# End-periodic homeomorphisms and volumes of mapping tori

Elizabeth Field, Heejoung Kim, Christopher Leininger, and Marissa Loving

###### Abstract.

Given an irreducible, end-periodic homeomorphism $f:S\to S$ of a surface with
finitely many ends, all accumulated by genus, the mapping torus, $M_{f}$, is
the interior of a compact, irreducible, atoroidal 3-manifold
$\overline{M}_{f}$ with incompressible boundary. Our main result is an upper
bound on the infimal hyperbolic volume of $\overline{M}_{f}$ in terms of the
translation length of $f$ on the pants graph of $S$. This builds on work of
Brock and Agol in the finite-type setting. We also construct a broad class of
examples of irreducible, end-periodic homeomorphisms and use them to show that
our bound is asymptotically sharp.

##  1\. Introduction

Infinite-type surfaces arise naturally as leaves of foliations of 3-manifolds.
In the case of taut, depth-one foliations of closed, hyperbolic $3$–manifolds,
any noncompact leaf is a surface $S$ of infinite genus with finitely many ends
(all accumulated by genus). The leaf $S$ is dense in an open submanifold which
is the mapping torus $M_{f}$ of an end-periodic homeomorphism $f\colon S\to
S$. As such, the study of homeomorphisms of infinite-type surfaces is
intimately related to the study of foliated 3-manifolds. The goal of this
paper is to relate the hyperbolic geometry of $M_{f}$ to the geometry and
dynamics of the monodromy $f$ in a way that mirrors the connections between
properties of pseudo-Anosov homeomorphisms of finite-type surfaces and the
geometry of their mapping tori.

The motivating result from the finite-type setting is due to Brock [Bro03b],
who related the translation length, $\tau(f)$, of a pseudo-Anosov
homeomorphism $f$ on the pants graph with the hyperbolic volume of its mapping
torus. In particular, he proved that there is some $K>0$ so that

$\frac{1}{K}\tau(f)\leq{\rm Vol}(M_{f})\leq K\tau(f),$

where $K$ depends only on the surface. Agol [Ago03] improved the upper bound
on volume, giving an explicit value of the constant, and moreover proved that
it is sharp.

For an end-periodic homeomorphism $f\colon S\to S$, the mapping torus $M_{f}$
is the interior of a compact, irreducible $3$–manifold $\overline{M}_{f}$ with
incompressible boundary (Proposition 3.1). Handel and Miller proved a Nielsen-
Thurston type classification theorem in the end-periodic setting, with the
notion of irreducibility serving as an analogue of being pseudo-Anosov; see
[CCF21] or Section 2.2. We prove that when $f$ is irreducible,
$\overline{M}_{f}$ is atoroidal (Proposition 3.4), and further define a notion
of strong irreducibility which ensures $\overline{M}_{f}$ is also acylindrical
(Lemma 3.5). Thurston’s Hyperbolization Theorem (see Theorem 2.20) then
reinforces the analogy with the Nielsen-Thurston classification in the finite-
type case, providing a _convex_ hyperbolic metric on $\overline{M}_{f}$ when
$f$ is irreducible, and one with totally geodesic boundary when $f$ is
strongly irreducible. Note that _every_ 3-manifold with boundary admits a non-
convex hyperbolic metric so it is important to make this distinction. We write
${\rm{V\underline{ol}}}(\overline{M}_{f})$ to denote the infimum of volumes of
all such hyperbolic metrics on $\overline{M}_{f}$. Our first theorem is an
analogue of Brock’s upper bound on volume and our approach follows Agol’s
proof in [Ago03].

######  Theorem 1.1.

For any irreducible, end-periodic homeomorphism $f\colon S\to S$ of a surface
with finitely many ends, all accumulate by genus, we have
${\rm{V\underline{ol}}}(\overline{M}_{f})\leq V_{\rm oct}\tau(f)$.

Here, $V_{\rm oct}$ denotes the volume of a regular ideal octahedron, and
$\tau(f)$ denotes the asymptotic translation length of $f$ on the pants graph
$\mathcal{P}(S)$. Throughout this paper, we assume that $S$ has finitely many
ends, all accumulated by genus. Although $\mathcal{P}(S)$ is disconnected for
such surfaces, the end-periodic assumption implies that there is a non-empty
subgraph, $\mathcal{P}_{f}(S)\subset\mathcal{P}(S)$, for which each component
is invariant by a positive power of $f$, and hence $\tau(f)$ is well-defined
(see Section 2.4 where we also discuss the path metric on the components of
$\mathcal{P}(S)$ we consider).

For any component $\Omega\subset\mathcal{P}_{f}(S)$ which is invariant by $f$,
we will construct a pants decomposition $P_{\Omega}$ of
$\partial\overline{M}_{f}$ for which $\overline{M}_{f}-P_{\Omega}$ is
acylindrical, and $P_{\Omega}$ depends only on the component $\Omega$; see
Proposition 4.3. Denoting the volume of the complete hyperbolic metric with
totally geodesic boundary on this manifold by ${\rm
Vol}(\overline{M}_{f}-P_{\Omega})$, which by a result of Storm [Sto07] is
equal to the infimum of volumes of all convex hyperbolic metrics on
$\overline{M}_{f}-P_{\Omega}$, and letting $\tau(f,\Omega)$ denote the
translation distance of $f$ on $\Omega$, we prove the following.

######  Theorem 1.2.

For any $f$–invariant component $\Omega\subset\mathcal{P}_{f}(S)$,

${\rm Vol}(\overline{M}_{f}-P_{\Omega})\leq V_{\rm oct}\tau(f,\Omega).$

The translation length of $f$ on various components of $\mathcal{P}(S)$ is
quite mysterious as the following corollary of Theorem 1.2 shows.

######  Corollary 1.3.

For any irreducible, end-periodic homeomorphism $f\colon S\to S$ and $R>0$,
there exists an $f$–invariant component $\Omega\subset\mathcal{P}_{f}(S)$ such
that $\tau(f,\Omega)\geq R$.

Theorem 1.2 also provides uniform lower bounds on asymptotic translation
length. To state the result in this case, we let $V_{\rm tet}$ denote the
volume of a regular ideal tetrahedron in $\mathbb{H}^{3}$ (which is also the
maximal volume of a tetrahedron in $\mathbb{H}^{3}$).

######  Corollary 1.4.

Given an irreducible, end-periodic homeomorphism $f$, we have

$\tau(f)\geq\frac{V_{\rm tet}\xi(\partial\overline{M}_{f})}{2V_{\rm oct}}.$

Here $\xi(\partial\overline{M}_{f})$ is the complexity of the boundary, see
Section 2.1. For a description of the lower bound which is more intrinsic to
the surface $S$ and homeomorphism $f$, see Corollary 4.4.

We end by describing a fairly robust construction for examples of both
strongly irreducible homeomorphisms and irreducible, but not strongly
irreducible, homeomorphisms.

######  Theorem 1.5.

Given any pure end-periodic homeomorphism $\phi\colon S\to S$ there exist both
irreducible and strongly irreducible homeomorphisms $f,f^{\prime}\colon S\to
S$, respectively, so that $f$ and $f^{\prime}$ agree with $\phi$ on the
complement of a compact set.

The construction is sufficiently robust to prove the following. Note that when
an end-periodic homeomorphism $f$ is strongly irreducible, $\overline{M}_{f}$
is acylindrical (see Proposition 3.1), and so the infimal volume is realized
by ${\rm Vol}(\overline{M}_{f})$, which denotes the volume of the complete
hyperbolic metric with totally geodesic boundary.

######  Theorem 1.6.

There is a sequence of strongly irreducible end-periodic homeomorphisms
$f_{k}\colon S\to S$ so that ${\rm Vol}(\overline{M}_{f_{k}})\to\infty$ and
$\frac{{\rm Vol}(\overline{M}_{f_{k}})}{\tau(f_{k})}\to V_{\rm oct}$ as
$k\to\infty$. In fact, $|{\rm Vol}(\overline{M}_{f_{k}})-V_{\rm
oct}\tau(f_{k})|$ is uniformly bounded, independent of $k$.

The examples in this theorem can also be taken to agree with any given end-
periodic homeomorphism $\phi\colon S\to S$ outside some compact set.

### Outline of the paper

In Section 2, we provide background on mapping class groups for finite-type
and infinite-type surfaces, end-periodic homeomorphisms, pants graphs,
decomposition spaces, and 3-manifolds. The mapping torus of an irreducible
end-periodic homeomorphism is shown to be the interior of a compact,
irreducible, atoroidal 3-manifold in Section 3. Furthermore, for a strongly
irreducible end-periodic homeomorphism, we show that the compactified mapping
torus is also acylindrical. In Section 4, we prove Theorem 1.2, Theorem 1.1,
and Corollary 1.4 as applications of Proposition 4.3, which is proved in
Section 5. Finally, we construct examples of irreducible (and strongly
irreducible) end-periodic homeomorphisms and prove Theorem 1.6 and Corollary
1.3 in Section 6.1.

###  1.1. Related and future work

Brock in [Bro03a] proved that the pants graph is quasi-isometric to the
Teichmüller space with its Weil-Petersson metric, which allows one to compare
volumes of mapping tori with Weil-Petersson translation lengths. A direct
proof of such a comparison was later given by Brock–Bromberg [BB16] and
Kojima–McShane [KM18]. Pants distance has also been related to volumes of
hyperbolic 3-manifolds in another setting by Cremaschi, Rodríguez-Migueles and
Yarmola [CRMY19], see also [RM20, CRM20]. In forthcoming work, Landry, Minsky,
and Taylor study stretch factors of end-periodic homeomorphisms arising from
depth one foliations [LMT21]. Finally, with Autumn Kent, the authors are
currently investigating lower bounds on volumes in terms of translation
lengths on the pants graph.

### Acknowledgements

The authors would like to thank Sergio Fenley for many useful discussions and
suggestions, and for pointing out an error in an earlier argument which
ultimately led to the definition of strong irreducibility. They would also
like to thank Ty Ghaswala for comments on an earlier draft and some
interesting and useful observations. The authors acknowledge support from NSF
grants DMS-1840190 (Field), DMS-1811518 (Leininger), and DMS-1902729 (Loving).

##  2\. Preliminaries

###  2.1. Surfaces and mapping class groups

Let $Y$ be any surface. We will consider the _mapping class group_ , ${\rm
Map}(Y)$, of $Y$, which is the group of orientation preserving homeomorphisms
of $Y$ up to isotopy. We do not distinguish the notation for a homeomorphism
and the mapping class it defines, as the context will make clear the intended
meaning.

Throughout this paper, we let $S$ denote a boundaryless, infinite-type surface
with $2\leq n<\infty$ ends, all accumulated by genus (in particular, we assume
$S$ has no planar ends). The mapping class group of an infinite-type surface
is an uncountably infinite group and is often called a _big mapping class
group_. Important subgroups of big mapping class groups include the pure
mapping class group, ${\rm PMap}(S)$, which is the subgroup of ${\rm Map}(S)$
consisting of mapping classes which fix each end of $S$, as well as the
_compactly supported mapping class group_ , ${\rm Map}_{c}(S)$. For more on
infinite-type surfaces and their associated big mapping class groups, see the
survey article on these topics by Aramayona and Vlamis [AV20].

We define the _complexity_ of a finite-type surface $\Sigma=\Sigma_{g,n}$ of
genus $g$ and with $n$ punctures to be $\xi(\Sigma_{g,n})=3g-3+n$. We then let
$\overline{\Sigma}=\overline{\Sigma}_{g,n}$ denote the compact surface of
genus $g$ with $n$ boundary components, which we declare to have the same
complexity as $\Sigma$. For example, we will call $\overline{\Sigma}_{0,3}$ a
_pair of pants_ (compact) and $\Sigma_{0,3}$ a _thrice-punctured sphere_ (non-
compact).

By a finite-type subsurface of $S$, we mean a connected, finite-complexity,
incompressible (i.e. $\pi_{1}$–injective) open subsurface $\Sigma\subset S$
with $\xi(\Sigma)\neq 0$. Such a surface $\Sigma$ is homeomorphic to the
interior of a compact surface $\overline{\Sigma}$ with boundary, and we assume
that the inclusion extends to a locally injective map of $\overline{\Sigma}$.
We abuse notation and write $\overline{\Sigma}\subset S$, even though it is
only assumed to be embedded on its interior.

Let $Y$ be a surface, either compact or non-compact, possibly of infinite
type. By a curve in $Y$, we mean the homotopy class of an essential (i.e. non-
null-homotopic and non-peripheral) simple closed curve in $Y$. A line in $Y$
is the proper homotopy class of a proper embedding of $\mathbb{R}$ into $Y$
which is essential (i.e. not homotopic via a proper homotopy into an arbitrary
neighborhood of an end). By _proper homotopy_ we mean that the homotopy itself
is a proper map this implies that it is also a homotopy through proper maps.
An arc in $Y$ is the relative homotopy class of an essential arc $(I,\partial
I)\to(Y,\partial Y)$ (i.e. an embedding not homotopic into $\partial Y$ by a
relative homotopy $h_{t}\colon(I,\partial I)\to(Y,\partial Y)$). We often
confuse curves, arcs, and lines with representatives of their
(relative/proper) homotopy classes. A multicurve, multiarc, or multiline in
$Y$ is the union of a collection of curves, arcs, or lines, respectively, in
$Y$, having pairwise disjoint representatives.

