Continuity of the bending map

Cyril LECUIRE May 2004

Abstract The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. As proved in [KeS] and [KaT], this map is continuous. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the space of measured geodesic laminations and shows that the quotient map obtained from the bending map is continuous.

Introduction

Let M be a compact, orientable 3-manifold with boundary. Assume that M is hyperbolic, namely that the interior of M is endowed with a complete metric σ of constant sectional curvature −1. Assume also that ∂M contains a surface with genus greater than 1. A fundamental subset of (M, σ) is its convex core N(σ). This core N(σ) is defined as the smallest non-empty closed subset of the interior of M which is locally convex and homotopically equivalent to M. Its boundary ∂N(σ) endowed with the intrinsic metric (given by rectifiable path length) is isometric to a hyperbolic surface of finite volume and can be embedded in a natural way into ∂M. This surface is bent along a geodesic lamination and the amount of bending is described by a measured geodesic lamination called the bending measured geodesic lamination of σ (cf. [Th] or [CEG]). This yields a bending map b which to a complete hyperbolic metric associates its bending measured geodesic lamination.

In [Bo2], F. Bonahon considers quasi-isometric deformations of a given metric σ on int(M), namely hyperbolic metrics σ′ on int(M) for which there exists a diffeomorphism (int(M), σ) → (int(M), σ′) whose differential is uniformly bounded. Let 𝒬𝒟 (σ) be the space of quasi-isometric deformations of a given metric σ, where we identify two deformations (int(M), σ) → (int(M), σ′) and (int(M), σ) → (int(M), σ′′) if they are isotopic. The continuity of the bending map b : 𝒬𝒟 (σ) → ℳ ℒ (∂M) is proved in [KaT], using ideas of Thurston. Its differentiability (in a weak sense) is proved in [Bo2].

A complete hyperbolic metric σ on int(M) is said to be convex cocompact if N(σ) is compact. If σ is convex cocompact, then 𝒬𝒟 (σ) is the set of convex cocompact metrics on int(M). In this case, the continuity of b : 𝒬𝒟 (σ) → ℳ ℒ (∂M) has been proved in [KeS] and its image has been described in [BoO] and [Le1]. A complete hyperbolic metric σ on int(M) is said to be geometrically finite if N(σ) has finite volume. When σ is a geometrically finite metric, 𝒬𝒟 (σ) is the set of geometrically finite having the same parabolic subgroups as σ. In the present paper, we are addressing the question of the continuity of the bending map on the whole set of geometrically finite metrics. In particular, we are interested in sequence converging to a limit with some new parabolics.

Since we want to consider metric which do not have the same parabolic subgroups, we have to allow deformations which are not quasi-isometric. We will consider the set 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 consider the set 𝒢 ℱ (M) of isotopy classes of geometrically finite hyperbolic metrics which are not fuchsian. We will topologize 𝒢 ℱ (M) 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 the ball 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. For a metric σ ∈ 𝒢 ℱ (M), 𝒬ℱ (σ) can be viewed as a subset of 𝒢 ℱ (M). The topology of 𝒬ℱ (σ) considered as a subset of 𝒢 ℱ (M) coincides with the topology given by quasi-isometric deformations

The bending map b_{𝒢 ℱ} (M) : 𝒢 ℱ (M) → ℳ ℒ (∂M) maps an isotopy class of geometrically finite metrics to its bending measured geodesic lamination. The image of b_{𝒢 ℱ} 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.

Taking a careful look at the behavior of the map b_{𝒢 ℱ} , we notice that it is not a continuous map. If a metric σ lies in 𝒢 ℱ (M) but has some rank one cusps, its measured geodesic laminations λ has some compact leaves with a weight equal to π. Let us denote by λ^{(p)} the union of the leaves of λ which have a weight equal to π. Using the result of [BoO], it is not hard to construct a sequence of metrics σ_{n} ∈ 𝒢 ℱ (M) with measured geodesic laminations λ_{n} such that the sequence (λ_{n}) converges to a measured geodesic lamination λ_{∞} which differs from λ only on λ^{(p)} and which have some leaves with a weight greater than π. Using some arguments of [Le1], we get that a subsequence of (σ_{n}) converges to a geometrically finite metric σ_{∞}. Since λ_{∞} does not satisfies condition a), it is not the bending measured geodesic lamination of σ_{∞}. Thus we get that b_{𝒢 ℱ} is not continuous on any neighborhood of a metric with some rank one cusps. To overcome this difficulty, we will quotient the space ℳ ℒ (∂M) of measured geodesic laminations by the following relationship :

Let λ, μ ∈ ℳ ℒ (∂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.

Let us endow ℳ ℒ (∂M) with the weak^{∗} topology and ℳ ℒ (∂M)/ℛ with the quotient topology. From b_{𝒢 ℱ} (M) we obtain a quotient map b_{ℛ} : 𝒢 ℱ (M) → 𝒫^{+}(M)/ℛ. We will prove the following result :

Theorem 1. The map b_{ℛ} from 𝒢 ℱ (M) to ℳ ℒ (∂M)/ℛ is a continuous map.

In [Le2], we show the reverse of this theorem. This gives rise to a criterion for the strong convergence of a sequence of geometrically finite representations ρ_{n} ∈ U(M).

The paper is organized as follows. In section 1, we state some definitions and we prove some facts about ℳ ℒ (∂M)/ℛ. In section 2, we study convex pleated surfaces and prove the continuity of the bending measured geodesic laminations of a converging sequence of convex pleated surfaces. In section 3 we use the results of section 2 to prove the continuity of b_{ℛ} .

1 Definitions

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 that appear in this way will be called representations associated to σ. The set of representations associated to σ is the set of all representations conjugated 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 [Th, 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 complementary 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.

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}. Such a homeomorphism will be said to be associated to σ.

Let σ ∈ 𝒞 𝒞 (M) and let ρ and h be respectively a representation and a homeomorphism associated to σ. Let h̃ : M̃ → ℍ^{3} be a lift of h, we will define the compactification \overline{M̃} of M̃ as being the closure of h̃(M̃) in the usual compactification of ℍ^{3} by the unit ball.

Let (σ_{n}) be a sequence of isotopy classes of complete hyperbolic metrics on the interior of M. The sequence (σ_{n}) converges algebraically when there is a sequence of representations ρ_{n} : π_{1}(M) → Isom(ℍ^{3}) associated to σ_{n} (as above) that converges algebraically; namely ρ_{n}(g) converges for any g ∈ π_{1}(M). We obtain a new representation ρ_{∞} : π_{1}(M) → Isom(ℍ^{3}) defined by ρ_{∞}(g) = lim_{n⟶∞} ρ_{n}(g) for any g ∈ π_{1}(M). This representation ρ_{∞} 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 (examples are given in [AnC]).

The sequence (σ_{n}) converges geometrically if there is a sequence of representations ρ_{n} : π_{1}(M) → Isom(ℍ^{3}) associated to the σ_{n} such that (ρ_{n}(π_{1}(M)) converges geometrically. The sequence of groups (ρ_{n}(π_{1}(M)) converges geometrically to a group Γ_{∞} ⊂ Isom(ℍ^{3}) if and only if :

- 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 sequence ρ_{n}(π_{1}(M)) converges strongly if there is a sequence of representations ρ_{n} associated to σ_{n} such that (ρ_{n}) converges algebraically to a representation ρ_{∞} and that (ρ_{n}(π_{1}(M))) converges geometrically to ρ_{∞}(π_{1}(M)). If (σ_{n}) converges to σ for the topology defined in the introduction, then (σ_{n}) converges strongly to σ_{∞} (cf. [CEG]).

2 Geodesic laminations and the relationship ℛ

A geodesic lamination L on ℍ^{2} is a closed subset which is the disjoint union of complete geodesics. A complete geodesic lying in L is a leaf of L.

A measured geodesic lamination λ is a transverse measure for some geodesic lamination |λ|. Any arc k ≈ [0, 1] embedded in S transversely to |λ|, such that ∂k ⊂ S − λ, is endowed with an additive measure dλ such that :

- the support of dλ_{|k} is |λ| ∩ k;

- if an arc k can be homotoped into k′ by a homotopy respecting |λ| then ∫_{k}dλ = ∫_{k′} dλ. We will denote by ℳ ℒ (ℍ^{2}) the space of measured geodesic lamination topologised with the topology of the weak^{∗} convergence.

Let S be a surface (which may not be compact) endowed with a complete hyperbolic metric with finite area. A geodesic lamination in S is a compact subset which is the disjoint union of simple complete geodesics. Using the fact that two complete hyperbolic metrics with finite area on S are quasi-isometric, this definition can be made independent of the chosen metric on S (see for example [Ot1]). A measured geodesic lamination is a transverse measure for some geodesic lamination as defined above. Let γ be a weighted simple closed geodesic with support |γ| and weight w and let λ be a measured geodesic lamination. The intersection number of γ and λ is defined by i(γ, λ) = w ∫_{|γ|} dλ. The weighted simple closed curves are dense in ℳ ℒ (S) and this intersection number extends continuously to a function i : ℳ ℒ (S) × ℳ ℒ (S) → ℝ (cf. [Bo1]).

A measured geodesic lamination λ ∈ ℳ ℒ (S) is arational if for any simple closed curve c which is not homotopic to a cusp, i(c, λ) = ∫_{c} dλ > 0.

Let us recall the definition of ℛ given in the introduction.

Let λ, μ ∈ ℳ ℒ (S) be two measured geodesic laminations. Let us denote by λ′ (resp. μ′) the measured geodesic lamination obtained by replacing by π the weights of the compact leaves of λ (resp. μ) which have a weight greater than π. We will say that λ is related to μ by the relationship ℛ, and we will write λℛμ, if and only if λ′ = μ′. We will denote by λ̇ the class of λ modulo ℛ and we will topologise ℳ ℒ (S) with the quotient topology of the weak^{∗} topology on ℳ ℒ (S).

