# A lower bound on volumes of
end-periodic mapping tori

Elizabeth Field, Autumn Kent,
Christopher Leininger, and Marissa Loving

###### Abstract.

We provide a lower bound on the volume of the compactified mapping torus of a
strongly irreducible end-periodic homeomorphism $f:S\to S$. This result,
together with work of Field, Kim, Leininger, and Loving [FKLL23], shows that
the volume of $\overline{M}_{f}$ is comparable to the translation length of
$f$ on a connected component of the pants graph $\mathcal{P}(S)$, extending
work of Brock [Bro03b] in the finite-type setting on volumes of mapping tori
of pseudo-Anosov homeomorphisms.

##  1\. Introduction

A central theme in the post-geometrization study of $3$-manifolds is to
clarify the relationship between geometric and topological features of a
manifold. Fibered hyperbolic $3$-manifolds provide a particularly rich class
of examples in this vein, as their topology is completely determined by the
monodromy homeomorphism $f\colon S\to S$ of their fiber surface $S$, which
realizes the manifold as a mapping torus $M_{f}$. When $S$ is of finite type,
the isotopy class of $f$ is an element of the mapping class group of $S$, and
the abundant collection of actions of this group provide a wealth of
information about the geometry of $M_{f}$. The current paper is motivated by
such a connection due to Brock [Bro03b], who showed that the hyperbolic volume
of $M_{f}$ is comparable to the translation length $\tau(f)$ of $f$ acting on
the pants graph $\mathcal{P}(S)$. More precisely, he shows that there are
constants $K_{1}$ and $K_{2}$, depending only on the topology of $S$, such
that

(1.1) $K_{2}\tau(f)\leq{\rm Vol}(M_{f})\leq K_{1}\tau(f).$

When $S$ has infinite type and $f\colon S\to S$ is a strongly irreducible end-
periodic homeomorphism, earlier work of Field, Kim, Leininger, and Loving
[FKLL23] provides an analogous upper bound on the hyperbolic volume of the
compactified mapping torus with its totally geodesic boundary structure.
Namely,

${\rm Vol}(\overline{M}_{f})\leq C_{1}\tau(f)$

for a controlled constant $C_{1}$. We complete the analogy with (1.1) here by
establishing the following lower bound.

######  Theorem 1.1.

For any surface $S$ with finitely many ends, each accumulated by genus, and
any strongly irreducible end-periodic homeomorphism $f\colon S\to S$, we have

$C_{2}\tau(f)\leq{\rm Vol}(\overline{M}_{f}),$

where the constant $C_{2}$ depends only on the capacity of $f$.

The capacity of $f$ is a pair of numbers that describes the topology of a
subsurface of $S$ where $f$ is “interesting” (i.e. where $f$ does not simply
act by translation)—see Section 2.3. Since the surface $S$ has infinite
topological type, the dependence of $C_{2}$ on the capacity of $f$ serves as a
substitute for the dependence of Brock’s $K_{2}$ on the topology of the
finite-type fiber.

######  Remark.

We note that this relationship between volumes of hyperbolic 3-manifolds and
distance in the pants graph has also been explored in a different setting by
Cremaschi–Rodríguez-Migueles–Yarmola [CRMY22] who give upper and lower bounds
analogous to those of Brock.

There are two pieces of data naturally associated to any $f$-invariant
component $\Omega\subset\mathcal{P}(S)$:

  1. (1)

the translation length of $f$ on $\Omega$, denoted $\tau_{\Omega}(f)$, and

  2. (2)

an induced pants decomposition $P_{\Omega}$ of $\partial\overline{M}_{f}$.

See Section 7 for a detailed description. The following theorem provides a
component that (coarsely) optimizes both of these, and ties the action of $f$
on $\mathcal{P}(S)$ to a bounded length pants decomposition of
$\partial\overline{M}_{f}$.

######  Theorem 1.2.

Given $f\colon S\to S$, a strongly irreducible end-periodic homeomorphism,
there is a component $\Omega\subset\mathcal{P}(S)$ and $E>0$ (depending on the
capacity), so that each curve in $P_{\Omega}\subset\partial\overline{M}_{f}$
has length at most $E$, and so that $\tau_{\Omega}(f)\leq E\tau(f)$.

We expect this theorem to be more generally useful in future analysis of the
hyperbolic geometry surrounding depth-one foliations.

### Historical notes and future directions

End-periodic homeomorphisms are an important class of homeomorphisms of
infinite-type surfaces, due in large part to their connection with depth-one
foliations of 3-manifolds. Indeed, after collapsing certain trivial product
foliation pieces, a co-oriented, depth-one foliation of a $3$-manifold is
obtained by gluing together finitely many compactified mapping tori of end-
periodic homeomorphisms. Such foliations (and more generally finite-depth
foliations) were studied in detail by Cantwell and Conlon [CC81], and arise in
Gabai’s analysis [Gab83, Gab83, Gab87] of the Thurston norm [Thu86b]. In
[Thu86b], Thurston also observed that depth-one foliations occur naturally as
limits of foliations by fibers in sequences of cohomology classes limiting to
the boundary of the cone on a fibered face of the Thurston norm ball.

In unpublished work, Handel and Miller began a systematic study of end-
periodic homeomorphisms using laminations in the spirit of the modern
interpretation of Nielsen’s approach to the Nielsen–Thurston Classification
(see Gilman [Gil81], Miller [Mil82], Handel–Thurston [HT85], and
Casson–Bleiler [CB88]). Some aspects of this work were described and developed
by Fenley in [Fen89, Fen97], and more recently expanded upon by Cantwell,
Conlon, and Fenley in [CCF21]. The analogy between strongly irreducible end-
periodic homeomorphisms and pseudo-Anosov homeomorphisms was further
strengthened by work of Patel–Taylor showing that many end-periodic
homeomorphisms admit loxodromic actions on various arc and curve graphs of
infinite-type surfaces [PT].

Recent work of Landry, Minsky, and Taylor [LMT] further studies the behavior
of Thurston’s depth-one foliations [Thu86b] arising from the boundaries of the
fibered faces. In particular, using how the lifts of a first return map act on
the boundary circle of a depth one leaf lifted to the universal cover, they
relate the invariant laminations for end-periodic homeomorphisms to the
laminations of the pseudo-Anosov flow associated to the fibered face.
Moreover, using veering triangulations [Ago11, Gué16], they show that any
compactified mapping torus appears in the boundary of some fibered face of
some fibered $3$–manifold.

Fenley [Fen92] provided the first connection between the hyperbolic geometry
of a $3$-manifold and its depth-one foliations, proving that when the end-
periodic monodromies are irreducible, the depth-one leaves admit
Cannon–Thurston maps from the compactified universal covers
$\overline{\mathbb{H}}^{2}\to\overline{\mathbb{H}}^{3}$. This is an analogue
of Cannon and Thurston’s seminal work in the finite-type case (circulated as a
preprint for decades before appearing in [CT07]). Unlike Cannon and Thurston’s
map, Fenley’s boundary map is not surjective, but rather surjects the limit
set of the compactified mapping torus, which is a Sierpinski carpet. In the
course of his arguments, Fenley provides a quasi-isometric comparison between
the hyperbolic metric and a (semi)-metric defined by the foliation, which also
parallels Cannon and Thurston’s approach.

The comparison between the hyperbolic metric and the metric defined by the
fibration, as studied by Cannon and Thurston, was greatly elaborated on by
Minsky [Min93] to provide uniform estimates depending on the injectivity
radius of the $3$-manifold and the genus of the fiber. Building on this, and
the deep machinery developed by Masur–Minsky [MM99, MM00], Minsky [Min10] and
Brock–Canary–Minsky [BCM12] constructed combinatorial, uniformly biLipschitz
models for fibered hyperbolic $3$-manifolds.

Brock’s volume estimates [Bro03b] above were used to prove his analogous
volume estimates in terms of the Weil–Petersson translation length on
Teichmüller space, but with less control over the constants. A more direct
proof of the Weil–Petersson upper bound, with explicit constants, was proved
by Brock–Bromberg [BB16] and Kojima–McShane [KM18] using renormalized volume
techniques.

The techniques developed here, and in [FKLL23], combined with forthcoming work
of Bromberg, Kent, and Minsky [BKM], provides the framework to prove volume
estimates for closed hyperbolic $3$-manifolds with depth-one foliations. One
might ultimately hope for a uniform biLipschitz model, but the tools needed to
guarantee one seem considerably more difficult. First steps in this direction
are taken by Whitfield in [Whi], where she extends a result of Minsky [Min00,
Theorem B] to the infinite-type setting to produce short curves whose lengths
are bounded in terms of subsurface projections. From a Teichmüller-theoretic
perspective, the fact that end-periodic homeomorphisms can be made to act
isometrically near the ends suggests the possibility of an action on a
Teichmüller space with finite translation length, and it is natural to wonder
on the relation of such a length to the volume.

Finally, some important motivation for this work comes from the study of big
mapping class groups. In particular, [AIM, Problem 1.7] asks for a
characterization of big mapping classes whose mapping tori admit complete
hyperbolic metrics. One hope is that better understanding the geometry of end-
periodic mapping tori will provide some insights into giving a complete
solution to this problem.

### Comparison with finite-type case

We briefly outline here Brock’s strategy for his lower bound [Bro03b], point
out the complications that arise when adapting the strategy to our setting,
and discuss how we address these challenges.

Brock’s proof involves controlling the number and location of bounded length
curves in the mapping torus, as each bounded length curve provides a definite
contribution to the volume [Bro03a, Lemma 4.8]. To produce bounded length
curves, Brock works in the infinite cyclic cover $S\times\mathbb{R}$ of the
mapping torus, and constructs an interpolation between a simplicial hyperbolic
surface [Can96, Bon86] homotopic to the inclusion of $S$ and the image of this
surface under the generator of the deck group. The deck group acts like $f$ on
the $S$ factor, and the interpolation produces a sequence of bounded length
pants decompositions, starting with one on the initial surface and ending with
its $f$–image in the translate. He then shows that the number of curves
arising in this sequence provides an upper bound on distance in the pants
graph, and hence a bound on the translation length of $f$ [Bro03a, Lemma 4.3].

The interpolation between the two simplicial hyperbolic surfaces can overlap
significantly with its translates by the deck transformation, leading to an
overestimate in the number of bounded length curves in the mapping torus (as
many curves may project to the same curve). To account for this, Brock first
situates a neighborhood of the entire interpolation inside some fixed, but
uncontrolled number $n_{0}$ of consecutive translates of a fundamental domain
for the covering action. The concatenation of any $j>0$ consecutive translates
of the interpolation produces a path between a pants decomposition and its
image under $f^{j}$, all of whose curves are of bounded length and situated
inside $j+n_{0}$ translates of the fundamental domain. The number of curves
that occur in the sequence of pants decompositions is an upper bound on $j$
times the translation length, and a lower bound on $j+n_{0}$ times the volume.
Thus, taking $j\to\infty$ and dividing both quantities by $j$, the $n_{0}$
term disappears, proving the required lower bound on volume in terms of pants
translation length.

$M_{f}$
---
$M_{\infty}$
---
$f$
---
$F$
---
$\overline{M}_{f}$
---
$\widetilde{M}_{\infty}$
---
$F$
---
Figure 1. In the figure on the right, when we “interpolate” from left to right
through $\widetilde{M}_{\infty}$ the “core” of $f$ slides along $S$ as we zip
along the bottom boundary and unzip along the top boundary. In both cases, $F$
is the covering transformation given by $F(x,t)=(f(x),t-1)$.

Our proof of Theorem 1.1 involves a similar strategy. Notably, Brock’s lower
bound on volume in terms of the number of bounded length curves still serves
as the primary mechanism for controlling the volume. We also carry out many of
the arguments in the infinite cyclic cover, though in our situation this cover
does not have finitely generated fundamental group. See Figure 1 for a cartoon
of the differences between the infinite cyclic covers in the two settings.

Several of the ideas in Brock’s proof break down in fundamental ways in our
setting, and we discuss these in turn. We make extensive use of pleated and
simplicial hyperbolic surfaces in the compactified mapping torus with totally
geodesic boundary as well as in its infinite cyclic cover. The infinite-type
setting demands some care, but no serious issues arise here.

The first real obstacle we encounter is that the bound on pants distance in
terms of the number of curves that appear in all of the pants is not
immediately applicable as it relies on the work of Masur and Minsky [MM99,
MM00], where the constants depend on the topological type of the surface.
While our pants decompositions contain infinitely many curves, a finite
sequence of pants moves takes place on a finite-type subsurface. Fixing the
capacity of $f$ provides an initial bound on the topological type of this
subsurface, but even under this condition, two additional issues arise. First,
iteration of the map increases the size of this subsurface linearly in the
power, and so the strategy of iterating and taking limits is not viable. The
second issue is that there is no universal bound on the length of curves in a
pants decomposition of a finite-type surface with boundary.

To address the second issue, we construct minimally well-pleated surfaces in
the compactified mapping torus, which send “as much of the (infinite-type)
surface as possible” into the boundary—see Section 3.2 and Section 4.
Appealing to Basmajian’s collar lemma [Bas94], the bound on capacity produces
a priori bounds on the length of the boundary of a minimal core—see Lemma 4.1.
After passing to a uniform power, we may enlarge the core to “support” the
interpolation through simplicial hyperbolic surfaces, which does have a bound
on the length of its boundary—see Section 5.3. This enlarged surface is of
bounded topological type, again thanks to bounded capacity, and, in this
setting, there is a uniform bound on the length of a bounded length pants
decomposition—see Theorem 2.9.

As we cannot iterate and take a limit as Brock does, we address the first
issue by essentially gaining control on the “uncontrolled” constant $n_{0}$ in
Brock’s proof. Interestingly, the feature of end-periodic homeomorphisms that
forces the capacity to grow linearly under powers is, along with strong-
irreducibility, what comes to the rescue. Namely, as all of our bounded length
curves are homotopic into the enlargement of the core, we can find a uniform
power (depending only on the capacity) so that for all higher powers the
images of these bounded length curves are not homotopic into this subsurface
(see Lemma 3.1). In particular, no two curves in our bounded length set
project to the same curve in the mapping torus for this uniform power, and
this guarantees that they all contribute to the volume.

### Outline of the paper

We begin in Section 2 with preliminaries on end-periodic homeomorphisms and
their mapping tori, including the definition of capacity. In Section 3 we
establish key topological features of a strongly irreducible end-periodic
homeomorphism acting on a core, and describe how the core sits in the
compactified mapping torus. The details for the pleated surface technology we
need, and the resulting uniform geometric features for a strongly irreducible
end-periodic homeomorphism are described in Section 4. The kinds of simplicial
hyperbolic surfaces we will use, as well as our applications of these, are
described in Section 5. We assemble all the ingredients into the proof of
Theorem 1.1 in Section 6. Finally, in Section 7, we prove Theorem 1.2.

##  2\. Preliminaries

In this section we set some notation, recall some of the facts that we will
need, particularly from [FKLL23], and define the notion of “capacity.”

###  2.1. End-periodic homeomorphisms

We restrict our attention to surfaces of infinite-type with finitely many
ends, each accumulated by genus, and without boundary. The interested reader
can find a more general discussion of end-periodic homeomorphisms in [Fen92,
Fen97, CCF21].

A homeomorphism of an infinite-type surface $S$ is end-periodic if there is an
$m>0$ such that, for each end $E$ of $S$, there is a neighborhood $U_{E}$ of E
so that either

  * (i)

$f^{m}(U_{E})\subsetneq U_{E}$ and the sets $\\{f^{nm}(U_{E})\\}_{n>0}$ form a
neighborhood basis of $E$; or

  * (ii)

$f^{-m}(U_{E})\subsetneq U_{E}$ and the sets $\\{f^{-nm}(U_{E})\\}_{n>0}$ form
a neighborhood basis of $E$.

