Bending map

and strong convergence

Cyril LECUIRE May 2004

Abstract The bending map of a hyperbolic 3-manifold with boundary maps a geometrically hyperbolic metric to its bending measured geodesic lamination. We show that the bending map is proper. Combining this we the result of [Le3], we deduce a criterion for the strong convergence of a sequence of geometrically finite representations to a geometrically representation. Using the properness of the bending map, we define a subset of ML(∂M) on which the action of the group of isotopy classes of homeomorphisms of M is properly discontinuous.

1 Introduction

Consider a compact, orientable 3-manifold M that is hyperbolic, namely the interior of M is endowed with a complete metric σ of constant sectional curvature −1. Assume that ∂M contains a surface of genus > 1. A fundamental subset of (M, σ) is its convex core N(σ), defined as the smallest non-empty closed subset of the interior of M that is locally convex and homotopy equivalent to M. Its boundary ∂N(σ) is a pleated surface, namely it is almost everywhere totally geodesic and it is bent along a geodesic lamination. The amount of bending is described by a measured geodesic lamination called the bending measured geodesic lamination of σ (cf. [Th1] or [CEG]). This yields a bending map b : H(M) → ML(∂M) which to a complete hyperbolic metric associates its bending measured geodesic lamination. In the present paper, we investigate some properness properties of this bending map.

We will consider isotopy classes of hyperbolic metric on the interior of M. Two metrics σ_{1} and σ_{2} are identified if there exists a diffeomorphism f : M → M isotopic to the identity such that σ_{1} = f^{∗}σ_{2}. We will topologize the set H(M) of isotopy classes of complete hyperbolic metrics in the following way. Let us choose a point x in int(M). A metric σ_{2} lies in a (k, r)-neighborhood of σ_{1} if there exists a diffeomorphism g : M → M isotopic to the identity such that the restriction of g to B(x, r) ⊂ (M, σ_{1}) is a k-quasi-isometry into its image in (M, σ_{2}). We obtain a basis of neighborhoods of σ_{1} by letting k tend to 1 and r tend to +∞. The topology defined in this way does not depend on the choice of the point x.

In the present paper we will only consider geometrically finite hyperbolic metric on int(M): a complete hyperbolic metric σ on int(M) is said to be geometrically finite if N(σ) has finite volume. The metric σ is fuchsian if int(N(σ)) = ∅. We will denote by GF(M) ⊂ H(M) the set of isotopy classes of geometrically finite non-fuchsian metrics on int(M). We will denote by CC(M) ⊂ GF(M) the set of isotopy classes of non-fuchsian geometrically finite hyperbolic metrics on int(M) whose only cusps are rank 2 cusps. On CC(M) the topology defined above is equivalent to the topology obtained when we consider quasi-isometric deformations of the whole metric on int(M). Let us denote by b_{CC}, b_{GF} the restrictions of b to CC(M) and to GF(M) respectively. The continuity and the differentiability of b_{CC} have been studied in [KeS] and [Bo2]. Its injectivity has been established in [BoO] in the special case of weighted multi-curves. The image of b_{GF} has been described in [BoO] and [Le1], it is the set P(M) of measured geodesic laminations satisfying the following conditions:

- a) no closed leaf of λ has a weight greater than π;

- b) ∃η > 0 such that, for any essential annulus E, i(∂E, λ) ≥ η;

- c) i(λ, ∂D) > 2π for any essential disc D.

In the present paper we will study properness properties of b_{GF}. As we will see, such properties are useful to establish some convergence criteria for geometrically finite metrics involving only bending measured geodesic laminations. The first result we will present is a generalization of the “lemme de fermeture” in [BoO]:

Theorem 1.1. Let M be a compact connected orientable hyperbolic 3-manifold with boundary and let σ_{n} be a sequence of geometrically finite non-fuchsian metrics on the interior of M. Denote by λ_{n} ⊂ ML(∂M) the bending measured geodesic lamination of σ_{n}. Suppose that λ_{n} converge in ML(∂M) to a measured geodesic lamination λ satisfying the following conditions:

- b) ∃η > 0 such that, for any essential annulus E, i(∂E, λ) ≥ η;

- c) i(λ, ∂D) > 2π for any essential disc D;

- d) there is no homeomorphism between M and an interval bundle over a pair of pants P such that the support of λ is the image of a section of the bundle over ∂P.

Then the sequence (σ_{n}) contains a subsequence converging to a geometrically finite non-fuchsian metric σ_{∞} on the interior of M.

Furthermore it follows from [Le3] that the bending measured geodesic lamination λ′ of σ_{∞} is obtained from λ by replacing by π the weights of the closed leaves which have a weight greater than π.

If we consider only convex cocompact metrics, we deduce from this theorem that b_{CC} is proper:

Corollary. The map b_{CC} from CC(M) to b(CC(M)) endowed with the weak^{∗} topology is proper.

Remark. Recall that CC(M) does not contain any fuchsian metric. If M admits some fuchsian metric, it is homeomorphic to an I-bundle over a compact surface S. Any hyperbolic metric on S yields a geometrically finite metric on int(M) with the following bending geodesic measured lamination λ : the support of λ is a section of the I-bundle over ∂S and each leaf has a weight equal to π. When S is not a pair of pants, the set of hyperbolic metric on S is not compact. Let us denote by F(M) the set of fuchsian metrics on M. This set F is not empty if M is a handlebody or an I-bundle over a closed surface. If F(M) is not empty, then it is not compact. Therefore the restriction of b to F(M) is not proper. For a discussion about non-fuchsian metrics tending to fuchsian ones, the reader should refer to [Se]. ⋄

When we deal with geometrically finite metrics, difficulties appear with leaves which have a weight greater than π. Especially, it follows from Theorem 1.1 that the bending map is discontinuous in the neighborhood of any geometrically finite metric having at least one rank 1 cusp. To overcome this difficulty, we quotient the set P^{+}(M) of measured geodesic laminations satisfying condition b), c) and d) by the following relation:

Let λ, μ ∈ ML(∂M) be two measured geodesic laminations and let us denote by λ′ (resp. μ′) the measured geodesic laminations obtained by replacing by π the weights of the leaves of λ (resp. μ) which have a weight greater than π. We set λRμ if and only if λ′ = μ′. We denote by λ̇ the class of λ modulo R.

Endow ML(∂M) with the weak^{∗} topology and ML(∂M)/R with the quotient topology. It is not hard to see that this space is a Hausdorff space (cf. [Le3, Lemma 2.3]). From b_{GF}(M) we obtain a quotient map b_{R} : GF(M) → P^{+}(M)/R. We can then give a general version of Theorem 1.1 :

Theorem 1.2. The map b_{R} from GF(M) to P^{+}(M)/R is proper.

Rather than considering isotopy classes of metrics, we could consider homotopy classes of metrics. Namely we quotient the set GF(M) by the group Mod_{0}(M) of diffeomorphisms of M which are homotopic to the identity. The interesting point is that this quotient GF (M) = GF(M)/Mod_{0}(M) is the same set as the set of conjugacy classes of geometrically finite representations ρ : π_{1}(M) → Isom(ℍ^{3}) such that ℍ^{3}/ρ(π_{1}(M)) is homeomorphic to the interior of M. We endow this set GF (M) with the quotient topology of the topology of GF(M). When we regard GF (M) as a set of representations, the topology thus defined is the topology of the strong convergence. We have to notice that if a representation ρ ∈ GF (M) is given (up to conjugacy) then its bending measured geodesic lamination is defined up to the action of the group Mod_{0}(M). Let us quotient the space ML(∂M) by R and by the action of Mod_{0}(M) : two measured geodesic laminations α and λ lie in the same class modulo Mod_{0}(M) ∗ R if and only if there is a diffeomorphism ϕ : M → M homotopic to the identity such that λ′ = ϕ(α′) where λ′ and α′ are the same as in the definition of R (namely λ′ and α′ are the representatives of λ̇ and α̇ whose closed leaves all have a weight less than or equal to π). As we will see below the action of Mod_{0}(M) on P^{+}(M) is properly discontinuous. Therefore P^{+}(M)/(Mod_{0}(M)∗R) is a Hausdorff space. In this context, we have the following corollary of Theorem 1.2 :

Corollary. Let (ρ_{n}) ∈ GF (M) be a sequence of representations of π_{1}(M), let λ̊_{n} ∈ ML(∂M)/Mod_{0}(M) be the bending measured geodesic lamination of ρ_{n} and let λ̈_{n} ∈ P^{+}(M)/(Mod_{0}(M) ∗ R) be the projection of λ̊_{n}. If the sequence (λ̈_{n}) converges to some λ̈_{∞} ∈ P^{+}(M)/(Mod_{0}(M) ∗ R), then (ρ_{n}) contains a subsequence that converges strongly to a geometrically finite representation ρ_{∞} ∈ AH(M).

It follows from the result of [Le3] that the bending measured geodesic lamination of ρ_{∞} is (modulo the action of Mod_{0}(M)) the representative of λ̈_{∞} whose closed leaves all have a weight less than or equal to π.

Let λ̊ ∈ P(M)/Mod_{0}(M) be a class of measured geodesic laminations and let B(λ̊) be the set of geometrically finite representations ρ ∈ AH(M) whose bending measured geodesic lamination is λ̊. Conjecturally B(λ̊) is a single point, this conjecture has been proved in some special cases, see [BoO] and [Bo5]. Combining the corollary above and the results of [Ta] and [Le3] we get the following criterion for strong convergence.

Theorem 1.3. Let λ̊ ∈ P(M)/Mod_{0}(M) be a class of measured geodesic laminations such that B(λ̊) is a single point ρ and let ρ_{n} ⊂ AH(M) be a sequence of conjugacy classes of representations. Then ρ_{n} converges strongly to ρ_{∞} if and only if :

1. there is N such that for n ≥ N, ρ_{n} is a geometrically finite representation with bending measured geodesic lamination λ̊_{n} ∈ P(M)/Mod_{0}(M);

2. the sequence λ̈_{n} ∈ P^{+}(M)/(Mod_{0}(M) ∗ R) converges to λ̈.

Theorem 1.1 may also be used to study the action of the modular group on ML(∂M). The modular group Mod(M) is the group of isotopy classes of diffeomorphisms of M.

A measured geodesic lamination λ ∈ ML(∂M) is doubly incompressible if ∃η > 0 such that i(∂E, λ) > η for any essential annulus or disc E. Let us denote by D(M) the set of doubly incompressible measured geodesic laminations. We have :

Theorem 1.4. The action of Mod(M) on D(M) is properly discontinuous.

The paper is organized as follows. In section 2, we state some definitions and facts about geodesic laminations, convergence of representations, P(M) and P^{+}(M). In sections 3 and 4 we recall some ideas of [BoO] and [Le1] and complete them to prove the properness of b_{R}. At this point, the proof of Theorems 1.1 and 1.2 is complete. In section 5, we deduce Theorem 1.4 from Theorem 1.1.

I would like to thank Francis Bonahon for discussions from which this paper originates, Jean-Pierre Otal for his advices and Young Eun Choi for useful comments.

2 Definitions

2.1 Geodesic laminations

Let ∂_{χ<0}M be the union of the components of ∂M with negative Euler characteristic. Assume that each component of ∂_{χ<0}M is endowed with a complete hyperbolic metric. A geodesic lamination on ∂M is a compact subset which is the disjoint union of complete simple geodesics embedded in ∂_{χ<0}M. The space of geodesic laminations on ∂M endowed with the Hausdorff topology is denoted by L(∂M). A geodesic lamination whose leaves are all closed is called a multi-curve. If each half-leaf of a geodesic lamination L is dense in L, then L is minimal. Such a minimal geodesic lamination is either a simple closed curve or an irrational lamination. A leaf l of a geodesic lamination L is recurrent if it lies in a minimal geodesic lamination. Any geodesic lamination is the disjoint union of finitely many minimal laminations and non-recurrent leaves. A leaf is said to be an isolated leaf if it is either a non-recurrent leaf or a compact leaf without any leaf spiraling toward it.

We say that two geodesic laminations L_{1} and L_{2} intersect transversely if a leaf of L_{1} intersects a leaf of L_{2} transversely. A geodesic lamination L is arational if any closed geodesic intersects L transversely.

It turns out that the definition of L(∂M) can be made independent of the choice of the metric on ∂_{χ<0}M. This is explained in the following note.

\underline{Note}. Let us consider two complete hyperbolic metrics s_{1} and s_{2} on a connected component S of ∂_{χ<0}M and let q : ℍ^{2} → S be a covering map. Let l ⊂ S be a s_{1}-geodesic and let l̂⊂ q^{−1}(l) be a lift of l. There is a unique s_{2}-geodesic Θ(l) such that there is a bounded homotopy between a lift Θ̂(l̂) ⊂ q^{−1}(Θ(l)) of Θ(l) and l̂. Here a bounded homotopy is a map F : ℝ × [0, 1] → S such that the lengths of the arcs F ({x} × [0, 1]) are uniformly bounded. So we get a homeomorphism Θ : {geodesics of (S, s_{1})} → {geodesics of (S, s_{2})} and the image Θ(L) of a s_{1}-geodesic lamination L is a s_{2}-geodesic lamination. We will say that Θ(L) is the geodesic lamination L for the metric s_{2}. Thus the geodesic lamination L is well defined for any hyperbolic metric on ∂_{χ<0}M.

Let L_{1} and L_{2} be two s_{1}-geodesic laminations and let x ⊂ L_{1}∩L_{2} be a transverse intersection. Let x̂ ⊂ ℍ^{2} be a lift of x. This point x̂ is the intersection of a leaf l̂_{1} of q^{−1}(L_{1}) and of a leaf l̂_{2} of q^{−1}(L_{2}). Since Θ̂(l̂_{1}) intersects Θ̂(l̂_{2}) transversely, there is a map L_{1} ∩ L_{2} → Θ(L_{1}) ∩ Θ(L_{2}) which to x associates q(Θ̂(l̂_{1}) ∩ Θ̂(l̂_{2})). This way, each point lying in the transverse intersection of two geodesic laminations is defined independently of the choice of the metric on ∂_{χ<0}M. ⋄

Let L be a connected geodesic lamination which is not a simple closed curve and let S̄(L) be the smallest surface with geodesic boundary containing L. This surface S̄(L) contains finitely many disjoint simple closed geodesics that are disjoint from L. We denote by ∂′S̄(L) the union of these geodesics (thus ∂S̄(L) ⊂ ∂′S̄(L)) . Removing annular neighborhoods of these curves from S̄(L) yields a surface which we call the surface embraced by L and which we denote by S(L)

(see [Le1] for details). When L is a closed curve we take an annular neighborhood for S(L) and we define ∂′S̄(L) to be L. When L is not connected we take S(L) to be the disjoint union of the surfaces embraced by the connected components of L.

