# Homoclinic leaves, Hausdorff limits, and homeomorphisms

Ian Biringer  Department of Mathematics
Boston College
140 Commonwealth Avenue
Chestnut Hill, MA 02467
USA<EMAIL_ADDRESS>and  Cyril Lecuire  Toulouse Mathematics Institute
Université Paul Sabatier
118, route de Narbonne
F-31062 Toulouse Cedex 9
France<EMAIL_ADDRESS>
###### Abstract.

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unpublished notes of A. Casson.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 $\partial H$ have powers that extend to
subcompressionbodies of $H$.

##  1\. Introduction

Let $H$ be a $3$-dimensional handlebody111The 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 $S\subset\partial M$ such that the multi-curve $\partial
S$ is incompressible in $M$. However, everything we do is just as interesting
in the handlebody case. with genus $g\geq 2$ and let $S:=\partial H$. 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 [12] for more information on laminations.

###  1.1. Homoclinic leaves

In J.P. Otal’s thesis [49] 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unpublished manuscript of A. Casson.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 [49, 35, 34, 31, 48, 26] and Long’s
earlier paper [36].

To define homoclinic, let $\tilde{H}$ be the universal cover of $H$, which is
homeomorphic to a thickened infinite tree. A path
$\ell:\mathbb{R}\longrightarrow\tilde{H}$ is called _homoclinic_ if there are
sequences $s_{i},t_{i}\in\mathbb{R}$ such that

$|s_{i}-t_{i}|\to\infty,\ \text{and}\
\sup_{i}d_{\tilde{H}}(\tilde{\ell}(s_{i}),\tilde{\ell}(t_{i}))<\infty.$

Here, distance in $\tilde{H}$ is measured using the lift of any Riemannian
metric on $H$. Since $H$ is compact, the choice of metric does not matter. A
path $\ell:\mathbb{R}\longrightarrow S$ is called _homoclinic_ if it has a
lift to $\partial\tilde{H}$ 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 $\partial\tilde{H}$, any lift of a periodic
parameterization of $m$ is also periodic, and therefore homoclinic. On the
other hand, if $\gamma$ is an essential simple closed curve that is not a
meridian, any periodic parameterization of $\gamma$ lifts to a properly
embedded biinfinite path in $\partial\tilde{H}$ 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.

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.

In his thesis [49], Otal showed that any Hausdorff limit of meridians has a
homoclinic leaf. (This statement was later extended by Lecuire [33] 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 $\lambda=m\cup\ell$ be a lamination with
two leaves, where $m$ is a nonseparating meridian and the two ends of $\ell$
spiral around $m$ in the same direction, but from opposite sides, as on the
left in Figure 1. Then $\lambda$ has a homoclinic leaf, but it is not a
Hausdorff limit of simple closed curves: any simple geodesic that approximates
$\ell$ closely is trapped and forced to spiral forever around $m$, so cannot
be closed.

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

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

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
$\mu_{1},\mu_{2}$ on $S$ are _commensurable_ if they contain a common
sublamination $\nu$ such that for both $i$, the difference
$\mu_{i}\setminus\nu$ is the union of finitely many isolated leaves. We say
$\mu_{1},\mu_{2}$ are _strongly commensurable_ if they contain a common $\nu$
such that for both $i$, the difference $\mu_{i}\setminus\nu$ 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 $\lambda$ in Figure 3
contains a meridian as a leaf, but it is not strongly commensurable to a limit
of meridians. Indeed, suppose that $\lambda^{\prime}$ is a Hausdorff limit of
meridians that contains $\lambda$. In each component $T\subset S\setminus m$,
there is a _unique_ homotopy class rel $m$ of $m$-waves in $T$. So
$\lambda^{\prime}$ contains a leaf $\ell$ 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 $\mu$. In either case, $\ell$ intersects
transversely the component of $\lambda$ 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 $S\setminus m$.

Figure 3. A lamination that contains a meridian as a leaf, but is not strongly
commensurable to a Hausdorff limit of meridians.

It turns out, though, that this is basically the only counterexample. Let’s
say that a lamination $\lambda$ on $S$ is _exceptional_ if $S$ has genus $2$,
there is a separating meridian $m$ on $S$ that is either disjoint from
$\lambda$ or is a leaf of $\lambda$, and there are minimal sublaminations of
$\lambda$ that fill the two components of $S\setminus m$.

######  Theorem 1.1 (A weak Casson Criterion, see Theorem 7.1).

If $\lambda$ is a geodesic lamination on $S$ that is not exceptional, then
$\lambda$ 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 $\lambda$ is minimal
filling on $S$ and contains a homoclinic leaf, to prove that $\lambda$ 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 $S\setminus\lambda$.

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
$\lambda_{-},\lambda_{+}\subset S$. Then either

  1. (1)

the two ends of $h$ are asymptotic on $S$,

  2. (2)

one of $\lambda_{-},\lambda_{+}$ is an intrinsic limit of meridians, or

  3. (3)

$\lambda_{-},\lambda_{+}$ are contained in incompressible subsurfaces
$S_{-},S_{+}\subset S$ that bound an essential interval bundle $B\subset H$
through which $\lambda_{-}$ and $\lambda_{+}$ are homotopic.

Figure 4. Examples of homoclinic geodesics (in grey) satisfying (3) in Theorem
1.2.

Here, a minimal lamination $\lambda\subset S$ 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
$S(\lambda)\subset S$ containing $\lambda$, see Proposition 5.11 for a number
of equivalent definitions. We refer the reader to 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, $\lambda_{-},\lambda_{+}$
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
$\mathbb{R}\times[-1,1]\subset\tilde{H}$ and the two ends of a lift of $h$ are
asymptotic to $\mathbb{R}_{+}\times\\{-1\\}$ and
$\mathbb{R}_{+}\times\\{1\\}$, respectively. On the right, we write
$H=Y\times[-1,1]$ where $Y$ is a genus two surface with one boundary
component. The laminations $\lambda_{\pm}$ are minimal (in the picture they
are drawn as ‘train tracks’) and fill $Z\times\\{\pm 1\\}$, where $Z\subset Y$
is a torus with two boundary components. Here, $B=Z\times[-1,1]$. 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 [40], and interval bundles appear frequently in Hamenstädt’s work on
the disk set, see e.g. [20, 21, 22]. We note that the interval bundles $B$
appearing in Theorem 1.2 may be twisted interval bundles over non-orientable
surfaces, in which case $\lambda_{-}=\lambda_{+}$ 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 $\lambda$ that is a
Hausdorff limits of meridians, one might expect to see minimal sublaminations
$\lambda$ 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 $\lambda\subset S$ is a nonexceptional geodesic lamination that
is a finite union of minimal components. Then $\lambda$ is strongly
commensurable to a Hausdorff limit of meridians if and only if either

  1. (1)

$\lambda$ is disjoint from a meridian on $S$,

  2. (2)

some component of $\lambda$ is an intrinsic limit of meridians, or

  3. (3)

there are components $\lambda_{\pm}\subset\lambda$ that fill incompressible
subsurfaces $S_{\pm}\subset S$, such that $S_{\pm}$ bound an essential
interval bundle $B\subset H$, the laminations $\lambda_{\pm}$ are essentially
homotopic through $B$, and there is a compression arc $\alpha$ for $B$ that is
disjoint from $\lambda$.

In (3), a _compression arc_ for $B$ is an arc from $\partial S_{-}$ to
$\partial S_{+}$ that is homotopic in $H$, keeping its endpoints in $\partial
S_{\pm}$, 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 $\lambda$ 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 [40].

###  1.3. Extension of reducible maps to compression bodies

As another application of our techniques, we consider extension properties of
homeomorphisms $f:S\longrightarrow S$. To motivate this, recall that a
_subcompression body_ of $H$ is a $3$-dimensional submanifold $C\subset H$
with $S\subset\partial C$ that is obtained by choosing a finite collection
$\Gamma$ of disjoint meridians on $S$, taking a regular neighborhood of $S$
and a collection of discs in $H$ with boundary $\Gamma$, and adding in any
complementary components that are topological $3$-balls. We say $C$ is
obtained by _compressing_ $\Gamma$. We usually consider subcompression bodies
only up to isotopy, and we allow the case that $\Gamma=\emptyset$, in which
case we recover the _trivial subcompression body_ , which is just a regular
neighborhood of $S$.

Two examples where $\Gamma$ 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
$\Gamma=\\{m_{1},m_{2}\\}$ gives the same subcompression body as compressing
$m_{2}$, because we fill in complementary components that are balls. See §2.4
for details.

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

Biringer-Johnson-Minsky [2] showed222Their condition was really that the
measured lamination $\lambda_{+}$ lies in the ‘limit set’ of the handlebody,
i.e. that it is a limit of meridians in $\mathcal{PML}(S)$, but that is
equivalent to being strongly commensurable to a Hausdorff limit of meridians.
See §2.8 for more information about the limit set. that the attracting
lamination $\lambda_{+}$ of a pseudo-Anosov map $f:S\longrightarrow S$ 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. [47] and [32].

Here, we show that extension of powers of a homeomorphism $f:S\longrightarrow
S$ 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
$S_{i}\subset S$, such that $f=id$ on $S_{id}:=S\setminus\cup_{i}S_{i}$, and
where for each $i$, if we set $f_{i}:=f|_{S_{i}}$, then either

  1. (1)

$S_{i}$ is an annulus and $f_{i}$ is a power of a Dehn twist, or

  2. (2)

$f_{i}$ is a pseudo-Anosov map on $S_{i}$.

It follows from the Nielsen-Thurston classification [16], 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 $f:S\longrightarrow S$ be a pure homeomorphism. Then $f$ has a power that
extends to a nontrivial subcompressionbody of $H$ if and only if either:

  1. (1)

there is a meridian in $S_{id}$,

  2. (2)

for some $i$, the map $f_{i}:S_{i}\longrightarrow S_{i}$ 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. (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 $f|_{S_{i}\cup
S_{j}}$ extends to $B$, and there is a compression arc $\alpha$ for $B$ whose
interior lies in $S_{id}$.

In (3), note that if (for simplicity) $B$ is a trivial interval bundle, then
$f|_{S_{i}\cup S_{j}}$ 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 $f|_{S_{i}\cup S_{j}}$ 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 [2] 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 $f:S\longrightarrow S$ 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 $H=Y\times[-1,1]$, where $Y$ is a
surface with boundary, and let $f:Y\longrightarrow Y$ be a pseudo-Anosov map
such that $f=id$ on $\partial Y$. Then

$S=Y\times\\{-1\\}\cup\partial Y\times[-1,1]\cup Y\times\\{+1\\}$

and if we let $F:S\longrightarrow S$ be $f\times id$ on $Y\times\\{\pm 1\\}$
and the identity map on the rest of $S$, while we let $G:S\longrightarrow S$
be $f\times id$ on $Y\times\\{1\\}$, and $f^{2}\times id$ on
$Y\times\\{-1\\}$, 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 $H$.

###  1.4. 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_ $\mathcal{D}(S,M)$ of all isotopy classes of
meridians in an essential subsurface $S\subset\partial M$ with $\partial S$
incompressible, where here $M$ is a compact, irreducible $3$-manifold with
boundary. We show in Proposition 3.1 that either $\mathcal{D}(S,M)$ 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 $\mathcal{D}(S,M)$ is _large_ , meaning that it has
infinite diameter in the curve complex $\mathcal{C}(S)$. 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 $\partial\mathbb{H}^{3}$
associated to hyperbolic metrics on $int(M)$. This picture was originally
known to Thurston [52], and was studied previously under more restrictive
assumptions by Walsh [53] and Lecuire [33]. 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 [53, 33]. 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 [2] to partial pseudo-Anosovs, and §9 contains the
proof of Theorem 9.2, which generalizes Theorem 1.4 above.

###  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. (1)

a simple closed geodesic $X$ on $S$, which is the degenerate case, or

  2. (2)

an immersed surface $X\longrightarrow S$ such that the restriction to $int(X)$
and to each component of $\partial X$ is an embedding, and where each
component of $\partial X$ 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 $X\subset S$, abusing notation.
We consider $X,Y\subset S$ to be _equal_ if they are either the same simple
closed geodesic, or if they are both immersions as in (2) 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 $T\subset S$ 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 $\pi_{1}T$ and $\pi_{1}X$ in
$\pi_{1}S$ 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
$\partial T$ is either geodesic or bounds a cusp in $S\setminus T$, and then
adding in any components of $S\setminus T$ that are cusp neighborhoods.
Alternatively, let $\tilde{T}$ be a component of the pre-image of $T$ in the
universal cover $\tilde{S}$, which is isometric to a convex subset of
$\mathbb{H}^{2}$, let $\Lambda_{T}\subset\partial\mathbb{H}^{3}$ be the set of
limit points of $\tilde{T}$ on $\partial_{\infty}\mathbb{H}^{2}$, and let
$\tilde{X}$ be the convex hull of $\Lambda_{T}$ within $\tilde{S}$. Then
$\tilde{X}$ projects to an $X$ as desired.

Conversely, suppose $X$ is a subsurface with geodesic boundary in $S$. Then
there is a compact essential subsurface $T\hookrightarrow S$, 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 $\mathcal{C}(S)$, 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 $\mathcal{C}(S)$ connect homotopy classes that admit disjoint
representatives.

Masur-Minsky [41] have shown that the curve complex is Gromov hyperbolic, when
considered with the path metric in which all edges have unit length. Klarreich
[30] (see also [19]) showed that the Gromov boundary
$\partial_{\infty}\mathcal{C}(S)$ 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 $S\subset\partial M$ is an essential subsurface of the boundary
of a compact, irreducible $3$-manifold $M$, and that $\partial S$ is
incompressible in $M$. An essential simple closed curve $\gamma$ on $M$ is
called a _meridian_ if it bounds an embedded disc in $M$. By the loop theorem,
$\gamma$ is a meridian if and only if it is homotopically trivial in $M$.

######  Definition 2.2.

The _disc set_ of $S$ in $M$, written $\mathcal{D}(S,M)$, is the (full)
subgraph of $\mathcal{C}(S)$ whose vertices are the meridians of $S$ in $M$.

When convenient, we will sometimes regard $\mathcal{D}(S,M)$ as a subset of
the space of projective measured laminations
${\mathcal{P}\mathcal{M}\mathcal{L}}(S)$, 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 $\partial M$.