We say that $E$ is an attracting end in the first case, and a repelling end in
the second. The neighborhoods $U_{E}$ are nesting neighborhoods of the ends,
and when convenient we assume (as we may) that we have chosen disjoint nesting
neighborhoods for distinct ends. We denote the union of the neighborhoods of
the attracting ends $U_{+}$ and write $U_{-}$ for the union of the
neighborhoods of the repelling ends. If $f^{\pm 1}(U_{\pm})\subset U_{\pm}$,
and $\partial\overline{U}_{\pm}$ is a union of simple closed curves, then we
say that $U_{\pm}$ are tight nesting neighborhoods. Every end-periodic
homeomorphism admits tight nesting neighborhoods. For instance, the good
nesting neighborhoods from [FKLL23] are a particular example of tight nesting
neighborhoods, with the additional assumption that each component of
$\overline{U}_{\pm}$ has a single boundary component.

A compact subsurface $Y\subset S$ is a core for $f$ if $S-Y$ is a disjoint
union of tight nesting neighborhoods $U_{+}$ and $U_{-}$. Given a core $Y$,
define the junctures $\partial_{+}Y$ and $\partial_{-}Y$ to be the boundary
components meeting $U_{+}$ and $U_{-}$, respectively. Note that there are
infinitely many choices of cores for $f$ (as a given core can always be
enlarged).

Given a core $Y$ for $f$, a hyperbolic metric on $S$ for which
$f|_{U_{+}}\colon U_{+}\to U_{+}$ and $f^{-1}|_{U_{-}}\colon U_{-}\to U_{-}$
are isometric embeddings is said to be compatible with $Y$. Adjusting $f$ by
an isotopy if necessary, there are always metrics which are compatible with a
given core $Y$, see [Fen97].

Define

$\mathcal{U}_{+}=\bigcup_{n\geq 0}f^{-n}(U_{+})\quad\mbox{ and
}\quad\mathcal{U}_{-}=\bigcup_{n\geq 0}f^{n}(U_{-}),$

which are the positive and negative escaping sets for $f$, respectively. We
note that any choice of nesting neighborhoods will give rise to the same
escaping sets $\mathcal{U}_{\pm}$ (depending only on the homeomorphism $f$).
With these assumptions, the restrictions $\langle
f\rangle|_{\mathcal{U}_{\pm}}$ act cocompactly on $\mathcal{U}_{\pm}$ with
quotients $S_{\pm}=\mathcal{U}_{\pm}/\langle f\rangle$, which may be
disconnected, see, e.g. [FKLL23, Lemma 2.4].

In the following, curve and line refer to proper homotopy classes of essential
simple closed curves and essential properly embedded lines, respectively. A
curve $\alpha$ is called _reducing_ with respect to an end-periodic
homeomorphism $f$ if there exists $m,n\in\mathbb{Z}$ with $m<n$ and such that
$f^{n}(\alpha)$ is contained in a nesting neighborhood of an attracting end
and $f^{m}(\alpha)$ is contained in a nesting neighborhood of a repelling end.

######  Definition 2.1 (Strong irreducibility).

An end-periodic homeomorphism, ${f:S\to S}$, is _strongly irreducible_ if it
has no _periodic curves_ , no _periodic lines_ , and no _reducing curves_.

###  2.2. Mapping tori and their compactifications

We define a partial compactification of $S\times(-\infty,\infty)$ inside
$S\times[-\infty,\infty]$ by

$\widetilde{M}_{\infty}=\\{(x,t)\in S\times[-\infty,\infty]\mid
x\in\mathcal{U}_{\pm}\mbox{ if }t=\pm\infty\\},$

and define $F\colon\widetilde{M}_{\infty}\to\widetilde{M}_{\infty}$ by
$F(x,t)=(f(x),t-1)$, where $\pm\infty-1=\pm\infty$. The group $\langle
F\rangle$ acts properly discontinuously and cocompactly on
$\widetilde{M}_{\infty}$, see, e.g. [FKLL23, Lemma 3.2]. The quotient
$p\colon\widetilde{M}_{\infty}\to\overline{M}_{f}=\widetilde{M}_{\infty}/\langle
F\rangle$ is a compact manifold with boundary naturally homeomorphic to
$S_{-}\cup S_{+}$ and whose interior is the mapping torus $M_{f}$ of $f$ which
we call the compactified mapping torus. The manifold $\overline{M}_{f}$ is
particularly nice when $f$ is strongly irreducible.

######  Theorem 2.2.

[FKLL23, Proposition 3.1] Let $f:S\to S$ be a strongly irreducible, end-
periodic homeomorphism of a surface with finitely many ends, all accumulated
by genus. Then $\overline{M}_{f}$, is a compact, irreducible, atoroidal,
acylindrical $3$-manifold, with incompressible boundary.

Together with Thurston’s Geometrization Theorem for Haken manifolds and Mostow
Rigidity [Thu86a, McM92, Mor84], the above result implies the following
theorem.

######  Theorem 2.3.

If $f:S\to S$ is a strongly irreducible, end-periodic homeomorphism of a
surface with finitely many ends, all accumulated by genus, then
$\overline{M}_{f}$ admits a convex hyperbolic metric $\sigma_{0}$ with totally
geodesic boundary, which is unique up to isometry.

Whenever discussing metric properties of $\overline{M}_{f}$, we will assume it
is equipped with the convex hyperbolic metric $\sigma_{0}$, and may simply
refer to it as the hyperbolic metric on $\overline{M}_{f}$ (due to the
uniqueness statement), without specific reference to its name. The metric
$\sigma_{0}$ pulls back to a complete hyperbolic metric on
$\widetilde{M}_{\infty}$ for which $\mathcal{U}_{\pm}\times\\{\pm\infty\\}$ is
totally geodesic.

Given a core $Y$ with tight nesting neighborhoods $U_{\pm}$, we choose a
hyperbolic metric $\mu$ on $S$ so that the “inclusions” $U_{\pm}\to
U_{\pm}\times\\{\pm\infty\\}$ into $\partial\widetilde{M}_{\infty}$ are
isometric embeddings. Then, after adjusting $f$ by an isotopy on $U_{\pm}$ if
necessary, $\mu$ is compatible with $Y$. We say that such a metric is induced
by the metric on $\widetilde{M}_{\infty}$.

###  2.3. Euler characteristic, complexity, and capacity

Given a compact surface $Z$ of genus $g$ with $n$ boundary components, there
are two measures of the “size” of $Z$; $\chi(Z)$, the Euler characteristic,
and the complexity $\xi(Z)=3g-3+n$. We note that $\xi(Z)$, when positive, is
the maximal number of essential, pairwise disjoint, pairwise non-isotopic
simple closed curves on $Z$, i.e. the number the of curves in a pants
decomposition of $Z$. Since $\chi(Z)=2-2g-n$, we have the following elementary
fact for all $Z$ with $\xi(Z)\geq 0$,

$|\chi(Z)|-1\leq\xi(Z)\leq\tfrac{3}{2}|\chi(Z)|.$

For $Z$ closed (and genus at least $1$) the second inequality is an equality.
We extend both of these quantities to disconnected surfaces, additively over
the components (which is natural for the Euler characteristic), and observe
that when all components have $\xi\geq 0$ (the only case of interest for us),
the inequality on the right still holds. We use all of this in what follows
without explicit mention.

Given an end-periodic homeomorphism $f\colon S\to S$, we define the _core
characteristic of $f$_ to be

$\chi(f)=\max_{Y\subset S}\chi(Y),$

where the maximum is taken over all cores $Y\subset S$ for $f$. Informally, a
core is a subsurface where curves from the repelling end get “hung up” under
forward iteration of $f$ (or where curves from the attracting end get hung up
under backward iteration). Thus, $\chi(f)$ measures the minimal size of the
subsurface where that behavior occurs. Any core $Y$ with $\chi(Y)=\chi(f)$
will be called a minimal core.

######  Remark.

Note that it is always possible to choose a core $Y$ so that each component of
$U_{+}$ or $U_{-}$ meets $Y$ in a single simple closed curve (see [FKLL23,
Corollary 2.5] and the discussion preceding it). In this case, connectivity of
$S$ implies connectivity of $Y$. However, cores (especially minimal cores)
need not be connected, as the example in Figure 2 illustrates.

$\rho$
---
$h$
---
Figure 2. The homeomorphism $\rho$ generates a covering action with quotient a
genus $6$ surface, and $h$ is a partial pseudo Anosov supported on the
subsurface which is the shaded (purple and gray) subsurface with genus $6$ and
four boundary components. The disconnected shaded subsurface (purple in the
figure) is a core for $f=h\circ\rho$, regardless of what $h$ is. Note that one
of the components of this core has a single boundary component that faces
$U_{+}$, and thus no component that faces $U_{-}$. We can remove this
component, and what remains is still a core for $f$. For $h$ “sufficiently
complicated”, $f=h\circ\rho$ will be strongly irreducible.

Given a core $Y$ for a strongly irreducible end-periodic homeomorphism
$f\colon S\to S$, some components of $Y$ may be disjoint from either
$\partial_{-}Y$ or $\partial_{+}Y$. We call such a component imbalanced, and
say that $Y$ is balanced if there are no imbalanced components. The next lemma
says that these imbalanced components can always be safely ignored. While the
existence of imbalanced components does not affect any of the arguments in
this paper, it can be helpful in developing intuition to assume there are
none.

######  Lemma 2.4.

If $f\colon S\to S$ is a strongly irreducible end-periodic homeomorphism and
$Y$ is a core for $f$, then there is a subsurface $Y^{\prime}\subset Y$ which
is balanced. In particular, any minimal core is balanced.

###### Proof.

Let $Y$ be a core for $f$ and suppose there is an imbalanced component
disjoint from $\partial_{-}Y$. We first show that we can remove at least one
such component to obtain a new core for $f$. To that end, let $Y_{0}\subset Y$
be the union of all imbalanced components with $\partial_{-}Y_{0}=\emptyset$.
Since $f(U_{+})\subset U_{+}$, and the components of
$\\{f^{k}(U_{+})\\}_{k>0}$ determine a neighborhood basis for the attracting
ends, it follows that $f^{-n}(\partial_{+}Y)$ has no transverse intersections
with $\partial_{+}Y$, for all $n\geq 1$. Now set $Y_{1}$ to be the (possibly
empty) intersection $Y_{1}=Y\cap f(Y_{0})$. Observe that
$f^{-1}(Y_{1})=f^{-1}(Y)\cap Y_{0}$, is a (possibly empty) subsurface of
$Y_{0}$ whose boundary components are either components of $\partial_{+}Y_{0}$
or else are contained in the interior of $Y_{0}$ and hence $\partial
f^{-1}(Y_{1})\subset f^{-1}(\partial_{+}Y)$. Consequently, $Y_{1}$ is a union
of components of $Y_{0}$.

We note that $Y_{1}$ is a proper subsurface of $Y_{0}$ since otherwise,
$f^{-1}$ would simply permute the boundary components of $Y_{0}$, creating a
periodic curve, contradicting the strong irreducibility of $f$. Continuing in
a similar way, we see that $Y_{2}=Y\cap f(Y_{1})$ is a union of components of
$Y_{1}$. In fact, this subsurface $Y_{2}$ must be a proper subsurface of
$Y_{1}$ by a similar argument as above. Continuing inductively, we find a
nested sequence

$Y_{0}\supset Y_{1}\supset Y_{2}\supset\cdots$

defined by $Y_{j+1}=Y\cap f(Y_{j})$, for all $j\geq 0$. Furthermore,
$Y_{j}\neq Y_{j+1}$ if $Y_{j}\neq\emptyset$.

As the areas of the subsurfaces in this sequence always decrease by a multiple
of $\pi$, there is some smallest $n\geq 1$ so that $Y_{n}=\emptyset$ (and then
$Y_{j}=\emptyset$ for all $j\geq n$). It follows that $Y_{n-1}\neq\emptyset$
is a union of components of $Y_{0}$ and $f(Y_{n-1})\cap Y=\emptyset$. Since
$f(\partial_{+}Y)\subset\overline{U}_{+}$, it follows that $f(Y_{n-1})\subset
U_{+}$. Therefore, $W=Y-Y_{n-1}$ must also be a core. Indeed,

$S-W=U_{-}\sqcup(U_{+}\cup Y_{n-1}),$

and since $f(Y_{n-1})\subset U_{+}$, we have

$f^{-1}(U_{-})\subset U_{-}\mbox{ and }f(U_{+}\cup Y_{n-1})\subset
U_{+}\subset U_{+}\cup Y_{n-1},$

meaning that $U_{-}$ and $U_{+}\cup Y_{n-1}$ are tight nesting neighborhoods.

We thus have a new core and have reduced the number of imbalanced components.
Repeating this procedure finitely many times we can remove all imbalanced
components with non-empty positive boundary. Likewise, repeating for the union
$Y_{0}^{\prime}$ of imbalanced components with
$\partial_{+}Y_{0}^{\prime}=\emptyset$, and replacing $f$ with $f^{-1}$ in the
arguments above, we can remove all imbalanced components. ∎

For additional intuition, we record the following.

######  Lemma 2.5.

If $Y\subset S$ is a balanced core for a strongly irreducible, end-periodic
homeomorphism and $\mathcal{U}_{0}\subset\mathcal{U}_{\pm}$ is a component
defined by a component $U_{0}\subset S-Y$, then $\partial\overline{U}_{0}$
separates $\mathcal{U}_{0}$ into two components, each a neighborhood of an end
of $\mathcal{U}_{0}$.

###### Proof.

Suppose $\mathcal{U}_{0}\subset\mathcal{U}_{+}$, for concreteness (the other
case follows by replacing $f$ with $f^{-1}$). By construction,
$U_{0}\subset\mathcal{U}_{0}-\partial U_{0}$ is an unbounded component which
is a neighborhood of an end of $\mathcal{U}_{0}$. Since $\mathcal{U}_{0}$ has
two ends, there is at least one other component
$U_{0}^{\prime}\subset\mathcal{U}_{0}-\partial U_{0}$, which is necessarily
unbounded. We must show that $\mathcal{U}_{0}-\partial U_{0}=U_{0}\cup
U_{0}^{\prime}$.

Suppose there is another component, $U_{1}\subset\mathcal{U}_{0}-\partial
U_{0}$, different from $U_{0},U_{0}^{\prime}$. Observe that $U_{1}$ is the
interior of a compact subsurface $\overline{U}_{1}$ with

$\partial\overline{U}_{1}\subset\partial U_{0}\subset\partial_{+}Y.$

We claim that $\partial Y-\partial U_{0}$ cannot intersect $\overline{U}_{1}$.
Indeed, it is disjoint from $\partial U_{0}$, so if it intersected
$\overline{U}_{1}$, it would necessarily be contained in it. But every
component $\alpha\subset\partial Y-\partial U_{0}$ faces a neighborhood of
another end of $S$ (different than $U_{0}$), and so $\alpha$ is contained in a
component $\mathcal{U}_{\pm}$ different than $\mathcal{U}_{0}$. The claim
implies $\overline{U}_{1}$ is a component of $Y$ with
$\partial\overline{U}_{1}\subset\partial U_{0}\subset\partial_{+}Y$; thus,
$\overline{U}_{1}$ is an imbalanced component, which is a contradiction. The
lemma follows. ∎

For any end-periodic homeomorphism $f\colon S\to S$, we also define the _end
complexity of $f$_ to be

$\xi(f)=\xi(\partial\overline{M}_{f}).$

Note that in [FKLL23], the right-hand side is the notation for this
complexity, but because it will appear often later, we have adopted this
short-hand.

