CONVERGENCE OF FREELY DECOMPOSABLE KLEINIAN GROUPS

INKANG KIM, CYRIL LECUIRE AND KEN’ICHI OHSHIKA

Abstract. We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressible, then the sequence has compact closure in the deformation space. As a consequence we generalise Thurston’s double limit theorem and solve his conjecture on convergence of function groups affirmatively.

Contents

1. Introduction  2
2. Preliminaries  6
2.1. Deformation space  6
2.2. ℝ-trees  6
2.3. Geodesic laminations  7
2.4. Some notations  8
2.5. Train tracks and their realisations  8
2.6. Compactification of M̃  9
2.7. Geodesic laminations on compressible surfaces  11
3. Realisations of doubly incompressible laminations  13
4. Mapping the train tracks to ϵ_{n}ℍ^{3}  19
5. Finding backtracking  24
6. Commuting elements  35
7. Conclusion  41
8. Appendix  42
References  45

12000 Mathematics Subject Classification. 51M10, 57S25. 2Key words and phrases. Doubly incompressible laminations, algebraic convergence,

action on ℝ−tree, Thurston compactification of Teichmüller space.

3The first author gratefully acknowledges the partial support of NRF Grant (NRF-2010-0024171).

1. Introduction

It is one of the most important topics in the theory of Kleinian group to study the topological structure of their deformation spaces. The quasiconformal deformation space of a geometrically finite Kleinian group G is fairly well understood by virtue of the work of Ahlfors, Bers, Kra, Marden and Sullivan, and others. To put it more concretely, for a geometrically finite Kleinian group G, it is known that there is a ramified covering map from the Teichmüller space of Ω_{G}/G to the quasi-conformal deformation space of G, where Ω_{G} denotes the region of discontinuity of G. On the other hand, in general it is difficult to understand how these coordinates on Teichmüller space relate to the full deformation space. In other words, it is a challenging task to determine which divergence sequence in Teichmüller space correspond to convergent sequences of Kleinian groups.

The first example of such a sufficient condition for convergence is the result of Bers in [Ber], which shows that the space of quasi-Fuchsian groups lying on a Bers slice is relatively compact. On the other hand, in the process of proving the uniformisation theorem for Haken manifolds, Thurston proved the double limit theorem for quasi-Fuchsian groups and the compactness of deformation spaces for acylindrical manifolds, in [Th1] and [Th2] respectively. These are generalised to give a convergence theorem for general freely indecomposable Kleinian groups in Ohshika [Oh1] and [Oh2]. The convergence in the deformation spaces for freely decomposable groups is more complicated and is harder to understand.

In [ThB], Thurston asked how one might generalise the double limit theorem to the setting of Schottky groups. This question was made into a more concrete conjecture using the notion of Masur domain, and then was generalised to function groups. Masur introduced in [Ma] an open set in the projective lamination space of the boundary of a handlebody on which the mapping class group of the handlebody acts properly discontinuously. This open set is what we call the Masur domain nowadays. This notion is generalised by Otal [Ot1] to the exterior boundary of a compression body. Thurston’s conjecture is paraphrased as follows: For a sequence in Teichmüller space converging in the Thurston compactification to a projective lamination lying in the Masur domain of the exterior boundary of a compression body M, the corresponding sequence of convex cocompact representations in AH(M) has a convergent subsequence. Otal in [Ot2] first proved that Thurston’s conjecture is true for rank-2 Schottky space provided that the limit lamination is arational, that is, any component of its complement is simply connected. Canary in [Ca] proved the conjecture for some special sequences in Schottky space. Ohshika in [Oh4] proved the conjecture for function groups which are isomorphic to the free products of two surface groups under the same assumption that the limit lamination is arational. The strongest result in this direction under the same assumption on the limit lamination was given by Kleineidam and Souto in [KlS] without any other assumption on compression bodies. Our main result, Theorem 1 yields a proof of this conjecture of Thurston in full generality without any extra assumption and generalises it to a slightly larger set than the Masur domain.

We need to introduce some notions and notations to state our main theorem. Consider a compact irreducible atoroidal 3-manifold M with boundary. By Thurston’s uniformisation theorem for atoroidal Haken manifolds, there is a representation ρ_{0} : π_{1}(M) → Isom(ℍ^{3}) such that ℍ^{3}/ρ_{0}(π_{1}(M)) is homeomorphic to Int(M) by a homeomorphism that induces ρ. Such a representation is said to uniformise M. Any quasi-conformal deformation of ρ_{0} also uniformises M. By the Ahlfors-Bers theory, when ρ_{0} is convexcocompact, the space QH(ρ_{0}) of quasi-conformal deformations of ρ_{0} up to conjugacy by elements of Isom(ℍ^{3}) is parametrised by the Teichmüller space of the boundary of M. More precisely, there is a (possibly ramified) covering map, called the Ahlfors-Bers map T (∂M) → QH(ρ_{0}) whose covering transformation group is the group of isotopy classes of diffeomorphisms of M which are homotopic to the identity.

The space QH(ρ_{0}) is a subspace of the deformation space AH(M). This deformation space AH(M) is the space of discrete faithful representations ρ : π_{1}(M) → Isom(ℍ^{3}) up to conjugacy by elements of P SL_{2}(ℂ). It is endowed with the quotient of the compact-open topology which is also called the algebraic topology. In the main theorem, we shall consider sequences of representations given by sequences in the Teichmüller space whose images under the Ahlfors-Bers map diverge in QH(ρ_{0}) and give a sufficient condition for their convergence in AH(M).

Thurston introduced in [Th3] the notion of doubly incompressible curves. This can be extended to measured geodesic laminations in the following way.

We say that a measured geodesic lamination λ ∈ ML(∂M) is doubly incompressible if and only if there exists η > 0 such that i(λ, ∂E) > η for any essential annulus or disc E in M. We denote by D(M) ⊂ ML(∂M) the set of doubly incompressible measured geodesic laminations and by PD(M) its projection in the projective lamination space PML(∂M). It is not hard to see that D(M) contains the Masur domain (see [Le2]). Our main theorem is the following.

Theorem 1. Let M be a compact orientable irreducible atoroidal 3-manifold with boundary and ρ_{0} : π_{1}(M)→P SL(2, ℂ) a convex cocompact representation that uniformises M. Let (m_{n}) be a sequence in the Teichmüller space T (∂M) that converges in the Thurston compactification to a projective measured lamination [λ] contained in PD(M). Let q : T (∂M) → QH(ρ_{0}) be the Ahlfors-Bers map, and suppose that (ρ_{n} : π_{1}(M)→G_{n} ⊂ P SL(2, ℂ)) is a sequence of discrete faithful representations corresponding to q(m_{n}). Then (ρ_{n}) has an algebraically convergent subsequence in AH(M).

It should be noted that our result here is closely related to the Bers-Thurston density conjecture. This conjecture states that every finitely generated Kleinian group is contained in the boundary of the quasi-conformal deformation space of geometrically finite Kleinian groups without rank-1 maximal parabolic subgroups. A special case has been proved by Bromberg and Brock-Bromberg [Brom, BB] using cone manifold deformation theory. Bromberg and Souto announced a complete prof of the general case using this approach and, in particular, avoiding an appeal to the ending lamination theorem ([BS]). The general case can be proved by combining the resolution of the tameness conjecture by Agol and Calegari-Gabai, the ending lamination conjecture by Brock-Canary-Minsky, and from some convergence theorems due to Thurston, Ohshika, Kleineidam-Souto and Lecuire, together with some topological argument due to Ohshika and Namazi-Souto(See [Ag], [CG], [Th3], [Oh4], [KlS], [Le2], [Min], [Oh5] and [NaS]).

Theorem 1 is a corollary of Theorem 2 below by the following argument. By Theorems of Thurston [Th2] and Canary [Ca], the convergence of (m_{n}) to [λ] implies that there is a sequence of weighted multi-curves (λ_{n} ∈ ML(∂M)) such that l_{ρ_{n}}(λ_{n}) tends to 0 and that the sequence (λ_{n}) converges in ML(∂M) to a measured geodesic lamination whose projective class is [λ]. Since λ lies in D(M), Theorem 1 is derived from the following theorem, whose proof occupies the main part of this paper.

Theorem 2. Let (ρ_{n} : π_{1}(M) → Isom(ℍ^{3})) be a sequence of convex cocompact representations that uniformise M and let (λ_{n}) ⊂ ML(∂M) be a sequence of measured geodesic laminations such that (l_{ρ_{n}}(λ_{n})) is a bounded sequence and that (λ_{n}) converges in ML(∂M) to a measured geodesic lamination λ ∈ D(M). Then the sequence (ρ_{n}) has a compact closure in AH(M), namely, any subsequence contains an algebraically convergent subsequence.

We prove Theorem 2 by contradiction using the following arguments. We can first assume that λ_{n} is a weighted simple closed curve since the set of weighted simple closed curves is dense in ML(S). Assuming that no subsequence of (ρ_{n}) converges, we use Culler-Morgan-Shalen compactification of the character variety ([MoS1]) and ideas of Otal [Ot3] to construct a sequence of train tracks τ_{n} carrying λ_{n} and maps f_{n} : τ_{n} → ℍ^{3}/ρ_{n}(π_{1}(M)) so that some parts of τ_{n} are mapped to long quasi-geodesic paths while the rest is mapped to relatively short paths. Then we construct a simplicial annulus joining f_{n}(λ_{n}) to its geodesic representative λ_{n}^{∗} ⊂ M_{n} = ℍ^{3}/ρ_{n}(π_{1}(M)). By Gauss-Bonnet Formula, this annulus is not large enough to cover the difference in length between f_{n}(λ_{n}) and λ_{n}^{∗}. It follows that f_{n}(λ_{n}) nearly backtracks along some long paths. We use this backtracking to construct a sequence of annuli whose boundary converges to a sublamination of λ. This contradicts the double incompressibility of λ.

We state Theorem 2 under the assumption that ρ_{n} is convex cocompact but it should hold without this assumption, namely when ρ_{n} is discrete and faithful. The presence of parabolics, in particular rank 2 cusps, would add extra technicalities to the proof that we wished to avoid. The presence of geometrically infinite ends would require very little change. In the same spirit, we state Theorem 1 under the assumption that ρ_{n} is convex cocompact but it should hold under the assumption that it is geometrically finite and minimally parabolic.

Plan of the paper:

In Section 2, we explain background materials and quote necessary results, some with proofs.

In Sections 3 and 4 we explain the construction of τ_{n} and h_{n}. In Section 3, we show that for a doubly incompressible measured lamination λ and a small minimal action of π_{1}(M) on ℝ-tree T , each component of λ is either realised in T , or is carried by train tracks with arbitrarily short branches in the following sense: there is a sequence of train tracks θ_{i} minimally carrying the component, such that the branches of θ_{i} are mapped to geodesic segments in T with lengths going to zero. This is proved in Lemma 3.1.

Using this result, we construct τ_{n} and f_{n} in Section 4. By a theorem of Morgan and Shalen ([MoS1]), if (ρ_{n}) does not have a compact closure in AH(M), a subsequence of (ρ_{n}) tends to a minimal small action of π_{1}(M) on an ℝ-tree T . Let L_{rec} be the union of the recurrent leaves of the Hausdorff limit L_{∞} of |λ_{n}| where the λ_{n} are measured laminations with (l_{ρ_{n}}(λ_{n})) bounded. We construct a sequence of train tracks τ_{n} = τ^{1} ∪τ_{n}^{2} ∪τ_{n}^{3} where τ^{1} carries realised components of L_{rec}, τ_{n}^{2} carries non-realised components of L_{rec} and τ_{n}^{3} carries L_{∞} − L_{rec} with small weight with respect to λ_{i}. We also construct a sequence of ρ_{n}-equivariant maps f̂_{n} from the universal cover τ̂_{n} of τ_{n} to ℍ^{3} which map the branches of τ̂_{1} to long geodesic segments and the branches of τ̂_{n}^{2} to comparatively short ones. This is proved in Lemma 4.1.

In Section 5 we show that there are long parts of f_{n}(λ_{n}) which nearly backtrack. We approximate λ_{n} by weighted simple closed curves c_{n}. By hypothesis, the geodesic representatives c_{n}^{∗} ⊂ M_{n} = ℍ^{3}/ρ_{n}(π_{1}(M)) of c_{n} have bounded lengths (taking the weight also into account). On the other hand, by construction, f_{n}(c_{n}) gets infinitely long with n. We construct an annulus between f_{n}(c_{n}) and c_{n}^{∗} whose area is controlled. In this annulus, it follows from the difference in length between f_{n}(c_{n}) and c_{n}^{∗} that, for large enough n, the path f_{n}(c_{n}) has a number of long segments in which it comes back nearly parallel to itself. This provides us with some long and thin strips connecting two segments of f_{n}(c_{n}).

In Section 6, we explain how these strips give rise to discs or annuli whose boundaries converge in the Hausdorff topology to a geodesic lamination which does not intersect λ transversely.

In Section 7 we deduce Theorem 2 and 1 from the results in the preceding sections.

Acknowledgements We thank referees for numerous comments to make the paper more readable.

2. Preliminaries

2.1. Deformation space. Let G be a finitely generated torsion-free Kleinian group, namely a (torsion-free and finitely generated) discrete subgroup of Isom^{+}(ℍ^{3}). The group G is convex cocompact if there is a compact subset C ⊂ ℍ^{3}/G that contains all the closed geodesics. Denote by Ω_{G} the domain of discontinuity for the action of G on ℂ̂. The group G is convex cocompact if and only ℍ^{3}/G ⊔ Ω_{G}/G is compact.

Let M be a compact orientable irreducible atoroidal manifold. Let AH(M) denote the set of faithful discrete representations from π_{1}(M) to P SL(2, ℂ) modulo conjugacy. We endow AH(M) with the topology induced from the representation space. The subspace of AH(M) consisting of convex cocompact representations is denoted by CC(M). This space CC(M) is not empty if and only if ∂M contains no tori and it may contain several connected components. The component consisting of representations ρ for which there is a homeomorphism from Int(M) to ℍ^{3}/ρ(π_{1}(M)) that induces ρ is denoted by CC_{0}(M). For such representations, the homeomorphism Int(M) → ℍ^{3}/ρ(π_{1}(M)) extends to a homeomorphism M → ℍ^{3}/ρ(π_{1}(M))⊔ Ω_{ρ}/ρ(π_{1}(M)). This produces a natural identification of Ω_{ρ}/ρ(π_{1}(M)) with ∂M. Notice that the homeomorphism M → ℍ^{3}/ρ(π_{1}(M)) ⊔ Ω_{ρ}/ρ(π_{1}(M)) is well defined up to composition by an element of Mod_{0}(M) (the set of diffeomorphisms of M that are homotopic to the identity map).

For a Kleinian group G, if there is a quasi-conformal automorphism f of S_{∞}^{2} such that fGf^{−1} is again a Kleinian group, then this group fGf^{−1} is said to be a quasi-conformal deformation of G. We denote by QH(G) the space of quasi conformal deformations of G up to conjugacy. By the theory of Ahlfors-Bers, there is a ramified covering map from T (Ω_{G}/G) to the space of quasi-conformal deformations of G modulo conjugacy.

Given a representation ρ_{0} ∈ CC_{0}(M), we have QH(ρ_{0}(π_{1}(M)) = CC_{0}(M). As we have seen, for any ρ ∈ CC_{0}(M), there is a natural identification of Ω_{ρ}/ρ(π_{1}(M)) with ∂M. Thus the theory of Ahlfors-Bers provides us with a covering map T (∂M) → CC_{0}(M) which we call the Ahlfors-Bers map.

2.2. ℝ-trees. An ℝ-tree T is a geodesic metric space in which any two points x, y can be joined by a unique simple arc. Let G be a group acting by isometries on an ℝ-tree T . The action is minimal if there is no proper invariant subtree and small if the stabilizer of any non-degenerate arc is virtually Abelian.

Morgan and Shalen [MoS1] made use of ℝ-trees to compactify deformation spaces. They used algebraic methods involving valuations, while the same result has been obtained by Paulin [Pa] and Bestvina [Bes] using a more geometrical approach. In this paper we shall adopt the point of view of Kapovich-Leeb [KaL] (see also [Ka, chapters 9 and 10]). Let (ρ_{n} ⊂ GF_{0}(M, P)) be a sequence of representations such that no subsequence of (ρ_{n}) converges algebraically. Let Γ ⊂ π_{1}(M) be a set of generators and let x̃_{n} ⊂ ℍ^{3} be a point realising the minimum ϵ_{n}^{−1} on ℍ^{3} of the function max{d(x̃, ρ_{n}(a)(x̃)), a ∈ Γ} (see for example [Pa, Lemma 6.5] for the existence of such a point). Since no subsequence of (ρ_{n}) converges algebraically, (ϵ_{n}^{−1}) tends to ∞. Choose a non-principal ultra-filter ω and denote by ϵ_{n}ℍ^{3} the hyperbolic space ℍ^{3} with the hyperbolic metric rescaled by ϵ_{n}. The ultra-limit (X_{ω}, x) = ω − lim(ϵ_{n}ℍ^{3}, x_{n}) of the sequence of rescaled spaces is defined as follows. Let Π_{n}(ϵ_{n}ℍ^{3}) be the infinite product of the spaces (ϵ_{n}ℍ^{3}). We define a function d_{ω} on Π_{n}(ϵ_{n}ℍ^{3}) by setting

d_{ω}(y, z) = ω − lim d_{ϵ_{n}ℍ^{3}}(ỹ_{n}, z̃_{n})

for any two points y = (ỹ_{n}) and z = (z̃_{n}) lying in Π_{n}(ϵ_{n}ℍ^{3}).

This function d_{ω} is a pseudo-distance in Π_{n}(ϵ_{n}ℍ^{3}) with values in [0, ∞] and we set (X_{ω}, d_{ω}) = (Π_{n}(ϵ_{n}ℍ^{3}), d_{ω})/ ∼ where we identify points with zero d_{ω}-distance. Let x = (x̃_{n}) denote the sequence of points x̃_{n} defined above. The metric space (X_{ω}, x) is the set of points of (X_{ω}) with a finite distance from x. This metric space is an ℝ-tree (cf. [KaL]). The action of ρ_{n}(π_{1}(M)) on ϵ_{n}ℍ^{3} gives rise to an action of π_{1}(M) on (X_{ω}, x) by isometries. This action is small (cf. [KaL]). Let T be the minimal invariant subtree of X_{ω} under this action. We say that (ρ_{n}) tends to the action of π_{1}(M) on T with respect to ω. For c ∈ π_{1}(M) let us denote by δ_{T} (c) the minimal translation distance of c on T . Then we have δ_{T} (c) = ω − lim ϵ_{n}l_{ρ_{n}}(c), where l_{ρ_{n}}(c) is the length in ℍ^{3}/ρ_{n}(π_{1}(M)) of the closed geodesic in the free homotopy class of c.

