Ian Biringer
email: biringer@bc.edu

Cyril Lecuire
email: cyril.lecuire@math.univ-toulouse.fr

We show that except for one exceptional case, a lamination on the boundary of a 3-dimensional handlebody H is a Hausdorff limit of meridians if and only if it is commensurable to a lamination with a 'homoclinic leaf'. This is a precise version of a philosophy called Casson's Criterion, which appeared in unpublished notes of A. Casson. Applications include a characterization of when a non-minimal lamination is a Hausdorff limit of meridians, in terms of properties of its minimal components, and a related characterization of which reducible self-homeomorphisms of ∂H have powers that extend to subcompressionbodies of H.

Introduction

Let H be a 3-dimensional handlebody1 with genus g ≥ 2 and let S := ∂H. A simple closed curve m on S is called a meridian if it bounds an embedded disk in H but not in S. Equip S with an arbitrary hyperbolic metric, and consider the set of geodesic laminations on S with the Hausdorff topology. We refer the reader to [START_REF] Casson | Automorphisms of surfaces after Nielsen and Thurston[END_REF] for more information on laminations.

1.1. Homoclinic leaves. In J.P. Otal's thesis [START_REF] Otal | Courants géodesique et produits libre[END_REF] the following is stated; it is attributed to an unpublished manuscript of A. Casson.

Statement 1.0.1 ('Casson's Criterion'). A geodesic lamination on S is a Hausdorff limit of meridians if and only if it has a homoclinic leaf.

We call it a 'statement' here instead of a theorem because it is not true as written, as we'll see later on. However, the connection between homoclinic leaves and meridians has been well studied, partly motivated by this statement, see for example the papers [START_REF] Otal | Courants géodesique et produits libre[END_REF][START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF][START_REF]An extension of the Masur domain, Spaces of Kleinian groups[END_REF][START_REF] Kleineidam | Algebraic convergence of function groups[END_REF][START_REF] Namazi | Non-realizability and ending laminations: Proof of the density conjecture[END_REF][START_REF] Jeon | Ken?ichi Ohshika, and Cyril Lecuire, Primitive stable representations of free kleinian groups[END_REF]] and Long's earlier paper [START_REF] Long | Discs in compression bodies[END_REF].

To define homoclinic, let H be the universal cover of H, which is homeomorphic to a thickened infinite tree. A path ℓ : R -→ H is called homoclinic m `m Figure 1. Laminations on the boundary of a genus two handlebody that have a meridian m as a leaf, but are not Hausdorff limits of meridians. if there are sequences s i , t i ∈ R such that

|s i -t i | → ∞, and sup i d H ( l(s i ), l(t i )) < ∞.
Here, distance in H is measured using the lift of any Riemannian metric on H. Since H is compact, the choice of metric does not matter. A path ℓ : R -→ S is called homoclinic if it has a lift to ∂ H that is homoclinic as above, and a complete geodesic on S is called homoclinic if it has a (possibly periodic, if the geodesic is closed) arc length parametrization that is homoclinic.

As an example, any geodesic meridian m on S is homoclinic, for as m lifts to a simple closed curve in ∂ H, any lift of a periodic parameterization of m is also periodic, and therefore homoclinic. On the other hand, if γ is an essential simple closed curve that is not a meridian, any periodic parameterization of γ lifts to a properly embedded biinfinite path in ∂ H that is invariant under a nontrivial deck transformation, and is readily seen to be non-homoclinic.

We should mention that the definition introduced above is not quite what Casson and Otal called 'homoclinic' (in French, 'homoclinique'), but rather what Otal calls 'faiblement homoclinique', or 'weakly homoclinic'. However, the definition above has been adopted in most subsequent papers. While some of the discussion below is incorrect if we use Casson's definition, it can all be modified to apply. In particular, the same counterexamples show that Statement 1.0.1 is still false using Casson's original definition. See §5.1.

In his thesis [START_REF] Otal | Courants géodesique et produits libre[END_REF], Otal showed that any Hausdorff limit of meridians has a homoclinic leaf. (This statement was later extended by Lecuire [START_REF] Lecuire | Structures hyperboliques convexes sur les variétés de dimension 3[END_REF] from handlebodies to more general 3-manifolds.) However, the converse is not true. First of all, any Hausdorff limit of meridians is connected, and there are disconnected laminations on S that have homoclinic leaves, e.g. the union of two disjoint simple closed curves, one of which is a meridian. There are also connected laminations with homoclinic leaves that are not Hausdorff limits of meridians. For example, let λ = m ∪ ℓ be a lamination with two leaves, where m is a nonseparating meridian and the two ends of ℓ spiral around m in the same direction, but from opposite sides, as on the left in Figure 1. Then λ has a homoclinic leaf, but it is not a Hausdorff limit of simple closed curves: any simple geodesic that approximates ℓ closely is trapped and forced to spiral forever around m, so cannot be closed.

As another example, let λ be the lamination on the right in Figure 1, which has three leaves, a meridian m and two leaves spiraling onto it. Then λ is a Hausdorff limit of simple closed curves, but it is not a limit of meridians. Indeed, given a simple closed curve µ on S, an arc α ⊂ µ is called an mwave disjoint from m if it is homotopic rel endpoints in H to an arc of m and int(α)∩m = ∅. Any meridian µ that intersects m has an m-wave disjoint from m: one looks for an 'outermost' arc of intersection with a disk bounded by µ in a disk bounded by m. However, no simple closed curve that is Hausdorff-close to λ has an m-wave disjoint from m, so λ is not a limit of meridians. See the discussion after the statement of Theorem 7.1 for more details.

In both of these examples, the problem lies with the spiraling isolated leaves. One way to address this is as follows. We say that two laminations µ 1 , µ 2 on S are commensurable if they contain a common sublamination ν such that for both i, the difference µ i \ ν is the union of finitely many isolated leaves. We say µ 1 , µ 2 are strongly commensurable if they contain a common ν such that for both i, the difference µ i \ ν is the union of finitely many isolated leaves, none of which are simple closed curves.

So, is Casson's Criterion at least true up to strong commensurability? It turns out the answer is still no: the lamination λ in Figure 3 contains a meridian as a leaf, but it is not strongly commensurable to a limit of meridians. Indeed, suppose that λ ′ is a Hausdorff limit of meridians that contains λ. In each component T ⊂ S \ m, there is a unique homotopy class rel m of m-waves in T . So λ ′ contains a leaf ℓ that either intersects T in an arc in this homotopy class, or is contained in T and is obtained by spinning an arc in this homotopy class around µ. In either case, ℓ intersects transversely the component of λ contained in T , a contradiction. One can also make similar examples of laminations that contain a meridian but are not even commensurable to a Hausdorff limit of meridians, by replacing the two curves on either side of m in Figure 3 with minimal laminations that fill the two components of S \ m.

It turns out, though, that this is basically the only counterexample. Let's say that a lamination λ on S is exceptional if S has genus 2, there is a separating meridian m on S that is either disjoint from λ or is a leaf of λ, and there are minimal sublaminations of λ that fill the two components of S \ m. Theorem 1.1 (A weak Casson Criterion, see Theorem 7.1). If λ is a geodesic lamination on S that is not exceptional, then λ is strongly commensurable to a Hausdorff limit of meridians if and only if it is strongly commensurable to a lamination with a homoclinic leaf. See Theorem 7.1 for a more general statement and for the proof. In our view, this is the strongest version of Casson's criterion that is likely to be true for arbitrary geodesic laminations. It may be, though, that the original Casson Criterion is true for minimal laminations. Our methods only work up to strong commensurability, though. For instance, if λ is minimal filling on S and contains a homoclinic leaf, to prove that λ is a limit of meridians one would have to ensure that the meridians produced in Lemma 5.8 do not run across any diagonals of the ideal polygons that are components of S \ λ.

The main tool in the proof of Theorem 1.1 is a complete characterization of the minimal laminations onto which the two ends of a homoclinic simple geodesic on S can accumulate. Theorem 1.2 (Limits of homoclinic geodesics, see Corollary 6.3). Suppose that h is a homoclinic simple biinfinite geodesic on S and that the two ends of h limit onto minimal laminations λ -, λ + ⊂ S. Then either [START_REF] Agol | Tameness of hyperbolic 3-manifolds[END_REF] the two ends of h are asymptotic on S, (2) one of λ -, λ + is an intrinsic limit of meridians, or

λ -, λ + are contained in incompressible subsurfaces S -, S + ⊂ S that bound an essential interval bundle B ⊂ H through which λ -and λ + are homotopic.

Here, a minimal lamination λ ⊂ S is an intrinsic limit of meridians if it is strongly commensurable to the Hausdorff limit of a sequence of meridians that are contained in the smallest essential subsurface S(λ) ⊂ S containing λ, see Proposition 5.11 for a number of equivalent definitions. We refer the reader to m Figure 3. A lamination that contains a meridian as a leaf, but is not strongly commensurable to a Hausdorff limit of meridians.

H = Y ⇥ [ 1, 1] B = Z ⇥ [ 1, 1] h h Figure 4.
Examples of homoclinic geodesics (in grey) satisfying (3) in Theorem 1.2. Theorem 6.1 and Corollary 6.3 for more precise and more general versions of the above that apply both to homoclinic biinfinite geodesics, and also to pairs of 'mutually homoclinic' geodesic rays on S.

Examples of (3) are shows in Figure 4. On the left, λ -, λ + are simple closed curves that bound an embedded annulus A in H and B is a regular neighborhood of A. The geodesic h is homoclinic since the annulus A lifts to an embedded infinite strip R × [-1, 1] ⊂ H and the two ends of a lift of h are asymptotic to R + × {-1} and R + × {1}, respectively. On the right, we write H = Y × [-1 , 1] where Y is a genus two surface with one boundary component. The laminations λ ± are minimal (in the picture they are drawn as 'train tracks') and fill Z × {±1}, where Z ⊂ Y is a torus with two boundary components. Here, B = Z × [-1 , 1]. Interval bundles are essential to the study of meridians on handlebodies, and it is no surprise that they appear in Theorem 1.2. For example, subsurfaces bounding such interval bundles are the 'incompressible holes' studied by Masur-Schleimer [START_REF] Masur | The geometry of the disk complex[END_REF], and interval bundles appear frequently in Hamenstädt's work on the disk set, see e.g. [START_REF]Hyperbolic relatively hyperbolic graphs and disk graphs[END_REF][START_REF]Asymptotic dimension and the disk graph i[END_REF][START_REF]Asymptotic dimension and the disk graph ii[END_REF]. We note that the interval bundles B appearing in Theorem 1.2 may be twisted interval bundles over non-orientable surfaces, in which case λ -= λ + and S -= S + . See §4.5 for background on interval bundles. 1.2. Hausdorff limits via their minimal sublaminations. The previous two theorems suggest that if a lamination λ that is a Hausdorff limits of meridians, one might expect to see minimal sublaminations λ that are intrinsic limits of meridians, or pairs of components that are homotopic through essential interval bundles in H. In fact, we show the following. Theorem 1.3 (see Theorem 7.1). Suppose that λ ⊂ S is a nonexceptional geodesic lamination that is a finite union of minimal components. Then λ is strongly commensurable to a Hausdorff limit of meridians if and only if either [START_REF] Agol | Tameness of hyperbolic 3-manifolds[END_REF] λ is disjoint from a meridian on S, (2) some component of λ is an intrinsic limit of meridians, or (3) there are components λ ± ⊂ λ that fill incompressible subsurfaces S ± ⊂ S, such that S ± bound an essential interval bundle B ⊂ H, the laminations λ ± are essentially homotopic through B, and there is a compression arc α for B that is disjoint from λ.

In (3), a compression arc for B is an arc from ∂S -to ∂S + that is homotopic in H, keeping its endpoints in ∂S ± , to a fiber of the interval bundle B. See §2.7 and Figure 6 for more explanation.

Note that Theorem 1.3 does not say anything interesting about which minimal filling laminations λ on S are strongly commensurable to Hausdorff limits of meridians, just that that happens if and only if (2) holds, which is trivial. Indeed, for minimal filling laminations, it is not clear that there should be an easy way to 'identify' Hausdorff limits of meridians. The point of Theorem 1.3, though, is that it reduces the characterization of Hausdorff limits of meridians to the minimal filling case. We note that it should be possible to replace the part of the proof of Theorem 1.3 that references homoclinic geodesics with arguments similar to those used in Masur-Schleimer's paper [START_REF] Masur | The geometry of the disk complex[END_REF]. 1.3. Extension of reducible maps to compression bodies. As another application of our techniques, we consider extension properties of homeomorphisms f : S -→ S. To motivate this, recall that a subcompression body of H is a 3-dimensional submanifold C ⊂ H with S ⊂ ∂C that is obtained by choosing a finite collection Γ of disjoint meridians on S, taking a regular neighborhood of S and a collection of discs in H with boundary Γ, and adding in any complementary components that are topological 3-balls. We say C is obtained by compressing Γ. We usually consider subcompression bodies only up to isotopy, and we allow the case that Γ = ∅, in which case we recover the trivial subcompression body, which is just a regular neighborhood of S.

Two examples where Γ contains a single meridian are drawn in Figure 5. On the left, we compress a separating meridian m 1 and obtain a subcompression body of H that has two 'interior' boundary components contained in int(H); these are the tori drawn in gray. On the right, we compress a nonseparating meridian m 2 and obtain a subcompressionbody with a single torus interior boundary component. Note that compressing Γ = {m 1 , m 2 } gives the same subcompression body as compressing m 2 , because we fill in complementary components that are balls. See §2.4 for details. Biringer-Johnson-Minsky [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF] showed2 that the attracting lamination λ + of a pseudo-Anosov map f : S -→ S is strongly commensurable to a Hausdorff limit of meridians if and only if f has a nonzero power that extends to a homeomorphism of some subcompression body of H. Both statements are 'genericity' conditions on f with respect to the structure of H and both were previously studied in the literature, see e.g. [START_REF] Namazi | Heegaard splittings and pseudo-anosov maps[END_REF] and [START_REF] Lackenby | Attaching handlebodies to 3-manifolds[END_REF].

Here, we show that extension of powers of a homeomorphism f : S -→ S to subcompression bodies can be detected by looking at extension of powers of its components in the Nielsen-Thurston decomposition. More precisely, recall that f is pure if there are disjoint essential subsurfaces S i ⊂ S, such that f = id on S id := S \ ∪ i S i , and where for each i, if we set f i := f | S i , then either (1) S i is an annulus and f i is a power of a Dehn twist, or (2) f i is a pseudo-Anosov map on S i . It follows from the Nielsen-Thurston classification [START_REF] Farb | A primer on mapping class groups[END_REF], that every homeomorphism of S has a power that is isotopic to a pure homeomorphism.

Theorem 1.4 (Partial extension of reducible maps, see Theorem 9.2). Let f : S -→ S be a pure homeomorphism. Then f has a power that extends to a nontrivial subcompressionbody of H if and only if either:

(1) there is a meridian in S id , (2) for some i, the map f i : S i -→ S i has a power that extends to a nontrivial subcompressionbody of H that is obtained by compressing a set of meridians in S i , or (3) there are (possibly equal) indices i, j such that S i , S j bound an essential interval bundle B in H, such that some power of f | S i ∪S j extends to B, and there is a compression arc α for B whose interior lies in S id .

In [START_REF] Biringer | Automorphisms of the compression body graph[END_REF], note that if (for simplicity) B is a trivial interval bundle, then f | S i ∪S j extends to B exactly when f i , f j become isotopic maps when S i , S j are identified through B. More generally, a power of f | S i ∪S j extends to B when f j is obtained from f i by multiplying by a periodic map that commutes with f i .

For the proof of Theorem 1.4, it is necessary to extend the theorem of Biringer-Johnson-Minsky [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF] referenced above to the case of pseudo-Anosovs on essential subsurfaces of S. More precisely, we show that (2) above holds exactly when the attracting lamination of f i is an intrinsic limit of meridians. This is done in Theorem 8.1; the proof is basically the same as theirs, although we reorganize it into separate topological and geometric arguments in a way that makes it clearer than the original version.

In contrast to the pseudo-Anosov case, one cannot determine when a pure homeomorphism f : S -→ S has a power that extends by looking at its attracting lamination. Here, the 'attracting lamination' of f is the union of all its twisting curves and all the attracting laminations of its pseudo-Anosov components. Indeed, set

H = Y × [-1, 1],
where Y is a surface with boundary, and let f : Y -→ Y be a pseudo-Anosov map such that f = id on ∂Y . Then

S = Y × {-1} ∪ ∂Y × [-1, 1] ∪ Y × {+1}
and if we let F : S -→ S be f × id on Y × {±1} and the identity map on the rest of S, while we let G : S -→ S be f × id on Y × {1}, and f 2 × id on Y × {-1}, and the identity map on the rest of S, then F, G have the same attracting lamination, but F extends to a homeomorphism of H, while G does not. In fact, it follows from Theorem 1.4 that no power of G extends to a nontrivial subcompression body of H.

Other results of interest.

There are two other theorems in this paper that we should mention in the introduction.

In §3 we study the disk set D(S, M ) of all isotopy classes of meridians in an essential subsurface S ⊂ ∂M with ∂S incompressible, where here M is a compact, irreducible 3-manifold with boundary. We show in Proposition 3.1 that either D(S, M ) is small, meaning that it is either empty, has a single element, or has a single non-separating element and infinitely many separating elements that one can explicitly describe, or D(S, M ) is large, meaning that it has infinite diameter in the curve complex C(S). This result will probably not surprise any experts, but we have never seen it in the literature.

In §4 we show how essential interval bundles in a compact 3-manifold with boundary M can be seen in the limit sets in ∂H 3 associated to hyperbolic metrics on int(M ). This picture was originally known to Thurston [START_REF] Thurston | Hyperbolic structures on 3-manifolds. III. Deformations of 3manifolds with incompressible boundary[END_REF], and was studied previously under more restrictive assumptions by Walsh [START_REF] Walsh | The bumping set and the characteristic submanifold[END_REF] and Lecuire [START_REF] Lecuire | Structures hyperboliques convexes sur les variétés de dimension 3[END_REF]. We need a more general theorem in the proof of Theorem 6.1: in particular, we need a version that allows accidental parabolics. This is Theorem 4.1. Our proof is also more direct and more elementary than those of [START_REF] Walsh | The bumping set and the characteristic submanifold[END_REF][START_REF] Lecuire | Structures hyperboliques convexes sur les variétés de dimension 3[END_REF]. See §4 for more context and details. 1.5. Outline of the paper. Section 2 contains all the necessary background for the rest of the paper. We discuss the curve complex, the disc set, compression bodies, interval bundles, the Jaco-Shalen and Johannson characteristic submanifold theory, compression arcs, and geodesic laminations. §3 and §4 are described in the previous subsection. §5 contains a discussion of homoclinic geodesics, intrinsic limits of meridians, and some of their basic properties. The main point of §6 is Theorem 6.1, which is the more precise and general version of Theorem 1.2 above. §7 is devoted to Theorem 7.1, which characterizes Hausdorff limits of meridians and combines Theorems 1.1 and 1.3 above. §8 contains our extension of Biringer-Johnson-Minksy [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF] to partial pseudo-Anosovs, and §9 contains the proof of Theorem 9.2, which generalizes Theorem 1.4 above.

an immersed surface X -→ S such that the restriction to int(X) and to each component of ∂X is an embedding, and where each component of ∂X maps to a simple closed geodesic on S. In (2), the point is that our surface is basically an embedding, except that we allow two boundary components of X to map to the same geodesic in S. We will usually suppress the immersion and write X ⊂ S, abusing notation. We consider X, Y ⊂ S to be equal if they are either the same simple closed geodesic, or if they are both immersions as in [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF] and the interiors of their domains have the same images. We say X, Y are essentially disjoint if either:

• X, Y are disjoint simple closed geodesics,

• X is a simple closed geodesic, Y is not, and X is disjoint from int(Y ), or vice versa with X, Y exchanged, or • X, Y have nonempty disjoint interiors. More generally, we define a (possibly disconnected) subsurface with geodesic boundary in S to be a finite union of essentially disjoint connected subsurfaces with geodesic boundary.

Any connected essential subsurface T ⊂ S that is not an annulus homotopic into a cusp of S determines a unique connected subsurface with geodesic boundary X such that the images of π 1 T and π 1 X in π 1 S are conjugate. Here, we say that X is obtained by tightening T . More generally, we can tighten a disconnected T to a disconnected X by tightening all its components.

Tightening is performed as follows. If T is an annulus, then we let X be the unique simple closed geodesic homotopic to the core curve of T . Otherwise, we obtain X by homotoping T so that every component of ∂T is either geodesic or bounds a cusp in S \ T , and then adding in any components of S \ T that are cusp neighborhoods. Alternatively, let T be a component of the pre-image of T in the universal cover S, which is isometric to a convex subset of H 2 , let Λ T ⊂ ∂H 3 be the set of limit points of T on ∂ ∞ H 2 , and let X be the convex hull of Λ T within S. Then X projects to an X as desired.

Conversely, suppose X is a subsurface with geodesic boundary in S. Then there is a compact essential subsurface T ↩→ S, unique up to isotopy and called a resolution of X, that tightens to X. When X is a simple closed geodesic, we take T to be a regular neighborhood of X. Otherwise, construct T by deleting half-open collar neighborhoods of all boundary components of X, and deleting open neighborhoods of all cusps of T .

Note that subsurfaces with geodesic boundary X, Y are essentially disjoint if and only if they admit disjoint resolutions.

2.2. The curve complex. Let S be a compact orientable surface, possibly with boundary, and assume that S is not an annulus.

Definition 2.1. The curve complex of S, written C(S), is the graph whose vertices are homotopy classes of nonperipheral, essential simple closed curves on S and whose edges connect homotopy classes that intersect minimally.

When S is a 4-holed sphere, minimally intersecting simple closed curves intersect twice, while on a punctured torus they intersect once. Otherwise, edges in C(S) connect homotopy classes that admit disjoint representatives.

Masur-Minsky [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF] have shown that the curve complex is Gromov hyperbolic, when considered with the path metric in which all edges have unit length. Klarreich [START_REF] Klarreich | The boundary at infinity of the curve complex and the relative teichmüller space[END_REF] (see also [START_REF] Hamenstädt | Train tracks and the Gromov boundary of the complex of curves, Spaces of Kleinian groups[END_REF]) showed that the Gromov boundary ∂ ∞ C(S) is homeomorphic to the space of ending laminations of S: i.e. filling, measurable geodesic laminations on S with the topology of Hausdorff super-convergence.

2.3.

The disc set. Suppose that S ⊂ ∂M is an essential subsurface of the boundary of a compact, irreducible 3-manifold M , and that ∂S is incompressible in M . An essential simple closed curve γ on M is called a meridian if it bounds an embedded disc in M . By the loop theorem, γ is a meridian if and only if it is homotopically trivial in M . Definition 2.2. The disc set of S in M , written D(S, M ), is the (full) subgraph of C(S) whose vertices are the meridians of S in M .

When convenient, we will sometimes regard D(S, M ) as a subset of the space of projective measured laminations PML(S), instead of as a graph.

The following is an extension of a theorem of Masur-Minsky [42, Theorem 1.1], which they prove in the case that S is an entire component of ∂M . To prove Theorem 2.3 as stated above, one follows the outline of [START_REF]Quasiconvexity in the curve complex[END_REF]: given a, b ∈ D(S, M ), the goal is to construct a well-nested curve replacement sequence from a = a 1 , . . . , a n = b consisting of meridians, which must be a quasi-geodesic by their Theorem 1.2. The sequence (a i ) is created by successive surgeries along innermost discs, and the only difference here is that one needs to ensure that none of the surgeries create peripheral curves. However, the surgeries create meridians and S has incompressible boundary.

Compression bodies.

We refer the reader to §2 of [START_REF] Biringer | Automorphisms of the compression body graph[END_REF] for a more detailed discussion of compression bodies, and state here only a few definitions that will be used later on.