A leaf l of a geodesic lamination L is homoclinic if any lift l̃ of l to the universal cover M̃ of M contains two sequences of points x̃_{n} and ỹ_{n} such that the distance between x̃_{n} and ỹ_{n} measured on l̃ tends to ∞ whereas their distance measured on M is uniformly bounded.

Let Λ be an ordered subgroup of ℝ^{p} (with the lexicographic order) of rank p. A Λ-measured lamination L is a transverse measure with value in Λ for some geodesic lamination |L|. Any arc k ≈ [0, 1] embedded in ∂M transversely to |L|, such that ∂k ⊂ ∂M − |L|, is endowed with a finitely additive measure dL with values in Λ such that :

- the support of dL_{|k} is |L| ∩ k;

- if an arc k′ can be homotoped into k by a homotopy respecting |L| then ∫_{k}dL = ∫_{k′} dL.

Let k be an embedded arc in S transverse to |L| satisfying ∂k ⊂ S − |L|. We will denote by [∫_{k}dL] the first (counting from left to right) non-zero coordinate of ∫_{k}dL ∈ Λ ⊂ ℝ^{p}.

If λ is an ℝ-measured lamination (we will use Greek letters to name ℝ-measured laminations), then its support |λ| does not contain any non-recurrent leaves. If γ is a weighted simple closed curve with weight w(γ), the intersection number i(λ, γ) is defined by i(λ, γ) = w(γ) ∫_{|γ|}dλ. This extends to a continuous function i : ML(∂M) × ML(∂M) → ℝ (cf. [Bo1]).

We will say that two Λ-measured laminations intersect transversely if their supports do so.

Let us now define the dual Λ-tree T_{L} of a Λ-measured lamination L ⊂ S (cf. [MoO]). Let q : ℍ^{2} → S be the covering projection. Let preT_{L} be the set whose points are the complementary regions of q^{−1}(L) in ℍ^{2}. Let us define a distance d : preT_{L} × preT_{L} → Λ in preT_{L}. Let R_{0} and R_{1} be complementary regions and choose a geodesic segment k ⊂ ℍ^{2} whose endpoints lie in R_{0} and R_{1} respectively. We define d(R_{0}, R_{1}) to be the q^{−1}(L)-measure of k. There is a unique (up to isometry) Λ-tree T_{L} and an isometric embedding e : preT_{L} → T_{L} such that any point of T_{L} lies in a segment with endpoints in e(preT_{L}) (cf. [GiS]). This tree T_{L} is the dual tree of L. The covering transformations yield an isometric action of π_{1}(M) on T_{L}. Let c be a simple closed curve and let us denote by δ_{L}(c) the distance of translation of an isometry of T_{L} corresponding to c. We have then δ_{L}(c) = inf{∫_{c′}dL/c′ is a closed curve freely homotopic to c and transverse to |L|}.

Consider now the case of an ℝ-measured lamination λ. This construction yields a natural projection ℍ^{2} − q^{−1}(λ) → T_{λ}. If λ does not have any closed leaves, this projection extends continuously to a map π_{λ} : ℍ^{2} → T_{λ}. Otherwise we replace the closed leaves of λ by foliated annuli endowed with uniform transverse measures, thus we get a continuous map π_{λ} : ℍ^{2} → T_{λ}.

Let us come back to the general case where Λ is a subgroup of ℝ^{n}. The following claim links δ_{L}(c) with the transverse measure of L for a simple closed curve c ⊂ |L|.

Claim 2.1. The measure δ_{T_{L}}(c) of a simple closed curve c ⊂ |L| is the half-sum of the measures supported by the half-leaves of L which spiral toward c.

Proof. Let ĉ be a lift of c in ℍ^{2} and let g be the element of π_{1}(S) that fixes ĉ. Denote by q : ℍ^{2} → S the covering projection. If c is an isolated leaf, then g fixes the components of ℍ^{2} − q^{−1}(L) whose boundary contains ĉ. Therefore δ_{T_{L}}(c) = 0. Otherwise let x̂ be a point lying on a component of ℍ^{2} − q^{−1}(|L|) whose closure contains an end ξ of ĉ. Denote by [x̂, gx̂] the geodesic segment joining x̂ to gx̂ and let d̂ = ⋃_{i∈ℤ}[g^{i}x̂, g^{i+1}x̂]. This curve d̂ is fixed by g. It is easy to see that the image of d̂− q^{−1}(|L|) under the projection ℍ^{2} − q^{−1}(|L|) → T_{L} lies on the axis of g. Thus the distance of translation of g in T_{L} is the measure of [x̂, gx̂]. The claim follows.

A train track τ in ∂M is a finite union of rectangles which are foliated by vertical segments and which meet only at non-degenerate segments lying in the vertical sides. The rectangles are called branches, the horizontal segments are called rails, the vertical segments ties and the connected components of the union of the vertical sides are the switches. A subtrack τ′ of a train track τ is a train track whose branches are all branches of τ. A geodesic lamination L is carried by a train track τ if L lies in τ and is transverse to the ties; a branch b of τ carries L if L ∩ b ≠ ∅. The train track τ minimally carries L if no proper subtrack of τ carries L.

We will conclude this subsection with some results about ML(∂M)/R. The proofs of these results are quite easy and can be found in [Le3].

Let us denote by ML(∂M) the space of ℝ-measured laminations (which we will simply call measured geodesic laminations) on ∂_{χ<0}M endowed with the weak^{∗} topology. Let R be the relation defined in the introduction. If λRμ, then λ and μ have the same support. Therefore we can define the support |λ̇| of an element λ̇ of ML(∂M)/R as the support of any representative of λ̇.

When ML(∂M)/R is endowed with the quotient topology, we have the following :

Claim 2.2. Let (λ_{n}) be a sequence of measured geodesic laminations such that (λ̇_{n}) tends to λ̇ and that |λ_{n}| tends to L in the Hausdorff topology. Then we have |λ̇| ⊂ L.

Claim 2.3. Let λ ∈ ML(∂M) and let λ′ be the representative of λ̇ whose compact leaves have all a weight less than or equal to π. Let k ⊂ ∂M be a simple arc such that the points of k ∩|λ̇| are transverse intersections. Then we have ∫_{k}dλ′ ≤ \underline{lim} ∫_{k}dλ_{n}. Furthermore, if k does not intersect a leaf of λ′ with a weight equal to π, then ∫_{k}dλ_{n} converges to ∫_{k}dλ.

2.2 Algebraic and strong convergence

Let σ be a hyperbolic metric (up to isotopy) on int(M). Given an isometry from the interior of M̃ to ℍ^{3}, the covering transformations yield a discrete faithful representation ρ : π_{1}(M) → Isom(ℍ^{3}). The representations which appear in this way will be called the representations associated to σ. The set of representations associated to σ is the set of all representations conjugate to ρ. The image ρ(π_{1}(M)) is a finitely generated torsion free Kleinian group and int(M) endowed with σ is isometric to ℍ^{3}/ρ(π_{1}(M)). The Nielsen core of ℍ^{3}/ρ(π_{1}(M)) is the quotient by ρ(π_{1}(M)) of the convex hull C(ρ) of the limit set L_{ρ} of ρ(π_{1}(M)) (see [Th1, chap 8] for details). This set N(ρ) and the convex core defined in the introduction are isometric and from now on we will identify them. The thick part, N(ρ)^{ep} of the Nielsen core is the complement of the cuspidal part of ℍ^{3}/ρ(π_{1}(M)) in N(ρ). The representation ρ is geometrically finite when N(ρ)^{ep} is compact (here it is equivalent to say that N(ρ) has finite volume) and convex cocompact when N(ρ) is compact.

The boundary of N(ρ)^{ep} is the union of tori and compact surfaces. Any such surface is the union of annuli lying in the boundary of the cuspidal part and of surfaces lying in the boundary of N(ρ). These last surfaces are bent along a geodesic lamination (cf. [Th1, chap 8]). The bending locus of ρ(π_{1}(M)) is the union of these geodesic laminations |λ^{i}| and of the cores of the annuli corresponding to the rank 1 cusps. This bending locus is endowed with a transverse measure measuring the amount of bending along the geodesic laminations |λ^{i}| and endowing the cores of the annuli with Dirac measures of weight π (cf. [EpM]). When ρ is geometrically finite and not fuchsian, the natural retraction from ℍ^{3}/ρ(π_{1}(M)) to N(ρ) associates to σ a homeomorphism (defined up to isotopy) h : M → N(ρ)^{ep}. Any homeomorphism M → N(ρ)^{ep} isotopic to h is said to be associated to σ. By this homeomorphism h, we may interpret the bending locus of ∂N(ρ) together with its transverse measure as a measured geodesic lamination b(σ) ∈ ML(∂M), depending only on the isotopy class of σ. This is the bending measured geodesic lamination of the geometrically finite metric σ.

Let (σ_{n}) be a sequence of complete hyperbolic metrics on the interior of M. This sequence (σ_{n}) converges algebraically when we can choose a sequence of representations ρ_{n} : π_{1}(M) → Isom(ℍ^{3}) associated to the σ_{n} (as above) that converges algebraically, namely ρ_{n}(g) converges for any g ∈ π_{1}(M). Defining ρ_{∞}(g) by ρ_{∞}(g) = lim_{n⟶∞} ρ_{n}(g), we obtain a new representation ρ_{∞} : π_{1}(M) → Isom(ℍ^{3}) which is discrete and faithful (cf. [Jor]). This representation defines a metric on a manifold homotopy equivalent to M. This manifold might not be homeomorphic to M (some examples are given in [AnC]).

The sequence (σ_{n}) converges geometrically if we can choose a sequence of representations ρ_{n} : π_{1}(M) → Isom(ℍ^{3}) associated to the σ_{n} so that (ρ_{n}(π_{1}(M)) converges geometrically. Namely there is Γ_{∞} ⊂ Isom(ℍ^{3}) such that

- for any sequence a_{n} ∈ ρ_{n}(π_{1}(M)), any accumulation point a of {a_{n}/n ∈ ℕ} lies in Γ_{∞}

- any element a of Γ_{∞} is the limit of a sequence a_{n} ∈ Γ_{n}. The limit manifold ℍ^{3}/Γ_{∞} can be different from int(M).

The sequence ρ_{n}(π_{1}(M)) converges strongly if we can choose the representations ρ_{n} so that ρ_{n} converges algebraically to some representation ρ_{∞} and so that (ρ_{n}(π_{1}(M))) converges geometrically to ρ_{∞}(π_{1}(M)). If (σ_{n}) converges to σ in the topology defined in the introduction, then (σ_{n}) converges strongly to σ_{∞} (cf. [CEG]).

2.3 Some relations between P(M), P^{+}(M) and D(M)

Let P(M) denote the set of measured geodesic laminations satisfying a), b) and c) and let P^{+}(M) denote the set of measured geodesic laminations satisfying b), c) and d). By definition P^{+}(M) is saturated in R. It is easy to see that a) and c) together implies d) and hence that P(M) ⊂ P^{+}(M). The injectivity of the map P(M) → P^{+}(M)/R is a direct consequence of condition a). Its surjectivity comes from the following lemma.

Lemma 2.4. Let λ ⊂ P^{+}(M) be a measured geodesic lamination. Then the measured geodesic lamination λ′ obtained by replacing by π the weights of leaves of λ which have a weight greater than π belongs to P(M).

Proof. This lemma is a straightforward consequence of the proof of [Le2, Lemma 3.4].

A measured geodesic lamination λ ∈ ML(∂M) is doubly incompressible if there is η > 0 such that i(λ, ∂F) > η for any essential annulus or disk F. We will denote by D(M) the set of doubly incompressible measured geodesic laminations. We have the following :

Lemma 2.5. Let λ be a measured geodesic lamination lying in P(M). For each irrational sublamination λ_{i} of λ, let us remove from λ its closed leaves lying in ∂′S̄(λ). Let γ be the geodesic measured lamination thus obtained. Then γ lies in D(M).

Proof. First assume that, up to extracting a subsequence, there is a sequence of essential discs D_{n} such that i(γ, ∂D_{n}) ⟶ 0. Extract a subsequence such that ∂D_{n} converge for the Hausdorff topology to a geodesic lamination L. We have i(γ, ∂D_{n}) ⟶ 0, so L does not intersect γ transversely. By Casson’s criterion (cf. [Ot3], [Le1, Theorem B1]), L contains a homoclinic leaf l. By [Le1, Lemma 3.4], l intersects λ transversely. Since l does not intersect γ transversely, l intersects λ − γ once or twice. Let l_{+} and l_{−} be the two unbounded connected components of l − λ. Since l is a homoclinic leaf, there is a bounded homotopy in M between l_{+} and l_{−}. Since l_{+} and l_{−} do not intersect λ transversely, it follows from [Le1, Proposition 3] that l_{+} and l_{−} are asymptotic on ∂M. Therefore there is an arc k ∈ λ − l such that k ∪ l_{+} ∪ l_{−} bounds an ideal triangle in ∂M. Set κ = l − (l_{+} ∪ l_{−}), then κ ∪ k bounds an essential disc and i(λ, κ ∪ k) = 0. This contradicts the fact that λ satisfies condition c). Thus we have proved that there is η > 0 such that for any essential disc D, i(λ, ∂D) ≤ η.

Assume now that there is a sequence of essential annuli A_{n} such that i(γ, ∂A_{n}) tends to 0. Let us extract a subsequence such that ∂A_{n} converge in the Hausdorff topology to a geodesic lamination L. This lamination L does not intersect γ transversely. Let p : M̃ → M be the covering projection. It follows from the proof of [Le1, Lemma C2] that p^{−1}(L) contains two leaves l̃_{1} and l̃_{2} such that there is a bounded homotopy in M between l̃_{1} and l̃_{2}. Since L does not intersect γ transversely, l̃_{1} and l̃_{2} intersect p^{−1}(λ) finitely many times (actually at most twice). For i = 1, 2, let l̃_{i}^{+} and l̃_{i}^{−} be the unbounded components of l̃_{i} − p^{−1}(λ). If κ̃_{i} = l̃_{i} − (l̃_{i}^{+} ∪ l̃_{i}^{−}) is not a point, its interior does not intersect λ. From the bounded homotopies between l̃_{1} and l̃_{2} we deduce bounded homotopies in M between l̃_{1}^{+} and l̃_{2}^{+} and between l̃_{1}^{−} and l̃_{2}^{−} respectively. It follows then from [Le1, Proposition 3] that l̃_{1}^{+} and l̃_{2}^{+} are asymptotic on ∂M̃ and that l̃_{1}^{−} and l̃_{2}^{−} are asymptotic in ∂M̃. This implies that there are two arcs k̃^{+}, k̃^{−} ∈ p^{−1}(λ) − p^{−1}(L) such that k̃^{+} ∪ l̃_{1}^{+} ∪ l̃_{2}^{+} and k̃^{−} ∪ l̃_{1}^{−} ∪ l̃_{2}^{−} bound two ideal triangles in ∂M̃. Since l̃_{1} and l̃_{2} are not homotopic in ∂M̃, neither κ̃_{1} nor κ̃_{2} is a point. The curve c̃ = k̃^{+} ∪ κ̃_{1} ∪ k̃^{−} ∪ κ̃_{2} bounds an essential disc and we have i(c, λ) = 0. This contradicts the assumption that λ ∈ P(M) and concludes the proof.