######  Theorem 2.3 (Masur-Minsky).

The subset $\mathcal{D}(S,M)$ of $\mathcal{C}(S)$ is quasiconvex.

To prove Theorem 2.3 as stated above, one follows the outline of [42]: given
$a,b\in\mathcal{D}(S,M)$, the goal is to construct a _well-nested curve
replacement sequence_ from $a=a_{1},\ldots,a_{n}=b$ 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.

###  2.4. Compression bodies

We refer the reader to §2 of [3] 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 $\pi_{1}$-surjective boundary component $\partial_{+}C$, called the
_exterior boundary_ of $C$. The complement $\partial
C\smallsetminus\partial_{+}C$ is called the _interior boundary_ , and is
written $\partial_{-}C$. Note that the interior boundary is incompressible.
For if an essential simple closed curve on $\partial_{-}C$ bounds a disk
$D\subset C$, then $C\smallsetminus D$ has either one or two components, and
in both cases, Van Kampen’s Theorem implies that $\partial_{+}C$, which is
disjoint from $D$, cannot $\pi_{1}$-surject.

Suppose $M$ is a compact irreducible $3$-manifold with boundary, let $\Sigma$
be a component of $\partial M$ and let $S\subset\Sigma$ is an essential
subsurface. A _subcompression body of $(M,S)$_ is a compression body $C\subset
M$ with exterior boundary $\Sigma$ that can be constructed as follows. Choose
a set $\Gamma$ of disjoint, pairwise nonhomotopic simple closed curves on $S$
that are all meridians in $M$. Let $C^{\prime}\subset M$ be the union of
$\Sigma$ with a set of disjoint disks in $M$ whose boundaries are the
components of $\Gamma$, and define $C\subset M$ to be the union of a regular
neighborhood of $C^{\prime}\subset M$ together with any components of the
complement of this neighborhood that are topological $3$-balls. Here, we say
that $C\subset M$ is obtained _by compressing $\Gamma$_. Note that the
irreducibility of $M$ implies that no component of $\partial C$ 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 $\Gamma$ above is empty, we obtain the _trivial subcompression
body_ of $(M,S)$, which is just a regular neighborhood of
$\Sigma\subset\partial M$. At the other extreme, we can compress a maximal
$\Gamma$, 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 $\Sigma$ is a
component of $\partial M$ and that $S\subset\Sigma$ is an essential subsurface
such that the multicurve $\partial S$ is incompressible in $M$. Then there is
a unique (up to isotopy) subcompression body

$C:=C(S,M)\subset M$

of $(M,S)$, called the _characteristic compression body_ of $(M,S)$, such that
a curve $\gamma$ 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 [5], 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 $3$-manifolds to problems about compression bodies and about
$3$-manifolds with incompressible boundary.

The reader can also compare the fact to Lemma 2.1 in [3], 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 $\Gamma$ be a maximal set of disjoint, pairwise nonhomotopic $M$-meridians
on $S$, and define $C$ by compressing $\Gamma$. We have to check that any
curve in $S$ that is an $M$-meridian is also a $C$-meridian. Suppose not, and
take an $M$-meridian $m\subset S$ that is not a $C$-meridian, and that
intersects $\Gamma$ minimally. Since $\Gamma$ is maximal, $m$ intersects some
component $\gamma\subset\Gamma$. Then there is an arc $\alpha\subset\gamma$
with endpoints on $m$ and interior disjoint from $m$, that is homotopic rel
endpoints in $M$ to the arcs $\beta^{\prime},\beta^{\prime\prime}\subset m$
with the same endpoints. (Here $\alpha$ is an ‘outermost’ arc of intersection
on a disk bounded by $\gamma$, where the intersection is with the disk bounded
by $m$, see e.g. Lemma 2.8 in [3].) Since $m$ is in minimal position with
respect to $\Gamma$, the curves $m^{\prime}=\alpha\cup\beta^{\prime}$ and
$m^{\prime\prime}=\alpha\cup\beta^{\prime\prime}$ are both essential, and are
$M$-meridians in $S$ that intersects $\Gamma$ fewer times than $m$. So by
minimality of $m$, both $m^{\prime},m^{\prime\prime}$ are $C$-meridians,
implying that $\alpha$ is homotopic rel endpoints to $\beta^{\prime}$ and
$\beta^{\prime\prime}$ 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

$\pi_{1}\Sigma\longrightarrow\pi_{1}C_{i}$

induced by inclusion are both normally generated by the set of all elements of
$\pi_{1}\Sigma$ that represent simple closed curves in $S$ that are merdians
in $M$. Hence, the disk sets $\mathcal{D}(\Sigma,C_{i})$ 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
$B\longrightarrow Y$, 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 $\partial I$-bundle over $Y$ the _horizontal boundary_ of
$B$, written $\partial_{H}B$. An interval bundle that is isomorphic to
$Y\times[-1,1]$ is called _trivial_ , and we often call nontrivial interval
bundles _twisted_.

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
$\pi:\hat{Y}\longrightarrow Y$ be its orientation cover. Then the mapping
cylinder

$B:=\hat{Y}\times[0,1]/\sim,\ \
(x,1)\sim(x^{\prime},1)\iff\pi(x)=\pi(x^{\prime})$

is orientable, and is a twisted interval bundle over $Y$, where the fiber over
$y\in Y$ is obtained by gluing together the two intervals $\\{x\\}\times[0,1]$
and $\\{x^{\prime}\\}\times[0,1]$ along $(x,1)$ and $(x^{\prime},1)$, where
$\pi^{-1}(y)=\\{x,x^{\prime}\\}$. The horizontal boundary $\partial_{H}B$ here
is $\hat{Y}\times\\{0\\}$, which is homeomorphic to the orientable surface
$\hat{Y}$. Note that $B$ is double covered by the trivial interval bundle
$\hat{Y}\times[-1,1]$.

######  Fact 2.5.

Suppose that $B\longrightarrow Y$ 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
$\partial_{H}B\subset\partial B$ 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 $\partial_{H}B\longrightarrow Y$. ∎

An interval bundle $B\longrightarrow Y$ comes with a _canonical involution_
$\sigma$, which is well defined up to isotopy, and which is defined as
follows. If $B\cong Y\times[0,1]$ is a trivial interval bundle, we define

$\sigma:Y\times[-1,1]\longrightarrow Y\times[-1,1],\ \ \sigma(y,t)=(y,-t).$

And if $B$ is the twisted interval bundle $B\cong\hat{Y}\times[0,1]/\sim$
above, we define

$\sigma:\hat{Y}\times[0,1]/\sim\longrightarrow\hat{Y}\times[0,1]/\sim,\ \
\sigma(\hat{y},t)=(\iota(\hat{y}),t)$

where $\iota$ is the nontrivial deck transformation of the orientation cover.
Note that $\sigma$ is always an orientation reversing involution of $B$, so in
particular, when we give the surface $\partial_{H}B$ its boundary orientation,
the restriction $\sigma|_{\partial_{H}B}$ 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 $S\times[-1,1]$ 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 $S\times[-1,1]$ has genus $2g+b-1$, since that is the rank of the
free group $\pi_{1}(S\times[-1,1])\cong\pi_{1}S$.

Finally, suppose $\pi:B\longrightarrow Y$ is an interval bundle and
$f:\partial_{H}B\longrightarrow\partial_{H}B$ is a homeomorphism. We say that
$f$ _extends to $B$_ if there is a homeomorphism $F:B\longrightarrow B$ such
that $F|_{\partial_{H}B}=f$. We leave the following to the reader.

######  Fact 2.7.

The following are equivalent:

  1. (1)

$f$ extends to $B$,

  2. (2)

$f\circ\sigma$ is isotopic to $f$ on $\partial_{H}B$,

  3. (3)

after isotoping $f$, there is a homeomorphism $\bar{f}:Y\longrightarrow Y$
such that $\pi\circ f=\bar{f}\circ\pi$,

  4. (4)

there is a homeomorphism from $B$ to either

$Y\times[-1,1]\ \ \text{or}\ \ \hat{Y}\times[0,1]/\sim,$

taking horizontal boundary to horizontal boundary, such that
$f=F|_{\partial_{H}B}$, and where either

$F:Y\times[-1,1]\longrightarrow Y\times[-1,1],\ \ F(y,t)=(\bar{f}(y),t),$

for some homeomorphism $\bar{f}:Y\longrightarrow Y$, or

$F:\hat{Y}\times[0,1]/\sim\ \longrightarrow\hat{Y}\times[0,1]/\sim,\ \
F(y,t)=(\bar{f}(y),t),$

for some homeomorphism $\bar{f}:\hat{Y}\longrightarrow\hat{Y}$ commuting with
the deck group of $\hat{Y}\longrightarrow Y$, 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
$S\subset\partial M$ is an incompressible subsurface. In the late 1970s, Jaco-
Shalen [25] and Johannson [27] 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 [24]).

There is a perfectly embedded Seifert pair $(X,\Sigma)\subset(M,S)$, unique up
to isotopy and called the _characteristic submanifold_ of $(M,S)$, such that
any nondegenerate map $(B,F)\longrightarrow(M,S)$ from a Seifert pair $(B,F)$
is homotopic as a map of pairs into $(X,\Sigma)$.

A _Seifert pair_ is $3$-manifold pair that is a finite disjoint union of
interval bundle pairs $(B,\partial_{H}B)$ and $S^{1}$-bundle pairs. Here, an
_$S^{1}$ -bundle pair_ $(B,F)$ is a $3$-manifold $B$ fibered by circles, where
$F\subset\partial B$ is a compact subsurface saturated by fibers. A Seifert
pair $(X,\Sigma)\subset(M,S)$ is _well embedded_ if $X\cap\partial
M=\Sigma\subset S$ and the frontier of $X$ in $M$ is a $\pi_{1}$-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 $f:(B,F)\longrightarrow(M,S)$
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 $\pi_{1}$-image is nontrivial, its
$\pi_{1}$-image is non-cyclic when $F=\emptyset$, 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. (1)

$(B,F)$ is a ‘fibered solid torus’, i.e. $B$ is an $S^{1}$-bundle over a disk,
and $F\subset\partial B\cong T^{2}$ is a collection of fibered parallel
annuli, or

  2. (2)

$(B,F)$ is a ‘thickened torus’, i.e. $B$ is an $S^{1}$-bundle over an annulus,
so is homeomorphic to $T^{2}\times[0,1]$, 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
$B\longrightarrow Y$ is an $S^{1}$-bundle, where $Y$ is a compact
$2$-orbifold, and the cyclic subgroup $Z\subset\pi_{1}B$ corresponding to a
regular fiber is normal in $\pi_{1}B$. In a hyperbolic $3$-manifold, any
subgroup of $\pi_{1}$ that has a cyclic normal subgroup is elementary, say by
a fixed point analysis on $\partial_{\infty}\mathbb{H}^{3}$. So, $\pi_{1}B$ is
either cyclic or isomorphic to $\mathbb{Z}^{2}$. 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, well-embedded
interval bundle pair $(B,\partial_{H}B)\hookrightarrow(M,S)$.

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 $Y^{\prime}\subset Y$ is obtained by deleting collar
neighborhoods of the boundary components, and we set $M=Y\times[-1,1]$, which
is a handlebody, then $(Y^{\prime}\times[-1,1],Y^{\prime}\times\\{-1,1\\})$ is
an essential interval bundle in $(M,\partial M)$, but is not perfectly
embedded. However, note that any essential interval bundle
$(B,\partial_{H}B)\hookrightarrow(M,S)$ is perfectly embedded in
$(M,\partial_{H}B)$.

###  2.7. Compression arcs

Suppose $(B,\partial_{H}B)\subset(M,S)$ is an essential interval bundle. An
arc $\alpha\subset S$ with endpoints on $\partial(\partial_{H}B)$ and interior
disjoint from $\partial_{H}B$ is called a _compression arc_ if it is homotopic
in $M$ to a fiber of $B$, while keeping its endpoints on
$\partial(\partial_{H}B)$. See Figure 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 $\overline{{\mathrm{F}r}(B)}$ is boundary compressible, see [24,
pp.36–37] for a definition.

Write our interval bundle as $\pi:B\longrightarrow Y$. Let $\alpha$ be a
compression arc for $B$. After isotoping the bundle map $\pi$, we can assume
that $\alpha$ is homotopic rel endpoints to a fiber $\pi^{-1}(y),$ where $y\in
Y.$ Suppose $c$ is an oriented, two-sided, 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 $\pi^{-1}(c)=c_{-}\cup c_{+}$, where
$c_{\pm}$ are disjoint simple closed oriented loops in $X$ based at $y_{\pm}$,
and where the orientations of $c_{\pm}$ project to that of $c$.

######  Claim 2.11.

The concatenation $m(c):=c_{-}\cdot\alpha\cdot c_{+}^{-1}\cdot\alpha^{-1}$ is
homotopic to a meridian on $S$.

So, a compression arc $\alpha$ allows one to make compressible curves on $S$
from essential curves on $Y$. See Figure 6.

###### Proof.

Since $\alpha$ is homotopic rel endpoints to the fiber $\pi^{-1}(y)$, the
curve $m(c)$ is homotopic in $M$ to a curve in $B$ that projects under $\pi$
to $c\cdot c^{-1}$, 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 $\partial_{H}B$ as oriented
curves. (This happens exactly when the curve $c\subset Y$ 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 $c_{\pm}$ are not freely homotopic in
$\partial_{H}B$ as oriented curves. If $m(c)$ is inessential, then $c_{\pm}$
_are_ freely homotopic on $S$, so $c_{\pm}$ are homotopic in $\partial_{H}B$
to boundary components $c_{\pm}^{\prime}\subset\partial_{H}B$ that bound an
annulus in $S\setminus\partial_{H}B$. In this case $c_{\pm}$ are peripheral,
so we may assume that $Y$ is either 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 $\partial_{H}B$ is a pair of disjoint annuli on
$S$, where $c_{\pm}^{\prime}$ lie in different components of $\partial_{H}B$.
Since $c_{\pm}^{\prime}$ bound an annulus in $S\setminus\partial_{H}B$, the
interval bundle $B$ is inessential, contrary to our assumption. ∎

In fact, more is true.