Let ∂_{χ<0}M be the union of the connected components of ∂M with negative Euler characteristic. To simplify the notations, we will note ℳ ℒ (∂M) for ℳ ℒ (∂_{χ<0}M).

Let (λ_{n}) be a sequence of measured geodesic laminations such that (λ̇_{n}) converges to λ̇ in ℳ ℒ (∂M)/ℛ and let λ ∈ ℳ ℒ (∂M) a representative of λ̇. Let us denote by λ^{(p)} the union of the compact leaves of λ with a weight greater than π. We have the following :

Claim 2.1. Let λ ∈ ℳ ℒ (∂M) and let λ′ be the representative of λ̇ whose compact leaves have all a weight less than or equal to π. If k ⊂ ∂M is 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.

Proof. Let k ⊂ ∂M be a simple arc such that the points of k ∩ |λ̇| are transverse intersections. The set 𝒱_{k,ε}(λ) = {γ ∈ ℳ ℒ (∂M)/| ∫_{k}dγ − ∫_{k}dλ| < ε} is a neighborhood of λ in ℳ ℒ (∂M). Since λ̇_{n} converges to λ̇, for any ε, there is n_{ε} such that for any n ≥ n_{ε}, ∃α_{n} with α_{n}ℛλ_{n} and α_{n} ∈ 𝒱_{k,ε}(λ). So, for n ≥ n_{ε}, we have | ∫_{k}dα_{n} − ∫_{k}dλ_{n}| < ε. We have ∫_{k}dλ′ ≤ ∫_{k}dα for any α ∈ ℳ ℒ (∂ℳ) satisfying αℛλ. So we get ∫_{k}dλ′ ≤ \underline{lim} ∫_{k}dλ_{n}.

If k does not intersect any leaf of λ′ with a weight equal to π, then we have ∫_{k}dα = ∫_{k}dλ for any α such that αℛλ. So we have ε > | ∫_{k}dα_{n} − ∫_{k}dλ| = | ∫_{k}dλ_{n} − ∫_{k}dλ| for any n ≥ n_{ε}. Letting n tend to ∞, we get that ∫_{k}dλ_{n} converges to ∫_{k}dλ.

Let λ, μ ∈ ℳ ℒ (∂M) be two measured geodesic laminations. If λℛμ, then λ and μ share the same support. Thus we can define the support |λ̇| of an element λ̇ of ℳ ℒ (∂M) as being the support of any representative of λ̇.

Claim 2.2. Let (λ_{n}) ∈ ℳ ℒ (∂M) be a sequence of measured geodesic laminations such that (λ̇_{n}) converges to λ̇ and that |λ_{n}| converges to some geodesic lamination L in the Hausdorff topology. We have |λ̇| ⊂ L.

Proof. Let x be a point of |λ̇|, let ε > 0 be a real number and let k be a geodesic arc intersecting |λ̇| transversely, with length ε such that x lies in the interior of k. Since ∫_{k} dλ′ > 0, we deduce

from 2.1 that, for n large enough, we have ∫_{k}dλ_{n} ≥ ^{\frac{∫_{k}dλ′}{2}} > 0. Therefore k intersects λ_{n} and x lies in an ε-neighborhood of |λ_{n}|. Considering a covering of |λ̇| by discs with radius ε and with centers lying in |λ̇|, we get that, for n large enough, |λ̇| lies in an ε-neighborhood of |λ_{n}|. Letting ε tend to 0, we get |λ̇| ⊂ L.

Claim 2.1 can also be used to prove that the space ℳ ℒ (∂M)/ℛ is a Hausdorff space.

Lemma 2.3. The space ℳ ℒ (∂M)/ℛ is a Hausdorff space.

Proof. Let λ̇ and μ̇ be two elements of ℳ ℒ (S)/ℛ such that any neighborhood of λ̇ intersects any neighborhood of α̇. So there is a sequence of measured geodesic laminations λ_{n} such that λ̇_{n} converges simultaneously to λ̇ and to μ̇. Let us denote by λ^{(p)} (resp. μ^{(p)}) the union of the compact leaves of λ (resp. μ) with a weight greater than π. Let λ′ (resp. μ′) be the representative of λ̇ (resp. μ̇) whose compact leaves have all a weight less than or equal to π. Let k ⊂ ∂M − λ^{(p)} be a simple arc intersecting |λ| and |μ| transversely so that ∫_{k}dλ < π. By Claim 2.1, we have ∫_{k}dμ′ = lim_{n⟶+∞} ∫_{k}dλ_{n} < π. It follows that μ^{(p)} ⊂ λ^{(p)}. Reversing the roles of λ and of μ, we get μ^{(p)} = λ^{(p)}. It follows also that we have the equality ∫_{k}dμ = ∫_{k}dλ for any arc k ⊂ ∂M − λ^{(p)}. This yields the conclusion λ̇ = μ̇.

The following variation of Claim 2.1 will be used in the present paper.

Claim 2.4. If c is a simple compact curve that does not intersect λ^{(p)} transversely, then the sequence i(c, λ_{n}) converges to i(c, λ).

Proof. If the points of c ∩ |λ̇| are transverse intersections, we get the conclusion by cutting c into two arcs and by applying Claim 2.1 to these two arcs.

Let us now consider the case where c ⊂ |λ̇|. We will show that any subsequence of (λ_{n}) contains a subsequence such that i(c, λ_{n}) converges to i(c, λ) = 0. Up to extracting a subsequence, we can assume that |λ_{n}| converges in the Hausdorff topology. Let τ be a train track carrying all the λ_{n} for n large enough (see [Bo1] and [Ot1] for details about train tracks). By Claim 2.2, λ is carried by a subtrack of τ. Let b_{1}, b_{2},..., b_{p} be the branches of τ. Given a measured geodesic lamination α carried by a subtrack of τ, the measure b_{i}(α) of a tie of b_{i} does not depend on the choice of the tie. By Claim 2.1, if b_{i} is a branch of τ not carrying a leaf of λ^{(p)}, then b_{i}(λ_{n}) → b_{i}(λ). Since c is a leaf of λ, it is carried by a subtrack τ_{c} of τ. For n large enough, we have i(c, λ_{n}) = \frac{1}{2} ∑_{j∈J} b_{j}(λ_{n}) where \{b_{j}| j ∈ J\} is the set of branches of τ − τ_{c} intersecting τ_{c}. Such a branch b_{j} does not carry a leaf of λ^{(p)}. It follows then from Claim 2.1 that b_{j}(λ_{n}) converges to b_{j}(λ) = 0. Finally, we have i(c, λ_{n}) ⟶ 0 = i(c, λ).

3 Convex pleated surfaces

A pleated surface in a complete hyperbolic 3-manifold M is a map f : S → M from a surface S to M with the following properties :

- the path metric obtained by pulling back the hyperbolic metric of M by f is a hyperbolic metric metric s on S;

- every point of S lies in the interior of some s-geodesic arc which is mapped into a geodesic arc in M;

- if c ∈ S is a simple closed curve lying in a cusp of S and if c does not bound a disc in S, then f(c) does not bound a disc in M.

A map f̂: ℍ^{2} → ℍ^{3} is a pleated map if any point of ℍ^{2} lies in the interior of a geodesic arc which is mapped by f̂ into a geodesic arc.

The pleating locus of a pleated map is the set of points of ℍ^{2} where the map fails to be an isometry. The pleating locus of a pleated map is a geodesic lamination (cf. [Th]).

An abstract pleated surface is a triple (f̂, Γ, ρ) where f̂ : ℍ^{2} → ℍ^{3} is a pleated map, Γ is a lattice in Isom(ℍ^{2}) and ρ : Γ → Isom(ℍ^{3}) is a discrete representation (which may not be faithful) such that for any a ∈ Γ, we have f̂ ∘ a = ρ(a) ∘ f̂ and that if a ∈ Γ is a parabolic isometry then ρ(a) is also a parabolic isometry.

Abstract pleated surfaces and pleated surfaces are related as follows:

When ρ(Γ) has no torsion, the abstract pleated surface (f̂, Γ, ρ) induces a pleated surface f : S → M where S ≈ ℍ^{2}/Γ, M ≈ ℍ^{3}/ρ(Γ) and where f is the quotient map from f̂.

If f : S → M is a pleated surface, consider isometric covering maps ℍ^{2} → S and ℍ^{3} → M. These maps yield representations r : π_{1}(S) → Isom(ℍ^{2}) and R : π_{1}(M) → Isom(ℍ^{3}) and by lifting f to a map f̂: ℍ^{2} → ℍ^{3} we get an abstract pleated surface (f̂, r(π_{1}(S)), R ∘ f_{∗}).

In the following we will omit the adjective abstract and assume that our (abstract) pleated surfaces are torsion free.

We will consider the following topology on the space of abstract pleated surfaces. A sequence (f̂_{n}, Γ_{n}, ρ_{n}) of pleated surfaces converge to a pleated surface (f̂, Γ, ρ) if and only if :

- (Γ_{n}) converges geometrically to Γ;

- for any sequence a_{n} ∈ Γ_{n} converging to a ∈ Γ, ρ_{n}(a_{n}) converges to ρ(a);

- f̂_{n} converges to f̂ on any compact set of ℍ^{2}.

Let (f̂, Γ, ρ) be a (abstract) pleated surface (without torsion), let L ⊂ ℍ^{2} be the pleating locus of f̂ and let P be a connected component of ℍ^{2} − L. The surface f̂(P) lies in a geodesic plane Π_{P}. Given an orientation of ℍ^{2}, Π_{P} inherits a natural orientation and we denote by H_{P}^{+} (resp. H_{P}^{−}) the half-space bounded by Π_{P} such that the union of a direct frame of Π_{P} and of the inward normal vector to Π_{P} is a direct frame of ℍ^{3} (resp. indirect).

A pleated surface (f̂, Γ, ρ) with pleating locus L is a convex pleated surface if :

1) there is ϵ ∈ {+, −} such that for any component P of ℍ^{2} − L, f̂(ℍ^{2}) lies in H_{P}^{ϵ};

2) the interior of Cf̂ = ⋂{H_{P_{i}}^{ϵ}/P_{i} is a connected component of ℍ^{2} − L̂} is not empty.