2.3. Geodesic laminations. A geodesic lamination L on a complete hyperbolic surface S is a compact set which is a disjoint union of complete embedded geodesics called leaves. It is known that this definition is independent of a chosen hyperbolic metric on S. The details can be found in [Ot3]. For a connected geodesic lamination L which is not a simple closed curve we denote by S̄(L) the smallest subsurface of S with compact geodesic boundary containing L. Inside S̄(L) there are finitely many closed geodesics (including the components of ∂S̄(L)) disjoint from L. These closed geodesics do not intersect each other (cf. [Le1] page 99) and we denote by ∂′S̄(L) ⊃ ∂S̄(L) their disjoint union. For example if a component of S̄(L)\L is an annulus, its core curve is contained in ∂′S̄(L) but not in ∂S̄(L). Removing a small open tubular neighbourhood of ∂′S̄(L) from S̄(L) we get a compact surface S(L). We call S(L) the surface embraced by the geodesic lamination L and ∂′S̄(L) the effective boundary of S(L). If L is a simple closed curve, we define S(L) to be an annular neighbourhood of L and we take ∂′S̄(L) = L. When L is not connected, S(L) is the disjoint union of the surfaces embraced by the connected components of L.

We say that a geodesic measured lamination L crosses another geodesic lamination L′ if at least one leaf of L intersects a leaf of L′ transversely.

A measured geodesic lamination λ is a geodesic lamination |λ| together with a transverse measure of full support. We denote by ML(S) the space of measured geodesic laminations on S endowed with the weak-∗ topology. To simplify the notations, we write ML(∂M) instead of ML(∂_{χ<0}M) for a compact 3-manifold M with boundary. The projective lamination space PML(∂M) is defined to be (ML(∂M) − {0})/ℝ_{+}^{∗} where 0 stands for the measured lamination with empty support. It should be noted that ML(∂M) contains measured laminations whose restriction to some component of ∂M is empty. The Teichmüller space T (∂M) denotes similarly T (∂_{χ<0}M). The boundary of the Thurston compactification of T (∂M) is equal to PML(∂M).

2.4. Some notations. When λ is a measured geodesic lamination, we denote by |λ| the support of λ. For an arc k whose intersections with |λ| are transverse, we will denote by ∫_{k} dλ the λ-measure of k.

Let (u_{n}) and (v_{n}) be two sequences of non-negative real numbers. We say that u_{n} is o(v_{n}) if for any ϵ there is N(ϵ) such that for n ≥ N(ϵ), we have u_{n} ≤ ϵv_{n}. We also write u_{n} = o(v_{n}).

We will say that u_{n} is O(v_{n}) if there are K, N > 0 such that for n ≥ N, we have u_{n} ≤ Kv_{n}.

We say that u_{n} is Θ(v_{n}) if u_{n} is O(v_{n}) and v_{n} is O(u_{n}).

2.5. Train tracks and their realisations. A train track τ in a hyperbolic surface S is a union of finitely many rectangles with a distinguished pair of vertical opposite sides. These rectangles meet each other only along nondegenerate segments contained in their vertical sides in such a way that every point on a vertical side of a rectangle lies in at least one other rectangle. The rectangles are called branches, and they are foliated by vertical segments called ties. A connected component of the intersection of the branches is called a switch. The branches are also foliated by horizontal segments, and a smooth arc which is a union of horizontal segments is called a rail or a train route. A geodesic lamination is carried by τ if it is isotopic to one which lies in τ in such a way that the leaves are transverse to the ties (see [Bo] or [Ot3] for more details about train tracks).

When τ is a train track and λ is a measured geodesic lamination whose support is carried by τ, we say that λ is carried by τ. For a branch b of τ, we define the number λ(b) to be the λ-measure of a tie of b. This number does not depend on the choice of a tie in b.

Consider an action of π_{1}(S) on an ℝ-tree T by isometries. A measured lamination λ is said to be realised in T if there is a π_{1}(S)-equivariant map ϕ : ℍ^{2}→T such that the restriction of ϕ to any lift of a leaf of λ in ℍ^{2} is a complete geodesic in the tree, and identifying this geodesic with ℝ gives a weakly monotonous unbounded function on the leaf. A train track τ ∈ S is said to be realised in T if there is an equivariant map ϕ : ℍ^{2}→T which maps each lift of a branch of τ to a non-trivial geodesic segment on T in such a way that each rail is mapped injectively, and that each tie collapses to a point. By [Ot3], λ is realised in T if and only if λ is carried by a train track τ which is realised in T .

We say that a measured lamination λ is collapsed by ϕ : ℍ^{2}→T when there is a train track τ carrying λ such that every component of τ̃, the preimage of τ in ℍ^{2}, is mapped to a point by ϕ. It is straightforward that there exists an equivariant map collapsing λ if and only if the action of i_{∗}(π_{1}(S(λ)) on T has a global fixed point.

For a measured lamination μ on a surface S, if it does not have atoms, the semidistance on ℍ^{2} induced by integrating the transverse measure μ̃ along paths is continuous with respect to the usual topology of ℍ^{2}. By replacing closed leaves by annuli foliated by parallel closed curves, we obtain a measured partial foliation F_{μ}. The quotient of ℍ^{2} under the semi-distance induced by F_{μ} gives rise to a dual tree T_{μ} with the projection π : ℍ^{2}→T_{μ}.

A morphism ϕ : T →T′ between ℝ-trees is a map with the property that every non-degenerate arc [p, q] ⊂ T contains a non-degenerate subarc [p, r] ⊂ [p, q] which is mapped isometrically onto ϕ[p, r] ⊂ T′. A morphism from T_{μ} to T is said to fold only at complementary regions if the only folding points are projections of complementary regions of F̃_{μ}, where p is a folding point if [p, q], [p, q′] ⊂ T_{μ}, [p, q] ∩ [p, q′] = {p} is mapped to the same segment in T . The following theorems [MoO] will be useful to us later. Theorem A [Morgan-Otal] Let (α_{1}, ⋯, α_{k}) be a collection of simple closed curves which define a pants decomposition of a surface S and let π_{1}(S) act on an ℝ-tree T . Then there is a measured lamination μ on S and an equivariant morphism

ϕ : T_{μ} → T

with δ_{T} (α_{i}) = δ_{T_{μ}}(α_{i}) = i(μ, α_{i}) for all i. Moreover ϕ folds only at complementary regions.

The above theorem is generalised by Skora [Sk]. Theorem B [Skora] Suppose the action of π_{1}(S) on an ℝ-tree T is minimal and small such that the action of each element representing ∂S has a fixed point in T . Then there is a unique measured lamination μ and an equivariant isometry

ϕ : T_{μ} → T .

2.6. Compactification of M̃. We denote by M̃ the universal covering of M and by p : M̃ → M the covering projection. We compactify M̃ in the following way: endow M with a geometrically finite hyperbolic metric σ with minimal parabolics which does not correspond to a Fuchsian representation, and let us denote by N(σ)^{thick} the complement in the convex core N(σ) of ϵ-thin neighbourhood of the cusps of σ for some ϵ smaller than the Margulis constant. Let us choose an isometry between the interior of M̃ and ℍ^{3}. Now we can consider Ñ(σ)^{thick} as a closed subset of ℍ^{3}. Since σ is geometrically finite, there is a natural homeomorphism between M and N(σ)^{thick}. Therefore we can regard M̃ as a closed subset of ℍ^{3}. The compactification \overline{M̃} of M̃ is the closure of this closed subset in the usual compactification of ℍ^{3} by the unit ball. If we replace σ by another geometrically finite metric σ′ with minimal parabolics, it follows from results of [Fl] that we get a compactification which is homeomorphic to the one obtained with σ. Therefore this definition is independent of the metric we chose. We call this the Floyd-Gromov compactification of M̃. To denote the ideal boundary in the Floyd-Gromov compactification, we use the symbol ∂_{∞}M̃.

A meridian is a simple closed curve c ⊂ ∂M which bounds a disc in M but not on ∂M. A compact surface Σ ⊂ ∂M is incompressible if it contains no meridians. When we consider the closure of a lift of an incompressible surface in \overline{M̃}, we have the following:

Lemma 2.1. Let Σ ⊂ ∂_{χ<0}M be a compact connected incompressible surface which does not contain any essential closed curve homotopic into ∂_{χ=0}M. Let Σ̂ be the universal covering of Σ, which is completed in the usual way to a closed disc \overline{Σ̂}. Then any lift of Σ to ∂M̃ is a disc whose closure in \overline{M̃} is homeomorphic to \overline{Σ̂} in an equivariant way.

Proof. This lemma was proved in [Ot1]. See also Lemme 2.4 of [Le1]. □

Let Σ ⊂ ∂M be a compact (possibly disconnected) incompressible surface. Johannson and Jaco-Shalen defined a characteristic submanifold W relative to Σ (cf. [Jo] and [JaS]). We say that an I-bundle N ⊂ M is essential in (M, Σ) if

(1) π_{1}(N) injects to π_{1}(M) by the homomorphism induced from the inclusion,

(2) N ∩ ∂M ⊂ Σ is the associated ∂I-bundle, and

(3) N cannot be homotoped into Σ by a homotopy fixing N ∩ Σ. Similarly a solid torus D^{2} × S^{1} is essential in (M, Σ) if it satisfies the first and the third conditions above.

When we are considering an atoroidal manifold M whose boundary contains no torus, a characteristic submanifold is a disjoint union of essential (possibly twisted) I-bundles over compact incompressible surfaces and essential solid tori. A disjoint union W of essential (possibly twisted) I-bundles and essential solid tori is said to be a characteristic submanifold if and only if it has the following two properties:

• any essential (possibly twisted) I-bundle and any essential solid torus in (M, Σ) can be homotoped in W;

• no connected component of W can be homotoped into another connected component of W.

By [Jo] and [JaS], if W and W′ are two characteristic submanifolds relative to Σ, then there is a diffeomorphism ψ : M → M isotopic to the identity relative to ∂M − Σ such that ψ(W) = W′ and that ψ(W ∩ Σ) = W′ ∩ Σ.

Such a characteristic submanifold can be found by looking only at \overline{M̃} −M̃.

Proposition 2.2 ([Le1], §2, paragraphs after Lemme 2.7). Let Σ and Σ′ ⊂ ∂_{χ<0}M be two compact, connected, incompressible surfaces which are disjoint or equal and do not contain any essential closed curve which can be homotoped into ∂_{χ=0}M. Let Σ̃ ⊂ ∂M̃ (resp. Σ̃′) be a connected component of the preimage of Σ (resp. Σ′) and let Γ ⊂ ρ(π_{1}(M)) (resp. Γ′) be the stabiliser of Σ̃ (resp. Γ′).

Then \overline{Σ̃} ∩ \overline{Σ̃}′ is either empty or equal to the limit set of Γ ∩ Γ′.

In the latter case, if Γ ∩ Γ′ is not cyclic, then it is the fundamental group of a (possibly twisted) I-bundle which is a connected component of a characteristic submanifold of (M, Σ∪Σ′). If Γ∩Γ′ is cyclic, then it is a finite index subgroup of a solid torus which is a connected component of a characteristic submanifold of (M, Σ ∪ Σ′).

See Appendix for a brief proof of Proposition 2.2.

2.7. Geodesic laminations on compressible surfaces. Let M be a compact 3-manifold with boundary, and let c ⊂ ∂M be a simple closed curve. A c-wave is a simple arc k, with k ∩ c = ∂k such that there is an arc κ in c with the simple closed curve k ∪ κ bounding a compressing disc in M. In some literature, a c-wave is allowed to intersect c in its interior. A simple innermost argument shows that if there is a c-wave in this generalised sense, there is one in our sense.

Let L be a geodesic lamination on ∂_{χ<0}M, and let c ⊂ ∂M be a multicurve. In the following, we always assume that simple closed curves or multicurves are geodesics for a fixed reference hyperbolic metric, hence there are no inessential intersections between them or with geodesic laminations. We say that L is in tight position with respect to c if L contains no c-waves and if every leaf of L intersects c transversely.

Following [Ot1, Theorem 1.6], we use cut-and-paste operations to construct a meridian m such that a given geodesic lamination contains no mwaves.

Claim 2.3. Let F ⊂ ∂M be a compressible compact surface, and let β ⊂ F be a measured geodesic lamination. Then either β intersects transversely a meridian m and contains no m-waves, or there is a sequence of meridians (m_{i} ⊂ F) converging in the Hausdorff topology to a geodesic lamination which does not cross β.

Proof. If β intersects no meridians transversely, then, since F is compressible, there is a meridian m ∈ F such that i(β, m) = 0. Setting m_{i} = m for every i, we get the conclusion.

Now we assume that β intersects a meridian m transversely. If β contains an m-wave k, let us “cut m along k” to get a new meridian m_{1} : let κ be the closure of a connected component of m−∂k such that ∫_{κ} dβ ≤ \frac{1}{2}i(β, m), and m_{1} the simple closed geodesic in the free homotopy class of k ∪ κ. We have i(β, m_{1}) ≤ \frac{1}{2}i(β, m) and m_{1} is a meridian. If β contains no m_{1}-waves, we are done. If β contains an m_{1}-wave k_{1}, then we cut m_{1} along k_{1} as above to get a new meridian m_{2} with i(m_{2}, β) ≤ \frac{1}{4}i(m, β). Repeating this operation, we get either a meridian m′ such that β contains no m′-waves or a sequence of meridians (m_{i}) such that i(m_{i}, β) ⟶ 0. In the latter case, we can extract a subsequence such that m_{i} converges in the Hausdorff topology to some geodesic lamination H. Since i(m_{i}, β) ⟶ 0, we see that H does not cross β. □

Much of the rest of this section parallels the argument in [Le1], and we refer the readers there for more details.

We shall now introduce homoclinic leaves. They have already proved to be a useful tool in the study of laminations on the compressible boundary of a hyperbolic 3-manifold (see for example [Ot1], [KlS] and [Le1]). They are also related to Culler-Morgan-Shalen compactification of Character Varieties of freely decomposable Kleinian groups (see [KlS] and [Le1]). They shall play an important role in different places in this paper.

Let l be a simple geodesic on ∂_{χ<0}M. Such a simple geodesic l is said to be homoclinic if a lift l̃ of l to the universal cover M̃ of M contains two sequences of point (x̃_{n}) and (ỹ_{n}) such that the distance between x̃_{n} and ỹ_{n} in M̃ is uniformly bounded whereas their distance measured on l̃ tends to ∞. By Lemma 2.1, an incompressible surface cannot contain a homoclinic geodesic.

Homoclinic leaves appear naturally in Hausdorff limits of sequences of meridians. This is illustrated in the following criterion of Casson whose proof can be found in [Ot1] and [Le1, Theorem B.1].

Lemma 2.4. Let (m_{n} ⊂ ∂M) be a sequence of meridians which converges to a geodesic lamination H in the Hausdorff topology. Then H contains a homoclinic leaf.

A simple half-geodesic is an embedded half-line in ∂M whose image is locally geodesic for some hyperbolic metric on ∂_{χ<0}M. Let l̃_{+} ⊂ ∂M̃ be a half-geodesic and let ¯ l̃_{+} be its closure in the Floyd-Gromov compactification of M̃. We say that l̃_{+} has a well-defined endpoint if ¯ l̃_{+} − l̃_{+} contains one point. We say that a geodesic l̃ ⊂ ∂M̃ has two well-defined endpoints if l̃ contains two disjoint half geodesics each having a well-defined endpoint. Notice that we allow the two endpoints to be the same.

As in the introduction, we say that a measured geodesic lamination λ ∈ ML(∂M) is doubly incompressible if and only if there exists η > 0 such that i(λ, ∂E) > η for every essential annulus, Möbius band or disc E in M.

We denote by D(M) ⊂ ML(∂M) the set of doubly incompressible measured geodesic laminations and by PD(M) its projection in the space PML(∂M) of projective measured laminations.

Some properties of this set D(M) are discussed in [Le2]. One can deduce the following from [Le1].

Lemma 2.5. Let λ ∈ D(M) be a measured geodesic lamination and l_{+}, l_{−} ⊂ ∂M two simple half-geodesics which do not intersect |λ| transversely. Then any lift of l_{+} (resp. l_{−}) to M̃ has a well-defined endpoint in ∂_{∞}M̃.

Furthermore, if a lift of l_{+} has the same endpoint as a lift of l_{−}, then they are asymptotic on ∂M̃.

Proof. The relation between doubly incompressible laminations and bending laminations is explained in [Le2, Lemma 3.5]. Knowing this relation, the proof of Lemma 2.5 can be deduced from [Le1] as follows.

The first property, namely that any lift of l_{+} (resp. l_{−}) to M̃ has a welldefined endpoint in ∂_{∞}M̃, can be deduced from the proofs of [Le1, Lemme 3.1] and [Le1, Lemme 3.3].

The proof of the second property, namely that if a lift of l_{+} has the same endpoint as a lift of l_{−} then they are asymptotic on ∂M̃, can be found in Lemme C5 and more specifically in the paragraph after [Le1, Affirmation C3]. □

From [Le2], we also get the following:

Lemma 2.6. Let λ ∈ D(M) be a measured geodesic lamination and h a homoclinic simple geodesic. Then the support |λ| of λ crosses h.

Proof. When M is not a genus-2 handlebody, this is [Le2, Lemma 3.6]. The case when M is a genus-2 handlebody is discussed in [Le2] in the remark following [Le2, Lemma 3.6]. □

By the same argument as the proof of Claim 2.3, we get the following.

Lemma 2.7. Let λ be a measured geodesic lamination in D(M), and let S ⊂ ∂M be a compressible surface. Then there is a meridian m in S such that S contains no m-waves disjoint from |λ|.

Proof. Take any meridian m in S. If there is an m-wave in S which is disjoint from |λ|, then by the same argument as the proof of Claim 2.3, we get a sequence of meridians (m_{i}) on S with i(m_{i}, λ) ⟶ 0. This contradicts the assumption that λ is doubly incompressible. □

3. Realisations of doubly incompressible laminations

Let π_{1}(M) ↷ T be a small minimal action of π_{1}(M) on an ℝ-tree for a compact irreducible atoroidal 3-manifold M. Let S be a connected component of ∂_{χ<0}M. Using the map i_{∗} : π_{1}(S) → π_{1}(M) induced by the inclusion, we get an action of π_{1}(S) on T . Therefore, if λ ∈ ML(S) is a measured geodesic lamination, it makes sense to ask whether or not it is realised in T . In this section, we shall discuss this question for the connected components of a measured lamination lying in D(M).