######  Fact 2.12 (Arcs that produce meridians).

Suppose $(B,\partial_{H}B)\subset(M,S)$ is an essential interval bundle and
let $\alpha\subset S$ be an arc with endpoints on $\partial_{H}B$ and interior
disjoint from $\partial_{H}B$. Let $X\subset S$ be a regular neighborhood of
$\alpha\cup\partial_{H}B$ within $S$. Then there is a meridian in $X$ if and
only if we have either:

  1. (1)

the endpoints of $\alpha$ lie on the same component $c$ of
$\partial(\partial_{H}B)$, and there is an arc $\beta\subset c$ such that
$\alpha\cup\beta$ is a meridian, or

  2. (2)

$\alpha$ is a compression arc.

Note that in the second case the endpoints of $\alpha$ lie on distinct
components of $\partial(\partial_{H}B)$, so in particular the two cases are
mutually exclusive.

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,
$Y^{\prime}\subset Y$ be obtained by deleting a collar neighborhood of
$\partial Y$, set $B=Y^{\prime}\times[-1,1]$ and $M=Y\times[-1,1]$, and let
$\alpha$ be a spanning arc of $B$ in $\partial M$.

Figure 6.  $Z\subset Y$ is a compact surface, the interval bundle
$B=Z\times[-1,1]$ embeds in $M=Y\times[-1,1]$, and $\alpha$ above is a
compression arc. Also pictured in light gray is a meridian as described in
Claim 2.11.

###### Proof.

The ‘if’ direction is immediate: in case (1) we are essentially given a
meridian in $X$, and in case (2) we can appeal to Claim 2.11.

We now work on the ‘only if’ direction. Write our regular neighborhood of
$\partial_{H}B\cup\alpha$ as $X=\partial_{H}B\cup R$ where $R$ is a rectangle
with two opposite ‘short’ sides on the boundary of $\partial_{H}B$. Let
$D\subset M$ be an essential disc whose boundary is contained in $X$, and
where $D$ intersects the frontier $\mathrm{Fr}(B)\subset M$ in a minimal
number of components. Let $a\subset D\cap\mathrm{Fr}(B)$ be an arc that is
‘outermost’ in $D$, i.e. there is some arc $a^{\prime}\subset\partial D$ with
the same endpoints as $a$ such that $a,a^{\prime}$ bound an open disk in $D$
that does not intersect $\mathrm{Fr}(B)$.

We claim that $a^{\prime}\subset R$. If not, then
$a^{\prime}\subset\partial_{H}B$, and bounds a disk in $B$ with the arc
$a\subset\partial B$. Writing the interval bundle as $\pi:B\longrightarrow Y$,
the projection $\pi(a\cup a^{\prime})$ in $Y$ is then also nullhomotopic, so
$\pi(a^{\prime})$ is homotopic rel endpoints into $\partial Y$. Lifting this
homotopy through the covering map $\partial_{H}B\longrightarrow Y$ we get that
$a^{\prime}$ is inessential in $\partial_{H}B$, i.e. is homotopic in
$\partial_{H}B$ rel endpoints into $\partial(\partial_{H}B)$. Lifting this
homotopy through the covering map $\partial_{H}Y\longrightarrow Y$
$\pi(a)\subset\partial Y$, it follows that $\pi(a^{\prime})$ is an inessential
arc in $Y$. We can then decrease the number of components of
$D\cap\mathrm{Fr}(B)$, contradicting that this number is minimal.

So, $a^{\prime}\subset R$. Again by minimality of the intersection, the
endpoints of $a^{\prime}$ lie on opposite short sides of $R$, so $\alpha$ is
homotopic to $a^{\prime}$ through arcs in $R$ with endpoints on
$\mathrm{Fr}(B)$. Since $a^{\prime}$ is homotopic rel endpoints to
$a\subset\mathrm{Fr}(B)$, it follows that $\alpha$ is homotopic rel endpoints
into $\mathrm{Fr}(B)$. If the two endpoints of $\alpha$ lie on the same
component of $\partial(\partial_{H}B)$, we are in case (1), and otherwise we
are in case (2). ∎

###  2.8. Laminations

We assume the reader is familiar with geodesic and measured laminations on
finite type hyperbolic surfaces. See e.g. [12, 28].

Suppose $\lambda$ is a connected geodesic lamination on a finite type
hyperbolic surface $S$ with geodesic boundary. We say that $\lambda$ _fills_
an essential subsurface $T\subset S$ if $\lambda\subset T$ and $\lambda$
intersects every essential, non-peripheral simple closed curve in $T$.

######  Fact 2.13.

For every connected $\lambda$, there is a unique subsurface with geodesic
boundary (as in §2.1) that is filled by $\lambda$, which we denote by
$S(\lambda)$. It is the minimal subsurface with geodesic boundary in $S$ that
contains $\lambda$.

Here, $S(\lambda)$ can be constructed by taking a component
$\tilde{\lambda}\subset\tilde{S}\subset\mathbb{H}^{2}$ of the preimage of
$\lambda$, letting $C\subset\mathbb{H}^{2}$ be the convex hull of the set of
endpoints of leaves of $\tilde{\lambda}$ in $\partial\mathbb{H}^{2}$, and
projecting $C$ into $S$.

Suppose that $M$ is a compact, orientable irreducible $3$-manifold let
$S\subset\partial M$ be an essential subsurface. The _limit set_ of $(S,M)$ is
the closure

$\Lambda(S,M)=\overline{\\{\text{meridians }\gamma\subset
S\\}}\subset{\mathcal{P}\mathcal{M}\mathcal{L}}(S),$

where ${\mathcal{P}\mathcal{M}\mathcal{L}}(S)$ is the space of projective
measured laminations on $S$. The limits set was first studied by Masur [39] in
the case that $M$ is a handlebody, with $S$ its entire boundary. In this case,
Kerckhoff [29] later proved that the limit set has measure zero in
${\mathcal{P}\mathcal{M}\mathcal{L}}(S)$, although a mistake in his argument
was later found and fixed by Gadre [18].

In some ways, $\Lambda(S,M)$ acts as a dynamical limit set. For instance, let
$\mathrm{Map}(S)$ be the mapping class group of $S$, and let
$\mathrm{Map}(S,M)\subset\mathrm{Map}(S)$ be the subgroup consisting of
mapping classes represented by restrictions of homeomorphisms of $M$. Then we
have:

######  Fact 2.14.

  1. (1)

If $\Lambda(S,M)$ is nonempty, it is the smallest nonempty closed subset of
${\mathcal{P}\mathcal{M}\mathcal{L}}(S)$ that is invariant under
$\operatorname{Mod}(S,H)$.

  2. (2)

If $\mathrm{Map}(S,M)$ contains a pseudo-Anosov map on $S$, then
$\Lambda(S,M)$ is the closure of the set of the attracting and repelling
laminations of pseudo-Anosov elements of $\mathrm{Map}(S,M)$.

Note that $\mathrm{Map}(S,M)$ contains a pseudo-Anosov map on $S$ if and only
if the disk set $\mathcal{D}(S,M)$ has infinite diameter in the curve complex
$\mathcal{C}(S)$, where the latter condition was discussed earlier in
Proposition 3.1. See also [2, 34].

###### Proof.

For the first part just note that Dehn twist $T_{m}$ around meridians
$m\subset S$ are in $\mathrm{Map}(M,S)$, so if
$A\subset{\mathcal{P}\mathcal{M}\mathcal{L}}(S)$ is nonempty and invariant,
$\lambda\in A$ and $m$ is a meridian, then $m=\lim_{i}T^{i}_{m}(\lambda)$ is
also in $A$, implying $\Lambda(M,S)\subset A$.

For the second part, take a pseudo-Anosov $f\in\mathrm{Map}(M,S)$ with
attracting lamination $\lambda_{+},$ say. If $m$ is a meridian in $S$, then
$T_{m}^{i}\circ f\circ T^{-i}_{m}$ are 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 hyperbolizable surface with boundary, and that
$B\longrightarrow Y$ is an interval bundle over $Y$. Endow $Y$ and the
horizontal boundary $\partial_{H}B$ with arbitrary hyperbolic metrics such
that the boundary components are all geodesic.

Suppose we have two geodesic laminations $\lambda_{\pm}$ on $\partial_{H}B$.

######  Definition 2.15.

We say that $\lambda_{\pm}$ are _essentially homotopic through $B$_ if there
is a lamination $\lambda$ and a homotopy $h_{t}:\lambda\longrightarrow B,\
t\in[-1,1]$ such that $h_{\pm 1}$ is a homeomorphism onto $\lambda_{\pm}$, and
where $(h_{t})$ is not homotopic into $\partial_{H}B$.

When $B$ is a trivial interval bundle, $\lambda_{\pm}$ are essentially
homotopic through $B$ if and only if we can write $B\cong Y\times[0,1]$ in
such a way that $\lambda_{\pm}=\lambda\times\\{\pm 1\\}$ 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 $\lambda_{\pm}$ are disjoint or equal geodesic laminations on
$\partial_{H}B$. Then the following are equivalent.

  1. (1)

$\lambda_{\pm}$ are essentially homotopic through $B$.

  2. (2)

$\lambda_{\pm}$ is isotopic on $\partial_{H}B$ to $\sigma(\lambda_{\mp})$,
where $\sigma$ is the canonical involution of $B$ discussed in §2.5.

Moreover, (1) and (2) imply

  1. $(3)$

There is a geodesic lamination $\bar{\lambda}$ on $Y$ such that
$\lambda_{-}\cup\lambda_{+}$ is isotopic on $\partial_{H}B$ to the preimage
$(\pi|_{\partial_{H}B})^{-1}(\bar{\lambda})$.

Here, (3) does not always imply (1,2), since it could be that $\bar{\lambda}$
has two components, $(\pi|_{\partial_{H}B})^{-1}(\bar{\lambda})$ has four, and
these components are incorrectly partitioned into the two laminations
$\lambda_{\pm}$. However, that’s the only problem, so for instance if
$\lambda_{\pm}$ 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 $\pi|_{\partial_{H}B}:\partial_{H}B\longrightarrow Y$ 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
$B^{\prime}\longrightarrow B$ that double covers $B$, giving homotopic
laminations $\lambda_{\pm}^{\prime}\subset\partial_{H}B^{\prime}$. (1) $\iff$
(2) follows since the canonical involution on $B^{\prime}$ covers that of $B$.
For (2) $\implies$ (3), note that since $\lambda_{\pm}$ are disjoint or equal
and differ by $\sigma$, their projections $\pi(\lambda_{\pm})\subset Y$ are
the same, and are a geodesic lamination $\bar{\lambda}$ on $Y$. ∎

##  3\. Large and small disk sets and compression bodies

Suppose that $S\subset\partial M$ is an essential subsurface of the boundary
of a compact, irreducible $3$-manifold $M$, and that $\partial S$ 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.

######  Proposition 3.1 (Diameters of disk sets).

With $M,S$ as above, either

  1. (1)

$\mathcal{D}(S,M)$ has infinite diameter in $\mathcal{C}(S)$,

  2. (2)

$S$ has one nonseparating meridian $\delta$, and every other meridian is a
band sum of $\delta$,

  3. (3)

$S$ has a single meridian, which is separating, or

  4. (4)

$\mathcal{D}(S,M)=\emptyset.$

In case (1), we will say that $\mathcal{D}(S,M)$ is _large_ , and in cases
(2)–(4), we will say that $\mathcal{D}(S,M)$ 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 $\mathcal{D}(S,M)$
is large or small. See also the discussion of small compression bodies in §3
of [3].

Here, recall that a _band sum_ of a meridian $\delta$ is the boundary of a
regular neighborhood of $\delta\cup\beta$, where $\beta$ is a simple closed
curve on $S$ that intersects $\delta$ once. Any such band sum must be a
meridian: for instance, as an element of $\pi_{1}M$ it is a commutator with a
trivial element. Also, (3) includes the case when $M$ is a solid torus and
$S=\partial M$, 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, $\gamma\subset S$ is a meridian on $S$ and
$\delta$ is a meridian that lies in a component $T\subset S\setminus\gamma$.
If $\gamma$ is not a band sum of $\delta$, there is a pseudo-Anosov
$f:T\longrightarrow T$ that extends to a homeomorphism of $M$.

The condition that $\gamma$ is not a band sum is necessary. For if $M$ is a
handlebody with $S=\partial M$, and $\gamma$ is a separating meridian that
bounds a compressible punctured torus $T\subset S$, then $T$ has only a single
meridian $\delta$. This $\delta$ is non-separating and $\gamma$ is a band sum
of $\delta$. Any map $T\longrightarrow T$ that extends to a homeomorphism of
$M$ must then fix $\delta$, so cannot be pseudo-Anosov.

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

###### Proof of Claim 3.2.

Suppose first that $\gamma$ is not separating. Any simple closed curve that
intersects $\gamma$ 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 $\alpha,\beta$ of band sums of $\gamma$
that fill $S\smallsetminus\gamma$. By a theorem of Thurston [17, III.3 in 13],
the composition of twists $T_{\alpha}\circ T_{\beta}^{-1}$ is pseudo-Anosov.
Each twist extends to $M$, because twist about meridians can be extended to
twists along the disks they bound.

Now suppose $\gamma$ separates $S$, and suppose that $R$ is the component of
$T\setminus(\gamma\cup\delta)$ adjacent to $\gamma$ and $\delta$. Any curve in
$R$ that bounds a pair of pants with $\gamma$ and $\delta$ is also a meridian.
Such curves are constructed as the boundary of a neighborhood of the union of
$\gamma,\delta$ and any arc in $R$ joining the two. Therefore, there is a pair
$\alpha,\beta$ of such curves that fills $R$. As before, $f=T_{\alpha}\circ
T_{\beta}^{-1}$ is a pseudo-Anosov on $R$ that extends to $M$.

Figure 7. The surfaces $R$ and $R^{\prime}$ fill $T$.

However, there was nothing special about $\delta$ in the above construction.
So if there is some (non-peripheral) meridian $\delta^{\prime}\subset T$ with
$\delta\neq\delta^{\prime}$, there is also a pseudo-Anosov $f^{\prime}$ on the
corresponding surface $R^{\prime}$, such that $f^{\prime}$ extends to $M$.
Since $R$ and $R^{\prime}$ fill $T$, [14, Theorem 6.1] says that for large $i$
the composition $f^{i}(f^{\prime})^{i}$ is a pseudo-Anosov on $T$. See Figure
7.