Given any core $Y$ for $f$ with tight nesting neighborhoods $U_{\pm}$, set

$\Delta_{\pm}=\overline{U_{\pm}-f^{\pm 1}U_{\pm}},$

which are compact subsurfaces in $\overline{U}_{\pm}$. We note that
$\Delta_{\pm}\subset\mathcal{U}_{\pm}$ serves as a fundamental domain for the
restricted action of $\langle f\rangle|_{{\mathcal{U}}_{\pm}}$, therefore
$\chi(\Delta_{\pm})=\chi(S_{\pm})$ (see the proof of [FKLL23, Corollary 2.5]).
Therefore,

$|\chi(\Delta_{\pm})|=|\chi(S_{\pm})|=\tfrac{2}{3}\xi(S_{\pm}),$

since $S_{\pm}$ are closed. From the same corollary, $\xi(S_{+})=\xi(S_{-})$,
and thus

$\xi(f)=\xi(\partial\overline{M}_{f})=\xi(S_{+})+\xi(S_{-})=2\xi(S_{+})=3|\chi(\Delta_{\pm})|.$

Thus, $\xi(f)$ can be thought of as measuring the amount of translation of $f$
on the ends of $S$; an alternative perspective on this is explained in
[FKLL23, Corollary 2.8] which connects to work of Aramayona-Patel-Vlamis
[APV20].

Taken together, $\chi(f)$ and $\xi(f)$ provide a measure of the size of the
subsurface where $f$ does something “interesting”. More precisely, $f$ is
“translating” from the negative end into the positive end, $\xi(f)$ measures
“how much” translation is happening and $\chi(f)$ measures how large of a
subsurface that translation is trying to pass through. These two quantities
thus serve as a substitute for the genus, Euler characteristic, or complexity
of a finite-type surface.

######  Definition 2.6.

For any end-periodic homeomorphism $f$, we call the pair $(\chi(f),\xi(f))$,
the capacity of $f$.

We also note that since $\xi(f)=\frac{3}{2}|\chi(\partial\overline{M}_{f})|$
and since for any $n>0$, $\overline{M}_{f^{n}}$ is an $n$–fold cover of
$\overline{M}_{f}$, we have

$\xi(f^{n})=\tfrac{3}{2}|\chi(\partial\overline{M}_{f^{n}})|=n\tfrac{3}{2}|\chi(\partial\overline{M}_{f})|=n\xi(f).$

Thus raising to powers increases end-complexity in a predictable way. On the
other hand, a core for $f$ is also a core for $f^{n}$, and hence core
characteristic is non-decreasing under raising to powers.

###  2.4. The pants graph and Bers pants decompositions

A pants decomposition on $S$ is a multicurve $P$ in $S$ such that $S-P$ is a
collection of three-holed spheres (i.e. pairs of pants). An elementary move on
a pants decomposition $P$ replaces a single curve in $P$ with a different one
intersecting it a minimal number of times, producing a new pants decomposition
$P^{\prime}$. There are two types of elementary moves corresponding to whether
the complexity one subsurface in which the elementary move takes place is a
one-holed torus or a four-holed sphere. This is illustrated in Figure 3.

Figure 3. Elementary moves on pants decompositions.

######  Definition 2.7.

The pants graph, $\mathcal{P}(S)$, is the graph whose vertices are (isotopy
classes of) pants decompositions on $S$, with edges between pants
decompositions that differ by an elementary move.

There is a path metric on (the components of) $\mathcal{P}(S)$ with respect to
which the action of ${\rm Map}(S)$ on $\mathcal{P}(S)$ is isometric. It is
defined as follows: an edge corresponding to an elementary move that occurs on
a one-holed torus has length $1$, and an edge corresponding to an elementary
move that occurs on a four-holed sphere has length $2$.

Brock proved in [Bro03b] that for finite-type surfaces $Z$, $\mathcal{P}(Z)$
is quasi-isometric to the Teichmüller space of $Z$, $\rm{Teich}(Z)$ equipped
with the Weil–Petersson metric. In the infinite-type setting, we no longer
have this correspondence between the pants graph and Teichmüller space, but,
as we shall see, the pants graph still encodes important geometric data.

######  Definition 2.8.

Given any end-periodic homeomorphism $f\colon S\to S$, we define the
asymptotic translation distance of $f$ on $\mathcal{P}(S)$ to be

$\tau(f)=\inf_{P\in\mathcal{P}(S)}\liminf_{n\to\infty}\frac{d(P,f^{n}(P))}{n},$

where this infimum is over all pants decompositions $P\in\mathcal{P}(S)$.
Observe that $\tau(f^{n})=n\tau(f)$ for all $n>0$.

Note that $\mathcal{P}(S)$ is necessarily disconnected when $S$ is of infinite
type (see Branman [Bra20] for more on the pants graphs of infinite-type
surfaces). In particular, for certain $P\in\mathcal{P}(S)$, $d(P,f^{n}(P))$ is
infinite for all $n>0$. Consequently, the infimum is effectively being taken
over the union of connected components for which this distance is finite for
some, hence infinitely many, $n>0$ (in [FKLL23], such $P$ were called
$f$–asymptotic pants decompositions).

Throughout the paper it will be necessary to produce pants decompositions of
bounded length. Bers proved that a closed hyperbolic surface of genus $g$
admits a pants decomposition for which all the curves have length bounded by a
constant depending only on the genus [Ber74, Ber85]. We will need a relative
version of Bers result for surfaces with boundary, a short proof of a very
concrete version of which was recently given by Parlier [Par].

######  Theorem 2.9.

[Par, Theorem 1.1] Let $X$ be a hyperbolic surface, possibly with geodesic
boundary, and of finite area. Then $X$ admits a pants decomposition where each
curve is of length at most

$L_{B}=\max{\\{\ell(\partial X),{\rm area}(X)\\}}.$

##  3\. Cores and topology

Let $f:S\to S$ be a strongly irreducible end-periodic homeomorphism. In this
section, we will prove some additional topological information about $f$, a
core $Y\subset S$ for $f$, and features of $Y$ that are reflected in
$\overline{M}_{f}$.

###  3.1. Uniform power bounds

After applying a sufficiently large power of $f$, some part of any curve
$\alpha\in Y$ must leave $Y$ (c.f. [Fen97, Theorem 2.7(iii)] and [LMT, Lemma
2.1]). We will need the following strengthening of this fact which provides a
uniform power for which that behavior occurs.

######  Lemma 3.1.

Given a core $Y\subset S$ for a strongly irreducible end-periodic $f\colon
S\to S$, then for all $k\geq 2\xi(Y)$ there are no closed, essential curves
contained in $f^{k}(Y)\cap Y$.

###### Proof.

We prove the equivalent statement that there are no essential curves in
$f^{-k}(Y)\cap Y$, since this introduces fewer total inverses in the proof.

We first make a definition and record an observation. We say that an
essential, possibly disconnected, subsurface $Z\subset S$ is lean if no
components are pants and no two annular components are homotopic. Given a lean
subsurface $Z\subset S$, define

$\zeta(Z)=(\xi(Z_{0}),y),$

where $Z_{0}\subset Z$ is the union of all non-annular components of $Z$,
$\xi(Z_{0})$ is the complexity of $Z_{0}$ (as defined in Section 2), and $y$
is the number of annular components of $Z$. Observe that $\xi(Z_{0})+y$ is the
number of pairwise disjoint, non-parallel curves in $Z$ (here the core curve
of an annulus is considered an essential curve in the annulus). We consider
such pairs as elements of $\mathbb{Z}^{2}$ with the dictionary order.

Suppose that $Z\subset Z^{\prime}$ are lean subsurfaces of $S$ (where we
assume that any annular component of $Z$ is either contained in an annular
component of $Z^{\prime}$, or in a non-annular component of $Z^{\prime}$ in
which it is non-peripheral). Then $\zeta(Z)\leq\zeta(Z^{\prime})$, with
equality if and only if there is a homeomorphism $h\colon S\to S$ isotopic to
the identity so that $h(Z)=Z^{\prime}$. Moreover, observe that if we write
$\zeta(Z)=(\xi,y)$ and $\zeta(Z^{\prime})=(\xi^{\prime},y^{\prime})$ and if
$Z^{\prime}$ is the union of $Z$ with the regular neighborhood of a
multicurve, then $\zeta(Z)\leq\zeta(Z^{\prime})$ implies that
$\xi+y\leq\xi^{\prime}+y^{\prime}$.

Now, suppose to the contrary that there is a curve $\alpha\subset
f^{-k}(Y)\cap Y$ for some $k\geq 2\xi(Y)$. Then, $f^{k}(\alpha)\subset Y$, and
since $S-Y=U_{+}\cup U_{-}$, it follows that $f^{j}(\alpha)\subset Y$ for all
$j=0,\ldots,k$, since $f(U_{+})\subset U_{+}$.

For each $j=0,\ldots,k$, let $Z_{j}\subset Y$ be the smallest lean subsurface
filled by

$\bigcup_{i=0}^{j}f^{i}(\alpha),$

and write $\zeta(Z_{j})=(\xi_{j},y_{j})$. Then
$\zeta(Z_{j})\leq\zeta(Z_{j+1})\leq\zeta(Y)=(\xi(Y),0)$ and
$\xi_{j}+y_{j}\leq\xi_{j+1}+y_{j+1}\leq\xi(Y)$ for all $0\leq j<k$. Observe
that for any $j=0,\ldots,k$, either $\zeta(Z_{j})=\zeta(Z_{j+1})$, or one of
the following strict inequalities must hold:

$\xi_{j}<\xi_{j+1}\quad\mbox{ or }\quad\xi_{j}+y_{j}<\xi_{j+1}+y_{j+1}.$

This implies that the sequence of $L^{1}$–norms of the pairs
$\\{(\xi_{j},\xi_{j}+y_{j})\\}_{j=0}^{k}$ is a non-decreasing sequence of
integers from $1$ to $2\xi_{k}+y_{k}\leq 2\xi(Y)$. But since $k\geq 2\xi(Y)$,
there must be consecutive pairs $(\xi_{j},\xi_{j}+y_{j})$ and
$(\xi_{j+1},\xi_{j+1}+y_{j+1})$ whose $L^{1}$–norms are equal, and so, for
this $j$, we have $\zeta(Z_{j})=\zeta(Z_{j+1})$. In particular, there is a
homeomorphism $h\colon S\to S$ isotopic to the identity so that
$h(Z_{j})=Z_{j+1}$.

Since $f(Z_{j})\subset Z_{j+1}$ and $\zeta(f(Z_{j}))=\zeta(Z_{j})$, there is a
homeomorphism $h^{\prime}\colon S\to S$ isotopic to the identity so that
$h^{\prime}(f(Z_{j}))=Z_{j+1}=h(Z_{j})$. Rewriting this, we have
$h^{-1}h^{\prime}f(Z_{j})=Z_{j}$. But $h^{-1}h^{\prime}f$ is isotopic to $f$,
and we conclude that $f$ preserves $Z_{j}$, and hence $\partial Z_{j}$, up to
isotopy, contradicting the strong irreducibility of $f$. ∎

###  3.2. Cores in the compactified mapping torus

The suspension flow on $M_{f}$ can be reparameterized and extended to a local
flow $(\psi_{s})$ on $\overline{M}_{f}$. Fixing a core $Y$ for $f$ with the
associated (tight) nesting neighborhoods, $U_{+}$ and $U_{-}$, we can define a
homotopy of the inclusion $S\to\overline{M}_{f}$ along the flowlines, by
flowing $U_{+}$ and $U_{-}$ forward and backward, respectively, until they
meet $\partial\overline{M}_{f}$. We do this, carrying along a small
neighborhood of $\partial Y$ in $Y$, but keeping the rest of $Y$ fixed. If we
let $h_{t}\colon S\to\overline{M}_{f}$, $t\in[0,1]$ denote the homotopy, then
we can assume that $h_{t}$ is injective for all $t\in[0,1)$. One way to think
about this construction is via spiraling neighborhoods of the boundary; see
e.g. [Fen97, §4],[LMT, §3.1]. We write
$Y_{1}=h_{1}(Y)\subset\overline{M}_{f}$, and call $Y_{1}$ the $Y$–viscera. It
is convenient to think of $h_{1}(S)$ as a branched surface in
$\overline{M}_{f}$, transverse to $(\psi_{s})$.

Since the first return map of $(\psi_{s})$ to $S$ is $f$, the result is a map
of $S$ into $\overline{M}_{f}$ which is embedded on the interior of $Y$, and
for which $U_{+}$ and $U_{-}$ map onto $\partial\overline{M}_{f}\cong
S_{+}\sqcup S_{-}$. After adjusting (precomposing) $f$ by an isotopy supported
in a small neighborhood of $\partial Y$, we may assume that
$f(\overline{U}_{+})\subset U_{+}$ and $f^{-1}(\overline{U}_{-})\subset
U_{-}$. Having done so, $Y_{1}\subset\overline{M}_{f}$ is then properly
embedded.

In what follows, we may need to adjust $f$ by an isotopy which is the identity
outside of $Y$. This does not affect the homeomorphism type of the pair
$(\overline{M}_{f},Y_{1})$, and we use the same notation to denote the new
pair.

A _boundary-compressing disk_ for $Y_{1}\subset\overline{M}_{f}$ (or more
generally, for any properly embedded surface) is an embedded disk
$D\subset\overline{M}_{f}$ such that $\partial D=\alpha\cup\beta$, where
$\alpha$ is a properly embedded essential arc in $Y_{1}$,
$\beta\subset\partial\overline{M}_{f}$, $\alpha$ and $\beta$ intersect
precisely in their endpoints, and the interior of $D$ is disjoint from
$Y_{1}\cup\partial\overline{M}_{f}$. We can perform a homotopy of $Y_{1}$, rel
$\partial Y_{1}$, “pushing across $D$” so that a neighborhood of $\alpha$ in
$Y_{1}$ is mapped into $\partial\overline{M}_{f}$. See Figure 4.

$\beta$
---
$\alpha$
---
$\partial\overline{M}_{f}$
---
$D$
---
$Y_{1}$
---
$\beta$
---
$\partial\overline{M}_{f}$
---
$Y_{1}$
---
Figure 4. A boundary-compressing disk $D$ (shaded purple) cobounded by an arc
$\alpha\subset Y_{1}$ and an arc $\beta\subset\partial\overline{M}_{f}$. On
the right of the figure is $Y_{1}$ after $D$ is compressed.

A flow-compressing disk for $Y_{1}$ is a boundary-compressing disk
$D\subset\overline{M}_{f}$ for $Y_{1}$ that is foliated by flowlines
transverse to $\alpha$ and $\beta$. Given a flow-compressing disk $D$, if the
arc $\beta$ lies in $S_{+}\subset\partial\overline{M}_{f}$, then
$\alpha^{\prime}=h_{1}^{-1}(\alpha)$ is an arc in $Y$, and
$\beta=h_{1}(f(\alpha^{\prime}))$. Since $h_{1}(f(\alpha^{\prime}))\subset
S_{+}$, it follows that $f(\alpha^{\prime})\subset U_{+}$. Consequently, a
flow-compressing disk is really determined by a properly embedded essential
arc $\alpha^{\prime}\subset Y$ that is mapped entirely into $U_{+}$ by $f$ (or
into $U_{-}$ by $f^{-1}$). In this case, the homotopy of $h_{1}(S)$ obtained
from the boundary compression of $Y_{1}$ along $D$ can be carried out
transverse to $(\psi_{s})$. See Figure 5. The part of the homotoped image of
$Y_{1}$ that remains in the interior $M_{f}$, also determines a core for $f$
which necessarily has larger Euler characteristic.