If a lamination is collapsed to a point by an equivariant map to a tree, then it is not realisable. Although the converse is not true, one may expect that non-realisability of a minimal component implies that the component is carried by a train track of bounded complexity whose edges are mapped to arbitrarily short arcs in the ℝ-tree. In the following lemma, we shall show that this expectation is fulfilled for a doubly incompressible lamination. The proof will occupy the rest of the section.

Lemma 3.1. Let π_{1}(M) ↷ T be a small minimal action of π_{1}(M) on an ℝ-tree. Let λ ∈ D(M) be a measured geodesic lamination, and α a minimal sublamination of λ. Then one of the following holds:

• the measured lamination α is realised in T ;

• there is a sequence of train tracks θ_{i} each of which minimally carries

α and has the following properties: – θ_{i} has only one switch κ_{i} and κ_{i} ⊃ κ_{i+1} for every i.

– There are a sequence η_{i} ⟶ 0, and a sequence of π_{1}(S)-equivariant maps ϕ_{i} : ℍ^{2} → T such that ϕ_{i} maps every branch of the preimage of θ_{i} to a geodesic segment (which may be a point) with length smaller than η_{i}.

– ϕ_{i}(κ_{i}) = x does not depend on i.

Furthermore there is at least one component of λ which is realised in T .

Note that, since θ_{i} has only one switch, the number of its branches is uniformly bounded.

Proof. When α is collapsed in T , it is easy to see that we are in the second situation. The difficult case here is when α is neither collapsed nor realised in T .

When α is a simple closed curve, either α is collapsed (when δ_{T} (α) = 0) or α is realised (when δ_{T} (α) > 0).

From now on, we assume that α is not a simple closed curve. As we will see later, the proof is easier when ∂M is incompressible. Bearing that in mind, we first get rid of the meridians that are disjoint from S(λ). We cut M along a maximal family of compressing discs disjoint from S(α). We denote by N the connected component of the resulting manifold that contains α on its boundary. By construction, the surface ∂N − S(α) is incompressible in N.

Using the homomorphism i_{∗} : π_{1}(N) → π_{1}(M) induced by the inclusion, we view π_{1}(N) as a subgroup of π_{1}(M). Thus we get a small action of π_{1}(N) on T . Let T_{N} be the minimal subtree of T that is invariant under the action of π_{1}(N) regarded as a subgroup of π_{1}(M).

We denote by F the component of ∂N that contains α. Since λ lies in D(M), no component of ∂S(α) bounds a disc in M. It follows that the measured lamination α can be regarded as an element of ML(F). Using the map i_{∗} : F → N induced by the inclusion, we get an action of π_{1}(F) on T_{N} (which is not small when ∂N is compressible). Let T_{F} be the minimal subtree of T_{N} that is invariant under the action of π_{1}(F). If T_{F} is trivial, then α is collapsed. From now on we assume that T_{F} is not trivial.

As is often the case for 3-manifolds, the case when ∂N is incompressible is easier and will give us important insights for the general case. Since we are only interested in α, we just need to assume that F is incompressible.

\underline{First case} : F is incompressible in N.

In this case, the action of π_{1}(F) on T_{F} is small. By Theorem B (Skora’s Theorem), there are a measured geodesic lamination β on F and an isomorphism ϕ : T_{β} → T_{F} from the dual tree T_{β} of β to T_{F}. If β crosses α, then α is realised in T (cf. [Ot3, Theorem 3.1.4]). If β and α are disjoint, then α is collapsed in T .

It remains to deal with the case when α is a connected component of β. Let (η_{i}) be a sequence of positive numbers converging to 0. Let κ_{i} ⊂ F be a sequence of segments intersecting α and β such that κ_{i+1} ⊂ κ_{i} and ∫_{κ_{i}} dβ ≤ \frac{1}{2}η_{i}. Let θ_{i} be a train track minimally carrying α and having only one switch, which is κ_{i} (refer to [BoO, § 3.2] for the construction of such a train track). Let p′ be a point of ⋂_{i} κ_{i}, and p̂′ ∈ ℍ^{2} a lift of p′. This point p̂′ corresponds to a point of T_{β} which we shall also denote by p̂′. Let p ∈ T be the image of p̂′ under ϕ. Let κ̂_{i} ⊂ ℍ^{2} be the lift of κ_{i} that contains p̂′, and θ̂_{i} the lift of θ_{i} that contains κ̂_{i}. Define ϕ_{i} on κ̂_{i} by ϕ_{i}(κ̂_{i}) = p. Extend ϕ_{i} to an equivariant map from the union of the switches of θ̂_{i} to T . If b̂ is a branch of θ̂_{i}, we define ϕ_{i}(b̂) to be the segment of T which connects the images of the vertical sides of b̂. Finally, extend ϕ_{i} to a π_{1}(S)-equivariant map ϕ_{i} : ℍ^{2} → T .

We have thus constructed θ_{i} and ϕ_{i}, and we need to check that they have the expected properties. Let b̂ be a branch of θ̂_{i}. Translating it by an element of π_{1}(S), we can assume that κ̂_{i} contains a vertical side of b̂. Since θ_{i} has only one switch, there is some g ∈ π_{1}(S) such that g(κ̂_{i}) contains the other vertical side of b̂. Let k̂_{1} be an arc joining κ̂_{i} to g(κ̂_{i}) whose projection k_{1} on S lies in b − |β|. Then we have ∫_{k_{1}} dβ = 0. Let k̂_{2} ⊂ κ̂_{i} be an arc joining p̂′ to k̂_{1} and let k̂_{3} ⊂ g(κ̂_{i}) be an arc joining g(p̂′) to k̂_{1}. The β-measures of k_{2} and k_{3} are less than ∫_{κ_{i}} dβ ≤ \frac{1}{2}η_{i}. Therefore we have ∫_{k_{1}∪k_{2}∪k_{3}} dβ ≤ η_{i}. This implies that the distance between p̂′ and g(p̂′) in T_{β} is less than η_{i}. It follows then from the construction of ϕ_{i} that the length of ϕ_{i}(b̂) is less than η_{i}.

Thus we conclude that when F is incompressible, one of the conclusions of Lemma 3.1 holds.

\underline{Second case} : F is compressible.

The difference from the previous case is that the action of π_{1}(F) on T_{F} is not small anymore, and hence we cannot use Theorem B directly. We shall show that if α is neither collapsed nor realised, the following conditions hold: S(α) is incompressible and each component d of ∂S(α) fixes a point in T (meaning that each element in the conjugacy class in π_{1}(N) associated to d has a fixed point). These conditions allow us to use Theorem B on S(α) and to construct θ_{i} as in the first case.

First we shall use Theorem A to associate a lamination to the action of π_{1}(F) on T .

Let (l_{n}) be a sequence of simple closed curves in S(α) converging to the support of α in the Hausdorff topology. By Theorem A (Morgan-Otal’s Theorem), there exist, for any n, a measured lamination β_{n} ∈ ML(S(α)) and a morphism Φ_{n} : T_{β_{n}}→T_{N}, where T_{β_{n}} is the ℝ-tree dual to β_{n}, such that δ_{T_{βn}} (l_{n}) = δ_{T_{N}} (l_{n}). The difference from the case when F is incompressible is that Φ_{n} may not be an isomorphism and that β_{n} depends on l_{n}. Let B be the Hausdorff limit of |β_{n}| after passing to a converging subsequence. If α is disjoint from B, then α is collapsed in T_{N}. If α intersects B transversely, then α is realised in T as was shown in [KlS, Lemma 11]. Hence we only need to deal with the case when |α| is contained in B.

As will be explained below, it follows from results of [KlS], that B can be extended with a homoclinic leaf h. Combining this with the assumption that α is a sublamination of a doubly incompressible lamination we shall prove that S(α) is incompressible. Then we will use h to construct an essential Ibundle W ⊂ N whose boundary contains S(α). Using W in the construction of β_{n} will give us enough control on β_{n} to guarantee that i(β_{n}, ∂S(α)) = 0. This will imply that each component of ∂S(α) fixes a point in T

Claim 3.2. There is a geodesic lamination H ⊃ B that contains a homoclinic leaf h.

Proof. By Claim 2.3, either β_{n} is in tight position with respect to some meridian in F or there is a homoclinic geodesic h_{n} ⊂ F which does not cross β_{n}. By the proof of [KlS, Proposition 2], if β_{n} intersects a meridian m and contains no m-waves, then |β_{n}| can be extended to a geodesic lamination with a homoclinic leaf h_{n}. Furthermore by the proof of [KlS, Proposition 1], there is a Hausdorff limit of meridians that does not cross β_{n}. (See Appendix, Proposition 8.2 and Lemma 8.3.) Thus we have found in every case a sequence of meridians in F converging in the Hausdorff topology to a geodesic lamination H which does not have a transverse intersection with B. By Casson’s criterion (Lemma 2.4), the lamination H contains a homoclinic leaf h. □

From now on we assume that |α| is a sublamination of B, since we have seen earlier that in the other cases, the conclusion is straightforward. Since α is a sublamination of a doubly incompressible lamination, h can not be a leaf of α (Lemma 2.6). As we saw in Lemma 2.1, an incompressible surface cannot contain a homoclinic geodesic. By construction F − S(α) is incompressible, hence the geodesic h does not lie in F − S(α). Since α is arational in S(α), this implies that there is a half-leaf h_{+} of h which is asymptotic to a half-leaf of α on ∂N. Up to cutting h_{+}, we may assume that h_{+} ⊂ S(α). Let h_{−} be another half-leaf of h which is disjoint from h_{+} and from ∂S(α). From the assumption that λ is doubly incompressible, we shall deduce that h_{−} is disjoint from S(α).

Claim 3.3. If |α| ⊂ B, then h_{−} ⊂ F − S(α).

Proof. Seeking a contradiction, we assume that h_{−} ⊂ S(α). We shall show that F − S(α) is compressible, which is impossible by construction.

We fix some geometrically finite hyperbolic metric on N and consider the Floyd-Gromov compactification of Ñ. Let h̃ be a lift of h to Ñ. Since h is homoclinic, the two endpoints of h̃ in ∂_{∞}Ñ coincide. Let h̃_{±} be the lift of h_{±} that lies in h̃.

Since α is a sublamination of λ which is doubly incompressible, then by Lemma 2.5, h̃_{+} and h̃_{−} are asymptotic on F̃. Take a short geodesic arc k̃ connecting h̃_{−} and h̃_{+} which does not lie on h̃ so that they form a triangle on ∂Ñ with one vertex at infinity in ∂_{∞}Ñ. Then any lift of a half-leaf of α entering this triangle must have the same endpoint in ∂_{∞}Ñ as the lifts of h̃_{−}, h̃_{+} since h does not intersect α transversely. Therefore such half-leaf of α is trapped between h̃_{−} and h̃_{+}. It follows that we can push k̃ towards the end of h̃_{±} without changing the intersection number with α̃, where α̃ is the preimage of α. Thus we obtain a sequence of geodesic arcs (k̃_{i}), whose lengths tend to zero and whose α-measure is ∫k̃_{i} dα̃ = ∫k̃ dα̃. We project it to ∂N and pass to a subsequence so that (k_{i}) converges to a point in the Hausdorff topology. Since the transverse measure of α is non-atomic, the only way this can happen is that ∫k̃ dα̃ = 0, i.e., α̃ lies outside the triangle and k is disjoint from α.

Let k̃′ be the segment on h̃ between the two endpoints of k̃ lying on h̃. Then k̃ is homotopic to k̃′ in Ñ since Ñ is simply connected. Let k, k′ be the projections of k̃, k̃′ to ∂N, and m be the closed geodesic homotopic to k ∪k′. We see that k is homotopic to k′ in N but not on ∂N since both of them are geodesic arcs. Therefore, m bounds a (possibly singular) disc in N, and it is disjoint from α since both h and k are disjoint from α. This contradicts the fact that F − S(α) is incompressible by the Loop Theorem. □

It is quite easy to deduce from this Claim that S(α) is incompressible.

Claim 3.4. If |α| ⊂ B, then S(α) is incompressible.

Proof. Let us assume the contrary. By Lemma 2.7, there is a meridian m ⊂ S(α) such that S(α) contains no m-waves disjoint from α. In particular h_{+} is in tight position with respect to m. Let (m̃_{n}) ⊂ Ñ, n ∈ ℕ, be the family of lifts of m which intersect h̃_{+}. Since h_{+} is in tight position with respect to m, each m̃_{n} intersects h̃_{+} once, the curves m̃_{n} are nested and converge to the endpoint of h̃_{+} ([Le1, Claim 3.4]). Since h̃_{+} and h̃_{−} have the same endpoint, h̃_{−} intersects m̃_{n} for n large enough. In particular h_{−} intersects m ⊂ S(α), contradicting Claim 3.3. □

We denote by A the minimal lamination contained in the closure of h_{−}. By Claim 3.3, S(A) is disjoint from S(α). Since F − S(α) is incompressible, so is S(A). We shall use Proposition 2.2, to construct an essential I-bundle whose boundary contains S(A) and S(α).

Claim 3.5. If |α| ⊂ B, then there is an essential I-bundle W ⊂ N whose associated ∂I-bundle is S(α)∪S(A). Moreover, W is homeomorphic to T ×I with T homeomorphic to S(α)

Proof. Let { \atop S(α)}, { \atop S(A)} be lifts of S(α), S(A) containing h̃_{+}, h̃_{−} respectively.

Since S(α) and S(A) are incompressible, by Lemma 2.1, the closures \overline{{ \atop S(α)}}, \overline{{ \atop S(A)}} of { \atop S(α)}, { \atop S(A)} in \overline{Ñ} are discs.

Since h̃_{+} and h̃_{−} have the same endpoint in ∂_{∞}Ñ, the two discs \overline{{ \atop S(α)}}

and \overline{{ \atop S(A)}} intersect in ∂_{∞}Ñ. By Proposition 2.2, there is an essential Ibundle W embedded in (N, S(α) ∪ S(A)), and h_{+} and h_{−} are contained in W ∩ (S(α) ∪ S(A)) (after cutting off arcs of finite lengths from h^{+} and h^{−}). Since α and A lie in the closures of h_{+} and h_{−} respectively, the corresponding ∂I-bundle contains S(α) ∪ S(A). Hence W ∩ ∂N = S(α) ∪ S(A). This can happen only when W is homeomorphic to T × I with T homeomorphic to S(α). □

Since |α| lies in the closure of h_{+} and A lies in the closure of h_{−}, for any leaf ã of the preimage of α in { \atop S(α)}, there is a leaf ã′ of the preimage of A with the same endpoints. It follows that we can isotope the bundle structure so that the projections of A and the support of α along the fibres of W coincide.

Now we can prove that each component of ∂S(α) has a fixed point in T_{N}.

Claim 3.6. If |α| ⊂ B, then δ_{T_{N}} (∂S(α)) = 0.

Proof. We choose a simple closed curve c ⊂ S(A) which is not homotopic to a component of ∂S(A). In the construction of β_{n}, we add the conditions that δ_{T_{βn}} (c) = δ_{T_{N}} (c) and that δ_{T_{βn}} (d) = δ_{T_{N}} (d) for any component d of ∂(T × {0} ∪ T × {1}). Notice that this has no effect on the proof of Claim 3.2 and all that follows (in particular Claim 3.5).

If β_{n} does not intersect ∂S(A) for some n, then δ_{T_{βn}} (∂S(A)) = 0 = δ_{T_{N}} (∂S(A)). By Claim 3.5, δ_{T_{N}} (∂S(α)) = δ_{T_{N}} (∂S(A)) = 0, and we are done.

Otherwise, B intersects S(A). Since h does not cross B and is asymptotic to A, this implies that A lies in B. In particular, any leaf of B intersecting ∂S(A) contains a half-leaf asymptotic to A. Such a half-leaf intersects the simple closed curve c ⊂ S(A) infinitely many times. This implies that i(β_{n}, ∂S(A)) is o(i(β_{n}, c)). On the other hand, by construction, we have i(β_{n}, c) = δ_{T} (c) for any n. Thus we get i(β_{n}, ∂S(A)) ⟶ 0. However, by assumption, i(β_{n}, ∂S(A)) = δ_{T_{N}} (∂S(A)) does not depend on n. Thus we have δ_{T_{N}} (∂S(A)) = 0 = δ_{T_{N}} (∂S(α)). □

It follows that the conjugacy class in π_{1}(F) represented by each component of ∂S(α) has a fixed point in T_{N}. This enables us to use Theorem B on the minimal subtree of T_{N} which is invariant under the action of π_{1}(S(α)). Then we can use the same arguments as in the First case to construct the train tracks θ_{i} and maps ϕ_{i} and show that the second alternative in the statement of Lemma 3.1 holds.

Thus we have proved the first part of Lemma 3.1. It only remains to show that at least one component of λ is realised in T . This was already proved in [Le2, Proposition 6.1]. Let us briefly review the proof. Let (L_{n}) be a sequence of multi-curves converging to λ in the Hausdorff topology. As we have already seen, by Theorem A there exist a measured geodesic lamination β_{n} ∈ ML(∂M) and a morphism ϕ_{n} : T_{β_{n}} → T from the dual tree of β_{n} to T such that for any simple closed curve l_{n} which lies in L_{n}, either δ_{T} (l_{n}) > 0 and the restriction of ϕ_{n} to the axis of l_{n} is an isometry or δ_{T} (l_{n}) = 0 and i(l_{n}, β_{n}) = 0. Extract a subsequence such that (β_{n}) converges in the Hausdorff topology to a geodesic lamination B. As we have seen above, any connected component of λ that intersects B transversely is realised in T .

If S(B) is compressible, then, by the proof of [KlS, Proposition 2], S(B) contains a homoclinic geodesic h which does not cross B. Such a homoclinic leaf crosses λ by Lemma 2.6. Thus, if S(B) is compressible, λ crosses B.

If S(B) is incompressible, then we can apply Skora’s Theorem B to each component of S(B). It follows that β_{n} does not depend on n for sufficiently large n. Denote by β this constant geodesic measured lamination β_{n}. We deduce then from [MoS2] that β is an annular lamination (see [BoO, Démonstration du Lemme 14]). Thus we see that λ crosses the support B of β in this case as well. □

4. Mapping the train tracks to ϵ_{n}ℍ^{3}

In this section, we shall begin the proof of Theorem 2. Here, we consider a situation more general than the setting in Theorem 2 as we shall explain below. We consider a sequence of measured laminations (λ_{n}) on S converging to λ. We also assume that |λ_{n}| converges to a geodesic lamination L_{∞} in the Hausdorff topology and that each minimal sublamination of L_{∞} satisfies the conclusion of Lemma 3.1. Let L_{rec} be the union of the recurrent leaves of L_{∞}.