If a pleated surface (f̂, Γ, ρ) satisfies 1) but not 2), namely if Cf̂ as empty interior, we will call it an even pleated surface.

In the following, for any convex pleated surface (f̂, Γ, ρ), we will choose the orientation of ℍ^{2} so that for any component P of ℍ^{2} − L, f̂(ℍ^{2}) lies in H_{P}^{+}.

Lemma 3.1. The set of convex and even pleated surfaces is a closed subset of the set of pleated surfaces.

Proof. Let (f̂_{n}, Γ_{n}, ρ_{n}) be a sequence of convex and even pleated surfaces converging to (f̂_{∞}, Γ_{∞}, ρ_{∞}). Let us follow [BoO, Lemmas 20 and 21], to show that f̂∞ is either a convex pleated surface or an even pleated surface. Let us denote by L̂_{n} the pleating locus of (f̂_{n}, Γ_{n}, ρ_{n}) and let us consider a geodesic lamination L̂_{∞} lying in the closure of {L̂_{n}/n ∈ ℕ} in the Hausdorff topology. By [CEG] the pleating locus of f̂_{∞} lies in L̂_{∞}. Let us consider a component P_{∞} of ℍ^{2} − L̂_{∞} and a component P_{n} of ℍ^{2} − L̂_{n} such that (P_{n}) tends to P_{∞}. Since f̂_{n}(P_{n}) converges to f̂_{∞}(P_{∞}), up to extracting a subsequence, H_{P_{n}}^{+} converges to a half-space H_{P_{∞}}^{+} such that f̂_{∞}(P_{∞}) ⊂ ∂H_{P_{∞}}^{+}. Since f̂_{n}(ℍ^{2}) converges to f̂_{∞}(ℍ^{2}), we have f̂_{∞}(ℍ^{2}) ⊂ H_{P_{∞}}^{+}. Doing this for any component of ℍ^{2} − L_{∞}, we conclude that f̂_{∞} satisfies 1).

Let x̂ be a point of ℍ^{2}, a support plane of f̂(ℍ^{2}) at f̂(x̂) is a hyperbolic plane Π_{f̂(x̂)} containing f̂(x̂) and such that f̂(ℍ^{2}) lies entirely in one of the two half-spaces bounded by Π_{f̂(x̂)}. We will denote this half-space by H_{f̂(x̂)}^{+}. Let k̂ ⊂ ℍ^{2} be a closed segment, a polygonal approximation to

f̂(k̂) is a finite family � = {(x̂_{i}, Π_{f̂(x̂_{i})})|i = 1, …, p} such that :

- the x̂_{i} are ordered points of k̂;

- Π_{f̂(x̂_{i})} is a support plane at f̂(x̂_{i});

- Π_{f̂(x̂_{i})} ∩ Π_{f̂(x̂_{i+1})} ≠ ∅ for any i = 1, …, p − 1;

- if x̂_{i−1} = x̂_{i} = x̂_{i+1}, then Π_{f̂(x̂_{i})} intersects the interior of H_{Π_{f̂(x̂_{i+1})}}^{+} − H_{Π_{f̂(x̂_{i−1})}}^{+} (the planes

Π_{f̂(x̂_{i})} are “ordered”);

- the projection of Π_{f̂(x̂_{i})} ∩ Π_{f̂(x̂_{i+1})} to f̂(ℍ^{2}) intersects the subarc of k̂ joining x̂_{i} to x̂_{i+1}.

The integer p is the length of the polygonal approximation. We will denote by θ(Π_{f̂(x̂_{i})}, Π_{f̂(x̂_{i+1})}) the internal angle of H_{Π_{f̂(x̂_{i})}}^{+} − H_{Π_{f̂(x̂_{i+1})}}^{+}.

The existence of a polygonal approximation to any arc f̂(k̂) intersecting at most once any leaf of f̂(L̂) is proved in [CEG].

The bending measure ∫k̂dλ̂ along k̂ is defined by ∫k̂dλ̂ = inf� ∑_{i=1}^{p} θ(Π_{f̂(x̂_{i})}, Π_{f̂(x̂_{i+1})}) where � runs over all polygonal approximations to f̂(k̂).

It is shown in [EpM, section 1.11] that this defines a transverse measure λ on the pleating locus of f.

A polygonal approximation � = {(x̂^{i}, Π^{i})} to a path f(k) is an (α, s)-approximation if

- max_{1≤i≤p} θ(Π_{i}, Π_{i+1}) < α and

- max d_{ℍ^{2}}(x̂_{i}, x̂_{i+1}) < s.

The existence of a (δ, ε)-approximation for any (δ, ε), and any arc k̂ is shown in [EpM]. In the sequel we will need to have (δ, ε)-approximations with bounded length. The following lemma shows there existence.

Lemma 3.2. Let δ and ε be two positive numbers such that 0 < ε < \frac{log 3}{2}. Let (f̂, Γ, ρ) be a convex pleated surface and let k̂ ⊂ ℍ^{2} be an arc that intersects the pleating locus |λ̂| transversely so that f̂(k̂) intersects at most once any leaf of f̂(|λ̂|). There is a (δ, ε)-approximation to f̂(k̂) with length less than \frac{4}{ε}(\frac{π}{δ} + 1)l(k) + 4(\frac{π}{δ} + 1) = B(ε, δ, l(k̂)).

Proof. Choose p + 2 points x̂_{0}, …, x̂_{p+1} in k̂ − |λ̂| such that we have \{x̂_{1}, x̂_{p+1}\} ⊂ ∂k̂, p ≤ \frac{l(k̂)}{ε} and d(x̂_{i}, x̂_{i+1}) ≤ ε for any i ∈ {0, …, p}. Choose also a support plane Π_{f̂(x̂_{i})} at f̂(x̂_{i}) for each i ∈ {0, …, p}. The first step of the proof will be to extend this family of support planes to obtain a polygonal approximation. There are three possible configuration for the positions of Π_{f̂(x̂_{i})} and Π_{f̂(x̂_{i+1})}. Let k̂_{i} be the subarc of k̂ joining x̂_{i} to x̂_{i+1}.

- First configuration : Π_{f̂(x̂_{i})} intersects Π_{f̂(x̂_{i+1})} and the projection of Π_{f̂(x̂_{i})} ∩ Π_{f̂(x̂_{i+1})} to f̂(ℍ^{2}) intersects the arc k̂_{i} ⊂ k̂. In this configuration, {(x̂_{i}, Π_{f̂(x̂_{i})}); (x̂_{i+1}, Π_{f̂(x̂_{i+1})})} is already a polygonal approximation to f̂(k̂_{i}).

- Second configuration : Π_{f̂(x̂_{i})} does not intersect Π_{f̂(x̂_{i+1})}. Let y be a point of k̂_{i} and let Π_{f̂(ŷ)} be a support plane at f̂(ŷ). The 3 half-spaces H_{f̂(ŷ)}^{−}, H_{f̂(x̂_{i})}^{−} and H_{f̂(x̂_{i+1})}^{−} intersect the ball

b(f̂(ŷ), ε) ⊊ b(f̂(ŷ), \frac{log 3}{2}). By [Ga] (see also [Br]) these 3 half-spaces are not disjoint, hence Π_{f̂(ŷ)} intersects either Π_{f̂(x̂_{i})} or Π_{f̂(x̂_{i+1})}. So any support plane at a point of f̂(k̂) intersects either Π_{f̂(x̂_{i})} or Π_{f̂(x̂_{i+1})}. The arc f̂(k̂_{i}) can be extended to an arc into the set of all the support planes at f̂(k̂_{i}). Therefore there is a point ŷ ⊂ int(k̂) and a support plane Π_{f̂(ŷ)} at f̂(ŷ) such that {(x̂_{i}, Π_{f̂(x̂_{i})}); (ŷ, Π_{f̂(ŷ)}); (x̂_{i+1}, Π_{f̂(x̂_{i+1})})} is a polygonal approximation to f̂(k̂_{i}) (cf. [KeS, §3.4]).

- Third configuration : Π_{f̂(x̂_{i})} intersects ∩Π_{f̂(x̂_{i+1})} but the projection of Π_{f̂(x̂_{i})} ∩ Π_{f̂(x̂_{i+1})} to f̂(ℍ^{2}) does not intersect f̂(k̂_{i}). If any support plane at a point of f̂(k̂_{i}) intersects either Π_{f̂(x̂_{i})} or Π_{f̂(x̂_{i+1})} then we can find, as in the preceding case, a point ŷ and a support plane Π_{f̂(ŷ)} such that {(x̂_{i}, Π_{f̂(x̂_{i})}); (ŷ, Π_{f̂(ŷ)}); (x̂_{i+1}, Π_{f̂(x̂_{i+1})})} is a polygonal approximation to f̂(k̂_{i}). Otherwise, there is a point ŷ ∈ k̂ and a support plane Π_{f̂(ŷ)} at f̂(ŷ) such that Π_{f̂(ŷ)} does not intersect Π_{f̂(x̂_{i})} nor Π_{f̂(x̂_{i+1})}. The planes Π_{f̂(x̂_{i})} and Π_{f̂(ŷ)} are in the second configuration, so there is a point ẑ ⊂ k̂_{i} between x̂_{i} and ŷ such that {(x̂_{i}, Π_{f̂(x̂_{i})}); (ẑ, Π_{f̂(ẑ)}); (ŷ, Π_{f(ŷ)})} is a polygonal approximation to the subarc of k̂ joining x̂_{i} to ŷ. Doing the same for Π_{f̂(ŷ)} and Π_{f̂(x̂_{i+1})} we get a polygonal approximation to f̂(k̂_{i}).