The only case left to consider is when $\delta$ is the only (non-peripheral)
meridian in $T$. Since new meridians usually can be created by joining
$\delta$ and $\gamma$ 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 $\gamma$ again. But then $\gamma$ is a band
sum of $\delta$. ∎

###### 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 $S\neq T^{2}$ below.

We first claim that if there are two meridians in $S$, neither of which is a
band sum of the other, then $\mathcal{D}(S,M)$ has infinite diameter in the
curve complex. To see this, suppose $\gamma_{1},\gamma_{2}$ are such
meridians. Claim 3.2 gives two pseudo-Anosov maps $f_{1},f_{2}$, each defined
on the component of $S\setminus\gamma_{i}$ that contains $\gamma_{j}$, where
$i\neq j$. Since the component of $S\setminus\gamma_{1}$ containing
$\gamma_{2}$ and the component of $S\setminus\gamma_{2}$ containing
$\gamma_{1}$ together fill $S$, for large $k$ the composition $f^{k}g^{k}$ is
a pseudo-Anosov map on the entire surface $S$, by [14, 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 [41], this implies that the set
of meridians has infinite diameter in $\mathcal{C}(S)$.

Starting now with the proof of the proposition, suppose there are _no
nonseparating meridians_ in $S$. If $\gamma,\delta$ are distinct (separating)
meridians, then an innermost disk surgery produces another separating meridian
$\gamma_{2}$ disjoint from $\gamma_{1}$, see [3, Lemma 2.8]. By the previous
paragraph, $\mathcal{D}(S,M)$ has infinite diameter in the curve complex. So,
the only other options are if $\mathcal{D}(S,M)=\emptyset$, or if the only
meridian is a single separating curve.

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

Figure 8. A surgery of a curve $\delta$ along a non-separating $\gamma$ cannot
produce a curve $\beta$ that is a band sum of $\gamma$.

##  4\. Windows from limit sets

Let $N=\Gamma\backslash\mathbb{H}^{3}$ be an orientable, geometrically finite
hyperbolic $3$-manifold, let $\Lambda\subset\partial\mathbb{H}^{3}$ be the
limit set of $\Gamma$, and let

$CC(N):=\Gamma\backslash CH(\Lambda)\subset N$

be the convex core of $N$. Equip $\partial CC(N)$ with its intrinsic length
metric, which is hyperbolic, see for instance [51, Prop 8.5.1].

Let $S_{\pm}$ be (possibly degenerate) incompressible subsurfaces with
geodesic boundary in $\partial CC(N)$ that are either equal or are essentially
disjoint, as in §2.1. Let

$\tilde{S}_{\pm}\subset\partial CH(\Lambda)\subset\mathbb{H}^{3}$

be lifts of $S_{\pm}$, where if $S_{-}=S_{+}$, we require that
$\tilde{S}_{-}\neq\tilde{S}_{+}$. Let $\Gamma_{\pm}\subset\Gamma$ be the
stabilizers of $\tilde{S}_{\pm}$, let
$\Lambda_{\pm}\subset\partial\mathbb{H}^{3}$ be their limit sets and
$\Delta=\Gamma_{+}\cap\Gamma_{-}$.

The lift $\tilde{S}_{\pm}$ is isometric to a convex subset of
$\mathbb{H}^{2}$. Let $\partial_{\infty}S_{\pm}\subset\partial\mathbb{H}^{2}$
be the boundary of $\tilde{S}_{\pm}$. By [45, Theorem 5.6], say, the inclusion
$\tilde{S}_{\pm}\hookrightarrow\mathbb{H}^{3}$ extends continuously to a
$\Gamma_{\pm}$-equivariant quotient map

$\iota_{\pm}:\partial_{\infty}\tilde{S}_{\pm}\longrightarrow\Lambda_{\pm}\subset\partial\mathbb{H}^{3}.$

######  Theorem 4.1 (Windows from limit sets).

$\Lambda_{-}\cap\Lambda_{+}=\Lambda_{\Delta}$. Next, suppose $\Delta$ is
nonempty and is not a cyclic group acting parabolically on either
$\tilde{S}_{-}$ or $\tilde{S}_{+}$, and let
$\tilde{C}_{\pm}\subset\tilde{S}_{\pm}$ be the convex hulls of the subsets
$\iota_{\pm}^{-1}(\Lambda_{\Delta})\subset\partial_{\infty}\tilde{S}_{\pm}.$
Then $\tilde{C}_{\pm}$ are $\Delta$-invariant, the quotients
$C_{\pm}:=\Delta\backslash\tilde{C}_{\pm}$ are (possibly degenerate)
subsurfaces with geodesic boundary in $S_{\pm}$, and there is an essential
homotopy from $C_{-}$ to $C_{+}$ in $CC(N)$ that is the projection of a
homotopy from $\tilde{C}_{-}$ to $\tilde{C}_{+}$.

Above, $C_{\pm}$ are (possibly degenerate) subsurfaces with geodesic boundary
in $S_{\pm}$, as defined in §2.1, but it follows from the above and Theorem
2.8 that there are ‘resolutions’ (see §2.1) $C_{\pm}^{\prime}\subset S_{\pm}$
such that $C_{\pm}^{\prime}$ bound an interval bundle in $CC(M)$. So
informally, the theorem says that the intersection
$\Lambda_{-}\cap\Lambda_{+}$ is exactly the limit set of the fundamental group
of some essential interval bundle in $(CC(M),S_{-}^{\prime}\cup
S_{+}^{\prime})$. The term ‘window’ comes from Thurston [52] 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 $\Delta$ is not cyclic and acting parabolically on either
$\tilde{S}_{\pm}$ is just for convenience in the statement of the theorem.
(Just to be clear, note that an element $\gamma\in\Delta$ can act
parabolically as an isometry of $\mathbb{H}^{3}$, but hyperbolically on the
convex subsets $\tilde{S}_{\pm}\subset\mathbb{H}^{2}$.) If $\Delta$ is cyclic
and acts parabolically on $\tilde{S}_{+}$ the subset $\tilde{C}_{+}$ 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 [52].
Precise statements for geometrically finite $N$ without accidental parabolics
were worked out in Lecuire’s thesis [33] and by Walsh [53]; 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 $\tilde{S}_{\pm}$, which are not
allowed in those theorems. Also, our proof is more direct and natural333In
both [53, 33], the authors focus on proving that the boundary components of
$\tilde{C}_{\pm}$ project to simple closed curves in $S_{\pm}$, but that isn’t
sufficient to say that $\tilde{C}_{\pm}$ projects to a subsurface with
geodesic boundary in $S_{\pm}$, which is what they then claim. E.g. in [53] 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. than those in [53, 33], 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
$\Gamma$. Essentially, the point is to use Canary’s covering theorem [9] to
show that degenerate NP-ends in the covers
$N_{\pm}:=\Gamma_{\pm}\backslash\mathbb{H}^{3}$ have neighborhoods that embed
in $N$, and then to use this to prove that geodesic rays in $\mathbb{H}^{3}$
that converge to points in $\Lambda_{-}\cap\Lambda_{+}$ cannot exit degenerate
ends in $N_{\pm}$. 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 $\Lambda_{-}\cap\Lambda_{+}=\Lambda_{\Delta}$.
For each $\xi\in\partial\mathbb{H}^{3}$, let
$\Gamma_{\pm}(\xi)\subset\Gamma_{\pm}$ be the stabilizer of $\xi$.

######  Claim 4.2.

Let $\xi\in\partial\mathbb{H}^{3}$ and suppose that $\Gamma_{-}(\xi)$ and
$\Gamma_{+}(\xi)$ are both nontrivial. Then they are equal.

###### Proof.

By the Tameness Theorem [1, 8], we can identify $CC(N)$ topologically with a
subset of a $3$-compact manifold with boundary $M$, where

(1) $CC(N)\supset int(M),\ \ CC(N)\cap\partial M=\partial CC(N),$

and where $\partial CC(N)$ is a collection of essential subsurfaces of
$\partial\bar{N}$. Let $\partial_{\chi=0}M$ be the union of all torus boundary
components of $M$, and let $(X,\Sigma)$ be the characteristic submanifold of
the pair $(M,S_{-}\cup S_{+}\cup\partial_{\chi=0}M),$ as in §2.6.

Since $\Gamma_{\pm}$ are both contained in a discrete group $\Gamma$, both
$\Gamma_{\pm}(\xi)$ are contained in the stabilizer $\Gamma(\xi)$, which is
either infinite cyclic, or rank $2$ parabolic.

Suppose first that $\Gamma(\xi)$ is rank $2$ parabolic. The groups
$\Gamma_{\pm}(\xi)$ are both cyclic, since $S_{\pm}$ are incompressible
hyperbolic surfaces, so their fundamental groups do not contain
$\mathbb{Z}^{2}$ subgroups. So, we can write
$\Gamma_{\pm}(\xi)=\langle\gamma_{\pm}\rangle$ for closed curves
$\gamma_{\pm}$ on $S_{\pm}$. Both $\gamma_{\pm}$ are homotopic into some fixed
component $T\subset\partial_{\chi=0}M$, the component whose fundamental group
can be conjugated to stabilize $\xi$. So, there is a component
$(X_{0},\Sigma_{0})\subset(X,\Sigma)$ of the characteristic submanifold such
that $\Sigma_{0}$ intersects $T$ and both $\gamma_{\pm}$ are homotopic on
$S_{\pm}$ into $\Sigma_{0}$. Since $M\not\cong T^{2}\times[0,x]$, the
component $(X_{0},\Sigma_{0})$ 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,
$\Sigma_{0}$ intersects each of $S_{\pm}$ in a fibered annulus, and these
annuli are disjoint, so they are parallel on a torus boundary component of
$X_{0}$, implying that $\gamma_{\pm}$ are homotopic in $M$, and hence
$\Gamma_{\pm}(\xi)$ are conjugate in $\Gamma$. But since $\Gamma_{\pm}(\xi)$
have the same fixed point at infinity, the conjugating element must fix $\xi$,
and therefore commute with the two groups, implying
$\Gamma_{-}(\xi)=\Gamma_{+}(\xi)$.

Now assume $\Gamma(\xi)$ is cyclic. Pick a basepoint $p\in S_{-}$, say, and
let $\gamma_{-}\subset S_{-}$ be a loop based at $p$ representing a generator
of $\Gamma_{-}(\xi)$. Represent a generator of $\Gamma_{+}(\xi)$ as
$\alpha\cdot\gamma_{+}\cdot\alpha^{-1}$, where $\alpha$ is an arc from $p\in
S_{-}$ to a point in $S_{+}$, and $\gamma_{+}$ is a loop in $S_{+}$. Since
$\Gamma_{\pm}$ stabilize distinct components $\tilde{S}_{\pm}$, the arc
$\alpha$ is not homotopic into $S_{-}\cup S_{+}$. So, $\alpha$ is a spanning
arc of an essential map from an annulus, where the boundary components of the
annulus map to powers of $\gamma_{\pm}$. It follows that the loops
$\gamma_{\pm}$ are homotopic on $S_{\pm}$ into $\Sigma_{0}$ for some component
$(X_{0},\Sigma_{0})\subset(X,\Sigma)$.

If $X_{0}$ is an $I$-bundle with horizontal boundary $\Sigma_{0}$, then as
$\gamma_{\pm}$ are not proper powers in $\pi_{1}S_{\pm}$, they are both
primitive in $\pi_{1}X_{0}$, and hence $\gamma_{\pm}$ (rather than their
powers) are homotopic in $X_{0}\subset M$. Similarly, if $(X_{0},\Sigma_{0})$
is a fibered solid torus, $\Sigma_{0}$ is a collection of parallel annuli on
$\partial X_{0}$, so since $\gamma_{\pm}$ are primitive in $\pi_{1}S_{\pm}$,
they are homotopic on $S_{\pm}$ to _simple_ closed curves in these annuli, and
hence are homotopic to each other in $X_{0}$.

It follows that there are generators for $\Gamma_{\pm}(\xi)$ that are
conjugate in $\Gamma$, but since these generators both fix $\xi$, they are
equal. ∎

######  Claim 4.3.

For all $\xi\in\Lambda_{-}\cap\Lambda_{+}$, we have
$\Gamma_{-}(\xi)=\Gamma_{+}(\xi)$. Moreover,

$\Lambda_{\Delta}=\Lambda_{-}\cap\Lambda_{+}.$

###### Proof.

Let $N_{\pm}\subset\mathbb{H}^{3}$ be the $1$-neighborhood of the convex hull
of $\Lambda_{\pm}$, and for small $\epsilon>0$, let
$T_{\pm}(\epsilon)\subset\mathbb{H}^{3}$ be the set of all points that are
translated less than $\epsilon$ by some parabolic element of $\Gamma_{\pm}$.
If $\epsilon$ is at most the Margulis constant $\epsilon_{0}$, then
$T_{\pm}(\epsilon)$ is a disjoint union of horoballs in $\mathbb{H}^{3}$.

The sets $N_{\pm}$ and $T_{\pm}(\epsilon)$ are $\Gamma_{\pm}$ invariant. Since
$\Gamma_{\pm}$ is a finitely generated subgroup of $\Gamma$, which is
geometrically finite, $\Gamma_{\pm}$ is geometrically finite as well by [9].
So, the action of $\Gamma_{\pm}$ on $N_{\pm}\setminus T_{\pm}(\epsilon)$ is
cocompact, see e.g. Theorem 3.7 in [43], implying that either the function

$D_{+}:\mathbb{H}^{3}\longrightarrow\mathbb{R}_{>0},\ \
D_{+}(x)=\min\\{d(x,\gamma(x))\ |\ \gamma\in\Gamma_{+}\text{ loxodromic}\\}$

is bounded above on $N_{+}\setminus T_{+}(\epsilon)$ by some $B(\epsilon)>0$,
or $\Gamma_{+}$ is elementary parabolic. A similar statement holds for $-$
instead of $+$. With $\epsilon_{0}$ the Margulis constant, the Margulis Lemma
then implies that _if $\epsilon>0$ is sufficiently small with respect to
$B(\epsilon_{0})$, and $\Gamma_{+}$ is not elementary parabolic, then_

(2) $T_{-}(\epsilon)\cap N_{+}\subset T_{+}(\epsilon_{0}),$

and similarly with $-,+$ exchanged. Indeed, if not then we have (say) a point
$p\in\mathbb{H}^{3}$ that is translated by less than $\epsilon$ by some
parabolic $\gamma_{-}\in\Gamma_{-}$ and by at most $B$ by some loxodromic
$\gamma_{+}\in\Gamma_{+}$. If $\epsilon$ is small with respect to $B$, then
both $\gamma_{-}$ and $[\gamma_{+},\gamma_{-}]$ translates $p$ by at most
$\epsilon_{0}$, so they generate an elementary discrete group by the Margulis
lemma applied to $\Gamma$, implying that $\gamma_{+}$ fixes the fixed point of
$\gamma_{-}$, which contradicts that they generate a discrete group.