We shall use the train tracks obtained in lemma 3.1 to construct a sequence of train tracks τ_{n} carrying λ and L_{∞} which are decomposed into three parts. The first part τ^{1} is independent of n and carries the realised part of L_{rec}. The second part τ_{n}^{2} carries the non-realised part of L_{rec}. The third part τ_{n}^{3}, which is τ_{n} −(τ^{1} ∪τ_{n}^{2}) and not a train track, carries the rest of L_{∞}, and has weights with smaller order. Let us denote by τ̂_{n} the preimage of τ_{n} under the covering projection from the universal cover of ∂M to ∂M.

In the process of constructing τ_{n}, we shall also build ρ_{n}-equivariant maps f̃_{n} from τ̂_{n} to ℍ^{3} which map the branches of τ̂_{n}^{1} to long segments and the branches of τ̂_{n}^{2} to much shorter ones. These properties make it possible to estimate the lengths of measured laminations carried by these train tracks.

Recall that when τ̂ ⊂ S̃ is the preimage of a train track τ on a component S of ∂M, we say that a map ĥ : τ̂→ℍ^{3} is ρ_{n}-equivariant if and only if for every g ∈ π_{1}(S) and x ∈ τ̂, we have ĥ(gx) = ρ_{n}(i_{∗}(g))ĥ(x), where i denotes the inclusion from S to M.

We shall work in the following setting: (ρ_{n}) is a sequence of geometrically finite representations uniformising M. There is an ultrafilter ω and ϵ_{n} ⟶ 0 such that the action of ρ_{n} on ϵ_{n}ℍ^{3} tends to a small minimal action of π_{1}(M) on an ℝ-tree T with respect to ω. We have a sequence (λ_{n}) of measured geodesic laminations converging to a measured lamination λ. Furthermore |λ_{n}| converges in the Hausdorff topology to a geodesic lamination L_{∞} and each minimal sublamination of L_{∞} satisfies the conclusion of Lemma 3.1.

We call these assumptions the light assumptions. We use the assumption that each minimal sublamination of L_{∞} satisfies the conclusion of Lemma 3.1 rather than the one that λ is doubly incompressible because it will not contradict the assumption that l_{ρ_{n}}(λ_{n}^{∗}) is bounded which we shall add in the next section. This way the statements of lemmas and claims in Sections 5 and 6 is non-empty, i.e. not based on contradictory hypothesis. To illustrate this statement, let us construct an example with these light assumptions which has bounded l_{ρ_{n}}(λ_{n}^{∗}).

Suppose that M is a handlebody which we regard as an I-bundle T × I over a compact surface with boundary T. We pick a convex cocompact representation ρ uniformising M, a meridian m, and two pseudo-Anosov diffeomorphisms φ, ψ : T → T. We extend φ and ψ to fibred diffeomorphisms φ_{M}, ψ_{M} : M → M. We set ρ_{n} = ψ_{M∗}^{n} ∘ ρ. This sequence tends to the action of π_{1}(M) ≈ π_{1}(T) on the ℝ-tree dual to the stable lamination of ψ. We set m_{n} = φ_{M}^{n}(m), pick a simple closed curve c ⊂ ∂M which is not entirely contained in T × {0} nor in T × {1}, and consider the sequence λ_{n} = D_{m_{n}}^{p_{n}} ∘ ψ_{M}^{n}(c) where D_{m_{n}} : ∂M → ∂M is the right Dehn twist along m_{n}. If p_{n} is large enough, compared to n, the Hausdorff limit L_{∞} of λ_{n} contains two minimal sublaminations both of which are projected along the fibres to the stable lamination of φ. Each of such sublaminations satisfies the conclusion of Lemma 3.1 (it is easy to extend each of them to a doubly incompressible lamination abandoning the other). On the other hand λ_{n} is homotopic to ψ_{M}^{n}(c), hence l_{ρ_{n}}(λ_{n}^{∗}) = l_{ρ}(c^{∗}) is bounded.

We detail the properties of τ_{n} and f̃_{n} in the following lemma. We shall work under the light assumptions which were introduced earlier. The rest of the section is devoted to the proof of Lemma 4.1, which consists in constructing τ_{n} and f̃_{n}.

Lemma 4.1. Under the light assumptions, after taking a subsequence of (ρ_{n}), there are train tracks τ_{n} carrying both L_{∞} and λ_{n} with decomposition τ_{n} = τ^{1} ∪ τ_{n}^{2} ∪ τ_{n}^{3} such that

• each minimal sublamination of L_{∞} is carried by τ^{1} or τ_{n}^{2};

• λ_{n} is minimally carried by τ_{n};

• τ^{1} and τ_{n}^{2} are disjoint subtracks of τ_{n} (we do not require τ_{n}^{3} to be a subtrack);

• τ_{n}^{3} is the union of all branches of τ_{n} that are not contained in τ^{1} ∪τ_{n}^{2};

• the switches of τ_{n} lie in τ^{1} ∪ τ_{n}^{2};

• the sum of the weights with which τ_{n} carries λ_{n} is bounded indepen-

dently of n; and there is a ρ_{n}-equivariant map f̃_{n} : τ̂_{n} → ℍ^{3} from the preimage of τ_{n} in the universal cover of ∂M to ℍ^{3}, which has decomposition τ̂_{n} = τ̂^{1} ∪ τ̂_{n}^{2} ∪ τ̂_{n}^{3} corresponding to the decomposition τ_{n} = τ^{1} ∪ τ_{n}^{2} ∪ τ_{n}^{3}, such that:

a) for any branch b̂ of τ̂_{n}, its image f̃_{n}(b̂) is either a geodesic segment or a point;

b) there are R > 0 and n(R) ∈ ℕ such that, for n ≥ n(R), if b̂ is a branch of τ̂^{1}, then l(f̃_{n}(b̂)) ≥ Rϵ_{n}^{−1}, where ϵ_{n} is the rescaling factor that appeared in the light assumptions;

c) there is a sequence of positive numbers δ_{n} ⟶ 0 such that, for any n ∈ ℕ, if b̂_{1}, b̂_{2} are two adjacent branches of τ̂^{1} which are separated by a switch, then the exterior angle between f̃_{n}(b̂_{1}) and f̃_{n}(b̂_{2}) is less than δ_{n};

d) there is a sequence of positive numbers η_{n} ⟶ 0 such that, for any n ∈ ℕ, if b̂ is a branch of τ̂_{n}^{2}, then we have ϵ_{n}l(f̃_{n}(b)) ≤ η_{n};

e) for any n ∈ ℕ, if b̂ is a branch of τ̂_{n}^{3}, then λ_{n}(b)(ϵ_{n}l(f̃_{n}(b̂))) is less than η_{n} for (η_{n}) given in (d).

Proof. It was proved by Thurston that the set of weighted simple closed curve is dense in ML(S) (cf. [FLP] and [Pe]), for every component S of ∂M. By approximating each (λ_{n}) by a sequence of such unions of weighted simple closed curves, and by a diagonal extraction, we get a sequence of unions of weighted simple closed curves satisfying the light assumption. Therefore we can assume that for each component S of ∂M, the intersection λ_{n} ∩ S is either empty or a weighted simple closed curve.

Since we have only to construct train tracks on each component of ∂M with non-empty intersection with L_{∞}, we can assume that L_{∞} is contained in a component S of ∂M. Let L be a minimal sublamination of L_{rec}.

Let us first consider the case when there is a train track θ minimally carrying L which is realised in T_{S} (recall that T_{S} is the minimal subtree of T invariant under i_{∗}π_{1}(S)). Let θ̂ be a lift of θ to the universal cover ℍ^{2} of S. There is a continuous π_{1}(S)-equivariant map ϕ_{n} : ℍ^{2} → T_{S} under ρ_{n} such that ϕ_{n} is constant on every tie of θ̂ and the restriction of ϕ_{n} to any rail is injective. Following [Ot2], we shall construct a ρ_{n}-equivariant map f̃_{n} : θ̂ → ℍ^{3}. Let κ_{1}, ..., κ_{p} be the switches of θ and κ̂_{1}, ..., κ̂_{p} ⊂ ℍ^{2} lifts of κ_{1}, ..., κ_{p}. Denote by x̃_{i,n} ∈ T_{S} ⊂ (X_{ω}, x) the point ϕ_{n}(κ̂_{i}). We first define f̃_{n} on {κ̂_{1}, ..., κ̂_{p}} by setting f̃_{n}(κ̂_{i}) = x̃_{i,n}. We extend this map to π_{1}(S)({κ̂_{1}, ..., κ̂_{p}}) by f̃_{n}(g(κ_{i})) = ρ_{n}(g) ∘ f̃_{n}(κ̂_{i}) for any g ∈ π_{1}(S) and any 1 ≤ i ≤ p. Let b̂ be a branch of θ̂. The vertical sides of b̂ lie in two switches κ̂ and κ̂′ whose images by f̃_{n} have already been defined. On b̂, we let f̃_{n} be the map which is constant on each tie of b̂, and which induces a parametrisation of the geodesic segment joining f̃_{n}(κ̂) to f̃_{n}(κ̂′) with constant speed on a horizontal side of b̂. Then for any branch b̂ of θ̂, we have

ϵ_{n}l(f̃_{n}(b̂)) ⟶ l_{T_{S}}(ϕ_{n}(b̂)) > 0.

Let θ′ be the first subdivision of θ as defined in [Ot3, Chapitre 4, §4.1]: that is, for each branch b_{j} of θ, starting from every endpoint of b_{j} ∩ b_{k} lying on the vertical side of b_{j}, we cut b_{j} along a horizontal arc up to its midpoint and reorganise the decomposition into branches keeping the condition that it carries λ_{n}. We let θ̂′ ⊂ θ̂ a lift of θ′. We shall deform the map f̃_{n} to one which is adapted to θ̂′. For a branch b̂ of θ̂′, its image by f̃_{n} is a broken geodesic segment which is the union of two geodesic segments. We deform f̃_{n} by a homotopy which is constant on the vertical sides of b̂ to a map which is constant on each tie of b̂ and takes b̂ to the geodesic segment joining the images under f̃_{n} of the vertical sides of b̂. By slightly abusing notation, we shall denote the deformed map by the same symbol f̃_{n}.

Since θ is realised in T_{S}, it follows from the argument of [Ot3, Chapitre 4] that f̃_{n} has the following properties:

b) there are R > 0 and n(R) such that, for n ≥ n(R), if b̂ is a branch of θ̂′, we have l(f̃_{n}(b̂)) ≥ Rϵ_{n}^{−1};

c) there is a sequence of positive numbers δ_{n} ⟶ 0 such that if b̂_{1}, b̂_{2} are two adjacent branches of θ̂ which are separated by a switch, then the external angle between f̃_{n}(b̂_{1}) and f̃_{n}(b̂_{2}) is smaller than δ_{n}.

We repeat the same construction for all the components of L_{rec} that are realised in T_{S}. Denote by τ^{1} the union of the train tracks θ′ thus obtained, by τ̂^{1} its preimage in ℍ^{2}, and by f̃_{n} : τ̂^{1} → ℍ^{3} the map which agrees with the map defined above on each connected component of τ̂^{1}. By Lemma 3.1, τ^{1} is not empty. We also see that λ_{n} passes through every branch of τ^{1} for every n after taking a subsequence.

When a component L of L_{rec} is not realised in T , Lemma 3.1 gives rise to a sequence of train tracks θ_{i} each carrying L minimally. We can assume that λ_{i} passes through every branch of θ_{i}. Let us denote by τ_{i}^{2} the union of the train tracks thus obtained from the components of L_{rec} which are not realised in T . Finally we add branches to τ^{1} ∪ τ_{i}^{2} to get a train track τ_{i} which minimally carries L_{∞} and λ_{i}.

We shall now extend the map f̃_{n} to the preimage τ̂_{i}^{2} of τ_{i}^{2}. Consider a connected component L of L_{rec} which is not realised in T . Consider the subtrack θ_{i} of τ_{i}^{2} which minimally carries L. We get from Lemma 3.1 that there are a point x ∈ T , a sequence η_{i} ⟶ 0, and a sequence of π_{1}(S)-equivariant maps ϕ_{i} : ℍ^{2} → T such that ϕ_{i} maps each branch of the preimage of θ_{i} to a geodesic segment (which may be a point) with length smaller than η_{i} and a lift of κ_{i} (the switch of θ_{i}) is mapped to x under ϕ_{i}. Set x = (x̃_{n}) ∈ Π_{n}(ϵ_{n}ℍ^{3}) (we pick an element in the equivalence class defined by x) and fix some i ∈ ℕ. Let θ̂_{i} ⊂ ℍ^{2} be a lift of θ_{i}, and κ̂_{i} ⊂ θ̂_{i} the lift of κ_{i} which is mapped to x by ϕ_{i}.

Let G_{i} ⊂ π_{1}(∂M) be a finite set consisting of all g ∈ π_{1}(∂M) such that if b̂ is a branch of θ̂_{i} and κ̂_{i} contains a vertical side of b̂, then the other vertical side of b̂ lies in g(κ̂_{i}). Recall that for each branch b̂, the length of ϕ_{i}(b̂) is less than η_{i}. Therefore, we have d(x, gx) ≤ η_{i} for any g ∈ G_{i}. Since T is the ω-limit of ϵ_{n}ℍ^{3}, we have ϵ_{n}d(x̃_{n}, ρ_{n}(g)(x̃_{n})) ≤ 2η_{i} for any g ∈ G_{i} for n large enough. For any g ∈ G_{i}∪{id}, we define f̃_{n}(g(κ̂_{i})) by f̃_{n}(g(κ̂_{i})) = ρ_{n}(g)(x̃_{n}). Let b̂ be a branch of θ̂_{i} with vertical sides lying in two switches κ̂_{i} and g(κ̂_{i}) for some g ∈ G_{i}. If f̃_{n}(κ̂_{i}) = f̃_{n}(ρ_{n}(g)(κ̂_{i})), then we set f̃_{n}(b̂) = f̃_{n}(κ̂_{i}). Otherwise, we set f̃_{n} to be the map which is constant on each tie of b̂ and takes b̂ to the geodesic segment joining f̃_{n}(κ̂_{i}) to f̃_{n}(ρ_{n}(g)(κ̂_{i})). Extend f̃_{n} to an equivariant map from θ̂_{i} to ℍ^{3}. For sufficiently large n and any branch b̂ of θ̂_{i}, we have ϵ_{n}l(f̃_{n}(b)) ≤ 2η_{i}. Furthermore, by construction, the sum of the weights with which θ_{i} carries λ_{n} is bounded by ∫_{κ_{i}} dλ_{n} ≤ ∫_{κ_{1}} dλ_{n} ⟶ ∫_{κ_{1}} dλ.

We do the same construction for all the components of L_{rec} that are not realised in T_{S}, and denote by f̃_{n} : τ̂^{1} ∪ τ̂_{i}^{2} → ℍ^{3} the maps whose restriction to each connected component of τ̂^{1} ∪ τ̂_{i}^{2} is the maps thus defined. It follows from the construction that there is N(i) such that for n ≥ N(i), if b̃ is a branch of τ̂_{i}^{2}, we have ϵ_{n}l(f̃_{n}(b̃)) ≤ 2η_{i}.

We set τ_{i}^{3} to be the closure of τ_{i} − (τ^{1} ∪ τ_{i}^{2}), and τ̂_{i}^{3} its preimage in ℍ^{2}. It remains to define f̃_{n} on the branches of τ̂_{i}^{3}. Let b̂ be a branch of τ̂_{i}^{3}. Let κ̂ and κ̂′ be the two vertical sides of b̂. Their projections κ and κ′ lie in τ^{1} ∪ τ_{i}^{2}. Hence their images by f̃_{n} are already defined, and there are two points x = (x̃_{n}), x′ = (x̃′_{n}) in T such that f̃_{n}(κ̄) = x̃_{n} and f̃_{n}(κ̄′) = x̃′_{n}. We set f̃_{n} to be the map which is constant on each tie of b̂ and takes b̂ to the geodesic segment joining x̃_{n} to x̃′_{n}. We then have ϵ_{n}d(x̃_{n}, x̃′_{n}) ⟶ d(x, x′). Furthermore, since λ is carried by τ^{1} ∪ τ_{i}^{2}, we have λ_{n}(b) ⟶ 0. Therefore, for n large enough, we have λ_{n}(b)(ϵ_{n}l(f̃_{n}(b̃))) ≤ 2η_{i}.

Thus we have proved that there is N(i) such that for n ≥ N(i), for a branch b of τ_{i} − τ^{1}, either b is a branch of τ_{i}^{2} and we have ϵ_{n}l(f̃_{n}(b̂)) ≤ 2η_{i} or b is a branch of τ_{i}^{3} and we have λ_{n}(b)(ϵ_{n}l(f̃_{n}(b̂))) ≤ 2η_{i}. Now by choosing N(i) such that N(i) < N(i + 1), and taking a subsequence λ_{N(n)} so that the n-th term is the original N(n)-th term, we obtain the desired train track. This concludes the proof of Lemma 4.1. □

5. Finding backtracking

In this section, we are going to show that for large enough n, the path f_{n}(c_{n}) has long segments in which it comes back nearly parallel to itself. Eventually these close returns will allow us to construct some long and thin strips connecting two segments of f_{n}(c_{n}). Here we use the adjective “long” not only for the rescaled metric of ϵ_{n}ℍ^{3} but also to mean combinatorially long in the sense that they go through many branches of τ_{n}. Let us fix some notations and definitions to put this idea into a precise statement.

We consider the train tracks τ_{n} and maps f̃_{n} which come from Lemma 4.1. Take a component S of ∂M such that S ∩ τ^{1} ≠ ∅. In this section, we only have to pay attention to the behaviour of λ_{n} on S. Therefore, for simplicity, we denote λ_{n} ∩ S by λ_{n}, and τ_{n} ∩ S by τ_{n}, etc. Furthermore, λ_{n} is assumed to be a weighted simple closed curve with support c_{n} and weight w_{n}.

Let f_{n} : τ_{n} ⟶ ℍ^{3}/ρ_{n}(π_{1}(M)) be the projection of f̃_{n}. By construction, τ_{n} carries c_{n}. We set c_{n}^{1} = c_{n} ∩ τ^{1}, c_{n}^{2} = c_{n} ∩ τ_{n}^{2} and c_{n}^{3} = c_{n} ∩ τ_{n}^{3}. We set c̄_{n}^{j} = f_{n}(c_{n}^{j}) for j = 1, 2, 3 and c̄_{n} = f_{n}(c_{n}) = c̄_{n}^{1} ∪ c̄_{n}^{2} ∪ c̄_{n}^{3}.