Let us do the construction above for all the components of k̂−{x̂_{1}, …, x̂_{p}}. In each component of k̂ − {x̂_{1}, …, x̂_{p}}, we have added at most 3 points. So the resulting polygonal approximation has a length smaller than 4p + 4. Let us denote by {(x̂_{i}, Π_{f̂(x̂_{i})})/i = 0, …, 4p + 4} this polygonal approximation and let us denote by k̂_{i} ⊂ k̂ the geodesic arc joining x̂_{i} to x̂_{i+1}. We have already seen that we can extend an arc f̂(k̂_{i}) to an arc in the set of all the support planes at f̂(k̂_{i}). This

implies that, for any δ, there are points ŷ_{j} ⊂ k̂_{i}, 0 ≤ j ≤ q + 1 with q ≤ \frac{θ(Π_{f̂(x̂_{i})},Π_{f̂(x̂_{i+1})})}{δ} ≤ \frac{π}{δ}, such that we have ŷ_{0} = x̂_{i}, ŷ_{q+1} = x̂_{i+1} and θ(Π_{f̂(ŷ_{j})}, Π_{f̂(ŷ_{j+1})}) ≤ δ. Choosing such points for

each arc k̂_{i}, 0 ≤ i ≤ 4p+3, we get a (δ, ε)-approximation with length smaller than (4p+4)(\frac{π}{δ} +1).

Since p ≤ \frac{l(k̂)}{ε}, this polygonal approximation satisfies the conclusion of the Lemma.

The following proposition of [KeS] gives an estimate of the error which is made when approximating the bending measure :

Proposition 3.3 (KeS, Proposition 4.8). There is a universal constant K, and a function s(α), 0 < s(α) < 1, such that if � is an (α, s(α))-approximation to a path f(k), where α < \frac{π}{2} , then we have

| ∑ ; �θ(Π^{i}, Π^{i+1}) − ∫_{k}dγ| < Kα l(k)

Now we will use this proposition to prove the continuity of the bending measured geodesic lamination of a converging sequence of convex pleated surfaces.

Lemma 3.4. Let (f̂_{n}, Γ_{n}, ρ_{n}) be a sequence of convex pleated surfaces converging to a pleated surface (f̂_{∞}, Γ_{∞}, ρ_{∞}) and let λ̂_{n} be the bending measured geodesic lamination of f̂_{n}. The sequence (λ̂_{n}) converges for the weak^{∗} topology to a measured geodesic lamination λ̂_{∞} and we have one of the following two situations :

- (f̂_{∞}, Γ_{∞}, ρ_{∞}) is a convex pleated surface and λ̂_{∞} is its bending measured geodesic lamination; - (f̂_{∞}, Γ_{∞}, ρ_{∞}) is an even pleated surface, |λ̂_{∞}| is the pleated locus of f̂_{∞} and λ̂_{∞} is obtained

by endowing each leaf of |λ̂_{∞}| with a Dirac mass with a weight equal to π.

Proof. Notice that when ℍ^{3}/ρ_{∞}(π_{1}(M)) is quasi-isometric to ℍ^{3}/ρ_{n}(π_{1}(M)) this result is a consequence of results of [Bo3]. By lemma 3.1, (f̂_{∞}, Γ_{∞}, ρ_{∞}) is an even or convex pleated surface. We will show that any subsequence of (f̂_{n}, Γ_{n}, ρ_{n}) contains a subsequence satisfying the conclusions of lemma 3.4, the conclusion follows.

Let us choose a subsequence such that (|λ̂_{n}|) converge in the Hausdorff topology to a geodesic lamination L̂_{∞}. By [CEG, §5.2], the pleating locus of f̂_{∞} lies in L̂_{∞}. Let k̂ ⊂ ℍ^{2} be an arc intersecting L̂_{∞} transversely such that f̂_{∞}(k̂) intersects at most once any leaf of f̂_{∞}(L̂_{∞}). Since (|λ̂_{n}|) tends to f̂(L̂_{∞}) in the Hausdorff topology and since (f̂_{n}) converge to f̂_{∞}, for n large enough f̂_{n}(k̂) intersects at most once any leaf of f̂(|λ̂_{n}|). Fix δ < \frac{π}{2} and ε < s(δ) and choose for each n a (δ, ε)-approximation (x̂_{i,n}, Π_{f̂_{n}(x̂_{i,n})}) to f̂_{n}(k̂) whose length is less than the number

B(δ, ε, l(k̂)) appearing in Lemma 3.2. Extract a subsequence such that for any i ≤ B(δ, ε, l(k̂)), the sequence (x̂_{i,n}, Π_{f̂_{n}(x̂_{i,n})}) converges and denote by (x̂_{i,∞}, Π_{f̂_{∞}(x̂_{i,∞})}) its limit. Thus we get a

(δ, ε)-approximation {(x̂_{i,∞}, Π_{f̂_{∞}(x̂_{i,∞})})} to f̂_{∞}(k̂) whose length is less than B(δ, ε, l(k̂)).

Claim 3.5. If (f̂_{∞}, Γ_{∞}, ρ_{∞}) is a convex pleated surface, then (λ̂_{n}) converge to the bending measured geodesic lamination λ̂_{∞} of (f̂_{∞}, Γ_{∞}, ρ_{∞}).

Proof. Assume that (f̂_{∞}, Γ_{∞}, ρ_{∞}) is a convex pleated surface and let us denote by λ̂_{∞} its bending measured geodesic lamination. Let us recall that λ̂_{n} tends to λ̂_{∞} if for any arc k̂ ⊂ ℍ^{2} transverse to |λ̂_{∞}|, (∫k̂dλ̂_{n}) tends to ∫k̂dλ̂_{∞}.

Let k ⊂ ℍ^{2} be an arc transverse to |λ̂_{∞}| and let {(x̂_{i,n}, Π_{f̂_{n}(x̂_{i,n})})} be the (δ, ε)-approximations defined above. For any i ≤ p, Π_{f̂_{n}(x̂_{i,n})} tends to Π_{f̂_{∞}(x̂_{i,∞})}. Therefore, for n large enough, we have ∑_{i=1}^{p} |θ(Π_{f̂_{∞}(x̂_{i,∞})}, Π_{f̂_{∞}(x̂_{i+1,∞})}) − θ(Π_{f̂_{n}(x̂_{i,n})}, Π_{f̂_{n}(x̂_{i+1,n})})| ≤ δ. It follows from Proposition

3.3 that we have | ∑_{i=1}^{q−1} θ(Π_{f̂_{n}(x̂_{i,n})}, Π_{f̂_{n}(x̂_{i+1,n})}) − ∫k̂dλ̂_{n}| < Kδl(k̂) for any n ∈ \overline{ℕ}. Hence we have

| ∫k̂dλ̂_{∞} − ∫k̂dλ̂_{n}| < 2Kδl(k̂) + δ. Letting δ tend to 0 yields ∫k̂dλ̂_{n} → ∫k̂dλ̂_{∞}. The same is true for any arc k̂ transverse to |λ̂_{∞}|, hence λ̂_{n} converges to λ̂_{∞}.

Next we will consider the case where (f̂_{∞}, Γ_{∞}, ρ_{∞}) is an even pleated surface. Let us first show that all the leaves of the pleating locus |λ̂_{∞}| of (f̂_{∞}, Γ_{∞}, ρ_{∞}) are isolated leaves, namely that their projections to ℍ^{2}/Γ_{∞} are isolated leaves.

Claim 3.6. When (f̂_{∞}, Γ_{∞}, ρ_{∞}) is an even pleated surface, the pleating locus |λ̂_{∞}| of f̂_{∞} contains only isolated leaves.

Proof. Choose two distinct successive points x̂_{i,∞} and x̂_{i+1,∞} and denote by k̂_{i} the subarc of k̂ joining x̂_{i,∞} to x̂_{i+1,∞}. Let ŷ be a point of int(k̂_{i}) and let (Π_{f̂_{n}(ŷ)}) be a sequence of support planes at f̂_{n}(ŷ) converging to a support plane Π_{f̂_{∞}(ŷ)} at f̂_{∞}(ŷ). For n large enough, ŷ lies in the subarc of k̂ joining x̂_{i,n} to x̂_{i+1,n}. Since {(x̂_{i,n}, Π_{f̂_{n}(x̂_{i,n})})} is a (δ, ε)-approximation, Π_{f̂_{n}(ŷ)} intersects both Π_{f̂_{n}(x̂_{i,n})} and Π_{f̂_{n}(x̂_{i+1,n})} and we have θ(Π_{f̂_{n}(x̂_{i,n})}, Π_{f̂_{n}(ŷ)}) ≤ δ and θ(Π_{f̂_{n}(ŷ)}, Π_{f̂_{n}(x̂_{i+1,n})}) ≤ δ. Letting n tend to ∞ yields θ(Π_{f̂_{∞}(ŷ)}, Π_{f̂_{∞}(x̂_{i,∞})}) ≤ δ and θ(Π_{f̂_{∞}(ŷ)}, Π_{f̂_{∞}(x̂_{i+1,∞})}) ≤ δ. Since

(f̂_{∞}, Γ_{∞}, ρ_{∞}) is an even pleated surface, the dihedral angle between two adjacent support planes lies in {0, π}. Hence we have θ(Π_{f̂_{∞}(ŷ)}, Π_{f̂_{∞}(x̂_{i,∞})}) = 0 and θ(Π_{f̂_{∞}(ŷ)}, Π_{f̂_{∞}(x̂_{i+1,∞})}) = 0. It follows

that for any arc κ̂ ⊂ int(k̂_{i}), i(κ̂, λ̂_{n}) tends to 0. From the proof of [Bo3, Prop 27] we deduce that int(k̂_{i}) does not intersect the pleating locus of f̂_{∞}.

So we have shown that the intersection between k̂ and the pleating locus of f̂_{∞} lies in {x̂_{i}| i = 1 … p}, in particular this intersection has a finite cardinal.

Endowing each leaf of |λ̂_{∞}| with a Dirac measure whose weight is equal to π yields a measured geodesic lamination λ̂_{∞}.