Given a curve $\alpha$ in $Y$, we let $T_{\alpha}$ denote the left Dehn twist
about the curve $\alpha$. A pseudo-Anosov mapping class of a finite-type
surface $Y$ is a mapping class with no periodic curves in $Y$. A partial
pseudo-Anosov mapping class of any surface $Y$ is a mapping class that has a
representative supported on a (finite-type) subsurface $\Sigma\subset Y$ which
is pseudo-Anosov on $\Sigma$. We call $\Sigma$ the support of such a partial
pseudo-Anosov. In general, we say that a mapping class is supported on a
subsurface $\Sigma$ if it has a representative which is the identity outside
of $\Sigma$. See [FM12] for more on mapping class groups of finite-type
surfaces and the Nielsen–Thurston classification.

###  2.2. End-periodic homeomorphims

We refer the reader to [Fen92, Fen97, CCF21, Gab84] for a more detailed
discussion of the theory of end-periodic homeomorphisms, examples, and their
relationship to depth-one foliations. We note that the exposition in these
references discusses end-periodic homeomorphisms of more general types of
surfaces than what we consider here.

######  Definition 2.1 (End-periodic homeomorphisms).

An _end-periodic homeomorphism_ of $S$ is a homeomorphism $f$ of $S$
satisfying the following. There exists $m>0$ 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 $E$.

In the first case, $E$ is said to be an attracting end for $f$, and in the
second case, $E$ is said to be a repelling end. We call such a neighborhood
$U_{E}$ a nesting neighborhood for $E$, and note that by definition, we may
always assume (as we will) that the closures of the nesting neighborhoods of
the ends are pairwise disjoint.

There are many examples of end-periodic homeomorphisms given by
Cantwell–Conlon in [CC16]. It is also straightforward to build more examples
using “handle shifts”, which were introduced by Patel–Vlamis in [PV18].

######  Definition 2.2.

Take a bi-infinite strip $\mathbb{R}\times[0,1]$ and remove disks of radius
$\tfrac{1}{4}$ centered at each point of $\mathbb{Z}\times\\{\tfrac{1}{2}\\}$.
To the boundary of each disk (now removed) glue the boundary of a one-holed
torus. Call this bi-infinite strip of handles $H$. Let $h$ be the
homeomorphism of $H$ given by shifting each handle over by one while tapering
to the identity in a neighborhood of the boundary (which consists of two
copies of $\mathbb{R}$). We will call the pair $(H,h)$ a _handle strip_ , as
shown in Figure 1. Note that $H$ has two ends accumulated by genus: one
towards which $h$ is shifting, called the attracting end, and one from which
$h$ shifts away, called the repelling end. Now embed $H$ into any surface $S$
with at least one end accumulated by genus and extend $h$ by the identity to a
homeomorphism, $\rho$, of $S$. The homeomorphism $\rho$ of $S$ is called a
_handle shift_.

Figure 1. A handle strip comprised of a bi-infinite strip of handles $H$ and a
shift map $h$. The left end of the strip is repelling, while the right end is
attracting.

######  Example 2.3.

Consider a finite number of handle strips $(H_{1},h_{1}),\ldots,(H_{k},h_{k})$
embedded in a ladder surface $L$ (that is an infinite-type surface with
exactly two ends, both accumulated by genus) so that the complement of
$\bigcup_{i=1}^{k}H_{i}$ in $L$ consists of $k$ bi-infinite strips and so that
the attracting ends of $H_{i}$ are sent into a single end of $L$ and the
repelling ends are sent into the other end of $L$. See Figure 2 for the case
of 2 handle strips. Let $\rho_{1},\ldots,\rho_{k}$ be the corresponding handle
shifts. We obtain an end-periodic homeomorphism $\rho$ by an isotopy of the
product of $\rho_{1},\ldots,\rho_{k}$. This is shown for 2 handle strips in
Figure 2. See 2.10 below for more examples constructed using handle shifts.

Figure 2. Products of pairwise commuting handle shifts as end-periodic
homeomorphisms.

In the example of the ladder surface just described, the end-periodic
homeomorphism generates a covering group acting with closed surface quotient
(of genus one more than the number of handle shifts involved). We now explain
the generalization of this behavior to arbitrary end-periodic homeomorphisms
of infinite-type surfaces.

Given an end-periodic homeomorphism $f\colon S\to S$, choose nesting
neighborhoods for each end and let $U_{+}$ be the union of nesting
neighborhoods of the attracting ends, and likewise $U_{-}$ the union of
nesting neighborhoods of repelling ends. Define

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

We call $\mathcal{U}_{+}$ the positive escaping set for $f$ and
$\mathcal{U}_{-}$ the negative escaping set for $f$. From Definition 2.1, any
choice of nesting neighborhoods will give rise to the same sets
$\mathcal{U}_{\pm}$. Our standing assumption that $S$ has only finitely many
ends, each accumulated by genus, together with the definition of nesting
neighborhoods leads to the following properties of the escaping sets.

######  Lemma 2.4.

Given an end-periodic homeomorphism $f\colon S\to S$, the escaping sets
$\mathcal{U}_{+}$ and $\mathcal{U}_{-}$ are each $f$–invariant, and the
infinite cyclic group $\langle f\rangle$ acts freely, properly
discontinuously, and cocompactly on each. Consequently, the quotients
$\mathcal{U}_{\pm}\to S_{\pm}=\mathcal{U}_{\pm}/\langle f\rangle$ are closed,
orientable surfaces.

###### Proof.

The $f$–invariance of each of $\mathcal{U}_{+}$ and $\mathcal{U}_{-}$ is
immediate from their definition.

Choose the nesting neighborhoods for the attracting ends so that their union
$U_{+}$ has the property that $f(U_{+})\subsetneq U_{+}$. Then
$K_{+}=\overline{U_{+}-f(U_{+})}$ is a compact subset of $\mathcal{U}_{+}$ and

$\mathcal{U}_{+}=\bigcup_{n\in\mathbb{Z}}f^{n}(K_{+}).$

From the fact that positive powers applied to nesting neighborhoods give a
neighborhood basis of the ends in Definition 2.1, we see that the
$f^{n}$–translates of $K_{+}$ form a locally finite cover of
$\mathcal{U}_{+}$. It follows that $\langle f\rangle$ acts properly
discontinuously and cocompactly on $\mathcal{U}_{+}$. Since $\langle f\rangle$
is torsion free, the action is free. The same proof is valid for the repelling
ends. ∎

Now note that $\mathcal{U}_{\pm}\to S_{\pm}$ is an infinite cyclic covering,
but there may be multiple components of $\mathcal{U}_{\pm}$ that project to a
single component of $S_{\pm}$. Fixing any component $\mathcal{U}$ of
$\mathcal{U}_{\pm}$ defining an end $E$ of $S$ and projecting to a component
$S_{\mathcal{U}}\subset S_{\pm}$, then $\mathcal{U}\to S_{\mathcal{U}}$ is a
connected, infinite cyclic cover of the connected, closed, orientable surface
$S_{\mathcal{U}}$. As such, there is a nonseparating simple closed curve
$\alpha\subset S_{\mathcal{U}}$ that lifts to $\mathcal{U}$ so that any lift
$\alpha_{0}\subset\mathcal{U}$ separates $\mathcal{U}$ into neighborhoods of
the two ends of $\mathcal{U}$. One of these is a nesting neighborhood $U_{E}$
of an end $E$ of $S$. Moreover, if $p_{E}$ is the period of $E$, then
$\overline{f^{p_{E}}(U_{E})}\subset U_{E}$: indeed, $\langle f^{p_{E}}\rangle$
generates the covering group of $\mathcal{U}\to S_{\mathcal{U}}$, and
$\alpha_{0}\cup f^{p_{E}}(\alpha_{0})$ bounds a subsurface that serves as a
fundamental domain. Note that

(2.1) $U_{E_{i}}=f^{i}(U_{E})\quad\mbox{ for each }\quad i=0,\ldots,p_{E}-1.$

are nesting neighborhoods for all the attracting ends
$E=E_{0},\ldots,E_{p_{E}-1}$ in the $f$–orbit of $E$ with the same properties
that $U_{E}$ has for $E$.

When the nesting neighborhood of an attracting end $E$ is chosen in this way,
and the neighborhoods of the other ends in the $f$–orbit of $E$ are given by
(2.1), then we will call these good nesting neighborhoods for the attracting
ends. We similarly define good nesting neighborhoods for the repelling ends.

Using the unions, $U_{+}$ and $U_{-}$, of good nesting neighborhoods for the
attracting and repelling ends, respectively, we can choose a hyperbolic metric
on $S$ so that the boundaries of these neighborhoods are geodesics, and
furthermore $f^{\pm 1}|_{U_{+}}$ and $f^{\pm 1}|_{U_{-}}$ are isometries onto
their images; see [Fen97].

######  Corollary 2.5.

The number of components of $S_{+}$ and $S_{-}$ is equal to the number of
$f$–orbits of components of $\mathcal{U}_{+}$ and $\mathcal{U}_{-}$,
respectively. Moreover, $\xi(S_{+})=\xi(S_{-})$.

###### Proof.

The first statement is clear since the preimage of any component of $S_{\pm}$
is an $f$–orbit of components of $\mathcal{U}_{\pm}$.

For the second claim, choose the unions $U_{+}$ and $U_{-}$ of good nesting
neighborhoods and a hyperbolic metric as above. Note that by our assumption on
$U_{+}$ and $U_{-}$, $\overline{U_{-}}\cap\overline{U_{+}}=\emptyset$. In
particular,

$\Sigma=\overline{S-(U_{+}\cup U_{-})}$

is a subsurface with geodesic boundary, as is $f(\Sigma)$. We will let
$K_{\pm}=\overline{U_{\pm}-f^{\pm}(U_{\pm})}$.

Now observe that $S_{+}$ admits a hyperbolic metric so that the restriction of
the covering projection $\mathcal{U}_{+}\to S_{+}$ to
$K_{+}\subset\mathcal{U}_{+}$ is a surjective local isometry, injective on the
interior. Combining this with the Gauss-Bonnet Theorem we have

$\tfrac{4\pi}{3}\xi(S_{+})={\rm Area}(S_{+})={\rm Area}(K_{+}).$

Likewise, there is a hyperbolic metric on $S_{-}$ so that

$\tfrac{4\pi}{3}\xi(S_{-})={\rm Area}(S_{-})={\rm Area}(K_{-}).$

On the other hand,

(2.2) ${\rm Area}(\Sigma\cup K_{-})={\rm Area}(\Sigma)+{\rm Area}(K_{-}),$

since $\Sigma$ and $K_{-}$ intersect along a finite union of simple closed
geodesics, which has area $0$. Similarly,

(2.3) ${\rm Area}(\Sigma\cup K_{+})={\rm Area}(\Sigma)+{\rm Area}(K_{+}).$

Now observe that $f$ maps $\Sigma\cup K_{-}$ to $f(\Sigma)\cup
f(K_{-})=\Sigma\cup K_{+}$, and so by Gauss-Bonnet the areas on the left hand
sides of (2.2) and (2.3) are equal. Thus, ${\rm Area}(K_{-})={\rm
Area}(K_{+})$, and so $\xi(S_{+})=\xi(S_{-})$, as required. ∎

In the following, recall that the terms “curve” and “line” actually mean a
(proper) homotopy class of such.

######  Definition 2.6 (Irreducibility/Strong irreducibility).

An end-periodic homeomorphism, $f:S\to S$, is _irreducible_ if it has

  * 1)

no periodic curves;

  * 2)

no AR-periodic lines, i.e. a line $\ell$ with one end in an attracting end of
$S$ and the other in a repelling end of $S$ so that there exists
$k\in\mathbb{Z}$ with $f^{k}(\ell)=\ell$; and

  * 3)

no reducing curves, i.e. a curve $\gamma$ so that there exists
$m,n\in\mathbb{Z}$ with $m<n$ and such that $f^{n}(\gamma)$ is contained in a
nesting neighborhood of an attracting end and $f^{m}(\gamma)$ is contained in
a nesting neighborhood of a repelling end.

An end-periodic homeomorphism $f$ is _strongly irreducible_ if it is
irreducible and contains no _periodic lines_ , i.e. a line $\ell$ in $S$ such
that $f^{k}(\ell)=\ell$ for some $k\in\mathbb{Z}$ (with no constraints on the
ends of the line).

######  Remark.

Note that the language used above to define periodic and AR-periodic lines
differs from that used in the literature. See, for example, [Fen97, Fen92,
CCF21].

In their unpublished work, Handel and Miller developed an analogue of the
Nielsen-Thurston classification/reduction theory for end-periodic
homeomorphisms. In that theory, irreducible end-periodic homeomorphisms play
the role of pseudo-Anosov homeomorphisms due to the analogous properties their
invariant laminations enjoy. See Cantwell–Conlon–Fenley [CCF21] for a detailed
exposition and expansion of this theory. The concept of strong irreducibility
is new, and arose in the current paper from geometric considerations of the
mapping torus; see Proposition 3.1.

In his thesis [Fen89], Fenley produces the irreducible end-periodic
homeomorphism shown in Figure 3. His proof of irreducibility relies on
building and analyzing the corresponding weighted train tracks. We will
provide a general construction of both irreducible and strongly irreducible
end-periodic homeomorphisms in Section 6.1. However, our proofs of
irreducibility rely on subsurface projection and curve complex geometry rather
than weighted train tracks.