Fix an orientation on c_{n}. Let s be a segment lying in c_{n}^{1} with the orientation induced by that on c_{n}. The train route b(1), ..., b(t) of s is the ordered finite sequence of branches of τ^{1} through which s passes: b(i) is an element of the set {b_{1}, ..., b_{p}} of branches of τ^{1}. We fix an orientation for each branch of τ^{1}. A branch b(i) in the train route of s is said to be positively oriented if its orientation coincides with the orientation of s and negatively oriented otherwise. The oriented train route bo(1), ..., bo(t) of s is the ordered finite sequence of oriented branches of τ^{1} through which s goes in this order with the assigned orientations: bo(i) is an element of {b_{1}, ..., b_{p}} × {+, −}. When (bo(i))_{i∈I} is an oriented train route, we shall denote by (b(i))_{i∈I} the corresponding non-oriented train route.

In the following lemma, we shall show that f_{n}(c_{n}) nearly backtracks along some long path. Using the terms of oriented train routes, this is expressed as follows.

Lemma 5.1. Under the light assumptions, if (l_{ρ_{n}}(λ_{n}^{∗})) is bounded, then there are two infinite oriented train routes bo, bo′ : ℕ → {b_{1}, ..., b_{p}} × {+, −} in τ^{1} and functions T, V : ℕ → ℕ such that for any n ∈ ℕ there are two disjoint segments s_{n}, s′_{n} ⊂ c_{n}^{1} satisfying the following:

• T is non-decreasing and unbounded;

• the oriented train routes of s_{n} and s′_{n} are (bo(i))_{0≤i≤T (n)} and (bo′(i))_{0≤i≤V (T (n))} respectively;

• there is a homeomorphism g_{n} : f_{n}(s_{n}) → f_{n}(s′_{n});

• g_{n}(f_{n}(v(i))) ∈ f_{n}(b′(V (i))) for any i ≤ T (n) where v(i) = b(i)∩b(i+ • any point x ∈ f_{n}(s_{n}) is connected to g_{n}(x) by an essential arc

ζ_{n}(x) ⊂ M_{n} with length less than 6ϵ;

• the simple closed curve f_{n}(s_{n}) ∪ ζ_{n}(f_{n}(∂s_{n})) ∪ f_{n}(s′_{n}) bounds a disc D_{n} containing all the arcs ζ_{n}(x) for x ∈ f_{n}(s_{n}).

Recall that λ_{n} is assumed to be a weighted simple closed curve with support c_{n} and weight w_{n}. We denote by c_{n}^{∗} the geodesic representative of c_{n} in M_{n} = ℍ^{3}/ρ_{n}(π_{1}(M)). Then we set l_{ρ_{n}}(λ_{n}^{∗}) = w_{n}l_{ρ_{n}}(c_{n}^{∗}) by definition.

Notice that we have adopted the conventions that 0 ∈ ℕ, and that if T (n) = 0, there are no segments s_{n} and s′_{n}.

We shall see that V (i) ≤ i\frac{R′}{R} + 1 for any i ∈ ℕ (equation (2)).

Proof. First we shall show that most points x in c̄_{n}^{1} are close to another component of c̄_{n}^{1} (i.e. not the component containing x). The proof goes roughly as follows: we construct a simplicial annulus A_{n} between c̄_{n} and c_{n}^{∗}. If a point x of c̄_{n}^{1} is not close in A_{n} to a point in another component of c̄_{n}^{1} then either x is close to a component of c̄_{n}^{2} or c̄_{n}^{3} or x is not close to any component of c̄_{n}^{j} for j = 1, 2, 3. Using the Gauss-Bonnet formula and the length comparison between the components of c̄_{n}^{j}, we shall show that this can happen only for the minority of the points of c̄_{n}^{j}.

Let us start the formal proof. The curve c̄_{n} = f_{n}(c_{n}) is a piecewise geodesic. We define the edges of c̄_{n} to be the images of the intersections of c_{n} with the branches of τ_{n}, and the vertices to be the images of the intersections of c_{n} with the switches of τ_{n}.

Let x_{n,1}, ..., x_{n,p_{n}} be the vertices of c̄_{n}, and choose the same number of points y_{n,1}, ..., y_{n,p_{n}} on c_{n}^{∗}. We shall make a strip with boundaries (x_{n,i}) and (y_{n,i}) and triangulate it by making each rectangle into a pair of triangles. To be more precise, for 1 ≤ i ≤ p_{n}, we consider the geodesic triangle with vertices y_{n,i}, x_{n,i}, x_{n,i+1} (with x_{n,p_{n}+1} = x_{n,1} and y_{n,p_{n}+1} = y_{n,1}) and the geodesic triangle with vertices x_{n,i+1}, y_{n,i}, y_{n,i+1}. The union of these triangles for i = 1, …, p_{n} is a simplicial annulus A_{n} = S^{1} × [0, 1] bounded by c_{n}^{∗} and c̄_{n}. The metric ν_{n} induced on this annulus by the lengths of paths is a hyperbolic metric with piecewise geodesic boundary. By the Gauss-Bonnet formula, the area of A_{n} is less than 2p_{n}π. By Lemma 4.1, the sequence (w_{n}p_{n}) = (∑_{b: the branches of τ_{n}} λ_{n}(b)) is bounded. We parametrise A_{n} by S^{1} × [0, 1] so that the projection of S^{1} × {1} to ℍ^{3}/ρ_{n}(π_{1}(M)) is c_{n}^{∗}.

For a positive number ϵ, which we shall specify later, and each point x ∈ c̄_{n}^{1}, we consider a geodesic arc a_{x} on (S^{1}×I, ν_{n}) perpendicular to S^{1}×{0} at x having length ϵ with respect to ν_{n}. If the perpendicular reaches S^{1} ×∂I before the length ϵ is attained, we define a_{x} to be the geodesic arc having both endpoints on S^{1} × ∂I.

We shall show that most arcs a_{x} issuing from a component of c̄_{n}^{1} intersect an arc a_{y} issuing from another component of c̄_{n}^{1}. For this purpose, we are going to estimate from below the length of the set of points x for which the a_{x} reach S^{1} × {1} without intersecting themselves or each other. Since the length of this set of points is bounded above by length(c_{n}^{∗}) (with respect to ν_{n}), we get an inequality, which will appear as the inequality (i) below. For that, we need to subtract from the length of c̄_{n}^{1} the lengths of (I) the set of points x for which a_{x} has self-intersection, (II) the set of points x for which a_{x} intersects a_{y} with x ≠ y, (III) the set of points x for which a_{x} has an endpoint on either c̄_{n}^{2} or c̄_{n}^{3}, and (IV) the set of points x which are neither of type (I) nor of type (II) and for which a_{x} has an endpoint in the interior of S^{1} × I. See Figure 1.

PSfrag replacements

c̄_{n}^{1}
c̄_{n}^{2}
c̄_{n}^{3}

x_{n,1} x_{n,2} y_{n,1} y_{n,2} c_{n}^{∗}

Figure 1. Annulus between c_{n}^{∗} and f_{n}(c_{n})

We first consider the contribution of the points of type (I) to the length, i.e., x for which a_{x} intersects itself transversely. By the Gauss-Bonnet formula, a geodesic loop formed by a subarc of a_{x} cannot be null-homotopic. Hence, there must be a loop formed by a subarc of a_{x} freely homotopic to S^{1} × {1}. It follows that if both of two perpendiculars a_{x_{1}}, a_{x_{2}} with x_{1} ≠ x_{2} have self-intersection, then a_{x_{1}} ∩ a_{x_{2}} ≠ ∅. Thus, the contribution of the set of x with self-intersecting a_{x} (i.e. of type (I)) to the length is absorbed in the contribution of x with a_{x} intersecting another a_{y}, that is, of type (II), which will be dealt with below.

We next consider the points x of type (II), i.e. those for which a_{x} intersects a_{y} for some y ∈ c̄_{n}^{1}. Let m be a point in the intersection a_{x} ∩ a_{y}, and let a′_{x}, a′_{y} be subarcs of a_{x}, a_{y} between x and m and y and m respectively. Let β be an arc on c̄_{n} to which a′_{x} ∪ a′_{y} is homotopic fixing the endpoints. Suppose that β is also contained in c̄_{n}^{1}. We then say that x is an inessential point of type (II) and that a′_{x} ∪ a′_{y} is an inessential arc. It was shown in [Bo, Lemme 5.11] that, in this situation, there is a constant ξ_{n} depending only on ϵ and the maximal exterior angle of the vertices on c̄_{n}^{1}, which is less than δ_{n} in our case, such that x is within distance ξ_{n} with respect to ν_{n} from a vertex of c̄_{n}^{1}. It was also shown in [Bo] that the constant ξ_{n} tends to 0 as either δ_{n} or ϵ goes to 0. If β does not lie on c̄_{n}^{1}, then we say that x is an essential point of type (II). See Figure 2.

PSfrag replacements

c̄_{n}^{1}
c̄_{n}^{2}

Figure 2. Inessential and essential points of type II.

We note that the length of an essential arc (such as a′_{x} ∪a′_{y}) is less than or equal to 2ϵ. Let c̄_{n}^{+} be the union of the essential points of type (II). We shall use the essential points of type (II) to construct the long and thin strips of Lemma 5.1.

Now we shall bound the length of the sets of points of type (III) and (IV). First we consider the points of type (III). The total length with respect to ν_{n} of the set of points x on c̄_{n}^{1} for which a_{x} reaches a point on c̄_{n}^{2} is bounded above by the length of c̄_{n}^{2}. Similarly, the total length of the set of points x for which a_{x} reaches a point on c̄_{n}^{3} is bounded above by the length of c̄_{n}^{3}.

Finally we consider the points of type (IV); the points x such that a_{x}\{x} is contained in the interior of S^{1} × I while a_{x} has neither self-intersection nor intersection with another a_{y}. Since the union of a_{x} for x of type (IV) has area bounded below by the length of the set of points x of type (IV) multiplied by sh(ϵ), we can bound the length from above by Area(A_{n})/sh(ϵ).

Putting all of these considerations together, we get an inequality:

(i) l_{ρ_{n}}(c̄_{n}^{1}) − 2p_{n}ξ_{n} − l_{ρ_{n}}(c̄_{n}^{+}) − l_{ρ_{n}}(c̄_{n}^{2}) − l_{ρ_{n}}(c̄_{n}^{3}) − Area(A_{n})/sh(ϵ) ≤ l_{ρ_{n}}(c_{n}^{∗}).

Notice that we have

w_{n}l_{ρ_{n}}(c̄_{n}^{2}) = ∑_{b: branches of τ_{n}^{2}} λ_{n}(b)l_{ρ_{n}}(f_{n}(b))

and

w_{n}l_{ρ_{n}}(c̄_{n}^{3}) = ∑_{b: branches of τ_{n}^{3}} λ_{n}(b)l_{ρ_{n}}(f_{n}(b)).

Therefore, by the property (d) of Lemma 4.1, we have w_{n}l_{ρ_{n}}(c̄_{n}^{2}) = o(ϵ_{n}^{−1}) and by the property (e), we have w_{n}l_{ρ_{n}}(c̄_{n}^{3}) = o(ϵ_{n}^{−1}).

By Lemma 4.1 (w_{n}p_{n}) is a bounded sequence. It follows that we have 2w_{n}p_{n}ξ_{n} ⟶ 0. This implies also that w_{n}Area(A_{n}) ≤ 2w_{n}p_{n}π is a bounded sequence and that we have ϵ_{n}w_{n}Area(A_{n}) ⟶ 0.

By assumption (w_{n}l_{ρ_{n}}(c_{n}^{∗})) = (l_{n}(λ_{n})) is a bounded sequence; hence ϵ_{n}w_{n}l_{ρ_{n}}(c_{n}^{∗}) tends to 0. Thus we have shown the following.

Claim 5.2. We have w_{n}ϵ_{n}(l_{ν_{n}}(c̄_{n}^{1}) − l_{ν_{n}}(c̄_{n}^{+})) ⟶ 0. □

Now that we know that c̄_{n}^{+} occupies the most part of c̄_{n}^{1}, we shall use c̄_{n}^{+} to construct maps g_{n} and strips D_{n}. First we define a discrete version of g_{n}. Let {σ_{1,n}, σ_{2,n}, … } ⊂ c̄_{n} be a maximal family of disjoint segments with diameter 6ϵ such that the midpoint x_{i,n} of σ_{i,n} lies in c̄_{n}^{+}. Consider a segment σ_{i,n} and its middle point x_{i,n}. In the family of essential arcs joining x_{i,n} to c̄_{n}^{1}, we take an arc a_{i,n} to be the shortest (with respect to ν_{n}). Since x_{i,n} lies in c̄_{n}^{+}, the length of a_{i,n} is less than 2ϵ.

If ∂a_{i,n} − x_{i,n} lies in some σ_{j,n}, we denote by ζ_{i,n} the geodesic segment in (S^{1} × I, ν_{i}) joining x_{i,n} to x_{j,n} which is homotopic to a_{i,n} relative to x_{i,n} ∪ σ_{j,n}. If ∂a_{i,n} − x_{i,n} is disjoint from ⋃ σ_{j,n}, we define ζ_{i,n} to be a_{i,n} (see Figure 3). The length of each segment ζ_{i,n} thus obtained is less than 5ϵ. Using the minimality of the lengths of the arcs a_{i,n}, we shall next show that the segments ζ_{i,n} have mutually disjoint interiors.

Consider two different segments ζ_{1,n} and ζ_{2,n} and assume that their interiors intersect. Then the interiors of a_{1,n} and a_{2,n} also intersect. Let y be a point in the intersection. Let [x_{ℓ,n}, y[, ℓ = 1, 2 be the connected component of a_{ℓ,n} − {y} containing x_{ℓ,n}. Let γ_{n} be the shortest of the two segments a_{1,n} − [x_{1,n}, y[ and a_{2,n} − [x_{2,n}, y[ (see Figure 3). Then, for ℓ = 1, 2, the length of the arc [x_{ℓ,n}, y] ∪ γ_{n} is less than or equal to the length of a_{ℓ,n}. Furthermore one of the two arcs [x_{ℓ,n}, y]∪γ_{n}, say [x_{1,n}, y]∪γ_{n} is not the shortest in its homotopy class relative to the endpoints. Let a′_{1} be the shortest arc homotopic to [x_{1,n}, y] ∪ γ_{n} relative to the endpoints. Then the length of the segment a′_{1} (with respect to ν_{n}) is less than the length of a_{1,n}. Recall that we chose a_{1,n} which is shortest among all essential arcs joining x_{1,n} to c̄_{n}^{1}. It follows that a′_{1} is not essential, i.e. there is a segment β ⊂ c̄_{n}^{1} homotopic to a′_{1} relative to the endpoints. The endpoints of β are x_{1,n} and another point which we call y_{1}. The distance (with respect to ν_{n}) between x_{1,n} and y_{1} is less than the length of a′_{1} which is less than 2ϵ. By the properties (b) and (c), each component of c̄_{n}^{1} (in particular the one containing β) is a union of long geodesic segments such that the exterior angle between two consecutive segments is small. By [CEG, Lemma 4.2.10] such a component of c̄_{n}^{1} is a (K, η)-quasi-geodesic with K→1, η→0 as n→∞. It follows that there is N (independent of β) such that for n ≥ N, the length of β is less than 3ϵ. This implies that y_{1} lies in σ_{1,n}. By our definition of ζ_{1,n}, it has an endpoint on x_{1,n}, not on y_{1} (see Figure 3). This contradicts our assumption that the interiors of ζ_{1,n} and ζ_{2,n} intersect.

PSfrag replacements x_{1,n} y_{1} y

x_{2,n} c̄_{n} σ_{1,n} σ_{2,n} c_{n}^{∗} a_{2,n} a_{1,n} ζ_{2,n} ζ_{1,n} γ_{n}

Figure 3. From a_{i,n} to ζ_{i,n}

Even if some segment a_{i,n} has a self-intersection, the same argument shows that ζ_{i,n} does not have any self-intersection.

Thus we have proved the following claim.

Claim 5.3. There are a map h_{n} : {x_{1,n}, x_{2,n}, ...} → c̄_{n}^{1} and a family (ζ_{i,n}) of essential segments with disjoint interiors such that the length of ζ_{i,n} is less than 5ϵ, ∂ζ_{i,n} = {x_{i,n}, h_{n}(x_{i,n})}, and {h_{n}(x_{1,n}), h_{n}(x_{2,n}), ...} ∩ ⋃_{i} σ_{i,n} ⊂ {x_{1,n}, x_{2,n}, ...}. □

We shall use those segments ζ_{i,n} as a skeleton to construct our strips D_{n}. To this end we need to fill the space between two consecutive segments ζ_{i,n} and ζ_{i+1,n} with a thin strip. In particular, we need to check that ζ_{i,n} and ζ_{i+1,n} are in the same homotopy class and that each point of c̄_{n}^{1} between x_{i,n} and x_{i+1,n} is close to a point on c̄_{n} between h_{n}(x_{i,n}) and h_{n}(x_{i+1,n}). Our proof will rely mainly on counting segments. Namely we start with sufficiently many segments and prove that at each step,there are only few among them which fail to satisfy the property that we need.

Let j be a positive integer, and cut c̄_{n}^{1} into disjoint segments s̄_{1,n}, s̄_{2,n}, ..., each containing j edges (if the number of edges of some component of c̄_{n}^{1} is not a multiple of j, then there are some edges of c̄_{n}^{1} not belonging to any one of these segments).

We shall evaluate the number of segments thus obtained using the following claim.

Claim 5.4. Let r_{n} be the number of the components of c̄_{n}^{1}. Then, w_{n}r_{n} ⟶ 0 as n → ∞.

Proof. Note that any train route on τ_{n} connecting a point in τ^{1} and a point in τ_{n}^{2} must pass through a point in τ_{n}^{3}. Therefore between any two distinct components of c_{n}^{1}, there is a component of c_{n}^{3}. Hence (w_{n}r_{n}) is bounded above by ∑_{b⊂τ_{n}^{3}} λ_{n}(b). Since |λ| ⊂ L_{rec} is carried by τ^{1} ∪ τ_{n}^{2}, the sum ∑_{b⊂τ_{n}^{3}} λ_{n}(b) tends to 0 as n → ∞. It follows that we have w_{n}r_{n} ⟶ 0. □

Since

(1) w_{n}(number of edges of c̄_{n}^{1}) = w_{n}( ∑_{b⊂τ^{1}} c_{n}(b)) ⟶ ∑_{b⊂τ^{1}} λ(b),

the number of edges of c̄_{n}^{1} is Θ(w_{n}^{−1}) = Θ(p_{n}) (see Section 2.4 for notations), where p_{n} was defined to be the number vertices on c̄_{n} (i.e. the number of times c_{n} crosses a switch of τ_{n}). The number of edges of c̄_{n}^{1} lying in none of the s̄_{i,n} is less than jr_{n} = o(p_{n}). It follows that the number of the segments s̄_{i,n} is Θ(p_{n}).