Fix $\xi\in\Lambda_{+}\cap\Lambda_{-}$. We claim that
$\Gamma_{-}(\xi)=\Gamma_{+}(\xi)$. By Claim 4.2 it suffices to show that
whenever $\Gamma_{-}(\xi)$ is nontrivial, say, so is $\Gamma_{+}(\xi)$.

First, assume that $\Gamma_{-}(\xi)$ is elementary parabolic. We claim that
$\Gamma_{+}(\xi)$ is elementary parabolic as well. Assume not, and let
$\alpha$ be a geodesic ray in $\mathbb{H}^{3}$ converging to $\xi$. Then
$\alpha(t)$ lies in $T_{-}(\epsilon)\cap N_{+}$ for large $t$, and therefore
in $T_{+}(\epsilon_{0})$ for large $t$ by (2), which implies $\xi$ is a
parabolic fixed point of $\Gamma_{+}$ as well, a contradiction.

Next, suppose that $\Gamma_{-}(\xi)$ is elementary loxodromic. If $\xi$ is a
parabolic fixed point of $\Gamma_{+}$, we are done, so let’s assume this isn’t
the case. Let $\alpha$ be the axis of $\Gamma_{-}(\xi)$, parametrized so
$\alpha(t)\to\xi$ as $t\to\infty$. Since $\xi$ is not a $\Gamma_{+}$ parabolic
fixed point, there are $t_{i}\to\infty$ such that $\alpha(t_{i})\not\in
T_{+}(\epsilon)$ for all $i$. Since the action of $\Gamma_{+}$ on
$N_{+}\setminus T_{+}(\epsilon)$ is cocompact, if $p\in\mathbb{H}^{3}$ is a
fixed basepoint, there are elements $\gamma_{i}^{+}\in\Gamma_{+}$ such that
$\sup_{i}d(\gamma_{i}^{+}(p),\alpha(t_{i}))<\infty$. Since the action of
$\Gamma_{-}(\xi)$ on $\alpha$ is cocompact, there are then elements
$\gamma_{i}^{-}\in\Gamma_{-}(\xi)$ with

$\sup_{i}d(\gamma_{i}^{+}(p),\gamma_{i}^{-}(p))<\infty.$

By discreteness of $\Gamma$, after passing to a subsequence we can assume
$\gamma_{i}^{+}=\gamma^{-}_{i}\circ g$ for some fixed $g\in\Gamma$. Hence, for
all $i$ we have

$\gamma_{i}^{+}\circ(\gamma_{1}^{+})^{-1}=\gamma_{i}^{-}\circ(\gamma_{1}^{-})^{-1}\in\Gamma_{+}\cap(\Gamma_{-}(\xi))\subset\Gamma_{+}(\xi),$

so we are done.

Finally, we want to show that $\Lambda_{-}\cap\Lambda_{+}=\Lambda_{\Delta}$.
The inclusion $\Lambda_{\Delta}\subset\Lambda_{-}\cap\Lambda_{+}$ is clear.
So, take $\xi\in\Lambda_{-}\cap\Lambda_{+}$. We can assume that
$\Gamma_{\pm}(\xi)=1$, since otherwise we’re in the cases handled above. Let
$\alpha$ be a geodesic ray in $\mathbb{H}^{3}$ converging to $\xi$. As in the
previous case, since $\xi$ is not a parabolic fixed point of $\Gamma_{+}$,
there are $t_{i}\to\infty$ such that $\alpha(t_{i})\not\in
T_{+}(\epsilon_{0})$ for all $i$. Discarding finitely many $i$, we have
$\alpha(t_{i})\in N_{+}$, so it follows from (2) that $\alpha(t_{i})\not\in
T_{-}(\epsilon)$. Fixing a base point $p\in\mathbb{H}^{3}$, as $\Gamma_{-}$
acts cocompactly on $N_{-}\setminus T_{-}(\epsilon)$ and $\Gamma_{+}$ acts
cocompactly on $N_{+}\setminus T_{+}(\epsilon_{0})$, there are elements
$\gamma_{i}^{\pm}\in\Gamma_{\pm}$ such that

$\sup_{i}d(\gamma_{i}^{\pm}(p),\alpha(t_{i}))<\infty.$

So passing to a subsequence, $\gamma_{i}^{+}=\gamma_{i}^{-}\circ g$ for some
fixed $g\in\Gamma$, and then

$\gamma_{i}^{+}\circ(\gamma_{1}^{+})^{-1}=\gamma_{i}^{-}\circ(\gamma_{1}^{-})^{-1}\in\Gamma_{+}\cap\Gamma_{-}=\Delta$

for all $i$. But applying this sequence to $p$ and letting $i\to\infty$ gives
a sequence of points in the orbit $\Delta(p)$ that converge to $\xi$, so
$\xi\in\Lambda_{\Delta}$.∎

Now assume that $\Delta\neq 1$. We want to construct the interval bundle $W$
mentioned in the statement of the theorem. After an isotopy on $\partial
CC(N)$, let’s assume that $S_{\pm}$ is a subsurface of $\partial CC(N)$ with
geodesic boundary. Consequently, we allow degenerate subsurfaces, where
$S_{\pm}$ is a simple closed geodesic, as well as subsurfaces where only the
interior is embedded and two boundary components can coincide. As $\partial
CC(N)$ may have cusps, we also must allow $S_{\pm}$ to be noncompact with
finite volume, rather than compact.

Recall that $\tilde{S}_{\pm}$ is isometric to a convex subset of
$\mathbb{H}^{2}$, and that if
$\partial_{\infty}S_{\pm}\subset\partial\mathbb{H}^{2}$ is the boundary of
$\tilde{S}_{\pm}$ the inclusion $\tilde{S}_{\pm}\hookrightarrow\mathbb{H}^{3}$
extends continuously to a $\Gamma_{\pm}$-equivariant quotient map

$\iota_{\pm}:\partial_{\infty}\tilde{S}_{\pm}\longrightarrow\Lambda_{\pm}\subset\partial\mathbb{H}^{3}.$

Moreover, if $\xi,\xi^{\prime}\in\partial_{\infty}\tilde{S}_{+}$, say, we have
$\iota_{+}(\xi)=\iota_{+}(\xi^{\prime})$ if and only if there is an element
$\gamma\in\Gamma_{+}$ that acts hyperbolically on
$\tilde{S}_{+}\cup\partial_{\infty}\tilde{S}_{+}$ with fixed points
$\xi,\xi^{\prime}\in\partial_{\infty}\tilde{S}_{+}$, but acts parabolically on
$\mathbb{H}^{3}$. By discreteness of the action
$\Gamma_{+}\curvearrowright\tilde{S}_{+}$, each
$\xi\in\partial_{\infty}\tilde{S}_{+}$ has the same image under $\iota_{+}$ as
_at most one_ other $\xi^{\prime}$. Similar statements holds with $-$ instead
of $+$. All this is a consequence (for instance) of Bowditch’s theory of the
boundary of a relatively hyperbolic group [6]: since the action
$\Gamma_{\pm}\curvearrowright\mathbb{H}^{3}$ is geometrically finite,
$\Lambda_{\pm}$ is a model for the Bowditch boundary of the group
$\Gamma_{\pm}$ relatively to its maximal parabolic subgroups, so the statement
above follows from Theorem 1.3 of [37], say444See also Theorem 5.6 of [45],
which says that there is a continuous equivariant extension $\iota_{\pm}$ of
the inclusion $\tilde{S}_{\pm}\hookrightarrow\mathbb{H}^{3}$ as above. This
theorem is stated in a much more general setting, though, and our statement is
a trivial case..

Let
$\iota_{\pm}^{-1}(\Lambda_{\Delta})\subset\partial_{\infty}\tilde{S}_{\pm}$.
Since $\Delta\neq 1$ and is not cyclic parabolic,
$\iota_{\pm}^{-1}(\Lambda_{\Delta})$ has at least two points, so it has a
well-defined convex hull $\tilde{C}_{\pm}\subset S_{\pm}$.

######  Claim 4.4 (Convex hulls).

One of the following holds.

  1. (1)

$\Delta$ is cyclic and acts hyperbolically on $\tilde{S}_{+}$. The convex hull
$\tilde{C}_{+}$ is its geodesic axis, which is precisely invariant under
$\Delta\subset\Gamma$, so that the quotient
$C_{+}:=\Delta\backslash\tilde{C}_{+}$ embeds as a simple closed geodesic in
$S_{+}$.

  2. (2)

$\tilde{C}_{+}$ is a subsurface of $\tilde{S}_{+}$ with geodesic boundary, the
interior $int(\tilde{C}_{+})$ is precisely invariant under
$\Delta\subset\Gamma$, and the quotient $C_{+}:=\Delta\backslash\tilde{C}_{+}$
is a generalized subsurface of $S_{+}$ with compact geodesic boundary.

A similar statement holds with $-$ instead of $+$.

###### Proof.

Let’s work with $+$, for concreteness. If $g\in\Delta$, then
$g(\Lambda_{\Delta})=\Lambda_{\Delta}$, so $g$ leaves
$\iota_{+}^{-1}(\Lambda_{\Delta})$ invariant by equivariance of $\iota_{+}$.
Hence $g$ leaves $\tilde{C}_{+}$ invariant.

Let’s suppose first that $\tilde{C}_{+}$ has nonempty interior, since that is
the more interesting case. We’ll address the case that $\tilde{C}_{+}$ is a
biinfinite geodesic at the end of the proof. Let
$g\in\Gamma_{+}\setminus\Delta$. We want to show

$g(int(\tilde{C}_{+}))\cap int(\tilde{C}_{+})=\emptyset.$

Assume this is not the case. By Claim 4.3, the fixed points of $g$ in
$\partial_{\infty}S_{+}$ lie outside $\iota_{+}^{-1}(\Lambda(\Delta))$. So, we
cannot have $g(\tilde{C}_{+})\subset\tilde{C}_{+}$, as then we’d have
$g^{n}(\tilde{C}_{+})\subset\tilde{C}_{+}$ for all $n$, contradicting that
points of $\tilde{S}_{+}$ converge to the fixed points of $g$ under iteration.
Considering backwards iterates, we also cannot have $\tilde{C}_{+}\subset
g(\tilde{C}_{+})$. Therefore, $\partial\tilde{C}_{+}$ and $\partial
g(\tilde{C}_{+})$ intersect transversely.

Since $\tilde{C}_{+}$ has nonempty interior, $\Delta$ is nonelementary, and
therefore the fixed points of loxodromic isometries of $\Delta$ are dense in
$\Lambda_{\Delta}$. Loxodromic fixed points of $\Delta$ are in particular
_not_ parabolic fixed points in $\Gamma_{\pm}$, so any biinfinite geodesic in
$\tilde{C}_{+}$ is a limit of biinfinite geodesics in $\tilde{C}_{+}$ whose
endpoints are _not_ fixed points of parabolic isometries of $\Gamma_{\pm}$. By
the previous paragraph, there are then biinfinite geodesics
$\alpha_{+},\beta_{+}$ in $\tilde{C}_{+}$ such that $g(\alpha_{+})$ and
$\beta_{+}$ intersect transversely, and where the endpoints of
$\alpha_{+},\beta_{+}$ project under $\iota_{+}$ to points
$\xi_{\alpha},\xi_{\alpha}^{\prime},\xi_{\beta},\xi_{\beta}^{\prime}\in\Lambda_{\Delta}$
that are not parabolic fixed points in $\Gamma_{\pm}$.

Let $\alpha_{-}$ be the geodesic in $\tilde{S}_{-}$ whose endpoints in
$\partial_{\infty}\tilde{S}_{-}$ map to the points
$\xi_{\alpha},\xi_{\alpha}^{\prime}$ under $\iota_{-}$. Define $\beta_{-}$
similarly. Then

$\alpha:=\alpha_{+}\cup\\{\xi_{\alpha},\xi_{\alpha}^{\prime}\\}\cup\alpha_{-},\
\ \beta:=\beta_{+}\cup\\{\xi_{\beta},\xi_{\beta}^{\prime}\\}\cup\beta_{-}$

are two _simple_ closed curves on the closure $cl(\partial
CH(\Lambda_{\Gamma}))\subset\mathbb{H}^{3}\cup\partial\mathbb{H}^{3}$, which
is homeomorphic to a sphere. For instance, the arcs $\alpha_{\pm}$ are
disjoint and $\xi_{\alpha}\neq\xi_{\alpha}^{\prime}$, since the endpoints of
$\alpha_{+}$ are not parabolic fixed points.

Now consider how the two simple closed curves $g(\alpha),\beta$ intersect. The
arcs $\beta_{-}$ and $g(\alpha_{+})$ are disjoint since
$\tilde{S}_{-}\neq\tilde{S}_{+}$. The arcs $g(\alpha_{-}),\beta_{-}$ are
disjoint since $g\not\in\Gamma_{-}$ and hence $g(\alpha_{-})$ lies on a
different translate of $\tilde{S}_{-}$ than $\beta_{-}\subset\tilde{S}_{-}$.
Moreover, since $g(\alpha_{+}),\beta_{+}$ intersect transversely in
$\tilde{S}_{+}$, the endpoints of $g(\alpha_{+})$ and $\beta_{+}$ are distinct
in $\partial_{\infty}\tilde{S}_{+}$, and since none of them are parabolic
fixed points, the points
$g(\xi_{\alpha}),g(\xi_{\alpha}^{\prime}),\xi_{\beta},\xi_{\beta}^{\prime}$
are all distinct. But by assumption, $g(\alpha_{+})$ intersects $\beta_{+}$
transversely in a single point! This shows that $g(\alpha)$ and $\beta$
intersect exactly once, transversely, which is a contradiction.