3 Convergence of bending laminations and algebraic convergence of metrics

We will begin by proving that the map b_{R} is proper in the algebraic topology.

Proposition 3.1. Let M be a compact, orientable, hyperbolic 3-manifold with boundary and let (σ_{n}) be a sequence of geometrically finite metrics on the interior of M. Let λ_{n} ⊂ ML(∂M) be the bending measured geodesic lamination of σ_{n}. Suppose that λ̇_{n} ∈ ML(∂M)/R tends to λ̇ ⊂ P^{+}(M)/R. Then a subsequence of (σ_{n}) converges algebraically.

Proof. Let us assume the contrary, namely that there is no sequence ρ_{n} of representations associated to the σ_{n} which contains an algebraically converging subsequence. By the theory of [MoS1], there exists a sequence (ε_{n}) of positive numbers tending to 0, a small minimal action of π_{1}(M) on an ℝ-tree T and a subsequence of (ρ_{n}) (which will also be denoted by (ρ_{n})) such that, for any g ∈ π_{1}(M), we have l_{T} (g) = lim_{n⟶∞} ε_{n}l_{ρ_{n}}(g). Here l_{ρ_{n}}(g) is the distance of translation of ρ_{n}(g) and l_{T} (g) is the distance of translation of g acting on T .

In order to prove Proposition 3.1, we will need some results about realizations of laminations in ℝ-trees.

3.1 Realizations of laminations in ℝ-trees

Let π_{1}(M) × T → T be a small minimal action on an ℝ-tree T , let L ⊂ ∂M be a geodesic lamination and let S be the connected component of ∂M containing L. The lamination L is realized in T if there is a continuous π_{1}(M) equivariant map Ŝ → T which is injective when restricted to any lift of a leaf of L to the universal cover Ŝ of S.

The following result comes from the proof of [Le1, Proposition 6] (cf. [Le2]).

Proposition 3.2 ([Le2], Proposition 6.1). Let T be an ℝ-tree, let π_{1}(M)×T → T be a small minimal action and let λ ∈ D(M). Then at least one connected component of λ is realized in T .

Let λ be a measured geodesic lamination. A geodesic lamination L is called an expansion of λ if |λ| ⊂ L and if L − |λ| is a union of finitely many non-recurrent leaves, each one going from an irrational component of |λ| to itself.

Using Lemma 2.5 and Proposition 3.2, we get the following :

Proposition 3.3. Let λ ⊂ P^{+}(M) and let E be an expansion of |λ|. Let π_{1}(M) × T → T be a small minimal action on an ℝ-tree T . Then there is at least one connected component E_{1} of E which is realized in T and which satisfies one of the following conditions:

- 1) E_{1} contains an irrational geodesic lamination;

- 2) E_{1} is a closed leaf and E_{1} does not lie in ∂′S̄(E_{i}) for any component E_{i} ≠ E_{1} of E.

Proof. Let E_{rec} be the union of the recurrent leaves of E. Let α be a measured geodesic lamination whose support is E_{rec}. Let us remove from α the closed leaves lying in ∂′S̄(α_{i}) for some irrational sublamination α_{i} of α and let γ be the resulting measured geodesic lamination. By Lemma 2.5, γ lies in D(M). By Proposition 3.2, at least one connected component γ_{1} of γ is realized in T . Let E_{1} be the connected component of E containing the support of γ_{1}. By [Ot2, chap. 3], E_{1} is realized in T . By construction E_{1} satisfies 1) or 2).

In the proof of Proposition 3.1, we are interested in actions of π_{1}(M) on T coming from a degenerating sequence of representations ρ_{n}(π_{1}(M)). In [Ot2], J.-P. Otal described, in the special case of handlebodies, the behavior of the lengths of measured geodesic laminations which are realized in T . A careful look at the proof yields the following statement.

Theorem 3.4 (Continuity Theorem [Ot2]). Let (ρ_{n}) be a sequence of geometrically finite representations of π_{1}(M) tending to a small minimal action of π_{1}(M) on an ℝ-tree T . Let ε_{n} ⟶ 0 be such that ∀g ∈ π_{1}(M), we have ε_{n}l_{ρ_{n}}(g) ⟶ l_{T} (g) and let L be a geodesic lamination which is realized in T . Then there exists a neighborhood V(L) and constants K, n_{0} such that for any simple closed curve c ⊂ V(L) and for any n ≥ n_{0},

ε_{n}l_{σ_{n}}(c^{∗}) ≥ Kl_{s_{0}}(c).

When c ⊂ ∂_{χ<0}M is a closed curve, l_{σ_{n}}(c^{∗}) denote the length of the closed geodesic of ℍ^{3}/ρ_{n}(π_{1}(M)) in the free homotopy class of c.

3.2 Proof of Proposition 3.1

We have assumed that (ρ_{n}) does not contain an algebraically converging subsequence and we have deduced from [MoS1] that a subsequence of ρ_{n} tends to an action of π_{1}(M) on an ℝ-tree T . Let s_{n} be the metric on ∂_{χ<0}M induced by the composition of the inclusion map and of a homeomorphism h_{n} : M → N(ρ_{n}) associated to σ_{n}. The following lemma which will be proved in section 3.4 is an important step in our proof of Proposition 3.1.

Lemma 3.5. Let (λ_{n}) be a sequence of geodesic measured laminations such that (λ̇_{n}) converges in ML(∂M)/R to λ̇. There is an expansion E ∈ L(∂M) of |λ| such that for any connected component E_{i} of E one of the two following situations occurs :

- i) there is a sequence of weighted multi-curves γ_{n}^{i} such that (|γ_{n}^{i}|) converges to E_{i} in the Hausdorff topology, that (γ_{n}^{i}) converges in ML(∂M) to a non trivial lamination γ_{∞}^{i} and that we have l_{s_{n}}(γ_{n}^{i}) ≤ l_{s_{n}}(λ_{n}) + i(λ_{n}, ∂′S̄(E_{i}))l_{s_{n}}(∂′S̄(E_{i}));

- ii) E_{i} is a closed leaf of |λ̇|.

Let E_{i} be a component of E satisfying i). We are going to prove that E_{i} is not realized in T .

By [MoS1], for large n, we have ε_{n}l_{σ_{n}}(∂′S̄(E_{i})^{∗}) ≤ l_{T} (∂′S̄(E_{i})) + 1.

Since |λ̇| ⊂ E, Claim 2.3 implies that i(λ_{n}, ∂′S̄(E_{i})) tends to 0. By the slight bending Lemma ([Le1, Lemma A.1] see also [Se]) this allows us to approximate l_{s_{n}}(∂′S̄(E_{i})) by l_{σ_{n}}(∂′S̄(E_{i})^{∗}), namely l_{s_{n}}(∂′S̄(E_{i})) ≤ A_{η_{n}}(l_{σ_{n}}(∂′S̄(E_{i})^{∗}) + B_{η_{n}}) for some A_{η_{n}} ⟶ 1 and B_{η_{n}} ⟶ 0. Thus we obtain ε_{n}l_{s_{n}}(∂′S̄(E_{i})) ≤ l_{T} (∂′S̄(E_{i})) + 1 for large enough n. Replacing in the inequality of i), we get

ε_{n}l_{s_{n}}(γ_{n}^{i}) ≤ ε_{n}l_{s_{n}}(λ_{n}) + i(λ_{n}, ∂′S̄(E_{i}))(l_{T} (∂′S̄(E_{i}) + 1).

As shown in [BoO] and [Le1], l_{s_{n}}(λ_{n}) is bounded :

Lemma 3.6. Under the hypothesis of Proposition 3.1 the lengths l_{σ_{n}}(λ_{n}) are bounded from above.

Proof. The proof of [Le1, Lemma 4.1] extends to this situation.

Thus we have ε_{n}l_{s_{n}}(γ_{n}^{i}) ≤ ε_{n}l_{s_{n}}(λ_{n}) + i(λ_{n}, ∂′S̄(E_{i}))(l_{T} (∂′S̄(E_{i}) + 1) ⟶ 0 (the fact i(λ_{n}, ∂′S̄(E_{i})) ⟶ 0 comes from Claim 2.3).

Since γ_{∞}^{i} is not trivial, we have l_{s_{0}}(γ_{n}^{i}) ⟶ l_{s_{0}}(γ_{∞}^{i}) > 0. It follows that ε_{n} \frac{l_{sn}(γ_{n}^{i})}{l_{s_{0}}(γ_{n}^{i})} tends to 0. We deduce from the Otal’s continuity theorem above that E_{i} is not realized in T .

By Proposition 3.2, at least one component E_{i} of E is realized in T and E_{i} satisfies condition 1) or 2) of Proposition 3.2. By the above, E_{i} is a closed leaf of |λ̇|. So E_{i} satisfies 2), namely E_{i} is not homotopic to a curve lying in the surface embraced by an irrational component of λ̇. Since E_{i} is realized in T , we have ε_{n}l_{σ_{n}}(E_{i}^{∗}) ⟶ l_{T} (E_{i}) > 0. The fact that E_{i} satisfies 2) allows us to find a surface F ⊂ ∂M with geodesic boundary such that E ∩ int(F) = E_{i}. Let c ⊂ F be a simple closed geodesic intersecting E_{i} in one or two points. This geodesic c satisfies the following:

Lemma 3.7. The simple closed geodesic c has the two following properties :

- \frac{l_{sn}(E_{i})}{l_{sn}(c)} ⟶ 0;

- ∃K such that l_{s_{n}}(c) ≤ Kl_{σ_{n}}(c^{∗}).

Before proving Lemma 3.7 let us conclude the proof of Proposition 3.1. By Lemma 3.7, we have \frac{l_{σn}(E_{i}^{∗})}{l_{σn}(c^{∗})} ≤ K \frac{l_{sn}(E_{i})}{l_{sn}(c)} ⟶ 0. By [MoS1], ε_{n}l_{σ_{n}}(c^{∗}) is bounded. Therefore we have ε_{n}l_{σ_{n}}(E_{i}^{∗}) ⟶ 0. This contradicts the fact that E_{i} is realized in T and concludes the proof of Proposition 3.1.

3.3 Proof of Lemma 3.7

To prove the first property we are going to study the behavior of the sequence (s_{n}). Since (λ̇_{n}) converges, up to extracting a subsequence, we may assume that the leaves of λ_{n} which have a weight equal to π does not depend on n. Let us denote by λ^{(p)} their union. Let S be the connected component of ∂M − λ^{(p)} containing E_{i} and let us now denote by s_{n} the complete hyperbolic metric induced on S by a homeomorphism h_{n} : M → N(ρ_{n}) associated to σ_{n}. Since E_{i} is realized in T , its length tends to infinity. So (s_{n}) is not bounded in Teichmüller space. By [MoS1] a subsequence of (s_{n}) tends to an action of π_{1}(S) on a Λ-tree T_{L}. The basic property of T_{L} is that for any two simple closed geodesics c_{1}, c_{2} such that l_{s_{n}}(c_{1}) ⟶ ∞, we have lim_{n⟶∞} \frac{l_{sn}(c_{2})}{l_{sn}(c_{1})} = \frac{δ_{T_{L}}(c_{2})}{δ_{T_{L}}(c_{1})}. By [MoO], this action is dual to a Λ-measured lamination L. In this context, we have the following version of the continuity theorem (see the proof in appendix).

Theorem A.1. Let (s_{n}) be a sequence of complete hyperbolic metrics of finite area on a surface S tending to a Λ-measured lamination L and let (λ_{n}) be a sequence of measured geodesic laminations such that (λ̇_{n}) converges to some λ̇ ∈ ML(S)/R and such that l_{s_{n}}(λ_{n}) is bounded. Then |λ̇| does not intersect |L| transversely.

Since l_{s_{n}}(λ_{n}) is bounded (Lemma 3.6), it follows that |λ̇| does not intersect |L| transversely. Since E_{i} is realized in T , we have l_{s_{n}}(E_{i}) ⟶ ∞. By the construction of the Λ-tree T , this implies that δ_{L}(E_{i}) > 0. Therefore, by Claim 2.1, |L| contains E_{i} and at least one leaf spiraling toward E_{i}. It follows that 0 < δ_{L}(E_{i}) < δ_{L}(c), hence we have \frac{l_{sn}(E_{i})}{l_{sn}(c)} ⟶ 0.

The second property will come from the fact that c can be cut into arcs with bounded bending measure and unbounded length.

Claim 3.8. For any sequence of arcs κ_{n} ⊂ c with \underline{lim} ∫_{κ_{n}} dλ_{n} > 0, we have l_{s_{n}}(κ_{n}) ⟶ ∞.

Proof. We are going to compare the behavior of the sequences (s_{n}) and (λ_{n}) near λ^{i}. Let λ′ be the representative of λ̇ lying in P(M). If E_{i} is a leaf of λ_{n}, then its weight in λ_{n} converges to its weight in λ′. Since l_{s_{n}}(E_{i}) tends to ∞ and l_{s_{n}}(λ_{n}) is bounded, we have E_{i} ⊄ |λ_{n}| for large n.

Let us consider a subsequence such that (|λ_{n}|) converges to some geodesic lamination L′ in the Hausdorff topology. Let V(E_{i}) be an annular neighborhood of E_{i}. By Claim 2.2, we have E_{i} ⊂ L′. Therefore, for large n, V(E_{i}) ∩ |λ_{n}| is the union of disjoint segments joining the two components of ∂V(E_{i}) and turning a lot of times toward E_{i} (see figure 1). By Claim 2.3, we have

PSfrag replacements

c
λ^{i}
λ_{n}
x_{n}
y_{n}
κ_{n}
c_{x_{n}y_{n}}

Figure 1: Picture of V(λ^{i})

i(λ_{n}, E_{i}) ⟶ 0. Therefore the total weight of the transverse measure carried by |λ_{n}| ∩ V(E_{i}) tends to 0.