Let t_{n} be the number of edges of c̄_{n}^{1} containing no one among the segments σ_{i,n} defined earlier. If an edge e contains no segment among the σ_{i,n}, then there is no point of c̄_{n}^{+} in e outside the 3ϵ-neighbourhood of ∂e. By the property (b) in Lemma 4.1, the total length of these edges is greater than t_{n}Rϵ_{n}^{−1} and smaller than l_{ν_{n}}(c̄_{n}^{1}) − l_{ν_{n}}(c̄_{n}^{+}) + 6t_{n}ϵ. By Claim 5.2 and the equation (1), we have w_{n}ϵ_{n}(l_{ν_{n}}(c̄_{n}^{1}) − l_{ν_{n}}(c̄_{n}^{+}) + 6t_{n}ϵ) ⟶ 0. Hence w_{n}ϵ_{n}t_{n}Rϵ_{n}^{−1} ⟶ 0 and t_{n} is o(w_{n}^{−1}) = o(p_{n}).

Thus we know that among the s̄_{i,n}, there are Θ(p_{n}) disjoint segments lying in c̄_{n}^{1} and containing j edges (where j is the number we have fixed when cutting c̄_{n}^{1}) each of which contains a segment among the σ_{i,n}. We shall denote these Θ(p_{n})-many segments again by s̄_{1,n}, s̄_{2,n}, . . ..

Next we shall show that most of these segments are joined to only one component of c̄_{n}^{1} through the arcs ζ_{i,n}.

Let s̄ ∈ {s̄_{1,n}, s̄_{2,n}, … } be a segment with the following property: there are at least two distinct components of c̄_{n}^{1} containing an endpoint of ζ_{i,n} for some x_{i,n} ∈ s̄. Let t′_{n} be the number of those with this property among the s̄_{i,n}. In each such segment s̄, we choose two points in {x_{1,n}, x_{2,n}, ...} ∩ s̄, say x_{1,n}, x_{2,n}, such that ζ_{1,n} and ζ_{2,n} connect s̄ to distinct components of c̄_{n}^{1} and the segment ]x_{1,n}, x_{2,n}[⊂ s̄ contains no x_{k,n}. We note that there may be ζ_{i,n} other than ζ_{1,n}, ζ_{2,n} which have x_{1,n} or x_{2,n} as an endpoint. There are two points y_{1,n} and y_{2,n} which lie in two distinct components of c̄_{n}^{1} such that y_{1,n} (resp. y_{2,n}) is connected to x_{1,n} or x_{2,n} by some ζ_{i,n} and that y_{1,n} and y_{2,n} are innermost in the following sense : if [y_{1,n}, y_{2,n}] is the segment of c̄_{n} joining y_{1,n} to y_{2,n} whose interior does not contain x_{1,n}, then there is no ζ_{i,n} connecting ]y_{1,n}, y_{2,n}[ to {x_{1,n}, x_{2,n}} (these y_{1,n}, y_{2,n} may or may not coincide with h_{n}(x_{1,n}), h_{n}(x_{2,n})). Let us embed A_{n} into a round disc in such a way that c̄_{n} is the boundary of the disc, and connect y_{1,n} to y_{2,n} by a geodesic segment with respect to the ordinary Euclidean metric on the disc.

Assume that for another segment s̄′ ∈ {s̄_{1,n}, … } with the same property, the resulting geodesic segment in the round disc intersects transversely the geodesic segment produced from s̄ above (i.e. the geodesic segment connecting y_{1,n} to y_{2,n}). Suppose that x_{3,n} and x_{4,n} are the points on s̄′ chosen in the same way as x_{1,n}, x_{2,n} for s̄. We number them so that the order in which the four points lie on the circle is x_{1,n}, x_{2,n}, x_{3,n}, x_{4,n}. Since ζ_{i,n} have disjoint interiors and y_{1,n} and y_{2,n} are innermost, we see that the only ways this can happen are the following two: (1) x_{3,n} = y_{1,n} and (2) x_{4,n} = y_{2,n}. Therefore for each s̄, there are only two configurations of s̄′ such that the geodesics in the round disc intersect transversely. See Figure 4.

Thus we can build at least \frac{1}{3}t′_{n} disjoint geodesic segments in the round disc, each connecting two distinct components of c̄_{n}^{1}. Furthermore, since the ζ_{i,n} have disjoint interiors, any pair of connected components of c̄_{n}^{1} is connected by at most two of these disjoint segments. Consider a map from c̄_{n} to a round circle which preserves the order and maps each connected component C_{i} of c̄_{n}^{1} to a point Q_{i}. Join two points Q_{i} and Q_{j} by a segment if and only if there is one of the segments constructed above which joins C_{i} and C_{j}. Thus we have constructed at least \frac{1}{6}t′_{n} segments with disjoint interiors. We can now add some geodesic segments in the disc to get a triangulation of the polygon with vertices Q_{1}, ..., Q_{r_{n}}. Such a triangulation has 2r_{n} − 3 edges (this can easily be computed with the Euler formula). Therefore we have \frac{1}{6}t′_{n} ≤ 2r_{n} − 3 = o(p_{n}).

Since we initially had Θ(p_{n}) segments {s̄_{1,n}, s̄_{2,n}, ...}, after excluding o(p_{n}) segments as above from them, there remains, by abusing notations again, Θ(p_{n}) disjoint segments {s̄_{1,n}, s̄_{2,n}, ...} in c̄_{n}^{1}, such that if s̄ is one of those segments, then we have:

- s̄ contains j edges;

- i) each edge of s̄ contains some x_{i,n};

- ii) there is a unique component C (depending on s̄) of c̄_{n}^{1} such that for any x_{i,n} ∈ s̄, we have h_{n}(x_{i,n}) ∈ C.

Let s̄ be one of these segments, C the associated component of c̄_{n}^{1}, and x a point of s̄∩ {x_{1,n}, ...}. Denote by ζ_{x} the corresponding segment ζ_{i,n}. Since A_{n} is an annulus and ζ_{x} is embedded, there are only two possibilities for the homotopy class of ζ_{x} relative to s̄ ∪ C. Therefore, taking 2j instead of PSfrag replacements

x_{2,n} x_{1,n}

y_{1,n} = x_{3,n} x_{4,n} y_{4,n} y_{2,n}

s̄

s̄′

y_{2,n} = x_{4,n} y_{1,n} x_{3,n} c̄_{n} c_{n}^{∗}

Case (1) Case (2)

Figure 4. A segment intersects at most two other segments

j at the beginning and cutting each segment into two groups, we get Θ(p_{n}) disjoint segments {s̄_{1,n}, ...} in c̄_{n}^{1}, each one containing j edges and satisfying (i), (ii) above and:

iii) for any x, y ∈ s̄ ∩ c̄_{n}^{+}, ζ_{x} and ζ_{y} are homotopic relative to s̄ ∪ C.

Now we have a sufficiently number of j consecutive segments σ_{i,n} with the properties (i), (ii) and (iii). Next we shall show that they lie in the boundary of the expected strips D_{n}.

Let s̄ be one of the segments produced above, and C the corresponding component of c̄_{n}^{1} in the property (ii). Let x and y be the extremal points of s̄ ∩ {x_{1,n}, ...}, and [x, y] the segment in s̄ joining x to y. The segment [x, y] contains at least (j − 2) edges. We have the following:

Lemma 5.5. There is N ∈ ℕ which does not depend on s̄ such that for any n ≥ N we have the following by homotoping A_{n} keeping ∂A_{n} and c̄_{n}^{i} unchanged: There is a homeomorphism g_{n} : [x, y] → [h_{n}(x), h_{n}(y)] such that for any n ≥ N and any z ∈ [x, y], the two points z and g_{n}(z) are connected by an essential arc whose length (with respect to the induced metric ν_{n} on A_{n}) is less than 6ϵ.

Proof. By the property (iii), the simple closed curve ζ_{x}∪[x, y]∪ζ_{y}∪[h_{n}(x), h_{n}(y)] bounds a disc in A_{n}. Since both [x, y] and [h_{n}(x), h_{n}(y)] lie in c̄_{n}^{1}, by the properties (b) and (c) in Lemma 4.1, they consist of long geodesic segments such that the external angles formed by two adjacent segments are less than δ_{n}. Let k be the geodesic segment in ℍ^{3}/ρ_{n}(π_{1}(M)) joining x to y which is homotopic to [x, y], and let k′ be the one in ℍ^{3}/ρ_{n}(π_{1}(M)) joining h_{n}(x) to h_{n}(y) which is homotopic to [h_{n}(x), h_{n}(y)]. We parametrise the arcs [x, y], [h_{n}(x), h_{n}(y)], k and k′ by their arc lengths.

By [CEG, Lemma 4.2.10], for sufficiently large n, we have d([x, y](t), k(t)) ≤ ϵ′_{n} and d([h_{n}(x), h_{n}(y)](t), k′(t)) ≤ ϵ′_{n} for any t with

ϵ′_{n} ⟶ 0. It follows that 1 ≤ \frac{l([x, y])}{l(k)} ≤ \frac{l(k) + ϵ′_{n}}{l(k)} ≤ 1 + ϵ′_{n} and that

1 ≤ \frac{l([h_{n}(x), h_{n}(y)])}{l(k′)} ≤ 1 + ϵ′_{n} for sufficiently large n, where l(.) denotes the

length in ℍ^{3}/ρ_{n}(π_{1}(M)). Therefore, we have d([x, y](\frac{l([x, y])}{l(k)}t), k(t)) ≤ 2ϵ′_{n}

for sufficiently large n. For the same reason, we have also

d([h_{n}(x), h_{n}(y)](\frac{l([h_{n}(x), h_{n}(y)])}{l(k′)}t), k′(t)) ≤ 2ϵ′_{n}.

By the property (iii), the simple closed curve ζ_{x} ∪k ∪ζ_{y} ∪k′ bounds a disc. Since k and k′ are geodesic segments, the function d(k(t), k′(\frac{l(k′)}{l(k)}t)) is convex.

Therefore we have d(k(t), k′(\frac{l(k′)}{l(k)}t)) ≤ 5ϵ for any t since d(x, h_{n}(x)) ≤ 5ϵ and d(y, h_{n}(y)) ≤ 5ϵ.

We define g_{n} : [x, y] → [h_{n}(x), h_{n}(y)] by the following formula g_{n}([x, y](\frac{l([x,y])}{l(k)}t)) = [h_{n}(x), h_{n}(y)](\frac{l([h_{n}(x),h_{n}(y)])}{l(k)}t). Setting z = [x, y](\frac{l([x,y])}{l(k)}t), the distance d(z, g_{n}(z)) is less than the following quantity

d([x, y](\frac{l([x, y])}{l(k)}t), k(t)) + d(k(t), k′(\frac{l(k′)}{l(k)}t))

+d(k′(\frac{l(k′)}{l(k)}t, [h_{n}(x), h_{n}(y)](\frac{l([h_{n}(x), h_{n}(y)])}{l(k)}t)).

Then we get d(z, g_{n}(z)) ≤ 5ϵ+4ϵ′_{n}. Now we conclude by taking N such that 4ϵ′_{n} ≤ ϵ for n ≥ N and by changing A_{n} by a homotopy so that the geodesic segment ζ_{n}(z) connecting z to g_{n}(z) lies in A_{n} for any z in s̄. □

Now we return to the proof of Lemma 5.1. We remove from s̄ its two extremal edges so that g_{n} is defined on the entire s̄ (and we say that s̄ originally had j + 2 edges so that it now has j edges). By the equation (1), there is a constant R′ such that for any branch b̂ of τ̂^{1}, we have l(f̃_{n}(b̂)) ≤ R′ϵ_{n}^{−1}. By Lemma 5.5 and since δ_{n} ⟶ 0 as n → ∞, we have l_{ρ_{n}}(g_{n}(s̄)) ≤ l_{ρ_{n}}(s̄) + 13ϵ ≤ jR′ϵ_{n}^{−1} + 13ϵ. Therefore if we let j′_{n} be the number of edges contained in g_{n}(s̄), we have

j′_{n} ≤ j(\frac{R′}{R}) + 1 (2)

for large n, where R is the constant in (b) of Lemma 4.1.

Since τ^{1} has only finitely many branches, once j is fixed, there are only finitely many possibilities for the oriented train routes of s̄ and of g_{n}(s̄). Thus we can find Θ(p_{n})-many disjoint segments {s̄_{1,n}, ...} with the properties (i), (ii) and (iii), having the same oriented train route, such that the segments g_{n}(s̄) also have the same oriented train route for all s̄ ∈ {s̄_{1,n}, … }.

Now we fix j and extract a subsequence so that the oriented train routes of s̄_{i,n} and g_{n}(s̄_{i,n}) do not depend on n.

By the same arguments, we can construct segments containing j+1 edges, j + 2 edges and so on. To avoid difficulties in the next part of the proof, we want to ensure that the segment containing j edges which we have chosen is contained in the segment with j + 1 edges as its first j segments. For that, to each s̄_{i,n}, we add the edge of c̄_{n} which is adjacent to the last (with respect to the orientation of s̄_{i,n}) edge of s̄_{i,n}. In this family of segments, take a maximal family of disjoint segments lying in c̄_{n}^{1}, which we denote by {s̄_{1,n}^{+}, s̄_{2,n}^{+}, ...}. By Claim 5.4 and the argument after that, there are only o(p_{n}) segments among {s̄_{1,n}, … } for which the added edge lies outside of c̄_{n}^{1}. Therefore the number of segments in {s̄_{1,n}^{+}, … } is Θ(p_{n}). It follows from the arguments we used for the segments s̄_{i,n} that among the segments s̄_{i,n}^{+} there are Θ(p_{n}) segments which have the properties (i), (ii) and (iii). The proof of Lemma 5.5 applies to these Θ(p_{n}) segments, yielding an homeomorphism g_{n} which can be chosen to coincide with the one defined on the segments s̄_{i,n} when restricted to them.

From this last family, we take Θ(p_{n}) segments s̄_{i,n}^{+} such that the oriented train routes of s̄_{i,n}^{+} and g_{n}(s̄_{i,n}^{+}) do not depend on i. Then we extract a subsequence (with respect to n) such that for sufficiently large n, the oriented train routes of s̄_{i,n}^{+} and of g_{n}(s̄_{i,n}^{+}) do not depend on n.

By doing this argument recursively, increasing j one by one, we complete the proof of Lemma 5.1. □

6. Commuting elements

In this section, we shall use the results of the preceding sections to construct a sequence of discs which cross λ less and less. This way, we shall obtain a homoclinic simple geodesic which does not cross λ.

Using Lemma 5.1, it is easy to observe that for j (the number of edges that s_{n} contains) large enough, s_{n} and s′_{n} come back simultaneously to a branch of τ_{1}. Still, the fact that the edges of c̄_{n}^{1} are very long makes it difficult to turn this observation into an actual construction. Instead we shall use Lemma 5.1 to construct two sequences a_{n}, a′_{n} of elements of π_{1}(∂M) which have nearly the same actions on some rather large part of f̃_{n}(τ̂_{1}). By results of Kapovich ([Ka]) this implies that the images of a_{n} and a′_{n} in π_{1}(M) commute and therefore correspond to an annulus. Then we shall construct discs as we want by adding some waves.

Recall that given an oriented train route (bo(i))_{i∈ℕ}, we denote by (b(i))_{i∈ℕ} the corresponding non-oriented train route. Since, by Lemma 5.1, for any t, the simple closed curve c_{n} goes through the oriented train routes (bo(i))_{i≤t} and (bo′(i))_{i≤t} in τ^{1} for n large enough, there are two half-leaves l_{+} and l′_{+} of the realised part of L_{rec} whose oriented train routes are (bo(i))_{i∈ℕ} and (bo′(i))_{i∈ℕ} respectively.

Let e ∈ ℕ be a natural number which we shall specify later, and let φ : ℕ → ℕ be an increasing function such that:

(ϕ1) bo(φ(i) + j) = bo(φ(0) + j) for any i ∈ ℕ and any 0 ≤ j ≤ e, i.e. the same train route (bo(φ(0) + j))_{0≤j≤e} is repeated starting from each φ(i).

(ϕ2) bo′(V (φ(i)) + j) = bo′(V (φ(0)) + j) for any i ∈ ℕ and any 0 ≤ j ≤

V (e), where V is the function which appeared in Lemma 5.1. (ϕ3) Suppose that l_{+} (resp. l′_{+}) is not a closed curve, and let k_{0,i} be a

sub-arc of l_{+} with train route (b(j))_{φ(0)≤j<φ(i)} (resp. k′_{0,i} the arc of l′_{+} with train route (b′(i))_{V (φ(0))≤i<V (φ(i))}). The two endpoints of k_{0,i} (resp k′_{0,i}) lie in the same switch of τ^{1} and the sequences (∂k_{0,i}) (resp. ∂k′_{0,i}) converges to a single point with respect to the Hausdorff topology as i ⟶ ∞.

The existence of such a function φ follows from the fact that the number of branches and switches of τ^{1} is finite. See Figure 5.

We define ψ : ℕ → ℕ as follows: ψ(i) is the largest integer such that φ(ψ(i)) ≤ i. Since φ is not surjective, it does not have an inverse function, and we use ψ instead. Set ψ_{n} = ψ(T (n) − e) for the function T which appeared in Lemma 5.1. Note that if we forget the condition (ϕ3), then ψ_{n} + 1 is the number of times that s_{n} comes back to the oriented train route (bo(k))_{φ(0)≤k≤e+φ(0)} and s′_{n} comes back to the oriented train route PSfrag replacements

k_{0,1}

bo(φ(1))
bo(φ(0))

bo(φ(1) + e)

bo(φ(0) + e)

Figure 5. Repeating the train route (bo(φ(0) + j))_{0≤j≤e}.

(bo′(k))_{V (φ(0))≤k≤V (φ(0))+V (e)} at the same time. Since φ is increasing and T is unbounded, (ψ_{n}) is unbounded.