$\alpha$
---
$\beta$
---
$\gamma$
---
Figure 3. The homeomorphism $f=T_{\alpha}T_{\beta}T_{\gamma}\rho$, where
$\rho$ is the end-periodic homeomorphism induced by a single handle strip, is
an irreducible end-periodic homeomorphism by work of Fenley.

###  2.3. End behavior

Given a homeomorphism $f\colon S\to S$ (or its associated mapping class
$f\in{\rm Map}(S)$), its “end behavior” describes the way in which $f$ acts on
neighborhoods of the ends. It is convenient to make this precise in two
different ways as follows.

First, we will say that two elements $f,f^{\prime}\in{\rm Map}(S)$ have the
same fine end behavior if $f^{-1}f^{\prime}\in{\rm Map}_{c}(S)={\rm
PMap}_{c}(S)$: that is, if $f^{-1}f^{\prime}$ is compactly supported. Thus,
for any end of $S$, $f$ and $f^{\prime}$, agree on some neighborhood of that
end. In Section 6.1, we will show that for any pure, end-periodic
homeomorphism $f\colon S\to S$, there are both irreducible and strongly
irreducible end-periodic homeomorphisms with the same fine end behavior as
$f$; see Theorem 1.5.

To construct examples in this level of generality, it is convenient to have a
notion of “coarse end behavior”. This coarse end behavior will also provide a
convenient description of the surfaces $S_{\pm}$ associated to an end-periodic
homeomorphism as described above. To formalize this, we will first introduce a
few more definitions and state a result of Aramayona–Patel–Vlamis [APV20].

Let $H_{1}^{sep}(S,\mathbb{Z})$ be the subgroup of $H_{1}(S,\mathbb{Z})$
spanned by homology classes of separating curves, and let
$H^{1}_{sep}(S,\mathbb{Z})\leq H^{1}(S,\mathbb{Z})$ be the subgroup naturally
identified with $\text{Hom}(H_{1}^{sep}(S,\mathbb{Z}),\mathbb{Z})$ by the
isomorphism $H^{1}(S,\mathbb{Z})\to\text{Hom}(H_{1}(S,\mathbb{Z}),\mathbb{Z})$
arising from the Universal Coefficient Theorem.

Given an end $E$ of $S$, let $\gamma$ be an oriented curve that separates $E$
from the other ends (with orientation such that the component of $S-\gamma$ to
the right of $\gamma$ is a neighborhood of $E$). Recall that we require that
$S$ is boundaryless and has at least two ends accumulated by genus. Then,
$\gamma$ defines a nonzero element $[\gamma]\in H_{1}^{sep}(S,\mathbb{Z})$,
canonically associated to $E$. When convenient, we denote this class
$v_{E}=[\gamma]\in H_{1}^{sep}(S,\mathbb{Z})$. Given $f\in{\rm PMap}(S)$,
$[\gamma]=[f(\gamma)]=f_{*}([\gamma])$ and Aramayona-Patel-Vlamis define an
integer $\varphi_{[\gamma]}(f)$ which can be viewed as the “signed genus”
between $\gamma$ and $f(\gamma)$. If, for example, $\gamma$ and $f(\gamma)$
are disjoint, then $|\varphi_{[\gamma]}(f)|$ is the genus of the subsurface
bounded by $\gamma$ and $f(\gamma)$ (with a negative sign if $f(\gamma)$ is to
the left of $\gamma$); see [APV20, Section 3] for a more precise description.
Furthermore, they show that $\varphi_{[\gamma]}\colon{\rm
PMap}(S)\to\mathbb{Z}$ is a well-defined homomorphism that depends only on the
homology class $[\gamma]$; see [APV20, Proposition 3.3].

According to [APV20, Proposition 4.4], this induces a surjective homomorphism

$\Phi^{*}\colon{\rm PMap}(S)\to H^{1}_{sep}(S,\mathbb{Z}),$

by $\Phi^{*}(f)([\gamma])=\varphi_{[\gamma]}(f)$. In other words,
$\Phi^{*}(f)$ describes “how much genus has been shifted” on each end by $f$.
It follows from [APV20, Theorem 5] and its proof that $\Phi^{*}$ is a
surjective homomorphism whose kernel is precisely $\overline{{\rm
Map}_{c}(S)}$, the closure of the compactly supported mapping class subgroup.
We define the coarse end behavior of $f\in{\rm PMap}(S)$ to be the cohomology
class $\Phi^{*}(f)\in H^{1}_{sep}(S,\mathbb{Z})$.

######  Remark.

We note that $\Phi^{*}(f)=\Phi^{*}(f^{\prime})$ if and only if
$f^{-1}f^{\prime}\in\overline{{\rm Map}_{c}(S)}$ (thanks to Ghaswala for
pointing this out). In particular, if $f$ and $f^{\prime}$ have the same fine
end behavior, then they have the same coarse end behavior (since ${\rm
Map}_{c}(S)<\overline{{\rm Map}_{c}(S)}$). Aramayona, Patel, and Vlamis
construct a section of $\Phi^{*}$ onto an abelian group generated by commuting
handle shifts, thus giving ${\rm PMap}(S)$ the structure of a semidirect
product; see [APV20, Theorem 3, Corollary 6]. Recall that we require that $S$
is boundaryless and has at least two ends accumulated by genus. Although we
will use handle shifts in 2.10, we will be using a different collection of
handle shifts. Since we will not appeal to this structure, we do not discuss
it further.

######  Lemma 2.7.

Suppose $f\colon S\to S$ is a pure, end-periodic homeomorphism, and let
$S_{\pm}$ be the surfaces obtained as quotients of the positive and negative
escaping sets. If $E$ is an end of $S$ with associated homology class
$v_{E}\in H_{1}^{sep}(S,\mathbb{Z})$, then the genus of the corresponding
component of $S_{\pm}$ is $1+|\Phi^{*}(f)(v_{E})|$.

###### Proof.

Since $f$ is pure, we may choose a simple closed curve $\gamma$ such that
$[\gamma]=v_{E}$, so that $\gamma$ cuts off a neighborhood $U$ of, say, an
attracting end $E$ and $f(\overline{U})\subset U$. In particular, $\gamma$ and
$f(\gamma)$ are disjoint and bound a subsurface of genus
$|\Phi^{*}(f)(v_{E})|$. This subsurface serves as a fundamental domain for the
action of $\langle f\rangle$ on the corresponding component of
$\mathcal{U}_{+}$, and hence the genus of the component of $S_{+}$ is one more
than $|\Phi^{*}(f)(v_{E})|$, as required. The case of a repelling end is
similar. ∎

Given any $f\in{\rm PMap}(S)$, let $E_{1},\ldots,E_{n}$ denote the ends of
$S$, $v_{i}=v_{E_{i}}$, for $i=1,\ldots,n$, and set

(2.4) $|\Phi^{*}(f)|=\sum_{i=1}^{n}|\Phi^{*}(f)(v_{i})|.$

With this definition, we have the following.

######  Corollary 2.8.

If $f\colon S\to S$ is an end-periodic homeomorphism with associated quotient
surfaces $S_{\pm}$ from the escaping sets, then

$\xi(S_{+})=\xi(S_{-})=\tfrac{3}{2}|\Phi^{*}(f)|.$

###### Proof.

By the Lemma 2.7, the genus of the component of $S_{\pm}$ corresponding to
$E_{i}$ is $|\Phi^{*}(f)(v_{i})|+1$. Therefore,

$\xi(S_{+})+\xi(S_{-})=\sum_{i=1}^{n}\left(3(1+|\Phi^{*}(f)(v_{i})|)-3\right)=3\sum_{i=1}^{n}|\Phi^{*}(f)(v_{i})|.$

By Corollary 2.5, $\xi(S_{+})=\xi(S_{-})=\tfrac{1}{2}(\xi(S_{+})+\xi(S_{-}))$,
completing the proof. ∎

To construct end-periodic homeomorphisms with a specified fine end behavior,
the next corollary shows that it suffices to construct an end-periodic
homeomorphism with the same coarse end behavior.

######  Corollary 2.9.

Let $f_{1},f_{2}:S\to S$ be pure, end-periodic homeomorphisms with the same
coarse end behavior. Then, there exists a mapping class $f_{1}^{\prime}$ with
the same fine end behavior as $f_{2}$ such that $f_{1}$ is conjugate to
$f_{1}^{\prime}$ by an element of ${\rm PMap}(S)$.

###### Proof.

For each $i=1,2$, choose good nesting neighborhoods $U_{\pm}^{i}$ for $f_{i}$
so that the neighborhoods $U_{E}^{i}$ of the end $E$ is bounded by a single
curve $v_{E}^{i}$ that projects to a nonseparating curve in the quotient
surface $S_{\pm}^{i}$. We further assume that the neighborhood are chosen so
that there is an orientation preserving homeomorphism

$h_{0}\colon\overline{S-(U_{+}^{1}\cup
U_{-}^{1})}\to\overline{S-(U_{+}^{2}\cup U_{-}^{2})}.$

The existence of such a homeomorphism follows from the classification of
surfaces, since we can increase the genus of either
$\overline{S-(U^{i}_{+}\cup U^{i}_{-})}$, $i=1,2$ by choosing smaller good
nesting neighborhoods. Moreover, we may assume $h_{0}(v_{E}^{1})=v_{E}^{2}$
for each end $E$. Since $f_{1}$ and $f_{2}$ have the same coarse end behavior,
by Lemma 2.7, for each end $E$, the image $S_{E}^{1}\subset S_{\pm}^{1}$ of
$U_{E}^{1}$ is a closed surface of the same genus as the image
$S_{E}^{2}\subset S_{\pm}^{2}$ of $U_{E}^{2}$. By the classification of
surfaces again there is an orientation preserving homeomorphism $S_{E}^{1}\to
S_{E}^{2}$ that sends the image of $v_{E}^{1}$ to the image of $v_{E}^{2}$.
This homeomorphism lifts to a homeomorphism $h_{E}\colon U_{E}^{1}\to
U_{E}^{2}$ which is equivariant with respect to the restriction of the
semigroup generated by $f_{1}$ and $f_{2}$ if $E$ is an attracting end, and
equivariant with respect to $f_{1}^{-1}$ and $f_{2}^{-1}$ if $E$ is a
repelling end. That is, on $U_{E}$ we have we have that
$f_{2}=h_{E}f_{1}h_{E}^{-1}$ or $f_{2}^{-1}=h_{E}f_{1}^{-1}h_{E}^{-1}$ if $E$
is attracting or repelling, respectively. Observe that in the repelling case,
on $f_{2}^{-1}(U_{E})$ we have $f_{2}=h_{E}f_{1}^{-1}h_{E}^{-1}$. Because all
of these homeomorphisms are orientation preserving, adjusting $h_{0}$ if
necessary, we may assume that $h_{E}$ agrees with $h_{0}$ on $v_{E}^{1}$. We
may therefore glue these together into a single pure homeomorphism $h\colon
S\to S$. Setting $f_{1}^{\prime}=hf_{1}^{-1}h^{-1}$ we see that
$f_{1}^{\prime}$ and $f_{2}$ agree on

$U_{+}^{2}\cup f_{2}^{-1}(U_{-}^{2}),$

the complement of which has compact closure. Thus, $f_{1}^{\prime}$ and
$f_{2}$ have the same fine end behavior. ∎

Note that for $v_{1},\ldots,v_{n}$ as above (e.g. in (2.4)), we have
$\sum_{i}v_{i}=0$ in $H_{1}^{sep}(S,\mathbb{Z})$, and any $n-1$ elements of
this set forms a basis. Given any $\omega\in H^{1}_{sep}(S,\mathbb{Z})$, we
thus have

$\sum_{i=1}^{n}\omega(v_{i})=0,$

and moreover, for any integers $w_{1},\ldots,w_{n}$ with $\sum w_{i}=0$, there
exists a unique $\omega\in H^{1}_{sep}(S,\mathbb{Z})$ so that
$\omega(v_{i})=w_{i}$ for all $i=1,\ldots,n$. Thus, specifying the coarse end
behavior is equivalent to specifying such a collection of integers
$w_{1},\ldots,w_{n}$ with sum zero.

For an end-periodic homeomorphism on a surface with $n$ ends, the integers
$w_{1},\ldots,w_{n}$ describing the coarse end behavior (as above, summing to
zero) are all nonzero. Conversely, for any prescribed such coarse end
behavior, in the following construction, we explain how to build examples of
end-periodic homeomorphisms with that coarse end behavior. Moreover, in
Section 6.1, we will show that many of the end-periodic homeomorphisms arising
from 2.10 are irreducible and strongly irreducible.

######  Construction 2.10.

As above, let $v_{1},\ldots,v_{n}$ be separating curves bounding pairwise
disjoint neighborhoods $U_{1},\ldots,U_{n}$ of the ends $E_{1},\ldots,E_{n}$
of $S$, respectively, and fix any nonzero integers $w_{1},\ldots,w_{n}$ with
$\sum_{i=1}^{n}w_{i}=0$. Next, properly embed pairwise-disjoint handle strips
$\\{(H_{j},h_{j})\\}_{1}^{m}$ into $S$ so that for each $w_{i}>0$, there are
exactly $w_{i}$ ends of the handle strips sent to $E_{i}$, all of which are
attracting, while if $w_{i}<0$, then there are $-w_{i}$ ends of the handle
strips sent to $E_{i}$, all of which are repelling. We assume the embeddings
are such that each $v_{i}$ meets each of the handle strips exiting the end
$E_{i}$ in a single separating arc in the handle strip like the red arc in the
top of Figure 1, and that in $U_{i}$ the complement of the handle strips is a
union of properly embedded half strips,
$[0,1]\times[0,\infty)\subset\overline{U}_{i}$; see Figure 4. As in Example
2.3, the associated handle shifts $\rho_{1},\ldots,\rho_{m}$ pairwise-commute,
and the map $\rho=\Pi_{1}^{m}\rho_{i}$ is isotopic to an end-periodic
homeomorphism with good nesting neighborhoods $U_{1},\ldots,U_{n}$. By
construction, $\Phi^{*}(\rho)(v_{i})=w_{i}$. In Section 6, we will construct
examples of irreducible and strongly irreducible end periodic homeomorphisms
with the same fine end behavior as $\rho$ by composing $\rho$ with a compactly
supported homeomorphism.