A compression body is a compact, orientable, irreducible 3-manifold C with a Suppose M is a compact irreducible 3-manifold with boundary, let Σ be a component of ∂M and let S ⊂ Σ is an essential subsurface. A subcompression body of (M, S) is a compression body C ⊂ M with exterior boundary Σ that can be constructed as follows. Choose a set Γ of disjoint, pairwise nonhomotopic simple closed curves on S that are all meridians in M . Let C ′ ⊂ M be the union of Σ with a set of disjoint disks in M whose boundaries are the components of Γ, and define C ⊂ M to be the union of a regular neighborhood of C ′ ⊂ M together with any components of the complement of this neighborhood that are topological 3-balls. Here, we say that C ⊂ M is obtained by compressing Γ. Note that the irreducibility of M implies that no component of ∂C is a 2-sphere, and hence that C is irreducible, and therefore a compression body. See [3, §2] for details about constructing compression bodies via compressions.

When the set Γ above is empty, we obtain the trivial subcompression body of (M, S), which is just a regular neighborhood of Σ ⊂ ∂M . At the other extreme, we can compress a maximal Γ, which gives the 'characteristic compression body' of (M, S), defined via the following fact.

Fact 2.4 (The characteristic compression body). Suppose M is an irreducible compact 3-manifold, that Σ is a component of ∂M and that S ⊂ Σ is an essential subsurface such that the multicurve ∂S is incompressible in M . Then there is a unique (up to isotopy) subcompression body

C := C(S, M ) ⊂ M
of (M, S), called the characteristic compression body of (M, S), such that a curve γ in S is a meridian in C if and only if it is a meridian in M .

Moreover, C can be constructed by compressing any maximal set of disjoint, pairwise nonhomotopic meridians in S. This is a version of a construction of Bonahon [START_REF] Bonahon | Cobordism of automorphisms of surfaces[END_REF], except that he only defines the characteristic compression body when S is an entire boundary component of M . In that case, the interior boundary components of M are incompressible in M , so Bonahon's construction can be used to reduce problems about 3manifolds to problems about compression bodies and about 3-manifolds with incompressible boundary.

The reader can also compare the fact to Lemma 2.1 in [START_REF] Biringer | Automorphisms of the compression body graph[END_REF], which is the special case of the fact where M is a compression body and S is its exterior boundary, so that C = M is obtained by compressing any maximal set of disjoint, nonhomotopic meridians in M .

Proof. Let Γ be a maximal set of disjoint, pairwise nonhomotopic M -meridians on S, and define C by compressing Γ. We have to check that any curve in S that is an M -meridian is also a C-meridian. Suppose not, and take an Mmeridian m ⊂ S that is not a C-meridian, and that intersects Γ minimally. Since Γ is maximal, m intersects some component γ ⊂ Γ. Then there is an arc α ⊂ γ with endpoints on m and interior disjoint from m, that is homotopic rel endpoints in M to the arcs β ′ , β ′′ ⊂ m with the same endpoints. (Here α is an 'outermost' arc of intersection on a disk bounded by γ, where the intersection is with the disk bounded by m, see e.g. Lemma 2.8 in [START_REF] Biringer | Automorphisms of the compression body graph[END_REF].) Since m is in minimal position with respect to Γ, the curves m ′ = α ∪ β ′ and m ′′ = α ∪ β ′′ are both essential, and are M -meridians in S that intersects Γ fewer times than m. So by minimality of m, both m ′ , m ′′ are C-meridians, implying that α is homotopic rel endpoints to β ′ and β ′′ in C. This implies m is a C-meridian, contrary to assumption.

For uniqueness, suppose we have two subcompression bodies C 1 , C 2 of (M, S) in which all curves in S that are meridians in M are also meridians in C 1 , C 2 . Since C 1 , C 2 are subcompression bodies of (M, S), the kernels of the maps All 3-manifolds in this paper are assumed to be orientable, but even when the total space B of an interval bundle is orientable, the base surface Y may not be. Indeed, let Y be a compact non-orientable surface and let π : Ŷ -→ Y be its orientation cover. Then the mapping cylinder

π 1 Σ -→ π 1 C i induced
B := Ŷ × [0, 1]/ ∼, (x, 1) ∼ (x ′ , 1) ⇐⇒ π(x) = π(x ′ )
is orientable, and is a twisted interval bundle over Y , where the fiber over y ∈ Y is obtained by gluing together the two intervals {x} × [0, 1] and {x ′ } × [0, 1] along (x, 1) and (x ′ , 1), where π -1 (y) = {x, x ′ }. The horizontal boundary ∂ H B here is Ŷ × {0}, which is homeomorphic to the orientable surface Ŷ . Note that B is double covered by the trivial interval bundle Ŷ × [-1, 1]. Fact 2.5. Suppose that B -→ Y is an interval bundle and B is orientable. If Y is orientable, then B is a trivial interval bundle. If Y is nonorientable, then B is isomorphic to the mapping cylinder of the orientation cover of Y .

Proof. If Y and B are orientable, so is the line bundle, so the bundle is trivial. If Y is nonorientable, the horizontal boundary ∂ H B ⊂ ∂B is an orientable surface that double covers Y , and from there it's easy to construct the desired isomorphism to the mapping cylinder of the projection ∂ H B -→ Y . □

An interval bundle B -→ Y comes with a canonical involution σ, which is well defined up to isotopy, and which is defined as follows.

If B ∼ = Y × [0, 1] is a trivial interval bundle, we define σ : Y × [-1, 1] -→ Y × [-1, 1], σ(y, t) = (y, -t).
And if B is the twisted interval bundle B ∼ = Ŷ × [0, 1]/ ∼ above, we define

σ : Ŷ × [0, 1]/ ∼-→ Ŷ × [0, 1]/ ∼, σ(ŷ, t) = (ι(ŷ), t)
where ι is the nontrivial deck transformation of the orientation cover. Note that σ is always an orientation reversing involution of B, so in particular, when we give the surface ∂ H B its boundary orientation, the restriction σ| ∂ H B is also orientation reversing.

We also recall the following well-known fact.

Fact 2.6. If S is a compact, orientable surface with nonempty boundary, The trivial interval bundle S × [-1, 1] is homeomorphic to a handlebody.

It's a nice topology exercise to visualize the homeomorphism. Regard S as the union of a polygon and a collection of bands (long, skinny rectangles), each of which is glued along its short sides to two sides of the polygon. Thickening, the picture becomes a ball with 1-handles attached.

Note that if S = S g,b has genus g and b boundary components, then the handlebody S × [-1, 1] has genus 2g + b -1, since that is the rank of the free group

π 1 (S × [-1, 1]) ∼ = π 1 S.
Finally, suppose π : B -→ Y is an interval bundle and f : ∂ H B -→ ∂ H B is a homeomorphism. We say that f extends to B if there is a homeomorphism F : B -→ B such that F | ∂ H B = f . We leave the following to the reader.

Fact 2.7. The following are equivalent:

(1) f extends to B, (2) f • σ is isotopic to f on ∂ H B, (3) after isotoping f , there is a homeomorphism f : Y -→ Y such that π • f = f • π, (4
F : Y × [-1, 1] -→ Y × [-1, 1], F (y, t) = ( f (y), t),
for some homeomorphism f : Y -→ Y , or

F : Ŷ × [0, 1]/ ∼ -→ Ŷ × [0, 1]/ ∼, F (y, t) = ( f (y), t),
for some homeomorphism f : Ŷ -→ Ŷ commuting with the deck group of Ŷ -→ Y , and hence covering a homeomorphism of Y .

2.6. The characteristic submanifold of a pair. Suppose that M is a compact, orientable 3-manifold and that S ⊂ ∂M is an incompressible subsurface. In the late 1970s, Jaco-Shalen [START_REF] Jaco | Seifert fibered spaces in 3-manifolds[END_REF] and Johannson [START_REF] Johannson | Homotopy equivalences of 3-manifolds with boundaries[END_REF] described a 'characteristic' submanifold of (M, S) that contains the images of all nondegenerate maps from interval bundles and Seifert fibered spaces.

Theorem 2.8 (see pg 138 of Jaco-Shalen [START_REF] Jaco | Lectures on three-manifold topology[END_REF]). There is a perfectly embedded Seifert pair (X, Σ) ⊂ (M, S), unique up to isotopy and called the characteristic submanifold of (M, S), such that any nondegenerate map (B, F ) -→ (M, S) from a Seifert pair (B, F ) is homotopic as a map of pairs into (X, Σ).

A Seifert pair is 3-manifold pair that is a finite disjoint union of interval bundle pairs (B, ∂ H B) and S 1 -bundle pairs. Here, an S 1 -bundle pair (B, F ) is a 3-manifold B fibered by circles, where F ⊂ ∂B is a compact subsurface saturated by fibers. A Seifert pair (X, Σ) ⊂ (M, S) is well embedded if X ∩ ∂M = Σ ⊂ S and the frontier of X in M is a π 1 -injective surface, and is perfectly embedded if it is well embedded, no component of the frontier of X in M is homotopic into S, and no component of X is homotopic into another component.

When (B, F ) is a connected Seifert pair, a map f : (B, F ) -→ (M, S) is essential if it is not homotopic as a map of pairs into S. Notice that this only depends on the image of f and not on f itself. One says f is nondegenerate if it is essential, its π 1 -image is nontrivial, its π 1 -image is non-cyclic when F = ∅, and no fiber of B is nullhomotopic in (M, S). For disconnected (B, F ), one says f is nondegenerate if its restriction to every component is nondegenerate.

The following is very well known. Fact 2.9. If int(M ) is hyperbolizable and (B, F ) is an S 1 -bundle pair that is perfectly embedded in (M, S), then either

(1) (B, F ) is a 'fibered solid torus', i.e. B is an S 1 -bundle over a disk, and

F ⊂ ∂B ∼ = T 2 is a collection of fibered parallel annuli, or (2)
(B, F ) is a 'thickened torus', i.e. B is an S 1 -bundle over an annulus, so is homeomorphic to T 2 × [0, 1], and each component of F is either a torus or a fibered annulus.

So in particular, the components of the characteristic submanifold of (M, S) are either interval bundles, solid tori, or thickened tori.

Proof. Suppose that (B, F ) is a perfectly embedded S 1 -bundle pair in M . Then B -→ Y is an S 1 -bundle, where Y is a compact 2-orbifold, and the cyclic subgroup Z ⊂ π 1 B corresponding to a regular fiber is normal in π 1 B. In a hyperbolic 3-manifold, any subgroup of π 1 that has a cyclic normal subgroup is elementary, say by a fixed point analysis on ∂ ∞ H 3 . So, π 1 B is either cyclic or isomorphic to Z 2 . It follows that Y is a disc, in which case B is a fibered solid torus, or Y is an annulus, in which case B is a thickened torus. □

In this paper we will mostly be interested in interval bundles. For brevity, we'll use the following terminology, which differs slightly from the terminology above used by Jaco-Shalen. Definition 2.10. An essential interval bundle in (M, S) is an essential, wellembedded interval bundle pair (B, ∂ H B) ↩→ (M, S).

Note that the horizontal boundary of any essential interval bundle is an incompressible subsurface of S.

The definition above differs from a well embedded interval bundle pair in that we are excluding boundary-parallel interval bundles over annuli, and differs from a perfectly embedded interval bundle pair in that we are allowing components of the frontier of an interval bundle over a surface that is not an annulus to be boundary parallel. For instance, if Y is a surface with boundary and Y ′ ⊂ Y is obtained by deleting collar neighborhoods of the boundary components, and we set 6. To link this definition with more classical ones, it is easy to see that there is a compression arc for B if and only if Fr(B) is boundary compressible, see [24, pp.36-37] for a definition.

M = Y × [-1, 1], which is a handlebody, then (Y ′ × [-1, 1], Y ′ × {-1, 1}) is an essential interval bundle in (M,
Write our interval bundle as π : B -→ Y . Let α be a compression arc for B. After isotoping the bundle map π, we can assume that α is homotopic rel endpoints to a fiber π -1 (y), where y ∈ Y. Suppose c is an oriented, twosided, essential, simple closed loop Y based at y, and suppose that either c is nonperipheral in Y , or that Y is an annulus or Möbius band. Write π -1 (c) = c -∪ c + , where c ± are disjoint simple closed oriented loops in X based at y ± , and where the orientations of c ± project to that of c.

= c -• α • c -1 + • α -1 is homotopic to a meridian on S.
So, a compression arc α allows one to make compressible curves on S from essential curves on Y . See Figure 6.

Proof. Since α is homotopic rel endpoints to the fiber π -1 (y), the curve m(c) is homotopic in M to a curve in B that projects under π to c • c -1 , and hence m(c) is nullhomotopic in M . Checking orientations, one can see that m(c) is homotopic to a simple closed curve on S. So, we only have to prove that m(c) is homotopically essential on S.

Suppose that c -, c + are freely homotopic on ∂ H B as oriented curves. (This happens exactly when the curve c ⊂ Y bounds a Möbius band in Y .) Then m(c) is homotopic to the commutator of two essential simple closed curves on S that intersect once, and hence is essential since S is not a torus.

We can now assume that that c ± are not freely homotopic in ∂ H B as oriented curves. If m(c) is inessential, then c ± are freely homotopic on S, so c ± are homotopic in ∂ H B to boundary components c ′ ± ⊂ ∂ H B that bound an annulus in S \ ∂ H B. In this case c ± are peripheral, so we may assume that Y is either Note that in the second case the endpoints of α lie on distinct components of ∂(∂ H B), so in particular the two cases are mutually exclusive.

N ↵ Z ⇥ {1} M = Y ⇥ [ 1, 1] Z ⇥ { 1} m(c) Figure 6. Z ⊂ Y is a compact surface, the interval bundle B = Z × [-1, 1] embeds in M = Y × [-1, 1],
The reason we say X 'contains a meridian' instead of 'is compressible' is that X may not be an essential subsurface of S, and we want to emphasize that the essential curve in X that is compressible in M is actually essential in S. For example, let Y be a compact surface with boundary, Y ′ ⊂ Y be obtained by deleting a collar neighborhood of ∂Y , set B = Y ′ × [-1, 1] and M = Y × [-1, 1], and let α be a spanning arc of B in ∂M .

Proof. The 'if' direction is immediate: in case (1) we are essentially given a meridian in X, and in case [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF] we can appeal to Claim 2.11.

We now work on the 'only if' direction. Write our regular neighborhood of ∂ H B ∪α as X = ∂ H B ∪R where R is a rectangle with two opposite 'short' sides on the boundary of ∂ H B. Let D ⊂ M be an essential disc whose boundary is contained in X, and where D intersects the frontier Fr(B) ⊂ M in a minimal number of components. Let a ⊂ D ∩ Fr(B) be an arc that is 'outermost' in D, i.e. there is some arc a ′ ⊂ ∂D with the same endpoints as a such that a, a ′ bound an open disk in D that does not intersect Fr(B).

We claim that a ′ ⊂ R. If not, then a ′ ⊂ ∂ H B, and bounds a disk in B with the arc a ⊂ ∂B. Writing the interval bundle as π : B -→ Y , the projection π(a ∪ a ′ ) in Y is then also nullhomotopic, so π(a ′ ) is homotopic rel endpoints into ∂Y . Lifting this homotopy through the covering map

∂ H B -→ Y we get that a ′ is inessential in ∂ H B, i.e. is homotopic in ∂ H B rel endpoints into ∂(∂ H B). Lifting this homotopy through the covering map ∂ H Y -→ Y π(a) ⊂ ∂Y , it follows that π(a ′
) is an inessential arc in Y . We can then decrease the number of components of D ∩ Fr(B), contradicting that this number is minimal.

So, a ′ ⊂ R. Again by minimality of the intersection, the endpoints of a ′ lie on opposite short sides of R, so α is homotopic to a ′ through arcs in R with endpoints on Fr(B). Since a ′ is homotopic rel endpoints to a ⊂ Fr(B), it follows that α is homotopic rel endpoints into Fr(B). If the two endpoints of α lie on the same component of ∂(∂ H B), we are in case [START_REF] Agol | Tameness of hyperbolic 3-manifolds[END_REF], and otherwise we are in case [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF]. □ 2.8. Laminations. We assume the reader is familiar with geodesic and measured laminations on finite type hyperbolic surfaces. See e.g. [START_REF] Casson | Automorphisms of surfaces after Nielsen and Thurston[END_REF][START_REF] Kapovich | Hyperbolic manifolds and discrete groups[END_REF]. Suppose λ is a connected geodesic lamination on a finite type hyperbolic surface S with geodesic boundary. We say that λ fills an essential subsurface T ⊂ S if λ ⊂ T and λ intersects every essential, non-peripheral simple closed curve in T . Fact 2.13. For every connected λ, there is a unique subsurface with geodesic boundary (as in §2.1) that is filled by λ, which we denote by S(λ). It is the minimal subsurface with geodesic boundary in S that contains λ.

Here, S(λ) can be constructed by taking a component λ ⊂ S ⊂ H 2 of the preimage of λ, letting C ⊂ H 2 be the convex hull of the set of endpoints of leaves of λ in ∂H 2 , and projecting C into S.

Suppose that M is a compact, orientable irreducible 3-manifold let S ⊂ ∂M be an essential subsurface. The limit set of (S, M ) is the closure

Λ(S, M ) = {meridians γ ⊂ S} ⊂ PML(S),
where PML(S) is the space of projective measured laminations on S. The limits set was first studied by Masur [START_REF]Measured foliations and handlebodies[END_REF] in the case that M is a handlebody, with S its entire boundary. In this case, Kerckhoff [START_REF] Steven | The measure of the limit set of the handlebody group[END_REF] later proved that the limit set has measure zero in PML(S), although a mistake in his argument was later found and fixed by Gadre [START_REF] Gadre | The limit set of the handlebody set has measure zero[END_REF].

In some ways, Λ(S, M ) acts as a dynamical limit set. For instance, let Map(S) be the mapping class group of S, and let Map(S, M ) ⊂ Map(S) be the subgroup consisting of mapping classes represented by restrictions of homeomorphisms of M . Then we have: Fact 2.14.

(1) If Λ(S, M ) is nonempty, it is the smallest nonempty closed subset of PML(S) that is invariant under Mod(S, H).

(2) If Map(S, M ) contains a pseudo-Anosov map on S, then Λ(S, M ) is the closure of the set of the attracting and repelling laminations of pseudo-Anosov elements of Map(S, M ).

Note that Map(S, M ) contains a pseudo-Anosov map on S if and only if the disk set D(S, M ) has infinite diameter in the curve complex C(S), where the latter condition was discussed earlier in Proposition 3.1. See also [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF][START_REF]An extension of the Masur domain, Spaces of Kleinian groups[END_REF].

Proof. For the first part just note that Dehn twist T m around meridians m ⊂ S are in Map(M, S), so if A ⊂ PML(S) is nonempty and invariant, λ ∈ A and m is a meridian, then m = lim i T i m (λ) is also in A, implying Λ(M, S) ⊂ A. For the second part, take a pseudo-Anosov f ∈ Map(M, S) with attracting lamination λ + , say. If m is a meridian in S, then

T i m • f • T -i m are
∼ = Y × [0, 1] in such a way that λ ± = λ × {±1}
for some geodesic lamination on Y . This is an easy consequence of the fact that on a surface, homotopic laminations are isotopic. In general: Fact 2.16. Suppose that λ ± are disjoint or equal geodesic laminations on ∂ H B. Then the following are equivalent.

(1) λ ± are essentially homotopic through B.

(

) λ ± is isotopic on ∂ H B to σ(λ ∓ ) 2
, where σ is the canonical involution of B discussed in §2.5.

Moreover, (1) and (2) imply

(3) There is a geodesic lamination λ on Y such that λ -∪ λ + is isotopic on

∂ H B to the preimage (π| ∂ H B ) -1 ( λ).
Here, (3) does not always imply (1,2), since it could be that λ has two components, (π| ∂ H B ) -1 ( λ) has four, and these components are incorrectly partitioned into the two laminations λ ± . However, that's the only problem, so for instance if λ ± are minimal then (1) -( 3) are equivalent.

While we have phrased things more generally in the section, we can always assume in proofs that our hyperbolic metrics have been chosen so that the covering map π| ∂ H B : ∂ H B -→ Y is locally isometric. Here, we're using the fact that given two hyperbolic metrics with geodesic boundary on a compact surface, a geodesic lamination with respect to one metric is isotopic to a unique geodesic lamination with respect to the other hyperbolic metric. In this case, we can remove the word 'isotopic' from ( 2) and (3).

Proof. The fact is trivial when B is a trivial interval bundle. When B is nontrivial, lift the homotopy to the trivial interval bundle 1) ⇐⇒ (2) follows since the canonical involution on B ′ covers that of B. For (2) =⇒ (3), note that since λ ± are disjoint or equal and differ by σ, their projections π(λ ± ) ⊂ Y are the same, and are a geodesic lamination λ on Y . □

B ′ -→ B that double covers B, giving homotopic laminations λ ′ ± ⊂ ∂ H B ′ . (

Large and small disk sets and compression bodies

Suppose that S ⊂ ∂M is an essential subsurface of the boundary of a compact, irreducible 3-manifold M , and that ∂S is incompressible in M . The following is probably known to some experts, but we don't think it appears anywhere in the literature, so we give a complete proof. In case (1), we will say that D(S, M ) is large, and in cases ( 2)-( 4), we will say that D(S, M ) is small. Similarly, if C(S, M ) is the characteristic compression body defined in Fact 2.4, then C(S, M ) is said to be large or small depending on whether D(S, M ) is large or small. See also the discussion of small compression bodies in §3 of [START_REF] Biringer | Automorphisms of the compression body graph[END_REF].

Here, recall that a band sum of a meridian δ is the boundary of a regular neighborhood of δ ∪ β, where β is a simple closed curve on S that intersects δ once. Any such band sum must be a meridian: for instance, as an element of π 1 M it is a commutator with a trivial element. Also, (3) includes the case when M is a solid torus and S = ∂M , in which case there is only one (nonseparating) meridian. When M is not a solid torus, though, every nonseparating curve has infinitely many band sums.

Before beginning the proof, we first establish the following:

Claim 3.2. Suppose S is not a torus, γ ⊂ S is a meridian on S and δ is a meridian that lies in a component T ⊂ S \ γ. If γ is not a band sum of δ, there is a pseudo-Anosov f : T -→ T that extends to a homeomorphism of M .
The condition that γ is not a band sum is necessary. For if M is a handlebody with S = ∂M , and γ is a separating meridian that bounds a compressible punctured torus T ⊂ S, then T has only a single meridian δ. This δ is nonseparating and γ is a band sum of δ. Any map T -→ T that extends to a homeomorphism of M must then fix δ, so cannot be pseudo-Anosov.

Similarly, if S is a torus and γ is a meridian, the complement of γ is an annulus, which does not admit any pseudo-Anosov map.

Proof of Claim 3.2. Suppose first that γ is not separating. Any simple closed curve that intersects γ once can be used to create a band sum. Now S is not a torus, and cannot be a punctured torus either, since then its boundary would be compressible. So, there are a pair α, β of band sums of γ that fill S 󰄀 γ. By a theorem of Thurston [START_REF] Laudenbach | Travaux de thurston sur les surfaces[END_REF]III.3 in 13], the composition of twists T α • T -1 β is pseudo-Anosov. Each twist extends to M , because twist about meridians can be extended to twists along the disks they bound. Now suppose γ separates S, and suppose that R is the component of T \ (γ ∪ δ) adjacent to γ and δ. Any curve in R that bounds a pair of pants with γ and δ is also a meridian. Such curves are constructed as the boundary of a neighborhood of the union of γ, δ and any arc in R joining the two. Therefore, there is a pair α, β of such curves that fills R. As before,

f = T α • T -1 β is a pseudo-Anosov on R that extends to M .
However, there was nothing special about δ in the above construction. So if there is some (non-peripheral) meridian δ ′ ⊂ T with δ ∕ = δ ′ , there is also a pseudo-Anosov f ′ on the corresponding surface R ′ , such that f ′ extends to M . Since R and R ′ fill T , [14, Theorem 6.1] says that for large i the composition f i (f ′ ) i is a pseudo-Anosov on T . See Figure 7.