Let κ_{n} ⊂ c be a sequence of segments such that \underline{lim} ∫_{κ_{n}} dλ_{n} > 0. Since c∩|λ̇| ⊂ E_{i}, it follows from Claim 2.3 that ∫_{κ_{n}−V(E_{i})} dλ_{n} tends to 0. Therefore it is sufficient to prove Claim 3.8 for κ_{n} ⊂ V(E_{i}). We may also assume that ∂κ_{n} ⊂ |λ_{n}|. So ∂κ_{n} lies in |λ_{n}| ∩ c and κ_{n} is well defined for any hyperbolic metric on ∂_{χ<0}M (see the note in section 1.1). Given an orientation of c we order the points of κ_{n} ∩ λ_{n} by the induced order. By the previous description of λ_{n} ∩ V(E_{i}), for large n, κ_{n} contains an arc κ′_{n} such that ∫_{κ′_{n}} dλ_{n} ≥ ^{\frac{∫_{κn} dλ_{n}}{3}} and such that any point x_{n} of κ′_{n}∩λ_{n} is joined to a point y_{n} > x_{n} of κ_{n} ∩λ_{n} by an arc ]x_{n}, y_{n}[⊂ λ_{n} −κ_{n}. Fix n and consider a point x_{n} in κ′_{n} ∩ λ_{n} and the associated point y_{n}. Let c_{x_{n}y_{n}} denote the simple closed curve which is the union of the arc [x_{n}, y_{n}] ⊂ λ_{n} and of the arc κ_{x_{n}y_{n}} ⊂ κ_{n} joining x_{n} to y_{n} (see figure 1). This curve c_{x_{n}y_{n}} is homotopic to E_{i}, hence l_{s_{n}}(c_{x_{n}y_{n}}) is greater than l_{s_{n}}(E_{i}). It follows that ∀n, ∀x_{n} ⊂ κ′_{n} ∩λ_{n}, we have l_{s_{n}}([x_{n}, y_{n}]) ≥ l_{s_{n}}(E_{i}) − l_{s_{n}}(κ_{n}). Since l_{s_{n}}(λ_{n}) ≥ (∫_{κ′_{n}} dλ_{n}) inf{l_{s_{n}}([x_{n}, y_{n}]) | x_{n} ⊂ κ′_{n} ∩ λ_{n}}, we have l_{s_{n}}(λ_{n}) ≥ (∫_{κ′_{n}} dλ_{n})(l_{s_{n}}(E_{i})−l_{s_{n}}(κ_{n})). Thus, we have l_{s_{n}}(κ_{n}) ≥ l_{s_{n}}(E_{i})−_{\frac{3}{∫_{κn} dλ_{n}}} l_{s_{n}}(λ_{n}). Since l_{s_{n}}(E_{i}) ⟶ ∞ and l_{s_{n}}(λ_{n}) is bounded, we get l_{s_{n}}(κ_{n}) ⟶ ∞.

Let 0 < ϵ < \frac{i(c,λ′)}{2}. It follows from the description of λ_{n} ∩ V(E_{i}) that, for large n, we may divide c into ⌊\frac{i(c,λ_{n})}{ϵ}⌋ segments κ_{n}^{j} such that ∀j ≤ ⌊\frac{i(c,λ_{n})}{ϵ}⌋, we have ϵ ≤ ∫_{κ_{n}^{j}} dλ_{n} < 2ϵ. By Claim 3.8, ∀j, l_{s_{n}}(κ_{n}^{j}) ⟶ ∞. Given ϵ such that 2ϵ ≤ \frac{π}{3} , let c_{n} be the union of the geodesic segments of (M, σ_{n}) joining the vertices of the segments κ_{n}^{j}. A consequence of the slight bending Lemma of [Le1] ([Le1, Lemma A.1]) is that ∃K_{ϵ} > 0 such that, l_{s_{n}}(c) ≤ K_{ϵ}l_{σ_{n}}(c_{n}). Moreover, by the Gauss-Bonnet formula, the geodesic arcs composing c_{n} have interior angles greater than π − 2\frac{π}{3} = \frac{π}{3} . Hence the curve c_{n} is the union of segments which are longer and longer (with respect to the metrics σ_{n}) and have interior angles greater than \frac{π}{3} . In this situation, it is a classical result (see [Ot3] for example) that ∃K′_{ϵ} such that we have l_{σ_{n}}(c_{n}) ≥ K′_{ϵ}l_{σ_{n}}(c^{∗}). Thus taking K = K′_{ϵ}K_{ϵ} yields the second property of Lemma 3.7. This concludes the proof of Lemma 3.7. □

3.4 Construction of L

Let S be a connected component of ∂_{χ<0}M. Fix n and consider a sequence (ν_{k}) of weighted simple closed curves converging in ML(S) to λ_{n} ∩ S. Thus we have l_{s_{n}}(ν_{k}) ⟶ l_{s_{n}}(λ_{n} ∩ S). Since l_{s_{n}}(λ_{n}) is bounded (Lemma 3.6), using a diagonal extraction, one constructs a sequence of weighted simple closed geodesics (ν_{n}) such that l_{s_{n}}(ν_{n}) is bounded and that (ν̇_{n}) tends to λ̇ ∩ S. To simplify the formulae, we will assume that we have l_{s_{n}}(ν_{n}) ≤ l_{s_{n}}(λ_{n}) and i(ν_{n}, ∂′S̄(λ^{i})) ≤ i(λ_{n}, ∂′S̄(λ^{i})) for any connected component λ^{i} of λ (this can be achieved by multiplying ν_{n} by a convenient constant C_{n} < 1 with C_{n} ⟶ 1).

Let λ^{i} be an irrational component of λ̇ and let S(λ^{i}) be the surface it embraces. We will change ν_{n} ∩ S(λ^{i}) as shown in figure 2. This will yield the sequence of weighted multi-curves (γ_{n}^{i}) that appears in Lemma 3.5. This construction easily provides the control on l_{s_{n}}(γ_{n}^{i}) that is needed in Lemma 3.5 but we have to take care to ensure that γ_{n}^{i} does not converge to the trivial measured geodesic lamination.

Let us extract a subsequence such that |ν̇_{n}| converges to some geodesic lamination N. By Claim 2.2, |λ̇| ⊂ N. It follows that k = N ∩ S(λ^{i}) is the union of λ^{i}, of simple geodesics asymptotic to λ^{i} and of simple half-geodesic which have an endpoint in ∂′S̄(λ^{i}) and which are asymptotic to λ^{i}. Let c ⊂ S(λ^{i}) be a simple closed curve. Any leaf of N intersecting S(λ^{i}) intersects c. Therefore there is a train track (with sidings) τ ⊂ S̄(λ^{i}) such that :

- τ minimally carries N ∩ S(λ^{i}), namely N ∩ S(λ^{i}) ⊂ τ and each branch of τ carries N;

- τ has one switch A ⊂ c and the remaining ones lie in ∂′S̄(λ^{i}) (they correspond to sidings);

- any two arcs k^{1}, k^{2} ⊂ k − A such that k̄^{1} and k̄^{2} are isotopic relatively to A ∪ ∂′S̄(λ^{i}) lie in the same rectangle of τ.

Such a train track may be constructed in the following way. Let τ′ ⊂ S̄(λ^{i}) be a train track (with sidings) carrying N ∩ S(λ^{i}) such that one switch A of τ′ lies in c and that the remaining PSfrag replacements

k_{n}

x_{1}
x_{2}
x_{3}
x_{4}

Figure 2: Constructing a multi-curve from k_{n}

switches lie in ∂′S̄(λ^{i}) (see [Bo1] or [Ot3] for the construction of τ′). Let b_{1} and b_{2} be two branches of τ′ containing two arcs k_{1} ⊂ k ∩ b_{1} and k_{2} ⊂ k ∩ b_{2} which are isotopic relatively to A ∪ ∂′S̄(λ^{i}). There are two arcs κ_{1}, κ_{2} ⊂ A ∪ ∂′S̄(λ^{i}) such that k_{1} ∪ κ_{1} ∪ k_{2} ∪ κ_{2} bounds a disc D ⊂ S̄(λ^{i}). Let k_{3} be a connected component of k ∩ D, then ∂k_{3} lies in κ_{1} ∪ κ_{2}. Since κ_{1}, κ_{2} and k_{3} are geodesic arcs, the two vertices of k_{3} do not lie in the same segment κ_{i}. Hence k_{3} is isotopic to k_{1} relatively to κ_{1} ∪ κ_{2} ⊂ A ∪ ∂′S̄(λ^{i}). Replace k_{2} by k_{3} and b_{2} by the branch of τ′ that contains k_{3} and repeat this operation until D − (b_{1} ∪ b_{2}) ∩ k = ∅. Then extend the foliation of b_{1} and b_{2} to a foliation of b_{1} ∪ D ∪ b_{2} and replace b_{1} and b_{2} by the new branch b_{1} ∪ D ∪ b_{2} thus constructed. Repeating this process finitely many times yields the desired train track τ.

Fix n large enough so that |ν_{n}| ∩ S̄(λ^{i}) is carried by τ and denote by k_{n} the union of two parallel copies of |ν_{n}| ∩ S̄(λ^{i}) carried by τ. Let d be a connected component of ∂′S̄(λ^{i}). The integer ♯(d ∩ k_{n}) is even. Choose x_{1} such that when x_{1}, x_{2}, ..., x_{2p} are the points of d ∩ k_{n} ordered along d, the arc [x_{1}, x_{2p}] ⊂ d joining x_{1} to x_{2p} and not containing x_{2} does not lie in τ. For each i ≤ p, add to k_{n} the arc of d joining x_{i} to x_{2p+1−i} and containing [x_{1}, x_{2p}] (cf. figure 3).

Repeating this operation for each connected component of ∂′S̄(λ^{i}) intersecting k_{n} yields a family k′_{n} of simple closed curves. Each component of k′_{n} is either homotopic to a simple closed geodesic or homotopic to a point. Let us denote by |γ_{n}^{i}| the union of the closed geodesics thus obtained, at this point we do not know whether or not |γ_{n}^{i}| is empty. Associating to each leaf of |γ_{n}^{i}| the half-weight of ν_{n} yields a weighted multi-curve γ_{n}^{i} ∈ ML(S(λ^{i})) such that l_{s_{n}}(γ_{n}^{i}) ≤ l_{s_{n}}(ν_{n}) + i(ν_{n}, ∂′S̄(λ^{i}))l_{s_{n}}(∂′S̄(λ^{i})) ≤ l_{s_{n}}(λ_{n}) + i(λ_{n}, ∂′S̄(λ^{i}))l_{s_{n}}(∂′S̄(λ^{i})) (the last inequality comes from our assumptions on l_{s_{n}}(ν_{n}) and on i(ν_{n}, ∂′S̄(λ^{i}))).

Claim 3.9. No component of k′_{n} − A can be homotoped into A.

Proof. Fix n and let κ be a component of k′_{n} − A. If κ is a component of k_{n} − A, then it is a geodesic segment. Since A is also a geodesic segment, κ can not be homotoped into A. If κ is not a component of k_{n} − A, it is the union of two components κ_{1} and κ_{2} of k_{n} − A and of an arc κ_{3} of ∂′S̄(λ^{i}) not lying in τ. If there is an arc κ_{4} ⊂ ∂′S̄(λ^{i}) − (κ_{1} ∪ κ_{2}) such that κ_{1} ∪ κ_{4} ∪ κ_{2} can be homotoped into A, then κ_{1} and κ_{2} are homotopic relative to A ∪ ∂′S̄(λ^{i}). So κ_{1} and κ_{2} lie in the same branch b of τ and κ_{4} lies in b. By construction κ_{3} ⊄ b hence κ_{1} ∪ κ_{3} ∪ κ_{2} can not be homotoped into A. The claim is thus proved.

Claim 3.10. There exists a simple closed curve c′ ⊂ S(λ^{i}) such that i(γ_{n}^{i}, c′) ≥ \frac{1}{2} ∫_{A}dλ^{i} for large n.

Proof. As we have already seen in the preceding proof, any connected component of k′_{n} − A is either homotopic to a rail of τ or to the union of two rails and of a segment of ∂′S̄(λ^{i}). There are

PSfrag replacements x_{1}

x_{2}
x_{6}
|ν_{n}|
k_{n}
A
k′_{n}

Figure 3: An example for the construction of k′_{n}

only finitely many possibilities for the homotopy class relative to A ∪ ∂′S̄(λ^{i}) of such an arc. On the other hand, there are infinitely many simple closed geodesics which pass near A. Therefore we can find a curve c′ and an arc A′ ⊂ c′ which is close to A so that no component of k′_{n} − A′ can be homotoped in c′−A′. By Claim 3.9, no component of k′_{n}−A′ can be homotoped into A′, hence no component of k′_{n} − A′ can be homotoped into c′. It follows that we have i(γ_{n}^{i}, c) ≥ ∫_{A′} dν_{n}^{i}. Since A lies in the surface embraced by λ^{i}, it does not intersect any closed leaf of λ̇. Therefore, by Claim 2.3, ∫_{A′} dν_{n}^{i} converges to ∫_{A′} dλ^{i}. Since A′ is close to A, we have ∫_{A′} dλ^{i} = ∫_{A}dλ^{i} and i(γ_{n}^{i}, c′) ≥ \frac{1}{2} ∫_{A}dλ^{i} for large n.

Since λ^{i} intersects A transversely, we have ∫_{A}dλ^{i} > 0. It follows from the claim above that, for large n, i(γ_{n}^{i}, c′) > 0, therefore γ_{n}^{i} is not trivial.

Let d ⊂ S(λ^{i}) be a simple closed curve. By Claim 2.3, i(d, ν_{n}) ⟶ i(d, λ). By construction i(d, γ_{n}^{i}) ≤ i(d, ν_{n}), hence, up to extracting a subsequence, γ_{n}^{i} converges in ML(∂M) to a measured geodesic lamination γ_{∞}^{i}. Since we have i(γ_{n}^{i}, c′) ≥ ∫_{A}dν_{n}^{i} ⟶ ∫_{A}dλ^{i} > 0, the measured geodesic lamination γ_{∞}^{i} is not trivial.

Let us consider a subsequence such that |γ_{n}^{i}| converges to a geodesic lamination L^{i} in the Hausdorff topology. Let us show that L^{i} is an expansion of λ^{i}. Since |γ_{∞}^{i}| lies in L^{i} ⊂ S(λ^{i}), this is equivalent to the following claim.

Claim 3.11. The geodesic lamination |γ_{∞}^{i}| does not intersect λ^{i} transversely.