$E_{1}$
---
$E_{2}$
---
$E_{3}$
---
$E_{4}$
---
$E_{5}$
---
$v_{1}$
---
$v_{2}$
---
$v_{3}$
---
$v_{4}$
---
$v_{5}$
---
Figure 4. The curves $v_{1},\ldots,v_{5}$ cut off neighborhoods
$U_{1},\ldots,U_{5}$ (in blue) of the ends $E_{1},\ldots,E_{5}$, respectively.

######  Remark.

The notion of coarse end behavior can be extended to an arbitrary element
$f\in{\rm Map}(S)$ by recording the permutation of the ends, together with the
cohomology class $\Phi^{*}(f^{m})$, where $m\geq 1$ is the order of the
permutation. The purpose of the discussion of end behaviors is to provide a
framework for illustrating the generality of the examples described above and
elaborated upon in Section 6.1. Expanding to non-pure end-periodic
homeomorphisms only serves to complicate the notation and obfuscate the ideas.
For this reason, we do not elaborate on, or further pursue end behaviors for
non-pure elements.

###  2.4. Graphs associated to surfaces

Graphs built from topological data associated to curves and arcs on a surface
have become an essential tool in studying surfaces and their mapping class
groups. Most notable among these is the curve graph, $\mathcal{C}(Y)$, of a
surface $Y$, whose vertices are curves on $Y$ with edges connecting any two
vertices that have disjoint representatives on $Y$.

When $Y$ has finite type, a germinal result of Masur–Minsky [MM99] with
numerous implications is that $\mathcal{C}(Y)$ is infinite diameter and
hyperbolic. Unfortunately, it is not hard to see that $\mathcal{C}(Y)$ has
diameter 2 when $Y$ has infinite type. The actions of big mapping class groups
on various other graphs of curves, simplicial complexes, and metric spaces
have been studied extensively. See, for just a few examples, Aramayona–Valdez
[AV18], Bavard [Bav16], Durham–Fanoni–Vlamis [DFV18], Lanier–Loving [LL21],
and Mann–Rafi [MR19].

Another useful graph is the arc and curve graph, $\mathcal{AC}(Y)$: the graph
whose vertices are curves and arcs on $Y$, and whose edges correspond to pairs
of curves and/or arcs having disjoint representatives. When $Y$ is an annulus,
the vertices of $\mathcal{AC}(Y)$ are also arcs (which recall means homotopy
classes of essential arcs), though in this setting we require the homotopies
to fix the endpoints (otherwise there would be only one vertex). In this case,
edges connect vertices with representatives having disjoint interiors.

Given $\Sigma\subset Y$ a non-annular, essential subsurface, the _subsurface
projection_

$\pi_{\Sigma}:\mathcal{C}(Y)\longrightarrow\mathcal{P}(\mathcal{AC}(\overline{\Sigma}))$

takes a vertex $\alpha\in\mathcal{C}(Y)$ to the collection of curves and arcs
in $\Sigma$ obtained by taking the union of all distinct homotopy classes
occurring in the intersection of $\alpha$ with $\overline{\Sigma}$, after
$\overline{\Sigma}$ and $\alpha$ are put in minimal position (we send points
in an edge to the union of the images of the endpoints of the edge). Note that
our definition differs slightly from the projections defined by Masur and
Minsky [MM00].

When $\Sigma$ is an annulus, we first consider the cover $Z_{\Sigma}$ of $Y$
corresponding to $\pi_{1}\Sigma$. This cover is naturally the interior of a
compact annulus, $\overline{Z}_{\Sigma}$, (obtained by adding the ideal
boundary) and we let $\pi_{\Sigma}(\alpha)$ be the union of the closures of
those components of the preimage of $\alpha$ in $Z_{\Sigma}$ that limit to
points on both boundary components of $\overline{Z}_{\Sigma}$.

We define the $\Sigma$-subsurface distance as

$d_{\Sigma}(\alpha,\beta)={\rm
diam}_{\mathcal{AC}(\overline{\Sigma})}(\pi_{\Sigma}(\alpha)\cup\pi_{\Sigma}(\beta)),$

whenever $\pi_{\Sigma}(\alpha),\pi_{\Sigma}(\beta)\neq\emptyset$. Observe that
in this case we have

(2.5) $d_{\Sigma}(\alpha,\beta)\geq 2\Leftrightarrow i(\alpha,\beta)\neq 0.$

If
$\pi_{\Sigma}(\alpha),\pi_{\Sigma}(\beta),\pi_{\Sigma}(\gamma)\neq\emptyset$,
then this distance satisfies the triangle inequality

$d_{\Sigma}(\alpha,\beta)\leq
d_{\Sigma}(\alpha,\gamma)+d_{\Sigma}(\gamma,\beta).$

Figure 5. Elementary moves on pants decompositions.

The graph of central interest in this paper is the pants graph of an infinite-
type surface $S$, which we now describe. First, a pants decomposition on $S$
is a multicurve in $S$ that cuts $S$ into 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 $P^{\prime}$. If $P^{\prime}$ is obtained from $P$ by an
elementary move and $Q=P\cap P^{\prime}$ is the maximal multicurve in common,
then $S-Q$ has one component $\Sigma$ which has complexity one. There are two
types of elementary moves, depending on the homeomorphism type of $\Sigma$,
and these are illustrated in Figure 5.

######  Definition 2.11.

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. We define a path metric on
(the components of) $\mathcal{P}(S)$ so that and 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 $2$. The group ${\rm Map}(S)$ still acts isometrically on
$\mathcal{P}(S)$ with this metric.

Brock proved in [Bro03b] that for finite-type surfaces $Y$, $\mathcal{P}(Y)$
is a coarse model for the Teichmüller space of $Y$ with the Weil–Petersson
metric. For infinite-type surfaces, the pants graph is not connected, and thus
the distance between pants decompositions may be infinite. See [Bra20] for an
alternative topology on this graph and other facts about it.

######  Definition 2.12.

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 $n>0$.

For a pants decomposition $P$ of $S$, it may be the case that for all $n>0$,
$P$ and $f^{n}(P)$ are not in the same component, and thus
$d(P,f^{n}(P))=\infty$ for all $n>0$. If this is the case, then the same is
necessarily true for all $P^{\prime}$ in the same component as $P$. For the
purposes of studying $\tau(f)$, such components are useless, and thus we let
$\mathcal{P}_{f}(S)$ be the subgraph of $\mathcal{P}(S)$ spanned by all
$f$–asymptotic pants decompositions; that is, those pants decompositions $P$
for which $P$ and $f^{n}(P)$ differ by a finite number of elementary moves for
some $n>0$ (and are thus a finite distance in $\mathcal{P}(S)$). So, in the
definition of $\tau(f)$, it suffices to consider only pants decompositions $P$
taken from $\mathcal{P}_{f}(S)$. Note that, since $\mathcal{P}_{f}(S)$ is non-
empty, $\tau(f)$ is finite.

######  Remark.

The behavior of end periodic homeomorphisms on the pants graph is reminicent
of the work of Funar and Kapoudjian in [FK09]. Indeed, their asymptotic
mapping class group of infinite genus is a subgroup of the mapping class group
of the blooming Cantor tree that preserves a certain component of its pants
graph. Thanks to Ghaswala for pointing out this reference.

Given a component $\Omega\subset\mathcal{P}_{f}(S)$, we define
$\tau(f,\Omega)$ by the same formula as $\tau(f)$, except the infimum is taken
over all $P\in\Omega$ (rather than over all $P\in\mathcal{P}(S)$). As such,
$\tau(f,\Omega)\geq\tau(f)$.

###  2.5. Decomposition spaces

Our proof of Theorem 1.1 will utilize some decomposition theory and a result
of Armentrout [Arm69].

Let $M$ be a 3-manifold. A decomposition $G$ of $M$ is a partition of $M$ into
disjoint sets. Given such a decomposition, there is a natural map $f:M\to M/G$
obtained by collapsing each element $X\in G$ to a point and endowing $M/G$
with the quotient topology. If $\pi:M\to N$ is any quotient map, then there is
a naturally associated decomposition of $M$ which comes from the preimages of
points in $N$ under $\pi$. As such, a quotient map may also be referred to as
a decomposition. For more details about decompositions; see [Dav86].

######  Definition 2.13 (Upper semicontinuous).

A decomposition $G$ of $M$ is said to be upper semicontinuous if each element
$X\in G$ is compact, and if for each $X\in G$ and each open subset $U\subseteq
M$ with $X$ contained in $U$, there exists a further open set $V\subseteq M$
containing $X$ such that for each $X^{\prime}\in G$ with $X^{\prime}\cap
V\neq\emptyset$, we have that $X^{\prime}\subseteq U$. A quotient map
$\pi:M\to N$ is said to be upper semicontinuous if the associated
decomposition space is upper semicontinuous.

The following result gives alternative characterizations of an upper
semicontinuous decomposition; see [Dav86, Proposition I.1.1].

######  Proposition 2.14.

Let $G$ be a decomposition of a 3-manifold $M$. The following are equivalent:

  1. (1)

The decomposition $G$ is upper semicontinuous.

  2. (2)

For each open set $U$ in $M$, let $U^{*}$ denote the union of all sets in $G$
which are contained in $U$. Then, the set $U^{*}$ is open in $M$.

  3. (3)

The associated quotient map $f:M\to M/G$ is closed.

######  Definition 2.15 (Cellular decomposition).

A subset $X$ of a 3-manifold $M$ is said to be cellular if there exists a
sequence $\\{B_{i}\\}$ of 3-cells in $M$ such that for each $i\geq 1$,
$B_{i+1}\subset\mathrm{Int}(B_{i})$, and $X=\cap_{i=1}^{\infty}B_{i}$.
Equivalently, a subset $X$ of a 3-manifold $M$ is cellular if and only if $X$
is compact and has arbitrarily small neighborhoods which are homeomorphic to
Euclidean 3-space. A decomposition $G$ of a 3-manifold $M$ is a cellular
decomposition if $G$ is an upper semicontinuous decomposition of $M$ and if
each element $X\in G$ is a cellular subset of $M$.

The following result of Armentrout [Arm69, Theorem 2] gives a sufficient
condition for when the quotient space is homeomorphic to the original space.
We apply this result to the double of $\overline{M}_{f}$ in the proof of
Proposition 4.3.

######  Theorem 2.16.

Suppose $M$ is a 3-manifold and $G$ is a cellular decomposition of $M$ such
that $M/G$ is a 3-manifold. Then, $M\to M/G$ is homotopic to a homeomorphism.

###  2.6. Surfaces in 3-manifolds and foliations

Suppose $\overline{M}$ is a compact, orientable, $3$-manifold with (possibly
empty) boundary. We say $\overline{M}$ is irreducible if every smoothly
embedded $2$–sphere in $\overline{M}$ bounds a $3$-ball in $\overline{M}$. An
incompressible surface in $\overline{M}$ is the image of a compact, connected,
orientable surface by an embedding
$(\overline{\Sigma},\partial\overline{\Sigma})\to(\overline{M},\partial\overline{M})$
such that $\overline{\Sigma}$ is not a sphere or disk, and such that the
induced map on fundamental groups is injective. We often write
$\overline{\Sigma}\subset\overline{M}$ for an incompressible surface, though
it will always come from an embedding of pairs.

An incompressible annulus $(A,\partial
A)\subset(\overline{M},\partial\overline{M})$ is boundary parallel if there is
an embedded solid torus $V\subset\overline{M}$ so that $\partial V$ is the
union of $A$ with an annulus $A^{\prime}=V\cap\partial\overline{M}$
intersecting each other along their common boundary $\partial A=\partial
A^{\prime}$. If $\overline{M}$ is irreducible, then by [Wal68, Lemma 5.3], an
incompressible annulus is boundary parallel if and only if the inclusion is
homotopic by a homotopy of pairs $h_{t}\colon(A,\partial
A)\to(\overline{M},\partial\overline{M})$ to a map of $A$ into
$\partial\overline{M}$. An essential annulus in $\overline{M}$ is an
incompressible annulus which is not boundary parallel. If no essential annulus
exists, we say that $\overline{M}$ is acylindrical. The manifold
$\overline{M}$ is atoroidal if for every incompressible torus, the inclusion
is homotopic into $\partial\overline{M}$.

Let $\mathcal{F}$ be a transversely oriented, codimension-one foliation of a
compact 3-manifold $\overline{M}$ for which the boundary is a union of leaves.
We say such a foliation is taut if for every leaf $\lambda$ of $\mathcal{F}$
there is a closed curve or embedded arc
$(\gamma_{\lambda},\partial\gamma_{\lambda})\subset(\overline{M},\partial\overline{M})$
which is transverse to $\mathcal{F}$ and intersects $\lambda$.