By precise invariance of the action on the interior, the quotient
$int(C_{+})=\Delta/int(\tilde{C}_{+})$ embeds in the finite volume surface
$S_{+}$, so $C_{+}$ has finite volume itself. So if $\partial C_{+}$ is non-
compact, it must have two noncompact boundary components that are asymptotic.
Lifting, we get two boundary components $\beta_{1},\beta_{2}$ of
$\tilde{C}_{+}$ that are asymptotic. Since $\tilde{C}_{+}$ is convex, it is
contained in the subset of $\mathbb{H}^{2}$ bounded by $\beta_{1},\beta_{2}$,
and hence the common endpoint of $\beta_{1},\beta_{2}$ is an isolated point of
$\Lambda_{\Delta}$, which is a contradiction since $\Delta$ is not elementary.

The case when $\tilde{C}_{+}$ is a biinfinite geodesic is similar. Here,
$\Delta$ must be cyclic, acting on $\tilde{S}_{+}$ with axis $\tilde{C}_{+}$,
and acting either parabolically or loxodromically on $\mathbb{H}^{3}$. In the
parabolic case, $\tilde{C}_{+}$ compactifies to a simple closed curve on the
sphere
$cl(CH(\Lambda_{\Gamma}))\subset\mathbb{H}^{3}\cup\partial\mathbb{H}^{3}$, so
no translate $g(\tilde{C}_{+}),g\in\Gamma_{+}$, can intersect $\tilde{C}_{+}$
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 $cl(\tilde{C}_{+}\cup\tilde{C}_{-})\subset
cl(\partial\tilde{M})$ and its $g$-image. So, $\tilde{C}_{+}$ is precisely
invariant under $\Delta\subset\Gamma$. The quotient
$C_{+}:=\Delta\backslash\tilde{C}_{+}$ is obviously compact, and is therefore
a simple closed geodesic in $S_{+}$. ∎

We claim that $C_{-}$ and $C_{+}$ are homeomorphic. If $C_{\pm}$ are isotopic
in $\partial CC(M)$ this is clear, and otherwise we argue as follows. The
subgroups $\pi_{1}C_{\pm}$ are both represented by $\Delta$, so are conjugate
in $\pi_{1}M$. The fact that every curve in $C_{-}$ is homotopic to a curve in
$C_{+}$ (and vice versa) implies that $C_{\pm}$ are isotopic to subsurfaces
$C_{\pm}^{\prime}\subset\Sigma$ in the boundary $\Sigma$ of a component
$(X,\Sigma)$ of the characteristic submanifold555Really, we need to be using
resolutions of our subsurfaces with geodesic boundary, as discussed in §2.1.
of $(CC(M),S_{-}\cup S_{+})$, see §2.6, and that even within $X$ every closed
curve in $C_{-}^{\prime}$ is homotopic to a closed curve in $C_{+}^{\prime}$,
and vice versa. When $X$ is a solid torus or thickened torus,
$C_{\pm}^{\prime}$ are annuli, while if $X$ is an interval bundle,
$C_{\pm}^{\prime}$ bound a vertical interval bundle in $X$, and are
homeomorphic.

So, let $f:C_{-}\longrightarrow C_{+}$ be a homeomorphism, lift $f$ to a
$\Delta$-equivariant homeomorphism
$\tilde{f}:\tilde{C}_{-}\longrightarrow\tilde{C}_{+}$ and let

$F:\tilde{C}_{-}\times[0,1]\longrightarrow CH(\Lambda)$

where $F(x,\cdot)$ parametrizes the geodesic from $x$ to $f(x)$. Then $F$ is
$\Delta$-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 $S=\partial_{\chi<0}M$ be the union of all non-torus
boundary components of $M$. When $\alpha,\beta\subset S$ 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
$A\subset M$ with $\partial A=\alpha\cup\beta$, see Scott [50].

More generally, equip $S$ with an arbitrary hyperbolic metric. An _essential
homotopy_ between two geodesic laminations $\lambda_{\pm}$ on $S$ is a map

$H:(\lambda\times[-1,1],\lambda\times\\{-1,1\\})\longrightarrow(M,S)$

where $\lambda$ is a lamination, such that $H$ maps $\lambda\times\\{\pm 1\\}$
homeomorphically onto $\lambda_{\pm}$, and where $H$ is not homotopic rel
$\lambda\times\\{-1,1\\}$ into $\partial M$.

Here is an ‘Annulus Theorem’ for minimal laminations.

######  Proposition 4.5 (An annulus theorem for laminations).

Let $\lambda_{-},\lambda_{+}$ be two minimal geodesic laminations on $S$ that
are either disjoint or equal, and assume that $S(\lambda_{\pm})$ are
incompressible in $M$. If $\lambda_{\pm}$ are essentially homotopic in
$(M,S)$, there is an essential interval bundle $(B,\partial_{H}B)\subset(M,S)$
such that $\lambda_{\pm}$ fill $\partial_{H}B$, and where $\lambda_{\pm}$ are
essentially homotopic through $B$, as in §2.9.

Here, $S(\lambda_{\pm})$ are the subsurfaces with geodesic boundary filled by
$\lambda_{\pm}$, as in §2.8. The assumption that they are incompressible
generalizes the assumption that $\alpha,\beta$ are homotopically essential in
$M$ in the Annulus Theorem.

###### Proof.

Identify $M\setminus\partial_{\chi=0}M$ with the convex core of a
geometrically finite hyperbolic $3$-manifold. Set $S_{\pm}:=S(\lambda_{\pm})$.
Lift the essential homotopy from $\lambda_{-}$ to $\lambda_{+}$ to a homotopy
from lifts $\tilde{\lambda}_{-}\subset\tilde{S}_{-}$ to
$\tilde{\lambda}_{+}\subset\tilde{S}_{+}$ in $\mathbb{H}^{3}$. Under the
homotopy, which has bounded tracks, corresponding leaves of
$\tilde{\lambda}_{\pm}$ have the same endpoints in $\partial\mathbb{H}^{3}$.
The endpoints of $\tilde{\lambda}_{\pm}$ are dense in
$\partial_{\infty}\tilde{S}_{\pm}$, so this means that the subsurfaces
$C_{\pm}\subset S_{\pm}$ constructed in Theorem 4.1 are just
$C_{\pm}=S_{\pm}$. Passing to disjoint or equal resolutions $S_{\pm}^{\prime}$
of $S_{\pm}$ and applying Theorem 2.8 gives an interval bundle $B$ where
$\lambda_{\pm}$ fill $\partial_{H}B=S_{-}^{\prime}\cup S_{+}^{\prime}$.

We claim that $\lambda_{\pm}$ are essentially homotopic through $B$. By Fact
2.16, it suffices to show that if $\sigma$ is the canonical involution of $B$,
as described in §4.5, then $\sigma(\lambda_{\pm})$ is isotopic to
$\lambda_{\mp}$ on $S_{\mp}^{\prime}$. Using the notation of Theorem 4.1,
$\sigma$ lifts to a $\Delta$-equivariant involution $\tilde{\sigma}$ of
$\tilde{B}$ that exchanges $\tilde{S}_{-}^{\prime}$ and
$\tilde{S}_{+}^{\prime}$, where here $\Delta=\Gamma_{-}\cap\Gamma_{+}$. By
equivariance, $\tilde{\sigma}$ extends continuously to the identity on
$\Lambda_{\Delta}$, so $\tilde{\sigma}(\tilde{\lambda}_{-})$ is a lamination
on $\tilde{S}_{+}$ with all the same endpoints at infinity as
$\tilde{\lambda}_{+}$, and hence equals $\tilde{\lambda}_{+}$. The claim
follows. ∎

##  5\. Laminations on the boundary

Suppose that $M$ is a compact, orientable $3$-manifold with hyperbolizable
interior and nonempty boundary $\partial M$. Equip $M$ with an arbitrary
Riemannian metric and lift it to a Riemannian metric on the universal cover
$\tilde{M}$. As in the introduction, a biinfinite path or ray $h$ on
$\partial\tilde{M}$ is called _homoclinic_ if there are points $s^{i},t^{i}$
with $|s^{i}-t^{i}|\to\infty$ such that

$\sup_{i}d_{\tilde{M}}(h(s^{i}),h(t^{i}))<\infty.$

Two rays $h_{+},h_{-}$ on $\partial\tilde{M}$ are called _mutually homoclinic_
if there are parameters $s_{\pm}^{i}\to\infty$ such that

$\sup_{i}d_{\tilde{M}}(h_{+}(s_{+}^{i}),h_{-}(s_{-}^{i}))<\infty.$

Here, a _ray_ is a continuous map from an interval $[a,\infty)$, and a
_biinfinite path_ is a continuous map from $\mathbb{R}$. We will also call
rays and paths on $\partial M$ (mutually) homoclinic if they have lifts that
are (mutually) homoclinic paths on $\partial\tilde{M}$. 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
$\tilde{M}$, 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. (1)

Suppose that $D$ is a properly embedded disc in $M$, and
$h:\mathbb{R}\longrightarrow\partial M$ is a path that covers $\partial
D\subset\partial M$. Then $h$ is homoclinic: indeed, $D$ lifts
homeomorphically to $\tilde{M}$, so $h$ lifts to a path in $\tilde{M}$ with
compact image.

  2. (2)

Suppose that
$\phi:(S^{1}\times[0,1],S^{1}\times\\{0,1\\})\longrightarrow(M,\partial M)$ is
an essential embedded annulus. Then rays covering the two boundary components
of the annulus are mutually homoclinic: indeed, $\phi$ lifts to

$\tilde{\phi}:\mathbb{R}\times[0,1]\longrightarrow\tilde{M},$

and we have
$\sup_{t\in\mathbb{R}}d(\tilde{\phi}(t,0),\tilde{\phi}(t,1))<\infty$, so
restricting to $t\in[0,\infty)$ we get two mutually homoclinic rays in
$\tilde{M}$.

It will be convenient below to work with a particular choice of metric on $M$.

######  Example 5.1 (An explicit metric on $M$).

Let $\partial_{\chi<0}M$ be the union of all components of $\partial M$ that
have negative Euler characteristic, i.e. are not tori. Thurston’s Haken
hyperbolization theorem, see [28], implies that there is a hyperbolic
$3$-manifold $N=\mathbb{H}^{3}/\Gamma$ homeomorphic to the interior of $M$,
where every component of $\partial_{\chi<0}M$ corresponds to a convex
cocompact end of $N$. A torus $T\subset\partial M$, 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 $N_{T}\subset N$ of the associated cusp that is the
quotient of a horoball in $\mathbb{H}^{3}$ by a $\mathbb{Z}^{2}$-action. Then

(3) $M\cong CC(N)\ \setminus\bigcup_{\text{tori }T\subset\partial M}N_{T},$

and we will identify $M$ with the right-hand side everywhere below. Then

  * •

$\tilde{M}\subset\mathbb{H}^{3}$ is obtained from the convex hull
$CH(\Gamma)\subset\mathbb{H}^{3}$ of the limit set of $\Gamma$ by deleting an
equivariant collection of horoballs, and

  * •

the path metric induced on $\partial_{\chi<0}M$ is hyperbolic [51, Proposition
8.5.1], and the path metric induced on every torus $T\subset\partial M$ is
Euclidean.

We now specialize to the case of paths that are _geodesics_ on $\partial M$.
Recall from Example (1) above that one can make homoclinic paths by running
around the boundaries of disks in $\partial M$. The following shows that discs
are essential in such constructions.

######  Fact 5.2.

Suppose that $S\subset\partial M$ is an essential subsurface. Then the
inclusion of any lift $\tilde{S}\subset\partial\tilde{M}$ is a quasi-isometric
embedding into $\tilde{M}$. Moreover if $S$ is incompressible then any pair of
mutually homoclinic infinite rays on $S$ are asymptotic and no biinfinite
geodesic $\gamma$ in $S$ is homoclinic.

###### Proof.

Think of $M$ as embedded in a complete hyperbolic $3$-manifold $N$ as in (3),
write $N=\Gamma\backslash\mathbb{H}^{3}$, and let
$\tilde{M}\subset\mathbb{H}^{3}$ be the preimage of $M$, so that $\tilde{M}$
is obtained from the convex hull $CH(\Gamma)$ be deleting an equivariant
collection of horoballs. Fix a subgroup $\Delta<\Gamma$ that represents the
conjugacy class associated to the image of the fundamental group of $S\subset
M$. To show that

$\tilde{S}\hookrightarrow\partial\tilde{M}$

is a quasi-isometric embedding, it suffices to show that $\Delta$ is
undistorted in $\Gamma$. But since $M$ is geometrically finite and $\Delta$ is
finitely generated, it follows from a result of Thurston (see Proposition 7.1
in [46]) that the group $\Delta$ is geometrically finite, and geometrically
finite subgroups of (say, geometrically finite) hyperbolic $3$-manifold groups
are undistorted, c.f. Corollary 1.6 in [23].

For the ‘moreover’ statement, assume $S$ is incompressible, so that
$\tilde{S}$ is simply connected, and consider a pair of infinite rays

$h^{\pm}:\mathbb{R}^{+}\to\tilde{S}$

that are geodesic for the induced hyperbolic metric and
$t^{\pm}_{n}\to+\infty$ such that
$d_{\tilde{M}}(h^{+}(t_{n}^{+}),h^{-}(t_{n}^{-}))$ is bounded. Since
$\tilde{S}\subset\partial\tilde{M}$ is a quasi-isometric embedding,
$d_{\tilde{S}}(h^{+}(t_{n}^{+}),h^{-}(t_{n}^{-}))$ is also bounded. Since
$\tilde{S}$ 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 $\tilde{S}$ can not be homoclinic. Taking $h^{+}\neq h^{-}$,
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 [28] and more generally [35], says that we can
choose the hyperbolic manifold $N$ so that $\partial
CC(N)\cong\partial_{\chi<0}M$ is totally geodesic. Hence, the preimage of
$\partial_{\chi<0}M$ in $\tilde{M}\subset\mathbb{H}^{3}$ is a collection of
hyperbolic planes. Any geodesic ray on $\partial_{\chi<0}M$ then lifts to a
geodesic in $\mathbb{H}^{3}$, and two geodesic rays on $\partial_{\chi<0}M$
are mutually homoclinic if and only if their geodesic lifts are asymptotic in
$\mathbb{H}^{3}$, which implies that they were asymptotic on
$\partial_{\chi<0}M$.