Let r be a lower bound for the set {d(x̂_{1}, x̂_{2})/x̂_{1} and x̂_{2} are two different points of k̂ ∩ |λ̂_{∞}|}. By the proof of Claim 3.6, we can choose r > 0. If {(x̂_{i,∞}, Π_{f̂_{∞}(x̂_{i,∞})})} is a (δ, ε)-approximation

satisfying ε < r, we have ∑_{i=1}^{q−1} θ(Π_{f̂_{∞}(x̂_{i,∞})}, Π_{f̂_{∞}(x̂_{i+1,∞})}) = ∫k̂dλ̂_{∞}.

Replacing the proposition 3.3 by this equality in the proof of Claim 3.5 we get the following claim which concludes the proof of Lemma 3.4 :

Claim 3.7. The sequence (λ̂_{n}) tends to λ̂_{∞} in ℳ ℒ (ℍ^{2}).

Let us notice that in the case where (f̂_{∞}, Γ_{∞}, ρ_{∞}) is even, we did not show that the projection of |λ̂_{∞}| to ℍ/Γ_{∞} is compact. We will see further in the text that this is actually true.

We will now improve the description of (f̂_{∞}, Γ_{∞}, ρ_{∞}) when it is an even pleated surface.

Lemma 3.8. Let (f̂_{n}, Γ_{n}, ρ_{n}) be a sequence of convex pleated surfaces converging to an even pleated surface (f̂_{∞}, Γ_{∞}, ρ_{∞}). There is a surface S with geodesic boundary, such that ℍ^{2}/Γ_{∞} is the double of S and such that the pleating locus |λ̂_{∞}| of f̂_{∞} project to ∂S ⊂ ℍ/Γ_{∞}. Furthermore ∂S is compact.

Proof. First the following claim will define S.

Claim 3.9. All the connected components of ℍ^{2} − |λ̂_{∞}| have the same image under f̂_{∞}.

Proof. Recall that since f̂_{∞} is a pleated surface, the restriction of f̂_{∞} to each connected component of ℍ^{2} − |λ̂_{∞}| is one to one.

Assume that Claim 3.9 is not true. There exist two connected components P̂ and P̂′ of ℍ^{2} − |λ̂_{∞}| whose closures intersect and whose images under f̂_{∞} are different. There is a point ŷ lying in the boundary of the closure of P̂ such that f_{∞}(ŷ) lies in f̂_{∞}(P̂′) or a point ŷ lying in the boundary of the closure of P̂′ such that f̂_{∞}(ŷ) lies in f_{∞}(P̂). This two cases are similar and we will only deal with the first one. Let us denote by l̂the leaf of |λ̂_{∞}| containing ŷ. By the proof of Claim 3.6, l̂ is an isolated leaf. Let k̂ ⊂ ℍ^{2} be a geodesic arc intersecting l̂ transversally so that k̂ ∩ |λ̂_{∞}| = \{y\}. For n ∈ \overline{ℕ}, let (x̂_{i,n}, Π_{f̂_{n}(x̂_{i,n})}) be the (δ, ε)-approximation to f̂_{n}(k̂) constructed

at the beginning of the proof of Lemma 3.4 and let us denote by p = B(δ, ε, l(k̂)) its length. By Claim 3.7, the bending measure on k̂ tends to a Dirac measure whose weight is equal to π and whose support is {y}. It follows that there is i such that (x̂_{i,n}) and (x̂_{i+1,n}) converge to y and such that we have the inequality \frac{π}{p} ≤ θ(Π_{f̂_{n}(x̂_{i,n})}, Π_{f_{n}(x̂_{i+1,n})}) ≤ δ for n large enough.

Let ẑ be the point of P̂′ such that we have f̂_{∞}(ẑ) = f̂_{∞}(y). For any n, ẑ lies either in a connected component P̂′_{n} of ℍ^{2}−|λ̂_{n}| or in the closures of two connected components of ℍ^{2}−|λ̂_{n}|, in this second case we will denote by P̂′_{n} the interior of the union of these two closures. Let us extract a subsequence such that P̂_{n} converges to an open subsurface P̂′_{∞} of P̂′ (for example a subsequence such that |λ̂_{n}| converges in the Hausdorff topology). Since f̂_{n} converge to f̂_{∞} on any compact set, the sequence f̂_{n}(P′_{n}) converges to f̂_{∞}(P′_{∞}) ⊂ f̂_{∞}(P′).

The point f̂_{∞}(ŷ) lies in f̂_{∞}(P̂′_{∞}), in Π_{f̂_{∞}(x̂_{i,∞})} and in Π_{f̂_{∞}(x̂_{i+1,∞})} and we have the inequality \frac{π}{p} ≤ θ(Π_{f̂_{n}(x̂_{i,n})}, Π_{f̂_{n}(x̂_{i+1,n})}) ≤ δ. Therefore, for n large enough, one plane among Π_{f̂_{n}(x̂_{i,n})}

and Π_{f̂_{n}(x̂_{i+1,n})} intersects f̂_{n}(P̂′_{n}) transversely. This contradicts the convexity of (f̂_{n}, Γ_{n}, ρ_{n}) and concludes this proof.

Next we will show that the quotient of ℍ^{2} − |λ̂_{∞}| by Γ_{∞} has two connected components. Let us extract a subsequence such that (|λ̂_{n}|) converges in the Hausdorff topology to a geodesic

lamination L̂_{∞} and let x̂, ŷ ⊂ ℍ^{2} − L̂_{∞} be two points whose images f̂_{∞}(x̂) and f̂_{∞}(ŷ) coincides. Denote by P̂x̂ and P̂ŷ the connected components of ℍ^{2} − |λ̂_{∞}| containing x̂ and ŷ and assume that H_{f̂_{∞}(P̂x̂)}^{+}is equal to H_{f̂_{∞}(P̂ŷ)}^{+}. We will show the existence of an element g_{∞} of Γ_{∞} such that x̂ = g_{∞}ŷ.

The half-spaces H_{f̂_{n}(P̂x̂)}^{+} and H_{f̂_{n}(P̂ŷ)}^{+} converge to H_{f̂_{∞}(P̂x̂)}^{+} and we have f̂_{n}(P̂ŷ) ⊂ H_{f̂_{n}(P̂x̂)}^{+}

and f̂_{n}(P̂x̂) ⊂ H_{f̂_{n}(P̂ŷ)}^{+}. It follows that the planes Π_{f̂_{n}(x̂)} and Π_{f̂_{n}(ŷ)} intersect for n large enough and that the sequence θ(Π_{f̂_{n}(x̂)}, Π_{f̂_{n}(ŷ)}) tends to 0. This implies that the distance d_{(Π_{f̂n(x̂)}∪Π_{f̂n(ŷ)})}(f̂_{n}(x̂), f̂_{n}(ŷ)) measured on Π_{f̂_{n}(x̂)} ∪ Π_{f̂_{n}(ŷ)} tends to 0. Since we have

d_{(Π_{f̂n(x̂)}∪Π_{f̂n(ŷ)})}(f̂_{n}(x̂), f̂_{n}(ŷ)) ≥ d_{f̂_{n}(ℍ^{2})}(f̂_{n}(x̂), f̂_{n}(ŷ)) and since f̂_{∞} is a local homeomorphism

on the complementary regions of |λ̂_{∞}|, there is, for n large enough, an isometry a_{n} ∈ Γ_{n} such that d_{ℍ^{2}}(a_{n}ŷ, x̂) ⟶ 0. The distance d_{ℍ^{2}}(ŷ, a_{n}ŷ) is bounded, hence, up to extracting a subsequence, (a_{n}) converges to some a_{∞} ∈ Γ_{∞}. We have then a_{∞}ŷ = x̂.

This implies that any point of f̂_{n}(ℍ^{2} − |λ̂_{∞}|) has at most two preimages in ℍ^{2}/Γ_{∞}. Combining this with Claim 3.9, we get the existence of a surface S such that ℍ^{2}/Γ_{∞} is the double of S and that the pleating locus |λ̂_{∞}| of f̂_{∞} projects to ∂S ⊂ ℍ^{2}/Γ_{∞}.

It remains to show that ∂S is compact. Assume the contrary, then ∂S contains two asymptotic half geodesics. Let k̂ ⊂ S be a geodesic arc joining these two half geodesic. The double of k̂ is a simple closed curve c ⊂ ℍ^{2}/Γ_{∞} bounding a cusp of ℍ^{2}/Γ_{∞}. The image of c under f_{∞} is f_{∞}(k̂) covered twice. It follows that the curve f_{∞}(c) is homotopic to a point in ℍ^{3}/ρ_{∞}(Γ_{∞}). This contradicts the assumption that the parabolic elements of Γ_{∞} are mapped to parabolic isometries by ρ_{∞}.

We will conclude this section with two lemmas which will be used in the next sections. They are proved in [Le1] (see also [Se]). We will call them slight bending Lemmas.

Lemma 3.10. Let (f̂, Γ, ρ) be a convex pleated surface, let x̂_{1} and x̂_{2} ∈ ℍ^{2}, let ĉ_{1} ⊂ ℍ^{2} be the geodesic segment joining x̂_{1} to x̂_{2} and let c̃_{2} ⊂ ℍ^{3} be the geodesic segment joining f̂(x̂_{1}) to f̂(x̂_{2}). If there exists ε < \frac{π}{2} such that the bending measure of f̂(ĉ_{1}) is smaller than ε, then ∃C_{ε} such that l(ĉ_{1}) ≤ C_{ε}l(c̃_{2}). Furthermore lim_{ε⟶0} C_{ε} = 0 and the sum of the exterior angles that f̂(ĉ_{1}) and c̃_{2} make at their vertices is smaller than ε.

Lemma 3.11 (Slightly bent curves are quasi-geodesics). Let (f̂, Γ, r) be a convex pleated surface, let λ̂ be its bending measured geodesic lamination, let c be a simple closed geodesic of ℍ^{2}/Γ and let c^{∗} be the geodesic of ℍ^{3}/r(Γ_{n}) in the homotopy class of f(c). For any ε < \frac{π}{2} there exists C_{ε} and A_{ε} such that if i(c, λ) ≤ ε then l(c) ≤ C_{ε}(l(c^{∗}) + A_{ε}). Moreover lim_{ε⟶0} C_{ε} = 1, and lim_{ε⟶0} A_{ε} = 0.