The only case left to consider is when δ is the only (non-peripheral) meridian in T . Since new meridians usually can be created by joining δ and γ with an arc and taking a regular neighborhood, the only possibility here is that T is a punctured torus, in which case this construction always just produces γ again. But then γ is a band sum of δ.

□ T = S S R = S R 0 = S 0 Figure 7. The surfaces R and R ′ fill T .
Proof of Proposition 3.1. When S is a torus, distinct curves have nonzero algebraic intersection number, so either there are no meridians or there is a single meridian. So, we assume S ∕ = T 2 below. We first claim that if there are two meridians in S, neither of which is a band sum of the other, then D(S, M ) has infinite diameter in the curve complex. To see this, suppose γ 1 , γ 2 are such meridians. Claim 3.2 gives two pseudo-Anosov maps f 1 , f 2 , each defined on the component of S \ γ i that contains γ j , where i ∕ = j. Since the component of S \ γ 1 containing γ 2 and the component of S \ γ 2 containing γ 1 together fill S, for large k the composition f k g k is a pseudo-Anosov map on the entire surface S, by [START_REF] Clay | The geometry of rightangled Artin subgroups of mapping class groups[END_REF]Theorem 6.1]. Any such composition extends to M , so maps meridians to meridians. As pseudo-Anosovs act with unbounded orbits on the curve complex [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF], this implies that the set of meridians has infinite diameter in C(S).

Starting now with the proof of the proposition, suppose there are no nonseparating meridians in S. If γ, δ are distinct (separating) meridians, then an innermost disk surgery produces another separating meridian γ 2 disjoint from γ 1 , see [START_REF] Biringer | Automorphisms of the compression body graph[END_REF]Lemma 2.8]. By the previous paragraph, D(S, M ) has infinite diameter in the curve complex. So, the only other options are if D(S, M ) = ∅, or if the only meridian is a single separating curve.

Suppose now that there is a non-separating meridian γ in S. By Claim 3.2, unless the disc set has infinite diameter in the curve complex, any meridian disjoint from γ must be a band sum of γ. So, either we are in case (2) of the proposition, or there is some meridian δ that intersects γ. Any innermost disk surgery of δ along γ must produce a band sum β of γ. However, this β must then bound a punctured torus T containing γ, and δ is then forced to lie inside T , which gives a contradiction, see Figure 8.

□ T ? Figure 8
. A surgery of a curve δ along a non-separating γ cannot produce a curve β that is a band sum of γ.

Windows from limit sets

Let N = Γ\H 3 be an orientable, geometrically finite hyperbolic 3-manifold, let Λ ⊂ ∂H 3 be the limit set of Γ, and let

CC(N ) := Γ\CH(Λ) ⊂ N
be the convex core of N . Equip ∂CC(N ) with its intrinsic length metric, which is hyperbolic, see for instance [START_REF] Thurston | The geometry and topology of 3-manifolds[END_REF]Prop 8.5.1].

Let S ± be (possibly degenerate) incompressible subsurfaces with geodesic boundary in ∂CC(N ) that are either equal or are essentially disjoint, as in

§2.1. Let S± ⊂ ∂CH(Λ) ⊂ H 3
be lifts of S ± , where if S -= S + , we require that S-∕ = S+ . Let Γ ± ⊂ Γ be the stabilizers of S± , let Λ ± ⊂ ∂H 3 be their limit sets and

∆ = Γ + ∩ Γ -.
The lift S± is isometric to a convex subset of H 2 . Let ∂ ∞ S ± ⊂ ∂H 2 be the boundary of S± . By [START_REF] Mj | Cannon-thurston maps for kleinian groups[END_REF]Theorem 5.6], say, the inclusion S± ↩→ H 3 extends continuously to a Γ ± -equivariant quotient map

ι ± : ∂ ∞ S± -→ Λ ± ⊂ ∂H 3 . Theorem 4.1 (Windows from limit sets). Λ -∩ Λ + = Λ ∆ .
Next, suppose ∆ is nonempty and is not a cyclic group acting parabolically on either Sor S+ , and let C± ⊂ S± be the convex hulls of the subsets ι -1

± (Λ ∆ ) ⊂ ∂ ∞ S± .
Then C± are ∆-invariant, the quotients C ± := ∆\ C± are (possibly degenerate) subsurfaces with geodesic boundary in S ± , and there is an essential homotopy from C -to C + in CC(N ) that is the projection of a homotopy from Cto C+ .

Above, C ± are (possibly degenerate) subsurfaces with geodesic boundary in S ± , as defined in §2.1, but it follows from the above and Theorem 2.8 that there are 'resolutions' (see §2.1) C ′ ± ⊂ S ± such that C ′ ± bound an interval bundle in CC(M ). So informally, the theorem says that the intersection Λ -∩ Λ + is exactly the limit set of the fundamental group of some essential interval bundle in (CC(M ), S ′ -∪ S ′ + ). The term 'window' comes from Thurston [START_REF] Thurston | Hyperbolic structures on 3-manifolds. III. Deformations of 3manifolds with incompressible boundary[END_REF] and refers to interval bundles; for example, one can 'see through' a trivial interval bundle from one horizontal boundary component to the other.

The assumption that ∆ is not cyclic and acting parabolically on either S± is just for convenience in the statement of the theorem. (Just to be clear, note that an element γ ∈ ∆ can act parabolically as an isometry of H3 , but hyperbolically on the convex subsets S± ⊂ H 2 .) If ∆ is cyclic and acts parabolically on S+ the subset C+ in the statement of the theorem will be empty. However, using the same proof one can construct a homotopy from a simple closed curve on S + bounding a cusp of S + to some simple closed curve on S -.

As mentioned in the introduction, a version of Theorem 4.1 was known to Thurston, see his discussion of the Only Windows Break Theorem in [START_REF] Thurston | Hyperbolic structures on 3-manifolds. III. Deformations of 3manifolds with incompressible boundary[END_REF]. Precise statements for geometrically finite N without accidental parabolics were worked out in Lecuire's thesis [START_REF] Lecuire | Structures hyperboliques convexes sur les variétés de dimension 3[END_REF] and by Walsh [START_REF] Walsh | The bumping set and the characteristic submanifold[END_REF]; note that Walsh uses the conformal boundary instead of the convex core boundary, but the two points of view are equivalent. However, for our applications in this paper, we need to allow accidental parabolics in S± , which are not allowed in those theorems. Also, our proof is more direct and natural 3 than those in [START_REF] Walsh | The bumping set and the characteristic submanifold[END_REF][START_REF] Lecuire | Structures hyperboliques convexes sur les variétés de dimension 3[END_REF], despite the extra complication coming from parabolics.

Finally, the assumption that N is geometrically finite is not really essential for the theorem statement. With a bit more work dealing with degenerate ends, one can prove the theorem for all finitely generated Γ. Essentially, the point is to use Canary's covering theorem [START_REF] Canary | A covering theorem for hyperbolic 3-manifolds and its applications[END_REF] to show that degenerate NP-ends in the covers N ± := Γ ± \H 3 have neighborhoods that embed in N , and then to use this to prove that geodesic rays in H 3 that converge to points in Λ -∩ Λ + cannot exit degenerate ends in N ± . After showing this, the proof of Claim 4.3 extends to the general case. However, we don't have an application for that theorem in mind, so we'll spare the reader the details.

Proof of Theorem 4.1. We first focus on proving that Proof. By the Tameness Theorem [START_REF] Agol | Tameness of hyperbolic 3-manifolds[END_REF][START_REF] Calegari | Shrinkwrapping and the taming of hyperbolic 3manifolds[END_REF], we can identify CC(N ) topologically with a subset of a 3-compact manifold with boundary M , where

Λ -∩ Λ + = Λ ∆ . For each ξ ∈ ∂H 3 , let Γ ± (ξ) ⊂ Γ ± be the stabilizer of ξ.
(1) CC(N ) ⊃ int(M ), CC(N ) ∩ ∂M = ∂CC(N ),
and where ∂CC(N ) is a collection of essential subsurfaces of ∂ N . Let ∂ χ=0 M be the union of all torus boundary components of M , and let (X, Σ) be the characteristic submanifold of the pair (M, S -∪ S + ∪ ∂ χ=0 M ), as in §2.6.

Since Γ ± are both contained in a discrete group Γ, both Γ ± (ξ) are contained in the stabilizer Γ(ξ), which is either infinite cyclic, or rank 2 parabolic.

Suppose first that Γ(ξ) is rank 2 parabolic. The groups Γ ± (ξ) are both cyclic, since S ± are incompressible hyperbolic surfaces, so their fundamental groups do not contain Z 2 subgroups. So, we can write Γ ± (ξ) = 〈γ ± 〉 for closed curves γ ± on S ± . Both γ ± are homotopic into some fixed component T ⊂ ∂ χ=0 M , the component whose fundamental group can be conjugated to stabilize ξ. So, there is a component (X 0 , Σ 0 ) ⊂ (X, Σ) of the characteristic submanifold such that Σ 0 intersects T and both γ ± are homotopic on

S ± into Σ 0 . Since M ∕ ∼ = T 2 × [0, x],
the component (X 0 , Σ 0 ) is either an interval bundle over an annulus (so, a fibered solid torus), or an S 1 -bundle pair, so by Fact 2.9, X 0 is either a fibered solid torus or a thickened torus. In either case, Σ 0 intersects each of S ± in a fibered annulus, and these annuli are disjoint, so they are parallel on a torus boundary component of X 0 , implying that γ ± are homotopic in M , and hence Γ ± (ξ) are conjugate in Γ. But since Γ ± (ξ) have the same fixed point at infinity, the conjugating element must fix ξ, and therefore commute with the two groups, implying Γ -(ξ) = Γ + (ξ). Now assume Γ(ξ) is cyclic. Pick a basepoint p ∈ S -, say, and let γ -⊂ S - be a loop based at p representing a generator of Γ -(ξ). Represent a generator of Γ + (ξ) as α • γ + • α -1 , where α is an arc from p ∈ S -to a point in S + , and γ + is a loop in S + . Since Γ ± stabilize distinct components S± , the arc α is not homotopic into S -∪ S + . So, α is a spanning arc of an essential map from an annulus, where the boundary components of the annulus map to powers of γ ± . It follows that the loops γ ± are homotopic on S ± into Σ 0 for some component

(X 0 , Σ 0 ) ⊂ (X, Σ).
If X 0 is an I-bundle with horizontal boundary Σ 0 , then as γ ± are not proper powers in π 1 S ± , they are both primitive in π 1 X 0 , and hence γ ± (rather than their powers) are homotopic in X 0 ⊂ M . Similarly, if (X 0 , Σ 0 ) is a fibered solid torus, Σ 0 is a collection of parallel annuli on ∂X 0 , so since γ ± are primitive in π 1 S ± , they are homotopic on S ± to simple closed curves in these annuli, and hence are homotopic to each other in X 0 .

It follows that there are generators for Γ ± (ξ) that are conjugate in Γ, but since these generators both fix ξ, they are equal.

□ Claim 4.3. For all ξ ∈ Λ -∩ Λ + , we have Γ -(ξ) = Γ + (ξ). Moreover, Λ ∆ = Λ -∩ Λ + .
Proof. Let N ± ⊂ H 3 be the 1-neighborhood of the convex hull of Λ ± , and for small 󰂃 > 0, let T ± (󰂃) ⊂ H 3 be the set of all points that are translated less than 󰂃 by some parabolic element of Γ ± . If 󰂃 is at most the Margulis constant 󰂃 0 , then T ± (󰂃) is a disjoint union of horoballs in H 3 . The sets N ± and T ± (󰂃) are Γ ± invariant. Since Γ ± is a finitely generated subgroup of Γ, which is geometrically finite, Γ ± is geometrically finite as well by [START_REF] Canary | A covering theorem for hyperbolic 3-manifolds and its applications[END_REF]. So, the action of Γ ± on N ± \ T ± (󰂃) is cocompact, see e.g. Theorem 3.7 in [START_REF] Matsuzaki | Hyperbolic manifolds and Kleinian groups[END_REF], implying that either the function

D + : H 3 -→ R >0 , D + (x) = min{d(x, γ(x)) | γ ∈ Γ + loxodromic}
is bounded above on N + \ T + (󰂃) by some B(󰂃) > 0, or Γ + is elementary parabolic. A similar statement holds forinstead of +. With 󰂃 0 the Margulis constant, the Margulis Lemma then implies that if 󰂃 > 0 is sufficiently small with respect to B(󰂃 0 ), and Γ + is not elementary parabolic, then

(2) T -(󰂃) ∩ N + ⊂ T + (󰂃 0 ),
and similarly with -, + exchanged. Indeed, if not then we have (say) a point p ∈ H 3 that is translated by less than 󰂃 by some parabolic γ -∈ Γ -and by at most B by some loxodromic γ + ∈ Γ + . If 󰂃 is small with respect to B, then both γ -and [γ + , γ -] translates p by at most 󰂃 0 , so they generate an elementary discrete group by the Margulis lemma applied to Γ, implying that γ + fixes the fixed point of γ -, which contradicts that they generate a discrete group.

Fix ξ ∈ Λ + ∩ Λ -. We claim that Γ -(ξ) = Γ + (ξ). By Claim 4.2 it suffices to show that whenever Γ -(ξ) is nontrivial, say, so is Γ + (ξ).
First, assume that Γ -(ξ) is elementary parabolic. We claim that Γ + (ξ) is elementary parabolic as well. Assume not, and let α be a geodesic ray in H 3 converging to ξ. Then α(t) lies in T -(󰂃) ∩ N + for large t, and therefore in T + (󰂃 0 ) for large t by [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF], which implies ξ is a parabolic fixed point of Γ + as well, a contradiction.

Next, suppose that Γ -(ξ) is elementary loxodromic. If ξ is a parabolic fixed point of Γ + , we are done, so let's assume this isn't the case. Let α be the axis of Γ -(ξ), parametrized so α(t) → ξ as t → ∞. Since ξ is not a Γ + parabolic fixed point, there are t i → ∞ such that α(t i ) ∕ ∈ T + (󰂃) for all i. Since the action of

Γ + on N + \ T + (󰂃) is cocompact, if p ∈ H 3 is a fixed basepoint, there are elements γ + i ∈ Γ + such that sup i d(γ + i (p), α(t i )) < ∞. Since the action of Γ -(ξ) on α is cocompact, there are then elements γ - i ∈ Γ -(ξ) with sup i d(γ + i (p), γ - i (p)) < ∞.
By discreteness of Γ, after passing to a subsequence we can assume γ + i = γ - i •g for some fixed g ∈ Γ. Hence, for all i we have

γ + i • (γ + 1 ) -1 = γ - i • (γ - 1 ) -1 ∈ Γ + ∩ (Γ -(ξ)) ⊂ Γ + (ξ)
, so we are done.

Finally, we want to show that

Λ -∩ Λ + = Λ ∆ . The inclusion Λ ∆ ⊂ Λ -∩ Λ + is clear. So, take ξ ∈ Λ -∩ Λ + .
We can assume that Γ ± (ξ) = 1, since otherwise we're in the cases handled above. Let α be a geodesic ray in H 3 converging to ξ. As in the previous case, since ξ is not a parabolic fixed point of Γ + , there are t i → ∞ such that α(t i ) ∕ ∈ T + (󰂃 0 ) for all i. Discarding finitely many i, we have α(t i ) ∈ N + , so it follows from (2) that α(t i ) ∕ ∈ T -(󰂃). Fixing a base point p ∈ H 3 , as Γ -acts cocompactly on N -\ T -(󰂃) and Γ + acts cocompactly on

N + \ T + (󰂃 0 ), there are elements γ ± i ∈ Γ ± such that sup i d(γ ± i (p), α(t i )) < ∞.
So passing to a subsequence, γ + i = γ - i • g for some fixed g ∈ Γ, and then

γ + i • (γ + 1 ) -1 = γ - i • (γ - 1 ) -1 ∈ Γ + ∩ Γ -= ∆ for all i.
But applying this sequence to p and letting i → ∞ gives a sequence of points in the orbit ∆(p) that converge to ξ, so ξ ∈ Λ ∆ . □ Now assume that ∆ ∕ = 1. We want to construct the interval bundle W mentioned in the statement of the theorem. After an isotopy on ∂CC(N ), let's assume that S ± is a subsurface of ∂CC(N ) with geodesic boundary. Consequently, we allow degenerate subsurfaces, where S ± is a simple closed geodesic, as well as subsurfaces where only the interior is embedded and two boundary components can coincide. As ∂CC(N ) may have cusps, we also must allow S ± to be noncompact with finite volume, rather than compact.

Recall that S± is isometric to a convex subset of H 2 , and that if

∂ ∞ S ± ⊂ ∂H 2 is the boundary of S± the inclusion S± ↩→ H 3 extends continuously to a Γ ± - equivariant quotient map ι ± : ∂ ∞ S± -→ Λ ± ⊂ ∂H 3 . Moreover, if ξ, ξ ′ ∈ ∂ ∞ S+ , say, we have ι + (ξ) = ι + (ξ ′ ) if
and only if there is an element γ ∈ Γ + that acts hyperbolically on S+ ∪ ∂ ∞ S+ with fixed points ξ, ξ ′ ∈ ∂ ∞ S+ , but acts parabolically on H 3 . By discreteness of the action Γ + ↷ S+ , each ξ ∈ ∂ ∞ S+ has the same image under ι + as at most one other ξ ′ . Similar statements holds withinstead of +. All this is a consequence (for instance) of Bowditch's theory of the boundary of a relatively hyperbolic group [START_REF] Brian | Relatively hyperbolic groups[END_REF]: since the action Γ ± ↷ H 3 is geometrically finite, Λ ± is a model for the Bowditch boundary of the group Γ ± relatively to its maximal parabolic subgroups, so the statement above follows from Theorem 1.3 of [START_REF] Manning | The bowditch boundary of (g,h) when g is hyperbolic[END_REF], say 4 .

Let ι -1 ± (Λ ∆ ) ⊂ ∂ ∞ S± . Since ∆ ∕ = 1 and is not cyclic parabolic, ι -1 ± (Λ ∆ ) has at least two points, so it has a well-defined convex hull C± ⊂ S ± . Claim 4.4 (Convex hulls). One of the following holds.

(1) ∆ is cyclic and acts hyperbolically on S+ . The convex hull C+ is its geodesic axis, which is precisely invariant under ∆ ⊂ Γ, so that the quotient C + := ∆\ C+ embeds as a simple closed geodesic in S + .

(2) C+ is a subsurface of S+ with geodesic boundary, the interior int( C+ ) is precisely invariant under ∆ ⊂ Γ, and the quotient C + := ∆\ C+ is a generalized subsurface of S + with compact geodesic boundary. A similar statement holds withinstead of +.

Proof. Let's work with +, for concreteness. If g ∈ ∆, then g(Λ ∆ ) = Λ ∆ , so g leaves ι -1 + (Λ ∆ ) invariant by equivariance of ι + . Hence g leaves C+ invariant. Let's suppose first that C+ has nonempty interior, since that is the more interesting case. We'll address the case that C+ is a biinfinite geodesic at the end of the proof. Let g ∈ Γ + \ ∆. We want to show

g(int( C+ )) ∩ int( C+ ) = ∅.
Assume this is not the case. By Claim 4.3, the fixed points of g in ∂ ∞ S + lie outside ι -1 + (Λ(∆)). So, we cannot have g( C+ ) ⊂ C+ , as then we'd have g n ( C+ ) ⊂ C+ for all n, contradicting that points of S+ converge to the fixed points of g under iteration. Considering backwards iterates, we also cannot have C+ ⊂ g( C+ ). Therefore, ∂ C+ and ∂g( C+ ) intersect transversely.

Since C+ has nonempty interior, ∆ is nonelementary, and therefore the fixed points of loxodromic isometries of ∆ are dense in Λ ∆ . Loxodromic fixed points of ∆ are in particular not parabolic fixed points in Γ ± , so any biinfinite geodesic in C+ is a limit of biinfinite geodesics in C+ whose endpoints are not fixed points of parabolic isometries of Γ ± . By the previous paragraph, there are then biinfinite geodesics α + , β + in C+ such that g(α + ) and β + intersect transversely, and where the endpoints of α + , β + project under ι + to points

ξ α , ξ ′ α , ξ β , ξ ′ β ∈ Λ ∆ that are not parabolic fixed points in Γ ± . Let α -be the geodesic in S-whose endpoints in ∂ ∞ S-map to the points ξ α , ξ ′ α under ι -. Define β -similarly. Then α := α + ∪ {ξ α , ξ ′ α } ∪ α -, β := β + ∪ {ξ β , ξ ′ β } ∪ β - are two simple closed curves on the closure cl(∂CH(Λ Γ )) ⊂ H 3 ∪ ∂H 3
, which is homeomorphic to a sphere. For instance, the arcs α ± are disjoint and ξ α ∕ = ξ ′ α , since the endpoints of α + are not parabolic fixed points. Now consider how the two simple closed curves g(α), β intersect. The arcs β -and g(α + ) are disjoint since S-∕ = S+ . The arcs g(α -), β -are disjoint since g ∕ ∈ Γ -and hence g(α -) lies on a different translate of Sthan β -⊂ S-. Moreover, since g(α + ), β + intersect transversely in S+ , the endpoints of g(α + ) and β + are distinct in ∂ ∞ S+ , and since none of them are parabolic fixed points, the points g(ξ α ), g(ξ ′ α ), ξ β , ξ ′ β are all distinct. But by assumption, g(α + ) intersects β + transversely in a single point! This shows that g(α) and β intersect exactly once, transversely, which is a contradiction.

By precise invariance of the action on the interior, the quotient int(C + ) = ∆/int( C+ ) embeds in the finite volume surface S + , so C + has finite volume itself. So if ∂C + is non-compact, it must have two noncompact boundary components that are asymptotic. Lifting, we get two boundary components β 1 , β 2 of C+ that are asymptotic. Since C+ is convex, it is contained in the subset of H 2 bounded by β 1 , β 2 , and hence the common endpoint of β 1 , β 2 is an isolated point of Λ ∆ , which is a contradiction since ∆ is not elementary.

The case when C+ is a biinfinite geodesic is similar. Here, ∆ must be cyclic, acting on S+ with axis C+ , and acting either parabolically or loxodromically on H 3 . In the parabolic case, C+ compactifies to a simple closed curve on the sphere cl(CH(Λ Γ )) ⊂ H 3 ∪ ∂H 3 , so no translate g( C+ ), g ∈ Γ + , can intersect C+ transversely, since if it did we'd get two simple closed curves on the sphere that intersect once. In the loxodromic case, we get a similar contradiction by looking at the simple closed curve cl( C+ ∪ C-) ⊂ cl(∂ M ) and its g-image. So, C+ is precisely invariant under ∆ ⊂ Γ. The quotient C + := ∆\ C+ is obviously compact, and is therefore a simple closed geodesic in S + . □

We claim that C -and C + are homeomorphic. If C ± are isotopic in ∂CC(M ) this is clear, and otherwise we argue as follows. The subgroups π 1 C ± are both represented by ∆, so are conjugate in π 1 M . The fact that every curve in C -is homotopic to a curve in C + (and vice versa) implies that C ± are isotopic to subsurfaces C ′ ± ⊂ Σ in the boundary Σ of a component (X, Σ) of the characteristic submanifold5 of (CC(M ), S -∪ S + ), see §2.6, and that even within X every closed curve in C ′ -is homotopic to a closed curve in C ′ + , and vice versa. When X is a solid torus or thickened torus, C ′ ± are annuli, while if X is an interval bundle, C ′ ± bound a vertical interval bundle in X, and are homeomorphic.