###  5.1. Alternate definitions of homoclinic

††margin:  Added this subsection

Above, we defined a path

$h:I\longrightarrow\partial\tilde{M}$

to be homoclinic if there is are $s^{i},t^{i}\in I$ with
$|s^{i}-t^{i}|\to\infty$ such that

$\sup_{i}d_{\tilde{M}}(h(s^{i}),h(t^{i}))<\infty.$

Some other papers use slight variants of this definition. For example, the
definition of _(faiblement) homoclinique_ in Otal’s thesis [49] is almost the
same as what is written above, except that distances are computed _in the
intrinsic metric on $\partial\tilde{M}$_ instead of in $\tilde{M}$. This is
equivalent to our definition, though: the nonobvious direction follows from
Fact 5.2, which says that boundary components of $\tilde{M}$ quasi-
isometrically embed in $\tilde{M}$. And in the definition of _homoclinique_ in
Lecuire’s earlier work [35], distances are computed not in $\tilde{M}$, but
within $\mathbb{H}^{3}$, with respect to a given identification of $\tilde{M}$
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
$\tilde{M}\hookrightarrow\mathbb{H}^{3}$ is not a quasi-isometric embedding,
but the following lemma says that $d_{\mathbb{H}^{3}}$ is bounded if and only
if $d_{\tilde{M}}$ is bounded, so Lecuire’s earlier definition is equivalent
to ours.

######  Lemma 5.3.

Whenever $x,y\in\tilde{M}$, we have

$d_{\mathbb{H}^{3}}(x,y)\leq d_{\tilde{M}}(x,y)\leq
e^{d_{\mathbb{H}^{3}}(x,y)/2}d_{\mathbb{H}^{3}}(x,y).$

###### Proof.

Set $N:=\Gamma\backslash\mathbb{H}^{3}$, so that $\tilde{M}$ is obtained from
the convex hull $CH:=CH(\Lambda(\Gamma))$ of the limit set of $\Gamma$ by
deleting horoball neighborhoods around all rank two cusps. Take a
$\mathbb{H}^{3}$-geodesic $\gamma$ from $x$ to $y$. Then $\gamma$ lies inside
$CH$, and it can only penetrate the deleted horoball neighborhoods to a depth
of $d(x,y)/2$. Now, whenever $B\supset B^{\prime}$ are horoballs in
$\mathbb{H}^{3}$ such that $d_{\mathbb{H}^{3}}(\partial B,B^{\prime})\leq
d(x,y)/2$, the closest point projection

$\pi:B\setminus B^{\prime}\longrightarrow\partial B$

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 $B^{\prime}$ as
the height $e^{d(x,y)/2}$ horoball, and then the claim is obvious.) So, the
parts of $\gamma$ above that penetrate the deleted horoballs can be projected
back into $\partial\tilde{M}$, 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
$\partial\tilde{M}\hookrightarrow\mathbb{H}^{3}$ as above, then a simple
geodesic $h:I\longrightarrow\partial\tilde{M}$ is called _homoclinic_ if when
we subdivide $h$ into two rays $h_{\pm}$, these rays limit onto subsets
$A_{\pm}\subset\mathbb{H}^{3}\cap\partial\mathbb{H}^{3}$ such that $A_{+}\cap
A_{-}\neq\emptyset$. This definition is stronger than all the ones mentioned
above: if $A_{+}\cap A_{-}$ contains a point on $\partial\tilde{M}$, rather
than at infinity, then the definition of homoclinic above is obviously
satisfied. Otherwise, $h_{\pm}$ have to have a common accumulation point in
$\partial\mathbb{H}^{3}$, which corresponds to an end $\xi$ of $\tilde{M}$,
and one can use the treelike structure of the universal cover $\tilde{M}$ of
the handlebody $M$ to say that $h_{\pm}$ have to both intersect a sequence of
meridians $(m_{i})$ on $\tilde{M}$ that cut off smaller and smaller
neighborhoods of $\xi$. The times $t^{i}_{p}m$ when $h_{\pm}$ intersects
$m_{i}$ then work in the definition of homoclinic above. In fact, Casson’s
definition is strictly stronger. For instance, if $h_{\pm}$ both spiral around
disjoint simple closed curves $\gamma_{\pm}\subset\partial\tilde{M}$, 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 $\partial M$, 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 $\partial M$, and let
$\gamma\subset\partial M$ be a simple geodesic ray or a simple biinfinite
geodesic. An _$m$ -wave_ is a segment $\beta\subset\gamma$ that has endpoints
on $m$, and is homotopic rel endpoints in $M$ to an arc of $m$. If $\gamma$
has no $m$-waves, and every infinite length segment of $\gamma$ intersects
$m$, then we say that $\gamma$ is in _tight position_ with respect to $m$.

Waves and tight position were discussed previously in [31, 35], for instance.
Note that in our definition, an $m$-wave $\beta$ 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 $\gamma$ 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 $\gamma$ can never be in tight position with respect
to another meridian $m$: taking discs with boundaries $\gamma$ and $m$ that
are transverse and intersect minimally, any arc of intersection of these disks
terminates in a pair of intersection points of $\gamma$ and $m$ that bound a
$m$-wave of $\gamma$.

More generally, we have the following fact.

######  Fact 5.5 (Tight position $\implies$ $\mathbb{H}^{3}$ quasi-geodesic).

Let $\gamma$ be a simple geodesic ray or biinfinite geodesic on $\partial M$.
If $\gamma$ is in tight position with respect to some meridian $m$ then any
lift $\tilde{\gamma}\subset\partial\tilde{M}$ of $\gamma$ is a quasigeodesic
in $\mathbb{H}^{3}$. In particular, $\gamma$ is an $\tilde{M}$-quasigeodesic,
and is not homoclinic.

###### Proof.

Intersecting with $m$ breaks $\gamma$ into a union of finite arcs. By
simplicity of $\gamma$, these arcs fall into only finitely many homotopy
classes rel $m$, and there is a universal upper bound $L=L(\gamma,m)$ on their
lengths. Let $D$ be a disc with boundary $m$ and let $\tilde{D}$ be the entire
preimage of $D$ in $\tilde{M}$. Tightness means that the path $\tilde{\gamma}$
intersects infinitely many components of $\tilde{D}$, and intersects no single
component more than once.

In the notation of Example 5.1, we have that $\tilde{M}\subset\mathbb{H}^{3}$
is obtained from $CH(\Gamma)$ by deleting an equivariant collection of
horoballs. Each component of $\tilde{D}$ separates $CH(\Gamma)$, so if
$\gamma^{\prime}$ is a segment of $\gamma$, any geodesic in $\mathbb{H}^{3}$
joining the endpoints of $\gamma^{\prime}$ must intersect each of the discs
that $\gamma^{\prime}$ intersects. Hence, if $\epsilon>0$ is the minimum
distance between any two components of $\tilde{D}$, then $\tilde{\gamma}$ is a
$(L/\epsilon,L)$-quasigeodesic. ∎

Figure 9. Surgering $A$ along $\alpha$ gives a meridian $c$.

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 $\lambda$ is a geodesic lamination on $\partial M$ and
$m=\sqcup_{i=1}^{n}m_{i}$ is a geodesic meridian multi-curve on $\partial M$,
and $\beta$ is an $m$-wave in $\lambda$ whose interior is disjoint from $m$.
Then the pair of points $\partial\beta$ separates some component $m_{i}$ of
$m$ into two arcs $m_{i}^{1},m_{i}^{2}$, both of which are homotopic to
$\beta$ rel endpoints in $M$. We perform a _$\lambda$ -surgery_ on $m$ by
replacing $m_{i}^{1}$ (say) with $\beta$, thus constructing a new multicurve
$m^{\prime}:=(\beta\cup m_{i}^{2})\sqcup\bigsqcup_{j\neq i}m_{j}$.

This notion of surgery appears in many other references, e.g. [3, 11, 31, 35].
We summarize its elementary properties here:

######  Fact 5.7.

Suppose that $\lambda$ is a geodesic lamination.

  1. (1)

If $m$ is a meridian and $\lambda$ has an $m$-wave, it also has an $m$-wave
whose interior is disjoint from $m$, so a $\lambda$-surgery can be performed.

  2. (2)

Any curve $m^{\prime}$ obtained by $\lambda$-surgery on a meridian $m$ as
above is a meridian.

  3. (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 $\lambda$-surgery on a component of $m$ is another cut system.

###### Proof.

For (1), suppose that $\lambda$ has an $m$-wave $\beta$. Let $\tilde{m}$ be
the entire preimage of $m$ in the cover $\partial\tilde{M}$, and lift $\beta$
to an arc $\tilde{\beta}$ starting and ending on some fixed component
$m_{0}\subset\tilde{m}$. Since each component of $\tilde{m}$ separates
$\partial\tilde{M}$, there is some “outermost” subarc $\tilde{\beta}^{\prime}$
that has endpoints on the same component of $\tilde{m}$, and that has interior
disjoint from $\tilde{m}$. This $\tilde{\beta}^{\prime}$ projects to an
$m$-wave of $\lambda$ whose interior is disjoint from $m$.

For (2), note that if $m^{\prime}:=\beta\cup m_{2}$ is obtained by
$\lambda$-surgery on $m$, as above, then $m,m^{\prime}$ are homotopic in $M$,
and hence $m^{\prime}$ is nullhomotopic. Also, if $m^{\prime}$ is inessential
in $\partial M$, then $\beta$ is homotopic on $\partial M$ to $m_{2}$,
implying that $\lambda$ and $m$ were not in minimal position, a contradiction
since they are both geodesic. Hence $m^{\prime}$ is a meridian.

For (3), consider an $m$-wave in $\lambda$ whose interior is disjoint from $m$
and say that $\partial\beta\subset m_{1}$. Then $\partial\beta$ separates
$m_{1}$ into two arcs $m_{1}^{1},m_{1}^{2}$. It is not difficult to see that
either $(\beta\cup m_{1}^{1})\sqcup\bigsqcup_{j\neq 1}m_{j}$ or $(\beta\cup
m_{1}^{2})\sqcup\bigsqcup_{j\neq 1}m_{j}$ 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 $\lambda$ is a geodesic lamination on $S=\partial M$ and $m$ is a
meridian multi-curve. Then either

  1. (1)

there exists a finite sequence of $\lambda$-surgeries on $m$ that terminates
in some meridian multi-curve $m^{\prime}$ where $\lambda$ has no
$m^{\prime}$-waves,

  2. (2)

$S(\lambda)$ contains a sequence of meridians $(\gamma_{i})$ such that
$i(\lambda,\gamma_{i})\to 0$, with respect to every transverse measure on
$\lambda$.

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

###### Proof of Lemma 5.8.

The two cases depend on whether $\lambda$ 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 $\lambda$, then a
finite sequence of $\lambda$-surgeries converts $m$ into a multi-curve
$m^{\prime}$ such that $\lambda$ has no $m^{\prime}$-waves, as each surgery
decreases the number of waves by at least one. If there are infinitely many
homotopy classes of $m$-waves in $\lambda$, then we can choose a sequence of
parameterized $m$-waves $\alpha_{i}:[0,1]\longrightarrow\mathbb{R}$ such that

  1. (1)

the two sequences of endpoints $(\alpha_{i}(0))$ and $(\alpha_{i}(1))$ both
converge, and if either sequence converges into a simple closed curve
$\gamma\subset\lambda$, then it approaches $\gamma$ from only one side,

  2. (2)

no $\alpha_{i}$ and $\alpha_{j}$ are homotopic keeping their endpoints on $m$,
for $i\neq j$.

To construct the desired sequence of meridians, let $\beta^{0}_{i}$ be the
shortest geodesic on $S$ from $\alpha_{i}(0)$ to $\alpha_{i+1}(0)$, and define
$\beta^{1}_{i}$ similarly. For large $i$, the union
$\beta^{0}_{i}\cup\alpha_{i}\cup\beta^{1}_{i}\cup\alpha_{i+1}$ is an essential
closed curve in $S(\lambda)$ that is nullhomotopic in $M$. It may not be
simple, since $\beta^{0}_{i}$ and $\beta^{1}_{i}$ may overlap, but it has at
most one self intersection. So by the Loop Theorem [24], one of the three
simple closed curves obtained by surgery on it is a meridian $\gamma_{i}$.

Now, the fact that the endpoints can approach a simple closed curve in
$\lambda$ only from one side implies that for large $i$, the curves
$\gamma_{i}$ do not intersect any simple closed curve contained in $\lambda$.
Since $\gamma_{i}$ only intersects $\lambda$ along the arcs $\beta^{0}_{i}$
and $\beta^{1}_{i}$, whose hyperbolic lengths converge to zero, it follows
that $i(\gamma_{i},\lambda)\to 0$ for any transverse measure on $\lambda$. ∎

Here is an important application of Lemma 5.8.

######  Lemma 5.9 (Quasigeodesic or a sequence of meridians).

Suppose $\lambda\subset\partial_{\chi<0}M$ is a minimal geodesic lamination
and that $\partial S(\lambda)$ is incompressible in $M$. Let $h\subset
S(\lambda)$ be a simple geodesic ray or biinfinite geodesic that is disjoint
from $\lambda$ or contained in $\lambda$. Then either

  1. (1)

any lift $\tilde{h}\subset\partial\tilde{M}$ of $h$ is a quasi-geodesic in
$\tilde{M}$.

  2. (2)

$S(\lambda)$ contains a sequence of meridians $(\gamma_{i})$ such that
$i(\lambda,\gamma_{i})\to 0$, with respect to every transverse measure on
$\lambda$.

In particular, if $h$ is homoclinic, then $\lambda$ 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 $\lambda$-surgeries until we obtain a new
cut system $m$ such that $\lambda\cup h$ has no $m$-waves. If $m$ intersects
$\lambda$, then $\lambda\cup h$ is in tight position with respect to $m$, so
(1) follows from Fact 5.5. Therefore, we can assume $m$ does not intersect
$\lambda$. Up to isotopy, we can also assume that $S(\lambda)$ does not
intersect $m$. Since $\partial S(\lambda)$ is assumed to be a collection of
incompressible curves, it follows that $S(\lambda)$ 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 $\lambda\subset\partial_{\chi<0}M$ is an
_intrinsic limit of meridians_ if there is a transverse measure666It 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. on $\lambda$ and
a sequence of meridians $(\gamma_{i})$ contained in $S(\lambda)$ such that
$\gamma_{i}\to\lambda$ in ${\mathcal{P}\mathcal{M}\mathcal{L}}(S(\lambda))$.