We will need the following “normal form” theorem for surfaces in taut foliated
$3$-manifolds due to Roussarie [Rou74, Théorèm 1] (see also [CC03, Theorem
9.5.5]). A strengthening of an absolute version of this theorem is due to
Thurston [Thu72] (see also [Has86]), but we will use the following relative
version.

######  Theorem 2.17.

Suppose $\mathcal{F}$ is a transversely oriented, taut foliation of a
3-manifold with boundary, $\overline{M}$, for which $\partial\overline{M}$ is
a union of leaves. Suppose $\overline{\Sigma}\subset\overline{M}$ is a
properly embedded, incompressible torus or annulus. After an isotopy of the
embedding which is the identity on the boundary (in the annulus case) we have:

  1. (1)

$\overline{\Sigma}$ is a torus and is either transverse to $\mathcal{F}$ or is
a leaf of $\mathcal{F}$; or

  2. (2)

$\overline{\Sigma}$ is an annulus and is transverse to $\mathcal{F}$, except
at finitely many circle tangencies occurring in the interior of
$\overline{M}$.

###  2.7. Dehn filling

Given a $3$-manifold $\overline{M}$ with a torus boundary component $T$, we
can _Dehn fill_ $\overline{M}$ along $T$ by gluing a solid torus $D\times
S^{1}$ to $\overline{M}$ by a homeomorphism $\varphi:D\times S^{1}\to T$. The
homeomorphism type of the Dehn filling is determined by the isotopy class of
$\beta\subset T$ to which $\partial D\times\\{x\\}$ is identified. We call
$\beta$ the Dehn filling coefficient, and write $\overline{M}(\beta)$ to
denote the $\beta$–Dehn filled manifold. Choosing a basis $\mu,\lambda$ for
$\pi_{1}(T)\cong\mathbb{Z}^{2}$, the isotopy class of $\beta$ on $T$ is given
by $\beta=\pm(p\lambda+q\mu)$, with $p$ and $q$ relatively prime. So, we may
also refer to this Dehn filling coefficient as $(p,q)$, writing
$\overline{M}(\beta)=\overline{M}(p,q)$. Additionally, we allow a Dehn filling
coefficient of “$\infty$”, which means that we do not glue in a solid torus at
all (i.e., we leave the boundary component unfilled). Any sequence of distinct
slopes is said to converge to $\infty$.

If $\overline{M}$ is a $3$-manifold with $k$ torus boundary components and
$\beta=(\beta_{1},\ldots,\beta_{k})$ is a $k$–tuple of Dehn filling
coefficients (some or all of which may be $\infty$), we may perform
$\beta_{i}$–Dehn filling on each boundary component, which we denote by
$\overline{M}(\beta_{1},\ldots,\beta_{k})$ or simply $\overline{M}(\beta)$.
Finally, we also consider Dehn filling on torus ends of a noncompact manifold.
These are ends which have neighborhoods homeomorphic to $T^{2}\times(0,1)$,
where $T^{2}$ is a torus. We perform Dehn filling on such ends by first
deleting a product open neighborhood of the end and then Dehn filling the
resultant boundary torus.

A link in a $3$–manifold, $L\subset\overline{M}$, is the image of an embedding
of a disjoint union of circles (a link with one component is a knot). Given a
link $L\subset\overline{M}$, we let $\overline{M}-L$ denote the exterior of
$L$ in $\overline{M}$ obtained by removing an open tubular neighborhood of $L$
from $\overline{M}$.

It is straightforward to see that $(1,n)$–Dehn filling the exterior of a knot
which is a simple closed curve in a surface fiber of a fibered $3$-manifold
(for an appropriate basis of the fundamental group of the torus boundary) has
the same effect as changing the monodromy by the $n^{th}$ power of a Dehn
twist; see e.g. [Sta78].

######  Proposition 2.18.

Let $\gamma$ be a curve in $Y$ and $K$ the associated knot in the mapping
torus $M_{f}$ for $f\in{\rm Map}(Y)$. Then $(M_{f}-K)(1,n)\cong M_{f\circ
T_{\gamma}^{n}}$.

More generally, if $L$ is a link consisting of curves
$\gamma_{1},\ldots,\gamma_{k}$ with $\gamma_{i}$ lying on fiber
$Y\times\\{\frac{i}{k+1}\\}\subset Y\times(0,1)\subset M_{f}$, for each $i$,
then

$(M_{f}-L)((1,n_{1}),\ldots,(1,n_{k}))\cong M_{f\circ
T_{\gamma_{k}}^{n_{k}}\circ\cdots\circ T_{\gamma_{1}}^{n_{1}}}.$

###  2.8. Pared 3-manifolds

The following definitions provide the appropriate topological framework for
stating Thurston’s Hyperbolization Theorem (see Theorem 2.20), which ensures
that the manifolds we are interested in admit hyperbolic structures. Given a
$3$-manifold $\overline{M}$ and closed subset
$\mathfrak{p}\subset\partial\overline{M}$, we say that
$(\overline{M},\mathfrak{p})$ is a pared $3$-manifold if:

  1. (1)

$\overline{M}$ is compact, connected, orientable, irreducible and
$\partial\overline{M}\neq\emptyset$;

  2. (2)

$\pi_{1}(\overline{M})$ is not virtually abelian;

  3. (3)

$\mathfrak{p}$ consists of incompressible annuli and tori, and contains all
tori in $\partial\overline{M}$;

  4. (4)

any incompressible annulus $(S^{1}\times I,S^{1}\times\partial
I)\to(\overline{M},\mathfrak{p})$ is properly homotopic into $\mathfrak{p}$
(i.e., homotopic as a map of pairs); and

  5. (5)

any non-cyclic abelian subgroup of $\pi_{1}(\overline{M})$ is conjugate into
the fundamental group of some component of $\mathfrak{p}$.

A pared manifold $(\overline{M},\mathfrak{p})$ has incompressible boundary if
$\partial\overline{M}-\mathfrak{p}$ is incompressible (when
$\mathfrak{p}=\emptyset$, this condition simply means that
$\partial\overline{M}$ is incompressible). An essential annulus in
$(\overline{M},\mathfrak{p})$ is an incompressible annulus $(S^{1}\times
I,S^{1}\times\partial I)\to(\overline{M},\partial\overline{M}-\mathfrak{p})$
which is not homotopic into $\partial\overline{M}$. One says that a pared
manifold $(\overline{M},\mathfrak{p})$ is acylindrical if it has
incompressible boundary and contains no essential annulus. If
$\mathfrak{p}=\emptyset$, this agrees with the definition of acylindrical
given above.

Observe that if $(\overline{M},\mathfrak{p})$ is a pared manifold with
incompressible boundary and $\mathfrak{p}$ meets each boundary component of
$\overline{M}$ of genus at least $2$ in annuli whose core curves define a
pants decomposition of these boundary components, then
$(\overline{M},\mathfrak{p})$ is acylindrical. To see this, observe that if
there were an incompressible annulus with boundary in
$\partial\overline{M}-\mathfrak{p}$, then since this surface is a union of
thrice-punctured spheres, the boundary of the annulus may be isotoped into
$\mathfrak{p}$, and hence the entire annulus is properly homotopic relative to
the boundary into $\mathfrak{p}$ by (4).

Suppose $\overline{M}$ is a compact, connected, orientable, irreducible
$3$-manifold with incompressible, nonempty boundary. If all components of
$\partial\overline{M}$ have genus at least $2$, then
$(\overline{M},\emptyset)$ is a pared manifold if $\overline{M}$ is atoroidal:
conditions (1)-(2) are immediate, (3)-(4) are vacuous, and (5) follows from
the Torus Theorem; see e.g. [JS79, Corollary IV.4.3]. Similarly, if
$\mathfrak{p}\neq\emptyset$ and contains all tori in $\partial\overline{M}$,
then $(\overline{M},\mathfrak{p})$ is pared if $\overline{M}$ is atoroidal and
not Seifert fibered, see [JS79, Theorem IV.4.1].

######  Remark.

It is sometimes convenient to replace some or all of the annuli in
$\mathfrak{p}$ by their core curves, so that $\mathfrak{p}$ is a union of
tori, annuli, and simple closed curves. Note that up to homeomorphism, this
does not affect the manifold $\overline{M}-\mathfrak{p}$ or its boundary
$\partial\overline{M}-\mathfrak{p}$.

A “pants block” is a pared manifold built as a quotient of
$\overline{\Sigma}\times I$, where $\overline{\Sigma}$ is a complexity-one
surface (so $\overline{\Sigma}\cong\overline{\Sigma}_{1,1}$ or
$\overline{\Sigma}\cong\overline{\Sigma}_{0,4}$) and $I$ an interval, say
$I=[0,1]$, together with some additional decoration. We make this precise with
the following.

######  Definition 2.19 (Pants block).

A pants block is a pair $({\mathcal{B}},\partial_{1}\mathcal{B})$, where
$\mathcal{B}$ is a handlebody of genus $2$ or $3$, and
$\partial_{1}\mathcal{B}\subset\partial\mathcal{B}$ is union of simple closed
curves defining a pants decomposition of the boundary of $\mathcal{B}$. We
further assume that $(\mathcal{B},\partial_{1}\mathcal{B})$ admits a quotient
map of pairs

$\varphi\colon(\overline{\Sigma}\times
I,\alpha_{0}\cup\alpha_{1}\cup(\partial\overline{\Sigma}\times
I))\to(\mathcal{B},\partial_{1}\mathcal{B})$

such that

  1. (a)

$\overline{\Sigma}$ is a compact, complexity-one surface;

  2. (b)

$\alpha_{i}\subset\overline{\Sigma}\times\\{i\\}$, for $i=0,1$ are curves of
the form $\alpha_{i}=\alpha^{i}\times\\{i\\}$, with
$\alpha^{0},\alpha^{1}\subset\overline{\Sigma}$ curves intersecting minimally
(thus intersecting once in $\overline{\Sigma}_{1,1}$ and twice in
$\overline{\Sigma}_{0,4}$); and

  3. (c)

$\varphi$ is 1-to-1, except on $\partial\overline{\Sigma}\times I$, where
$\varphi^{-1}(\varphi(x,t))=\\{x\\}\times I$, for all
$(x,t)\in\partial\overline{\Sigma}\times I$.

We sometimes refer to a pants block simply as a block.

Up to homeomorphism (ignoring the pared locus), there are exactly two distinct
pants blocks, as shown in Figure 6. It is straightforward to see that pants
blocks are acylindrical pared manifolds using the product structure, since

$\mathcal{B}-\partial_{1}\mathcal{B}\cong\overline{\Sigma}\times
I-(\alpha_{0}\cup\alpha_{1}\cup(\partial\overline{\Sigma}\times I)).$

Figure 6. Pants blocks corresponding to the one-holed torus and 4-holed
sphere. The curve $\alpha_{0}$ is shown in blue, the curve $\alpha_{1}$ is
shown in red, and the image (under $\varphi$) of $\partial\overline{\Sigma}$
is shown in grey. The meridian disks for the handlebodies are drawn for
illustrative purposes only.

It will be convenient to keep track of where the components of
$\partial_{1}\mathcal{B}$ came from. As such, we use the following notation
for the components of $\partial_{1}\mathcal{B}$:

$\partial_{1}^{-}\mathcal{B}=\varphi(\alpha_{0})\quad\partial_{1}^{+}\mathcal{B}=\varphi(\alpha_{1})\quad\partial_{1}^{v}\mathcal{B}=\varphi(\partial\overline{\Sigma}\times
I).$

We also write
$\partial_{2}\mathcal{B}=\partial\mathcal{B}-\partial_{1}\mathcal{B}$, which
is a disjoint union of thrice-punctured spheres, $\Sigma_{0,3}$, (two for
$\overline{\Sigma}_{1,1}$ and four for $\overline{\Sigma}_{0,4}$), and set

$\partial_{2}^{-}\mathcal{B}=\partial_{2}\mathcal{B}\cap\varphi(\overline{\Sigma}\times\\{0\\})\mbox{
and
}\partial_{2}^{+}\mathcal{B}=\partial_{2}\cap\varphi(\overline{\Sigma}\times\\{1\\}).$

We often denote a pants block simply as $\mathcal{B}$, suppressing the
decoration and the quotient map, though they are in fact part of the
structure. When we want to distinguish between the two types of pants blocks
built from $\overline{\Sigma}_{0,4}$ and $\overline{\Sigma}_{1,1}$, we write
$\mathcal{B}^{S}$ and $\mathcal{B}^{T}$, respectively.

######  Remark.

We note that $(\mathcal{B},\partial_{1}\mathcal{B})$ is a pared $3$–manifold:
everything except condition (4) in the definition is immediate. Condition (4)
follows from an identification of the fundamental group as a free group, in
which all components of $\partial_{1}\mathcal{B}$ are distinct homotopy
classes. The product structure implies $\partial_{2}\mathcal{B}$ is
incompressible, and by the comments above,
$(\mathcal{B},\partial_{1}\mathcal{B})$ is acylindrical.

###  2.9. Geometry of hyperbolic 3-manifolds

By a convex hyperbolic $3$-manifold, we mean a $3$-manifold which is the
quotient of a closed, convex subset of $\mathbb{H}^{3}$ by a discrete,
torsion-free subgroup $\Gamma<{\rm PSL}_{2}(\mathbb{C})$. A convex hyperbolic
metric on a $3$-manifold, $M$, is a metric which makes $M$ isometric to a
convex hyperbolic $3$-manifold. A special case of Thurston’s Geometrization
Theorem for Haken manifolds is the following; see [Thu86, McM92, Mor84].