Proof. Let w(ν_{n}) be the weight of ν_{n}. Since each leaf of γ_{n} is homotopic to a component of k′_{n} and has a weight equal to w(ν_{n}), we have i(γ_{n}, ν_{n}) ≤ w(ν_{n}) ∫_{k′_{n}} dν_{n}. Since k_{n} is the union of two parallel copies of |ν_{n}| ∩ S(λ^{i}), we have ∫_{k_{n}} dν_{n} = 0. To obtain k′_{n}, we have added to k_{n} some arcs lying in ∂′S̄(λ^{i}). Since the intersection number of each of these arc with ν_{n} is less than i(∂′S̄(λ^{i}), ν_{n}), we have ∫_{k′_{n}} dν_{n} ≤ ♯(|ν_{n}| ∩ ∂′S̄(λ^{i}))i(∂′S̄(λ^{i}), ν_{n}). Thus we have i(γ_{n}, ν_{n}) ≤ w(ν_{n})♯(|ν_{n}|∩∂′S̄(λ^{i}))i(∂′S̄(λ^{i}), ν_{n}) = i(∂′S̄(λ^{i}), ν_{n})^{2}. By Claim 2.3, i(∂′S̄(λ^{i}), ν_{n}) ⟶ 0, hence i(γ_{n}^{i}, ν_{n}) ⟶ 0.

It is easy now to conclude that |γ_{∞}^{i}| does not intersect λ^{i} transversely.

Doing this construction for each irrational component of λ and adding the result to the closed leaves of λ yields a geodesic lamination L expanding λ which satisfies the conditions of Lemma 3.5.

4 Properness of b_{CC} and b_{R}

Let M be a compact orientable hyperbolic 3-manifold, let (σ_{n}) be a sequence of geometrically finite metrics on int(M) and let λ_{n} ⊂ ML(∂M) be the bending measured geodesic lamination of σ_{n}. In the previous section, we have shown that if λ̇_{n} converges to λ̇ in P^{+}(M)/R, then there is a sequence ρ_{n} : π_{1}(M) → Isom(ℍ^{3}) of representations associated to (σ_{n}) such that a subsequence of (ρ_{n}) tends to a representation ρ_{∞} : π_{1}(M) → Isom(ℍ^{3}). By [Jor], ρ_{∞} is discrete and faithful. Next, we are going to show that ρ_{∞} is geometrically finite and corresponds to a metric σ_{∞} on the interior of M. Finally we will show that a subsequence of (σ_{n}) converges to σ_{∞} and conclude that b_{R} is proper.

Let us first keep apart the closed curves of ∂M that have lengths tending to 0, in the same way as in [BoO].

Lemma 4.1. Up to extracting a subsequence, there is a multi-curve L^{0} ⊂ ∂_{χ<0}M with the two following properties :

- Any connected component c of L^{0} satisfies l_{s_{n}}(c) ⟶ 0;

- If c ⊂ ∂_{χ<0}M is a simple closed curve which is not in the homotopy class of any connected component of L^{0}, then the sequence l_{s_{n}}(c) is bounded from below by a strictly positive number (depending on c).

Proof. See [BoO, Lemme 15].

It will follow from the continuity of b_{R} ([Le3]) that L^{0} is in fact the union of the closed leaves of λ′ with a weight equal to π. For the moment, let us just show that L^{0} does not intersect |λ̇| transversely.

Claim 4.2. The multi-curve L^{0} does not intersect |λ̇| transversely.

Proof. We will follow [BoO, lemme 16]. Let c be a connected component of L^{0} and let λ′ be the representative of λ̇ that lies in P(M). If c intersects |λ̇| transversely then for large n, we have i(c, λ_{n}) ≥ \frac{1}{2}i(c, λ′) (cf. Claim 2.3). It follows that c is not homotopic to a leaf of λ_{n}, and in particular c is not homotopic to a cusp of s_{n}. Therefore c is homotopic to a geodesic of s_{n} and, up to extracting a subsequence, we have l_{s_{n}}(c) → 0. Therefore c is the core of a Margulis tube of ∂M and c is far from the boundary of this tube. Since i(c, λ_{n}) ≥ \frac{1}{2}i(c, λ′) > 0, this implies that l_{s_{n}}(λ_{n}) tends to ∞, contradicting Lemma 3.6.

Let c be component of L^{0}. Since c does not intersect λ̇ ∈ P^{+}(M)/R, c represents a nontrivial element i_{∗}(c) of π_{1}(M). Since ρ_{n} converges to ρ_{∞} and l_{σ_{n}}(c) ≤ l_{s_{n}}(c) → 0, it follows that ρ_{∞} ∘ i_{∗}(c) is a parabolic isometry.

4.1 The metrics in the boundary

Since λ̇_{n} tends to λ̇, up to extracting a subsequence, the closed leaves of λ_{n} with a weight equal to π do not depend on n. Let us denote their union by λ^{(p)} ⊂ L^{0}. Let h_{n} : M − λ^{(p)} → N(ρ_{n}) be a homeomorphism associated to σ_{n}. The map f_{n} = h_{n} ∘ i : ∂_{χ<0}M − λ^{(p)} → ℍ^{3}/ρ_{n}(π_{1}(M)) is a convex pleated surface whose bending measured geodesic lamination is λ_{n} − λ^{(p)}. We will show that this sequence of pleated surfaces tends to a pleated surface f_{∞} : ∂M − L^{0}. As in [BoO], we will then see that f_{∞} is a connected component of ∂N(ρ_{∞}) and use a homological argument to conclude that ρ_{∞} is geometrically finite.

Let S be a connected component of ∂_{χ<0}M − L^{0} and let Ŝ be the universal cover of S. Given an isometry from Ŝ endowed with the metric lifted from s_{n} to a subset of ℍ^{2}, the covering transformations yields a discrete faithful representation r_{n} : π_{1}(S) → Isom(ℍ^{2}) associated to s_{n}. We say that the restriction s_{n}^{S} of s_{n} to S converges algebraically if we can choose representations r_{n} associated to the s_{n}^{S} which converge algebraically.

Lemma 4.3. The sequence (s_{n}^{S}) contains an algebraically converging subsequence.

Proof. Let us assume the contrary. By the definition of L^{0}, l_{s_{n}}(∂S̄) ⟶ 0. So we can use the theory of [MoS1]. By [MoS1] and [MoO] a subsequence of the sequence (s_{n}^{S}) tends to a Λ-measured geodesic lamination L. Since l_{s_{n}}(λ_{n}) is bounded, by Theorem A.1, |λ̇| does not intersects |L| transversely.

Claim 4.4. The support of L is a multi-curve.

Proof. Assume first that |L| contains an irrational lamination. Let η > 0, since |λ̇| does not intersect |L| transversely, there is a simple closed curve c_{η} such that i(λ′, c_{η}) ≤ η and δ_{L}(c_{η}) > 0. By Claim 2.3, for large n, we have i(λ_{n}, c_{η}) ≤ 2η. Since δ_{L}(c_{η}) > 0, then l_{s_{n}}(c_{η}) ⟶ ∞. By the slight bending Lemma, ∃C_{η} such that l_{s_{n}}(c_{η}) ≤ C_{η}l_{σ_{n}}(c_{η}^{∗}). Since σ_{n} converges algebraically, l_{σ_{n}}(c_{η}^{∗}) is bounded and the case studied here does not occur, namely L does not contain any irrational lamination.

Let c be a closed leaf of |L| such that δ_{L}(c) > 0. By [MoS1], we have l_{s_{n}}(c) ⟶ ∞. Since |λ̇| does not intersect |L| transversely, we have i(c, λ′) = 0. By Claim 2.3, i(c, λ_{n}) tends to 0. Therefore by the slight bending Lemma ([Le1, Lemma A.1]), there exists C such that for large n, we have l_{s_{n}}(c) ≤ Cl_{σ_{n}}(c^{∗}). This yields the same contradiction as in the preceding paragraph. Thus for any closed leaf c of L, we have δ_{L}(c) = 0.

By Claim 2.1, it follows that |L| is a multi-curve.

By the construction of the Λ-tree T_{L}, the sequence s_{n}^{S} is bounded when restricted to any component of S − |L|. Therefore there is C_{1} > 0 such that for any n we have inf{l_{s_{n}}(c)/ c is a closed geodesic of S − |L|} > C_{1}. By the definition of L^{0}, there is C_{2} > 0 such that we have l_{s_{n}}(c) > C_{2} for any n and for any component c of |L|. Since |L| is a multi-curve, we have δ_{L}(|L|) = 0. Hence l_{s_{n}}(L) is bounded from above. It follows that ∃C_{3} > 0 such that, for any n, inf{l_{s_{n}}(c)/ c intersects L transversely} > C_{3}. Now we have inf{l_{s_{n}}(c)/ c is a closed geodesic of S} > min{C_{1}, C_{2}, C_{3}} > 0. Therefore, by Mumford’s Lemma (cf. [CEG, Proposition 3.2.13]), the sequence s_{n}^{S} is bounded in the modular space, namely there are diffeomorphisms ψ_{n} such that the sequence (ψ_{n∗}s_{n}^{S}) is bounded in the algebraic topology.

Since the restriction of s_{n}^{S} to each component of S − |L| is bounded, we may assume that ψ_{n} is the composition of Dehn twists along the leaves of |L|.

Claim 4.5. The number of turns of the Dehn twists composing ψ_{n} is bounded.

Proof. For a closed curve c ∈ ∂M and n ∈ ℕ∪{∞} we will denote by ρ_{n}(c) the corresponding element of ρ_{n}(π_{1}(M)). The isometry ρ_{n}(c) depends only on the free homotopy class of c ∈ ∂M. Let c be a connected component of L. Since (ρ_{n}) converges algebraically, ρ_{n}(c) converges to ρ_{∞}(c). Let us prove that ρ_{∞}(c) is not parabolic. Assume the contrary, then we have l_{σ_{n}}(c^{∗}) ⟶ 0. Since i(λ_{n}, c) ⟶ 0, it follows from the slight bending Lemma that we have l_{s_{n}}(c) ⟶ 0. Hence c lies in L^{0} contradicting the fact that |L| lies in S ⊂ ∂M − L^{0}. So ρ_{∞}(c) is hyperbolic and up to extracting a subsequence, ρ_{n}(c) is also hyperbolic.

Let us assume that ψ_{n} turns p_{n} times toward c with p_{n} ⟶ ∞. Let F^{1} and F^{2} be the connected components (which may coincide) of S − |L| whose boundary contains c. Since |L| does not intersect λ′ transversely, for any ε > 0, F^{j} (with j = 1, 2) contains a simple closed curve c_{j,ε} with i(c_{j,ε}, λ′) ≤ ε. We have λ′ ∈ P(M), let us fix ε < min{2π, η} (here η comes from condition b)). The condition c) implies that i_{∗}(c_{j,ε}) ∈ π_{1}(M) is not trivial. Fix a point x ∈ F^{1} and consider a loop a ∋ x homotopic to c_{1,ε} and a loop b ∋ x homotopic to c_{2,ε}. We are in the situation of figure 4.

Since ψ_{n∗}s_{n}^{S} is bounded, l_{s_{n}^{S}}(ψ_{n}(ab)) is bounded and l_{σ_{n}}(ψ_{n}(ab)^{∗}) is also bounded. Therefore

PSfrag replacements

a b x c F^{1} F^{2}

Figure 4: Controlling the number of Dehn twists

we can extract a subsequence such that ρ_{n}(ψ_{n}(ab)) = ρ_{n}(ac^{p_{n}}bc^{−p_{n}}) converges. Since ρ_{n}(a) converges, it follows that ρ_{n}(c^{p_{n}}bc^{−p_{n}}) = ρ_{n}(c)^{p_{n}}ρ_{n}(b)ρ_{n}(c)^{−p_{n}} contains a converging subsequence.

Let l_{n} ⊂ ℍ^{3} be the axis of ρ_{n}(c), let x_{n} ∈ l_{n} be a sequence converging to x_{∞} ∈ l_{∞} and let us extract a subsequence such that ρ_{n}(c)^{p_{n}}ρ_{n}(b)ρ_{n}(c)^{−p_{n}}(x_{n}) converges. Since ρ_{n}(c) tends to ρ_{∞}(c) which is hyperbolic, ρ_{n}(c)^{−p_{n}}(x_{n}) tends to a fixed point of ρ_{n}(c). Now, we know that ρ_{n}(b) tends to ρ_{∞}(b), that ρ_{n}(c)^{−p_{n}}(x_{n}) tends to the boundary at infinity and that d_{ℍ^{3}}(ρ_{n}(c)^{−p_{n}}(x_{n}), ρ_{n}(b)ρ_{n}(c)^{−p_{n}}(x_{n})) = d_{ℍ^{3}}(x_{n}, ρ_{n}(c)^{p_{n}}ρ_{n}(b)ρ_{n}(c)^{−p_{n}}(x_{n})) is bounded. This implies that ρ_{n}(c)^{−p_{n}}(x_{n}) tends to a fixed point of ρ_{∞}(b). Therefore ρ_{∞}(b) and ρ_{∞}(c) have a common fixed point. Since ρ_{∞}(π_{1}(M)) is a discrete group, ρ_{∞}(b) and ρ_{∞}(c) are multiples of the same indivisible element of ρ_{∞}(π_{1}(M)), namely there exist d ∈ π_{1}(M) and p, q ∈ ℕ with b = d^{p} and c = d^{q}. Fix n and choose a homeomorphism (M̃, σ_{n}) → ℍ^{3} ∪ Ω(ρ_{n}). Recall that c_{2,ε} is freely homotopic to b. Let c̃ and c̃_{2,ε} ∈ Ω(ρ_{n}) be lifts of c and c_{2,ε}, respectively, which are left invariant by ρ_{n}(d^{pq}). The family {c̃, ρ_{n}(d)(c̃), …, ρ_{n}(d^{q−1})(c̃)} cuts ∂ℍ̄^{3} ≈ S^{2} into q crescents Z_{1}, …, Z_{q}. Since c_{2,ε} does not intersect c, for any i ∈ ℤ, ρ_{n}(d^{i})(c̃_{2,ε}) lies in one Z_{j}. Furthermore Z_{j} = ρ_{n}(d)(Z_{j−1}), hence the number of translates of c̃_{2,ε} that lie in Z_{j} does not depend on j. It follows that q divides p. By exchanging the roles of c and c̃_{2,ε}, we get that p divides q. Thus we get q = p. So c and c̃_{2,ε} are homotopic. It follows from [ShW] that c_{2,ε} ∪ b bounds an essential annulus. Since i(λ′, c ∪ c_{ε}^{2}) ≤ ε < η, this contradicts condition c).