$Y_{1}$
---
$\partial\overline{M}_{f}$
---
$\beta$
---
$\alpha$
---
$D$
---
Figure 5. If a boundary-compressing disk $D$ is foliated by flowlines
transverse to $\alpha$ and $\beta$, as shown here, then we call it a flow-
compressing disk. Here $\alpha=h_{1}(\alpha^{\prime})$ and
$\beta=h_{1}(f(\alpha^{\prime}))$.

######  Lemma 3.2.

If $D\subset\overline{M}_{f}$ is a flow-compressing disk for the $Y$–viscera,
then there is a core $Y^{\prime}\subset Y$ for $f$ such that
$\chi(Y^{\prime})>\chi(Y)$.

###### Proof.

Suppose that $D$ intersects $S_{+}$ in the arc $\beta$ (the proof in the case
that $\beta\subset S_{-}$ is identical except with $f$ replaced by $f^{-1}$).

There is a small neighborhood $N$ of $D$ which is also a union of segments of
flowlines, so that $X:=N\cap Y_{1}$ is a regular neighborhood of $\alpha$. Let
$Y^{\prime\prime}=\overline{Y_{1}-X}$ and set
$Y^{\prime}=h_{1}^{-1}(Y^{\prime\prime})$, where $h_{t}$ is the homotopy
defining $Y_{1}$. Observe that $\chi(Y^{\prime})>\chi(Y)$ since $Y$ is
obtained from $Y^{\prime}$ by adding a $1$–handle. We can use $N$ to push $X$
along flowlines until it lies entirely inside $\partial\overline{M}_{f}$.
Concatenating this homotopy with $h_{t}$, we get a homotopy $g_{t}\colon
S\to\overline{M}_{f}$ pushing along flowlines, so that
$g_{1}^{-1}(\partial\overline{M}_{f})=\overline{S-Y^{\prime}}$.

Now observe that $Y^{\prime}$ is a core for $f$. Indeed, let $W_{+}$ and
$W_{-}$ be the unions of components of $S-Y^{\prime}$ containing $U_{+}$ and
$U_{-}$, respectively. Given $x\in W_{+}$, consider its maximal forward
$(\psi_{s})$–flowline in $\overline{M}_{f}$,

$\ell_{x}^{+}=\bigcup_{s\geq 0}\psi_{s}(x),$

where the union is over all $s\geq 0$ for which $\psi_{s}(x)$ is defined.
Every point of $\ell_{x}^{+}\cap S$ is mapped by $g_{1}$ into $S_{+}$, and in
particular, these points all lie in $W_{+}$. On the other hand,
$\ell_{x}^{+}\cap S$ is the union of forward images of $x$ by $f$ (since the
first return map of $(\psi_{s})$ is $f$), and thus $f(x)\in W_{+}$. Therefore,
$f(W_{+})\subset W_{+}$. By similarly analyzing a backward flowline
$\ell_{x}^{-}$, we can see that $f^{-1}(W_{-})\subset W_{-}$, proving that
$W_{+}$ and $W_{-}$ are tight nesting neighborhoods of the ends, and thus
$Y^{\prime}$ is a core for $f$. ∎

The next lemma says that we can promote an arbitrary $\partial$–compressing
disk to a flow-compressing disk.

######  Lemma 3.3.

If the $Y$–viscera in $\overline{M}_{f}$ is boundary compressible, then after
adjusting $f$ by an isotopy which is the identity outside $Y$, there is a
flow-compressing disk $D\subset\overline{M}_{f}$.

###### Proof.

The local flow $(\psi_{s})$ defines a transverse orientation to $Y_{1}$.
Suppose $D$ is a compressing disk with boundary arcs $\alpha\subset Y_{1}$ and
$\beta\subset\partial\overline{M}_{f}$. We assume that $\beta\subset S_{+}$,
with the case $\beta\subset S_{-}$ proved by replacing $f$ with $f^{-1}$.
Since $D$ is a compressing disk, $D\cap Y_{1}=\alpha$, and so $D$ must either
be on the positive or negative side of $Y$ near $\alpha$.

If $D$ is on the negative side, we observe that there are arcs
$\alpha^{\prime},\beta^{\prime}\subset S$ meeting in their endpoints such that
$h_{1}(\alpha^{\prime})=\alpha$ and $h_{1}(\beta^{\prime})=\beta$. See Figure
6. We can piece together a map of a disk $D^{\prime}\to\overline{M}_{f}$ from
the homotopy $h_{t}$ and the disk $D$ so that the boundary of $D^{\prime}$
maps homeomorphically to $\alpha^{\prime}\cup\beta^{\prime}$. We lift this to
a map $D^{\prime}\to\widetilde{M}_{\infty}$ so that the boundary of
$D^{\prime}$ maps homeomorphically to
$(\alpha^{\prime}\cup\beta^{\prime})\times\\{0\\}$ in $S\times\\{0\\}$.
Projecting $\widetilde{M}_{\infty}$ onto the first factor, this in turn
defines a homotopy from $\alpha^{\prime}$ to $\beta^{\prime}$ in $S$, rel
endpoints. However, $\beta^{\prime}$ is an arc in $\overline{U}_{+}$, while
$\alpha^{\prime}$ is an essential arc in $Y$. Since an essential arc in $Y$
cannot be homotoped outside of $Y$, this is a contradiction.

$Y_{1}$
---
$\partial\overline{M}_{f}$
---
$\beta$
---
$\alpha$
---
Figure 6. The arcs $\alpha\subset Y_{1}$ and
$\beta\subset\partial\overline{M}_{f}$ when $D$ is on the negative side of $Y$
near $\alpha$

From the previous paragraph, we may assume that $D$ is on the positive side of
$Y$ near $\alpha$. See Figure 7. We can again find arcs
$\alpha^{\prime},\beta^{\prime}\subset S$ so that
$h_{1}(\alpha^{\prime})=\alpha$ and $h_{1}(\beta^{\prime})=\beta$, but now
these arcs do not meet at their endpoints. Instead, we can find such arcs so
that the endpoints of $\beta^{\prime}$ are the first return points of the
endpoints of $\alpha^{\prime}$ by $(\psi_{s})$, i.e. the $f$–image of the
endpoints of $\alpha^{\prime}$. Since flowing $\alpha^{\prime}$ forward until
it hits $S$, the image is precisely $f(\alpha^{\prime})$, we see that
$f(\alpha^{\prime})$ is an arc with the same endpoints as $\beta^{\prime}$.

$Y_{1}$
---
$\partial\overline{M}_{f}$
---
$\beta$
---
$\alpha$
---
$D$
---
Figure 7. A boundary-compressing disk $D$, which is not foliated by flowlines.

Observe that if $f(\alpha^{\prime})\subset U_{+}$, then
$h_{1}(\alpha^{\prime}\cup f(\alpha^{\prime}))$ is the boundary of a flow
compressing disk, and we are done. Because we have not imposed any constraints
on the behavior of $f$ inside of $Y$, we may not have
$f(\alpha^{\prime})\subset U_{+}$, in which case we claim we can adjust $f$ by
an isotopy supported in $Y$ so that the new homeomorphism does send
$\alpha^{\prime}$ into $U_{+}$.

To find the required isotopy, we first observe that using the disk $D$, we can
construct a homotopy, rel endpoints, from $f(\alpha^{\prime})$ to
$\beta^{\prime}$. To do this, we consider the “rectangle” which is a union of
the flowlines between $\alpha^{\prime}$ and $f(\alpha^{\prime})$ in
$\overline{M}_{f}$, and drag this along via the homotopy $h_{t}$ to define a
disk $D^{\prime}\subset\overline{M}_{f}$ whose boundary is the union of the
two arcs $h_{1}(\alpha^{\prime})=\alpha$ and $h_{1}(f(\alpha^{\prime}))$, and
whose interior is contained in a component of the complement of
$\partial\overline{M}_{f}\cup Y_{1}$. Since both disks $D$ and $D^{\prime}$
have their interiors in the same complementary component of
$\overline{M}_{f}\cup Y_{1}$, their union $D\cup D^{\prime}$ can be pushed
forward via the flow into the branched surface $h_{1}(S)$, and then lifted
back to $S$ to define the homotopy, rel endpoints from $f(\alpha^{\prime})$ to
$\beta^{\prime}$.

Now we assume, as we may, that $f(\partial Y)$ meets $\partial Y$ transversely
and minimally. Then since $\beta^{\prime}\subset U_{+}$ is an arc disjoint
from $\partial Y$ and it is homotopic, rel endpoints, to $f(\alpha^{\prime})$
which meets $f(\partial Y)$ only in its endpoints, it follows that we may
postcompose $f$ with an isotopy that is the identity outside $f(Y)$, so that
$f(\alpha^{\prime})$ is disjoint from $\partial Y$, and thus contained in
$U_{+}$. This is equivalent to precomposing $f$ by an isotopy that is the
identity outside $Y$. Replacing $f$ with this isotopic homeomorphism,
$f(\alpha^{\prime})\subset U_{+}$, and thus $\alpha=h_{1}(\alpha^{\prime})$
and $h_{1}(f(\alpha^{\prime}))$ defines a flow-compressing disk, as required.
∎

From the two lemmas above, we deduce the following.

######  Proposition 3.4.

If $Y\subset S$ is a minimal core for a strongly irreducible end-periodic
homeomorphism and $Y_{1}\subset\overline{M}_{f}$ is the $Y$–viscera, then $Y$
is boundary incompressible.

###### Proof.

If $Y_{1}$ were boundary compressible, Lemma 3.3 would providuce a flow-
compressing disk, then Lemma 3.2 would produce a new core $Y^{\prime}\subset
Y$ with $\chi(Y^{\prime})>\chi(Y)$, which contradicts minimality of $Y$. ∎

##  4\. Pleated surfaces

For the remainder of this section, we assume that $f\colon S\to S$ is a
strongly irreducible end-periodic homeomorphism. Throughout, $Y\subset S$ will
denote a core for $f$.

A pants–lamination on a hyperbolic surface $X$ is a geodesic lamination whose
leaves are the curves of a pants decomposition together with isolated leaves
such that each complementary component is an ideal triangle spiraling into all
three cuffs of its pair of pants, see Figure 8. We also make a technical
assumption that for each pants curve, the leaves that spiral in towards that
curve do so “in the same direction” on both sides of the curve. This is
necessary to ensure that in Section 5.2 we can find simplicial hyperbolic
surfaces limiting to our choice of pleated surface without adding many
additional vertices. See Figure 9 for an illustration of this behavior.

Figure 8. An ideal triangulation of a pair of pants. The three biinfinite
geodesics shown are examples of the isolated leaves found in a
pants–lamination.  Figure 9. The blue pants curve in the middle has leaves of
the pants-lamination spiraling in towards it “in the same direction” on both
sides, i.e. they both turn to the right before begining to spiral.

Let $M$ be a hyperbolic 3-manifold. A pleated surface is an arc-length
preserving map $\varphi:(X,\sigma)\to M$ from a surface $X$ with a complete
hyperbolic metric $\sigma$ such that: (a) $\varphi$ maps leaves of some
$\sigma$–geodesic lamination $\lambda\subset X$ to geodesics, and (b)
$\varphi$ is totally geodesic in the complement of $\lambda$. We say that
$\lambda$ is the pleating locus of $\varphi$ if it is the smallest lamination
satisfying conditions (a) and (b). See [CEG87, Section 5] for more details.

We introduce the following definition as an adaption of pleated surfaces to
our infinite-type setting. A well-pleated surface adapted to the core $Y$ in
$\overline{M}_{f}$ is a pleated surface
$\varphi\colon(S,\sigma)\to\overline{M}_{f}$, homotopic to the inclusion of
$S$ into $\overline{M}_{f}$, such that

  1. (1)

the pleating locus is contained in a pants–lamination, and

  2. (2)

$\varphi(S-Y)\subset\partial\overline{M}_{f}$.

Observe that “the inclusion” of $S$ into $\overline{M}_{f}$ is really only
well-defined up to precomposing with powers of $f$. We will later avoid this
issue by passing to the cover $\widetilde{M}_{\infty}\subset
S\times[-\infty,\infty]$, where this ambiguity disappears. We also note that
the metric $\sigma$ on $S$ is compatible with $Y$ and induced by the metric on
$\widetilde{M}_{\infty}$.

One may follow [CEG87, Theorem 5.3.6] to construct a well-pleated surface
representing the inclusion, adapted to any given core $Y$. For this, construct
a homotopy $h_{t}\colon S\to\overline{M}_{f}$ as in Section 3.2. We replace
$h_{1}$ with a homotopic map $\varphi^{\prime}\colon S\to\overline{M}_{f}$ so
that $\varphi^{\prime}$ sends each component of $\partial Y$ to a closed
geodesic in $\partial\overline{M}_{f}$. Since $h_{1}$ was a local
homeomorphism from the neighborhoods of the ends $S-Y$ onto
$\partial\overline{M}_{f}$, we can assume that $\varphi^{\prime}$ is as well.
We choose a hyperbolic structure on $S-Y$ so that $\varphi^{\prime}|_{S-Y}$ is
a local isometry. Now we choose any pants lamination $\lambda$ containing
$\partial Y$, and observe that we are now reduced to constructing a pleated
surface $Y\to\overline{M}_{f}$ homotopic to $\varphi^{\prime}|_{\partial Y}$,
rel $\partial Y$, realizing the finite lamination $\lambda\cap Y$. We can view
this restriction $\varphi^{\prime}|_{Y}\colon(Y,\partial
Y)\to(\overline{M}_{f},\partial\overline{M}_{f})$ as a pleated surface
representative of the $Y$–viscera, $(Y_{1},\partial
Y_{1})\subset(\overline{M}_{f},\partial\overline{M}_{f})$, realizing the
finite lamination $h_{1}(\lambda\cap Y)\subset Y_{1}$.

We let

$\Omega(f,Y)=\\{\varphi\colon(S,\sigma)\to\overline{M}_{f}\mid\varphi\mbox{ is
well-pleated adapted to }Y\\}$

denote the set of all well-pleated surfaces adapted to $Y$. A well pleated
surface adapted to some core will simply be called a well-pleated surface and
we denote the set of all such by $\Omega(f)$.

In the proof of the following lemma, we will make use of the theorem of
Basmajian [Bas94, Theorem 1.1] that there is a constant $\beta>0$, depending
only on $\xi(f)$, so that the $\beta$–neighborhood of
$\partial\overline{M}_{f}$ is a product,
$\partial\overline{M}_{f}\times[0,\beta]$.

######  Lemma 4.1 (Bounding the boundary).

Suppose that $Y$ is a minimal core of the strongly irreducible, end-periodic
homeomorphism $f\colon S\to S$ and suppose
${(\varphi\colon(S,\sigma)\to\overline{M}_{f})\in\Omega(f,Y)}$, then the total
$\sigma$–length of the boundary of $\partial Y$ satisfies
$\ell_{\sigma}(\partial Y)\leq\frac{2\pi|\chi(Y)|}{\beta}$.

