility is that A is a regular neighbourhood of g(d 1 ). □ In the other cases, we shall prove the proposition by contradiction. Assume that M is not an I-bundle, that m(σ(m i ), d, µ) -→ ∞, and that {m(m i , d j , µ)} is bounded for every j = 2, . . . , p. By Lemma 5.2, after re-marking and passing to a subsequence, we may assume that {ρ i = q(m i )} satisfies: ) -→ 0. We note that by Lemma 5.2, {m(m i , d j , µ)} is bounded for every j = 2, . . . , p and m(σ(m i ), d, µ) -→ ∞ even after re-marking.

Taking a further subsequence, we can also assume that for any essential annulus E of M , either length ρ i (∂E) -→ 0 or length ρ i (∂E) is bounded away from 0. Let A = ∪ k A k be a maximal family of pairwise disjoint non-isotopic essential annuli such that

Lemma 3 . 1 .

 31 1. Let p : M → M be a finite-sheeted regular covering. Then p induces the covering map between the boundaries p ∂ : ∂ M → ∂M . This map induces a proper embedding between Teichmüller spaces, p * ∂ : T(∂M ) → T(∂ M ) which is obtained by pulling back the conformal structures by p ∂ . Also the involution ι lifts to an orientation-reversing involution ι : ∂ M → ∂ M taking each component to another one. Since M is also an atoroidal boundary-irreducible Haken manifold, we can consider the skinning map σ : T(∂ M ) → T( ∂ M ). If (ι • σ) n has bounded image for some n ∈ N, then so does (ι • σ) n . Proof. The map p * ∂ properly embeds T(∂M ) into T(∂ M ). By the definition of the maps σ and ι, we have p

Theorem 4 . 5 .

 45 Let M be an atoroidal boundary-irreducible Haken 3manifold all of whose characteristic I-pairs are product I-bundles. Let {m i } be a sequence in T(∂M ), and {ρ i : π 1 (M ) → PSL 2 (C)} a sequence of representations corresponding to {q(m i )}. Let µ ⊂ ∂M be a full and clean marking, and W ⊂ M a submanifold with paring locus P which is a union of disjoint non-parallel essential annuli on ∂W . (See Definition 2.1 for the definition of pared manifolds.) We assume the following: (a) The closure of ∂W \ ∂M is a union of essential annuli contained in P . (b) For any non-contractible simple closed curve c in P , length ρ i (c) is bounded as i -→ ∞. (c) For any essential annulus E ⊂ W disjoint from P , there is a component c of ∂E such that {m(m i |S, c, µ)} is bounded for the component S of ∂M on which c lies. (d) If M is an I-bundle, then P ̸ = ∅.

Lemma 5 . 2 .

 52 Let d be an essential simple closed curve on ∂M , and let d 1 , . . . , d p be disjoint simple closed curves on ∂M representing the homotopy classes of simple closed curves on ∂M homotopic to d in M , where d 1 = d.

  By our definition of d 1 , . . . , d p , the boundary of Y ′ i contains some d k with k ≥ 2. Then, {m(m i , d k , µ)} is bounded by the assumption (*), and ψ i (d k ) = d k by the property (b). In particular

  on the union of which the restriction of Π d is injective.If p = 1, then T = ∅ by definition, and hence W = V d . We can take a cusp neighbourhood corresponding to ρ ∞ (d) intersecting g W (W ) along an annulus in the homotopy class of d. Since p = 1, such an annulus is isotopic on ∂V d to an annular neighbourhood of d 1 = d.

5. 3 . 1 .

 31 Completion of the proof of Proposition 5.Proof of Proposition 5.1. If M is an I-bundle, then m(σ(m i ), d, µ) = m(m i , d 2 , µ) and the conclusion follows.

( 1 )

 1 for any simple closed curve c ⊂ ∂M , either {m(m i , c, µ)} (resp. {m(σ(m i ), c, µ)}) is bounded or m(m i , c, µ) -→ ∞ (resp. m(σ(m i ), c, µ) -→ ∞); (2) if A ⊂ M isan essential annulus such that ∂A does not intersect d (hence any of d j ) and m(m i , ∂ j A, µ) -→ ∞ for both boundary components ∂ 1 A and ∂ 2 A of A, then length ρ i (∂A * 1

  [i] the length of the core curve of each annulus A k tends to 0 (length ρ i (∂A k ) -→ 0 for any k), [ii] ∂A does not intersect d, and [iii] no component of A contains a curve homotopic to d. Denote by V d the component of M \N (A) containing d, where N (A) denotes a thin regular neighbourhood of A. Let P be the closure of V d \ ∂M , which is a union of annuli. Next we shall control the geometry of V d and the length of d.