Now we can conclude that, up to extracting a subsequence, ∃n_{1} such that the diffeomorphisms ψ_{n} ∘ ψ_{n_{1}}^{−1} are isotopic to the identity. This concludes the proof of Lemma 4.3.

Using the convergence of the metrics on the boundary, we are going to construct a convex pleated surface f_{∞} : S → ℍ^{3}/ρ_{∞}(π_{1}(M)).

Lemma 4.6. Let S be a connected component of ∂M − L^{0}. Then there exists a convex pleated surface f_{∞} : S → ℍ^{3}/ρ_{∞}(π_{1}(M)) which is homotopic to the inclusion map.

Proof. Let r_{n} : π_{1}(S) → Isom(ℍ^{2}) be a representation associated to s_{n} such that (r_{n}) converges algebraically to a representation r_{∞} : π_{1}(S) → Isom(ℍ^{2}).

To extract from (f_{n}) a converging subsequence, we will follow the ideas of [CEG]. For this, we need to show that the f_{n}(S) all intersect the image in ℍ^{3}/ρ_{n}(π_{1}(M)) of a compact subset of ℍ^{3}. Let x_{n} be the image in ℍ^{3}/ρ_{n}(π_{1}(M)) of the base point o ∈ ℍ^{3}, then we have :

Lemma 4.7 ([BoO], lemma 17). There is a sequence of arcs k_{n} joining f_{n}(S) to the base point x_{n} with uniformly bounded length.

Let S′ be the component of ∂M −λ^{(p)} that contains S and let y_{n} be the vertex of k_{n} that is not x_{n}. Using Lemmas 4.3 and 4.7 and the arguments of [CEG, §5.2], we get that, up to extracting a subsequence, the sequence of pleated surfaces f_{n} : S′ → ℍ^{3}/ρ_{n}(π_{1}(M)) with basepoint y_{n} converges to a pleated surface f_{∞} : S → ℍ^{3}/ρ_{∞}(π_{1}(M)). Since all the f_{n} are homotopic to the inclusion map, their limit f_{∞} is also homotopic to the inclusion map.

By [BoO, Lemme 21] (see also [Le3, Lemma 3.1]), either f_{∞}(S) is a convex surface, namely there is a convex set C_{f_{∞}} ⊂ ℍ^{3}/ρ_{∞}(π_{1}(M)) whose boundary is f_{∞}(S), or f_{∞}(S) lies in a totally geodesic surface ℍ^{3}/ρ_{∞}(π_{1}(M)). It follows also from the proof of [BoO, Lemme 21] (see also [Le3, Lemma 3.1]) that the convex core N(ρ_{∞}) of ρ_{∞}(π_{1}(M)) lies either in Cf̂_{∞} (when f_{∞}(S) is convex) or in f_{∞}(S) (in the other case). If f_{∞}(S) is not a convex surface (as defined above), then ρ_{∞}(π_{1}(M)) is a fuchsian group and the restriction of f_{∞} to S − |λ̇_{∞}| is a two-sheeted covering of the interior of f_{∞}(S) considered as a two dimensional surface with boundary (see [Le3, Lemma 3.8] for more details). It follows that M is an interval bundle over a closed surface I × F and that |λ̇_{∞}| is a section of the bundle ∂S × I. This contradicts the fact that λ̇_{∞} lies in P^{+}(M)/R. Therefore f_{∞}(S) is a convex surface and f_{∞} is a covering onto its image (cf. [EpM]). It follows from the same arguments that f_{∞} is a one-sheeted covering, namely a homeomorphism into its image (see [BoO, Lemme 21]).

Since f_{∞} is a pleated surface, we have f_{∞}(S) ⊂ N(ρ_{∞}). Since N(ρ_{∞}) ⊂ C_{f_{∞}}, the surface f_{∞}(S) is a component of the boundary of N(ρ_{∞}).

Let us do the same construction for each component of ∂_{χ<0}M − L_{0} and denote by f_{∞} : ∂_{χ<0}M − L_{0} → ∂N(ρ_{∞}) the resulting map. By the above, the image of f_{∞} is a union of components of ∂N(ρ_{∞}) and f_{∞} is a local homeomorphism.

Claim 4.8. The map f_{∞} : ∂_{χ<0}M − L^{0} → ∂N(ρ_{∞}) is injective.

Proof. Assume the contrary. We have shown above that the restriction of f_{∞} to any component of ∂_{χ<0}M − L_{0} is a homeomorphism into its image. Hence there are two connected components S and S′ of ∂_{χ<0}M − L_{0} such that f_{∞}(S) ∩ f_{∞}(S′) ≠ ∅. Since f_{∞}(S) and f_{∞}(S′) are connected components of ∂N(ρ_{∞}), they intersect if and only if they are equal. Since N(ρ_{n}) lies between f_{n}(S) and f_{n}(S′), this is possible only if f_{∞}(S) and f_{∞}(S′) are totally geodesic and if ρ_{∞}(π_{1}(M)) is fuchsian (see [BoO, Lemma 21]). In this case, M is homeomorphic to S × I and |λ̇_{∞}| lies in a section of the bundle ∂S × I contradicting again the fact that λ̇_{∞} lies in P^{+}(M)/R.

We can now prove, following [BoO], that ρ_{∞} is geometrically finite and is associated to a complete hyperbolic on int(M).

Lemma 4.9. There is a geometrically finite metric σ_{∞} on int(M) such that ρ_{∞} is a representation associated to σ_{∞}.

Proof. To each leaf c of L^{0} correspond two cusps of ∂_{χ<0}M − L^{0}. The images of these two cusps under f_{∞} are two totally geodesic open annuli tending to a rank one cusp of ℍ^{3}/ρ_{∞}(π_{1}(M)). Remove these two annuli from f_{∞}(∂_{χ<0} − L^{0}) and join the two boundary components of the remaining surface by a compact annulus. Doing this for each leaf of L^{0}, we get a compact surface F_{∞} ⊂ ℍ^{3}/ρ_{∞}(π_{1}(M)). The same construction in M yields a surface S_{∞} such that S_{∞} ∪ ∂_{χ=0}M bounds a compact subset of M. Let us denote by F′_{∞} the surface obtained by adding to F_{∞} the boundary of a neighborhood of the rank 2 cusps of ℍ^{3}/ρ_{∞}(π_{1}(M)). From f_{∞}, we get easily a homeomorphism f : S_{∞} ∪∂_{χ=0}M → F′_{∞} which is homotopic to the inclusion. Since S_{∞} ∪∂_{χ=0}M bounds a compact set in M, F′_{∞} is homologous to 0 and therefore bounds a finite volume cycle. Adding some finite volume sets (which are rank 2 cusps and “slices” of rank 1 cusps) to this cycle, we get a new cycle C_{∞} which is bounded by f_{∞}(∂_{χ<0}M − L^{0}) and has finite volume.

By Claim 4.8, f_{∞} is a homeomorphism into the union of some components of ∂N(ρ_{∞}). If some component of ∂N(ρ_{∞}) did not lie in f_{∞}(∂_{χ<0}M−L^{0}) then the corresponding component b of ℍ^{3}/ρ_{∞}(π_{1}(M)) − N(ρ_{∞}) would lie in C_{∞}. But, since the projection ℍ^{3}/ρ_{∞}(π_{1}(M)) − N(ρ_{∞}) ⟶ ∂N(ρ_{∞}) does not increase distances, b has infinite volume. So b does not lie in C_{∞}. It follows that we have f_{∞}(∂_{χ<0}M − L^{0}) = ∂N(ρ_{∞}) and that ρ_{∞} is geometrically finite.

Since the homeomorphism f_{∞} : ∂_{χ<0}M −L^{0} → ∂N(ρ_{∞}) is homotopic to the identity, there is a homotopy equivalence M −(L^{0} ∪∂_{χ=0}M) → N(ρ_{∞}) which agrees with f_{∞} on ∂_{χ<0}M −L^{0}. By [Wa], this homotopy equivalence can be realized by a homeomorphism. Composing this homeomorphism with the natural homeomorphism int(N(ρ_{∞})) → ℍ^{3}/ρ_{∞}(π_{1}(M)) yields the desired geometrically finite metric σ_{∞} on int(M).

4.2 Convergence of (σ_{n}) to σ_{∞}

To conclude that b_{R} is proper, it remains to show that a subsequence of (σ_{n}) converges to σ_{∞}. First, we are going to show that, up to extracting a subsequence, ρ_{n}(π_{1}(M)) converges strongly to ρ_{∞}(π_{1}(M)). Let Ω(ρ_{n}) be the domain of discontinuity of ρ_{n}(π_{1}(M)). We will use a result of [JoM] (see also [MaT]):

Theorem 4.10. Assume that Ω(ρ_{∞}) is not empty. If (Ω(ρ_{n})) converges to Ω(ρ_{∞}) in the sense of Carathéodory, then ρ_{n} converges strongly to ρ_{∞}.

Let us recall that (Ω(ρ_{n})) converges to Ω(ρ_{∞}) in the sense of Carathéodory if and only if (Ω(ρ_{n})) satisfies the two following conditions :

- every compact subset K ⊂ Ω(ρ_{∞}) lies in Ω(ρ_{n}) for all sufficiently large n;

- every open set O which lies in Ω(ρ_{n}) for infinitely many n also lies in Ω(ρ_{∞}).

We are going to show that Ω(ρ_{n}) and Ω(ρ_{∞}) satisfy these two conditions.

Lemma 4.11. Under the hypothesis of Proposition 3.1, a subsequence of (ρ_{n}) converges strongly to ρ_{∞}.

Proof. Since ρ_{∞} is geometrically finite, Ω(ρ_{∞}) is not empty.

Let K ⊂ Ω(ρ_{∞}) be a compact connected subset and let Ω(ρ_{∞})_{1} be the connected component of Ω(ρ_{∞}) that contains K. Let Σ̃_{1} ⊂ ∂Ñ(ρ_{∞}) and K′ ⊂ Σ̃_{1} be the images of Ω(ρ_{∞})_{1} and of K, respectively, under the retraction map Ω(ρ_{∞})_{1} → ∂C(ρ_{∞}). Let f̂_{n} : ℍ^{2} → ℍ^{3} be a lift of f_{n} and let λ̂_{n} be the bending geodesic measured lamination of f̂_{n}. Let us extract a subsequence such that |λ̂_{n}| converges for the Hausdorff topology to a geodesic lamination L̂ ⊂ ℍ^{2}. Let x̂_{1}, ..., x̂_{∞} ∈ ℍ^{2} − L̂ be a sequence of points such that any connected component of ℍ^{2} − L̂ contains one of these points and let Π_{i} be the support plane at f̂_{∞}(x̂_{i}). The boundary of Π_{i} in ∂\overline{ℍ}^{3} is a circle. This circle bounds a disc D_{i} ⊂ ∂\overline{ℍ}^{3} such that int(D_{i}) ⊂ Ω(ρ_{∞}). We have Ω(ρ_{∞})_{1} = ⋃_{i} int(D_{i}). Let i ∈ ℕ; for n large enough, x̂_{i} lies in ℍ^{2} − |λ̂_{n}|. Let Π_{i,n} be the support plane at f̂_{n}(x̂_{i}). The boundary of Π_{i,n} in ∂\overline{ℍ}^{3} bounds a disc D_{i,n} ⊂ ∂\overline{ℍ}^{3} such that int(D_{i,n}) ⊂ Ω(ρ_{n}). We have ⋃_{i} int(D_{i,n}) ⊂ Ω(ρ_{n}).

Since (f̂_{n}) converges to f̂_{∞}, it follows that, for any i ∈ ℕ, (D_{i,n}) converges to D_{i} in the Hausdorff topology. Since K is a compact set lying in Ω(ρ_{∞}), it is covered by finitely many compact sets K_{j} such that each K_{j} lies in the interior of some D_{i}. For n large enough, each K_{j} lies in the interior of the corresponding disc D_{i,n}, hence K lies in Ω(ρ_{n}).

Let O be an open set lying in Ω(ρ_{n}) for infinitely many n and let w ∈ O ∩ L_{ρ_{∞}}. Since the fixed points of hyperbolic isometries are dense in L_{ρ_{∞}} and since (ρ_{n}) converges to ρ_{∞}, there is a sequence of points (w_{n}) converging to w such that w_{n} ∈ L_{ρ_{n}}. For n large enough, we have w_{n} ⊂ O ∩ L_{ρ_{n}}, contradicting our hypothesis that O ⊂ Ω(ρ_{n}) for infinitely many n.

Let n ∈ \overline{ℕ}, let M_{n} = ℍ^{3}/ρ_{n}(π_{1}(M)) ≈ int(M), let x_{n} ∈ M_{n} be the projection of the origin o ∈ ℍ^{3} and let h_{n} : M − (λ_{n}^{(p)} ∪ ∂_{χ=0}M) → N(ρ_{n}) be a homeomorphism associated to σ_{n}.

Since ρ_{n}(π_{1}(M)) converges strongly to ρ_{∞}(π_{1}(M)), there are sequences k_{n} ⟶ 1 and r_{n} ⟶ ∞ and a sequence of maps ũ_{n} : B(o, r_{n}) ⊂ ℍ^{3} → ℍ^{3} with the following properties (cf. [BeP] for example) :

- the map ũ_{n} is a k_{n}-bilipschitz diffeomorphism onto its image, u_{n}(o) = o and ũ_{n} converges to the identity map on any compact set;

- the map ũ_{n} is equivariant, namely for any point x ∈ B(o, r_{n}) and any element a ∈ π_{1}(M) such that ρ_{∞}(a)(x) ∈ B(o, r_{n}), we have u_{n} ∘ ρ_{∞}(a)(x) = ρ_{n}(a) ∘ u_{n}(x).

It follows from this second property that ũ_{n} yields a quotient map u_{n} : B(x_{∞}, r_{n}) ⊂ M_{∞} → M_{n} which is also a k_{n}-bilipschitz diffeomorphism onto its image.

Since M_{∞} is geometrically finite, for r_{n} large enough, B(x_{∞}, r_{n}) ⊂ M_{∞} is homeomorphic to M and the inclusion map B(x_{∞}, r_{n}) ⊂ M_{∞} ≈ int(M) is a homotopy equivalence. This implies that u_{n}(B(x_{∞}, r_{n})) is homeomorphic to M. Since ũ_{n} is equivariant, the inclusion map u_{n}(B(x_{∞}, r_{n})) ⊂ M_{n} ≈ int(M) is a homotopy equivalence. By a result of [MMS], M − B(x_{∞}, r_{n}) and M − u_{n}(B(x_{∞}, r_{n})) are homeomorphic to ∂M × (0, 1] and we can extend the map u_{n} : B(x_{∞}, r_{n}) ⊂ int(M) → int(M) to a homeomorphism v_{n} : M → M homotopic to the identity. To show that (σ_{n}) converges to σ_{∞}, it remains to show that v_{n} is isotopic to the identity for large n.