######  Theorem 2.20.

Suppose $(\overline{M},\mathfrak{p})$ is a pared $3$-manifold and
$\partial\overline{M}-\mathfrak{p}\neq\emptyset$ is incompressible. Then
$\overline{M}-\mathfrak{p}$ admits a convex hyperbolic metric $\sigma$. If
$(\overline{M},\mathfrak{p})$ is additionally assumed to be acylindrical and
non-degenerate, then there is a unique (up to isometry) convex hyperbolic
structure $\sigma_{min}$ on $\overline{M}-\mathfrak{p}$ for which
$\partial\overline{M}-\mathfrak{p}$ is totally geodesic.

The non-degeneracy assumption on $(\overline{M},\mathfrak{p})$ rules out the
case that the structure degenerates to a lower-dimensional manifold. Because
of acylindricity, this simply means that $(\overline{M},\mathfrak{p})$ is not
a pair of pants times an interval, with $\mathfrak{p}$ being the three annuli
which are the boundary curves of the pants times the interval. We also note
that $\mathfrak{p}$ is allowed to be empty in the theorem above, though
$\partial\overline{M}$ is not. The uniqueness of $\sigma_{min}$ in Theorem
2.20 is a consequence of Mostow-Prasad Rigidity by doubling
$\overline{M}-\mathfrak{p}$ over the boundary.

We will write ${\rm Vol}(\overline{M}-\mathfrak{p},\sigma)$ for the total
volume of the manifold $\overline{M}-\mathfrak{p}$ with respect to the metric
$\sigma$. In the special case that $(\overline{M},\mathfrak{p})$ is
acylindrical, we will write ${\rm Vol}(\overline{M}-\mathfrak{p})$ to mean
${\rm Vol}(\overline{M}-\mathfrak{p},\sigma_{min})$, where $\sigma_{min}$ is
the metric given by Theorem 2.20.

A pants block $(\mathcal{B},\partial_{1}\mathcal{B})$ is an acylindrical,
pared manifold, as described in the remark above; thus Theorem 2.20 gives a
unique hyperbolic structure on
$\mathring{\mathcal{B}}=\mathcal{B}-\partial_{1}\mathcal{B}$ with totally
geodesic boundary. In fact, this can be constructed explicitly from regular
ideal octahedra: one octahedron in the case of $\mathcal{B}^{T}$ and two in
the case of $\mathcal{B}^{S}$; see e.g. Agol [Ago03, Lemma 2.3].

######  Proposition 2.21.

The manifold $\mathring{\mathcal{B}}$ has a convex hyperbolic metric
$\sigma_{\mathcal{B}}$ with totally geodesic boundary consisting of a union of
thrice-punctured spheres. The metric $\sigma_{\mathcal{B}}$ is obtained by
gluing together half the faces of a regular ideal octahedron for
$\mathcal{B}=\mathcal{B}^{T}$ and two regular ideal octahedra for
$\mathcal{B}=\mathcal{B}^{S}$. Consequently,

${\rm Vol}(\mathring{\mathcal{B}}^{T},\sigma_{\mathcal{B}^{T}})=V_{\rm
oct}\quad\mbox{ and }\quad{\rm
Vol}(\mathring{\mathcal{B}}^{S},\sigma_{\mathcal{B}^{S}})=2V_{\rm oct},$

where $V_{\rm oct}$ is the volume of a regular ideal octahedron.

The notation $\sigma_{min}$ is justified by the following theorem due to Storm
[Sto07, Theorem 3.1]; see also Storm [Sto02] and Agol–Storm–Thurston [AST07].

######  Theorem 2.22.

Given a pared manifold $(\overline{M},\mathfrak{p})$ which is acylindrical and
non-degenerate with $\partial\overline{M}-\mathfrak{p}\neq\emptyset$ and given
any convex hyperbolic metric $\sigma$ on $\overline{M}-\mathfrak{p}$, we have
${\rm Vol}(\overline{M}-\mathfrak{p},\sigma_{min})\leq{\rm
Vol}(\overline{M}-\mathfrak{p},\sigma)$ with equality precisely when
$\sigma_{min}=\sigma$.

Suppose $(\overline{M},\mathfrak{p})$ is a pared $3$-manifold and
$\partial\overline{M}-\mathfrak{p}\neq\emptyset$ is incompressible. In this
situation, we define

(2.6) ${\rm{V\underline{ol}}}(\overline{M}-\mathfrak{p})=\inf\\{{\rm
Vol}(\overline{M}-\mathfrak{p},\sigma)\mid\sigma\mbox{ a convex hyperbolic
metric on }\overline{M}-\mathfrak{p}\\}.$

If $\mathfrak{p}=\emptyset$, we also write
${\rm{V\underline{ol}}}(\overline{M})={\rm{V\underline{ol}}}(\overline{M}-\emptyset)$.

By Theorem 2.22, ${\rm{V\underline{ol}}}(\overline{M}-\mathfrak{p})$ is
realized as ${\rm Vol}(\overline{M}-\mathfrak{p},\sigma_{min})$ when
$(\overline{M},\mathfrak{p})$ is acylindrical. The main fact we will need
about the general case is the second half of the result of Storm [Sto07,
Theorem 3.1] referenced above.

######  Theorem 2.23.

If $(\overline{M},\mathfrak{p})$ is a pared $3$-manifold with
$\partial\overline{M}-\mathfrak{p}\neq\emptyset$ incompressible, then
${\rm{V\underline{ol}}}(\overline{M}-\mathfrak{p})$ is equal to half of the
simplicial volume of the double of $\overline{M}$ over
$\overline{\partial\overline{M}-\mathfrak{p}}$. In particular, for any degree
$k<\infty$ covering,
$(\overline{M}^{\prime},\mathfrak{p}^{\prime})\to(\overline{M},\mathfrak{p})$,
one has
${\rm{V\underline{ol}}}(\overline{M}^{\prime}-\mathfrak{p}^{\prime})=k{\rm{V\underline{ol}}}(\overline{M}-\mathfrak{p})$.

It will also be useful to relate the infimal volume of a pared manifold to the
infimal volume of the manifold after forgetting the paring locus; see Storm
[Sto07, Corollary 1.2, Corollary 2.10, and Theorem 3.2].

######  Theorem 2.24.

Suppose $(\overline{M},\mathfrak{p})$ is a pared acylindrical $3$-manifold and
that $\partial\overline{M}$ is incompressible, nonempty, and all components
have genus at least $2$. Then
${\rm{V\underline{ol}}}(\overline{M})\leq{\rm{V\underline{ol}}}(\overline{M}-\mathfrak{p})$.

Since $(\overline{M},\mathfrak{p})$ is pared and $\partial\overline{M}$
contains no tori we see that $(\overline{M},\emptyset)$ is also pared and
hence admits a convex hyperbolic structure by Theorem 2.20 so
${\rm{V\underline{ol}}}(\overline{M})$ is well-defined.

Storm’s proof of Theorem 2.24 exploits the behavior of volume (and simplicial
volume) under Dehn filling. The behavior of the hyperbolic and simplicial
volumes is part of Thurston’s Dehn Surgery Theorem; see Thurston [Thu78,
Theorem 5.8.2, Theorem 6.5.4, and Theorem 6.5.6], as well as [NZ85, Theorem
1A] and [Mar16, 15.4.1]. The version of Thurston’s theorem, which we state
here, follows by doubling over the totally geodesic boundary and appealing to
[MR92]. The notion of a sequence of slopes going to $\infty$ is natural from
the perspective of Thurston’s Hyperbolic Dehn Surgery Theorem.

######  Theorem 2.25.

Let $(M,\sigma)$ be a finite volume hyperbolic $3$-manifold with totally
geodesic boundary and $k$ torus cusps. Suppose
$\beta^{n}=(\beta_{1}^{n},\ldots,\beta_{k}^{n})$ are a sequence of Dehn
filling coefficients so that for each $i$, either $\beta_{i}^{n}=\infty$ for
all $n$, or $\beta_{i}^{n}\to\infty$. Then for $n$ sufficiently large,
$M(\beta^{n})$ admits a complete hyperbolic structure $\sigma(\beta^{n})$ with
totally geodesic boundary and

${\rm Vol}(M(\beta^{n}),\sigma(\beta^{n}))\to{\rm Vol}(M,\sigma).$

Moreover,

${\rm Vol}(M(\beta^{n}),\sigma(\beta^{n}))\leq{\rm Vol}(M,\sigma)$

with equality if and only if $\beta^{n}=(\infty,\ldots,\infty)$.

##  3\. Compactified mapping tori

Given an end-periodic homeomorphism $f\colon S\to S$, we let

$M_{f}=S\times[0,1]/(x,1)\sim(f(x),0)$

denote the mapping torus of $f$. We will write $(\psi_{s})$ to denote the
suspension flow on $M_{f}$, defined as the descent to $M_{f}$ of the local
flow on $S\times[0,1]$ given by $\psi_{s}(x,t)=(x,t+s)$, when $0\leq t\leq
t+s\leq 1$. The goal of this section is to prove the following result.

######  Proposition 3.1.

Let $f\colon S\to S$ be an irreducible, end-periodic homeomorphism of a
surface with finitely many ends, all accumulated by genus. Then the mapping
torus, $M_{f}$, is the interior of a compact, irreducible, atoroidal
$3$-manifold, $\overline{M}_{f}$, with incompressible boundary. If $f$ is
strongly irreducible, then $\overline{M}_{f}$ is also acylindrical.

The fact that $M_{f}$ is the interior of a compact, irreducible manifold with
boundary is well-known: the boundary points are obtained by adding “endpoints
at infinity” to each positive (respectively, negative) ray of the suspension
flow that passes through $\mathcal{U}_{+}\times\\{0\\}$ (respectively,
$\mathcal{U}_{-}\times\\{0\\}$); see [Fen97]. As we will make explicit use of
this compactification in our discussion, we discuss this in more detail.

Suppose $f$ is any end-periodic homeomorphism of $S$. We begin by describing
the compactification of $M_{f}$ in a way that is well-suited to our
discussion. As a starting point, we identify the infinite cyclic cover of
$M_{f}$ dual to the fibration over $S^{1}$ by

$p\colon S\times(-\infty,\infty)\to M_{f}$

together with the action of the covering group isomorphic to $\mathbb{Z}$,
generated by the homeomorphism $F$ given by

$F(x,t)=(f(x),t-1).$

This is a covering action with quotient $M_{f}$. Indeed, $S\times[0,1]$ is a
fundamental domain, and the only points that are nontrivially identified are
of the form $(x,1)\sim F(x,1)=(f(x),0)$.

Recall that $\mathcal{U}_{+},\mathcal{U}_{-}\subset S$ are the positive and
negative escaping sets, respectively, as defined in Section 2.2. We define a
partial compactification of $S\times(-\infty,\infty)$ inside
$S\times[-\infty,\infty]$ by

$\widetilde{M}_{\infty}=S\times(-\infty,\infty)\,\,\sqcup\,\,\mathcal{U}_{+}\times\\{\infty\\}\,\,\sqcup\,\,\mathcal{U}_{-}\times\\{-\infty\\}.$

Since $\mathcal{U}_{+}$ and $\mathcal{U}_{-}$ are $f$–invariant, we may extend
the action of $\langle F\rangle$ by defining

$F(x,\pm\infty)=(f(x),\pm\infty)$

whenever $x\in\mathcal{U}_{\pm}$. By Lemma 2.4, $\langle
f\rangle|_{\mathcal{U}_{\pm}}$ acts cocompactly, with quotient closed (not
necessarily connected) surfaces $S_{+}=\mathcal{U}_{+}/\langle f\rangle$ and
$S_{-}=\mathcal{U}_{-}/\langle f\rangle$.

######  Lemma 3.2.

Suppose $f$ is end-periodic and $\langle F\rangle$ acts on
$S\times(-\infty,\infty)$ as above. Then the action of $\langle
F\rangle\cong\mathbb{Z}$ extends to a properly discontinuous, cocompact action
on $\widetilde{M}_{\infty}$ by $F(x,\pm\infty)=(f(x),\pm\infty)$. The
quotient, $\overline{M}_{f}$, is thus a compact manifold with
$\partial\overline{M}_{f}\cong S_{+}\sqcup S_{-}$.

###### Proof.

Recall that

$\mathcal{U}_{\pm}=\bigcup_{k=1}^{\infty}f^{\mp k}U_{\pm}$

where $U_{\pm}$ are good nesting neighborhoods. Since $\mathcal{U}_{\pm}$ is
invariant by $f$, it follows that the action of $F$ extends to an action by
the given formula.

By considering an appropriate open covering of $\widetilde{M}_{\infty}$, we
note that for any compact set $K\subset\widetilde{M}_{\infty}$, there is some
$n\in\mathbb{N}$ and a decomposition $K=K_{+}\cup K_{-}\cup K_{0}$ where

$K_{+}\subset(f^{-n}(U_{+})-f^{n}(U_{+}))\times[0,\infty],$
$K_{-}\subset(f^{n}(U_{-})-f^{-n}(U_{-}))\times[-\infty,0],$

and

$K_{0}\subset(S-(f^{n}(U_{+})\cup f^{-n}(U_{-})))\times[-n,n].$

Note that only finitely many $\langle F\rangle$–translates of each of $K_{0}$,
$K_{+}$, or $K_{-}$ can intersect themselves (the case of $K_{0}$ by
inspection of the second coordinate, and $K_{\pm}$ by inspection of the first
coordinate). Therefore, we must show that only finitely many $\langle
F\rangle$–translates of any one of these sets can nontrivially intersect
another one. And for this, it suffices to consider the $\langle
F\rangle$–translates of $K_{0}$ with either $K_{+}$ or $K_{-}$ and the
$\langle F\rangle$–translates of $K_{+}$ with $K_{-}$.