4 The continuity of b_{ℛ}

Next we will use the results of the previous section to show the continuity of b_{ℛ} . But first let us precise the definition of the bending measured lamination of a geometrically finite metric σ ∈ 𝒢 ℱ (M). Let ρ : π_{1}(M) → Isom(ℍ^{3}) be a representation associated to σ and let N(ρ) be the Nielsen core of ρ. There are a multi-curve λ^{(p)} and a natural (relative to σ) homeomorphism f : ∂_{χ<0}M − λ^{(p)} → ∂N(ρ). This homeomorphism f is constructed by using the retraction map from the domain of discontinuity to the boundary of the convex core of the limit set. It is well-defined up to isotopy. The surface ∂N(ρ) is the image of a convex pleated surface and therefore, it has a bending measured lamination. Let us denote by λ ∈ ℳ ℒ (∂M) the image under f^{−1} of this bending measured lamination. Adding the leaves of λ^{(p)} endowed with Dirac masses with weights equal to π we get the bending measured geodesic lamination of σ. This measured lamination λ does not depend on the choice of ρ (among the representations associated to σ).

Given a simple closed geodesic c and a hyperbolic metric s on ∂_{χ<0}M, let l_{s}(c) be the length of the corresponding s − geodesic. If γ is a weighted simple closed geodesic with a weight w(γ) we define l_{s}(γ) by l_{s}(γ) = w(γ)l_{s}(|γ|). This function l_{s} extends continuously to a function l_{s} : ℳ ℒ (∂M) → ℝ.

Given a simple closed curve c ∈ ∂_{χ<0}M, and a hyperbolic metric σ on int(M), we denote by c^{∗} the closed σ-geodesic in the free homotopy class of c and by l_{σ}(c^{∗}) its σ-length.

Proposition 4.1. Let (σ_{n}) be a sequence of geometrically finite metrics on the interior of M converging to a non fuchsian geometrically finite metric σ_{∞} and let λ_{n} be the bending measured geodesic lamination of σ_{n}. The sequence λ̇_{n} converges to λ̇ ∈ 𝒫^{+}(M)/ℛ and the bending measured geodesic lamination of σ_{∞} is the representative λ′ of λ̇ that lies in 𝒫(M).

Proof. We are going to show that any subsequence of (σ_{n}) contains a subsequence satisfying the conclusion of the proposition. Let us begin by a result about the curves whose length tends to 0 when n tends to ∞.

Lemma 4.2. Let (σ_{n}) be a sequence of geometrically finite metrics on the interior of M converging algebraically and let (s_{n}) be the sequence of metrics induced on ∂M by a homeomorphism h_{n} : M → N(ρ_{n})^{ep} associated to σ_{n}. Then, there is K > 0 such that if c ⊂ ∂M is a simple closed curves that bounds an essential disc, we have l_{s_{n}}(c) ≥ K.

Proof. Let D ⊂ M be an essential disc and let c ∈ ∂M be a simple closed curve which does not bound an essential disc. We will say that c intersects ∂D essentially if the ends of any lift of e to M̃ are separated by a lift of ∂D. As in the introduction 𝒫(M) contains the measured geodesic laminations satisfying conditions a), b) and c).

Claim 4.3. Let γ ⊂ 𝒫(M) be a weighted multi-curve and let D ⊂ M be an essential disc; then at least one leaf of γ intersects ∂D essentially.

Proof. Let ∂D̃ be a lift of ∂D to ∂M̃. This lift separates ∂\overline{M̃} ≈ S^{2} into two discs C and C′. Let us denote by I_{i}, 1 ≤ i ≤ p (resp. I′_{i}, 1 ≤ i ≤ p′) the connected components of p^{−1}(γ) − ∂D̃ lying in C (resp. C′) whose endpoints lie in ∂D̃. These arcs I_{i} (resp. I′_{i}) cut C (resp. C′) in p+1 discs C_{i}, 0 ≤ i ≤ p (resp. in p′+1 discs C′_{i}, 0 ≤ i ≤ p′). The curves ∂C_{i} and ∂C′_{i} ⊂ ∂M̃ bound essential discs in M̃. It follows from conditions a) and c) that for any i, we have ♯{∂C_{i} ∩ p^{−1}(γ)} ≥ 3 and ♯{∂C′_{i} ∩ p^{−1}(γ)} ≥ 3. If no leaf of γ intersects ∂D essentially, then, for any i, each point of ∂C_{i} ∩ ∂D̃ ∩ p^{−1}(γ) is the endpoint of an I′_{j} and each point of ∂C′_{i} ∩ ∂D̃ ∩ p^{−1}(γ) is the endpoint of an I_{j}. This implies that 3(p + 1) = 2p′ and that 3(p′ + 1) = 2p. We get p + p′ = −6. This contradiction shows that at least one leaf of γ intersects ∂D essentially.

Let (c_{n}) be a sequence of simple closed curves such that each c_{n} bounds an essential disc and that l_{s_{n}}(c_{n}) ⟶ 0. For large n, c_{n} is the core of a wide Margulis tube T_{n} ⊂ ∂M. Let c_{1,n} and c_{2,n} be the boundary components of T_{n} and let us choose T_{n} so that we have l_{s_{n}}(c_{1,n}) = l_{s_{n}}(c_{2,n}) ⟶ 0 and d_{s_{n}}(c_{i,n}, c_{n}) ⟶ ∞. Let γ ∈ 𝒫(M) be a weighted multi-curve and let us extract a subsequence such that a leaf e of γ intersects all the c_{n} essentially. For each n, let e_{n}^{∗} be the σ_{n}-geodesic in the homotopy class of e, let ẽ_{n}^{∗}, c̃_{n} and T̃_{n} be lifts of e_{n}^{∗}, c_{n}, T_{n} respectively. Let Π be a geodesic plane containing ẽ_{n}^{∗}, let c̃_{1,n} and c̃_{2,n} be the components of ∂T̃_{n}. Let k̃_{n} and k̃′_{n} be the two connected components of T̃_{n} ∩ Π and let κ̃_{n} and κ̃′_{n} be the geodesic segment of Π joining the endpoints of k̃_{n} and k̃′_{n}, see figure 1. Assume that ∂k̃_{n} and ∂k̃′_{n} does not lie in the preimage of λ_{n}. Let d̃_{n} and d̃′_{n} be the intersections of Π and of PSfrag replacements

k̃_{n} κ̃_{n}

c̃^{∗}
κ̃′_{n}

k̃′_{n} d̃_{n} d̃′_{n} α_{2} α_{1}

Figure 1: The section Π ∩ C(ρ_{n})

support planes at k̃_{n} ∩ c̃_{1,n} and at k̃′_{n} ∩ c̃_{1,n} respectively. The section C(ρ_{n}) ∩ Π lies between d̃_{n} and d̃′_{n}. Since ẽ_{n}^{∗} lies in C(ρ_{n}), ẽ_{n}^{∗} does not intersect d̃_{n} nor d̃′_{n}. The arcs κ̃_{n} and κ̃′_{n} are very close, this implies that the angle α_{1} between d̃_{n} and κ̃_{n} is small. The same considerations apply for the angle α_{2} on the other vertex of κ̃_{n}. Let α < \frac{π}{2} be an upper bound for α_{1} and α_{2} and set r_{n} = max{d(x̃, κ̃_{n})/x̃ ∈ k̃_{n}} ≤ max{d(x̃, ẽ_{n}^{∗})/x̃ ∈ k̃_{n}}. Any point x̃ lying in k̃_{n} lies in a simple closed curve c̃_{x,n} ⊂ T̃_{n} homotopic to c̃_{n} such that l_{s_{n}}(c_{x,n}) ≤ l_{s_{n}}(c_{1,n}). This curve c̃_{x,n} bounds a disc with diameter less than l_{s_{n}}(c_{x,n}) which intersects ẽ_{n}^{∗}. It follows that r_{n} ≤ l_{s_{n}}(c_{1,n}) ⟶ 0. A computation using Fermi coordinates (compare with [Le1, Lemme A.2]) yields l_{σ_{n}}(k̃_{n}) ≤ \sqrt{1 + tan^{2} α} (chr)^{2}l_{σ_{n}}(κ̃_{n}). Since the width of T_{n} tends to infinity, we have l_{σ_{n}}(k̃_{n}) ⟶ ∞. So we get l_{σ_{n}}(κ̃_{n}) ⟶ ∞. Since any point of κ_{n} is close to the geodesic e_{n}^{∗}, we have l_{σ_{n}}(e_{n}^{∗}) ⟶ ∞. But this contradicts the algebraic convergence of (σ_{n}).

Since (σ_{n}) converges to σ_{∞}, we may choose representations ρ_{n} : π_{1}(M) → Isom(ℍ^{3}) associated to σ_{n} such that (ρ_{n}) converges algebraically to a representation ρ_{∞} associated to σ_{∞} and that (ρ_{n}(π_{1}(M))) converges geometrically to ρ_{∞}(π_{1}(M)). For n ∈ \overline{ℕ}, let λ_{n} be the bending measured geodesic lamination of σ_{n}. Denote by λ_{n}^{(p)} the union of the leaves of λ_{n} which have a weight equal to π. Let M_{n} = ℍ^{3}/ρ_{n}(π_{1}(M)) and let h_{n} : M − λ_{n}^{(p)} → N(ρ_{n}) be a homeomorphism associated to σ_{n}.