Let n ∈ ℕ, the map f_{n} : ∂_{χ<0}M −L^{0} → ∂N(ρ_{n}) ⊂ M_{n} ≈ int(M) is the restriction of an homeomorphism h_{n} associated to ρ_{n}. Therefore f_{n} is isotopic to the inclusion map ∂_{χ<0}M −L^{0} → M. Since u_{n} is a k_{n}-bilipschitz map, for n large enough, B(x_{n}, r_{n} − 1) lies in u_{n}(B(x_{∞}, r_{n})). Let S be a connected component of ∂_{χ<0}M − L^{0}. By Lemma 4.3 and 4.7, for n large enough, the thick part of f_{n}(S) lies in B(x_{n}, r_{n} − 1) ⊂ u_{n}(B(x_{∞}, r_{n})). In the proof of Lemma 4.6 we have shown that f_{n} : S → ∂N(ρ_{n}) converges to f_{∞} : S → ∂N(ρ_{∞}) and we already know that ũ_{n} converges to the identity on any compact set. It follows that u_{n}^{−1} ∘ f_{n} = v_{n}^{−1} ∘ f_{n} converges to f_{∞} on any compact set. Hence, for n large enough, the restrictions of v_{n}^{−1} ∘ f_{n} and of f_{∞} to S are isotopic. Since f_{n} and f_{∞} are isotopic to the inclusion map S → M, the restriction of v_{n} to S ⊂ ∂M is isotopic to the inclusion map S → M.

Let V(L^{0}) be an annular neighborhood of L^{0}. Since λ̇ ∈ P^{+}(M)/R and since L^{0} does not intersect |λ̇| transversely, V(L^{0}) ∪ ∂_{χ=0}M does not contain any essential disc nor annulus. We have already shown that v_{n} is homotopic to the identity and that its restriction to ∂_{χ<0}M −V(L^{0}) is isotopic to the inclusion map ∂_{χ<0}M − V(L^{0}) → M. It follows from [Joh] that v_{n} is isotopic to the identity, concluding the proof that b_{R} is proper.

5 Consequences of Theorem 1.2

In this section, we will explain the proofs of Theorems 1.3 and 1.4.

Theorem 1.3. Let λ̊ ∈ P(M)/Mod_{0}(M) be a class of measured geodesic laminations such that B(λ̊) is a single point ρ and let ρ_{n} ⊂ AH(M) be a sequence of conjugacy classes representations. Then ρ_{n} converges strongly to ρ_{∞} if and only if :

- there is N such that for n ≥ N, ρ_{n} is a geometrically finite representation with bending measured geodesic lamination λ̊_{n} ∈ P(M)/Mod_{0}(M);

- the sequence λ̈_{n} ∈ P^{+}(M)/(Mod_{0}(M) ∗ R) converges to λ̈.

Proof. Let λ̊ ∈ P(M)/Mod_{0}(M) be a class of measured geodesic laminations such that B(λ) is a single point ρ, and let λ ∈ P(M) be a representative of λ̊.

Let ρ_{n} ⊂ AH(M) be a sequence of representations with the following properties : there is N such that for n ≥ N, ρ_{n} is a geometrically finite representation with bending measured geodesic lamination λ̊_{n} ∈ P(M)/Mod_{0}(M) and the sequence λ̈_{n} ∈ P^{+}(M)/(Mod_{0}(M) ∗ R) converges to λ̈. For n ≥ N, let λ_{n} ∈ P(M) be a representative of λ̊_{n} such that λ̇_{n} converges to λ̇. For n ≥ N, there is a geometrically finite metric σ_{n} with associated representation ρ_{n} and with bending measured geodesic laminations λ_{n}. By Theorem 1.2, any subsequence of σ_{n} contains a subsequence converging to a geometrically finite metric σ_{∞}. By [Le3], the bending measured geodesic lamination of σ_{∞} is λ. Since B(λ̊) is a single point ρ, the representation ρ is associated to σ_{∞}. By Lemma 4.11, the corresponding subsequence of ρ_{n} converges strongly (up to conjugacy) to ρ. We have thus proved that any subsequence of ρ_{n} contains a subsequence that converges strongly to ρ. It follows that ρ_{n} converges strongly to ρ.

Let ρ_{n} ∈ AH(M) be a sequence of representation converging strongly to ρ. Let σ ∈ GF(M) be a geometrically finite representation whose bending measured geodesic lamination is λ. By [Ta], there is N such that for n ≥ N, ρ_{n} is geometrically finite. Therefore there is a sequence of metrics σ_{n} ∈ GF(M) with associated representations ρ_{n} such that σ_{n} converges to σ in GF(M). Let λ_{n} be the bending measured geodesic lamination of σ_{n}, λ_{n} is a representative of the bending measured lamination λ̊_{n} ∈ P(M)/Mod_{0}(M) of ρ_{n}. By [Le3], the sequence λ̇_{n} converges to λ̇. Therefore the sequence λ̈_{n} ∈ P^{+}(M)/(Mod_{0}(M) ∗ R) converges to λ̈.

Theorem 1.4. The action of Mod(M) on D(M) is properly discontinuous.

Proof. If we assume the contrary, there are λ ∈ D(M), (λ_{n}) ∈ D(M) and (ϕ_{n}) ∈ Mod(M) such that (λ_{n}) and (ϕ_{n}(λ_{n})) converge simultaneously to λ in ML(∂M) and that for any n ≠ m, ϕ_{n} is not isotopic to ϕ_{m}.

First assume that there is no homeomorphism between M and an I-bundle over a pair of pants P such that |λ| is mapped to a section of the bundle over ∂P. Since λ ∈ D(M), there is η > 0 such that i(λ, ∂D) > η for any essential disc D. Let \frac{2π}{η} λ be the measured geodesic lamination obtained by rescaling the measure of λ by \frac{2π}{η}. Let λ^{i} be a compact leaf of \frac{2π}{η} λ with a weight greater than or equal to π. If, up to extracting a subsequence, λ^{i} is a compact leaf of all the measured geodesic laminations λ_{n}, replace, in \frac{2π}{η} λ and in all \frac{2π}{η} λ_{n}, λ^{i} by a leaf with weight equal to π. Let λ′_{∞} and λ′_{n} be the measured geodesic laminations obtained by doing this operation for all the leaves of \frac{2π}{η} λ with a weight greater than or equal to π. We remark that λ′_{∞} may have some leaves with a weight greater than π but that for n large enough, the compact leaves of λ′_{n} all have a weight less than or equal to π. Note also that (λ′_{n}) and (ϕ_{n}(λ′_{n})) converge simultaneously to λ′_{∞} in ML(∂M). By [Le2, Lemma 3.4], λ′_{∞} and λ′_{n} satisfy conditions b), c). For n large enough, the λ′_{n} also satisfy condition a) hence, by [Le1], there is a geometrically finite metric σ_{n} ∈ GF(M) whose bending measured lamination is (λ′_{n}). The bending measured geodesic lamination of ϕ_{n∗}(ρ_{n}) is ϕ_{n}(λ′_{n}) and by construction ϕ_{n}(λ′_{n}) ⟶ λ′_{∞}. By Theorem 1.2, up to extracting a subsequence (σ_{n}) and (ϕ_{n∗}(σ_{n})) converge in GF(M).

Therefore we can choose representations ρ_{n} : π_{1}(M) → Isom(ℍ^{3}) associated to σ_{n} such that ρ_{n} and ϕ_{n∗} ∘ ρ_{n} both converge algebraically. It is a classical result that the action of Mod(M)/Mod_{0}(M) on the subset of AH(M) made up by the geometrically finite representations is properly discontinuous. It follows that, up to extracting a subsequence, there is n_{0} such that ϕ_{n_{0}}^{−1} ∘ ϕ_{n} is an element of Mod_{0}(M).

Since (λ′_{n}) converges to λ′_{∞}, up to extracting a subsequence, the union λ^{(p)} of the compact leaves of λ′_{n} with a weight equal to π does not depend on n. Since (ϕ_{n}(λ′_{n})) converges to λ′_{∞}, up to extracting a subsequence, (ϕ_{n}(λ^{(p)})) does not depend on n. Up to changing n_{0} and extracting a subsequence, we may assume that ϕ_{n_{0}}^{−1} ∘ ϕ_{n} fixes λ^{(p)} and that ϕ_{n_{0}}^{−1} ∘ ϕ_{n} maps each component of ∂M − λ^{(p)} to itself.

Let S be a connected component of ∂M − λ^{(p)}. Let us denote by s_{n}^{S} the metric induced on S ⊂ ∂M by a homeomorphism h_{n} : M −(λ^{(p)}∪∂_{χ=0}M) → N(σ_{n}) associated to σ_{n}. We have seen in Lemma 4.3 that, up to extracting a subsequence, the sequence (s_{n}^{S}) converges algebraically. Since ϕ_{n_{0}}^{−1} ∘ ϕ_{n}(λ′_{n}) converges to ϕ_{n_{0}}^{−1}(λ′_{∞}), it follows from the proof of Lemma 4.3 that the sequence (ϕ_{n_{0}}^{−1} ∘ ϕ_{n})_{∗}s_{n}^{S} converges algebraically. It is well known that the action of Mod(S) on the isotopy classes of hyperbolic metrics on S is properly discontinuous. It follows that there is n_{1} such that, up to extracting a subsequence, the restrictions of ϕ_{n_{0}}^{−1} ∘ ϕ_{n} and of ϕ_{n_{0}}^{−1} ∘ ϕ_{n_{1}} to S ⊂ ∂M are isotopic.

All the homeomorphisms ϕ_{n_{0}}^{−1} ∘ ϕ_{n} are homotopic to the identity, hence they are homotopic to each other. In particular, ϕ_{n_{0}}^{−1} ∘ ϕ_{n} is homotopic to ϕ_{n_{0}}^{−1} ∘ ϕ_{n_{1}} for any n. We have seen in the previous paragraph that the restriction of ϕ_{n_{0}}^{−1} ∘ ϕ_{n} to each component of ∂M − λ^{(p)} is isotopic to the restriction of ϕ_{n_{0}}^{−1} ∘ ϕ_{n_{1}}. Let V(λ^{(p)}) be an annular neighborhood of λ^{(p)}, since λ ∈ D(M), (M, V(λ^{(p)})) does not contain any annulus nor essential disc. It follows from [Joh, Proposition 27.1] that, up to extracting a subsequence and changing n_{1}, ϕ_{n_{0}}^{−1} ∘ ϕ_{n} is isotopic to ϕ_{n_{0}}^{−1} ∘ ϕ_{n_{1}} on the whole manifold M. This contradicts our hypothesis (that for any n ≠ m, ϕ_{n} is not isotopic to ϕ_{m}) and concludes the proof in this case.

Finally assume that there is a homeomorphism between M and an I-bundle over a pair of pants P such that |λ| is mapped to a section of the bundle over ∂P. Let η > 0 be such that i(∂D, λ) > η for any essential disc D. Let \frac{2π}{η} λ be the measured geodesic lamination obtained by rescaling the measure of λ by \frac{2π}{η}. If a homoclinic leaf l intersects |λ| only in one point then l contains two disjoint half leaves both spiraling in the same direction toward a leaf of λ. Such a leaf can not lie in a Hausdorff limit of simple closed curve. Furthermore i(∂E, \frac{2π}{η} λ) > 2π for any essential annulus E. It follows that the proof of Theorem 1.2 still works in this case yielding the following conclusion : if σ_{n} ∈ GF(M) is a sequence of metrics whose bending geodesic measured laminations λ_{n} converges to \frac{2π}{η} λ then there is a subsequence such that σ_{n} converges to the fuchsian metric whose parabolics are the leaves of λ. Now we can apply the same arguments as in the other case.

A Compactification of Teichmüller space and Λ-measured laminations

Let S be a hyperbolic surface with finite area and let (s_{n}) be a sequence of complete hyperbolic metrics on S. Assume that no subsequence of (s_{n}) converges algebraically. Then by [MoS1], there is a subsequence (which we will also denote by (s_{n})) which tends to a small minimal action of π_{1}(S) on a Λ-tree T . The basic property of this action is that for any simple closed curves c_{1}, c_{2} such that l_{s_{n}}(c_{1}) ⟶ ∞, we have lim_{n⟶∞} \frac{l_{sn}(c_{2})}{l_{sn}(c_{1})} = \frac{δ_{L}(c_{2})}{δ_{L}(c_{1})}. By [MoO], this action of π_{1}(S) on T is dual to a Λ-measured geodesic lamination L.

In this appendix, we will prove the continuity theorem for Λ-measured geodesic laminations which was used in the present paper.

Theorem A.1. Let (s_{n}) be sequence of complete hyperbolic metrics of finite volume on a surface S tending to a Λ-measured lamination L and let (λ_{n}) be a sequence of measured geodesic laminations such that (λ̇_{n}) converges to some λ̇ ∈ ML(S)/R and such that l_{s_{n}}(λ_{n}) is bounded. Then |λ̇| does not intersect |L| transversely.

Proof. Let (s_{n}), L, (λ_{n}), λ be as in the statement and suppose that |λ| intersects |L| transversely. Fix a reference hyperbolic metric on S and denote by λ^{i} a component of λ which intersects L transversely.

First assume that λ^{i} is an irrational lamination. Endow S with a complete hyperbolic metric and let c ⊂ S(λ^{i}) be a simple closed geodesic. Since λ^{i} is arational in S(λ^{i}), any leaf of L intersecting c transversely intersects λ^{i} transversely. Therefore, there is a train track τ with the following properties :

- τ carries λ^{i};

- τ has only one switch A ⊂ c − L;

- for any arc k ⊂ τ transverse to the ties such that k ∩ A = ∂k, we have ∫_{k} dL > 2 ∫_{c} dL.

Let k ⊂ λ^{i} be an arc, parametrized by [0, 1], such that k ∩ A = ∂k. Since |λ^{i}| and c are uniquely defined for any hyperbolic metric on S and since ∂k ⊂ |λ^{i}| ∩ c, this arc k is well-defined for any hyperbolic metric on S (see 1.1). Let us denote by b the branch of τ containing k and by b(λ) the measure of a tie of b.