So, let f : C --→ C + be a homeomorphism, lift f to a ∆-equivariant homeomorphism f : C--→ C+ and let

F : C-× [0, 1] -→ CH(Λ)
where F (x, •) parametrizes the geodesic from x to f (x). Then F is ∆-equivariant, and projects to an essential homotopy from C -to C + , as desired.

4.1. An annulus theorem for laminations. Suppose M is a compact, orientable, hyperbolizable 3-manifold with nonempty boundary and let S = ∂ χ<0 M be the union of all non-torus boundary components of M . When α, β ⊂ S are disjoint simple closed curves that are essential and homotopic in M , but not homotopic in S, the Annulus Theorem says that there is an essential embedded annulus A ⊂ M with ∂A = α ∪ β, see Scott [START_REF] Scott | Strong annulus and torus theorems and the enclosing property of characteristic submanifolds of 3-manifolds[END_REF].

More generally, equip S with an arbitrary hyperbolic metric. An essential homotopy between two geodesic laminations λ ± on S is a map

H : (λ × [-1, 1], λ × {-1, 1}) -→ (M, S)
where λ is a lamination, such that H maps λ × {±1} homeomorphically onto λ ± , and where

H is not homotopic rel λ × {-1, 1} into ∂M .
Here is an 'Annulus Theorem' for minimal laminations.

Proposition 4.5 (An annulus theorem for laminations). Let λ -, λ + be two minimal geodesic laminations on S that are either disjoint or equal, and assume that S(λ ± ) are incompressible in M . If λ ± are essentially homotopic in (M, S), there is an essential interval bundle (B, ∂ H B) ⊂ (M, S) such that λ ± fill ∂ H B, and where λ ± are essentially homotopic through B, as in §2.9.

Here, S(λ ± ) are the subsurfaces with geodesic boundary filled by λ ± , as in §2.8. The assumption that they are incompressible generalizes the assumption that α, β are homotopically essential in M in the Annulus Theorem.

Proof. Identify M \ ∂ χ=0 M with the convex core of a geometrically finite hyperbolic 3-manifold. Set S ± := S(λ ± ). Lift the essential homotopy from λ - to λ + to a homotopy from lifts λ-⊂ Sto λ+ ⊂ S+ in H 3 . Under the homotopy, which has bounded tracks, corresponding leaves of λ± have the same endpoints in ∂H 3 . The endpoints of λ± are dense in ∂ ∞ S± , so this means that the subsurfaces C ± ⊂ S ± constructed in Theorem 4.1 are just C ± = S ± . Passing to disjoint or equal resolutions S ′ ± of S ± and applying Theorem 2.8 gives an interval bundle B where λ ± fill ∂ H B = S ′ -∪ S ′ + . We claim that λ ± are essentially homotopic through B. By Fact 2.16, it suffices to show that if σ is the canonical involution of B, as described in §4.5, then σ(λ ± ) is isotopic to λ ∓ on S ′ ∓ . Using the notation of Theorem 4.1, σ lifts to a ∆-equivariant involution σ of B that exchanges S′ -and S′ + , where here ∆ = Γ -∩ Γ + . By equivariance, σ extends continuously to the identity on Λ ∆ , so σ( λ-) is a lamination on S+ with all the same endpoints at infinity as λ+ , and hence equals λ+ . The claim follows. □

Laminations on the boundary

Suppose that M is a compact, orientable 3-manifold with hyperbolizable interior and nonempty boundary ∂M . Equip M with an arbitrary Riemannian metric and lift it to a Riemannian metric on the universal cover M . As in the introduction, a biinfinite path or ray h on ∂ M is called homoclinic if there are points s i , t i with

|s i -t i | → ∞ such that sup i d M (h(s i ), h(t i )) < ∞. Two rays h + , h -on ∂ M are called mutually homoclinic if there are parameters s i ± → ∞ such that sup i d M (h + (s i + ), h -(s i -)) < ∞.
Here, a ray is a continuous map from an interval [a, ∞), and a biinfinite path is a continuous map from R. We will also call rays and paths on ∂M (mutually) homoclinic if they have lifts that are (mutually) homoclinic paths on ∂ M . We refer the reader to §5.1 for some comments on alternate definitions of homoclinic that exist in the literature.

Note that if we divide a biinfinite homoclinic path into two rays, then either one of the two rays is itself homoclinic, or the two rays are mutually homoclinic. Also, these definitions are metric independent: since M is compact, any two Riemannian metrics on M lift to quasi-isometric metrics on M , and a path is homoclinic or mutually homoclinic with respect to one metric if and only if it is with respect to the other metric.

Here are some examples.

(1) Suppose that D is a properly embedded disc in M , and h : R -→ ∂M is a path that covers ∂D ⊂ ∂M . Then h is homoclinic: indeed, D lifts homeomorphically to M , so h lifts to a path in M with compact image. (2) Suppose that φ : (S 1 × [0, 1], S 1 × {0, 1}) -→ (M, ∂M ) is an essential embedded annulus. Then rays covering the two boundary components of the annulus are mutually homoclinic: indeed, φ lifts to φ : R × [0, 1] -→ M , and we have sup t∈R d( φ(t, 0), φ(t, 1)) < ∞, so restricting to t ∈ [0, ∞) we get two mutually homoclinic rays in M . It will be convenient below to work with a particular choice of metric on M .

Example 5.1 (An explicit metric on M ). Let ∂ χ<0 M be the union of all components of ∂M that have negative Euler characteristic, i.e. are not tori.

Thurston's Haken hyperbolization theorem, see [START_REF] Kapovich | Hyperbolic manifolds and discrete groups[END_REF], implies that there is a hyperbolic 3-manifold N = H 3 /Γ homeomorphic to the interior of M , where every component of ∂ χ<0 M corresponds to a convex cocompact end of N . A torus T ⊂ ∂M , on the other hand, determines a cusp of N . So, in other words, N is 'minimally parabolic': the only parabolics come from torus boundary components of M . For each T , pick an open neighborhood N T ⊂ N of the associated cusp that is the quotient of a horoball in H 3 by a Z 2 -action. Then

(3)

M ∼ = CC(N ) \ 󰁞 tori T ⊂∂M N T ,
and we will identify M with the right-hand side everywhere below. Then

• M ⊂ H 3 is obtained from the convex hull CH(Γ) ⊂ H 3 of the limit set of Γ by deleting an equivariant collection of horoballs, and

• the path metric induced on ∂ χ<0 M is hyperbolic [51, Proposition 8.5.1],
and the path metric induced on every torus T ⊂ ∂M is Euclidean.

We now specialize to the case of paths that are geodesics on ∂M . Recall from Example (1) above that one can make homoclinic paths by running around the boundaries of disks in ∂M . The following shows that discs are essential in such constructions. Fact 5.2. Suppose that S ⊂ ∂M is an essential subsurface. Then the inclusion of any lift S ⊂ ∂ M is a quasi-isometric embedding into M . Moreover if S is incompressible then any pair of mutually homoclinic infinite rays on S are asymptotic and no biinfinite geodesic γ in S is homoclinic.

Proof. Think of M as embedded in a complete hyperbolic 3-manifold N as in (3), write N = Γ\H 3 , and let M ⊂ H 3 be the preimage of M , so that M is obtained from the convex hull CH(Γ) be deleting an equivariant collection of horoballs. Fix a subgroup ∆ < Γ that represents the conjugacy class associated to the image of the fundamental group of S ⊂ M . To show that S ↩→ ∂ M is a quasi-isometric embedding, it suffices to show that ∆ is undistorted in Γ. But since M is geometrically finite and ∆ is finitely generated, it follows from a result of Thurston (see Proposition 7.1 in [START_REF] Morgan | On thurston's uniformization theorem for three-dimensional manifolds[END_REF]) that the group ∆ is geometrically finite, and geometrically finite subgroups of (say, geometrically finite) hyperbolic 3-manifold groups are undistorted, c.f. Corollary 1.6 in [START_REF] Hruska | Relative hyperbolicity and relative quasiconvexity for countable groups[END_REF].

For the 'moreover' statement, assume S is incompressible, so that S is simply connected, and consider a pair of infinite rays h ± : R + → S that are geodesic for the induced hyperbolic metric and t ± n → +∞ such that

d M (h + (t + n ), h -(t - n )) is bounded. Since S ⊂ ∂ M is a quasi-isometric embed- ding, d S (h + (t + n ), h -(t - n )
) is also bounded. Since S is simply connected and hyperbolic, this is possible only if h + and h -are asymptotic on S. Taking h + = h -we get that a geodesic ray on S can not be homoclinic. Taking h + ∕ = h -, we get that any pair of mutually homoclinic infinite rays on S are asymptotic. In particular two disjoint geodesic rays in a homoclinic geodesic should be asymptotic. This is impossible for a geodesic in a simply connected hyperbolic surface. □

Example (2) above shows how embedded annuli in M can be used to create mutually homoclinic rays. In analogy to Fact 5.2, one can show that annuli are essential in such a construction. For instance, suppose M is acylindrical. Then work of Thurston, see [START_REF] Kapovich | Hyperbolic manifolds and discrete groups[END_REF] and more generally [START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF], says that we can choose the hyperbolic manifold N so that ∂CC(N ) ∼ = ∂ χ<0 M is totally geodesic. Hence, the preimage of ∂ χ<0 M in M ⊂ H 3 is a collection of hyperbolic planes. Any geodesic ray on ∂ χ<0 M then lifts to a geodesic in H 3 , and two geodesic rays on ∂ χ<0 M are mutually homoclinic if and only if their geodesic lifts are asymptotic in H 3 , which implies that they were asymptotic on ∂ χ<0 M .

Added this subsection

5.1. Alternate definitions of homoclinic. Above, we defined a path

h : I -→ ∂ M to be homoclinic if there is are s i , t i ∈ I with |s i -t i | → ∞ such that sup i d M (h(s i ), h(t i )) < ∞.
Some other papers use slight variants of this definition. For example, the definition of (faiblement) homoclinique in Otal's thesis [START_REF] Otal | Courants géodesique et produits libre[END_REF] is almost the same as what is written above, except that distances are computed in the intrinsic metric on ∂ M instead of in M . This is equivalent to our definition, though: the nonobvious direction follows from Fact 5.2, which says that boundary components of M quasi-isometrically embed in M . And in the definition of homoclinique in Lecuire's earlier work [START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF], distances are computed not in M , but within H 3 , with respect to a given identification of M with the convex core of some minimally parabolic hyperbolic 3-manifold, as discussed in Example 5.1. When M has tori in its boundary, the inclusion M ↩→ H 3 is not a quasi-isometric embedding, but the following lemma says that d H 3 is bounded if and only if d M is bounded, so Lecuire's earlier definition is equivalent to ours.

Lemma 5.3. Whenever x, y ∈ M , we have

d H 3 (x, y) ≤ d M (x, y) ≤ e d H 3 (x,y)/2 d H 3 (x, y).
Proof. Set N := Γ\H 3 , so that M is obtained from the convex hull CH := CH(Λ(Γ)) of the limit set of Γ by deleting horoball neighborhoods around all rank two cusps. Take a H 3 -geodesic γ from x to y. Then γ lies inside CH, and it can only penetrate the deleted horoball neighborhoods to a depth of d(x, y)/2. Now, whenever B ⊃ B ′ are horoballs in H 3 such that d H 3 (∂B, B ′ ) ≤ d(x, y)/2, the closest point projection

π : B \ B ′ -→ ∂B
is well defined and e d(x,y)/2 -lipschitz. (Indeed, it suffices to take B as the height 1 horoball in the upper half space model and B ′ as the height e d(x,y)/2 horoball, and then the claim is obvious.) So, the parts of γ above that penetrate the deleted horoballs can be projected back into ∂ M , and if we do this the resulting path has length at most e d(x,y)/2 d(x, y). □

We should mention the version of homoclinic defined in Casson's original unpublished notes. There, M is a handlebody, and if we regard ∂ M ↩→ H 3 as above, then a simple geodesic h : I -→ ∂ M is called homoclinic if when we subdivide h into two rays h ± , these rays limit onto subsets

A ± ⊂ H 3 ∩ ∂H 3 such that A + ∩ A -∕ = ∅.
This definition is stronger than all the ones mentioned above: if A + ∩ A -contains a point on ∂ M , rather than at infinity, then the definition of homoclinic above is obviously satisfied. Otherwise, h ± have to have a common accumulation point in ∂H 3 , which corresponds to an end ξ of M , and one can use the treelike structure of the universal cover M of the handlebody M to say that h ± have to both intersect a sequence of meridians (m i ) on M that cut off smaller and smaller neighborhoods of ξ. The times t i p m when h ± intersects m i then work in the definition of homoclinic above. In fact, Casson's definition is strictly stronger. For instance, if h ± both spiral around disjoint simple closed curves γ ± ⊂ ∂ M , then h is homoclinic by our definition but not by Casson's. However, Statement 1.0.1 still fails using Casson's original definition, due to the examples in Figure 1.

5.2.

Waves, Tight position, and Intrinsic limits. As in the previous section, let M be a compact, orientable hyperbolizable 3-manifold with nonempty boundary ∂M , which we think of as the convex core of a hyperbolic 3-manifold with horoball neighborhoods of its rank 2 cusps deleted. Definition 5.4 (Waves and tight position). Suppose that m is a meridian multicurve on ∂M , and let γ ⊂ ∂M be a simple geodesic ray or a simple biinfinite geodesic. An m-wave is a segment β ⊂ γ that has endpoints on m, and is homotopic rel endpoints in M to an arc of m. If γ has no m-waves, and every infinite length segment of γ intersects m, then we say that γ is in tight position with respect to m. Waves and tight position were discussed previously in [START_REF] Kleineidam | Algebraic convergence of function groups[END_REF][START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF], for instance. Note that in our definition, an m-wave β can intersect m in its interior. More generally, an m-wave of a lamination is an m-wave of one of its leaves, and a lamination is in tight position with respect to m if all of its leaves are. Note that from this perspective, if a geodesic γ is in tight position with respect to some multicurve m (regarded as a lamination), then it is in tight position with respect to some component of m.

As an example, a meridian γ can never be in tight position with respect to another meridian m: taking discs with boundaries γ and m that are transverse and intersect minimally, any arc of intersection of these disks terminates in a pair of intersection points of γ and m that bound a m-wave of γ.

More generally, we have the following fact. In the notation of Example 5.1, we have that M ⊂ H 3 is obtained from CH(Γ) by deleting an equivariant collection of horoballs. Each component of D separates CH(Γ), so if γ ′ is a segment of γ, any geodesic in H 3 joining the endpoints of γ ′ must intersect each of the discs that γ ′ intersects. Hence, if 󰂃 > 0 is the minimum distance between any two components of D, then γ is a (L/󰂃, L)-quasigeodesic.

□

We now describe how to create systems of meridians with respect to which a given lamination is in tight position. Definition 5.6 (Surgery). Suppose that λ is a geodesic lamination on ∂M and m = ⊔ n i=1 m i is a geodesic meridian multi-curve on ∂M , and β is an m-wave in λ whose interior is disjoint from m. Then the pair of points ∂β separates some component m i of m into two arcs m 1 i , m 2 i , both of which are homotopic to β rel endpoints in M . We perform a λ-surgery on m by replacing m 1 i (say) with β, thus constructing a new multicurve m ′ := (β ∪ m 2 i ) ⊔ 󰁉 j∕ =i m j . This notion of surgery appears in many other references, e.g. [START_REF] Biringer | Automorphisms of the compression body graph[END_REF][START_REF] Aj Casson | Algorithmic compression of surface automorphisms[END_REF][START_REF] Kleineidam | Algebraic convergence of function groups[END_REF][START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF]. We summarize its elementary properties here: Fact 5.7. Suppose that λ is a geodesic lamination.

(1) If m is a meridian and λ has an m-wave, it also has an m-wave whose interior is disjoint from m, so a λ-surgery can be performed.

(2) Any curve m ′ obtained by λ-surgery on a meridian m as above is a meridian.

(3) If m is a cut system for M , i.e. a multi-curve of meridians bounding discs that cut M into balls and 3-manifolds with incompressible boundary, then some λ-surgery on a component of m is another cut system.

Proof. For (1), suppose that λ has an m-wave β. Let m be the entire preimage of m in the cover ∂ M , and lift β to an arc β starting and ending on some fixed component m 0 ⊂ m. Since each component of m separates ∂ M , there is some "outermost" subarc β′ that has endpoints on the same component of m, and that has interior disjoint from m. This β′ projects to an m-wave of λ whose interior is disjoint from m. For (2), note that if m ′ := β ∪ m 2 is obtained by λ-surgery on m, as above, then m, m ′ are homotopic in M , and hence m ′ is nullhomotopic. Also, if m ′ is inessential in ∂M , then β is homotopic on ∂M to m 2 , implying that λ and m were not in minimal position, a contradiction since they are both geodesic. Hence m ′ is a meridian.

For [START_REF] Biringer | Automorphisms of the compression body graph[END_REF], consider an m-wave in λ whose interior is disjoint from m and say that ∂β ⊂ m 1 . Then

∂β separates m 1 into two arcs m 1 1 , m 2 1 . It is not difficult to see that either (β ∪ m 1 1 ) ⊔ 󰁉 j∕ =1 m j or (β ∪ m 2 1 ) ⊔ 󰁉 j∕ =1 m j is a cut system. □
The following lemma is a modification of a result of Kleineidam-Souto [31, Lemmas 7 and 8] that is essential for everything below. Lemma 5.8 (No waves, or a sequence of meridians). Suppose λ is a geodesic lamination on S = ∂M and m is a meridian multi-curve. Then either

(1) there exists a finite sequence of λ-surgeries on m that terminates in some meridian multi-curve m ′ where λ has no m ′ -waves, (2) S(λ) contains a sequence of meridians (γ i ) such that i(λ, γ i ) → 0, with respect to every transverse measure on λ.

Here, (2) makes sense even when λ admits no transverse measure of full support. Note that if λ is a minimal lamination and ∂S(λ) is incompressible, then (2) implies that λ is an intrinsic limit of meridians.

Proof of Lemma 5.8. The two cases depend on whether λ contains infinitely many homotopy classes of m-waves, or not. Here, our homotopies are through arcs on S, keeping their endpoints on m.

If there are only finitely many classes of m-waves in λ, then a finite sequence of λ-surgeries converts m into a multi-curve m ′ such that λ has no m ′ -waves, as each surgery decreases the number of waves by at least one. If there are infinitely many homotopy classes of m-waves in λ, then we can choose a sequence of parameterized m-waves α i : [0, 1] -→ R such that

(1) the two sequences of endpoints (α i (0)) and (α i (1)) both converge, and if either sequence converges into a simple closed curve γ ⊂ λ, then it approaches γ from only one side, (2) no α i and α j are homotopic keeping their endpoints on m, for i ∕ = j. To construct the desired sequence of meridians, let β 0 i be the shortest geodesic on S from α i (0) to α i+1 (0), and define β 1 i similarly. For large i, the union

β 0 i ∪ α i ∪ β 1 i ∪ α i+1 is an essential closed curve in S(λ) that is nullhomotopic in M .
It may not be simple, since β 0 i and β 1 i may overlap, but it has at most one self intersection. So by the Loop Theorem [START_REF] Jaco | Lectures on three-manifold topology[END_REF], one of the three simple closed curves obtained by surgery on it is a meridian γ i . Now, the fact that the endpoints can approach a simple closed curve in λ only from one side implies that for large i, the curves γ i do not intersect any simple closed curve contained in λ. Since γ i only intersects λ along the arcs β 0 i and β 1 i , whose hyperbolic lengths converge to zero, it follows that i(γ i , λ) → 0 for any transverse measure on λ. □

Here is an important application of Lemma 5.8.

Lemma 5.9 (Quasigeodesic or a sequence of meridians). Suppose λ ⊂ ∂ χ<0 M is a minimal geodesic lamination and that ∂S(λ) is incompressible in M . Let h ⊂ S(λ) be a simple geodesic ray or biinfinite geodesic that is disjoint from λ or contained in λ. Then either

(1) any lift h ⊂ ∂ M of h is a quasi-geodesic in M .
(2) S(λ) contains a sequence of meridians (γ i ) such that i(λ, γ i ) → 0, with respect to every transverse measure on λ. In particular, if h is homoclinic, then λ satisfies (2).

Proof. Assume that (2) does not hold. Given a cut system m for M , Lemma 5.8 and Fact 5.7 (3) say that we can perform λ-surgeries until we obtain a new cut system m such that λ ∪ h has no m-waves. If m intersects λ, then λ ∪ h is in tight position with respect to m, so (1) follows from Fact 5.5. Therefore, we can assume m does not intersect λ. Up to isotopy, we can also assume that S(λ) does not intersect m. Since ∂S(λ) is assumed to be a collection of incompressible curves, it follows that S(λ) is itself incompressible, so (1) follows from Fact 5.2.

□

We now come to the central definition of the section.

Definition 5.10. A minimal geodesic lamination λ ⊂ ∂ χ<0 M is an intrinsic limit of meridians if there is a transverse measure6 on λ and a sequence of meridians (γ i ) contained in S(λ) such that γ i → λ in PML(S(λ)).

Using Lemma 5.9, we can prove the following proposition, which gives several equivalent characterizations of intrinsic limits. Proposition 5.11 (Intrinsic limits). Suppose λ ⊂ S = ∂M is a minimal geodesic lamination and ∂S(λ) is incompressible. The following are equivalent:

(1) λ is an intrinsic limit of meridians, (2) given (some/any) transverse measure on λ, there is a sequence of meridians (γ i ) in S(λ) such that i(γ i , λ) → 0, (3) there is a biinfinite homoclinic geodesic in S(λ) that is either a leaf of λ, or is disjoint from λ, (4) given any transverse measure on λ, there is a sequence of essential (possibly non-simple) closed curves (γ i ) in S(λ) such that each γ i is nullhomotopic in M , and i(γ i , λ) → 0.

Note that when we say ∂S(λ) is incompressible, we mean that no closed curve that is a boundary component of S(λ) is nullhomotopic in M . This condition is mainly here to make statements and proofs easier. For instance, without this assumption our proof of (4) =⇒ (2) may produce peripheral meridians, but peripheral meridians can't be used in (2) =⇒ (1).

Proof. (2) =⇒ [START_REF] Agol | Tameness of hyperbolic 3-manifolds[END_REF]. Fix some transverse measure on λ. By (2), i(γ i , λ)/ length(γ i ) → 0, so after passing to a subsequence we can assume that (γ i ) converges to a measured lamination µ in S(λ) that does not intersect λ transversely. As λ fills S(λ), µ is supported on λ.

(1) =⇒ (3). After passing to a subsequence, we can assume that (γ i ) converges in the Hausdorff topology to some lamination, which must then be an extension of λ by finitely many leaves. (3) follows from an unpublished criterion of Casson, see Lecuire [START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF]Théorème B.1] for a proof, that states that any Hausdorff limit of meridians has a homoclinic leaf.

(3) =⇒ (2). This is an immediate corollary of Lemma 5.9.

(4) ⇐⇒ (2). The direction ⇐ is immediate, so suppose (γ i ) is a sequence of essential closed curves in S(λ) that are nullhomotopic in H and i(λ, γ i ) → 0. By Stallings' version of the Loop Theorem, for each i there is a meridian γ ′ i that is obtained from γ i by surgery at the self intersection points. Such surgeries can only decrease the intersection number with λ, so (2) follows. □

We will also need the following criterion in the next section.

Lemma 5.12 (Intrinsic limits of annuli). Suppose λ ⊂ ∂ χ<0 M is a minimal lamination such that S(λ) is compressible but ∂S(λ) is incompressible, and that there is a sequence (A i ) of essential embedded annuli in (M, S(λ)) with i(∂A i , λ) → 0. Then λ is an intrinsic limit of meridians.