  The case of n > 1 follows by induction.□ For any natural number n, there is an (possibly empty) incompressible surface Σ n ⊂ X ∩ ∂M such that each component of Σ n is ntime vertically extendible, no component of Σ n can be isotoped on ∂M into another component of Σ n and every n-time vertically extendible surface can be isotoped into Σ n .Proof. If there is no surface that is n-time vertically extendible, we set Σ n to be ∅. Otherwise, let Σ ⊂ X be an n-time vertically extendible incompressible surface. If every n-time vertically extendible surface can be isotoped into Σ, we are done, by taking Σ n = Σ.

	Corollary 3.8.

  3) For every d j among d 1 , . . . , d p defined above, (i) ψ i (d

j ) = d j for every i and j, (ii) for every j = 1, . . . , p, {m(ψ i * (m i ), d j , µ)} is bounded if and only if {m(m 1 , d j , µ)} is bounded, and

  is an embedding and its image Π F j •h j (F j ) lies on the outward side of the cusp corresponding to ρ∞ (d) = θ ∞ (d) when the latter is a rank-2 cusp.Proof. This is a relative version of [2,Lemma 4.5].Let W ′ and W ′′ be the components of V d \ T intersecting F j (we set W ′ = W ′′ if there is only one such component), and set F

Continuation of Proof of Claim 5.5. Set ϑ i = ρ i •ψ -1 i * •I * : π 1 (S) → PSL 2 (C) where I * : π 1 (S) → π 1 (M ) is the homomorphism induced by the inclusion. Following [START_REF] Minsky | Kleinian groups and the complex of curves[END_REF], we denote by C 0 (ϑ i , K) the set of simple closed curves on S whose translation lengths with respect to ϑ i are less than or equal to K. By the claim above, we see that both d 1,i and d k 0 ,i lie in C 0 (ϑ i , K) and that

This also concludes the proof of Lemma

End invariants and wrapping. In this subsection, we shall discuss how algebraic limits projects to geometric limits and how this is reflected in the behaviour of the end invariants.

Let us now fix the assumptions and notations which will be used in most results of this section. We also assume that we have a submanifold V d of M whose frontier consists of incompressible annuli and which has the following three properties:

(i) V d contains all the curves d j (j = 1, . . . , p), and Proof. Let us first assume that m(m i , d 1 , µ) -→ ∞, and verify the hypotheses of Theorem 4.5 with W = V d . The hypothesis (a) follows from the construction of V d . The hypothesis (b) follows from the property (2) above. By Lemma 4.2, {m(m i , c, µ)} is bounded for any simple closed curve c intersecting d. This observation combined with the assumption that {m(m i , d j , µ)} is bounded for any j = 2, . . . p, the property (2) above and the maximality of A yields the hypothesis (c). Now by Theorem 4.5 we can take a subsequence in such a way that the restrictions {ρ i|π 1 (V d ) } converge.

If length m i (d) -→ 0, we are done. Otherwise, since we are assuming that m(m i , d, µ) -→ ∞, there is a sequence of subsurfaces [START_REF] Minsky | The classification of Kleinian surface groups. I. Models and bounds[END_REF]Short Curve Theorem]. We add to P a thin regular neighbourhood of d on ∂V d and we can verify as above that the hypotheses of Theorem 4.5 are satisfied for (V d , P ). □ Now we are in the situation of Setting 5.7, and we use its notations. By Lemma 5.12, g(F 1 ) lies on the inward side of the cusp corresponding to ρ ∞ (d), and g(F j ) lies on the outward side for every j = 2, . . . , p. Then Lemma 5.10 implies that {m(σ(m i ), d 1 , µ)} is bounded. This contradicts our assumption. □

6. The proof of Theorem 1.1

Now we shall complete the proof of Theorem 1.1. By Lemmas 3.1 and 3.4, we can assume that every M is strongly untwisted. Let L be the number provided by Lemma 3.9, and consider a sequence