Let us denote by b′ the branch bo′(V (φ(0))). By Lemma 5.1, for any i ≤ ψ_{n}, we have g_{n}∘f_{n}∘v(φ(i)) ∈ f_{n}(b′(V (φ(i))) = f_{n}(b′). It follows that, for any n large, there are i_{n}, j_{n} ≤ ψ_{n} such that the distance between g_{n}∘f_{n}∘v(φ(i_{n})) and g_{n} ∘ f_{n} ∘ v(φ(j_{n})) measured on f_{n}(b′) is at most \frac{1}{ψ_{n}} l_{ρ_{n}}(f_{n})(b′) = o(ϵ_{n}^{−1}). Allowing g_{n} ∘ f_{n} ∘ v(φ(i_{n})) and g_{n} ∘ f_{n} ∘ v(φ(j_{n})) to be a bit further from each other, we can still keep their distance to be o(ϵ_{n}^{−1}) while assuming that j_{n} − i_{n} ⟶ ∞.

We pick such sequences (i_{n}) and (j_{n}), i.e. for each n, we take two indices i_{n} < j_{n} ≤ ψ_{n} such that j_{n} − i_{n} ⟶ ∞ and the distance between g_{n} ∘ f_{n} ∘ v(φ(i_{n})) and g_{n} ∘ f_{n} ∘ v(φ(j_{n})) measured on f_{n}(b′) is o(ϵ_{n}^{−1}). We denote by I_{n} ⊂ s_{n} the segment between v(φ(i_{n})) and v(φ(j_{n})) and by J_{n} be the sub-segment of s_{n} consisting of the e vertices following v(φ(j_{n})).

Let s̃_{n} ⊂ ℍ^{3} be a lift of f_{n}(s_{n}), and let ṽ(φ(i_{n})), ṽ(φ(j_{n})), Ĩ_{n} and J̃_{n} be lifts of f_{n} ∘ v(φ(i_{n})), f_{n} ∘ v(φ(j_{n})), f_{n}(I_{n}) and f_{n}(J_{n}) respectively, lying on s̃_{n}. We lift the map g_{n} to a map g̃_{n} from s̃_{n} to a lift s̃′_{n} of f_{n}(s′_{n}). Let ρ_{n}(a_{n}) ∈ ρ_{n}(π_{1}(M)) be the covering translation which takes f̃_{n}(b̃(φ(i_{n})) to f̃_{n}(b̃(φ(j_{n})). Since bo(φ(i_{n}) + j) = bo(φ(j_{n}) + j) for all j ≤ e, the isometry ρ_{n}(a_{n}) acts as a translation on Ĩ_{n} ∪ J̃_{n}.

Let Ĩ′_{n} ⊂ s̃′_{n} be the piecewise geodesic segment between g̃_{n} ∘ ṽ(φ(i_{n})) and g̃_{n} ∘ ṽ(φ(j_{n})), and let J̃′_{n} ⊂ s̃′_{n} be the segment between g̃_{n} ∘ ṽ(φ(i_{n})) and g̃_{n} ∘ ṽ(φ(j_{n}) + e). Let ρ_{n}(a′_{n}) ∈ ρ_{n}(π_{1}(M)) be the covering translation which takes f̃_{n}(b̃′(V (φ(i_{n}))) to f̃_{n}(b̃′(V (φ(j_{n}))). From the assumption that bo′(V (φ(i_{n})) + j) = bo′(V (φ(j_{n})) + j) for any 0 ≤ j ≤ V (e), it follows that ρ_{n}(a′_{n}) acts as a translation on Ĩ′_{n} ∪ J̃′_{n}. See Figure 6.

By our choice of i_{n}, j_{n}, we have

d(g̃_{n} ∘ ṽ(φ(j_{n})), ρ_{n}(a′_{n}) ∘ g̃_{n} ∘ ṽ(φ(i_{n})) = o(ϵ_{n}^{−1}).

From this together with the facts that ρ_{n}(a_{n}) acts as a translation on Ĩ_{n} ∪J̃_{n} and that ρ_{n}(a′_{n}) acts as a translation on Ĩ′_{n}∪J̃′_{n}, we shall deduce the following claim:

PSfrag replacements

ṽ(φ(i_{n}))

ṽ(φ(j_{n}))
Ĩ_{n}
J̃_{n}
s̃_{n}
g̃_{n}
ρ_{n}(a_{n})

g̃_{n}(ṽ(φ(i_{n})))

g̃_{n}(ṽ(φ(j_{n})))

s̃′_{n}
ρ_{n}(a′_{n})
Ĩ′_{n}
J̃′_{n}

Figure 6. Commuting isometries

Claim 6.1. For R > 0, let V_{R}(J̃_{n}) be the R-neighbourhood of J̃_{n}, then for any sequence of points z̃_{n} ∈ V_{R}(J̃_{n}), we have that d(z̃_{n}, ρ_{n}(a_{n}^{−1}a′_{n})(z̃_{n})) is o(ϵ_{n}^{−1}).

Proof. It is sufficient to prove this claim for any sequence (z̃_{n}) lying on J̃_{n}. Since ρ_{n}(a_{n}^{−1}) acts as a translation on Ĩ_{n} ∪ J̃_{n}, the point ρ_{n}(a_{n}^{−1})(z̃_{n}) is the point z̃′_{n} ∈ Ĩ_{n} with

d(z̃_{n}, ṽ(φ(j_{n}))) = d(z̃′_{n}, ṽ(φ(i_{n}))), (3)

where d denotes the distance measured on Ĩ_{n} ∪ J̃_{n}. The point z̃′′_{n} = ρ_{n}(a′_{n}) ∘ g̃_{n}(z̃′_{n}) ∈ J̃′_{n} is at the distance d(g̃_{n}(z̃′_{n}), g̃_{n}∘ṽ(φ(i_{n})) from ρ_{n}(a′_{n})∘g̃_{n}∘ṽ(φ(i_{n})) (measured on Ĩ′_{n} ∪ J̃′_{n}). As we saw above, d(g̃_{n} ∘ ṽ(φ(j_{n})), ρ_{n}(a′_{n}) ∘ g̃_{n} ∘ ṽ(φ(i_{n}))) = o(ϵ_{n}^{−1}). Therefore we have d(z̃′′_{n}, g̃_{n} ∘ ṽ(φ(j_{n}))) = d(g̃_{n}(z̃′_{n}), g̃_{n} ∘ ṽ(φ(i_{n}))) + o(ϵ_{n}^{−1}).

By Lemma 5.1, g̃_{n} moves each point within distance 6ϵ, and hence we get the equality

d(z̃′′_{n}, g̃_{n} ∘ ṽ(φ(j_{n}))) = d(z̃′_{n}, ṽ(φ(i_{n}))) + o(ϵ_{n}^{−1}).

Using equation 3, we get

d(z̃′′_{n}, g̃_{n} ∘ ṽ(φ(j_{n}))) = d(z̃_{n}, ṽ(φ(j_{n})) + o(ϵ_{n}^{−1}).

Since J̃_{n} and J̃′_{n} are within the Hausdorff distance 6ϵ from each other, we have

d(z̃′′_{n}, z̃_{n}) = o(ϵ_{n}^{−1}).

By the triangle inequality

d(ρ_{n}(a′_{n}a_{n}^{−1})(z̃_{n}), z̃_{n}) ≤ d(z̃_{n}, z̃′′_{n}) + d(z̃′′_{n}, ρ_{n}(a′_{n}a_{n}^{−1})(z̃_{n})).

and by z̃′′_{n} = ρ_{n}(a′_{n}) ∘ g̃_{n} ∘ ρ_{n}(a_{n}^{−1})(z̃_{n}) we get

d(z̃′′_{n}, ρ_{n}(a′_{n}a_{n}^{−1})z̃_{n}) = d(ρ_{n}((a′_{n})^{−1})z̃′′_{n}, ρ_{n}(a_{n}^{−1})z̃_{n})
= d(g̃_{n} ∘ ρ_{n}(a_{n}^{−1})(z̃_{n}), ρ_{n}(a_{n}^{−1})(z̃_{n})) = o(ϵ_{n}^{−1}).

Thus we finally get d(ρ_{n}(a′_{n}a_{n}^{−1})(z̃_{n}), z̃_{n}) = o(ϵ_{n}^{−1}). □

We note that, as can be seen in the proof, the o(ϵ_{n}^{−1}) is “uniform”, namely there is a sequence δ_{n} ⟶ 0 independent of (z̃_{n}) such that d(ρ_{n}(a′_{n}a_{n}^{−1})(z̃_{n}), z̃_{n}) ≤ δ_{n}ϵ_{n}^{−1}.

We shall use this claim to prove the following lemma.

Lemma 6.2. There is N ∈ ℕ such that for n ≥ N, ρ_{n}(a_{n}^{−1}a′_{n}) = id.

The main argument in the proof is the following.

Lemma 6.3. Let [P_{n}, Q_{n}] ⊂ ℍ^{3} be a sequence of geodesic segments between P_{n} and Q_{n} such that l([P_{n}, Q_{n}]) is Θ(ϵ_{n}^{−1}) and let (δ_{n}), (δ′_{n}) ⊂ π_{1}(M) be two sequences such that the distances d(P_{n}, ρ_{n}(δ_{n})(P_{n})), d(P_{n}, ρ_{n}(δ′_{n})(P_{n})), d(Q_{n}, ρ_{n}(δ_{n})(Q_{n})) and d(Q_{n}, ρ_{n}(δ′_{n})(Q_{n})) are all o(ϵ_{n}^{−1}). Then there is N such that for n ≥ N, [ρ_{n}(δ_{n}), ρ_{n}(δ′_{n})] = id.

Proof. This comes directly from the arguments which Kapovich used in [Ka, Theorem 10.24] to prove that the minimal action of π_{1}(M) on ℝ-tree is small (see [Ka, p.239]). □

Proof of Lemma 6.2. Set e = 2p + 1 where p is the number of the branches of τ^{1}. If we fix some n ∈ ℕ, by our choice of e, we can find two different integers i_{1} and i_{2} between φ(j_{n}) and φ(j_{n}) + e such that bo(i_{1}) = bo(i_{2}). Let K_{n} ⊂ J_{n} be the segment of s_{n} with train route (b(k))_{i_{1}≤k≤i_{2}}, and let δ_{n} ∈ π_{1}(M) be an element which takes ṽ(i_{1}) to ṽ(i_{2}). Since i_{1}, i_{2} do not depend on n, the element δ_{n} does not depend on n either. Let us denote it by g. The isometry ρ_{n}(g) acts as a translation on the lift K̃_{n} of f_{n}(K_{n}) which lies in s̃_{n}.

Let K̃_{n−} and K̃_{n+} be the two extremal edges of K̃_{n}, such that ρ_{n}(g) takes K̃_{n−} to K̃_{n+}. By Claim 6.1, the translation length of ρ_{n}(a_{n}^{−1}a′_{n}) is o(ϵ_{n}^{−1}) on Ĩ_{n} ∪ J̃_{n}. By the same arguments, the translation length of ρ_{n}(a_{n}^{′−1}a_{n}) is also o(ϵ_{n}^{−1}) on Ĩ_{n} ∪ J̃_{n}. It follows that:

• for any sequence z̃_{n} ∈ V_{R}(K̃_{n+}), d(z̃_{n}, [ρ_{n}(a_{n}^{−1}a′_{n}), ρ_{n}(g)](z̃_{n})) =

o(ϵ_{n}^{−1});

• for any sequence z̃_{n} ∈ V_{R}(K̃_{n−}), d(z̃_{n}, [ρ_{n}(g^{−1}), ρ_{n}(a_{n}^{−1}a′_{n})](z̃_{n})) = o(ϵ_{n}^{−1});

• for any sequence z̃_{n} ∈ V_{R}(K̃_{n+}), d(z̃_{n}, [ρ_{n}(g), ρ_{n}(a_{n}^{′−1}a_{n})](z̃_{n})) = o(ϵ_{n}^{−1}).

Since K̃_{n+} and K̃_{n−} are edges of c̄_{n}^{1}, their lengths are Θ(ϵ_{n}^{−1}). Applying Lemma 6.3 to the segments K̃_{n−} and K̃_{n+}, we see that for sufficiently large n, [ρ_{n}(a_{n}^{−1}a′_{n}), ρ_{n}(g)], [ρ_{n}(g^{−1}), ρ_{n}(a_{n}^{−1}a′_{n})] and [ρ_{n}(g), ρ_{n}(a_{n}^{′−1}a_{n})] commute with ρ_{n}(a_{n}^{−1}a′_{n}). Therefore they belong to an elementary subgroup of ρ_{n}(π_{1}(M)). By [Ka, p.239], it follows that the group generated by ρ_{n}(a_{n}^{−1}a′_{n}) and ρ_{n}(g) is elementary.

Since i_{1} ≠ i_{2}, the distance of translation of ρ_{n}(g) is Θ(ϵ_{n}^{−1}). Since the group generated by ρ_{n}(a_{n}^{−1}a′_{n}) and ρ_{n}(g) is elementary, there are d ∈ π_{1}(M), t, t_{n} ∈ ℕ such that g = d^{t} and a_{n}^{−1}a′_{n} = d^{t_{n}}. Since the translation distance of ρ_{n}(g) is Θ(ϵ_{n}^{−1}), so is the translation distance of ρ_{n}(d). By Lemma 6.1 however, the translation distance of ρ_{n}(a_{n}^{−1}a′_{n}) is o(ϵ_{n}^{−1}). Therefore we have ρ_{n}(a_{n}^{−1}a′_{n}) = id for sufficiently large n. □

Now we are ready to construct the homoclinic geodesic which was announced at the beginning of the section.

Lemma 6.4. Under the light assumptions, assume that l_{ρ_{n}}(λ_{n}^{∗}) is bounded, then there is a homoclinic simple geodesic which does not cross |λ|.

Proof. Recall that, earlier in this section, we have picked a lift s̃_{n} ⊂ ℍ^{3} of f_{n}(s_{n}), and lifts ṽ(φ(i_{n})), ṽ(φ(j_{n})) of f_{n} ∘ v(φ(i_{n})) and f_{n} ∘ v(φ(j_{n})) respectively, lying on s̃_{n}. We have also lifted the map g_{n} to a map g̃_{n} from s̃_{n} to a lift s̃′_{n} of f_{n}(s′_{n}) and we have defined a_{n}, a′_{n} ∈ π_{1}(M).

To simplify the notations, set x̃_{n} = ṽ(φ(i_{n}) and ỹ_{n} = ṽ(φ(j_{n}). Recall that ỹ_{n} = ρ_{n}(a_{n})(x̃_{n}). By Lemma 6.2, ρ_{n}(a_{n})(g̃_{n}(x̃_{n})) = ρ_{n}(a′_{n})(g̃_{n}(x̃_{n})), which implies that ρ_{n}(a_{n})(g̃_{n}(x̃_{n})) lies on the same component of c̄_{n}^{1} as g̃_{n}(ỹ_{n}). It follows that we can change A_{n} so that ρ_{n}(a_{n})(ζ̃_{n}(x̃_{n})) (recall that ζ̃_{n}(x_{n}) is a short segment in Ã_{n} which joins x̃_{n} to g̃_{n}(x̃_{n})) lies in Ã_{n}. For the sake of simplicity, we set ζ̃_{n}(ỹ_{n}) = ρ_{n}(a_{n})(ζ̃_{n}(x̃_{n})).

Denote by k̃_{n} the segment of c̃_{n} which is mapped by f̃_{n} to an arc joining x̃_{n} to ỹ_{n}, namely ∂f̃_{n}(k̃_{n}) = {x̃_{n}, ỹ_{n}}. Similarly denote by k̃′_{n} the segment of c̃_{n} which is mapped by f̃_{n} to an arc joining ζ̃_{n}(x̃_{n}) to ζ̃_{n}(ỹ_{n}). By construction, k̃_{n} and k̃′_{n} lie in lifts of s_{n} and s′_{n} respectively and goe through the train routes (bo(i))_{φ(i_{n})≤i<φ(j_{n})} and (bo′(i))_{V (φ(i_{n}))≤i<V (φ(j_{n}))} respectively.

The arcs f̃_{n}(k̃_{n}), f̃_{n}(k̃′_{n}), ζ̃_{n}(x̃_{n}) and ζ̃_{n}(ỹ_{n}) bound a disc R̃_{n} in Ã_{n} (by Lemma 5.1) and all of their endpoints lie on the same component f̃_{n}(c̃_{n}) of ∂Ã_{n}. It follows that one component of Ã_{n} − R̃_{n} is a disc D̃_{n}. The boundary of this disc is the union of an arc d̃_{n} ⊂ f̃_{n}(c̃_{n}) and ζ̃_{n}(x̃_{n}) or ζ̃_{n}(ỹ_{n}). See Figure 7.

PSfrag replacements g̃_{n}(x̃_{n})

g̃_{n}(ỹ_{n})
x̃_{n}
ỹ_{n}
d̃_{n}
c̄_{n}
c̃_{n}^{∗}
D̃_{n}

Figure 7. The innermost disc D̃_{n}

Let us fix a reference hyperbolic metric on ∂M. The endpoints of k_{n} (resp. k′_{n}) are connected by an arc κ_{n} (resp. κ′_{n}) which lies on a switch of τ^{1}. By Lemma 6.2, the closed curves k_{n} ∪ κ_{n} and k′_{n} ∪ κ′_{n} are homotopic in M. In particular they bound an annulus E_{n} in M.

Consider the projection d_{n} ⊂ c_{n} of the arc which is mapped to d̃_{n} by f̃_{n}. The closed curve m_{n} = κ_{n}k_{n}d_{n}k_{n}^{′−1}κ_{n}^{′−1}d_{n}^{−1} (with appropriate choices of orientation) bounds a (possibly singular) disc which is the union of E_{n} and two copies of the projection D_{n} of D̃_{n} (see Figure 8).

Since both j_{n} − i_{n} and i_{n} can be assumed to go to ∞, it follows from the property (ϕ3) that the length of κ_{n} tends to 0 (with respect to our reference metric on ∂M). It follows that ∫_{m_{n}} dλ_{n} ⟶ 0. By the proof of the Loop Theorem (see [He] for example), there is a meridian m′_{n} such that i(m′_{n}, λ_{n}) ⟶ 0. By Casson’s criterion (Theorem 2.4) this concludes the proof of Lemma 6.4. □

PSfrag replacements κ_{n}

k_{n}
d_{n}
m_{n}

Figure 8. The curve m_{n}

7. Conclusion

In this section, we shall complete the proofs of Theorem 2 and 1.

Proof of Theorem 2. Now we complete the proof of Theorem 2. Let ρ_{n}, λ_{n} and λ be as in Theorem 2. If no subsequence of (ρ_{n}) converges algebraically, it follows from Lemma 4.1 that ρ_{n} and λ_{n} satisfy the light assumptions. Moreover l_{ρ_{n}}(λ_{n}) is bounded by assumption. By Lemma 6.4, there is a homoclinic geodesic which does not intersect |λ| transversely. By Lemma 2.6, this contradicts the assumption that λ ∈ D(M). □

We shall now deduce Theorem 1 from Theorem 2.