###### Proof.

Take $\epsilon>0$ maximal such that the open $\epsilon$–neighborhood (with
respect to the metric $\sigma$) of $\partial Y$ are annuli. The boundary of
this neighborhood meets itself at some point, producing an essential arc of
length $2\epsilon$ in $Y$. The area of the $\epsilon$–neighborhood is no more
than the total area of $(Y,\sigma)$, which is $-2\pi\chi(Y)=2\pi|\chi(Y)|$ by
the Gauss–Bonnet formula. On the other hand, the area of this
$\epsilon$–neighborhood is $\sinh(\epsilon)\ell_{\sigma}(\partial
Y)>\epsilon\ell_{\sigma}(\partial Y)$, and so

$\epsilon\ell_{\sigma}(\partial Y)<2\pi|\chi(Y)|.$

If $\ell_{\sigma}(\partial Y)>\frac{2\pi|\chi(Y)|}{\beta}$, then
$\epsilon<\beta$. In this case, the closure of the $\epsilon$–neighborhood of
$\partial Y$ would map into the $\beta$–neighborhood of
$\partial\overline{M}_{f}$. It follows that $\alpha$ is mapped into the collar
neighborhood of $\partial\overline{M}_{f}$, and is hence properly homotopic
into $\partial\overline{M}_{f}$. But $\varphi|_{Y}$ is a pleated surface
representative of $Y_{1}$, and thus [AR93, Theorem 2.2] implies that $Y_{1}$
is boundary-compressible. This contradicts Proposition 3.4, since $Y$ was a
minimal core. Therefore, $\ell_{\sigma}(\partial
Y)\leq\frac{2\pi|\chi(Y)|}{\beta}$, as required. ∎

A minimally well-pleated surface is any well-pleated surface
$\varphi\in\Omega(f,Y)$ for which $Y$ is a minimal core.

######  Corollary 4.2.

If $\varphi\in\Omega(f)$ is minimally well-pleated and $\beta(\xi(f))>0$ is
Basmajian’s constant, then

$\ell_{\sigma}(\partial\Sigma_{\varphi})\leq\frac{2\pi|\chi(f)|}{\beta(\xi(f))}.$

##  5\. Interpolation and pants

In this section, we describe how to produce paths of pants decompositions that
we will use to prove the required lower bounds on volume. In what follows, we
typically assume $Y$ is a minimal core for a strongly irreducible end-periodic
$f$, so that $\chi(Y)=\chi(f)$.

###  5.1. Pants and cores

Suppose $(\varphi\colon(S,\sigma)\to\overline{M}_{f})\in\Omega(f,Y)$ is any
minimally well-pleated surface (i.e. $Y$ is a minimal core), and let
$U_{+}\cup U_{-}=S-Y$ be the associated tight nesting neighborhoods. Applying
an isotopy if necessary, we assume that for any component $\alpha$ of
$\partial_{-}Y$, $f^{-1}(\alpha)$ is either an essential curve of $U_{-}$ or
else a different component of $\partial_{-}Y$.

We also set

$\Delta_{+}=\overline{U_{+}-f(U_{+})},$

as in Section 2.3, which is a fundamental domain for the action of $f$ on
$\mathcal{U}_{+}$. Note that $\Delta_{+}$ is the subsurface bounded by (some
components of) $\partial_{+}Y$ and $f(\partial_{+}Y)$, by Lemma 2.5, since $Y$
is balanced by minimality.

Now we define

$\Sigma=Y\cup\Delta_{+},$

(see Figure 10) and observe that from the discussion in Section 2.3 we have

(5.1)
$\xi(\Delta_{+})\leq\tfrac{3}{2}|\chi(\Delta_{+})|=\tfrac{3}{4}|\chi(\partial\overline{M}_{f})|=\tfrac{1}{2}\xi(f).$

$Y$
---
$\Delta_{+}$
---
$\Sigma$
---
$U_{-}$
---
$U_{+}$
---
Figure 10. A schematic of the surface $S$ with $Y$, $\Delta_{+}$, and $\Sigma$
labelled. Here $\Delta_{+}$ is a fundamental domain for the action of $f$ on
$\mathcal{U}_{+}$, and $\Sigma$ is the union of the minimal core, $Y$, with
$\Delta_{+}$.

We are now ready to construct a bounded-length, $f$-asymptotic pants
decomposition of $S$.

######  Lemma 5.1.

There exists a constant $L=L(\chi(f),\xi(f))$ so that for any minimally well-
pleated surface
${(\varphi\colon(S,\sigma)\to\overline{M}_{f})\in\Omega(f,Y)}$, there exists a
pants decomposition $P$ of $S$ such that

  1. (1)

$\partial\Sigma\subset P$;

  2. (2)

$P$ and $f(P)$ differ only on $\Sigma$; and

  3. (3)

each curve in $P$ has $\sigma$–length at most $L$.

###### Proof.

Fix $(\varphi\colon(S,\sigma)\to\overline{M}_{f})\in\Omega(f,Y)$ minimally
well-pleated and adapted to a core $Y$, and continue with the notation as
above, so that $U_{\pm}$ are the tight nesting neighborhoods defined by $Y$.
Since $\varphi(U_{\pm})$ is entirely contained in $\partial\overline{M}_{f}$,
it follows that, with respect to the metric $\sigma$, after an isotopy we can
assume that $f$ isometrically maps $U_{+}$ into itself and $f^{-1}$
isometrically maps $U_{-}$ into itself.

We now construct the desired pants decomposition $P$ of $S$. We start with the
curves $\partial Y$. Note that $|\chi(Y)|=|\chi(f)|$ and

$\ell_{\sigma}(\partial
Y)\leq\frac{2\pi|\chi(Y)|}{\beta(\xi(f))}\leq\frac{2\pi|\chi(f)|}{\beta(\xi(f))},$

by Corollary 4.2. Since $\xi(Y)\leq\frac{3}{2}|\chi(Y)|$, Theorem 2.9 then
guarantees that we can choose a pants decomposition of $Y$ so that each pants
curve has length bounded by some $L_{0}=L_{0}(\chi(f),\xi(f))$. We require $P$
to contain this pants decomposition of $Y$ (including the boundary curves
$\partial Y$).

From Equation 5.1, $\xi(\Delta_{-})\leq\frac{1}{2}\xi(f)$. Since
$\ell_{\sigma}(\partial Y)$ is uniformly bounded and since $f$ isometrically
maps $U_{+}$ into itself, we see that there is also a uniform bound on the
length of $\partial\Delta_{+}$ in terms of $\chi(f)$ and $\xi(f)$. So again
applying Theorem 2.9, there is a pants decomposition $P_{+}$ of $\Delta_{+}$
(including the boundary curves, $\partial\Delta_{+}$) such that each pants
curve has length bounded by some $L_{1}=L_{1}(\chi(f),\xi(f))$. Then, since

$U_{+}=\bigcup_{j=0}^{\infty}f^{j}(\Delta_{+})$

with any two distinct translates $f^{j}\Delta_{+}$ and
$f^{j^{\prime}}\Delta_{+}$ intersecting at most in their boundary curves, we
can extend $P$ over $U_{+}$ so that

$U_{+}\cap P=\bigcup_{j=0}^{\infty}f^{j}(P_{+}),$

We similarly construct $P$ on $U_{-}$ with each curve’s length bounded by some
constant $L_{2}=L_{2}(\chi(f),\xi(f))$.

Now set $L=\max\\{L_{0},L_{1},L_{2}\\}$. The lemma follows by construction:
Item 3 is by definition, and Item 1 and Item 2 are a consequence of the fact
that $\Sigma=Y\cup\Delta_{+}$. ∎

######  Convention 5.2.

For the remainder of this section, fix
$(\varphi\colon(S,\sigma)\to\overline{M}_{f})\in\Omega(f,Y)$ with
$\chi(Y)=\chi(f)$ and $L=L(\chi(f),\xi(f))$, and $P$ as in Lemma 5.1. Note
that $P$ contains $\partial Y$, and so we may change our pleated surface
$\varphi$ to assume it realizes a pants lamination containing $P$. The
conclusions of the lemma are still satisfied for this well-pleated surface for
the same $P$.

Next, fix the lift $\widetilde{\varphi}\colon
S\to\widetilde{M}_{\infty}\subset S\times[-\infty,\infty]$ of $\varphi$ so
that $\widetilde{\varphi}$ is homotopic to the identity after projecting onto
the first factor.

Composing with the covering transformation $F$ on $\widetilde{M}_{\infty}$, we
get $F\circ\widetilde{\varphi}\colon S\to\widetilde{M}_{\infty}$, which is
another lift of $\varphi$, but rather than being homotopic to the identity on
$S$ (after projecting onto the first factor) it is homotopic to $f$, since
$F(x,t)=(f(x),t-1)$. Therefore,

$F\circ\widetilde{\varphi}\circ f^{-1}\colon S\to\widetilde{M}_{\infty}$

is homotopic to the identity (again, after projecting to the first factor).

Pulling back the metric by $f^{-1}$ gives a new (lift of a) well-pleated
surface

$F\circ\widetilde{\varphi}\circ
f^{-1}\colon(S,(f^{-1})^{*}\sigma)\to\widetilde{M}_{\infty}.$

Since $P$ has length at most $L$ with respect to $\sigma$, $f(P)$ has length
at most $L$ with respect to $(f^{-1})^{*}\sigma$.

By Lemma 5.1, $P$ and $f(P)$ differ only in $\Sigma$, and we define

$P_{\alpha}=P\cap\Sigma\quad\mbox{ and }\quad P_{\omega}=f(P)\cap\Sigma.$

These are pants decompositions of $\Sigma$ such that

(5.2) $d_{\mathcal{P}(S)}(P,f(P))\leq
d_{\mathcal{P}(\Sigma)}(P_{\alpha},P_{\omega}),$

where

(5.3)
$|\chi(\Sigma)|=|\chi(\Delta_{-})|+|\chi(Y)|=\tfrac{1}{3}\xi(f)+|\chi(f)|<\xi(f)+|\chi(f)|,$

and each curve of $\partial\Sigma$ has length at most $L$.

To simplify the notation, we write
$\phi_{\alpha}\colon\Sigma\to\widetilde{M}_{\infty}$ to denote the restriction
of $\widetilde{\varphi}$ to $\Sigma$ and
$\phi_{\omega}\colon\Sigma\to\widetilde{M}_{\infty}$ to denote the restriction
of $F\circ\widetilde{\varphi}\circ f^{-1}$ to $\Sigma$. As the notation
suggests, $\phi_{\alpha}$ maps the curves of $P_{\alpha}$ to geodesics of
length at most $L$ and $\phi_{\omega}$ maps the curves of $P_{\omega}$ to
geodesics of length at most $L$. We write $\sigma_{\alpha}$ and
$\sigma_{\omega}$ to denote the hyperbolic structures so that
$\phi_{\alpha}\colon(\Sigma,\sigma_{\alpha})\to\widetilde{M}_{\infty}$ and
$\phi_{\omega}\colon(\Sigma,\sigma_{\omega})\to\widetilde{M}_{\infty}$ are
pleated surfaces (representing the $\Sigma$–viscera). Since both
$\phi_{\alpha}$ and $\phi_{\omega}$ map $\partial\Sigma$ to geodesics, by
precomposing one of these with an isotopy of the identity, we assume (as we
may) $\phi_{\alpha}$ and $\phi_{\omega}$ agree on the boundary and that they
are homotopic by a homotopy that is stationary on $\partial\Sigma$.

###  5.2. Simplicial hyperbolic surfaces

To produce continuous families of “good” representatives of a homotopy class
of $\phi_{\alpha}\colon\Sigma\to\widetilde{M}_{\infty}$ (and hence of
$\phi_{\omega}$) we use simplicial hyperbolic surfaces, following [Can96].

We fix once and for all $k_{0}=|\partial\Sigma|$ points, one on each component
of $\partial\Sigma$. Let $k\geq k_{0}$. A $k$–simplicial pre-hyperbolic
surface in $\widetilde{M}_{\infty}$ is a map
$\eta\colon\Sigma\to\widetilde{M}_{\infty}$ that satisfies the following:

  * •

There is a triangulation111This is not a triangulation in the classical sense,
but rather a $\Delta$–complex structure in the sense of [Hat02]
${\mathcal{T}}$ of $\Sigma$ with $k$ vertices, exactly one on each boundary
component at the fixed $k_{0}$ points, such that $\eta$ takes each triangle to
a non-degenerate totally geodesic triangle in $\widetilde{M}_{\infty}$.

  * •

The restriction of $\eta$ to each component of $\partial\Sigma$ is a closed
geodesic.

  * •

The map $\eta$ is homotopic to $\phi_{\alpha}$ through maps that are
stationary on $\partial\Sigma$.

Note that, for such a surface, each component of $\partial\Sigma$ is
parameterized by a single edge of the triangulation under $\eta$. Furthermore,
the hyperbolic metrics on the triangles induce a singular hyperbolic metric
$\sigma_{\eta}$ on $\Sigma$. In this metric, the boundary is a smooth
geodesic, except possibly at the vertices. The $k$–simplicial pre-hyperbolic
surface is a $k$–simplicial hyperbolic surface if the cone angles in the
interior are all at least $2\pi$, and those on the boundary are at least
$\pi$. The set of all $k$–simplicial hyperbolic surfaces is denoted
${\mathcal{SH}}_{k}$, and the set of all simplicial hyperbolic surfaces by
${\mathcal{SH}}=\cup_{k}{\mathcal{SH}}_{k}$, and we equip both with the
compact–open topology. We will be primarily interested in $k$–simplicial
hyperbolic surfaces when $k=k_{0}$ or $k=k_{0}+1$.

The universal cover $\widetilde{M}$ of $\widetilde{M}_{\infty}$ may be
identified with a convex subset of $\mathbb{H}^{3}$ whose frontier is a union
of totally geodesic hyperbolic planes. This allows the identification of
$\overline{M}_{f}$ as the quotient of such a subset. We choose such an
identification, as well as an equivariant lift
$\widetilde{\Sigma}\to\widetilde{M}$ of $\phi_{\alpha}$, which also gives us
an equivariant lift $\widetilde{\eta}$ of any $k$–simplicial hyperbolic
surface $\eta$ by lifting the homotopy to $\phi_{\alpha}$.

We let ${\mathcal{SH}}({\mathcal{T}})\subset{\mathcal{SH}}$ be the subspace
consisting of all simplicial hyperbolic surfaces whose underlying
triangulation is ${\mathcal{T}}$.

######  Lemma 5.3.

For any triangulation ${\mathcal{T}}$ of $\Sigma$ with $k_{0}$ vertices, one
vertex at each of the fixed points on $\partial\Sigma$, the space
${\mathcal{SH}}({\mathcal{T}})$ is non-empty and any two elements of
${\mathcal{SH}}({\mathcal{T}})$ differ by precomposing with a homeomorphism
isotopic to the identity by an isotopy that is stationary on the boundary.

###### Proof.

We will first show that ${\mathcal{SH}}(\mathcal{T})$ is nonempty by
constructing an element $\eta:\Sigma\to\widetilde{M}_{\infty}$ of
${\mathcal{SH}}({\mathcal{T}})$ inductively over the skeleta.

We begin by declaring that $\eta$ agrees with $\phi_{\alpha}$ on the boundary
of $\Sigma$, and hence the vertices of ${\mathcal{T}}$, which is required for
any element of ${\mathcal{SH}}(\mathcal{T})$. Let $e$ be an edge of
${\mathcal{T}}$. If the endpoints of $e$ are distinct, then, by lifting to
$\mathbb{H}^{3}$ and using a straight–line homotopy, we may homotope
$\phi_{\alpha}\big{|}_{e}$ relative to its endpoints to a locally geodesic
segment. If this is the case, we define $\eta$ to send $e$ to this local
geodesic segment which necessarily has positive length: the endpoints are
either on distinct boundary components of $\partial\widetilde{M}_{\infty}$, or
on disjoint closed geodesics in a single component of
$\partial\widetilde{M}_{\infty}$. If $e$ is a loop, then it is homotopically
nontrivial, and hence it is carried by $\phi_{\alpha}$ to an essential loop,
and we homotope $\phi_{\alpha}$ by a straight–line homotopy again to $\eta$
mapping $e$ to the geodesic representative of the based homotopy class of
loops (if the loop is a boundary component, it is already geodesic and $\eta$
and $\phi_{\alpha}$ agree there). The boundary $\partial T$ of any triangle
$T$ in ${\mathcal{T}}$ is null–homotopic in $\Sigma$, and so it also lifts to
the universal cover where we can extend our straight line homotopy to a
homotopy of $\phi_{\alpha}|_{T}$ to $\eta|_{T}$ mapping $T$ to a (possibly
degenerate) geodesic triangle immersed in $\widetilde{M}_{\infty}$. Having
done this for every triangle $T$, we have defined
$\eta:\Sigma\to\widetilde{M}_{\infty}$ and the homotopy from $\phi_{\alpha}$.