Proof. Pick a meridian m ⊂ S(λ). For each i, let T i : M -→ M be the Dehn twist along the annulus A i . Then for any sequence n i ∈ Z, the curves T n i i (m) are meridians, and if n i grows sufficiently fast, then i(T n i i (m), λ)/ length(T n i i (m)) → 0. Hence, after passing to a subsequence T n i i (m) converges to a lamination λ ′ supported in S(λ) with zero intersection number with λ, implying λ ′ and λ have the same support, so λ is an intrinsic limit of meridians. □

Limits of homoclinic rays

In this section we characterize the laminations onto which pairs of disjoint mutually homoclinic rays can accumulate. Theorem 6.1 (Mutually homoclinic rays). Let M be a compact orientable hyperbolizable 3-manifold and equip ∂ χ<0 M with an arbitrary hyperbolic metric. Let h ± be two disjoint, mutually homoclinic simple geodesic rays on ∂ χ<0 M that accumulate onto (possibly equal) minimal laminations λ ± , and that the multicurve ∂S(λ ± ) is incompressible in M . Then one of the following holds:

(1) one of λ + or λ -is an intrinsic limit of meridians, (2) h + and h -are asymptotic on ∂ χ<0 M , and either (a) any two mutually homoclinic lifts h± to ∂ M are asymptotic on ∂ M , or (b) λ := λ -= λ + is a simple closed curve that is homotopic in M to a nontrivial power γ n , n > 1 of some closed curve γ in M , (3) h ± are not asymptotic on ∂ χ<0 M , and there is an essential (possibly nontrivial) interval bundle B ⊂ M such that λ ± each fill a component of ∂ H B, and λ ± are essentially homotopic through B, as in §2.9. The proof of Theorem 6.1 is given in §6.1.

One can construct examples of mutually homoclinic rays in each of cases (1)-(3). For concreteness, suppose that M is a handlebody. For (1), pick two meridians λ -, λ + on M and let h ± spiral onto m ± . One can also produce similar examples by letting λ ± be arbitrary laminations in disjoint subsurfaces S(λ ± ) that are spheres with at least 4 boundary components, all of which are compressible in M , and letting h ± accumulate onto λ ± . For (2) (a), take λ to be any simple closed curve on ∂M that is essential in M but has no nontrivial roots in π 1 M , and let h ± spiral around λ in the same direction. We'll discuss 2 (b) in Remark 6.2. For (3), write M = S × [-1, 1] where S is a surface with boundary, let λ be a lamination on S, and let h ± be corresponding leaves of λ ± := λ × {±1}. Then M ∼ = S × [-1, 1], so there are lifts of h ± that are mutually homoclinic. One can also construct similar examples of (3) where the interval bundle B is twisted.

In case (1), we expect it is possible that S(λ -) is incompressible, say, while λ + is an intrinsic limit of meridians. For instance, suppose C is a compression body with connected, genus-at-least-two interior boundary ∂ -C, and exterior boundary ∂ + C. Let f : C -→ C be a homeomorphism such that f | ∂ + C and f | ∂ -C are both pseudo-Anosov, with attracting laminations λ + , λ -, respectively. We expect that there are rays ℓ ± ⊂ λ ± that are mutually homoclinic. But λ + is an intrinsic limit of meridians, while S(λ -) is incompressible.

The assumption that ∂S(λ ± ) is incompressible is necessary in Theorem 6.1. For instance, suppose M is a compression body with exterior boundary a genus 3 surface S, where the only meridian on S is a single separating curve γ. Let T be the component of S \ γ that is a punctured genus 2 surface. Then there are distinct minimal geodesic laminations λ, λ ′ ⊂ T , each of which fills T , that are properly homotopic in M : just take distinct laminations that are identified when we cap off the puncture of T to get a closed genus 2 surface. Corresponding ends of corresponding leaves of λ, λ ′ are mutually homoclinic rays that accumulate onto λ, λ ′ , respectively, but none of (2)-(3) hold. One could write down a version of Theorem 6.1 that omits the assumption that ∂S(λ ± ) is incompressible, but the conclusion would be relative to capping off S(λ ± ), and the statement would be more complicated. Remark 6.2. The reader may be wondering about (2) (b) in Theorem 6.1, and how it differs from (2) (a). A relevant example of two asymptotic rays that have mutually homoclinic nonasymptotic lifts to ∂ M is given in Figure 10. On the left we have a solid torus that is a boundary-connect-summand of M , which (say) is a handlebody. The biinfinite geodesic h is homoclinic and its two ends are mutually homoclinic rays that both spiral onto a simple closed curve λ, the (2, 1)-curve on the solid torus. Then λ is homotopic to the square of the core curve of the solid torus. Although the two ends of h are asymptotic on ∂M , any lift h in ∂ M will have ends that are mutually homoclinic, but nonasymptotic. On the right, we have drawn the preimage λ of λ, and two lifts h1 , h2 of h. Note that one end of h1 is asymptotic to an end of h2 .

When M is a compression body, Casson-Gordon prove in [13, Theorem 4.1] that any simple closed curve λ ⊂ ∂M that has a nontrivial root in π 1 M lies on the boundary of a solid torus that is a boundary connect summand of M , exactly as in Figure 10. When M has incompressible boundary, such λ come from components of the characteristic submanifold of M , see §2.6, that are either solid tori or twisted interval bundles over nonorientable surfaces.

Here is a slightly more refined version of Theorem 6.1 that applies to homoclinic biinfinite geodesics on ∂ χ<0 M . Corollary 6.3 (Homoclinic biinfinite geodesics). Suppose that M is as in Theorem 6.1, that h is a homoclinic biinfinite simple geodesic on some component S ⊂ ∂ χ<0 M , that h ± are the two ends of h, that h ± limit onto λ ± , and that ∂S(λ ± ) is incompressible in M .

Then one of (1)-(3) in Theorem 6.1 holds. Moreover, in case (2), if λ := λ ± is not an intrinsic limit of meridians then either (i) after reparametrizing h, there is some s such that h(-s) and h(s) are joined by a geodesic segment c with h∩int(c) = ∅, such that c, h| (-∞,-s) and h| [s,∞) bound an embedded geodesic triangle ∆ ⊂ S with one ideal vertex, and c ∪ h([-s, s]) is a meridian in M , or (ii) λ is a simple closed curve on S, the two ends of h spiral around λ in the same direction, and any neighborhood of the union h ∪ λ ⊂ S contains a meridian. And in case (3), we can choose the interval bundle B such that h contains a subarc α ⊂ h that is a compression arc for B.

Proof. Let h be a homoclinic lift of h on ∂ M . By Lemma 5.9, either one of λ ± is an intrinsic limit of meridians in M , in which case we're in case (1) and are done, or both ends of h are quasi-geodesic in M . Since h is homoclinic, it follows that its two ends are mutually homoclinic, so we're in the setting of Theorem 6.1 and one of ( 2)-(3) holds. Assume we're in case (2) of Theorem 6.1, and set λ := λ ± . Assume first that λ := λ ± is a simple closed curve. Since the two ends of h are asymptotic, they spiral around λ in the same direction. Let U be a neighborhood of h ∪ λ on ∂ χ<0 M . Then h is a homoclinic geodesic contained in U , so Lemma 5.2 implies that U is compressible as desired in (ii). Now suppose that λ is not a simple closed curve, in which case we're in case (2) (a) of Theorem 6.1. We show (i) holds. Let's start by constructing the desired geodesic triangle. Parametrize h, pick a universal covering map H 2 -→ S and lift h to a ĥ in H 2 , and let

ξ = lim t→+∞ ĥ(t) ∈ ∂ ∞ H 2 .
Since the two ends of h are asymptotic on S, there is a deck transformation γ : H 2 -→ H 2 such that ξ = lim t→-∞ γ • ĥ(t). It follows that if we use a particular arc-length parametrization of h, we may assume that for each t ∈ R, the points ĥ(t), γ • ĥ(-t) lie on a common horocycle tangent to ξ. Fix some large s such that the geodesic segment ĉ joining ĥ(s) and γ • ĥ(-s) is shorter than the injectivity radius of S, and therefore projects to a simple geodesic segment c in S.

Let ∆ ⊂ H 2 be the triangle bounded by ĉ and the two rays ĥ([s, ∞)) and γ • ĥ((-∞, s]). Let g : H 2 -→ H 2 be a deck transformation. We claim that g • ĥ(R) ∩ int( ∆) = ∅. If not, then since ∆ has geodesic sides, two of which are disjoint from g • ĥ, it follows that one of the two endpoints of g • ĥ is ξ. If it's the positive endpoint, then g fixes ξ, and the axis of g projects to a (simple) closed curve on S, around which the two ends of h spiral, contradicting that λ isn't a simple closed curve. If the negative endpoint of g • ĥ is ξ, then g • γ -1 fixes ξ and we get a similar contradiction.

Next, we claim that we have g(int( ∆)) ∩ int( ∆) = ∅ as long as g ∕ = id. Suppose that for some g ∕ = id the intersection is nonempty. Then g(ξ) ∕ = ξ, since otherwise we have a contradiction as in the previous paragraph. The previous paragraph implies that the sides of the triangles g( ∆), ∆ that are lifts of rays of h do not intersect the interior of the other triangle. So, the only way the interiors of g( ∆), ∆ can intersect is if ĉ and g(ĉ) intersect. However, this does not happen since we chose s large enough so that ĉ projects to a simple geodesic segment in S.

The previous two paragraphs imply that ∆ projects to an embedded geodesic triangle ∆ in S whose interior is disjoint from h, as desired in (i). By construction, c and h([-s, s]) are simple geodesic segments and, since g • ĥ(R) ∩ int( ∆) = ∅ for any g ∕ = id, they are disjoint. It follows that c ∪ h([-s, s]) is an essential simple closed curve on S. Note that c ∪ h([-s, s]) is an essential simple closed curve on S, since it is the concatenation of two geodesic segments with disjoint interiors. We need to show it is nullhomotopic in M . Now, if h is a lift of h to ∂ M , its ends are mutually homoclinic, and hence are asymptotic on ∂ M by the assertion in case (2) of Theorem 6.1. Therefore, after choosing compatible lifts, the projection ∆ -→ ∆ factors through a geodesic triangle ∆ ⊂ ∂ M bounded by h([s, ∞)), h((-∞, -s]) and a geodesic segment c. The curve c ∪ h([-s, s]) is the projection of the closed curve c ∪ h([-s, s]) ⊂ M , and therefore is nullhomotopic in M . Now assume we are in case [START_REF] Biringer | Automorphisms of the compression body graph[END_REF]. Let S ± be the components of ∂ H B containing λ ± . We may assume that h is in minimal position with respect to ∂S ± . Since h is simple and the ends of h limit onto minimal laminations that fill S ± , we have that h intersects ∂S -∪ ∂S + at most twice. Furthermore, in the case that S -= S + , the homoclinic geodesic h cannot be contained entirely in the incompressible surface S ± , by Fact 5.2. So, h is the concatenation of two rays in S ± and an arc α such that int(α) lies outside S ± .

Let X ⊂ S be the union of S ± and a regular neighborhood of α. Since h is homoclinic, there is a meridian on X by Fact 5.2. If the two endpoints of α lie on different boundary components of ∂ H B, then α is a compression arc for B by Fact 2.12. So, we may assume that the two endpoints of α lie on the same boundary component c of ∂ H B. Fact 2.12 then says that α is homotopic rel endpoints in M to an arc of c. So, if we make a new path h ′ ⊂ ∂ H B from h by replacing α with that arc of c, then h ′ is still homoclinic, so it cannot be boundedly homotopic to a geodesic in ∂ H B by Fact 5.2, which implies that its ends h ± are asymptotic, a contradiction to the assumption in (3). □ 6.1. Proof of Theorem 6.1. The proof proceeds in a few cases. As in Example 5.1, we identify M with the convex core CC(N ) of a geometrically finite hyperbolic 3-manifold N = H 3 /Γ, and we identify the universal cover M with the preimage of CC(N ) in H 3 , which is the convex hull of the limit set of Γ.

Note that the closure of M in H 3 ∪ ∂H 3 is a ball. There are four cases to consider:

(A) Both λ ± are simple closed curves. We show that either (1) or (2) holds. (B) Both λ ± are distinct, in which case the surfaces S(λ ± ) are disjoint, but one of these surfaces is compressible, say S(λ + ). We show (1). (C) At least one of λ ± is not a simple closed curve, and both S(λ ± ) are incompressible. We show (2) (a) or (3) holds. (D) λ -= λ + , which is not a simple closed curve, and S(λ ± ) is compressible.

We show either (1) or (2) (a) holds. (A) and (B) above are the easiest. Our proof in case (C) involves a hyperbolic geometric interpretation of the characteristic submanifold of a pair, as discussed in §3 of [START_REF] Lecuire | Structures hyperboliques convexes sur les variétés de dimension 3[END_REF] and in Walsh [START_REF] Walsh | The bumping set and the characteristic submanifold[END_REF]; our argument is a bit more complicated than theirs, since we have to deal with accidental parabolics. In case (D), our argument adapts and fills some gaps in a surgery argument of Kleineidam-Souto [START_REF] Kleineidam | Algebraic convergence of function groups[END_REF] and Lecuire [START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF]Appendix C].

Proof of (A). Assume that both λ ± are simple closed curves. If one of λ ± is a meridian, we are in case (1) and are done. So, we may assume that both λ ± are incompressible in M . If λ -∕ = λ + , then we are in case (3) by the Annulus Theorem. So we may assume the two curves are the same, and write λ := λ ± .

We claim that h ± spiral around λ in the same direction, so that they are asymptotic on ∂M . Suppose not, and pick mutually homoclinic lifts h± in M . Then hand h+ are asymptotic to lifts λ and α( λ) of λ, where α ∈ Γ is a deck transformation. Any lift of λ is a quasi-geodesic in M , and hence in H 3 , so h± are quasi-geodesic rays, and therefore have well-defined endpoints in ∂H 3 , which must be the same since the two rays are mutually homoclinic. Since h ± spiral around λ in opposite directions, this means that α ∈ Γ takes one endpoint of λ in ∂H 3 to the other endpoint of λ. Since λ is stabilized by a loxodromic isometry in Γ, and Γ is torsion-free and discrete, this is impossible.

Suppose we are not in case (2) (a), so there are mutually homoclinic lifts h± that are not asymptotic on ∂ M . As in the previous paragraph, we may assume that hand h+ are asymptotic to lifts λ and α( λ) for some deck transformation α ∈ Γ. Since h± are not asymptotic, λ ∕ = α( λ). As before, α fixes the common endpoint of h± in ∂H 3 , which is a fixed point of the cyclic group 〈β〉 ⊂ Γ of loxodromic isometries fixing λ. As Γ is discrete and torsion-free, and α ∕ ∈ 〈β〉, we have that α is a root of β or β -1 in Γ, and (2) (b) follows.

Proof of (B). Suppose that λ ± are distinct, in which case the surfaces S(λ ± ) are disjoint, but that one of these surfaces is compressible, say S(λ + ). We claim that λ + is an intrinsic limit of meridians, in which case (1) holds and we are done. If not, take a meridian m ⊂ S(λ + ) and apply Lemma 5.8. We obtain a new meridian m ′ ⊂ S(λ + ) such that λ + has no m-waves. Since λ + fills S(λ + ) and the boundary components ∂S(λ ± ) are incompressible, it follows that λ + is in tight position with respect to m. So after possibly restricting the domains, h + is in tight position with respect to m ′ , while h -never intersects m ′ . This contradicts the fact that h ± are mutually homoclinic, since if h± are homoclinic lifts in M , for large t the point h+ (t) is separated from the image of hby arbitrarily many lifts of m ′ .

Proof of (C). Assume that at least one of λ ± is not a simple closed curve, and that S ± := S(λ ± ) are incompressible. Note that S ± are equal or have disjoint interiors. We want to prove that we're in case [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF] or [START_REF] Biringer | Automorphisms of the compression body graph[END_REF]. Lift h ± to mutually homoclinic rays h± ⊂ ∂ M . Fact 5.2 implies that each inclusion S± ↩→ M is a quasi-isometric embedding, so if S-= S+ , then the two mutually homoclinic rays h± are actually asymptotic on ∂ M . If this is true for all lifts h± , we are in case (2) (a) and are done. So, we can assume below that S-∕ = S+ . Note that it may still be that λ -= λ + and S -= S + .

Let Γ ± ⊂ Γ be the stabilizer of S± and let Λ ± ⊂ ∂H 3 be the limit set of Γ ± . Since Γ ± acts cocompactly on S± , the inclusion S± ↩→ H 3 extends continously to a map ∂ ∞ S± -→ Λ ± ⊂ ∂H 3 , by the main result of [START_REF] Mj | Cannon-thurston maps for kleinian groups[END_REF]. In particular, h± have well defined endpoints in ∂H 3 , and since they're mutually homoclinic, they have the same endpoint ξ ∈ Λ -∩ Λ + ⊂ ∂H 3 .

We now apply Theorem 4.1. Since ξ ∈ Λ -∩ Λ + , using the notation of Theorem 4.1, the rays h± are either eventually contained in the convex hulls C± ⊂ S± , or are asymptotic onto their boundaries. But C± project to (possibly degenerate) generalized subsurfaces C ± with geodesic boundary in S ± , and the rays h ± limit onto filling laminations in S ± , so it follows that actually C ± = S ± , and that there is a homotopy from S -to S + in M that is the projection of a homotopy from Sto S+ . Since one of λ ± is not a simple closed curve, this means they are both not simple closed curves and the (a priori degenerate) subsurfaces with geodesic boundary S ± are not simple closed geodesics.

Let S ′ ± ⊂ int(S ± ) be obtained by deleting small collar neighborhoods of ∂S ± , so that S ′ ± are both actually embedded, still contain λ ± , and are either disjoint or equal. Since S ′ ± are incompressible and homotopic in M , Theorem 2.8 implies that they bound an essential interval bundle B ⊂ M . Moreover, the fact that the homotopy from S -to S + is the projection of a homotopy from Sto S+ means that we can assume that there is a component B ⊂ M of the preimage of B that intersects ∂ M in S′ ± . Note that B is invariant under ∆ = Γ -∩ Γ + , since any element of ∆ preserves S′ ± , and hence B. We claim that λ ± are essentially homotopic through B. By Fact 2.16, it suffices to show that if σ is the canonical involution of B, as described in §4.5, then σ(λ ± ) is isotopic to λ ∓ on S ′ ∓ . Well, σ lifts to a ∆-equivariant involution σ of B that exchanges S′ -and S′ + , where here ∆ = Γ -∩ Γ + . By equivariance, σ extends continuously to the identity on Λ ∆ , so in particular its extension fixes ξ, and hence σ(h ± ) is properly homotopic to h ∓ on S ∓ , which implies σ(λ ± ) is isotopic to λ ∓ as desired.

If h ± are not asymptotic on ∂M , then we are in case (3) and are done. So, assume h ± are asymptotic. Then there is some γ ∈ Γ such that γ( h-) ⊂ S′ + and is asymptotic to h+ . This γ fixes the endpoint ξ ∈ ∂H 3 of h± . Moreover, γ( B) is a component of the preimage of B that contains S ′ + , and therefore equals B. So, γ exchanges S′ ± , and therefore γ 2 ∈ ∆. But then h± are asymptotic to the axes of γ 2 ↷ S± , implying that h ± accumulate onto simple closed curves in ∂M , contradicting our assumption in (C).

Proof of (D). Assume that λ -= λ + , write λ = λ ± for brevity, assume that λ is not a simple closed curve, and that S(λ) is compressible. We want to prove that either λ is an intrinsic limit of meridians, or h ± are asymptotic, as are any pair of mutually homoclinic lifts h± .

If λ is an intrinsic limit of meridians, we are done, so since S(λ) is compressible with incompressible boundary, by Lemma 5.8 we can choose a meridian m ⊂ S(λ) with respect to which λ is in tight position. Let m be its full preimage in ∂ M , and let h± be any pair of mutually homoclinic lifts in ∂ M .

Truncating if necessary, we can assume that h ± are in tight position with respect to m, and hence the lifts h± are quasigeodesic rays in H 3 , by Fact 5.5. Since they are mutually homoclinic, hand h+ converge to the same point ξ ∈ ∂ ∞ H 3 , and tightness further implies that after restricting to appropriate subrays, hand h+ intersect exactly the same components of m, in the same order. Reparametrizing, we have

h± : [0, ∞) -→ ∂ M , h+ (i), h-(i) ∈ mi , ∀i ∈ N,
where each mi is a component of m, and where h± (t) ∕ ∈ m when t ∕ ∈ N. Let

d i := d m󰀃 h+ (i), h-(i) 󰀄
be the distance along m between h+ (i) and h-(i).

Claim 6.4.

There is some uniform 󰂃 > 0, independent of the particular chosen lifts h± , such that either

(1) h± are asymptotic on ∂ M , and hence h ± are asymptotic on ∂M , or (2) lim inf i d i ≥ 󰂃.

Proof. Let's assume that h+ and hare not asymptotic on ∂ M , and write d = lim inf i d i . Fix some transverse measure on λ. If d is small, we will construct meridians γ ⊂ S(λ) with very small intersection number with λ. Since λ is not an intrinsic limit of meridians, there is some fixed lower bound for such intersection numbers, which will give a contradiction for small d.

Suppose d is small and pick 0 << i < j such that We first claim that after fixing i, it is possible to choose j such that γij is homotopically essential on ∂ M . Assume not, let S ⊂ ∂ M be the component containing h± , fix a universal covering map

d i , d j <
H 2 -→ S,
and lift the rays h± | [i,∞) to rays

h ± : [i, ∞) → H 2
in such a way that b i lifts to a segment connecting h -(i) to h + (i). Now, there are infinitely many j > i with d j < 2d. For each such j, we know that γij is homotopically inessential on ∂ M , so the points h -(j) to h + (j) are at most 2d away from each other in H 2 . This gives a sequences of points exiting the rays h ± that are always at most 2d apart, so h -is asymptotic to h + . Hence, his asymptotic to h+ , contrary to our assumption.

We now fix large i, j such that d i , d j < 2d and γ := γij is homotopically essential on ∂ M . Then γ projects to a homotopially essential loop γ ⊂ ∂M that is homotopically trivial in M . Note that if i, j are chosen large enough and d is small, then γ ⊂ S(λ). Furthermore, since the segments b i , b j are the only parts of γ that intersect λ, and these segments have hyperbolic length less than 2d, the intersection number i(γ, λ) is small when d is small. (Recall that λ is a minimal lamination that is not a simple closed curve, so no leaves have positive weight, and hence hyperbolic length can be compared to intersection number.) But λ is not an intrinsic limit of meridians, so Proposition 5.11 (4) says that there is some positive lower bound for the intersection numbers of λ with essential curves that are nullhomotopic in M . Hence, we get a contradiction if d is small. □ Suppose we have two pairs {a, b} and {c, d} of points in m, all four of which are distinct. We say the two pairs are unlinked in m if in the induced cyclic ordering on {a, b, c, d} ⊂ m, a is adjacent to b and c is adjacent to d, and we say that the two pairs are linked otherwise. Claim 6.5. If i, j ∈ N, i < j, then the pairs {h + (i), h -(i)} and {h + (j), h -(j)} are unlinked in m. For an example where the pairs are linked, see Figure 12. The proof below works in general whenever h ± are simple geodesic rays on ∂M in tight position with respect to m, where neither h + nor h -spirals onto a simple closed curve.

Proof. The essential observation used in the proof is that the closure

cl(∂ M ) ⊂ H 3 ∪ ∂H 3
is homeomorphic to a sphere: indeed, the closure of M in H 3 ∪ ∂H 3 is a ball, since M ⊂ H 3 is convex with nonempty interior, and the closure of the boundary is the boundary of the closure. We obtain the unlinking property above by exploiting separation properties of arcs and curves on cl(∂ M ).