In the first case, we observe that for any integer $m>n$, $F^{m}(K_{0})\cap
K_{+}=\emptyset$, by considering the second coordinates. On the other hand,
for any integer $m<-2n$, inspection of the first coordinate shows that
$F^{m}(K_{0})\cap K_{+}=\emptyset$. Thus $F^{m}(K_{0})\cap K_{+}=\emptyset$
for all but at most $3n+1$ values of $m\in\mathbb{Z}$. Similar reasoning
applies to show that at most $3n+1$ $\langle F\rangle$–translates of $K_{0}$
nontrivially intersect $K_{-}$.

Next observe that for all integers $m<0$, $F^{m}(K_{+})\cap K_{-}=\emptyset$,
by considering the second coordinates. Finally, note that for all $k>n$,
$f^{k}(U_{+})\cap f^{n}(U_{-})=\emptyset$, and so if $m>2n$, then
$F^{m}(K_{+})\cap K_{-}=\emptyset$. Consequently, there are at most $2n+1$
values of $m\in\mathbb{Z}$ so that $F^{m}(K_{+})\cap K_{-}\neq\emptyset$.
Therefore, the action is properly discontinuous, and hence
$\overline{M}_{f}=\widetilde{M}_{\infty}/\langle F\rangle$ is a $3$-manifold
with boundary $S\pm$.

All that remains is to prove that the action is cocompact. For convenience,
define

$C_{n}=S-(f^{n}(U_{+})\cup f^{n}(U_{-}))\quad\mbox{ and
}\quad\Delta_{\pm}=\overline{U}_{\pm}-f^{\pm 1}(U_{\pm}).$

We observe that $C_{n}$ is compact for all $n>0$, and $\Delta_{\pm}$ is a
compact fundamental domain for $\langle f\rangle$ on $\mathcal{U}_{\pm}$.

Now we suppose $\\{z_{k}\\}_{k=1}^{\infty}\subset\overline{M}_{f}$ is any
sequence, and show that after passing to a subsequence, it converges. Since
$\partial\overline{M}_{f}=S_{+}\cup S_{-}$ is compact, without loss of
generality we may assume $z_{k}\in M_{f}$, for all $k$.

For each $k\geq 1$, let $(x_{k},t_{k})$ be the unique point in
$p^{-1}(z_{k})\cap S\times[0,1)$. Since each $C_{n}$ is compact, we may assume
that for all $n$, only finitely many $x_{k}$ lie in $C_{n}$. After passing to
a subsequence, we may assume that for any $n$, all but finitely many $x_{k}$
lie in $f^{n}(U_{+})$ or in $f^{-n}(U_{-})$. Assuming the first case (the
second is similar) and passing to a further subsequence, we can choose an
increasing sequence $\\{n_{k}\\}_{k=1}^{\infty}$ so that $x_{k}\in
f^{n_{k}}(\Delta_{+})$. Now observe that

$F^{-n_{k}}(x_{k},t_{k})=(f^{-n_{k}}(x_{k}),t_{k}+n_{k})\in\Delta_{+}\times[0,\infty].$

The set on the right is compact, and so the sequence converges. It follows
that $\\{z_{k}\\}$ converges, as required. ∎

Note that the foliation of $\widetilde{M}_{\infty}$ by fibers of the
projection $\widetilde{M}_{\infty}\to[-\infty,\infty]$ is invariant by
$\langle F\rangle$, and so descends to a transversely oriented foliation
$\mathcal{F}$ of $\overline{M}_{f}$ for which $\partial\overline{M}_{f}$ is a
union of (compact) leaves.

######  Lemma 3.3.

Suppose $f$ is an end-periodic homeomorphism of $S$ and $\mathcal{F}$ the
foliation on $\overline{M}_{f}$ just defined. Then, $\mathcal{F}$ is a taut
foliation.

###### Proof.

Any transversal of the foliation of $\widetilde{M}_{\infty}$ connecting a
point of $\mathcal{U}_{-}\times\\{-\infty\\}$ to a point of
$\mathcal{U}_{+}\times\\{\infty\\}$, projects to a transversal of
$\mathcal{F}$ on $\overline{M}_{f}$. Doing this for each component of
$\mathcal{U}_{+}$ and $\mathcal{U}_{-}$, produces enough transversals to
intersect every leaf of $\mathcal{F}$, as required. ∎

We let $D\overline{M}_{f}$ be the double of $\overline{M}_{f}$ over its
boundary. This induces a foliation $\mathcal{F}_{D}$ on $D\overline{M}_{f}$
which is also taut.

######  Proposition 3.4.

Suppose $f$ is an irreducible, end-periodic homeomorphism. Then
$\overline{M}_{f}$ is a compact, irreducible, atoroidal $3$-manifold with
incompressible boundary.

###### Proof.

By Lemma 3.2, we know that $\overline{M}_{f}$ is a compact $3$-manifold with
boundary. Since $M_{f}$ is a mapping torus, it follows that $M_{f}$ (and thus
$\overline{M}_{f}$) is irreducible. Since each component of
$\partial\overline{M}_{f}$ is covered by a component of the
$\pi_{1}$–injective subsurface
$\mathcal{U}_{\pm}\times\\{\pm\infty\\}\subset\widetilde{M}_{\infty}$, it
follows that $\partial\overline{M}_{f}$ is incompressible.

To prove that $\overline{M}_{f}$ is atoroidal, we suppose that
$T\subset\overline{M}_{f}$ is an incompressible torus and derive a
contradiction. By Theorem 2.17, after an isotopy of the inclusion, we can
assume that $T$ is transverse to $\mathcal{F}$ (since $\mathcal{F}$ has no
torus leaves).

Cutting $M_{f}$ open along a fiber, we obtain a product $S\times[0,1]$ and the
torus is cut into a union of annuli connecting a finite, disjoint union of
curves in $S\times\\{0\\}$ to a finite disjoint union of curves in
$S\times\\{1\\}$ (the torus cannot be disjoint from any fiber since
$S\times[0,1]$ contains no incompressible tori). Since $M_{f}$ is obtained by
gluing $S\times\\{1\\}$ to $S\times\\{0\\}$ by $f$, this produces a finite
$f$–invariant union of curves, and thus an $f$–periodic curve. This is a
contradiction to the irreducibility assumption, and thus $\overline{M}_{f}$ is
atoroidal. ∎

Suppose $f$ is an irreducible, end-periodic homeomorphism. Since the boundary
components of $\overline{M}_{f}$ are closed surfaces of genus at least $2$,
Proposition 3.4 and Theorem 2.20 imply that $\overline{M}_{f}$ admits a convex
hyperbolic metric, since $(\overline{M}_{f},\emptyset)$ is pared; see the
discussion following Theorem 2.20.

######  Lemma 3.5.

Suppose $f$ is an irreducible, end-periodic homeomorphism and $(A,\partial
A)\subset(\overline{M}_{f},\partial\overline{M}_{f})$ is an essential annulus.
Then both boundary components of $A$ are in $S_{+}$ or both are in $S_{-}$,
and their preimages in $\widetilde{M}_{\infty}$ are lines in the surface
$\mathcal{U}_{+}\times\\{\infty\\}$ or $\mathcal{U}_{-}\times\\{-\infty\\}$,
respectively.

If $f$ is strongly irreducible, then $\overline{M}_{f}$ is acylindrical.

###### Proof.

An application of Theorem 2.17 implies that, after an isotopy of the inclusion
which is the identity on $\partial A$, we may assume $A$ is transverse to
$\mathcal{F}$, except possibly at finitely many circle tangencies occurring in
$M_{f}\cap A$. In particular, $\mathcal{F}$ induces a foliation
$\mathcal{F}_{A}$ on $A$ whose leaves are components of intersection with the
leaves of $\mathcal{F}$ (consequently, each component of $\partial A$ is a
leaf of $\mathcal{F}_{A}$).

If there is a circle leaf $\alpha$ of $\mathcal{F}_{A}$ in the interior of $A$
(this happens, for example, if there is a circle tangency) then $\alpha$ is
also a curve in some fiber $S$ of $M_{f}$. Since $\alpha$, viewed as a loop on
$A$, generates $\pi_{1}(A)$ and lies on $S$, it follows that the inclusion of
$A$ lifts to an embedding into $\widetilde{M}_{\infty}$. In particular, the
preimage of $A$ in this covering space is a disjoint union of copies of $A$
permuted by the covering transformation $F$.

Suppose one boundary component of $A$ lies on $S_{-}$ and the other on
$S_{+}$. Then any lift of $A$ has one boundary component in
$\mathcal{U}_{-}\times\\{-\infty\\}$ and the other in
$\mathcal{U}_{+}\times\\{\infty\\}$. This means there is an $f$-reducing curve
which corresponds to $(S\times\\{0\\})\cap A$, contradicting irreducibility.

Therefore, both boundary components of $A$ lie in $S_{+}$ or both lie in
$S_{-}$. In $\widetilde{M}_{\infty}\subset S\times[-\infty,\infty]$, any lift
of $A$ is therefore boundary parallel, implying the same for $A$ in
$\overline{M}_{f}$, contradicting the fact that $A$ is essential.

From the contradictions above, we conclude that $A$ has no closed leaves
except its boundary curves. In this case, all leaves of $\mathcal{F}_{A}$
spiral in toward the boundary components of $A$. Let
$\widetilde{A}\subset\widetilde{M}_{\infty}$ be any component of the preimage
of $A$. Note that the foliation $\mathcal{F}_{A}$ lifts to a foliation
$\widetilde{\mathcal{F}}_{A}$ on $\widetilde{A}$ whose non-boundary leaves are
intersections with fibers $S\times\\{t\\}$.

######  Claim 3.6.

$\widetilde{A}$ is the universal covering of $A$ and some power of $F$
preserves $\widetilde{A}$ and generates the covering group.

###### Proof.

Either some positive power of $F$ preserves $\widetilde{A}$, or no positive
power does. In the former case, the smallest such positive power generates an
infinite order covering action on $\widetilde{A}$ with quotient $A$. In this
case, $\widetilde{A}$ must be the universal cover and we are done. If there is
no positive power that preserves $\widetilde{A}$, then $\widetilde{A}$ must be
a lift of $A$. In this case, the boundary of $\widetilde{A}$ is a union of
curves in
$\mathcal{U}_{+}\times\\{\infty\\}\cup\,\mathcal{U}_{-}\times\\{-\infty\\}$.
Since the non-boundary leaves of $\widetilde{\mathcal{F}}_{A}$ limit onto the
boundary curves of $\widetilde{A}$, we see that some fiber $S\times\\{t\\}$
contains a path (the leaf) that accumulates on
$\mathcal{U}_{\pm}\times\\{\pm\infty\\}$, which is impossible. This
contradiction means we are in the first case that $\widetilde{A}$ is the
universal cover, as required. ∎

Let $F^{m}$ be the smallest positive power of $F$ that preserves
$\widetilde{A}$ and thus generates the covering group of $\widetilde{A}\to A$.
Let $\ell\subset(S\times\\{t\\})\cap\widetilde{A}$ be a leaf of
$\widetilde{\mathcal{F}}_{A}$ and consider the strip in $\widetilde{A}$
between $\ell$ and $F^{m}(\ell)$, which is contained in $S\times[t,t+m]$. This
is a union of leaves of $\widetilde{\mathcal{F}}_{A}$ which are slices of a
product structure on the strip. Note that in the covering to $A$ (or
$\overline{M}_{f}$), the restriction to each leaf in the strip injects to a
leaf of $\mathcal{F}_{A}$ and spirals in toward $\partial A$. In particular,
the two ends of the strip must exit ends of $S\times[t,t+m]$ (since the only
sequences in $S\times[t,t+m]$ which when projected to $\overline{M}_{f}$
accumulate on $\partial\overline{M}_{f}$ are those that exit the ends). Since
$F(x,s)=(f(x),s-1)$, it follows that after projecting out the
$\mathbb{R}$–direction, $\ell$ projects to a line $\ell_{0}$ in $S$ and the
strip projects to a proper homotopy from $\ell_{0}$ to $f^{m}(\ell_{0})$. That
is, $\ell_{0}$ is an $f$–periodic line. We note that $\ell_{0}$ is in fact an
essential line. For, if not, then it would bound an infinite monogon
(topological half-plane) in $S$, and hence each leaf of
$\widetilde{\mathcal{F}}_{A}$ would also bound a monogon in the surface slice
$S\times\\{s\\}$. The union of these would project to the interior of a solid
torus in $\overline{M}_{f}$ whose boundary is the union of $A$ and an annulus
in $\partial\overline{M}_{f}$, contradicting the fact that $A$ is essential.
If $f$ is strongly irreducible, this is a contradiction. Therefore, in this
case, there is no essential annulus and $\overline{M}_{f}$ is acylindrical.