Let x_{∞} be a point in ∂N(ρ_{∞}), let ∂_{x_{∞}}N(ρ_{∞}) be the connected component of ∂N(ρ_{∞}) containing x_{∞} and let S be a connected component of ∂M − λ_{∞}^{(p)} satisfying h_{∞}(S) = ∂_{x_{∞}}N(ρ_{∞}). Let x̃_{∞} ∈ C(ρ_{∞}) be a lift of x_{∞} and let ∂x̃_{∞}C(ρ_{∞}) be the connected component of ∂C(ρ_{∞}) containing x̃_{∞}. By [Ta], the convex hull C(ρ_{∞}) of L_{ρ_{∞}} is the limit of (C(ρ_{n})) with respect to the Hausdorff topology. Hence there exists x̃_{n} ⊂ ∂C(ρ_{n}) such that (x̃_{n}) converges to x̃_{∞}. Let us denote by ∂x̃_{n}C(ρ_{n}) the connected component of ∂C(ρ_{n}) containing x̃_{n} and by ∂x̃_{n}N(ρ_{n}) its projection to N(ρ_{n}). There is a connected component S_{n} of ∂M − λ_{n}^{(p)} and a pleated map g_{n} : S_{n} → M onto ∂x̃_{n}C(ρ_{n})/ρ_{n}(π_{1}(M)). Let (ĝ_{n}, Γ_{n}, r_{n}) be the pleated surface that lifts g_{n}, namely the map ĝ_{n} : ℍ^{2} → ∂x̃_{n}C(ρ_{n}) is onto, r_{n} : Γ_{n} → ρ_{n}(π_{1}(M)) is onto the stabilizer of ∂x̃_{n}C(ρ_{n}) and the quotient map ℍ^{2}/ ker(r_{n}) → ∂x̃_{n}C(ρ_{n}) is a homeomorphism. By [CEG, Theorem 5.2.2] and Lemma 4.2 a subsequence of the sequence of pleated surfaces (ĝ_{n}, Γ_{n}, r_{n}) converges to a pleated surface (ĝ_{∞}, Γ_{∞}, r_{∞}).

By Lemma 3.1, (ĝ_{∞}, Γ_{∞}, r_{∞}) is a convex or even pleated surface. Since x̃_{∞} ∈ ĝ_{∞}(ℍ^{2}), then ĝ_{∞}(ℍ^{2}) ⊂ ∂x̃_{∞}C(ρ_{∞}) and r_{∞}(Γ_{∞}) ⊂ ρ_{∞}(π_{1}(M)) is a subgroup of the stabilizer of ∂x̃_{∞}C(ρ_{∞}).

Lemma 4.4. The quotient map g_{∞} : ℍ^{2}/Γ_{∞} → ∂_{x_{∞}}N(ρ_{∞}) is a homeomorphism.

Proof. Set Cĝ_{n} = ⋂{H_{P_{i}}^{ϵ}/P_{i} is a connected component of ℍ^{2} − L̂}. For any n ∈ ℕ, we have C(ρ_{n}) ⊂ Cĝ_{n}. Since σ_{n} converges to σ_{∞}, it follows from [Ta] that C(ρ_{n}) converges to C(ρ_{∞}) in the Hausdorff topology. So we get C(ρ_{∞}) ⊂ Cĝ_{∞}. Since ρ_{∞} is not fuchsian, we have int(C(ρ_{∞})) ≠ ∅. Therefore int(Cĝ_{∞}) ≠ ∅ and (ĝ_{∞}, Γ_{∞}, r_{∞}) is a convex pleated surface. By [EpM], this implies that g_{∞} is a covering map.

Let γ̂_{n} be the bending geodesic lamination of ĝ_{n}. Extract a subsequence such that (|γ̂_{n}|) converges to some geodesic lamination L̂_{∞} in the Hausdorff topology.

Assume that g_{∞} is not a homeomorphism. There are two points z ≠ y ∈ (ℍ^{2} − L̂_{∞})/r_{∞}(Γ_{∞}) such that g_{∞}(z) = g_{∞}(y). If we lift the situation to ℍ^{2}, we get ẑ, ŷ ⊂ ℍ^{2} − L̂_{∞} and ρ_{∞}(a) ∈ ρ_{∞}(π_{1}(M)) such that ẑ ∉ Γ_{∞}ŷ and that ĝ_{∞}(ẑ) = ρ_{∞}(a) ∘ ĝ_{∞}(ŷ). The sequences z̃_{n} = ĝ_{n}(ẑ) and ỹ_{n} = ρ_{n}(a) ∘ ĝ_{n}(ŷ) converge together to ĝ_{∞}(ẑ). Let Πz̃_{n} and Πỹ_{n} be support planes of C_{n}(ρ_{n}) at z̃_{n} and ỹ_{n}. The half-spaces (H_{Πz̃n}^{+} ) and (H_{Πỹn}^{+} ) tend to half-spaces H+ ; ẑ and H+ ; ŷ respectively. These two half-spaces H+ ; ẑ and H+ ; ŷ are bounded by the support plane Π_{ĝ_{∞}(ẑ)} at ĝ_{∞}(ẑ). Thus we

have either H+ ; z̃ ∩ H+ ; ỹ = Π_{ĝ_{∞}(ẑ)}, or H+ ; z̃ = H+ ; ỹ.

Since C(ρ_{∞}) ⊂ Cĝ_{∞} ⊂ H+ ; z̃_{n} ∩ H+ ; ỹ_{n} and since ρ_{∞}(π_{1}(M)) is not fuchsian, we have H+ ; z̃ ∩ H+ ; ỹ ≠ Π_{ĝ_{∞}(ẑ)}. So we have H+ ; z̃ = H+ ; ỹ. This implies that for large n, if Πz̃_{n} are Πỹ_{n} disjoint then we have either H_{Πz̃n}^{+} ⊊ H_{Πỹn}^{+} or H_{Πỹn}^{+} ⊊ H_{Πz̃n}^{+} . We get a contradiction with the fact that we have ỹ_{n} ⊂ C(ρ_{n}) ⊂ H_{Πz̃n}^{+} and z̃_{n} ⊂ C(ρ_{n}) ⊂ H_{Πỹn}^{+} . We deduce that, up extracting a subsequence, Πz̃_{n} intersects Πỹ_{n}. It follows that, for n large enough, ĝ_{n}(ℍ^{2}) intersects ρ_{n}(a) ∘ ĝ_{n}(ℍ^{2}). Therefore ĝ_{n}(ℍ^{2}) and ρ_{n}(a) ∘ ĝ_{n}(ℍ^{2}) coincide. Since H_{Πỹ}^{+} is equal to H_{Πz̃}^{+}, the dihedral angle θ(Πẑ_{n}, Πŷ_{n}) tends to 0. This implies that the distance between z̃_{n} and ỹ_{n} measured on ĝ_{n}(ℍ^{2}) tends to 0. Since ĝ_{∞} is a covering map, there is a neighborhood 𝒱 (ẑ) ⊂ ℍ^{2} of ẑ such that, for n ∈ \overline{ℕ} large enough, the map ĝ_{n|𝒱 (ẑ)} : 𝒱 (ẑ) → ĝ_{n}(𝒱 (ẑ)) is a homeomorphism. For n large enough, ŷ_{n} lies in g_{n}(𝒱 (ẑ)). Hence there is ŷ′_{n} ∈ 𝒱 (ẑ) such that ĝ_{n}(ŷ′_{n}) = ỹ_{n}. Since ĝ_{n}(ŷ′_{n}) = ρ_{n}(a) ∘ ĝ_{n}(ŷ) and since r_{n}(Γ_{n}) is the sabilizer of ĝ_{n}(ℍ^{2}) in ρ_{n}(π_{1}(M)), there is a_{n} ∈ Γ_{n} such that a_{n}ŷ = ŷ′_{n}. The point ŷ′_{n} lies in 𝒱 (ẑ), so a_{n} moves ŷ a bounded distance. Therefore there is a subsequence such that (a_{n}) converge to an isometry a_{∞} ∈ Γ_{∞}. Moreover ĝ_{n}(ŷ′_{n}) tends to ĝ_{∞}(ẑ), hence (ŷ′_{n}) tend to ẑ. Thus we get a_{∞}ŷ = ẑ. This yields a contradiction with the assumption that ẑ ∉ Γ_{∞}ŷ and concludes the proof of Lemma 4.4.

Let F be a compact subset of S such that the connected components of S − F are infinite annuli. Since Γ_{n} converges geometrically to Γ_{∞}, by Mumford Lemma (cf. [CEG]) there are ε > 0 and maps l_{n} : F → ℍ^{2}/Γ_{n} with the following properties:

- l_{n} is a homeomorphism onto its image;

- l_{n}(F) is a connected component of the ε-thick part of ℍ^{2}/Γ_{n};

- the induced representations l_{n∗} : π_{1}(F) → Γ_{n} converge to an isomorphism l_{∞∗} : π_{1}(F) → Γ_{∞};

- g_{∞} ∘ l_{∞} coincides with h_{∞} on F ⊂ ∂M.

Let us show that, for large n, g_{n} ∘ l_{n} : F → ∂N(ρ_{n}) is isotopic to h_{n}. Let F̂ be the universal cover of F and let l̂_{n} : F̂ → ℍ^{2} be a lift of l_{n}. Since (l_{n∗}) converges algebraically, we can choose the l̂_{n} such that they converge to l̂_{∞} on compact sets. Since (ĝ_{n}) converges to ĝ_{∞}, the sequence (ĝ_{n} ∘ l̂_{n}) converges to ĝ_{∞} ∘ l̂_{∞} on compact sets.

Let x ∈ int(M). Since (σ_{n}) converges to (σ_{∞}), there are diffeomorphisms u_{n} : M → M isotopic to the identity which satisfie the following : for any k > 1 and r > 0, there exists n(k, r) such that for n ≥ n(k, r), the restriction of u_{n} to B(x, r) ⊂ (M, σ_{∞}) is a k-quasi-isometry into its image in (M, σ_{∞}). For n ∈ \overline{ℕ}, set M_{n} = ℍ^{3}/ρ_{n}(π_{1}(M)) and let x_{n} ∈ M_{n} be the projection of the origin o ∈ ℍ^{3}. For n ∈ \overline{ℕ}, the metric σ_{n} yields an identification between (int(M), σ_{n}) and M_{n} such that x is identified with x_{n}. Thus we can consider the restriction of u_{n} to int(M) as an homeomorphism u_{n} : M_{∞} → M_{n}.