Assume that \frac{dk}{dt}(0) and \frac{dk}{dt}(1) point to the same side of A. Denote by κ the subsegment of A joining k(0) to k(1) and by c_{b} the simple closed geodesic homotopic to k ∪ κ. Let d̂ be a lift of k ∪ κ to ℍ^{2}, we may assume that A has been chosen small enough (with respect to the reference metric) so that d̂ is a quasi-geodesic. Then there are a lift ĉ_{b} of c_{b} and a number ε > 0 such that d̂ lies in an ε-neighborhood of ĉ_{b}. Moreover ĉ_{b} contains a family of segments [a_{i}, a_{i+1}], i ∈ ℤ such that the ε-neighborhood of each [a_{i}, a_{i+1}] contains one lift of k. Therefore any leaf of the preimage of L which intersects d̂ intersects d̂ only once and also intersects ĉ_{b}. Then we have δ_{L}(c_{b}) = δ_{L}(k) > 2δ_{L}(c). Hence l_{s_{n}}(c_{b}) − 2l_{s_{n}}(c) tends to ∞. So we have

l_{s_{n}}(k) ≥ l_{s_{n}}(c_{b}) − l_{s_{n}}(A) ≥ l_{s_{n}}(c_{b}) − l_{s_{n}}(c) ⟶ ∞.  (1)

This inequality (1) holds for any arc k of λ^{i} − A lying in b. Up to extracting a subsequence, the laminations λ_{n} are carried by a train track containing τ as a subtrack and inequality (1) holds for any arc of λ_{n} − A lying in b. So we have l_{s_{n}}(λ_{n}) ≥ b(λ_{n})(l_{s_{n}}(c_{b}) − l_{s_{n}}(c)). Since b(λ_{n}) tends to b(λ), for large n, we have l_{s_{n}}(λ_{n}) ≥ \frac{1}{2}b(λ)(l_{s_{n}}(c_{b}) − l_{s_{n}}(c)) ⟶ ∞. This contradicts the assumption that l_{s_{n}}(λ_{n}) is bounded and shows that \frac{dk}{dt}(0) and \frac{dk}{dt}(1) do not point to the same side of A.

Assume now that for any arc k ⊂ τ transverse to the ties such that k ∩ A = ∂k, the tangent vectors \frac{dk}{dt}(0) and \frac{dk}{dt}(1) point to different sides of A. Let k^{1} ⊂ τ and k^{2} ⊂ τ be two arcs transverse to the ties such that we have k^{j} ∩ A = ∂k^{j} and k^{1} ∩ k^{2} = k^{1}(1) = k^{2}(0). Denote by b^{j} the branch of τ containing k^{j}. By our assumption, we have b^{1} ≠ b^{2}. Let [k^{1}(0), k^{2}(1)] ⊂ A denote the segment joining k^{1}(0) to k^{2}(1). If [k^{1}(0), k^{1}(1)] ∩ (k^{1} ∪ k^{2}) is empty, let c_{b} be the simple closed geodesic isotopic to k^{1} ∪ k^{2} ∪ [k^{1}(0), k^{1}(1)]. Otherwise let c_{b} be the simple closed geodesic isotopic to k^{1} ∪ [k^{1}(1), k^{2}(1)] ∪ (k^{2})^{−1} ∪ [k^{2}(0), k^{1}(0)]. By the same arguments as before we have l_{s_{n}}(λ_{n}) ≥ \frac{1}{2}(∫_{A∩b^{1}∩b^{2}} dλ)(l_{s_{n}}(c_{b}) − 2l_{s_{n}}(c)) ⟶ ∞. This yields the same contradiction as before implying that λ^{i} is not an irrational lamination.

Thus λ^{i} is a closed leaf. Since l_{s_{n}}(λ^{i}) is bounded, λ^{i} is not a leaf of λ_{n} for large n. Endow S with a complete hyperbolic metric and consider two pairs of pants P^{1} ⊂ ∂M and P^{2} ⊂ ∂M such that λ^{i} ⊂ int(P^{1} ∪ P^{2}). We will only deal with the case where P^{1} and P^{2} are distinct since only slight modifications are required to deal with the other case. Remove from P^{j} an annular neighborhood of ∂P^{j} − λ^{i}, for j = 1, 2, and denote by S the surface P^{1} ∪ P^{2}. We have ∂S = ∂P^{1} ∪ ∂P^{2} − λ^{i} and the points of ∂S ∩ |L| are transverse intersections. Let c ⊂ S be a simple closed geodesic which intersects λ^{i} in two points. Changing c by Dehn twists along λ^{i} if necessary, we may assume that c ⊄ |L|. Let us denote by |L^{i}| the union of the components of |L| ∩ S which intersect λ^{i} and by |L^{r}| the union of the remaining components of |L| ∩ S. The transverse Λ-measure of L induces transverse Λ-measures on |L^{i}| and on |L^{r}| and we have δ_{L}(c) = δ_{L^{i}}(c)+δ_{L^{r}}(c). Since any component of |L|∩P^{j} is compact, there is m ∈ ℕ such that for j = 1, 2 and for any component l of |L| ∩ P^{j}, we have ♯\{l ∩ c\} ≤ \frac{m}{2}. Therefore δ_{L^{i}}(c) ≤ mδL(λ^{i}).

By [Th3], there is a sequence of measured geodesic laminations (ν_{n}) such that for any simple closed curve d ⊂ S, ∃C_{d} such that i(ν_{n}, d) ≤ l_{s_{n}}(d) ≤ i(ν_{n}, d)+C_{d}. Let us divide ν_{n}∩S into ν_{n}^{i}∪ν_{n}^{r} where ν_{n}^{i} is the union of the components which intersect λ^{i}. We have i(ν_{n}, c) = i(ν_{n}^{i}, c)+i(ν_{n}^{r}, c). Any connected component of |ν_{n}^{r}| intersects c once or twice, depending on whether its vertices lie in different components of ∂S or not. Hence there are a_{1}, a_{2}, a_{3}, a_{4} ∈ \{\frac{1}{2}\}ℕ and ϵ ∈ \{0, 1\} such that i(ν_{n}^{r}, c) = ∑_{k=1}^{4} a_{k}i(ν_{n}, ∂^{k}S) − ϵi(ν_{n}, λ^{i}) where ∂^{k}S, k = 1, 2, 3, 4 are the connected components of ∂S. Let us explain how a_{1} and a_{2} are computed, considering that we have ∂^{1}S, ∂^{2}S ⊂ ∂P^{1} (a_{3} and a_{4} are computed in the same way):

- if we have i(ν_{n}, ∂^{1}S) + i(ν_{n}, ∂^{2}S) ≥ i(ν_{n}, λ^{i}), i(ν_{n}, ∂^{2}S) + i(ν_{n}, λ^{i}) ≥ i(ν_{n}, ∂^{1}S) and i(ν_{n}, ∂^{1}S) + i(ν_{n}, λ^{i}) ≥ i(ν_{n}, ∂^{2}S) then the total weight carried by the components of ν_{n}^{r} is \frac{1}{2}(i(ν_{n}, ∂^{1}S) + i(ν_{n}, ∂^{2}S) − i(ν_{n}, λ^{i})), therefore we set a_{1} = a_{2} = \frac{1}{2} and ϵ = 1 (see fig 5, case 1);

- if we have i(ν_{n}, ∂^{1}S) + i(ν_{n}, ∂^{2}S) ≤ i(ν_{n}, λ^{i}) then we set a_{1} = a_{2} = ϵ = 0 (see fig 5, case 2);

- if we have i(ν_{n}, ∂^{2}S) + i(ν_{n}, λ^{i}) ≤ i(ν_{n}, ∂^{1}S) then we set a_{1} = 1, a_{2} = 0 and ϵ = 1 (see fig 5, case 3). If we have i(ν_{n}, ∂^{1}S) + i(ν_{n}, λ^{i}) ≤ i(ν_{n}, ∂^{2}S) then we set a_{1} = 0, a_{2} = 1 and ϵ = 1.

If i(ν_{n}, ∂^{1}S), i(ν_{n}, ∂^{2}S) and i(ν_{n}, λ^{i}) satisfy an equality in the triangle inequalities, then we are in case 1 and in case 2 or in case 1 and in case 3 and we can choose the a_{k} and ϵ which are the most convenient for our purpose.

Up to extracting a subsequence, we may assume that the a_{k} and ϵ do not depend on n.

Let (κ_{n}) be a sequence of arcs of c such that \underline{lim} ∫_{κ_{n}} dλ_{n} > 0 and that (κ_{n}) converges for the Hausdorff topology to a point of λ^{i}. Let V(|λ^{i}|) be an annular neighborhood of |λ^{i}| and recall the description of λ_{n} ∩ V(|λ^{i}|) that we gave in the proof of Claim 3.8 : |λ_{n}| ∩ V(|λ^{i}|) is a family of disjoint segments joining the two components of ∂V(|λ^{i}|) and turning a lot of times toward |λ^{i}| (by Claim 2.2) and the total weight of the transverse measure carried by |λ_{n}|∩V(|λ^{i}|) tends to 0 (by Claim 2.3). Given an orientation of c we order the points of κ_{n} ∩ λ_{n} by the induced order. For large n, κ_{n} contains an arc κ′_{n} such that we have ∫_{κ′_{n}} dλ_{n} ≥ ^{\frac{∫_{κn} dλ_{n}}{3}} and that any point x_{n} of κ′_{n} ∩ λ_{n} is joined to a point y_{n} > x_{n} of κ_{n} ∩ λ_{n} by an arc ]x_{n}, y_{n}[⊂ λ_{n} − κ_{n}. Fix n, consider a point x_{n} in κ′_{n} ∩ λ_{n} and the associated point y_{n}. Let c_{x_{n}y_{n}} be the simple

PSfrag replacements

case 1 case 2 case 3 c ∩ P^{1}

|ν_{n}^{i}| ∩ P^{1} |ν_{n}^{r}| ∩ P^{1}

λ^{i}

∂^{1}S
∂^{2}S

Figure 5: Computation of a_{k} and ϵ

closed curve which is the union of [x_{n}, y_{n}] ⊂ λ_{n} and of the arc κ_{x_{n}y_{n}} ⊂ κ_{n} joining x_{n} to y_{n}. This curve c_{x_{n}y_{n}} can be homotoped in λ^{i}, hence we have i(ν_{n}, c_{x_{n},y_{n}}) ≥ i(ν_{n}, λ^{i}). We have then i(ν_{n}, λ_{n}) ≥ (∫_{κ′_{n}} dλ_{n}) min{∫_{[x_{n},y_{n}]} dν_{n}/x_{n} ∈ κ′_{n} ∩ λ_{n}} ≥ (∫_{κ′_{n}} dλ_{n})(i(ν_{n}, λ^{i}) − ∫_{κ′_{n}} dν_{n}) hence ∫_{κ_{n}} dν_{n} ≥ ∫_{κ′_{n}} dν_{n} ≥ i(ν_{n}, λ^{i}) − _{\frac{3}{∫_{κn} dλ_{n}}} i(ν_{n}, λ_{n}). Let Q be the weight of λ^{i} as a sublamination

of λ′ and choose a number E with 0 < E < \frac{Q}{m+3}. By the previous description of λ_{n} ∩ V(|λ^{i}|), c contains m + 3 disjoint segments κ_{n}^{j} such that ∫_{κ_{n}^{j}} dλ_{n} ≥ E and such that for any 1 ≤ j ≤ m + 3, (κ_{n}^{j}) converges to a point x ⊂ c ∩ λ^{i} (which does not depend on j). It follows also from the description of λ_{n} ∩ V that any leaf of |ν_{n}| ∩ S intersecting a κ_{n}^{j} lies in |ν_{n}^{i}|. Thus we have i(ν_{n}^{i}, c) ≥ ∑_{j} ∫_{κ_{n}^{j}} dν_{n} ≥ (m + 3)i(ν_{n}, λ^{i}) − _{\frac{3(m+3)}{∫_{κ^{j}_{n}} dλ_{n}}} i(ν_{n}, λ_{n}).

Now we have i(ν_{n}, c) ≥ ∑_{k=1}^{4} a_{k}i(ν_{n}, ∂^{k}S) + (m + 3 − ϵ)i(ν_{n}, λ^{i}) − \frac{3(m+3)}{E}i(ν_{n}, λ_{n}) ≥ ∑_{k=1}^{4} a_{k}i(ν_{n}, ∂^{k}S) + (m + 2 − ϵ)i(ν_{n}, λ^{i}). This yields the inequalities : l_{s_{n}}(c) ≥ ∑_{k=1}^{4} a_{k}(l_{s_{n}}(∂^{k}S) − C_{∂^{k}S}) + (m + 2)(l_{s_{n}}(λ^{i}) − C_{λ^{i}}) ≥ ∑_{k=1}^{4} a_{k}l_{s_{n}}(∂^{k}S) + (m + 1)l_{s_{n}}(λ^{i}). By [MoS1], this implies the inequality : δ_{L}(c) ≥ ∑_{k=1}^{4} a_{k}δ_{L}(∂^{k}S) + (m + 1)δ_{L}(λ^{i}).

As for the (ν_{n}) there exist a_{1}, a_{2}, a_{3}, a_{4} ∈ \{\frac{1}{2}\}ℕ and ϵ ∈ \{0, 1, 2\} such that we have δ_{L^{r}} = ∑_{k=1}^{4} a_{k}δ_{L}(∂^{k}S) − ϵδ_{L}(λ^{i}). If i(ν_{n}, ∂^{j}S) tens to ∞, then we have lim_{n⟶∞} \frac{i(ν_{n},∂^{i}S)}{i(,ν_{n},∂^{j}S)} = \frac{δ_{L}(∂^{i}S)}{δ_{L}(∂^{j}S)}. Therefore, for large n, δ_{L}(∂^{k}S) and δ_{L}(λ^{i}) satisfy the same inequalities as i(ν_{n}, ∂^{k}S) and i(ν_{n}, λ^{i}). Hence the a_{k} are the same as before. Thus we have δ_{L}(c) ≤ ∑_{k=1}^{4} a_{k}δ_{L}(∂^{k}S) + mδ_{L}(λ^{i}). This contradicts the inequality δ_{L}(c) ≥ ∑_{k=1}^{4} a_{k}δ_{L}(∂^{k}S) + (m + 1)δ_{L}(λ^{i}) and concludes the proof of Theorem A.1.

Remark. An informal but more visual way to understand what is happening here is the following. The sequence of measured foliations ν_{n} tends to L, namely for any simple closed curves c_{1} and c_{2} such that i(ν_{n}, c_{1}) ⟶ ∞, we have lim_{n⟶∞} \frac{i(ν_{n},c_{2})}{i(,ν_{n},c_{1})} = \frac{δ_{L}(c_{2})}{δ_{L}(c_{1})}. The fact that l_{s_{n}}(λ_{n}) is bounded forces λ_{n} to follow ν_{n}, which implies that λ does not intersect L transversely. ⋄

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