Using Lemma 5.9, we can prove the following proposition, which gives several
equivalent characterizations of intrinsic limits.

######  Proposition 5.11 (Intrinsic limits).

Suppose $\lambda\subset S=\partial M$ is a minimal geodesic lamination and
$\partial S(\lambda)$ is incompressible. The following are equivalent:

  1. (1)

$\lambda$ is an intrinsic limit of meridians,

  2. (2)

given (some/any) transverse measure on $\lambda$, there is a sequence of
meridians $(\gamma_{i})$ in $S(\lambda)$ such that $i(\gamma_{i},\lambda)\to
0,$

  3. (3)

there is a biinfinite homoclinic geodesic in $S(\lambda)$ that is either a
leaf of $\lambda$, or is disjoint from $\lambda$,

  4. (4)

given any transverse measure on $\lambda$, there is a sequence of essential
(possibly non-simple) closed curves $(\gamma_{i})$ in $S(\lambda)$ such that
each $\gamma_{i}$ is nullhomotopic in $M$, and $i(\gamma_{i},\lambda)\to 0.$

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

###### Proof.

$(2)\implies(1)$. Fix some transverse measure on $\lambda$. By $(2)$,

$i(\gamma_{i},\lambda)/\operatorname{length}(\gamma_{i})\to 0,$

so after passing to a subsequence we can assume that $(\gamma_{i})$ converges
to a measured lamination $\mu$ in $S(\lambda)$ that does not intersect
$\lambda$ transversely. As $\lambda$ fills $S(\lambda)$, $\mu$ is supported on
$\lambda$.

$(1)\implies(3)$. After passing to a subsequence, we can assume that
$(\gamma_{i})$ converges in the Hausdorff topology to some lamination, which
must then be an extension of $\lambda$ by finitely many leaves. $(3)$ follows
from an unpublished criterion of Casson, see Lecuire [35, Théorème B.1] for a
proof, that states that any Hausdorff limit of meridians has a homoclinic
leaf.

$(3)\implies(2)$. This is an immediate corollary of Lemma 5.9.

$(4)\iff(2)$. The direction $\Leftarrow$ is immediate, so suppose
$(\gamma_{i})$ is a sequence of essential closed curves in $S(\lambda)$ that
are nullhomotopic in $H$ and $i(\lambda,\gamma_{i})\to 0$. By Stallings’
version of the Loop Theorem, for each $i$ there is a meridian
$\gamma_{i}^{\prime}$ that is obtained from $\gamma_{i}$ by surgery at the
self intersection points. Such surgeries can only decrease the intersection
number with $\lambda$, so $(2)$ follows. ∎

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

######  Lemma 5.12 (Intrinsic limits of annuli).

Suppose $\lambda\subset\partial_{\chi<0}M$ is a minimal lamination such that
$S(\lambda)$ is compressible but $\partial S(\lambda)$ is incompressible, and
that there is a sequence $(A_{i})$ of essential embedded annuli in
$(M,S(\lambda))$ with $i(\partial A_{i},\lambda)\to 0$. Then $\lambda$ is an
intrinsic limit of meridians.

###### Proof.

Pick a meridian $m\subset S(\lambda)$. For each $i$, let
$T_{i}:M\longrightarrow M$ be the Dehn twist along the annulus $A_{i}$. Then
for any sequence $n_{i}\in\mathbb{Z}$, the curves $T_{i}^{n_{i}}(m)$ are
meridians, and if $n_{i}$ grows sufficiently fast, then

$i(T_{i}^{n_{i}}(m),\lambda)/\operatorname{length}(T_{i}^{n_{i}}(m))\to 0.$

Hence, after passing to a subsequence $T_{i}^{n_{i}}(m)$ converges to a
lamination $\lambda^{\prime}$ supported in $S(\lambda)$ with zero intersection
number with $\lambda$, implying $\lambda^{\prime}$ and $\lambda$ have the same
support, so $\lambda$ is an intrinsic limit of meridians. ∎

##  6\. 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
$\partial_{\chi<0}M$ with an arbitrary hyperbolic metric. Let $h_{\pm}$ be two
disjoint, mutually homoclinic simple geodesic rays on $\partial_{\chi<0}M$
that accumulate onto (possibly equal) minimal laminations $\lambda_{\pm}$, and
that the multicurve $\partial S(\lambda_{\pm})$ is incompressible in $M$. Then
one of the following holds:

  1. (1)

one of $\lambda_{+}$ or $\lambda_{-}$ is an intrinsic limit of meridians,

  2. (2)

$h_{+}$ and $h_{-}$ are asymptotic on $\partial_{\chi<0}M$, and either

    1. (a)

any two mutually homoclinic lifts $\tilde{h}_{\pm}$ to $\partial\tilde{M}$ are
asymptotic on $\partial\tilde{M}$, or

    2. (b)

$\lambda:=\lambda_{-}=\lambda_{+}$ is a simple closed curve that is homotopic
in $M$ to a nontrivial power $\gamma^{n},\ n>1$ of some closed curve $\gamma$
in $M$,

  3. (3)

$h_{\pm}$ are not asymptotic on $\partial_{\chi<0}M$, and there is an
essential (possibly nontrivial) interval bundle $B\subset M$ such that
$\lambda_{\pm}$ each fill a component of $\partial_{H}B$, and $\lambda_{\pm}$
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 $\lambda_{-},\lambda_{+}$ on $M$ and let $h_{\pm}$ spiral onto
$m_{\pm}$. One can also produce similar examples by letting $\lambda_{\pm}$ be
arbitrary laminations in disjoint subsurfaces $S(\lambda_{\pm})$ that are
spheres with at least 4 boundary components, all of which are compressible in
$M$, and letting $h_{\pm}$ accumulate onto $\lambda_{\pm}$. For (2) (a), take
$\lambda$ to be any simple closed curve on $\partial M$ that is essential in
$M$ but has no nontrivial roots in $\pi_{1}M$, and let $h_{\pm}$ spiral around
$\lambda$ in the same direction. We’ll discuss 2 (b) in Remark 6.2. For (3),
write $M=S\times[-1,1]$ where $S$ is a surface with boundary, let $\lambda$ be
a lamination on $S$, and let $h_{\pm}$ be corresponding leaves of
$\lambda_{\pm}:=\lambda\times\\{\pm 1\\}$. Then
$\tilde{M}\cong\tilde{S}\times[-1,1]$, so there are lifts of $h_{\pm}$ 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 $S(\lambda_{-})$ is incompressible,
say, while $\lambda_{+}$ is an intrinsic limit of meridians. For instance,
suppose $C$ is a compression body with connected, genus-at-least-two interior
boundary $\partial_{-}C$, and exterior boundary $\partial_{+}C$. Let
$f:C\longrightarrow C$ be a homeomorphism such that $f|_{\partial_{+}C}$ and
$f|_{\partial_{-}C}$ are both pseudo-Anosov, with attracting laminations
$\lambda_{+},\lambda_{-}$, respectively. We expect that there are rays
$\ell_{\pm}\subset\lambda_{\pm}$ that are mutually homoclinic. But
$\lambda_{+}$ is an intrinsic limit of meridians, while $S(\lambda_{-})$ is
incompressible.

The assumption that $\partial S(\lambda_{\pm})$ 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 $\gamma$. Let $T$ be the component of $S\setminus\gamma$ that
is a punctured genus $2$ surface. Then there are distinct minimal geodesic
laminations $\lambda,\lambda^{\prime}\subset T$, 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
$\lambda,\lambda^{\prime}$ are mutually homoclinic rays that accumulate onto
$\lambda,\lambda^{\prime}$, respectively, but none of (2)–(3) hold. One could
write down a version of Theorem 6.1 that omits the assumption that $\partial
S(\lambda_{\pm})$ is incompressible, but the conclusion would be relative to
capping off $S(\lambda_{\pm})$, and the statement would be more complicated.

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

######  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 $\partial\tilde{M}$ 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 $\lambda$, the $(2,1)$-curve on the solid torus. Then $\lambda$ is
homotopic to the square of the core curve of the solid torus. Although the two
ends of $h$ are asymptotic on $\partial M$, any lift $\tilde{h}$ in
$\partial\tilde{M}$ will have ends that are mutually homoclinic, but
nonasymptotic. On the right, we have drawn the preimage $\tilde{\lambda}$ of
$\lambda$, and two lifts $\tilde{h}_{1},\tilde{h}_{2}$ of $h$. Note that one
end of $\tilde{h}_{1}$ is asymptotic to an end of $\tilde{h}_{2}$.

When $M$ is a compression body, Casson-Gordon prove in [13, Theorem 4.1] that
any simple closed curve $\lambda\subset\partial M$ that has a nontrivial root
in $\pi_{1}M$ 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 $\lambda$ 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 $\partial_{\chi<0}M$.

######  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 $S\subset\partial_{\chi<0}M$, that $h_{\pm}$
are the two ends of $h$, that $h_{\pm}$ limit onto $\lambda_{\pm}$, and that
$\partial S(\lambda_{\pm})$ is incompressible in $M$.

Then one of (1)–(3) in Theorem 6.1 holds. Moreover, in case (2), if
$\lambda:=\lambda_{\pm}$ is not an intrinsic limit of meridians then either

  1. (i)

after reparametrizing $h$, there is some $s$ such that $h(-s)$ and $h(s)$ are
joined by a geodesic segment $c$ with $h\cap int(c)=\emptyset$, such that $c$,
$h|_{(-\infty,-s)}$ and $h|_{[s,\infty)}$ bound an embedded geodesic triangle
$\Delta\subset S$ with one ideal vertex, and $c\cup h([-s,s])$ is a meridian
in $M$, or

  2. (ii)

$\lambda$ is a simple closed curve on $S$, the two ends of $h$ spiral around
$\lambda$ in the same direction, and any neighborhood of the union
$h\cup\lambda\subset S$ contains a meridian.

And in case (3), we can choose the interval bundle $B$ such that $h$ contains
a subarc $\alpha\subset h$ that is a compression arc for $B$.

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

###### Proof.

Let $\tilde{h}$ be a homoclinic lift of $h$ on $\partial\tilde{M}$. By Lemma
5.9, either one of $\lambda_{\pm}$ is an intrinsic limit of meridians in $M$,
in which case we’re in case (1) and are done, or both ends of $\tilde{h}$ are
quasi-geodesic in $\tilde{M}$. Since $\tilde{h}$ 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 $\lambda:=\lambda_{\pm}$.
Assume first that $\lambda:=\lambda_{\pm}$ is a simple closed curve. Since the
two ends of $h$ are asymptotic, they spiral around $\lambda$ in the same
direction. Let $U$ be a neighborhood of $h\cup\lambda$ on
$\partial_{\chi<0}M$. 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 $\lambda$ 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
$\mathbb{H}^{2}\longrightarrow S$ and lift $h$ to a $\hat{h}$ in
$\mathbb{H}^{2}$, and let

$\xi=\lim_{t\to+\infty}\hat{h}(t)\in\partial_{\infty}\mathbb{H}^{2}.$

Since the two ends of $h$ are asymptotic on $S$, there is a deck
transformation $\gamma:\mathbb{H}^{2}\longrightarrow\mathbb{H}^{2}$ such that
$\xi=\lim_{t\to-\infty}\gamma\circ\hat{h}(t).$ It follows that if we use a
particular arc-length parametrization of $h$, we may assume that for each
$t\in\mathbb{R}$, the points $\hat{h}(t),\gamma\circ\hat{h}(-t)$ lie on a
common horocycle tangent to $\xi$. Fix some large $s$ such that the geodesic
segment $\hat{c}$ joining $\hat{h}(s)$ and $\gamma\circ\hat{h}(-s)$ is shorter
than the injectivity radius of $S$, and therefore projects to a simple
geodesic segment $c$ in $S$.

Let $\hat{\Delta}\subset\mathbb{H}^{2}$ be the triangle bounded by $\hat{c}$
and the two rays $\hat{h}([s,\infty))$ and $\gamma\circ\hat{h}((-\infty,s])$.
Let $g:\mathbb{H}^{2}\longrightarrow\mathbb{H}^{2}$ be a deck transformation.
We claim that $g\circ\hat{h}(\mathbb{R})\cap int(\hat{\Delta})=\emptyset$. If
not, then since $\hat{\Delta}$ has geodesic sides, two of which are disjoint
from $g\circ\hat{h}$, it follows that one of the two endpoints of
$g\circ\hat{h}$ is $\xi$. If it’s the positive endpoint, then $g$ fixes $\xi$,
and the axis of $g$ projects to a (simple) closed curve on $S$, around which
the two ends of $h$ spiral, contradicting that $\lambda$ isn’t a simple closed
curve. If the negative endpoint of $g\circ\hat{h}$ is $\xi$, then
$g\circ\gamma^{-1}$ fixes $\xi$ and we get a similar contradiction.

Next, we claim that we have $g(int(\hat{\Delta}))\cap
int(\hat{\Delta})=\emptyset$ as long as $g\neq id$. Suppose that for some
$g\neq id$ the intersection is nonempty. Then $g(\xi)\neq\xi$, since otherwise
we have a contradiction as in the previous paragraph. The previous paragraph
implies that the sides of the triangles $g(\hat{\Delta}),\hat{\Delta}$ that
are lifts of rays of $h$ do not intersect the interior of the other triangle.
So, the only way the interiors of $g(\hat{\Delta}),\hat{\Delta}$ can intersect
is if $\hat{c}$ and $g(\hat{c})$ intersect. However, this does not happen
since we chose $s$ large enough so that $\hat{c}$ projects to a simple
geodesic segment in $S$.

The previous two paragraphs imply that $\hat{\Delta}$ projects to an embedded
geodesic triangle $\Delta$ 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 $g\circ\hat{h}(\mathbb{R})\cap
int(\hat{\Delta})=\emptyset$ for any $g\neq id$, they are disjoint. It follows
that $c\cup h([-s,s])$ is an essential simple closed curve on $S$. Note that
$c\cup h([-s,s])$ is an essential simple closed curve on $S$, since it is the