Since h ± are in tight position with respect to m, both lifts h± cross mi exactly once. Since h± limit to the same point in ∂H 3 , they must then cross mi in the same direction. In other words, the tangent vectors h + (i) ′ , h -(i) ′ point to the same side of m. The same statement holds for j. This allows us to break into the following two cases:

(a) the arcs h ± | [i,j] start and end on the same side of m, i.e. the vectors h ± (i) ′ point to the opposite side of m as the vectors h ± (j) ′ , or (b) the arcs h ± | [i,j] start and end on different sides of m, i.e. all four velocity vectors h ± (i) ′ , h ± (j) ′ point to the same side of m, see Figure 13.

First, assume we're in case (a). Let

α ± := h± | [i,j] ,
which we regard as oriented arcs in ∂ M starting on mi and ending on mj . Let γ : M -→ M be the deck transformation taking mj to mi . Then the arcs

β ± := γ • h± | [i,j]
start on γ( mi ) and end on γ( mj ) = mi , and since we're in case (a) they end on the same side of mi as the arcs α ± start. Note that γ( mi ) is not mi or mj . Indeed, if γ( mi ) = mi then we'd have γ = id, contradicting that γ( mj ) = mi . And if γ( mi ) = mj , then γ 2 would leave mi invariant, implying that γ 2 = id, which is impossible since π 1 M has no torsion. We claim that the interiors of the arcs β ± do not intersect mi or mj , and the arcs α ± do not intersect γ( mi ). Indeed, the interiors of β ± don't intersect mi because the arcs β ± end on mi and intersect each component of m at most once, by tight position. The interiors of β ± don't intersect mj because any arc from mi to mj intersect at least ji + 1 components of m (counting mi and mj ), while any proper subarc of β ± intersects at most ji components of m. Here, for the ji + 1 bound we are using tight position of h ± , the definitions of mi , mj , and the fact that each component of m separates ∂ M . The fact that the arcs α ± don't intersect γ( mi ) is similar: any arc from m i to γ( mi ) must pass through at least ji + 1 components of m, while any proper subarc of α ± intersects at most ji components, and α ± do not end on γ( mi ) ∕ = mj .

Let A ⊂ cl(∂ M ) ∼ = S 2 be the annulus that is the closure of the component of cl(∂ M ) \ ( mi ∪ mj ) that contains the side of mi on which the arcs α ± start and the arcs β ± end. Then α ± are two disjoint arcs in A that join mi to mj , and therefore α ± separate A into two rectangles. The component γ( mi ) on which the arcs β ± start is contained in the interior of one of these two rectangles. Therefore, the two arcs β ± must lie in the same component of A \ (α + ∪ α -). Looking at endpoints, this means the pairs { h+ (i), h-(i)} and {γ • h+ (j), γ • h-(j)} are unlinked in mi , and the claim follows. Now assume that we're in case (b). The curve mi separates ∂ M , and we let X ⊂ ∂ M be the closure of the component of ∂ M \ mi into which the velocity vectors h′

± (i) and (γ • h± ) ′ (j) all point. The closure cl(X) ⊂ H 3 ∪ ∂H 3
is homeomorphic to a disk, since cl(∂ M ) is a sphere. As before, we let γ : M -→ M be the deck transformation taking mj to mi . Then the rays

α ± := h± ([i, ∞)), β ± := γ • h± ([j, ∞))
are all contained in X. Note that α ± both limit to a point ξ ∈ ∂H 3 , while β ± limit to γ(ξ) ∈ ∂H 3 . The union α -∪α + compactifies to an arc in cl(X), since the two rays limit to the same point in H 3 . The same is true for β -∪β + . Hoping for a contradiction, suppose that the points in the statement of the claim are linked. Then the pairs of endpoints of α -∪ α + and β -∪ β + are also linked on mi = ∂cl(X). We now have two arcs on the disk cl(X) with linked endpoints on ∂cl(X), so the two arcs must intersect. As α ± , β ± are all disjoint, the only intersection can be on ∂H 3 , so their endpoints at infinity must all agree, i.e. γ(ξ) = ξ.

Since γ(ξ) = ξ, all the rays γ k • h+ limit to ξ, where k ∈ Z. Hence, all these (quasi-geodesic) rays are pairwise mutually homoclinic, and for each pair k, l, the rays γ k • h+ and γ l • h+ eventually intersect the same components of m, in the same order, although their initial behavior may be different. In analogy with the setup of Claim 6.4, let d k,l be the liminf of the distances from γ k • h+ to γ l • h+ along the components of m that they both intersect.

We claim that there are k, l such that d k,l < 󰂃, where 󰂃 is the constant from Claim 6.4. Indeed, for N > length(m)/󰂃, it is impossible to pack N points at least 󰂃 apart in any component of m. So if we let k range over a set F ⊂ Z of size N , whenever a component of m intersects all γ k • h+ , k ∈ S, two such intersections must be within 󰂃 of each other. There are infinitely many such components of m and F is finite, so we can pick k, l ∈ S such that γ k • h+ and γ l • h+ are within 󰂃 on infinitely many such components.

Finally, γ k • h+ and γ l • h+ are mutually homoclinic lifts of h+ , and d k,l < 󰂃, so the exact same argument as in Claim 6.4 shows that γ k • h+ and γ l • h+ are asymptotic on ∂ M . It follows that h + spirals onto a (simple) closed curve in ∂M in the homotopy class of (a primitive root of) γ l-k . (Indeed, γ l-k lifts to a deck transformation of the universal cover H 2 -→ ∂M , and the axis of this deck transformation is asymptotic to suitably chosen lifts of both γ k • h+ and γ l • h+ .) This is a contradiction, though, since h + limits onto λ, which is not a simple closed curve. □ Assume now that our mutually homoclinic rays h± are not asymptotic on ∂ M , in which case we're in case (2) of the theorem and are done. By Claim 6.4, there is some 󰂃 > 0 such that d mi ( h+ (i), h-(i)) ≥ 󰂃 for all i. We will show that λ is an intrinsic limit of annuli, in the sense of Lemma 5.12, which says that then λ is an intrinsic limit of meridians.

h + h h + (i) h + (j) h (i) h (j) h + h h + (i) h (j) h (i) h + (j) I + I I I + Figure 14. Above, the horizontal curve is always m.
The proof is an adaptation and correction of a surgery argument of Lecuire [START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF]Affirmation C.3]. As there are two gaps 7 in Lecuire's earlier argument, we give the proof in full detail below, without many citations of [START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF]. Claim 6.6. Given 0 < δ < 󰂃, there are choices of i < j such that either (I) The points h + (i) and h + (j) bound a segment I + ⊂ m of length less than δ, and similarly withinstead of +. The four velocity vectors

h ′ + (i), h ′ + (j), h ′ -(i), h ′ - (j
) all point to the same side of m, and the four segments h + ([i, j]), h -([i, j]), I + and I -have disjoint interiors. So, the curves γ ± := h ± ([i, j]) ∪ I ± ⊂ ∂M are simple and disjoint. (II) The points h + (i) and h -(j) bound a segment I + ⊂ m of length less than δ, and similarly the points h -(i) and h + (j) bound a segment I -⊂ m of length less than δ. The four velocity vectors

h ′ + (i), h ′ + (j), h ′ -(i), h ′ - (j
) all point to the same side of m, and the four segments h + ([i, j]), h -([i, j]), I -and I + have disjoint interiors. So, the curve γ ⊂ ∂M obtained by concatenating all four segments is simple.

See Figure 14 for a very useful picture. Note that in the picture, the velocity vectors of all paths intersecting m point to the same side of m, i.e. 'up', and all 4-tuples of points are unlinked as in Claim 6.5.

Proof. Start by fixing a circular order on m. Define 'the right' to be the direction in m that is increasing with respect to the circular order, and define 'the left' similarly. Since λ is minimal and not a simple closed curve, it has 7 The first gap is that the sentence "Quitte à extraire, la suite (gh -1 ) 2n g( l1 ) converge vers une géodésique γ ⊂ p -1 (α 1 ) dont la projection l ⊂ ∂M est une courbe fermé." at the end of the proof of Affirmation C.3 isn't adequately justified; this is fixed in Claim 6.4. The second is that the assumption d(l 2 + (y i ), l 2 + (y j )) < ε ′ in the statement of Affirmation C.3 is never actually verified, and does not come trivially from a compactness argument. This is fixed in Claim 6.6.

infinitely many leaves ℓ that are not boundary leaves. Fix some such ℓ, making sure that h ± ∕ ⊂ ℓ if the given rays happen to lie inside the lamination λ. The ray h + accumulates onto both sides of ℓ, so if we fix p ∈ ℓ ∩ m, the set h + (N) accumulates onto p from both sides, and similarly withinstead of +. Fix an interval J ⊂ m of length δ centered at p, and write J = J l ∪ J r as the union of the closed subintervals to the 'left' and to the 'right' of p. Note that p ∕ ∈ h ± (N), so each intersection of h ± with J lies in exactly one of J l or J r .

Let's call an index i left-closest if either h + (i) or h -(i) lies in J l and is closer to p than any previous h ± (k), k < i, that lies in J l . Right-closest is defined similarly using J r , and we call an index i closest if it is either left or right closest. Note that since δ < 󰂃 we can never have both h + (i), h -(i) in J simultaneously, so no i is both right-closest and left-closest at the same time. Since there are infinitely many i of both types, at some point there will be a transition where some i l is left-closest, some i r > i l is right-closest, and there are no closest indices in between.

Let i c be the smallest closest index that is bigger than i r . (Here, c stands for 'center', since the corresponding point on J will lie between the points we get from the indices i l and i r .) We now have three points in J, so two of the corresponding velocity vectors point to the same side of m. Let i, j ∈ {i l , i r , i c } be the two corresponding indices, and for concreteness, let's assume for the moment that h + (i) and h + (j) are the corresponding points in J, deferring a discussion of the other cases to the end of the proof. Note that since the rays h ± are mutually homoclinic and are in tight position with respect to m, the velocity vectors h ′ -(i) and h ′ -(j) point to the same sides of m as h ′ + (i) and h ′ + (j), respectively, and so all four vectors point to the same side. That is, (a)

h ′ + (i), h ′ + (j), h ′ -(i), h ′ - (j
) all point to the same side of m. (b) the segment I + ⊂ J bounded by h + (i) and h + (j) contains no element

h + (k) or h -(k)
where k is between i and j.

Let I -⊂ m \ J be the segment that is bounded by the points h -(i) and h -(j). Suppose for a moment that we knew that I -had length less than δ. Then for each k, it is impossible that both h + (k) or h -(k) lie in I -, as we're assuming that corresponding intersections of h ± with m stay at least 󰂃 > δ apart. In particular, if k is between i, j and we apply the unlinking condition of Claim 6.5 twice, once to i, k and once to j, k, we get from this and (b) above that neither element h + (k) or h -(k) is contained in I -. So, the four segments h + ([i, j]), h -([i, j]), I + and I -have disjoint interiors, and we're in the situation of case (I) in the claim, as desired.

As constructed above, however, there is unfortunately no reason to believe that the interval I -has length less than δ. To rectify this, recall that λ actually has infinitely many non-boundary leaves ℓ n . For each such ℓ n and p n ∈ ℓ n ∩ m, we can repeat the above construction using constants δ n → 0, producing points (say) h + (i n ), h + (j n ) that lie within the length δ n -interval J n ∋ p n and that satisfy properties (a) and (b) above. Choose the sequence p n ∈ ℓ n ∩ m so that it is monotonic in the circular order induced on m, and let δ n → 0 fast enough so that the associated intervals I n + are all disjoint, so that in the circular order on m we have

h + (i 1 ) < h + (j 1 ) < h + (i 2 ) < h + (j 2 ) < • • • < h + (i n ),
and where each

I n + is the interval [h + (i n ), h + (j n )]
, rather than the complementary arc on m that has endpoints h

+ (i n ), h + (j n ). Then Claim 6.5 implies that h -(i 1 ) > h -(j 1 ) > h -(i 2 ) > h -(j 2 ) > • • • > h -(i n ).
Discarding finitely many n, we can assume all the point h + (i n ), h + (j n ) lie in an interval U ⊂ m of length less than δ. Since the points h -(i n ), h -(j n ) are at least 󰂃 > δ away from the corresponding + points, they all lie in m \ U . Then since the interval I n -is defined to be disjoint from I n + ⊂ U , we must have

I n -= [h -(j n ), h -(i n )]
, rather than the other interval with those endpoints. It follows that at least for large n, all the intervals I n -are disjoint. Since m as compact, we can then pick some n where I n -has length less than δ, as desired. Therefore, we are in case (I) in the statement of the claim, and are done.

In the argument above we have simplified the notation by assuming that we have points h + (i n ), h + (j n ) ∈ J n satisfying conditions (a) and (b), which put us in case (I) at the end. Up to exchanging +, -, the only other relevant case is when, our chosen points are h -(i n ), h + (j n ) ∈ J n . After passing to a subsequence in n, if we are not in the case already addressed, then we may assume that our chosen points are h -(i n ), h + (j n ) ∈ J n for all n. And after exchanging + withand passing to a further subsequence, we may assume

h -(i 1 ) < h + (j 1 ) < h -(i 2 ) < h + (j 2 ) < • • • < h -(i 1 )
in the circular order on m, and that the interval

I n + = [h -(i n ), h + (j n )].
Everything from then on works exactly as above: if we set I n -to be the interval bounded by h

+ (i n ), h -(j n ) that is disjoint from I n
+ , then for some n we have that the length of I n -is less than δ, and it is easy to verify that we are in case (II) of the claim. □

We now finish the proof of Theorem 6.1. Suppose we are in case (I) of Claim 6.6. Then the two simple closed curves γ ± drawn on the left in Figure 14 are the projections to M of the paths in M obtained by concatenating the arcs h± | [i,j] with lifts Ĩ± ⊂ mj of the intervals I ± ⊂ m, see Figure 15. We can homotope one path to the other in M while preserving the fact that the endpoints are points on mi , mj that differ by the unique deck transformation taking mi to mj . So projecting down, the simple closed curves γ ± are freely On the left, the two paths drawn in heavy ink project to the two simple closed curves γ ± in Claim 6.6 (I), shown on the left in Figure 14. The shaded region is a rectangle embedded in M that projects to an embedded annulus A ↩→ M with boundary γ -∪ γ + . On the right, the union of the two paths projects to the simple closed curve γ on M of Claim 6.6 (II), and the shaded region projects to a Möbius band B ↩→ M with boundary γ. homotopic in M , and hence bound an annulus A ↩→ M . See the left part of Figure 15.

There is a uniform lower bound (depending on λ, m) for the angle at any intersection point of any leaf of λ with m, and the points h ± (i) are at least 󰂃 away from each other in m. This implies that there is a uniform lower bound for the Hausdorff distance between h ± | [i,j] on ∂M . As long as the bound δ on the lengths of I ± is small enough, the geodesics in the homotopy classes of γ ± stay very close to h ± | [i,j] , and are therefore distinct. So, the curves γ ± are not homotopic in ∂M , and hence bound an essential annulus A ↩→ M .

Choosing i, j to be large and δ to be very small, ∂A is contained in S(λ) and its intersection number with λ is small. Hence, λ is an intrinsic limit of annuli, in the sense of Lemma 5.12, so we're done.

Case (II) is similar. Here, the single simple closed curve γ described in Claim 6.6 (II) bounds a Möbius band B ↩→ M , see the right side of Figure 15. Since ∂M is orientable, B is not boundary parallel, and hence by JSJ theory the boundary of a regular neighborhood of B is an essential annulus A ↩→ M whose boundary consists of two disjoint curves that are both homotopic to γ on ∂M . As in case (I), we can make the intersection number of ∂A with λ arbitrarily small, so λ is an intrinsic limit of annuli, and we are done. Remark 6.7. The proof (D) above is quite delicate. Most of this delicacy comes from Claims 6.4 and 6.6, which are needed to ensure that the annuli approximating λ that are produced immediately afterward are embedded. But ↵ Figure [START_REF] Farb | A primer on mapping class groups[END_REF]. The two curves α and β are homotopic through the handlebody pictured, and therefore bound an essential singular annulus. The only annuli one can produce from surgery are inessential, but one can surger to obtain the 'obvious' disc in the picture that separates the handlebody into two solid tori.

while we are able to prove these claims using arguments involving the planarity of the closure of ∂ M in H 3 ∪ ∂ ∞ H 3 , one would not have to worry about these annuli being embedded if there was a strong 'Annulus Theorem' guaranteeing that any essential singular annulus in an irreducible 3-manifold M can be surgered to give an essential embedded annulus. If this were true, Claims 6.4 and 6.6 could be replaced by a one paragraph compactness argument. Here, a singular annulus is a map f : (A, ∂A) -→ (M, ∂M ) where A = S 1 × [0, 1]. We say f is essential if it is not homotopic rel ∂A into ∂M .

Such an annulus theorem follows from the JSJ decomposition when M has incompressible boundary. When M has compressible boundary, there is a similar theorem as long as the original singular annulus has a spanning arc that is not homotopic rel ∂ into ∂M , see Cannon-Feustel [START_REF] James | Essential embeddings of annuli and möbius bands in 3-manifolds[END_REF]. However, our proof above does not provide such annuli, and indeed such annuli do not exist in compression bodies (the M of most interest to us), since any proper arc in a compressionbody M is homotopic rel ∂ into ∂M .

In fact, in a general M , one cannot always surger essential singular annuli to produce embedded essential annuli. For instance, the two curves in Figure 16 bound an essential singular annulus that cannot be surgered to give an embedded essential annulus. However, in that example, one can surger to get a meridian in the handlebody, so maybe an essential singular annulus can always be surgered to give either a meridian or an essential embedded annulus? This also turns out not to be true. Suppose P is a pair of pants and let M = P ×[0, 1], which is homeomorphic to a genus two handlebody. If γ -→ P is an immersed figure-8 whose image forms a spine of P , then the singular annulus γ × [0, 1] -→ P × [0, 1] is essential, but the three embedded annuli that one can obtain from it by surgery are all inessential, and no meridian can be created by surgery either. However, we expect this is the only counterexample. The first author of this paper has spent considerable time trying to prove this with a tower argument, but pushing down the tower is very subtle, since if the obvious constructions fail, one has to characterize the figure 8 example.

Hausdorff limits of meridians

Let M be an orientable hyperbolizable compact 3-manifold, equip ∂ χ<0 M with a hyperbolic metric. In [START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF]Theorem B.1], Lecuire showed that every lamination λ on ∂ χ<0 M that is a Hausdorff limit of meridians contains a homoclinic leaf that is a homoclinic geodesic. The converse is not true: certainly in order to be a Hausdorff limit of meridians µ needs to be connected, and as explained in Figure 1 in the introduction there are even connected laminations that contain homoclinic leaves but are not Hausdorff limits of meridians.

We say that two laminations µ 1 , µ 2 are commensurable if they contain a common sublamination ν such that for both i, the difference µ i \ ν is the union of finitely many leaves. µ 1 and µ 2 are strongly commensurable if they contain a common ν such that for both i, the difference µ i \ ν is the union of finitely many leaves, none of which are simple closed curves.

Theorem 7.1 (Hausdorff limits of meridians). Suppose that S ⊂ ∂ χ<0 M is a connected subsurface with geodesic boundary, such that ∂S is incompressible, and that the disc set D(S, M ) is 'large', i.e. it has infinite diameter in the curve graph C(S). Let λ be a geodesic lamination in int(S) that is a finite union of minimal laminations, and assume that the following does not hold:

(󰂏) S is a closed, genus two surface, there is a separating meridian µ that does not intersect λ transversely, and λ intersects transversely the two nonseparating meridians disjoint from µ. Then λ is strongly commensurable to a Hausdorff limit of meridians in S if and only if λ is strongly commensurable to a lamination containing a homoclinic leaf, and this happens if and only if one of the following holds:

(1) λ is disjoint from a meridian on S, (2) some component of λ is an intrinsic limit of meridians, or

there is an essential (possibly nontrivial) interval bundle B ⊂ M over a compact surface Y that is not an annulus or Möbius band, and there are components λ ± ⊂ λ that each fill a component of ∂ h B (possibly the same component, if ∂ h B is connected), such that λ ± are essentially homotopic through B, as in §2.9, and there is a compression arc α for B that is disjoint from λ.

Recall from Proposition 3.1 that when D(S, M ) does not have infinite diameter in C(S), it is either empty, consists of a single separating meridian, or consists of a single non-separating meridian m and all separating curves that are band sums of m. In these cases, it is obvious what the Hausdorff limits of meridians are. For instance, in the last case a finite union λ of minimal laminations in S is strongly commensurable to a Hausdorff limit of meridians if and only if either λ = m or λ ⊂ S \ m. For the 'if' direction, note that if λ ⊂ S \ m then it can be approximated by an arc with endpoints on opposite sides of m, and doing a band sum with m gives a curve that approximates λ. For the 'only if' direction, just note that all meridians are either equal to m or are contained in S \ m.

The case (󰂏) above really is exceptional. In that case, (1) holds, and at least when µ ⊂ λ we have that λ contains a homoclinic leaf, but λ is not commensurable to a Hausdorff limit of meridians. Indeed, let T ± ⊂ S \ µ be the two components of S \ µ and hoping for a contradiction, take a sequence of meridians (m n ) that Hausdorff converges to λ ′ ⊃ λ. We can assume after passing to a subsequence that m n has an µ-wave in T + (say) for all n. Since T + is a compressible punctured torus, there is a unique homotopy class rel µ of µ-wave in T + , so λ ′ contains a leaf ℓ that either intersects T + in an arc in this homotopy class, or is contained in T + and is obtained by spinning an arc in this homotopy class around µ. But then ℓ intersects nontrivially every nonperipheral minimal lamination in T + other than the unique nonseparating meridian µ + of T + , so λ is disjoint from µ + , contrary to assumption. Lemma 7.2. Suppose λ ⊂ S is a lamination, there is a meridian µ that does not intersect λ transversely, and that if S is a closed surface of genus 2 then µ is nonseparating. Then λ is strongly commensurable to a Hausdorff limits of meridians on S.

The proof of Lemma 7.2 uses some ideas that the first author developed with Sebastian Hensel, whom we thank for his contribution.

Proof. We may assume that λ is a finite union of minimal components. It suffices to assume µ is not a leaf of λ, as long as we prove the conclusion both for such a λ and for λ ∪ µ.

Assume first that µ is nonseparating in S. Let (c i ) be a sequence of simple closed curves on S that Hausdorff-converges to λ. One can do this by constructing for each component λ 0 ⊂ λ a simple closed geodesic approximating λ 0 , by taking an arc that runs along a leaf of λ 0 for a long time, and then closing it up the next time it passes closest to its initial endpoint in the correct direction. Let α be a simple closed curve on S that intersects µ once, and intersects all the components of λ. For each k, let γ k i be the geodesic homotopic to the 'band sum' of µ and T k c i (α), where T c i is the twist around c i and a band sum of two curves intersecting once is the boundary of a regular neighborhood of their union. Note that γ k i is a meridian for all i, k. If (k i ) is a sequence that increases quickly enough, (γ k i i ) converges to a lamination strongly commensurable to λ. And if we pick a meridian β on S that intersects both µ and λ, then

T i µ • T i γ k i i (β)
Hausdorff converges to a lamination strongly commensurable to λ ∪ µ. Now suppose µ is separating. We claim that there is another separating meridian in S that is disjoint from µ. Let m be a maximal multicurve of meridians in S that contains µ as a component. Since D(S, M ) is large, m ∕ = µ. If m has a separating component distinct from µ, we are done. So, suppose we have a nonseparating component m 0 ⊂ m. We can make a (separating) band sum of m 0 that is disjoint from µ unless m 0 lies in a punctured torus component of S \ µ. So, we assume the latter is true. Since D(S, M ) is large, it cannot be that m = µ ∪ m 0 , since then all meridians are disjoint from m 0 . So, there is another component m 1 of m, which we can assume is also nonseparating. This m 1 must lie on the opposite side of µ from m 0 , and as before we're done unless the component of S \ µ containing m 1 is also a punctured torus. But in this case, S is a genus two surface contrary to assumption.