We claim that every geodesic triangle $\eta|_{T}$ is in fact non-degenerate.
This means that the lift $\widetilde{\eta}|_{T}$ to $\mathbb{H}^{3}$ is an
embedding of a geodesic triangle. Suppose that this is not the case. First,
observe that every edge of $T$ in $\Sigma$ either connects distinct vertices
or is a non-null homotopic loop, so the image of each edge is a non-degenerate
segment. Thus, the only degeneracy that may occur in $\widetilde{\eta}|_{T}$
is that all three vertices lie on a single geodesic segment. All three of
these points lie on the boundary of $\widetilde{M}$, which we recall is a
convex subset of $\mathbb{H}^{3}$ bounded by hyperbolic planes. Therefore, the
entire segment must lie in the boundary.

Now, projecting the segment back to $\widetilde{M}_{\infty}$, we obtain a
segment in $\partial\widetilde{M}_{\infty}$ passing through three vertices in
$\eta(\partial\Sigma)$. Note that the segment cannot be entirely contained in
$\eta(\partial\Sigma)$ since then all three edges of $T$ are mapped to the
homotopy class of the boundary loop, which is not allowed in the
triangulation. Therefore, the segment defines a geodesic path in a component
of $\partial\widetilde{M}_{\infty}$, and hence in either
$\mathcal{U}_{+}\times\\{\infty\\}$ or $\mathcal{U}_{-}\times\\{-\infty\\}$.
Moreover, this segment meets the geodesic boundary $\eta(\partial\Sigma)$
transversely. A subsegment of the path between two of the vertices enters one
of $U_{+}\times\\{\infty\\}$ or $U_{-}\times\\{-\infty\\}$ from the vertex.
Projecting $\widetilde{M}_{\infty}$ onto the first factor $S$, this subsegment
projects to an essential path in $S$ which cannot be homotoped entirely
contained in $\Sigma$. This is impossible because the path is homotopic to an
edge of the triangulation, and hence an essential arc in $\Sigma$. Therefore,
the triangle $T$ is non-degenerate.

The link of a vertex $v$ in ${\mathcal{T}}$ defines a path in the sphere
$T^{1}_{\widetilde{\eta}(v)}(\mathbb{H}^{3})$ joining antipodal points—namely,
the two tangent vectors to the boundary geodesic—and thus has length at least
$\pi$, proving that the cone angle at the vertices is at least $\pi$ (c.f. the
NLSC [not locally strictly convex] property and [Can96, Lemma 4.2]).
Therefore, after reparameterizing if necessary, we conclude that $\eta$ is the
desired $k_{0}$–simplicial hyperbolic surface and so,
${\mathcal{SH}}(\mathcal{T})$ is non-empty.

Now, given any $\eta,\eta^{\prime}\in{\mathcal{SH}}(\mathcal{T})$, the two
maps are homotopic to $\phi_{\alpha}$ by a homotopy that is stationary on
$\partial\Sigma$, they are also homotopic to each other by such a homotopy.
Since each edge $e$ of $\mathcal{T}$ must be sent to the geodesic in the
relative homotopy class of $\phi_{\alpha}\big{|}_{e}$, it follows that $\eta$
and $\eta^{\prime}$ differ on each edge by a reparameterization constant on
the endpoints. Therefore, $\eta$ and $\eta^{\prime}$ differ by
reparameterization that maps simplicies to simplices. Since each triangle maps
to a non-degenerate triangle, the reparameterization is necessarily a
homeomorphism preserving the triangulation which is the identity on the
vertices. It follows that this reparameterizing homeomorphism is isotopic to
the identity rel the vertices, and thus $\eta$ and $\eta^{\prime}$ differ by
precomposing by a homeomorphism isotopic to the identity rel the vertices. ∎

For each component $\gamma\subset\partial\Sigma$,
$\phi_{\alpha}(\gamma)=\phi_{\omega}(\gamma)$ is a closed geodesic in the
totally geodesic boundary $\partial\widetilde{M}_{\infty}$ of length at most
$L$. Consider the annular cover of the component of
$\partial\widetilde{M}_{\infty}$ to which this curve lifts. This annulus is
divided by (the lift of the image of) $\gamma$ into two half-open annuli with
boundary $\gamma$, and we let $\breve{\Sigma}\supset\Sigma$ be obtained by
gluing these half-open annuli to each boundary component of $\Sigma$. For any
$\eta\colon(\Sigma,{\mathcal{T}})\to\widetilde{M}_{\infty}\in{\mathcal{SH}}$,
we have $\eta|_{\gamma}=\phi_{\alpha}|_{\gamma}$, and so we can extend $\eta$
to a map

$\breve{\eta}\colon\breve{\Sigma}\to\widetilde{M}_{\infty},$

whose restriction to the added half-open annuli is the restriction of the
covering map to $\partial\widetilde{M}_{\infty}$. The singular hyperbolic
metric $\sigma_{\eta}$ extends to a singular hyperbolic metric of the same
name on $\breve{\Sigma}$, so that $\breve{\eta}$ is a local isometry on each
half-open annulus. Then let $\sigma_{\eta}^{{\mathcal{H}}}$ be the hyperbolic
uniformization of the conformal structure on $(\breve{\Sigma},\sigma_{\eta})$,
considered as a point in the Teichmüller space $\mathcal{T}(\breve{\Sigma})$.
We write $\ell_{\sigma_{\eta}^{\mathcal{H}}}(\gamma)$ for the length of the
$\sigma_{\eta}^{\mathcal{H}}$–geodesic representative of any essential closed
curve $\gamma$ in $\breve{\Sigma}$.

######  Theorem 5.4.

For any $\eta\colon(\Sigma,{\mathcal{T}})\to\widetilde{M}_{\infty}$ in
${\mathcal{SH}}$, the identity map

${\rm
id}\colon(\breve{\Sigma},\sigma_{\eta}^{\mathcal{H}})\to(\breve{\Sigma},\sigma_{\eta})$

is $1$–Lipschitz. Furthermore, for each component
$\gamma\subset\partial\Sigma$, the
$\ell_{\sigma_{\eta}^{\mathcal{H}}}(\gamma)\leq 2L$.

###### Proof.

The first statement is a consequence of the Ahlfors–Schwartz–Pick Theorem
[Ahl38]. For the second statement, we will use a modulus argument (see [EM06,
Theorem 2.16.1] for a precise statement of the correspondence between modulus
of an annulus and the length of its core curve). Note that the length of
$\phi_{\alpha}(\gamma)=\eta(\gamma)$ in $\widetilde{M}_{\infty}$ is at most
$L$, and so the annular cover of the component of
$\partial\widetilde{M}_{\infty}$ containing this curve has modulus at least
$\frac{\pi}{L}$. Consequently, the interior of the half-open annulus (which is
half of this annular cover) has modulus at least $\frac{\pi}{2L}$. But this
annulus lifts to the annular cover of $\breve{\Sigma}$ to which $\gamma$
lifts, and hence this cover has modulus at least $\frac{\pi}{2L}$ by
monotonicity of moduli of annuli. This in turn implies that
$\ell_{\sigma_{\eta}^{\mathcal{H}}}(\gamma)\leq 2L$, as required. ∎

###  5.3. Interpolation and pants paths

Two triangulations ${\mathcal{T}}$ and ${\mathcal{T}}^{\prime}$ differ by a
flip move if there are edges $e$ of ${\mathcal{T}}$ and $e^{\prime}$ of
${\mathcal{T}}^{\prime}$ so that $e$ and $e^{\prime}$ intersect transversely
in a single point and their complements in the $1$–skeleta agree:

${\mathcal{T}}^{(1)}-e={\mathcal{T}}^{\prime(1)}-e^{\prime}.$

In this case, let ${\mathcal{T}}*{\mathcal{T}}^{\prime}$ be the triangulation
obtained from ${\mathcal{T}}$ by adding a vertex at $e\cap e^{\prime}$,
subdividing each of $e$ and $e^{\prime}$ and adding the subdivided
$e^{\prime}$ to the $1$–skeleton. This is the “minimal common subdivision” of
${\mathcal{T}}$ and ${\mathcal{T}}^{\prime}$ which is illustrated in Figure
11.

$e$$e^{\prime}$ Figure 11. The minimal common subdivision
${\mathcal{T}}*{\mathcal{T}}^{\prime}$ (shown on the right) of ${\mathcal{T}}$
and ${\mathcal{T}}^{\prime}$ (shown on the left) where $e\subset{\mathcal{T}}$
is represented by the solid black diagonal edge and
$e^{\prime}\subset{\mathcal{T}}^{\prime}$ is represented by the dashed purple
diagonal edge. The four pink dotted edges in
${\mathcal{T}}*{\mathcal{T}}^{\prime}$ are the new edges introduced by
subdividing $e$ and $e^{\prime}$

Given $\eta\colon(\Sigma,{\mathcal{T}})\to\widetilde{M}_{\infty}$ in
${\mathcal{SH}}_{k_{0}}$ and a triangulation ${\mathcal{T}}^{\prime}$
differing from ${\mathcal{T}}$ by a flip, we may reparameterize $\eta$ by
precomposing with a homeomorphism isotopic to the identity so that it is also
a $(k_{0}+1)$–simplicial hyperbolic surface

$\eta\colon(\Sigma,{\mathcal{T}}*{\mathcal{T}}^{\prime})\to\widetilde{M}_{\infty}.$

The proof of [Can96, Lemma 5.3] can be applied to prove the following.

######  Lemma 5.5.

Suppose ${\mathcal{T}}$ and ${\mathcal{T}}^{\prime}$ differ by a flip and that
$\eta\colon(\Sigma,{\mathcal{T}})\to\widetilde{M}_{\infty}$ and
$\eta^{\prime}\colon(\Sigma,{\mathcal{T}}^{\prime})\to\widetilde{M}_{\infty}$
lie in ${\mathcal{SH}}_{k_{0}}$. Then there is a $1$–parameter family

$\\{\eta_{t}\colon(\Sigma,{\mathcal{T}}*{\mathcal{T}}^{\prime})\to\widetilde{M}_{\infty}\mid
t\in[0,1]\\}\subset{\mathcal{SH}}_{k_{0}+1}$

such that $\eta_{0}=\eta$ and $\eta_{1}=\eta^{\prime}$, up to
reparameterization by homeomorphisms isotopic to the identity by an isotopy
that is stationary on the boundary. ∎

Rather than repeat all the details of the proof as in [Can96], we explain the
basic idea. By further reparameterization if necessary, the two maps $\eta$
and $\eta^{\prime}$ will agree outside the “square” with diagonals $e$ and
$e^{\prime}$, and the interpolation takes place entirely within this square.
Lifting the restrictions of $\eta$ and $\eta^{\prime}$ to the square, the two
original triangles of $\mathcal{T}$ in this square, together with the two
triangles of $\mathcal{T}^{\prime}$ define a tetrahedron in $\mathbb{H}^{3}$.
Connecting the image of $e$ via the lift of $\eta$ to the image of
$e^{\prime}$ via the lift of $\eta^{\prime}$ by a geodesic segment, the one-
parameter family is essentially obtained by “sliding” the new vertex along
this geodesic, and then projecting back to $\widetilde{M}_{\infty}$.

We will also need the following result; see e.g. Hatcher [Hat91].

######  Lemma 5.6.

Let ${\mathcal{T}}$ and ${\mathcal{T}}^{\prime}$ be two triangulations of
$\Sigma$ with $k_{0}$ vertices, one on each component of $\partial\Sigma$.
Then there is a sequence
${\mathcal{T}}_{0}={\mathcal{T}},{\mathcal{T}}_{1},\ldots,{\mathcal{T}}_{m}$
of triangulations, each differing from the previous one by a flip so that
$\mathcal{T}_{m}=\mathcal{T}^{\prime}$, up to isotopy which is stationary on
$\partial\Sigma$. ∎

We will use these two lemmas together with Lemma 5.3 to prove the following
corollary.

######  Corollary 5.7.

Suppose $\eta\colon(\Sigma,{\mathcal{T}})\to\widetilde{M}_{\infty}$ and
$\eta^{\prime}\colon(\Sigma,{\mathcal{T}}^{\prime})\to\widetilde{M}_{\infty}$
are two $k_{0}$–simplicial hyperbolic surfaces. Then there is a $1$–parameter
family

$\\{\eta_{t}\colon(\Sigma,{\mathcal{T}}_{t})\to\widetilde{M}_{\infty}\mid
t\in[0,1]\\}\subset{\mathcal{SH}}$

such that $\eta_{0}=\eta$ and $\eta_{1}=\eta^{\prime}$, up to
reparameterization by homeomorphisms isotopic to the identity by an isotopy
that is stationary on the boundary.

###### Proof.

Let $\mathcal{T}=\mathcal{T}_{0},\ldots,\mathcal{T}_{m}=\mathcal{T}^{\prime}$
be the sequence of flips from Lemma 5.6. By Lemma 5.3, there is a sequence of
simplicial hyperbolic surfaces

$\\{\eta_{j}\colon(\Sigma,\mathcal{T}_{j})\to\widetilde{M}_{\infty}\\}_{j=0}^{m}$

and $\eta_{0}=\eta$, $\eta_{m}=\eta^{\prime}$, up to reparameterization
(isotopic to the identity by an isotopy that is stationary on the boundary).
By Lemma 5.5, for each $j=1,\ldots,m$, we can interpolate between $\eta_{j-1}$
and $\eta_{j}$ by a one-parameter family of simplicial hyperbolic surfaces.
Concatenating these one-parameter families, and reparameterizing by
precomposing by isotopies between these families whenever necessary, produces
the required 1-parameter family from $\eta$ to $\eta^{\prime}$, as required. ∎

We are now able to define a continuous path in Teichmüller space given by the
hyperbolic structures obtained from uniformization of the 1-parameter family
of simplicial hyperbolic surfaces given to us in Corollary 5.7.

######  Lemma 5.8.

Given the family
$\\{\eta_{t}\colon(\Sigma,{\mathcal{T}}_{t})\to\widetilde{M}_{\infty}\mid
t\in[0,1]\\}\subset{\mathcal{SH}}$ from Corollary 5.7, the map
$[0,1]\to\rm{Teich}(\breve{\Sigma})$, given by
$t\mapsto\sigma_{\eta_{t}}^{\mathcal{H}}$ defines a (continuous) path in
$\rm{Teich}(\breve{\Sigma})$.

###### Proof.

The family in Lemma 5.5 defines a continuous path in
$\rm{Teich}(\breve{\Sigma})$ since the shapes of the hyperbolic triangles in
the interpolation vary continuously; see [Can96]. The terminal point of the
path from $\eta_{j-1}$ to $\eta_{j}$ and the initial point of the path from
$\eta_{j}$ to $\eta_{j+1}$ differ by reparameterization by a homeomorphism
isotopic to the identity. This isotopy defines a constant path in
$\rm{Teich}(\breve{\Sigma})$ since the cone metrics are all obtained by
pulling back the same cone metric by the homeomorphisms throughout the
isotopy. Therefore, the paths can be concatenated to produces a path from
$\sigma_{\eta}^{\mathcal{H}}$ to $\sigma_{\eta^{\prime}}^{\mathcal{H}}$, as
required. ∎

Recall that $P_{\alpha},P_{\omega}$ are pants decompositions on $\Sigma$ so
that with respect to $\sigma_{\alpha}$ and $\sigma_{\omega}$, the lengths of
each component of $P_{\alpha}$ and $P_{\omega}$, respectively are at most
$L=L(\chi(f),\xi(f))$, from Lemma 5.1. The next lemma says we can find
simplicial hyperbolic surfaces whose cone metrics have hyperbolic
uniformizations in which $P_{\alpha}$ and $P_{\omega}$ are also bounded
length.