Let ũ_{n} : ℍ^{3} → ℍ^{3} be a lift of u_{n} such that ũ_{n}(o) = o. Since (ρ_{n}(π_{1}(M))) converges geometrically to ρ_{∞}(π_{1}(M)), ũ_{n} : ℍ^{3} → ℍ^{3} converges to the identity on compact sets (see [BeP]).

Since (ĝ_{n} ∘ l̂_{n}) converges to ĝ_{∞} ∘ l̂_{∞} on compact sets, the sequence ũ_{n}^{−1} ∘ ĝ_{n} ∘ l̂_{n} converges to ĝ_{∞} ∘ l̂_{∞} on compact sets. Furthermore, F has a compact fondamental domain. Hence u_{n}^{−1} ∘ g_{n} ∘ l_{n} : F → M_{∞} converges uniformly to g_{∞} ∘ l_{∞} which coincides with h_{∞}. It follows that for large n, u_{n}^{−1} ∘ g_{n} ∘ l_{n} is isotopic to h_{∞|F}. Therefore g_{n} ∘ l_{n} is isotopic to u_{n} ∘ h_{∞|F}. Since u_{n} is isotopic to the identity, u_{n} ∘ h_{∞|F} is isotopic to h_{n}. Thus we can change l_{n} by an isotopy such that g_{n} ∘ l_{n} coincides with h_{n} on F.

Let γ̂_{n} be the bending measured geodesic lamination of ĝ_{n} and let γ_{n} ∈ ℳ ℒ (ℍ^{2}/Γ_{n}) be its projection. Let k ⊂ F be an arc such that l_{∞}(k) is transverse to |γ_{∞}| and let k̂ ⊂ F̂ be a lift of k. The arcs l̂_{n}(k̂) converge to l̂_{∞}(k̂). Since the convex pleated surfaces (ĝ_{n}, Γ_{n}, r_{n}) converge to (ĝ_{∞}, Γ_{∞}, r_{∞}), Lemma 3.4 yields ∫_{l̂_{n}(k̂)} dγ̂_{n} ⟶ ∫_{l̂_{∞}(k̂)} dγ̂_{∞}. Since, for any n ∈ \overline{ℕ}, g_{n} ∘ l_{n} coincides with h_{n} on F, we have ∫_{k} dλ_{n} = ∫_{l̂_{n}(k̂)} dγ̂_{n} where λ_{n} is the bending measured geodesic lamination of σ_{n}. Thus we get ∫_{k} dλ_{n} ⟶ ∫_{k} dλ_{∞}.

Doing the same for each component of ∂M −λ_{∞}^{(p)}, we conclude that for any arc k ⊂ ∂M −λ_{∞}^{(p)} transverse to |λ_{∞}|,we have ∫_{k} dλ_{n} ⟶ ∫_{k} dλ_{∞}. It follows that (λ̇_{n}) converges in ℳ ℒ (∂M)/ℛ and that the limit λ̇ differs from λ̇_{∞} only on λ^{(p)}, namely if λ is a representative of |λ̇| then removing from λ the closed leaves which lie in λ^{(p)} yields the same measured geodesic lamination as removing λ^{(p)} from λ_{∞}. Thus, if we show that the leaves of λ^{(p)} are leaves of λ with a weight greater than or equal to π, we can conclude that λ̇_{n} converges to λ̇_{∞}.

Claim 4.5. Let λ be a representative of λ̇; any leave c of λ^{(p)} is a leave of λ and has a weight greater than or equal to π (as a leaf of λ).

Proof. Let c be a leaf of |λ^{(p)}| ⊂ |λ_{∞}|. Since λ̇_{n} converges to λ̇ which satisfies |λ̇| ⊂ |λ_{∞}|, then either c is a leaf of λ and we will denote its weight by w(c), or c is a simple closed curve disjoint from λ and we will take w(c) = 0.

We will prove the claim by assuming that w(c) < π and by ending in a contradiction.

Let S be a component of ∂M − λ^{(p)} whose closure contains c. From the fact that r_{n}(π_{1}(S)) converges geometrically to r_{∞}(π_{1}(S)) we deduce that l_{s_{n}}(c) ⟶ 0.

In the case where, up to extracting a subsequence, i(c, λ_{n}) is equal to 0 for all n, it follows from [BoO, Lemma 19] that c is a leaf of λ with a weight equal to π.

Let us consider the other case and let us extract a subsequence such that i(c, λ_{n}) > 0. Let η > 0 be a number such that π − η is less than w(c). The curve c lies in the boundaries of two surfaces F^{1} and F^{2} (which may coincide) such that the following holds : c ⊂ int(F^{1} ∪ F^{2}) and either F^{1} (resp. F^{2}) is a pair of pants satisfying int(F^{1}) ∩ |λ| = ∅ or there is connected component of λ which is an arational geodesic lamination in F^{1} (resp. F^{2}). We will only deal with the case where F^{1} and F^{2} are distinct, the other case is handled in the same way. Let d be a simple closed geodesic intersecting c in two points such that i(d, λ − c) ≤ η. We have i(c, λ) ≤ 2π − η. Consider a point x_{n}^{1} (respectively x_{n}^{2}) of d ∩ F^{1} − λ_{n} (respectively d ∩ F^{2} − λ_{n}) lying in the thick part of (F^{1}, s_{n}) (respectively (F^{2}, s_{n})). We may choose the x_{n}^{j} so that if k_{n}^{1} and k_{n}^{2} are the connected components of d − \{x_{n}^{1}, x_{n}^{2}\} then we have ∫_{k_{n}^{j}} dλ ⟶ \frac{i(d,λ)}{2}. The existence of such points x_{n}^{j} comes from the fact that in a neighborhood 𝒱 (c) of c, λ_{n} ∩ 𝒱 (c) is a family of compact arcs each one spiraling many times toward c (because c lies in any limit of λ_{n} for the Hausdorff topology, see Claim 2.2) and carrying a small measure (since, by Claim 2.4, we have i(λ_{n}, c) ⟶ 0)) Let y_{n}^{j} ⊂ k_{n}^{j} be a point such that if κ_{n}^{j} is the segment of k_{n}^{j} joining x_{n}^{1} and y_{n}^{j} (cf. figure 2), then we have ∫_{κ_{n}^{j}} dλ_{n} ⟶ \frac{i(d,λ)}{4}. Let κ_{n}^{3} and κ_{n}^{4} be the segments joining x_{n}^{1} to

y_{n}^{2} and y_{n}^{1} to x_{n}^{2} respectively. We have ∫_{κ_{n}^{j}} dλ_{n} ⟶ \frac{i(c,λ)}{4} ≤ \frac{π}{2} − \frac{η}{4} for any j ∈ \{1, 2, 3, 4\}. For j ∈ {1, 2, 3, 4}, consider a lift κ̃_{n}^{j} ⊂ C(ρ_{n}) ⊂ ℍ^{3} of κ_{n}^{j} such that ⋃_{j} κ̃_{n}^{j} is connected. Let d̃_{n}^{j} ⊂ ℍ^{3} PSfrag replacements

κ_{n}^{1}

κ_{n}^{2}
κ_{n}^{3}
κ_{n}^{4}
x_{n}^{1}
x_{n}^{2}
y_{n}^{1}
y_{n}^{2}
λ_{n}

Figure 2: Cutting c

be the geodesic segment joining the vertices of κ̃_{n}^{j}.

If there is j_{0} such that l_{s_{n}}(κ_{n}^{j_{0}}) is bounded, then κ_{n}^{j_{0}} is entirely contained in the thick part of (F^{1} ∪ F^{2}, s_{n}). Since f_{n}(F^{1}) and f_{n}(F^{2}) tend to pleated surfaces whose bending measured geodesic laminations are λ ∩ F^{j}, we have lim_{n⟶∞} ∫_{κ_{n}^{j_{0}}} dλ_{n} ≤ i(d, λ − c) ≤ η. Furthermore,

we have ∫_{κ_{n}^{j_{0}}} dλ_{n} ⟶ \frac{i(c,λ)}{4}, hence i(c, λ) ≤ 4η. Since d intersects c transversally, and since l_{s_{n}}(d) ⟶ 0, l_{s_{n}}(d) tends to ∞. Taking η < \frac{π}{8} , we get from the slight bending lemmas (Lemma 3.11) that l_{σ_{n}}(d^{∗}) ≥ C_{4η}(l_{s_{n}}(d) + A_{4η}). Thus l_{σ_{n}}(d^{∗}) tends to ∞ contradicting the algebraic convergence of ρ_{n}.

Thus we have ∀j, l_{s_{n}}(κ_{n}^{j}) ⟶ ∞. The slight bending Lemmas (Lemma 3.10) says that there is C > 0 such that l_{s_{n}}(κ̃_{n}^{j}) ≤ Cl_{σ_{n}}(d̃_{n}^{j}) and that the interior angle between two adjacent segments d̃_{n}^{j} is greater than \frac{η}{2}. Let a be the element of π_{1}(M) such that ρ_{n}(a) fixes d^{∗} and let d̃_{n} = ⋃_{i∈ℤ;j=1,2,3,4} a^{i}(κ_{n}^{j}). The curve d̃_{n} is the union of long geodesic segments with interior angles greater than \frac{η}{2}. It follows from a classical result (see [Ot2] for example) that there is K such that l(d̃_{n}/a) is smaller than Kl_{σ_{n}}(d^{∗}). Thus we get l_{s_{n}}(d) ≤ Cl(d̃_{n}/a) ≤ CKl_{σ_{n}}(d^{∗}). Since l_{s_{n}}(d) ⟶ ∞, this yields a contradiction with the algebraic convergence of σ_{n}.

It follows that λ̇ = λ̇_{∞}. Thus Proposition 4.1 is proved.

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