Let T ⊂ S \ µ be a component that contains a nonperipheral separating meridian, which we call µ ′ . Let V be the other component. Write λ = λ T ∪λ V , where λ T ⊂ T and λ V ⊂ S \ T . Let C be the compression body with exterior boundary equal to the component of ∂M that contains S, that one obtains by compressing the meridian µ.

We claim that there are sequences of simple closed curves (α i ), (β i ) in T such that the following two properties hold:

• (α i ) and (β i ) both Hausdorff converge to a geodesic lamination strongly commensurable to λ T , • for all i, α i and β i bound an essential annulus in C.

To construct these sequences, start by picking a simple closed curve α in T such that α and each component of λ T together fill T . Let β be a simple closed curve on T such that α, β, µ bound a pair of pants in T . In C, we can compress the boundary component µ of this pair of pants, so α, β bound an annulus in C. Moreover, this annulus is essential, since otherwise α, β bound an annulus in T , implying T is torus with the one boundary component µ, contradicting the fact that there is a separating nonperipheral meridian in T . Then find a sequence (c i ) of simple closed curves in T that Hausdorff converge to a geodesic lamination strongly commensurable to λ T , take k i to be a fast increasing sequence and set α i = T k i c i (α) and β i := T k i c i (β). Since α fills with every component of λ T , the curve β intersects every component of λ T . It follows that (α i ) and (β i ) Hausdorff converge to a geodesic lamination strongly commensurable to λ T . And since each c i is nonperipheral in T , each component of c i bounds an annulus in C with a curve on the interior boundary of C, so the twist T c i extends to C, implying that α i , β i bound an annulus in C as desired above. Now let C ′ be the compression body obtained by compressing both µ and µ ′ , so C ⊂ C ′ ⊂ M . Note that since both curves are separating and are disjoint, Proposition 3.1 says that D(S, C ′ ) is large, so we can pick a meridian m ∈ D(S, C ′ ) that intersects µ and every component of λ. Fix a sequence of geodesic muticurves (d i ) in V that Hausdorff converges to λ V . As with the twists T c i in C, the twists T d i extend to C ′ . And the compositions

T α i • T -1 β i extend to C ′ because the curves bound annuli in C ⊂ C ′ . We then define γ i := (T α i • T -1 β i ) k i • T k i d i (m)
for some fast increasing k i → ∞. These γ i are all meridians and Hausdorff converge to a lamination strongly commensurable to λ. To obtain λ ∪ µ instead, hit γ i with high powers of twists around µ. □

Here is a more powerful version of Lemma 7.2. The idea of the proof is more or less the same, but more complicated. Proposition 7.3 (Promoting Hausdorff limits). Suppose that ν, η are disjoint geodesic laminations on S that are finite unions of minimal components, and set λ = ν ∪ η. Suppose also that no component of ν is a meridian.

Let X be the union of the subsurfaces with geodesic boundary that are filled by the components of ν. Suppose that there are disjoint, nonhomotopic meridians µ, µ ′ on S that are disjoint from η, and a sequence of homeomorphisms

f i : S -→ S, f i | S\int(X) = id
such that µ i := f i (µ) and µ ′ i := f i (µ ′ ) are both sequences of meridians that Hausdorff converge to laminations strongly commensurable to ν. Then λ = ν ∪ η is strongly commensurable to a Hausdorff limit of meridians in S.

Before proving the proposition, we record the following application.

Corollary 7.4. Suppose that λ is a geodesic lamination on S that is a finite union of minimal components. If either

• some component ν ⊂ λ that is not a simple closed curve is an intrinsic limit of meridians, • there are (possibly equal) components λ ± ⊂ λ, neither of which is a simple closed curve, and where each fills a component of the horizontal boundary (possibly the same component if

∂ h B is connected) of an essential interval bundle (B, ∂ H B) ↩→ (M, S),
where λ ± are essentially homotopic through B, and where there is a compression arc α for B that is disjoint from λ, then λ is strongly commensurable to a Hausdorff limit of meridians.

Proof. Suppose some component ν ⊂ λ that is not a simple closed curve is an intrinsic limit of meridians. Setting X := S(λ) we can take (µ i ) to be any sequence of meridians in X that Hausdorff converges to a lamination strongly commensurable to ν. Moreover, since ν fills X and is a limit of meridians, the disc set D(X, M ) is large, so for each i there is some meridian µ ′ i disjoint from µ i . Since there are only finitely many topological types of pairs of disjoint curves in X up to the pure mapping class group of X, after passing to a subsequence we can assume that all µ i , µ ′ i are of the form in the proposition. The desired conclusion follows.

In the second case, we let X be the subsurface with geodesic boundary obtained by tightening ∂ H B and set ν = λ -∪ λ + . Write the interval bundle as π : B -→ Y , where Y is a compact surface with boundary. We can assume without loss of generality that α is a strict compression arc, i.e. that it is homotopic rel endpoints to a fiber π -1 (y), y ∈ ∂Y . Note that since λ ± are not simple closed curves, Y is not an annulus or Möbius band.

Since λ ± are essentially homotopic through B, Fact 2.16 says that if our reference hyperbolic metrics are chosen appropriately, we have that λ -∪ λ + = (π| ∂ H B ) -1 ( λ) for some geodesic lamination λ on Y . Since λ ± together fill ∂ H B, the lamination λ is minimal and fills Y . So in particular, it has no closed, onesided leaves, and therefore if we pick a nonzero transverse measure on λ, we have that λ is the projective limit of a sequence of two-sided nonperipheral simple closed curves (c i ) in Y , by Theorem 1.2 of [START_REF] Viveka Erlandsson | Mapping class group orbit closures for non-orientable surfaces[END_REF]. Homotope the c i on Y to based simple loops at y ∈ ∂Y , let m(c i ) be the associated compressible curves on S constructed in Claim 2.11, and let µ i be the geodesic meridians on S in their homotopy classes. Then (µ i ) Hausdorff converges to a lamination strongly commensurable to λ -∪ λ + . After passing to a subsequence, we can assume that all the c i differ by pure homeomorphisms of Y , in which case the meridians µ i are as required in Proposition 7.3, for some µ, f i . Note that since our compression arc is assumed to be disjoint from λ, all the µ i are disjoint from η := λ\λ ± , and hence so is our µ. We create disjoint meridians µ ′ i similarly, by taking some c ′ i on Y disjoint from c i , and letting µ ′ i be the geodesic meridian homotopic to m(c ′ i ). It then follows from Proposition 7.3 that λ is strongly commensurable to a Hausdorff limit of meridians as desired.

□

We now prove the proposition.

Proof of Proposition 7.3. Assume that µ, µ ′ are disjoint meridian on S that are disjoint from η, that f i : S -→ S are homeomorphisms that are the identity outside of X, and that µ i := f i (µ) and µ ′ i := f i (µ ′ ) are sequences of meridians that Hausdorff converge to laminations strongly commensurable to ν.

We want to show that ν ∪ η is strongly commensurable to a Hausdorff limit of meridians on S. We now basically repeat the argument in Lemma 7.2, so the reader should make sure that they understand that argument before continuing here. Suppose µ (say) is nonseparating in S. Choose a simple closed curve α on S that intersects µ once and intersects essentially each component of η. Let α i := f i (α), and note that α i intersects µ i once, and also intersects essentially each component of η. Let (c i ) be a geodesic multi-curve that Hausdorff converges to η, and let γ k i be the geodesic homotopic to the 'band sum' ( 4)

B(µ i , T k c i (α i )) = T k c i (B(µ i , α i )) = T k c i • f i (B(µ, α))
, where here B(•, •) takes in two simple closed curves that intersect once and returns the boundary of the regular neighborhood of their union. If one of the inputs in a band sum is a meridian, then so is the output, so γ k i is a meridian for all i, k. The given equalities are true at least for large i. The first equality holds because µ is disjoint from η, f i = id on the subsurfaces filled by the components of η, and therefore µ i is disjoint from c i for large i. The second equality is obvious from the definitions of µ i , α i .

Let (k i ) be a fast increasing sequence. After passing to a subsequence, we can assume that (γ k i i ) Hausdorff converges to a lamination λ. We claim that λ is strongly commensurable to ν ∪ η.

First, using the second term in (4), if k i is huge with respect to i, then c i is contained in a small neighborhood of γ k i i , and so since (c i ) Hausdorff converges to η, we have λ ⊃ η.

We claim that each γ k i i essentially intersects each component X 0 ⊂ X. If not, then from the third term in (4) it follows that B(µ, α) is disjoint from X 0 . But µ essentially intersects X 0 , since otherwise the Hausdorff limit of the µ i will not contain the associated component ν 0 ⊂ ν. So, µ, α and X 0 all lie in the punctured torus T ⊂ S bounded by B(µ, α). But since α intersects every component of η, we have that η intersects T as well, in a collection of arcs disjoint from µ. Since X 0 is disjoint from η, X 0 = µ, so ν 0 = µ is a meridian, contrary to our standing assumption.

It now follows that the Hausdorff limit λ essentially intersects each component of X. Since γ k i i is disjoint from µ i and (µ i ) Hausdorff converges to a lamination containing ν, The laminations λ, ν cannot intersect transversely. Since each component of X is filled by a component of ν, we have λ ⊃ ν.

Finally, if Y is the union of all the subsurfaces with geodesic boundary that are filled by the components of η, then as f i = id outside X and c i ⊂ Y for large i, the intersection of γ k i i with S \ (X ∪ Y ) is properly homotopic to the intersection of B(µ, α), which is independent of i. It follows that λ \ (ν ∪ η) is a finite collection of non-closed leaves, so we are done.

We can now assume that both µ, µ ′ are separating, so that µ i , µ ′ i are also separating for all i. Let T i ⊂ S \ µ i be the component containing µ ′ i , and let V i be the other component. Note that T i is not a punctured torus, since it contains a nonperipheral separating curve. Since ∂T i ∩ η = ∅, we have

η = η T ⊔ η V ,
where the first term is the intersection of η with T i , and the second term is defined similarly. Note that since f i = id on S \ X, all the µ i induce the same two-element partition of the components of S \ X, so at least after passing to a subsequence the decomposition of η above is actually independent of i, which is why we have omitted the i in the notation.

Let C be the compression body whose exterior boundary is the component of ∂M containing S, and which is obtained by compressing the curve µ. Let C ′ be similarly obtained by compressing both µ and µ ′ , so that C ⊂ C ′ ⊂ M . Since C ′ admits two nonhomotopic disjoint separating meridians, the disc set D(S, C) is large by Proposition 3.1, so we can pick a meridian m ∈ D(S, C) that intersects every component of ν ∪ η, as well as µ, µ

′ . Let C i ⊂ C ′ i ⊂ M be the compression bodies obtained by compressing µ i , µ ′ i . Then f i extends to a map C ′ -→ C ′ i , implying that m i := f i (m) is a meridian in C ′ i .
As in the proof of Lemma 7.2, we can pick sequences (α i ), (β i ) of simple closed curves in T i such that (α i ) and (β i ) both Hausdorff converge to η T , and where α i , β i bound an essential annulus in C i for all i. As in the lemma,

T α i • T -1 β i extends to C ′ i .
Let (c i ) be a sequence of multicurves in V i that Hausdorff converges to η V . Each component of c i bounds an annulus in C ′ i with a curve on the interior boundary of C ′ i , and hence the multitwist T c i extends to a homeomorphism of C ′ i . For any given k, set

γ k i := (T α k • T -1 β k ) k • T k c k (m i ).
We claim that for fast increasing k i , the curves γ k i i Hausdorff converge to a lamination that is strongly commensurable to ν ∪ η as desired. This is proved using the same types of arguments we employed in the nonseparating case above. In particular, recall that m was selected to intersect all components of ν ∪ η. Since f i is supported on subsurfaces filled by components of ν, all the m i = f i (m) intersect all components of ν ∪ η, and hence for large k i they intersect α k i , β k i . So, γ k i i is twisted many times around α k i , β k i , and hence its Hausdorff limit contains ν. Similarly, the m i intersect c k i for large i. Since c k lies in V k , it is disjoint from α k ⊂ T k and β k ⊂ T k , and thus the Hausdorff limit of γ k i i contains η. Finally, the Hausdorff limit has no other minimal components because α k , β k , c k are contained in subsurfaces filled by components of ν ∪ η, and m i = f i (m) is constant outside this subsurfaces. □

We can now start the proof of the theorem.

Proof of Theorem 7.1. Suppose that λ ⊂ S is a lamination and (󰂏) does not hold, so that it is not the case that S is a genus two surface and λ is disjoint from a separating meridian µ, but intersects the two nonseparating meridians disjoint from µ. We want to show that λ is strongly commensurable to a Hausdorff limit of meridians if and only if it is strongly commensurable to a lamination containing a homoclinic leaf, which happens if and only if either (1) λ is disjoint from a meridian, (2) some component of λ is an intrinsic limit of meridians, or (3) there is an essential (possibly nontrivial) interval bundle B ⊂ M over a compact surface Y that is not an annulus or Möbius band, and there are components λ ± ⊂ λ that each fill a component of ∂ H B, such that λ ± are essentially homotopic through B, as in §2.9, and there is a compression arc α for B that is disjoint from λ.

Hausdorff limit =⇒ homoclinic leaf. Suppose first that λ is strongly commensurable to a Hausdorff limit of meridians λ ′ . Then by [START_REF]Plissage des variétés hyperboliques de dimension 3[END_REF]Theorem B.1], there is a homoclinic leaf h ⊂ λ ′ , so λ is strongly commensurable to a lamination with a homoclinic leaf as desired.

Homoclinic leaf =⇒ (1), ( 2) or [START_REF] Biringer | Automorphisms of the compression body graph[END_REF]. We now assume we have a homoclinic leaf h in some lamination strongly commensurable to λ. The two ends of h limit onto (possibly equal) components λ ± ⊂ λ. If one of S(λ ± ) has compressible boundary, there is a meridian disjoint from λ, so we are in case (1) and are done. So, ∂S(λ ± ) is incompressible, and we're in the situation of Theorem 6.1 and Corollary 6.3. We now break into cases.

If one of λ ± is an intrinsic limit of meridians, we're in case (2) and are done. If we're in case (3) of Theorem 6.1 and Corollary 6.3, we're in case (3) of the theorem and are done, unless the given interval bundle B -→ Y is over an annulus or Möbius band. But in that case, letting c be a boundary component of Y , we can consider the geodesic meridian µ on S homotopic to the m(c) constructed in Claim 2.11, using the compressing arc given by Corollary 6.3. This µ is disjoint from λ, so we're in case (1) of the theorem.

Finally suppose that the two ends of h are asymptotic on S, so that λ -= λ + . Let's separate further into the cases (i) and (ii) in Corollary 6.3. In case (i), using the notation of the corollary, the curve c ∪ h([-s.s]) is a meridian disjoint from λ. So, we're in case (1) of the theorem. In case (ii), let T be a neighborhood of h ∪ λ ± that is either a punctured torus or a pair of pants, depending on whether the two ends of h limit onto opposite sides of λ ± , or onto the same side. Because we're in case (ii), there is a meridian in T . Hence, one of the boundary components of T is a meridian, and is disjoint from λ so we're done.

(1), ( 2) or (3) =⇒ Hausdorff limit. Suppose (1), ( 2) or (3) holds. We want to show λ is strongly commensurable to a Hausdorff limit of meridians. If λ is disjoint from a meridian, then we're done by Lemma 7.2. If a component of λ is an intrinsic limit of meridians, we're done by the first part of Corollary 7.4. In case (3) above, we're done by the second part of Corollary 7.4. □

Extending partial pseudo-Anosovs to compression bodies

Let M be a compression body with exterior boundary Σ. Let S ⊂ Σ be an essential subsurface such that ∂S is incompressible. In this section, we prove: Theorem 8.1 (Extending partial pseudo-Anosovs). Suppose that f : Σ -→ Σ is a partial pseudo-Anosov supported on S. Then f has a power that extends to a nontrivial subcompressionbody of (M, S) if and only if the attracting lamination of f is a projective limit of meridians that lie in S.

When S = Σ, this is a theorem of Biringer-Johnson-Minsky [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF]. The proof of Theorem 8.1 is basically the same as their proof, but we need to go through it anyway, to note the places that parabolics appear, and to deal with the fact that we are looking at subcompression bodies of (M, S) rather than of M . Also, before starting on the bulk of the proof in §8.2, we isolate part of the argument into a separate purely topological subsection, §8.1. This separation of the argument into distinct topological and geometric parts makes it more understandable than the original version, we think.

8.1. Dynamics on the space of marked compression bodies. Let Σ be a closed, orientable surface, and let S ⊂ Σ be an essential subsurface. The space of marked S-compression bodies is defined to be

CBod(S) = {(C, h : Σ → ∂ + C)}/ ∼,
where here C is a compression body, h is a homeomorphism, and

• the multicurve h(∂S) ⊂ ∂ + C is incompressible,
• there is a multicurve on S such that h(S) is a cut system for C, i.e. h(S) bounds a collection of disks that cut C into balls and trivial interval bundles over the interior boundary components. We declare (C i , h i : S → ∂ + C i ), i = 1, 2 to be equivalent (written ∼ as above) if there is a homeomorphism φ : C 1 → C 2 that respects the boundary markings: that is, φ • h 1 and h 2 are homotopic maps S → ∂ + C 2 .

We write

(C 1 , h 1 ) ⊂ (C 2 , h 2 ) if there is an embedding φ : (C 1 , ∂ + C 1 ) ↩→ (C 2 , ∂ + C 2 )
that respects the boundary markings. This gives a partial ordering on CBod(S). We often identify Σ with ∂ + C instead of specifying the boundary marking, and simply write C for an element of CBod(S). So CBod(S) is the set of all compression bodies with exterior boundary Σ that one obtains by compressing curves in S (without compressing boundary curves) up to the obvious equivalence. Fixing M ∈ CBod(S) and f as above, let A be the set of accumulation points in CBod(S) of the f -orbit of M , and let

A min = {C ∈ A | ∄D ∈ A such that D ⊊ C}
be the subset consisting of all minimal elements of A.

Theorem 8.4 (Existence of maximal subcompression body).

A min is a finite f -orbit that contains a single element C f such that C f ⊂ M . Moreover, C f is the unique maximal element of CBod(S) such that C f ⊂ M and a power of f extends to C f . This result was proved in [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF] when S = Σ. Our proof follows the same general lines, but is topological instead of hyperbolic geometric.

We proceed with a series of lemmas.

Lemma 8.5. A min is nonempty, finite and f -invariant.

Proof. A is nonempty, since CBod(S) is compact. This implies that A min is nonempty, for example since the 'height' of a compression body is nonnegative and decreases under strict containment, see §3 of [START_REF] Biringer | Automorphisms of the compression body graph[END_REF]. By Claim 8.2, A min is discrete. But A min is closed in A, which is closed in CBod(S), which is compact. So, A min is compact, and must be finite. Finally, A min is f -invariant since A is and the f -action respects containment. □ Lemma 8.6. Suppose that for i = 1, 2 we have C i ∈ CBod(S) with C i ⊂ M , and that

f i (C 1 ) = C 1 while f j (C 2 ) = C 2 . Then there is an element C ∈ A min such that C 1 , C 2 ⊂ C ⊂ M .
Proof. Every element of A is the image under a power of f of an accumulation point of the sequence f nij (M ), so since A min is f -invariant there is some

C ′ ∈ A min to which a subsequence of f nij (M ) limits. As C 1 , C 2 ⊂ f nij (M ) for all n, we must have C 1 , C 2 ∈ C ′ . By Claim 8.2, there is some n such that f nij (M ) ⊃ C ′ . Then C := f -nij (C ′ ) ∈ A min is contained in M and must contain C 1 , C 2 as well. □ Lemma 8.7.
There is a unique element C f ∈ A min that is contained in M , and A min is an f -orbit.

Proof. Applying the previous lemma to two copies of the trivial compression body Σ × I shows that A min has an element that is contained in M . So, suppose that C, D ∈ A min are both contained in M . By the previous lemma, there is another element of A min that contains them both, which contradicts the minimality assumption unless C = D. Therefore, there is a unique element C f ∈ A min that is contained in M .

To show that A min is an f -orbit, suppose that C ∈ A min . Since C is an accumulation point, there is some

n such that f n (M ) ⊃ C. Then f -n (C) ⊂ M , implying that f -n (C) = C f by uniqueness. □
This finishes the proof of Theorem 8.4, since Lemma 8.5 shows that a power of f extends to C f and Lemmas 8.6 and 8.7 imply that any subcompression body of M to which a power of f extends is contained in C f . 8.2. The proof of Theorem 8.1. Let S ⊂ Σ = ∂M + be a compact essential subsurface, with ∂S incompressible in M , and let f : Σ -→ Σ be a pseudo-Anosov map on S.

The 'only if' direction of the theorem is trivial. Namely, suppose that some power f k extends to a nontrivial subcompressionbody C of (M, S). Pick a meridian m ⊂ S for C. Then (f k (m)) is a sequence of meridians in M that lie in S, and converge to the attracting lamination of f .

For the 'if' direction of the theorem, assume that no nonzero power of f extends to a nontrivial subcompression body of (M, S). We must show that the attracting lamination λ + is not in the limit set Λ(S, M ). The argument is similar to the proof of the main theorem in [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF]. As such, we will sketch the argument in places and refer to [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF] for details.

Consider the sequence M n = f -n (M ) of marked S-compression bodies, where we consider the exterior boundary of each M n as identified with the surface Σ. Fix a base point [X] ∈ T (Σ) and give the interior of each M n a geometrically finite hyperbolic metric such that the end adjacent to the exterior boundary Σ = ∂ + M is convex cocompact, and when its conformal boundary is identified with Σ, the conformal structure is [X]. Let

ρ n : π 1 Σ -→ PSL 2 C, N n := H 3 /ρ n (π 1 Σ)
be a representation (unique up to conjugacy) uniformizing the interior of M n and compatible with our markings, in the sense that ρ n is the composition of the map π 1 Σ -→ π 1 M n ∼ = π 1 N n induced by inclusion and a faithful uniformizing representation of π 1 N n . Note that the kernel of ρ n is ker(ρ n ) = f -n * (ker(π 1 Σ → π 1 M )). By Theorem 8.4 and the assumption that no power of f extends to a nontrivial subcompression body of M , the only minimal accumulation point of (M n ) in CBod(S) is the trivial compression body. So in particular, we can choose a subsequence (M n j ) that converges to the trivial compression body. By the compactness of generalized Bers slices (see [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF]Theorem 4.3]), we may assume after appropriate conjugations and passing to a further subsequence that (ρ n j ) converges algebraically to a representation

ρ ∞ : π 1 Σ -→ PSL 2 C, N ∞ := H 3 /ρ ∞ (π 1 Σ)
and that N ∞ can be identified with the interior of a compression body M ∞ with exterior boundary Σ in such a way that the end of N ∞ adjacent to Σ is convex cocompact with conformal structure [X] and the representation ρ ∞ is compatible with the marking in the same way as before.

The disk set D(S, M ∞ ) consists of all simple closed curves on S represented by elements γ ∈ π 1 Σ with ρ ∞ (γ) = 1. By Chuckrow's Theorem (see [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF]Lemma 2.11]), ρ ∞ (γ) = 1 if and only if ρ n j (γ) = 1 for all sufficiently large i. Since (M n j ) converges to the trivial compression body in the topology of CBod(S), it follows that the surface S ⊂ Σ = ∂ + M ∞ is incompressible in M ∞ .

Claim 8.8. The repelling lamination λ -of f is unrealizable in N ∞ .

Proof. The proof is almost identical to that of [2, Lemma 6.2], so we offer a sketch and we refer the reader to their paper for details.