######  Lemma 5.9.

There exists $L_{1}=L_{1}(\chi(f),\xi(f))$ and a pair of simplicial hyperbolic
surfaces
$\eta_{\alpha}\colon(\Sigma,{\mathcal{T}}_{\alpha})\to\widetilde{M}_{\infty}$
and
$\eta_{\omega}\colon(\Sigma,{\mathcal{T}}_{\omega})\to\widetilde{M}_{\infty}$
in ${\mathcal{SH}}_{k_{0}}$ such that each component of $P_{\alpha}$ and
$P_{\omega}$ have length at most $L_{1}$ in
$\sigma_{{\mathcal{T}}_{\alpha}}^{\mathcal{H}}$ and
$\sigma_{{\mathcal{T}}_{\omega}}^{\mathcal{H}}$, respectively.

######  Remark.

We will ultimately use an interpolation through simplicial hyperbolic surfaces
to find a path in the pants graph between $P_{\alpha}$ and $P_{\omega}$ via
Lemma 5.8. In the finite-type case [Bro03b], Brock similarly constructs such a
path, though in his situation, the initial and terminal pants decompositions
are defined from a simplicial hyperbolic surface and its image under a power
of the monodromy. In our case, $\Sigma$ is not invariant by any nontrivial
power of $f$. However, Lemma 5.9 allows us to choose simplicial hyperbolic
surfaces that are adapted to our existing pants decompositions $P_{\alpha}$
and $P_{\omega}$ (as opposed to choosing these pants decompositions from the
simplicial hyperbolic surfaces themselves). Our construction in Lemma 5.9 is
guided by the minimally well-pleated surfaces $\phi_{\alpha}$ and
$\phi_{\omega}$ that realize $P_{\alpha}$ and $P_{\omega}$, respectively.

###### Proof of Lemma 5.9.

We carry out the proof for $P_{\alpha}$ with the one for $P_{\omega}$ being
identical. Recall that $P_{\alpha}$ is the pants decomposition in $\Sigma$ so
that the pleating locus of $\phi_{\alpha}$ is contained in a pants lamination
$\lambda$ containing $P_{\alpha}$. Choose a triangulation $\mathcal{T}$ of
$\Sigma$ so that the arcs of intersection of the $1$–skeleton with each pair
of pants connect every pair of distinct boundary curves in those pants. To do
this, we can first find a collection of pairwise disjoint essential arcs with
this property, slide the endpoints to lie on the $k_{0}$ vertices, then extend
to a triangulation.

For each closed curve $\delta$ in $P_{\alpha}\cup\partial\Sigma$, the non-
compact leaves spiraling in toward $\delta$ spiral to the left or to the right
(from both sides when $\delta\subset P_{\alpha}$, by assumption; see Section
4). Let $D\colon\Sigma\to\Sigma$ be the multitwist obtained by applying a
right-handed Dehn twist around those $\delta\subset
P_{\alpha}\cup\partial\Sigma$ for which the spiraling is to the right, and a
left-handed Dehn twist around those $\delta$ for which the spiraling is to the
left, and set $\mathcal{T}_{j}=D^{j}(\mathcal{T})$, for all integers $j\geq
0$. We also let
$\eta_{j}\colon(\Sigma,\mathcal{T}_{j})\to\widetilde{M}_{\infty}$ be the
simplicial hyperbolic surface given by Lemma 5.3.

Consider the sequence of triangulations $\mathcal{T}_{j}$ on $\Sigma$,
straightened to be geodesic with respect to $\sigma_{\alpha}$. Lift
$\phi_{\alpha}$ to the universal covers
$\tilde{\phi}_{\alpha}\colon\widetilde{\Sigma}\to\widetilde{M}\subset\mathbb{H}^{3}$.
We write $\widetilde{\mathcal{T}}_{j}$ for the lifted triangulation.

######  Claim 5.10.

Given $\epsilon>0$, there exists $J>0$ so that for all $j\geq J$ and every
edge $e$ of $\widetilde{\mathcal{T}}_{j}$, $\tilde{\phi}_{\alpha}\big{|}_{e}$
is a $(1+\epsilon,\epsilon)$–quasi-geodesic.

###### Proof.

First, we observe that as $j$ tends to infinity, we have Hausdorff convergence
of the $1$–skeleta, $\mathcal{T}_{j}^{(1)}\to\lambda$. This is because all
angles of intersection with $P_{\alpha}\cup\partial\Sigma$ tend to zero, so
the limit is a lamination containing $P_{\alpha}\cup\partial\Sigma$. See
Figure 12. Further, in any pair of pants, our original choice of $\mathcal{T}$
guarantees that there are non-compact leaves that spiral between any two
boundary components. Finally, our choice of $D$ ensures that the spiraling
toward all the curves is in the correct direction. These conditions uniquely
determine the pants lamination $\lambda$. If $\tilde{\lambda}$ is the lifted
lamination to $\widetilde{\Sigma}$, then we also have
$\widetilde{\mathcal{T}}_{j}^{(1)}\to\tilde{\lambda}$ as $j\to\infty$. Thus,
given $c,r>0$ there exists $J>0$ so that for all $j>J$, every edge $e$ of
$\widetilde{\mathcal{T}}_{j}$, and every segment $\delta\subset e$ of length
at most $r$ and there is a segment $\delta^{\prime}$ of a leaf of $\lambda$
such that the endpoints of $\delta^{\prime}$ and the endpoints of $\delta$ are
at most $c$ apart.

Since $\tilde{\phi}_{\alpha}$ maps each leaf of $\tilde{\lambda}$ to a
geodesic and is $1$–Lipschitz, it follows that the endpoints of
$\tilde{\phi}_{\alpha}(\delta)$ are within $c$ of the endpoints of the
geodesic $\tilde{\phi}_{\alpha}(\delta^{\prime})$. Therefore, if $x,y$ are the
endpoints of $\delta$, we have

$d(\tilde{\phi}_{\alpha}(x),\tilde{\phi}_{\alpha}(y))\geq\ell(\tilde{\phi}_{\alpha}(\delta^{\prime}))-2c=\ell(\delta^{\prime})-2c\geq\ell(\delta)-4c=d(x,y)-4c.$

Since $\tilde{\phi}_{\alpha}\big{|}_{e}$ is $1$–Lipschitz, it follows that
this path is an $r$–local, $(1,4c)$–quasi-geodesic (i.e. every segment of
length less than $r$ is a $(1,4c)$–quasi-geodesic). Using hyperbolic geometry,
we can find $r$ sufficiently large and $c$ sufficiently small so that such a
path is also $(1+\epsilon,\epsilon)$–quasi-geodesic. Specifically, take $r=4$,
and consider consecutive points $x_{0},x_{1},\ldots,x_{n}$ along $e$ with
$1\leq d(x_{j},x_{j+1})\leq 2$. Taking $c$ sufficiently small, the angle at
$\tilde{\phi}_{\alpha}(x_{j})$ between geodesic segments
$[\tilde{\phi}_{\alpha}(x_{j-1}),\tilde{\phi}_{\alpha}(x_{j})]$ and
$[\tilde{\phi}_{\alpha}(x_{j}),\tilde{\phi}_{\alpha}(x_{j+1})]$ in
$\widetilde{M}\subset\mathbb{H}^{3}$ can be made arbitrarily close to $\pi$
(depending on $c$), and then we can apply [CEG87, Theorem 4.2.10] to see that
the concatenation of geodesic segments is arbitrarily close to the geodesic.
Then $\tilde{\phi}_{\alpha}\big{|}_{e}$ is also as close to a geodesic as we
like, and we can promote the local quasi-geodesic to a global quasi-geodesic.
Taking $J$ large enough to find such an $r$ and $c$, completes the proof of
the claim. ∎

Figure 12. By “spinning” the triangulation on the left around each boundary
component—more precisely, applying powers of Dehn twists about curves parallel
to the boundary—we obtain the pants lamination in the limit, as shown on the
right.

######  Claim 5.11.

Given $\epsilon>0$, there exists $J>0$ so that for all $j\geq J$:

  1. (1)

For every component $\gamma\subset P_{\alpha}$, we have
$\ell_{\sigma_{\eta_{j}}}(\gamma)\leq\ell_{\sigma_{\alpha}}(\gamma)+\epsilon$,
and

  2. (2)

all cone angles in the boundary of $\Sigma$ for $\mathcal{T}_{j}$ are at most
$\pi+\epsilon$.

Note that because $\sigma_{\alpha}$ maps any component $\gamma\subset
P_{\alpha}$ to a geodesic, and since $\eta_{j}$ is $1$–Lipschitz with respect
to $\sigma_{\eta_{j}}$, we have
$\ell(\phi_{\alpha}(\gamma))=\ell_{\sigma_{\alpha}}(\gamma)\leq\ell_{\sigma_{\eta_{j}}}(\gamma)$,
so part (1) in the claim really says that $\ell_{\sigma_{\alpha}}(\gamma)$ and
$\ell_{\sigma_{\eta_{j}}}(\gamma)$ are nearly the same.

###### Proof.

Let $m$ be the maximum geometric intersection number between (the union of
arcs in) $\mathcal{T}^{(1)}$ and any component of $P_{\alpha}$. Since
$\mathcal{T}^{(1)}_{j}=D^{j}(\mathcal{T}^{(1)})$, this maximum, $m$, also
bounds the geometric intersection number between $\mathcal{T}^{(1)}_{j}$ and
any component of $P_{\alpha}$.

Now, let
$\tilde{\eta}_{j}\colon\widetilde{\Sigma}\to\widetilde{M}\subset\mathbb{H}^{3}$
be the lift of the simplicial hyperbolic surface
$\eta_{j}\colon(\Sigma,\mathcal{T}_{j})\to\widetilde{M}_{\infty}$. From 5.10,
we deduce that, for $j$ sufficiently large, and for every edge $e$ of
$\widetilde{\mathcal{T}}_{j}$, $\tilde{\eta}_{j}\big{|}_{e}$ and
$\tilde{\phi}_{\alpha}|_{e}$ can be made arbitrarily close (depending on $j$).

For any component $\gamma\subset P_{\alpha}$ pick a lift
$\widetilde{\gamma}\subset\widetilde{\Sigma}$ and a segment $\bar{\gamma}$
that serves as a fundamental domain for the action of $\langle g\rangle$, the
stabilizer of $\widetilde{\gamma}$ in $\pi_{1}\Sigma$. Then, for any $j$
consider the set of (at most $m$) edges of $\widetilde{\mathcal{T}}_{j}$ that
cross $\widetilde{\gamma}$ at a point of $\bar{\gamma}$. For $j$ sufficiently
large, the $\tilde{\phi}_{\alpha}$-image of these edges are arbitrarily close
to the geodesic $\tilde{\phi}_{\alpha}(\widetilde{\gamma})$ on arbitrarily
long segments (depending on $j$), thus the same is true for the
$\tilde{\eta}_{j}$ images of these edges.

Next, pick the edge $e\subset\widetilde{\mathcal{T}}_{j}$ intersecting
$\bar{\gamma}$ as close to the initial point of $\bar{\gamma}$ as possible and
assume that $j$ is sufficiently large so that all the edges between $e$ and
$g\cdot e$ (ordered by the intersection with $\widetilde{\gamma}$) are mapped
within $\frac{\epsilon}{2(m+1)}$ of
$\tilde{\phi}_{\alpha}(\widetilde{\gamma})$ on segments of length at least
$\ell_{\sigma_{\alpha}}(\gamma)+\epsilon$ centered at the points of
intersection with $\widetilde{\gamma}$.

$g\cdot e$$e$$\bar{\gamma}$$g$$\widetilde{\gamma}$ Figure 13. The edges of
$\widetilde{\mathcal{T}}_{j}$ through $\widetilde{\gamma}$ between $e$ and
$g\cdot e$ are mapped close to $\tilde{\phi}_{\alpha}(\widetilde{\gamma})$ for
a long time by $\eta_{j}$. The blue segment connects $e\cap\widetilde{\gamma}$
to $g\cdot e\cap\widetilde{\gamma}$, and projects to a closed loop in
$\Sigma$.

From this we can construct a path in $\widetilde{\Sigma}$ from
$e\cap\widetilde{\gamma}$ to $g\cdot e\cap\widetilde{\gamma}$ built from a
segment of $e$ of length at most
$\ell_{\sigma_{\alpha}}(\gamma)+\frac{\epsilon}{2(m+1)}$ and at most $m$ short
segments of length at most $2\frac{\epsilon}{2(m+1)}$ between consecutive
edges of $\widetilde{\mathcal{T}}_{j}$. Thus the $\sigma_{\eta_{j}}$–length of
this path is at most

$\ell_{\sigma_{\alpha}}(\gamma)+\frac{\epsilon}{2(m+1)}+m\left(\frac{\epsilon}{(m+1)}\right)<\ell_{\sigma_{\alpha}}(\gamma)+\epsilon.$

See Figure 13. This path then projects to a loop in $\Sigma$ homotopic to
$\gamma$ of $\sigma_{\eta_{j}}$–length at most
$\ell_{\sigma_{\alpha}}(\gamma)+\epsilon$. This proves part (1) of the claim.

For part (2), we similarly observe that for any vertex $v$, any edge $e$
adjacent to $v$ has $\eta_{j}$–image that stays arbitrarily close to the
geodesic image of the boundary component $\eta_{j}(\gamma)$ for an arbitrarily
long time (depending on $j$). Now the boundary component $\gamma$ gives two
adjacencies to $v$ pointing in opposite directions, and all other edges
adjacent to $v$ have $\eta_{j}$–image making arbitrarily small angle with the
$\eta_{j}$–image of exactly one of these. It follows that there is one angle
equal to at most $\pi$, and at most $m$ angles that can be made arbitrarily
small, depending on $j$. It follows that for $j$ sufficiently large, the angle
can be made at most $\pi+\epsilon$, proving part (2). ∎

To complete the proof, we now observe that since the
$\sigma_{\eta_{j}}$–length of each boundary curve is the same as its length
with respect to $\sigma_{\alpha}$, it is bounded by $L$. Further, since the
cone angle is less than $\pi+\epsilon$, then provided $\epsilon<\pi$, we can
enlarge $\Sigma$ to an open surface, $\Sigma\subset\Sigma^{\circ}$, and extend
$\sigma_{\alpha}$ to a complete non-singular hyperbolic metric. The convex
core $\bar{\Sigma}$ of $\Sigma^{\circ}$ will contain $\Sigma$; see Figure 14.

$\Sigma^{\circ}$
---
$\bar{\Sigma}$
---
$\Sigma$
---
Figure 14. Enlarging $\Sigma$ to $\Sigma^{\circ}$, with convex core
$\bar{\Sigma}$.

Given any $\epsilon^{\prime}>0$, there exists $\epsilon>0$ so that if
$\gamma\subset\partial\Sigma$ is any component and if the cone angle at the
vertex on $\gamma$ has angle less than $\pi+\epsilon$, then
$\bar{\Sigma}\subset N_{\epsilon^{\prime}}(\Sigma)$ on $\Sigma^{\circ}$ with
respect to $\sigma_{\eta_{j}}$, for any $j\geq J$ as in 5.11. We note that the
choice of $\epsilon$ depends not only on $\epsilon^{\prime}$, but also on a
lower bound on
$\ell_{\sigma_{\eta_{j}}}(\gamma)=\ell_{\sigma_{\alpha}}(\gamma)$.