Theorem 1. Let M be a compact irreducible atoroidal 3-manifold with boundary. Let (m_{n}) be a sequence in the Teichmüller space T (∂M) which converges in the Thurston compactification to a projective lamination [λ] contained in PD(M). Let q : T (∂M) → GF_{0}(M, P) be the Ahlfors-Bers map, and suppose that (ρ_{n} : π_{1}(M)→P SL(2, ℂ)) is a sequence of discrete faithful representations corresponding to (q(m_{n})). Then passing to a subsequence, (ρ_{n}) converges in AH(N).

Proof. For a simple closed curve c ⊂ ∂M, we denote by l_{m_{n}}(c) the length of c with respect to the metric m_{n} and by l_{ρ_{n}}(c) the length of the closed geodesic of ℍ^{3}/ρ_{n}(π_{1}(M)) in the free homotopy class of c. By [Th2, Theorem 2.2] (see also [FLP]), there is a sequence of simple closed curves c_{n} ⊂ ∂M whose

projective classes converge to [λ] in PML(∂M) such that \frac{l_{m_{n}}(c_{n})}{l_{m_{0}}(c_{n})} tends to

0 as n → ∞.

Using the following result of [BrC], we shall get an upper bound for the sequence (l_{ρ_{n}}(λ_{n})).

Theorem C [Bridgeman-Canary] For any Q > 0, there is a constant K > 0 depending only on Q with the following properties. Let Γ be a finitely generated Kleinian group without torsion such that the shortest length of the meridians with respect to the compatible hyperbolic structure on Ω_{Γ}/Γ is greater than Q. Let C(Γ) be the convex core of ℍ^{3}/Γ, and consider the nearest point retraction r : Ω_{Γ}/Γ → ∂C(Γ). Then r is K-Lipschitz and has a homotopically inverse K-Lipschitz map.

Let us first verify that we are considering a situation where the hypothesis of this theorem is fulfilled.

Lemma 7.1. There is a positive number Q such that l_{m_{n}}(d) ≥ Q for any meridian d of ∂M.

Proof. Assuming the contrary, we have a sequence of meridians (d_{n}) such that (l_{m_{n}}(d_{n})) tends to 0. Let us extract a subsequence so that (d_{n}) converges with respect to the Hausdorff topology to a geodesic lamination D ⊂ M. By Casson’s criterion, D contains a homoclinic leaf. Since [λ] ∈ PD(M), the lamination D intersects the support of [λ] transversely (Lemma 2.6). It follows that the sequence i(λ, d_{n}) is bounded away from 0. This implies that the sequence l_{m_{n}}(d_{n}) tends to ∞. Thus we get a contradiction. □

It is clear that the length l_{ρ_{n}}(c_{n}^{∗}) is less than the length of any curve in ∂C(ρ_{n}(π_{1}(M))) which is freely homotopic to c_{n}. Thus, applying Theorem

C, we see that \frac{l_{ρ_{n}}(c_{n}^{∗})}{l_{m_{0}}(c_{n})} tends to 0.

Let us denote by λ_{n} the measured geodesic lamination obtained by endowing c_{n} with a Dirac measure whose weight is equal to l_{m_{0}}(c_{n})^{−1}. The sequence λ_{n} converges in ML(∂M) to a measured geodesic lamination λ

which lies in the projective class [λ]. Since \frac{l_{ρ_{n}}(c_{n}^{∗})}{l_{m_{0}}(c_{n})} tends to 0, we have

l_{ρ_{n}}(λ_{n}) ⟶ 0. Since λ lies in the projective class [λ] ∈ PD(M), the measured geodesic lamination λ lies in D(M). Applying Theorem 2, we see that a subsequence of (ρ_{n}) converges algebraically. □

8. Appendix

For the sake of completeness, we shall give brief proofs of some propositions which we cited in previous sections. Proposition 2.2 [[Le1], §2 paragraphs after Lemme 2.7] Let Σ and Σ′ ⊂ ∂_{χ<0}M be two compact, connected, incompressible surfaces which are disjoint or equal and do not contain any essential closed curve which can be homotoped into ∂_{χ=0}M. Let Σ̃ ⊂ ∂M̃ (resp. Σ̃′) be a connected component of the preimage of Σ (resp. Σ′) and let Γ ⊂ ρ(π_{1}(M)) (resp. Γ′) be the stabiliser of Σ̃ (resp. Γ′).

Then \overline{Σ̃} ∩ \overline{Σ̃}′ is either empty or equal to the limit set of Γ ∩ Γ′.

In the latter case, if Γ ∩ Γ′ is not cyclic, then it is the fundamental group of a (possibly twisted) I-bundle which is a connected component of a characteristic submanifold of (M, Σ∪Σ′). If Γ∩Γ′ is cyclic, then it is a finite index subgroup of a solid torus which is a connected component of a characteristic submanifold of (M, Σ ∪ Σ′).

Proof. Suppose that ξ belongs to the limit set of Γ∩Γ′. Since we are considering only geometrically finite groups, both Γ and Γ′ are convex cocompact. Let l be a geodesic ray from the origin O ∈ ℍ^{3} to ξ. Let F be a fundamental domain of the convex core of Γ containing O, and F′ a fundamental domain of the convex core of Γ′ containing O. Since Γ and Γ′ are convex cocompact, the diameters of F and F′ are bounded. Choose g_{n} ∈ Γ, g′_{n} ∈ Γ′ such that g_{n}F ∩ l ≠ ∅, g′_{n}F′ ∩ l ≠ ∅ and g_{n}F ∩ g′_{n}F′(≠ ∅) ⟶ ξ. Since the diameters of F and F′ are bounded, we have

d(g_{n}O, g′_{n}O) = d(O, g_{n}^{−1}g′_{n}O) ≤ K

for all n and a fixed K. By discreteness of ρ(π_{1}(M)), after passing to a subsequence, g_{n}^{−1}g′_{n} = g for all n. Then g_{n}^{−1}g′_{n} = g_{m}^{−1}g′_{m}, i.e., g_{m}g_{n}^{−1} = g′_{m}g_{n}^{′−1} ∈ Γ ∩ Γ′. Since Γ and Γ′ are convex cocompact, these elements are hyperbolic and the limit set of Γ ∩ Γ′ contains at least two points. Let h ∈ Γ ∩ Γ′ be a hyperbolic element. Then the invariant geodesics c̃, c̃′ of h in Σ̃, Σ̃′ descend to closed geodesics c and c′ in Σ and Σ′. Hence c and c′ bound an annulus A (not necessarily embedded) which is not homotoped into ∂M. Hence A must be contained in some characteristic submanifold of (M, Σ ∪ Σ′). If Σ = Σ′, then c = c′ and A is a Möbius band.

Let F = \overline{Σ̃} ∩ \overline{Σ̃}′ and C, C′ be the projections of the convex hulls of F in Σ̃ and Σ̃′. Then π_{1}(C) = Γ ∩ Γ′ and C ∪ C′ is the boundary of an I-bundle C × I which is essential in (M, Σ ∪ Σ′). If Σ = Σ′, then it is a twisted I-bundle with the boundary C. □

Next we include some proofs from [KlS] for the reader’s convenience.

Proposition 8.1. Suppose that the action of π_{1}(M) on an ℝ-tree T is minimal and small. Let S ⊂ ∂M be a compact compressible surface which has (possibly empty) geodesic boundary with respect to a hyperbolic metric on ∂M. Suppose that ϕ : T_{μ}→T is a π_{1}(M)-equivariant morphism which folds only at complementary regions. If μ ⊂ S is in tight position with respect to a meridian m ⊂ S, then |μ| can be extended to a geodesic lamination which contains a homoclinic leaf h in S.

Proof. Fix a hyperbolic metric on ∂M. The universal cover S̃ ⊂ ℍ^{2} is a convex subset of ℍ^{2}. Let m̄ represent an element of π_{1}(S) corresponding to the meridian m, which leaves invariant a lift m̃ ⊂ S̃. By equivariance, for x ∈ m̃,

ϕ(π(m̄x)) = ϕ(π(x))

since m̄ is trivial in π_{1}(M), where π : S̃→T_{μ} is an equivariant map. Since ϕ folds [π(x), π(m̄x)] only finitely many times, one can find segments Ĩ_{1}, Ĩ_{2} ⊂ m̃ such that Ĩ_{1} ∩ Ĩ_{2} = y ∈ m̃ and ϕ folds π(Ĩ_{1}) and π(Ĩ_{2}) along π(y). For x ∈ m ∩ μ, let μ_{x}^{+} denote a half leaf of μ starting from x to a chosen positive direction. Then using Skora’s idea, Kleineidam and Souto showed [KlS] (Proposition 3) that there are z_{i} ∈ I_{i} ∩ μ such that the lifts of μ_{z_{1}}^{+} and μ_{z_{2}}^{+} to ∂M̃ have the same endpoints. Let C be the complementary region of μ̃ in S̃, which contains the folding point y. Let μ̃_{1}, μ̃_{2} be boundary leaves of C. Up to reversing the orientation, we can assume that μ̃_{1}^{+}, μ̃_{2}^{+} are not asymptotic in S̃. Since the lifts of μ_{z_{1}}^{+} and μ_{z_{2}}^{+} to ∂M̃ have the same end points, by shrinking the intervals, we can see that the projection of μ̃_{1}^{+}, μ̃_{2}^{+} to ∂M̃ have the same endpoint. Let l be the geodesic in S̃ joining the end points of μ̃_{1}^{+}, μ̃_{2}^{+}. The projection of l to S becomes a homoclinic leaf disjoint from μ. □

Proposition 8.2. Let S be a compressible surface in ∂M, which contains a homoclinic leaf h. Then there is a sequence of meridians whose Hausdorff limit does not cross h.

Proof. Since a homoclinic leaf cannot be contained in an incompressible surface by Lemma 2.1, S(h̄) must contain a meridian m.

If h contains infinitely many homotopy classes of m-waves. Then there are (x_{i}), (y_{i}) ⊂ ℝ such that h(x_{i}), h(y_{i}) ∈ m and h[x_{i}, y_{i}] are non-homotopic mwaves. Since m is compact, after passing to a subsequence, we may assume that h(x_{i}) and h(y_{i}) converge. Hence for any ϵ > 0, we can choose i, j such that the lengths of segments [h(x_{i}), h(x_{j})], [h(y_{i}), h(y_{j})] ⊂ m are less than ϵ. Then h[x_{i}, y_{i}] ∪ h[x_{j}, y_{j}] ∪ [h(x_{i}), h(x_{j})] ∪ [h(y_{i}), h(y_{j})] is a meridian whose geodesic representative lies nearby the homoclinic leaf.

If h contains only finitely many homotopy classes of m-waves. then there is a meridian m and two disjoint half-leaves h^{+} and h^{−} of h such that h^{+} and h^{−} are in tight position with respect to m. Considering the intersections of m and h^{+} and h^{−} respectively, one obtains a picture similar to Figure 8, namely there is an arc k ⊂ h^{+} and an arc k′ ⊂ h^{−} which nearly bounds an annulus and a wave between k and k′. These arcs can be used to construct a sequence of meridians whose Hausdorff limit does not cross h as explained in the proof of [KlS, Proposition 1]. □

With more work, one could probably prove that h̄ is a Hausdorff limit of meridians. On the other hand, in all the situations we have used Proposition 8.2, with only little changes, we could have replaced it with the following weaker result whose proof is easier.

Lemma 8.3. Let S be a compressible surface which contains a homoclinic leaf h, and let β be a measured lamination which does not cross h. Then there is a sequence of meridians whose Hausdorff limit does not cross β.

Proof. Using cut-and-paste operation as in Claim 2.3, we construct a sequence of meridians m_{i} such that i(m_{i}, β) ⟶ 0. Start with a meridian m ⊂ S(h̄). Since h is homoclinic, it contains an m-wave. Using this wave as in the proof of Claim 2.3, we get a meridian m_{1} such that i(m_{1}, β) ≤ \frac{1}{2}i(m, β). Then we do the same again on m_{1}. Repeating this, we get a sequence of meridians m_{i} such that i(m_{i}, β) ≤ \frac{1}{2^{i}} i(m, β). □

References

[Ag] I. Agol, Tameness of hyperbolic 3-manifolds, preprint, arXiv, math.GT/0405568

[Ber] L. Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570–600.

[Bes] M. Bestvina, Degenerations of the hyperbolic space. Duke Math. J. 56 (1988), 143–161.

[Bo] F. Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), 71–158.

[BoO] F. Bonahon, J.-P. Otal, Laminations mesurées de plissage des variétés hyperboliques de dimension 3, Ann. of Math. (2) 160 (2004), 1013–1055.

[BrC] M. Bridgeman and R. D. Canary, From the boundary of the convex core to the conformal boundary. Geom. Dedicata 96 (2003), 211-240.

[Brom] K. Bromberg, Projective structures with degenerate holonomy and the Bers’ density conjecture, Annals of Math. 166 (2007), 77–93.

[BB] J. Brock and K. Bromberg, On the density of geometrically finite hyperbolic 3-manifolds, Acta Math. 192 (2004), 33–93.

[BS] K. Bromberg and J. Souto, The density conjecture: A prehistoric approach, in preparation.

[BCM] J. Brock, R. Canary and Y. Minsky, The classification of Kleinian surface groups, II: The Ending Lamination Conjecture, arXiv math.GT/0412006

[CG] D. Calegari and D. Gabai, Shrinkwrapping and the taming of hyperbolic 3manifolds. J. Amer. Math. Soc. 19 (2006), 385–446.

[Ca] R. Canary, Algebraic convergence of Schottky groups, Trans. Amer. math. Soc. 337 (1993), 235–258.

[CEG] R.D. Canary, D.B.A. Epstein, P. Green, Notes on notes of Thurston, Analytical and Geometrical Aspects of hyperbolic Space (1987), 3–92.

[FLP] A. Fathi, F. Laudenbach, V. Ponéaru, Travaux de Thurston sur les surfaces. Séminaire Orsay, Astérisque No.66-67 (1979).

[Fl] W. J. Floyd, Group completions and limit sets of kleinian groups, Invent. Math. 57 (1980), 205–218.

[JaS] W. Jaco, P.B. Shalen, Seifert fibered spaces in 3-manifolds, Memoirs Amer. Math. Soc. 220 (1979).

[Jo] K. Johannson, Homotopy equivalence of 3-manifolds with boundary, Lectures Notes in Mathematics 761 (1979).

[He] J. Hempel, 3-Manifolds, Annals of Mathematics Studies 86 (1976).

[Ka] M. Kapovich, Hyperbolic manifolds and discrete groups, Progress in Math.,183 (2000) Birkhäuser.

[KaL] M. Kapovich, B. Leeb, On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds, Geom. Funct. Anal. 5 (1995), 582–603.

[KlS] G. Kleineidam, J. Souto, Algebraic convergence of function groups, Comment. Math. Helv. 77 (2002), 244–269.

[Le1] C. Lecuire, Plissage des variété hyperboliques de dimension 3, Invent. Math. 164 (2006), no. 1, 85–141.

[Le2] C. Lecuire, An extension of Masur domain, Spaces of Kleinian groups, London Math. Soc. Lec. Notes 329, 49–73.

[Ma] H. Masur, Measured foliations and handlebodies, Ergodic Theory Dynam. Systems 6 (1986), 99–116.

[Min] Y. Minsky, The classification of Kleinian surface groups, I: Models and bounds, Ann. of Math. (2) 171 (2010), 1–107.

[MoO] J.W. Morgan, J.-P. Otal, Relative growth rates of closed geodesics on surfaces under varying hyperbolic structures, Comment. Math. Helv., 68 (1993), 171–208.

[MoS1] J.W. Morgan, P.B. Shalen, Degenerations of Hyperbolic structures I: Valuations, trees and surfaces, Ann. of Math., 120 (1984), 401–476.

[MoS2] J.W. Morgan, P.B. Shalen, Degenerations of Hyperbolic structures III: actions of 3-manifolds groups on trees and Thurston’s compactness theorem, Ann. of Math. 127 (1988), 457–519.

[NaS] H. Namazi and J. Souto, Non-realizability, ending laminations and the density conjecture, preprint

[Oh1] K. Ohshika, On limits of quasi-conformal deformations of Kleinian groups, Math. Z. 201 (1989),167–176.

[Oh2] K. Ohshika, Limits of geometrically tame Kleinian groups, Invent. Math. 99, (1990), 185–203.

[Oh3] K. Ohshika, Ending laminations and boundaries for deformation spaces of Kleinian groups. J. London Math. Soc. (2) 42 (1990), 111–121.

[Oh4] K. Ohshika, A convergence theorem for Kleinian groups which are free products. Math. Ann. 309 (1997), no. 1, 53–70.

[Oh5] K. Ohshika, Realising end invariants by limits of minimally parabolic groups, Geometry & Topology, 15 (2011) 827–890.

[Ot1] J.-P. Otal, Courants géodésiques et produits libres, Thèse d’Etat, Université Paris-Sud, Orsay (1988).

[Ot2] J.-P. Otal, Sur la dégénérescence des groupes de Schottky, Duke Math. J. 74 (1994), 777–792.

[Ot3] J.-P. Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996).

[Pa] F. Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbre réels, Invent. Math., 94 (1988), 53–80.

[Pe] J.L. Harer, R.C. Penner, Combinatorics of train tracks, Annals of Mathematics Studies 125 (1992).

[Sk] R.K. Skora, Splitting of surfaces, J. Amer. Math. Soc. 9 (1996), 605–616.

[ThB] W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381.

[Th1] W. P. Thurston, Hyperbolic structures on 3-manifolds. I: Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), 203–246.

[Th2] W.P. Thurston, Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, preprint arXiv:math.GT/9801045 (1998).

[Th3] W.P. Thurston, Hyperbolic structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary, preprint arXiv:math.GT/9801058 (1998).

Inkang Kim School of Mathematics KIAS, Hoegiro 85, Dongdaemun-gu Seoul, 130-722, Korea e-mail: inkang@kias.re.kr

Cyril Lecuire CNRS Institut de Mathématiques de Toulouse Université Paul Sabatier 118 route de Narbonne 31062 Toulouse Cedex 4 e-mail: lecuire@math.ups-tlse.fr

Ken’ichi Ohshika Department of Mathematics Graduate School of Science Osaka University Toyonaka, Osaka 560-0043, Japan email: ohshika@math.sci.osaka-u.ac.jp