Fixing an M -meridian γ ⊂ S, the sequence f -n j (γ) converges in the Hausdorff topology to a lamination λ M that is the union of λ -and finitely many leaves spiraling onto it. It suffices by [START_REF] Brock | Continuity of Thurston's length function[END_REF]Theorem 2.3] to show that λ M is unrealizable. So, hoping for a contradiction, assume λ M is realizable; then λ M is carried by a train track τ that maps nearly straightly into N ∞ (see [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF]).

By algebraic convergence, τ also maps nearly straightly into N n j when i is large. Since f -n j (γ) → λ M , for large i the curve f -n j (γ) is carried by τ . This implies that f -n j (γ) is geodesically realizable in N n j for large i, contradicting the fact that it is homotopically trivial. □ By work of Thurston [51, Prop 9.7.1], the π 1 -injective surface S ⊂ ∂ + M ∞ is isotopic into a degenerate end of N ∞ with ending lamination λ -. In particular, the peripheral curves of S represent cusps in N ∞ and every non-peripheral curve on S has hyperbolic type in N ∞ . Any pair of disjoint non-peripheral simple closed curves on S can then be realized geodesically by a pleated surfaces S -→ N ∞ in the given homotopy class, and Thurston's compactness of pleated surfaces (see [START_REF] Matsuzaki | Hyperbolic manifolds and Kleinian groups[END_REF]Lemma 6.13]) implies the following. Lemma 8.9 (compare with Lemma 6.3, [START_REF] Biringer | Extending pseudo-Anosov maps into compression bodies[END_REF]). Let α ⊂ S be a simple closed curve. Then for every k, there is some K such that for any other simple closed curve β in S, we have

d C(S) (α, β) ≤ k =⇒ d N∞ (α ∞ , β ∞ ) ≤ K,
where α and β ∞ are the geodesics in N ∞ in the homotopy classes of α and β.

Hoping for a contradiction, suppose now that λ + ∈ Λ(S, M ). When regarded as an element of ∂ ∞ C(S), the support of λ + is then an accumulation point of D(S, M ) ⊂ C(S). If α ∈ C(S), then for n = 1, 2, . . . the sequence (f n (α)) is a quasi-geodesic path that limits to λ + ∈ ∂ ∞ C(S), see [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF]. Since D(S, M ) is a quasi-convex subset (see Theorem 2.3, due to Masur-Minsky), there is a constant C and for each n a meridian γ i ∈ D(S, M ) with

d C(S) (f n (α), γ i ) ≤ C.
Translating the points f n (α) and γ i by f -n , this becomes:

d C(S) (α, f -n (D(S, M )) ≤ C.
By Lemma 8.9, an element γ n j ∈ f -n j (D(S, M )) at distance at most C from α in C(S) can be geodesically realized in some fixed compact subset A ⊂ N ∞ . Algebraic convergence implies that for sufficiently large j this geodesic can be pulled back and tightened to a geodesic in N n j . But by construction, γ n j is a meridian in M n j , so it cannot possibly be realized geodesically in N n j , which is a contradiction.

Extending reducible maps to compression bodies

We present here a generalization of [2, Theorem 1.1] that characterizes which (possibly reducible) mapping classes of the boundary of a 3-manifold M have powers that extend to sub-compression bodies.

In what follows, let M be a compression body with exterior boundary ∂ + M . Let S ⊂ ∂ + M be an essential subsurface such that ∂S is incompressible. Let f : ∂ + M -→ ∂ + M be a homeomorphism that is 'supported' in S, meaning that f = id on ∂ + M \ S. Definition 9.1. We say that f is pure if there are disjoint, compact, essential f -invariant subsurfaces S i ⊂ S, i = 1, . . . , n, such that f = id on S id := S \ ∪ i S i , and where for each i, if we set f i := f | S i , then either (1) S i is an annulus and f i is a power of a Dehn twist, or (2) f i is a pseudo-Anosov map on S i .

It follows from the Nielsen-Thurston classification, see [START_REF] Farb | A primer on mapping class groups[END_REF], that every f has a power that is isotopic to a pure homeomorphism.

When f is pure, with associated restrictions f i : S i -→ S i as above, we define a geodesic lamination λ = ∪ i λ i on S, where λ i ⊂ S i as follows. If f i is pseudo-Anosov, we let λ i be the support of the attracting lamination of f i . If f i is a Dehn twist, we let λ i be the core curve of the annulus S i . So defined, λ is called the attracting lamination of the pure homeomorphism f . Theorem 9.2. Suppose that S ⊂ ∂ + M is an essential subsurface such that the multicurve ∂S is incompressible. Let f : ∂ + M -→ ∂ + M be a pure homeomorphism supported in S. Then f has a power that extends to a nontrivial subcompressionbody of (M, S) if and only if either:

(1) there is a meridian in S id , (2) for some i, the map f i has a power that extends to a nontrivial subcompressionbody of (M, S i ), or (3) there are (possibly equal) indices i, j such that S i , S j bound an essential interval bundle B in M , such that some power of f | S i ∪S j extends to B, and there is a compression arc α for B whose interior lies in S id .

Recall from §2.7 that a 'subcompressionbody of (M, S) is a compression body obtained from ∂ + M by compressing some meridian multicurve in S.

Theorem 8.1 says that (2) is equivalent to asking that the attracting lamination (say) of f is a projective limit of meridians in S i . In (3), the condition that a power of f | S i ∪S j extends to B is easier to check. Indeed, if σ : ∂ H B -→ ∂ H B is the canonical involution, as defined in §2.5, then by Fact 2.7 we have that f k | ∂ H B extends to B exactly when σ •f k i is isotopic to f k j . When B is a twisted interval bundle, f i , f j are both pseudo-Anosovs on ∂ H B and this means that as mapping classes we have f j = g • f i for some finite order g commuting with both f i , f j , see e.g. McCarthy's thesis [START_REF] Mccarthy | Normalizers and centralizers of pseudo-anosov mapping classes[END_REF]. When B is a trivial interval bundle, σ indentifies S i and S j , and we have similarly that f j = g • σ(f i ) for some g commuting with both.

Proof of Theorem 9.2. Let's start with the 'if' direction, since that's easier. If there is a meridian in S id , then f extends to the compression body obtained by compressing that meridian. Suppose (2) holds, so that some power f k i extends to a nontrivial subcompression body C of (M, S i ). Then f also extends to C, since all the S j , where j ∕ = i, bound trivial interval bundles with subsurfaces of the interior boundary of C. So we're done.

The only interesting case is if (3) holds, so that some S i , S j bound an essential interval bundle B in M such that some power of f k | S i ∪S j extends to B, and there is a compression arc α for B whose interior lies in S id . Here, let S ′ ⊂ S be the smallest essential subsurface containing S i , S j and α; so, S ′ is obtained from a regular neighborhood of the union of these three subsets of S by capping off any inessential boundary components with discs. Let C be the characteristic compression body of the pair (M, S ′ ), as defined in Fact 2.4.

We claim that f k extends to C. To see this, note that we can construct C as follows. For concreteness, first assume that the boundary components of S i , S j that contain the endpoints of α bound an annulus A ⊃ α on S. Then S ′ = S i ∪ S j ∪ A, the annulus A is parallel in M to component A ′ ⊂ Fr(B) that is an annulus with the same boundary curves as A, and C is the union of the interval bundle B, the solid torus bounded by A, A ′ , and a trivial interval bundle over ∂ + M \ S ′ . We can then extend f k to C by letting it be the given extension of f k | S i ∪S j on B, the identity on the solid torus, and the obvious fiber preserving extension of f k | ∂ + M \S ′ to the adjacent interval bundle. The case that the boundary components of S i , S j that contain the endpoints of α do not bound an annulus on S is similar, except that instead of the solid torus above we take a thickened disk bounded by a rectangular neighborhood of α on S \ (S i ∪ S j ), and a rectangular neighborhood of the homotopic arc on the frontier of B.

We now work on the 'only if' direction. Passing to a power, suppose that f extends to a nontrivial subcompressionbody C of (M, S). We may assume that there is no proper, f -invariant essential subsurface S ′ ⊂ S such that f | S ′ extends to a nontrivial subcompression body of (M, S ′ ). If there were such a subsurface S ′ , we could replace S by a minimal such S ′ , therefore reducing the argument to the minimal case we are assuming we are in above.

If f = id on S, the fact that there is a nontrivial subcompressionbody of (M, S) means there is a meridian in S = S id , so we're in case [START_REF] Agol | Tameness of hyperbolic 3-manifolds[END_REF] and are done. This case may seems silly, but observe that if f is some complicated pure homeomorphism where there's a meridian in S id , the associated 'minimal' case that we pass to in the previous paragraph is where S is an annular neighborhood of some such meridian, and f = id on S.

Assume from now on that f is not the identity map of S.

We claim that every meridian m ∈ D(C, S) intersects every component of λ. Indeed, suppose some λ i is disjoint from some such m and let S ′ be the component of S \ S i containing m. Since f extends to C and S ′ ⊂ ∂C + is f k -invariant, f extends to the characteristic subcompressionbody C ′ of the pair (C, S ′ ), defined by starting with ∂ + M and compressing all meridians of C that lie in S ′ , see Fact 2.4. Since m is a meridian in C ′ , this C ′ is nontrivial, which contradicts the mimimality assumption in the first paragraph.

Pick a meridian m ∈ D(C, S). Since m intersects all components of λ, the sequence of meridians m i := f i (m) Hausdorff converges to a lamination λ ′ strongly commensurable to λ. Applying Theorem 7.1 to the pair (C, S) and using that all meridians in C intersect all components of λ, we have that either:

• some component λ a is an intrinsic limit of meridians lying in S(λ a ), in which case Proposition 8.1 (applied to f a : S a -→ S a ) says we're in case (2), or • there are indices a, b such that S a , S b bound an essential interval bundle B ⊂ C, where λ a , λ b are essentially homotopic in B, and where there is a compression arc α ⊂ S for B that is disjoint from λ, and hence can be isotoped so that its interior lies in S id . Let's assume we're in the last case, since otherwise we're done. We want to show that some power of f a ∪ f b : ∂ H B -→ ∂ H B extends to B.

First, suppose that B is a twisted interval bundle, so that S a = S b , f a = f b , λ a = λ b . Using just the index a from now on, if σ is the canonical involution of B, then Fact 2.16 implies that σ(λ a ) is isotopic to λ a . Let A ⊂ T (S) be the axis of f a on the Teichmüller space T (S). By Theorem 12.1 of [START_REF] Laudenbach | Travaux de thurston sur les surfaces[END_REF] and Theorem 2 of [START_REF] Masur | Uniquely ergodic quadratic differentials[END_REF], we have that A, σ(A) are asymptotic, so as they are both pseudo-Anosov axes they must be equal by discreteness of the action of the mapping class group. Since σ has finite order, it then fixes A pointwise. Now the group Γ = 〈σ, f a 〉 ⊂ Mod(∂ H B) is isomorphic to the direct product of a finite group fixing A pointwise and a cyclic group of pseudo-Anosovs, so for some positive k we have σ • f k a = f k a in Mod(∂ H B). By Theorem 3 of [START_REF] Birman | On isotopies of homeomorphisms of riemann surfaces[END_REF] we may isotope f k a , σ so that they commute, while preserving that σ 2 = id; we can then alter the bundle map π : B -→ Y so that the new σ is still the canonical involution. It follows that f k a is a lift to ∂ H B of a pseudo-Anosov map g : Y -→ Y , and hence f k a extends to B as desired, see Fact 2.7. Next, assume B is a trivial interval bundle, with canonical involution σ that switches S a , S b . As in the previous paragraph, we have that σ(λ b ) is isotopic to λ a , so Γ = 〈f a , σ • f b • σ -1 〉 ⊂ Mod(S a ) is a direct product Γ = F × 〈φ〉 of a finite group F and a cyclic group generated by a pseudo-Anosov φ, where if we quotient by F then f a and σ • f b • σ -1 both project to positive powers of φ. It suffices to show that they project to the same positive power of φ, for then we are done by the same argument as in the previous paragraph. For this, recall that all meridians of C intersect S a , S b , so these surfaces are 'holes' for the disk set of C, as discussed in [START_REF] Masur | The geometry of the disk complex[END_REF]. So with m i = f i (m) the sequence of meridians in C constructed above, Lemma 12.20 of [START_REF] Masur | The geometry of the disk complex[END_REF] says that for each i, the distance in the arc complex of S a between m i ∩ S a = f i a (m ∩ S a ) and σ(m i ∩ S b ) = (σ • f b • σ -1 ) i σ(m ∩ S b ) is at most 6. However, if f a and σ • f b • σ -1 project to different positive powers of φ, their stable translation lengths on the arc complex of S a are different, which is a contradiction. □

Figure 2 .

 2 Figure 2. Two meridians on a genus two handlebody. The two arcs of µ that lie on the right side of m are m-waves.

m 1 m 2 Figure 5 .

 25 Figure 5. Compression bodies inside a genus 2 handlebody.

Theorem 2 . 3 (

 23 Masur-Minsky). The subset D(S, M ) of C(S) is quasiconvex.

π 1 -

 1 surjective boundary component ∂ + C, called the exterior boundary of C. The complement ∂C 󰄀 ∂ + C is called the interior boundary, and is written ∂ -C. Note that the interior boundary is incompressible. For if an essential simple closed curve on ∂ -C bounds a disk D ⊂ C, then C 󰄀 D has either one or two components, and in both cases, Van Kampen's Theorem implies that ∂ + C, which is disjoint from D, cannot π 1 -surject.

  ) there is a homeomorphism from B to eitherY × [-1, 1] or Ŷ × [0, 1]/ ∼,taking horizontal boundary to horizontal boundary, such that f = F | ∂ H B , and where either

Claim 2 . 11 .

 211 The concatenation m(c) :

Proposition 3 . 1 (

 31 Diameters of disk sets). With M, S as above, either (1) D(S, M ) has infinite diameter in C(S), (2) S has one nonseparating meridian δ, and every other meridian is a band sum of δ, (3) S has a single meridian, which is separating, or (4) D(S, M ) = ∅.

Claim 4 . 2 .

 42 Let ξ ∈ ∂H 3 and suppose that Γ -(ξ) and Γ + (ξ) are both nontrivial. Then they are equal.

Figure 9 .

 9 Figure 9. Surgering A along α gives a meridian c.

Fact 5 . 5 (

 55 Tight position =⇒ H 3 quasi-geodesic). Let γ be a simple geodesic ray or biinfinite geodesic on ∂M . If γ is in tight position with respect to some meridian m then any lift γ ⊂ ∂ M of γ is a quasigeodesic in H 3 . In particular, γ is an M -quasigeodesic, and is not homoclinic.Proof. Intersecting with m breaks γ into a union of finite arcs. By simplicity of γ, these arcs fall into only finitely many homotopy classes rel m, and there is a universal upper bound L = L(γ, m) on their lengths. Let D be a disc with boundary m and let D be the entire preimage of D in M . Tightness means that the path γ intersects infinitely many components of D, and intersects no single component more than once.

Figure 10 .

 10 Figure 10. An example of (2) (b) in Theorem 6.1, see Remark 6.2.

Figure 11 .

 11 Figure 11. Homoclinic geodesics h as in cases (i) and (ii) in Corollary 6.3, respectively.

  2d, let b i be the (unique) shortest path on m from h-(i) to h+ (i), and define b j similarly. Let γij be the loop on ∂ M obtained by concatenating the four segments h+ ([i, j]), h-([i, j]), b i and b j in the obvious way.

Figure 12 .

 12 Figure 12. Two rays spiraling onto a simple closed curve (which is not allowed below), where the points in Claim 6.5 are linked.

Figure 13 .

 13 Figure 13. The cases (a) and (b) in the proof of Claim 6.5.

  Figure 15. On the left, the two paths drawn in heavy ink project to the two simple closed curves γ ± in Claim 6.6 (I), shown on the left in Figure14. The shaded region is a rectangle embedded in M that projects to an embedded annulus A ↩→ M with boundary γ -∪ γ + . On the right, the union of the two paths projects to the simple closed curve γ on M of Claim 6.6 (II), and the shaded region projects to a Möbius band B ↩→ M with boundary γ.

7. 1 .

 1 The proof of Theorem 7.1. Most of the proof of Theorem 7.1 is contained in the following results. Assume that S ⊂ ∂ χ<0 M is a connected subsurface with geodesic boundary, ∂S is incompressible, and D(S, M ) is large.

Lemma 8 . 3 .

 83 A marked S-compression body (C, h) has a disk set D(C) ⊂ C(S), where a simple closed curve γ ∈ C(S) lies in the disc set if h(γ) is a meridian in C. In fact, the disk set D(C) determines (C, h) up to equivalence, say by an argument similar to the last paragraph of the proof of Fact 2.4. The set CBod(S) can then be identified with the 'set of all disk sets' in C(S). It then inherits a topology as a subset of the power set P(C(S)), wherein D n → D if and only if for every c ∈ C(S), we have either c ∈ D and c ∈ D n for all large n, or c ∕ ∈ D and c ∕ ∈ D n for all large n. Lemma 8.2. If C n → C in CBod(S), then C ⊂ C n for large n.Proof. Suppose that C is obtained by compressing a finite set Γ of disjoint simple closed curves on S. For large n, we haveΓ ⊂ D(C n ), so C ⊂ C n .□ CBod(S) is compact.Proof. As P(C(S)) is compact, we want to show that CBod(S) is closed. Suppose C n is a sequence of marked compression bodies with disk setsD n = D(C n ) ⊂ C(S),and that D n → D ⊂ C(S). Let Γ be a maximal set of disjoint, pairwise nonhomotopic elements of D. Compressing Γ yields a marked compression body C. Since Γ is finite, Γ ⊂ D n for large n, so D(C) ⊂ D n . Thus, D(C) ⊂ D. It therefore suffices to show D ⊂ D(C). Suppose this is not the case, and pick β ∈ D \ D(C) such that the intersection number of β and Γ is minimal. By maximality of Γ, this intersection number cannot be zero. Since β ∈ D, if n is large we have β ∈ D(C n ). By an outermost disk argument, if γ ∈ Γ is a component that intersects β, there is an arc c ⊂ γ with endpoints on β and interior disjoint from β, that is homotopic rel endpoints in C n to the two arcs b 1 , b 2 ⊂ β into which β is cut by ∂c. Passing to a subsequence, we can assume that c, b 1 , b 2 are independent of n. Then c ∪ b 1 and c ∪ b 2 are both meridians in C n for all large n, and hence lie in D. Since they intersect Γ fewer times than β does, they lie in D(C). But then β (which is a band sum of the two curves) also lies in D(C), a contradiction. □ Let f : Σ -→ Σ be a homeomorphism with f = id on Σ \ S. Then f acts on CBod(S) by f • (C, h) = (C, h • f -1 ). When we regard marked S-compression bodies as compression bodies with exterior boundary equal to Σ, we'll just write C and f (C) for a marked compression body and its image. Note that f (C) = C if and only if f extends to a homeomorphism of C.

  1.6. Acknowledgements. The authors would like to thank Jeff Brock, Juan Souto and Sebastian Hensel for useful conversations. The first author was partially supported by NSF grant DMS-1308678.

2. Preliminaries 2.1. Subsurfaces with geodesic boundary. Suppose S is a finite type hyperbolic surface with geodesic boundary. A connected subsurface with geodesic boundary in S is by definition either (1) a simple closed geodesic X on S, which is the degenerate case, or

  by inclusion are both normally generated by the set of all elements of π 1 Σ that represent simple closed curves in S that are merdians in M . Hence, the disk sets D(Σ, C i ) are the same for i = 1, 2. It follows that C 1 , C 2 are isotopic in M , say by Corollary 2.2 of [3]. □ 2.5. Interval bundles. In this paper, an interval bundle always means a fiber bundle B -→ Y , where Y is a compact surface with boundary, and where all fibers are closed intervals I. Regarding the fibers as 'vertical', we call the associated ∂I-bundle over Y the horizontal boundary of B, written ∂ H B. An interval bundle that is isomorphic to Y × [-1, 1] is called trivial, and we often call nontrivial interval bundles twisted.

  ∂M ), but is not perfectly embedded. However, note that any essential interval bundle (B, ∂ H B) ↩→ (M, S) is perfectly embedded in (M, ∂ H B). Compression arcs. Suppose (B, ∂ H B) ⊂ (M, S) is an essential interval bundle. An arc α ⊂ S with endpoints on ∂(∂ H B) and interior disjoint from ∂ H B is called a compression arc if it is homotopic in M to a fiber of B, while keeping its endpoints on ∂(∂ H B). See Figure

	2.7.

  and α above is a compression arc. Also pictured in light gray is a meridian as described in Claim 2.11. an annulus or a Möbius band. If Y is a Möbius band, we are in the situation of the previous paragraph and are done. So, Y is an annulus, and ∂ H B is a pair of disjoint annuli on S, where c ′ ± lie in different components of ∂ H B. Since c ′

	± bound an annulus in S \ ∂ H B, the interval bundle B is inessential, contrary to our assumption. □
	In fact, more is true.
	Fact 2.12 (Arcs that produce meridians). Suppose (B, ∂ H B) ⊂ (M, S) is an essential interval bundle and let α ⊂ S be an arc with endpoints on ∂ H B and interior disjoint from ∂ H B. Let X ⊂ S be a regular neighborhood of α ∪ ∂ H B within S. Then there is a meridian in X if and only if we have either:
	(1) the endpoints of α lie on the same component c of ∂(∂ H B), and there
	is an arc β ⊂ c such that α ∪ β is a meridian, or (2) α is a compression arc.

  We say that λ ± are essentially homotopic through B if there is a lamination λ and a homotopy h t : λ -→ B, t ∈ [-1, 1] such that h ±1 is a homeomorphism onto λ ± , and where (h t ) is not homotopic into ∂ H B.

		pseudo-
	Anosov maps on S and their attracting laminations converge to m, and then
	the argument finishes as before.	□
	2.9. Laminations on interval bundles. Suppose that Y is a compact hy-
	perbolizable surface with boundary, and that B -→ Y is an interval bundle over Y . Endow Y and the horizontal boundary ∂ H B with arbitrary hyperbolic
	metrics such that the boundary components are all geodesic.
	Suppose we have two geodesic laminations λ ± on ∂ H B.
	Definition 2.15.

When B is a trivial interval bundle, λ ± are essentially homotopic through B if and only if we can write B

The body of this paper is written in greater generality, with the pair (H, S) replaced by a compact, orientable, 3-manifold M with hyperbolizable interior, together with an essential connected subsurface S ⊂ ∂M such that the multi-curve ∂S is incompressible in M . However, everything we do is just as interesting in the handlebody case.

Their condition was really that the measured lamination λ + lies in the 'limit set' of the handlebody, i.e. that it is a limit of meridians in PML(S), but that is equivalent to being strongly commensurable to a Hausdorff limit of meridians. See §2.8 for more information about the limit set.

In both[START_REF] Walsh | The bumping set and the characteristic submanifold[END_REF][START_REF] Lecuire | Structures hyperboliques convexes sur les variétés de dimension 3[END_REF], the authors focus on proving that the boundary components of C± project to simple closed curves in S ± , but that isn't sufficient to say that C± projects to a subsurface with geodesic boundary in S ± , which is what they then claim. E.g. in[START_REF] Walsh | The bumping set and the characteristic submanifold[END_REF] it is stated that under a covering map, the boundary of a subset goes to the boundary of the image, but this isn't true.

See also Theorem

5.6 of[START_REF] Mj | Cannon-thurston maps for kleinian groups[END_REF], which says that there is a continuous equivariant extension ι ± of the inclusion S± ↩→ H 3 as above. This theorem is stated in a much more general setting, though, and our statement is a trivial case.

Really, we need to be using resolutions of our subsurfaces with geodesic boundary, as discussed in §2.1.

It is currently unknown whether the particular transverse measure matters: we might suspect that a measured lamination is a projective limit of meridians if and only if the same is true for any other measured lamination with the same support, but there could also very well be a counterexample.