For any $0<\epsilon<1$, $j\geq J$ as in 5.11, and any component $\gamma\subset
P_{\alpha}$, $\ell_{\sigma_{\eta_{j}}}(\gamma)<L+\epsilon<1$. By the collar
lemma, there exists $w=w(L+1)$ such that the $w$–neighborhood of $\gamma$ is
an embedded annulus in $\bar{\Sigma}$. Thus, if
$\epsilon^{\prime}<\frac{w}{2}$, and $\epsilon$ is chosen as in the previous
paragraph, then the collar of width $w/2$ is entirely contained in
$\Sigma\subset\breve{\Sigma}$. This gives a lower bound on the
$\sigma_{\eta_{j}}$–moduli of $\gamma$, and hence an upper bound $L_{1}$ on
$\ell_{\sigma_{\eta_{j}}^{\mathcal{H}}}(\gamma)$ depending on $L$ and $w$,
which in turn both depend only on $\chi(f)$ and $\xi(f)$, as required. ∎

###  5.4. Pants distances and bounded length curves

Write $\mathcal{T}_{2L}(\breve{\Sigma})$ to denote the subspace of
$\mathcal{T}(\breve{\Sigma})$ consisting of complete hyperbolic structures for
which the (geodesic representatives of) components of $\partial\Sigma$ have
length at most $2L$. Set

(5.4) $\Xi=\Xi(f)=\tfrac{3}{2}\left(\xi(f)+|\chi(f)|\right).$

From Equation 5.3 we have that
$\xi(\Sigma)\leq\frac{3}{2}|\chi(\Sigma)|\leq\Xi$. Now set

$L_{2}=L_{2}(\chi(f),\xi(f))=\max\\{L_{1},L_{B}(\Xi,2L)\\},$

where $L_{B}$ is as in Theorem 2.9.

For any pants decomposition $P_{0}$ of $\Sigma$, let

$V(P_{0},L_{2})=\\{\sigma\in\mathcal{T}_{2L}(\Sigma)\mid\ell_{\sigma}(\gamma)<L_{2}\mbox{
for each component }\gamma\subset P_{0}\\}.$

By Theorem 2.9, the set $\\{V(P_{0},L_{2})\\}_{P_{0}\in{\mathcal{P}}(\Sigma)}$
is an open cover of $\mathcal{T}_{2L}(\Sigma)$.

Now let

$\\{(\eta_{t}\colon(\Sigma,{\mathcal{T}}_{t})\to\widetilde{M}_{\infty})\mid
t\in[0,1]\\}\subset{\mathcal{SH}}$

be the path of simplicial hyperbolic surfaces from Lemma 5.8 connecting the
$k_{0}$–simplicial hyperbolic surfaces
$\eta_{\alpha}\colon(\Sigma,{\mathcal{T}}_{\alpha})\to\widetilde{M}_{\infty}$
to
$\eta_{\omega}\colon(\Sigma,{\mathcal{T}}_{\omega})\to\widetilde{M}_{\infty}$
from Lemma 5.9. By compactness (and Lemma 5.9), there is a partition

$0=t_{0}<t_{1}<\cdots<t_{n}=1$

and pants decompositions $P_{j}$, for $j=1,\ldots,n$, so that
$P_{1}=P_{\alpha}$, $P_{n}=P_{\omega}$, and

$\sigma_{t}\in V(P_{j},L_{2})$

for all $t\in[t_{j-1},t_{j}]$ and $j=1,\ldots,n$.

Since $P_{j}$ and $P_{j+1}$ both have length at most $L_{2}$ with respect to
$\sigma_{\eta_{t_{j}}}^{\mathcal{H}}$, Lemma 3.3 of [Bro03a] implies that
$d_{{\mathcal{P}}}(P_{j},P_{j+1})\leq\kappa=\kappa(L_{2},\Xi)$.

Write $\mathcal{S}(P_{1},\ldots,P_{n})$ to denote the union of all curves in
all pants decompositions $P_{1},\ldots,P_{n}$. The next result is also due to
Brock [Bro03a, Lemma 4.3].

######  Lemma 5.12.

There exists $\mathcal{K}$ depending on $\kappa>0$ and $\xi(\Sigma)$ with the
following property. Let $P_{1},\ldots,P_{n}$ be a sequence of pants
decompositions of $\Sigma$ such that

$d_{{\mathcal{P}}}(P_{j},P_{j+1})\leq\kappa,$

for all $j=1,\ldots,n-1$. Then

$d_{{\mathcal{P}}}(P_{1},P_{n})\leq\mathcal{K}|\mathcal{S}(P_{1},\ldots,P_{n})|.$

For us, the key application of this lemma is the following.

######  Corollary 5.13.

Let $P_{\alpha}=P_{1},\ldots,P_{n}=P_{\omega}$ be as above and $\mathcal{K}$
as in Lemma 5.12 (which depends only on $\xi(f)$ and $\chi(f)$). Then

$d_{{\mathcal{P}}}(P_{\alpha},P_{\omega})\leq\mathcal{K}|\mathcal{S}(P_{1},\ldots,P_{n})|.$

Finally, from Theorem 5.4 we have the following.

######  Lemma 5.14.

Let $P_{\alpha}=P_{1},\ldots,P_{n}=P_{\omega}$ be as above. Then the geodesic
representative in $\widetilde{M}_{\infty}$ of each curve in
$\mathcal{S}(P_{1},\ldots,P_{n})$ has length bounded above by $L_{2}$. ∎

##  6\. Bounding volume

We are almost ready to prove the main theorem. We will need one more result,
again due to Brock [Bro03a, Lemma 4.8]. Suppose $M$ is compact, convex,
hyperbolic $3$-manifold, $\mathcal{L}>0$, and let
$\mathcal{G}_{\mathcal{L}}(M)$ denote the set of closed geodesics in $M$ with
length less than $\mathcal{L}$.

######  Proposition 6.1.

Given $\mathcal{L}>0$ greater than the Bers constant for closed surfaces of
complexity $\xi$, there is a constant
$\mathcal{V}=\mathcal{V}(\xi,\mathcal{L})>0$ with the following property.
Given a compact hyperbolic $3$-manifold $M$ with totally geodesic boundary and
$\xi(\partial M)\leq\xi$, then

${\rm Vol}(M)\geq\mathcal{V}|\mathcal{G}_{\mathcal{L}}(M)|.$

######  Remark.

Brock’s statement in [Bro03a] involves an additive error as well that depends
on $\chi(\partial M)$ (and not $\mathcal{L}$). However, it does not require
$\mathcal{L}$ sufficiently large (in our statement, greater than the Bers
constant). Because we assume $M$ is acylindrical in this statement, there is a
uniform lower bound to the volume (approximately 6.452…) by a result of Kojima
and Miyamoto [KM91], and, because we have assumed $\mathcal{L}$ is greater
than the Bers constant, $\mathcal{G}_{\mathcal{L}}(M)$ is nonempty.
Consequently, we can absorb the additive constant into the multiplicative one,
arriving at the version that is most useful for our purposes.

###### Proof of Theorem 1.1.

Let $P_{\alpha}=P_{1},\ldots,P_{n}=P_{\omega}$ be the sequence of pants
decomposition constructed in §5.4 on $\Sigma$ and
$\mathcal{S}(P_{1},\ldots,P_{n})$ the set of all curves in all pants
decompositions $P_{1},\ldots,P_{n}$. By Lemma 5.14,

$\mathcal{S}(P_{1},\ldots,P_{n})\subset\mathcal{G}_{L_{2}}(\widetilde{M}_{\infty}).$

By Lemma 3.1, setting $N=2\xi(\Sigma)\leq 2\Xi(f)$, we have that $\Sigma$ and
$f^{m}(\Sigma)$ have no closed curves in common for all $m\geq N$. In
particular, no two curves in $\mathcal{S}(P_{1},\ldots,P_{n})$ differ by an
element of $\langle f^{N}\rangle$. Consequently, no two elements of
$\mathcal{S}(P_{1},\ldots,P_{n})$ project to the same homotopy class in
$\overline{M}_{f^{N}}$. In particular, we have

(6.1)
$|\mathcal{S}(P_{1},\ldots,P_{n})|\leq|\mathcal{G}_{L_{2}}(\overline{M}_{f^{N}})|.$

Now observe that

$\displaystyle\tau(f)$ $\displaystyle\leq$ $\displaystyle
d_{{\mathcal{P}}}(P,f(P))\leq
d_{{\mathcal{P}}}(P_{\alpha},P_{\omega})\leq\mathcal{K}|\mathcal{S}(P_{1},\ldots,P_{n})|$
$\displaystyle\leq$
$\displaystyle\mathcal{K}|\mathcal{G}_{L_{2}}(\overline{M}_{f^{N}})|\leq\frac{\mathcal{K}}{\mathcal{V}}{\rm
Vol}(\overline{M}_{f^{N}})=\frac{N\mathcal{K}}{\mathcal{V}}{\rm
Vol}(\overline{M}_{f}).$

The first inequality is by definition. The second follows from Equation 5.2.
The third is by Corollary 5.13. The fourth follows from Equation 6.1. The
fifth inequality comes from Proposition 6.1 (note that we must adjust
$\mathcal{V}$ because of the power $N$, but this is also uniform depending on
the capacity). The final equality comes from the fact that
$\overline{M}_{f^{N}}$ is an $N$–fold cover of $\overline{M}_{f}$. Since
$N,\mathcal{K},\mathcal{V}$ all depend only on $\xi(f)$ and $\chi(f)$, this
completes the proof. ∎

##  7\. Bounded length invariant components

Any $f$–invariant component $\Omega\subset\mathcal{P}(S)$ determines a pants
decomposition $P_{\Omega}$ of $\partial\overline{M}_{f}$ (see [FKLL23]). More
precisely, if $P\in\Omega$ is any pants decomposition representing a vertex in
this component then, after identifying $\partial\overline{M}_{f}$ with the
quotient $S_{+}\sqcup S_{-}=(\mathcal{U}_{+}\sqcup\mathcal{U}_{-})/\langle
f\rangle$, the preimage of $P_{\Omega}$ in
$\mathcal{U}_{+}\sqcup\mathcal{U}_{-}$ agrees with $P$ on neighborhoods of the
attracting and repelling ends of $\mathcal{U}_{+}$ and $\mathcal{U}_{-}$,
respectively. Given an $f$–invariant component $\Omega\subset\mathcal{P}(S)$,
the pants decomposition $P_{\Omega}$ can be constructed by first observing
that for any $P\in\Omega$ there are good nesting neighborhoods $U_{\pm}$ so
that $P$ defines a pants decomposition $P|_{U_{\pm}}$ of $U_{\pm}$ (in
particular, $\partial U_{\pm}$ is a union of curves in $P$) and so that
$f^{\pm 1}(P|_{U_{\pm}})\subset P|_{U_{\pm}}$. It follows that

$\bigcup_{k=0}^{\infty}f^{\mp k}(P|_{U_{\pm}})$

is a pants decomposition of $\mathcal{U}_{\pm}$ which is $\langle
f\rangle$–invariant, and hence, descends to a pants decomposition $P_{\Omega}$
on $S_{\pm}$.

The construction of the pants decomposition $P$ in the proof of Lemma 5.1
defines such an $f$–invariant component $\Omega_{0}\subset\mathcal{P}(S)$, and
can be explicitly described as follows. The subsurface
$\Delta_{+}\subset\overline{U_{+}}\subset\mathcal{U}_{+}$ is a fundamental
domain for the action of $\langle f\rangle$, and the chosen pants
decomposition $P_{+}$ from that proof projects to the components of
$P_{\Omega_{0}}$ contained in $\partial_{+}\overline{M}_{f}$. A similar
statement is true for the components of $P_{\Omega_{0}}$ in
$\partial_{+}\overline{M}_{f}$. We note that the components of
$P_{\Omega_{0}}$ have uniformly bounded length, depending only on the capacity
by Lemma 5.1.

Every $f$–invariant component $\Omega\subset\mathcal{P}(S)$ has its own
translation length

$\tau_{\Omega}(f)=\inf_{P\in\Omega}\lim_{k\to\infty}\frac{d_{\mathcal{P}}(P,f^{k}(P))}{k}.$

As was shown in [FKLL23], for any strongly irreducible end-periodic
homeomorphisms $f$, there is always a sequence of $f$–invariant components
$\Omega_{n}$ so that $\tau_{\Omega_{n}}(f)\to\infty$ as $n\to\infty$. On the
other hand, by definition

$\tau(f)=\inf_{k,\Omega}\frac{\tau_{\Omega}(f^{k})}{k},$

where the infimum is taken over all $k\geq 1$ and $f^{k}$–invariant components
$\Omega$.

Thus there are two measures of efficiency for a component
$\Omega\subset\mathcal{P}(S)$ with respect to a strongly irreducible end-
periodic homeomorphism $f\colon S\to S$. The first is that $P_{\Omega}$ has
bounded length, which is a geometric condition in terms of the hyperbolic
geometry of $\overline{M}_{f}$. The second is purely
topological/combinatorial, and is that $\tau_{\Omega}(f)$ approximates
$\tau(f)$. The next result says that these can be achieved simultaneously.

Theorem 1.2 Given $f\colon S\to S$, a strongly irreducible end-periodic
homeomorphism, there is a component $\Omega\subset\mathcal{P}(S)$ and $E>0$
(depending on the capacity), so that each curve in
$P_{\Omega}\subset\partial\overline{M}_{f}$ has length at most $E$, and so
that $\tau_{\Omega}(f)\leq E\tau(f)$.

###### Proof.

We claim that the component $\Omega_{0}$ defined by $P$ from Lemma 5.1
satisfies the conditions of the proposition. Since $U_{+}\cup U_{-}$ projects
locally isometrically to $\partial\overline{M}_{f}$, and since the components
of $P$ have length bounded by $L=L(\xi(f),\chi(f))$, the component of
$P_{\Omega_{0}}$ are similarly bounded by $L$.

To see that $\tau_{\Omega_{0}}(f)$ is bounded by a uniform constant multiple
of $\tau(f)$, we first observe that the proof of Theorem 1.1 in fact shows
that

$\tau_{\Omega_{0}}(f)\leq\frac{1}{C_{2}}{\rm Vol}(\overline{M}_{f}),$

where $C_{2}$ was explicitly shown to be given by
$\frac{\mathcal{V}}{N\mathcal{K}}$. On the other hand, ${\rm
Vol}(\overline{M}_{f})\leq V_{\rm oct}\tau(f)$ by the main result of [FKLL23].
Therefore,

$\tau_{\Omega_{0}}(f)\leq\frac{V_{\rm oct}}{C_{2}}\tau(f).$

Setting $E=\max\left\\{L,\frac{V_{\rm oct}}{C_{2}}\right\\}$ proves the
theorem. ∎

### Acknowledgments

Field and Loving were supported in part by NSF Mathematical Sciences
Postdoctoral Research Fellowships. The authors were also supported in part by
NSF grants DMS-1840190 and DMS-2103275 (Field), DMS-1904130 and DMS-2202718
(Kent), DMS-2106419 and DMS-2305286 (Leininger), and DMS-2231286 (Loving). The
authors would like to thank Yair Minsky for useful conversations, Heejoung Kim
for her collaboration on the paper [FKLL23], which motivated this one, and
Hugo Parlier for pointing us towards his proof of the relative Bers lemma.
They are grateful to Sam Taylor for a suggestion that simplified the proof of
Theorem 1.2 and the exposition more generally. Finally, they would like to
thank Michael Landry, Chi Cheuk Tsang, and Brandis Whitfield for their
comments on a draft of this paper.

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