In general, the boundary of $\widetilde{A}$ consists of two lines in
$\mathcal{U}_{\pm}\times\\{\pm\infty\\}$. If one of these lines lies in
$\mathcal{U}_{-}\times\\{-\infty\\}$ and the other lies in
$\mathcal{U}_{+}\times\\{\infty\\}$, then $\ell_{0}$ is an AR-periodic line
with one end in an attracting end of $S$ and the other in a repelling end, a
contradiction. It follows that both components of the boundary of
$\widetilde{A}$ are contained in $\mathcal{U}_{+}\times\\{\infty\\}$ or both
are contained in $\mathcal{U}_{-}\times\\{-\infty\\}$. This proves the first
claim of the lemma, and since we have already proved the second claim, we are
done. ∎

###### Proof of Proposition 3.1.

This is immediate from Proposition 3.4 and Lemma 3.5. ∎

Finally, we note that the suspension flow $(\psi_{s})$ on $M_{f}$ can be
reparameterized to a local flow so that the flow lines that exit every compact
set limit to a point on $\partial\overline{M}_{f}$ in finite time. We do this
so that it extends to a local flow $(\bar{\psi}_{s})$ on $\overline{M}_{f}$.
This lifts to a local flow on $\widetilde{M}_{\infty}\subset
S\times[-\infty,\infty]$ and has flow lines given by
$\\{x\\}\times[-\infty,\infty]$, with either or both endpoints missing if $x$
is not in one or both of $\mathcal{U}_{\pm}\times\\{\pm\infty\\}$. Moreover,
we can assume that $(\bar{\psi}_{s})$ on $\overline{M}_{f}$, and its negative
on the reflected copy of $\overline{M}_{f}$ in the double $D\overline{M}_{f}$,
glue together to a well-defined flow $(\psi_{s}^{D})$ on $D\overline{M}_{f}$.

##  4\. A quotient from a path of pants decompositions

In this section, we begin by describing a method for decomposing $3$-manifolds
that will be useful in proving bounds on volumes (see the remark below for
other instances of such decompositions in the literature). We will use this
idea to prove Theorems 1.1 and 1.2 as well as the corollaries described in the
introduction. In our situation, the decomposition will arise naturally from a
quotient map with various properties; see Proposition 4.3.

###  4.1. Block decompositions

First, recall from 2.19 that a pants block is a compact $3$-manifold,
$\mathcal{B}$, with pared boundary
$\partial_{1}\mathcal{B}\subset\partial\mathcal{B}$, where
$\partial_{1}\mathcal{B}$ is a disjoint union of three or six embedded
circles, depending on whether the block comes from $\Sigma_{1,1}$ or
$\Sigma_{0,4}$, respectively. We view $\mathcal{B}$ as a stratified space (for
bookkeeping purposes only), with strata being $\partial_{1}\mathcal{B}$,
$\partial_{2}\mathcal{B}=\partial\mathcal{B}-\partial_{1}\mathcal{B}$, and the
interior ${\rm{Int}}(\mathcal{B})$. We also recall that
$\mathring{\mathcal{B}}=\mathcal{B}-\partial_{1}\mathcal{B}$ admits a complete
hyperbolic structure with totally geodesic, thrice-punctured sphere boundary
components.

Suppose $N$ is a compact $3$-manifold, possibly with boundary, and $L\subset
N$ a link. This is the image of a disjoint union of finitely many circles by a
(smooth) embedding. We require each component of $L$ to be contained in the
interior of $N$ or the boundary of $N$ (that is, if a component intersects the
boundary, then it is contained in it).

######  Definition 4.1.

A block map to $N$ relative to $L$ is a map of pairs
$(\mathcal{B},\partial_{1}\mathcal{B})\to(N,L)$ which is injective on each
component of each stratum of $\mathcal{B}$, and so that the image of each
component of $\partial_{2}\mathcal{B}$ is either disjoint from $\partial N$ or
contained in it (but distinct components may have the same image). We also
require that the image of $\mathcal{B}$ meets $L$ precisely in the image of
$\partial_{1}\mathcal{B}$. Given a block map, it is convenient to identify the
block with its image $\mathcal{B}\subset N$, with the map itself implicit,
thus allowing us to identify the components of each stratum in $N$ (some may
be equal).

A block decomposition of $N$ relative to $L$ (or a block decomposition of
$(N,L)$) is a finite set of block maps $\\{\mathcal{B}_{i}\\}_{i=1}^{n}$ to
$N$ relative to $L$, so that (after identifying the blocks with their images)
we have

  1. (1)

the interiors of the $\mathcal{B}_{i}$ are pairwise disjoint;

  2. (2)

the union of the blocks is all of $N$; and

  3. (3)

whenever two blocks intersect, they do so in a union of components of strata.

Thus, we can view $N$ as being built from a finite number of blocks. The
blocks are glued together in pairs along closures of three-punctured spheres
in their boundaries. The link $L$ is the union of the images of the
$1$–boundaries, $\bigcup\partial_{1}\mathcal{B}_{i}$.

Note that if $(N,L)$ admits a block decomposition, then $L\cap\partial N$
defines a pants decomposition of $\partial N$ (which may be empty).

######  Remark.

Using block decompositions to build $3$-manifolds is very similar to the
construction of Minsky’s “model manifold” defined by Minsky [Min10] and used
by Brock–Canary–Minsky [BCM12] in the proof the Ending Lamination Conjecture.
Our particular definition includes as a special case the construction Agol
used in [Ago03] for mapping tori of finite-type surfaces.

An immediate consequence of Proposition 2.21 is the following result. Recall
that $V_{\rm oct}$ is the volume of the regular ideal octahedron in hyperbolic
$3$-space.

######  Corollary 4.2.

Suppose $L\subset N$ is a link in a compact $3$-manifold $N$ with a block
decomposition of $N$ relative to $L$ that consists of $n_{T}$ blocks of type
$\mathcal{B}^{T}$ and $n_{S}$ blocks of type $\mathcal{B}^{S}$. Then, $N-L$
admits a complete hyperbolic metric with totally geodesic thrice-punctured
sphere boundary components and volume $V_{\rm oct}(n_{T}+2n_{S})$.

###  4.2. A block decomposition from a path in $\mathcal{P}(S)$

Now assume $f$ is an irreducible, end-periodic homeomorphism of $S$. Choose a
pants decomposition $P$ in any $f$–invariant component
$\Omega\subset\mathcal{P}_{f}(S)$ and a path
$P=P_{0},P_{1},\ldots,P_{n}=f^{-1}(P)$ in $\Omega$. We fix representatives of
the pants decompositions and of the isotopy class of $f$ (all denoted by the
same names) so that $f(P_{n})=P_{0}$. We also assume that our representatives
are such that $P_{k}$ and $P_{k+1}$ agree on the complement of a complexity
$1$ (open) subsurface $Z_{k}$. We further assume that on $Z_{k}$, $P_{k}$ and
$P_{k+1}$ meet in simple closed curves $\alpha_{k}^{-},\alpha_{k}^{+}$,
respectively, that intersect each other minimally: once or twice depending on
the homeomorphism type of $Z_{k}$. We let $n_{T}$ and $n_{S}$ denote the
number of surfaces $Z_{k}$ homeomorphic to $\Sigma_{1,1}$ and $\Sigma_{0,4}$,
respectively, with $n_{S}+n_{T}=n$. As usual, $\overline{Z}_{k}\subset S$ is
the compactified subsurface (which may only be embedded on its interior; see
Section 2.1).

For each $0\leq k\leq n$, we also view $P_{k},Z_{k}\subset S$ as a subset of
the slice

$P_{k},Z_{k}\subset S\times\left\\{\tfrac{k}{n}\right\\}\subset S\times[0,1]$

via the inclusion in the first factor. Gluing $S\times\\{1\\}$ to
$S\times\\{0\\}$ via $f$, we produce $M_{f}$. Each of $P_{k}$ and $Z_{k}$ are
well-defined subsets of $M_{f}\subset\overline{M}_{f}$, where we identify
$P_{n}$ with $P_{0}$ via the gluing map $f$, and similarly identify $Z_{n}$
with $Z_{0}$. Define $L=\cup_{k}P_{k}\subset M_{f}$.

For each $0\leq k<n$, set

$W_{k}=Z_{k}\times\left(\tfrac{k}{n},\tfrac{k+1}{n}\right),\quad\overline{W}_{k}=\overline{Z}_{k}\times\left[\tfrac{k}{n},\tfrac{k+1}{n}\right],\quad
W=\bigsqcup_{k=0}^{n-1}W_{k},\quad\mbox{ and
}\quad\overline{W}=\bigcup_{k=0}^{n-1}\overline{W}_{k}.$

The set $W$ and each $W_{k}$ is naturally an open subset of $M_{f}$, and by
another abuse of notation we also view $\overline{W}$ and each
$\overline{W}_{k}$ as subsets of $M_{f}$ (again, only the interior is actually
embedded).

Each $\overline{W}_{k}$ admits a natural map to a block
$\phi_{k}\colon\overline{W}_{k}\to\mathcal{B}_{k}$ (by definition of a block).
Here, $\alpha_{k}^{-}$, $\alpha_{k}^{+}$, and
$\partial\overline{Z}_{k}\times\left[\tfrac{k}{n},\tfrac{k+1}{n}\right]$ map
to $\partial_{1}^{-}\mathcal{B}_{k}$, $\partial_{1}^{+}\mathcal{B}_{k}$, and
$\partial_{1}^{v}\mathcal{B}_{k}$, respectively. We write
$\partial_{2}\overline{W}_{k}$ and $\partial_{2}^{\pm}\overline{W}_{k}$ for
the unions of $3$-punctured spheres mapping to $\partial_{2}\mathcal{B}_{k}$
and $\partial_{2}^{\pm}\mathcal{B}_{k}$, respectively. We identify
$\partial_{2}\overline{W}_{k}$, and $\partial_{2}^{\pm}\overline{W}_{k}$ with
their images in $M_{f}$ when convenient. We also write
$\partial^{v}\overline{W}_{k}=\partial\overline{Z}_{k}\times\left[\tfrac{k}{n},\tfrac{k+1}{n}\right]$
to denote vertical boundary annuli which project to
$\partial_{1}^{v}\mathcal{B}_{k}$.

To state the main proposition of this section, recall from Corollary 2.8 that
$\xi(\partial\overline{M}_{f})=\xi(S_{+})+\xi(S_{-})=3|\Phi^{*}(f)|$.

######  Proposition 4.3.

Given a sequence of elementary moves
$P=P_{0},\ldots,P_{n}=f^{-1}(P)\subset\Omega\subset\mathcal{P}_{f}(S)$,
together with the notation and assumptions above, there is a manifold
$\widehat{M}_{f}$ and a quotient map
$h\colon\overline{M}_{f}\to\widehat{M}_{f}$ with the following properties:

  1. (1)

$h$ restricts to a homeomorphism from
$\partial\overline{M}_{f}\to\partial\widehat{M}_{f}$;

  2. (2)

$h$ is homotopic (rel boundary) to a homeomorphism
$H\colon\overline{M}_{f}\to\widehat{M}_{f}$;

  3. (3)

$\hat{L}=h(L)$ is a link with $n+\tfrac{3}{2}|\Phi^{*}(f)|$ components;

  4. (4)

$h|_{\overline{W}_{k}}\colon\overline{W}_{k}\to\widehat{M}_{f}$ is the
composition of $\phi_{k}$ and a block map
$\mathcal{B}_{k}\subset\widehat{M}_{f}$ relative to $\hat{L}$. Moreover,
$\\{\mathcal{B}_{k}\\}_{k=0}^{n-1}$ is a block decomposition of
$(\widehat{M}_{f},\hat{L})$;

  5. (5)

the pants decomposition
$P_{\Omega}=h^{-1}(\hat{L}\cap\partial\widehat{M}_{f})=H^{-1}(\hat{L}\cap\partial\widehat{M}_{f})\subset\partial\overline{M}_{f}$
depends only on the component $\Omega\subset\mathcal{P}(S)$ containing $P$, up
to isotopy; and

  6. (6)

$(\overline{M}_{f},P_{\Omega})$ is a pared acylindrical manifold.

The pants decomposition $P_{\Omega}$ of $\partial\overline{M}_{f}$ is the one
alluded to in the introduction, and so by Proposition 4.3 (6) together with
Theorem 2.20, $\overline{M}_{f}-P_{\Omega}$ admits a complete hyperbolic
structure, $\sigma(f,\Omega)$, with totally geodesic, thrice-punctured sphere
boundary components. So, by Theorem 2.22,

${\rm{V\underline{ol}}}(\overline{M}_{f}-P_{\Omega})={\rm
Vol}(\overline{M}_{f}-P_{\Omega})={\rm
Vol}(\overline{M}_{f}-P_{\Omega},\sigma(f,\Omega)).$

###  4.3. Applications of Proposition 4.3

We will prove the proposition in the next section, but first we use it to
deduce the first two theorems from the introduction and their corollaries.

Theorem 1.2. For any $f$–invariant component
$\Omega\subset\mathcal{P}_{f}(S)$,

${\rm Vol}(\overline{M}_{f}-P_{\Omega})\leq V_{\rm oct}\tau(f,\Omega).$

###### Proof.

Observe that for any $N>0$, $\overline{M}_{f^{N}}$ is an $N$–fold cover of
$\overline{M}_{f}$. We will apply Proposition 4.3 to $\overline{M}_{f^{N}}$
for an appropriate choice of $N$, which we now specify. Fix any $f$–invariant
component $\Omega\subset\mathcal{P}_{f}(S)$ and let $\epsilon>0$. Let $N>0$
and $P=P_{0},\ldots,P_{n}=f^{-N}(P)\subset\Omega$ be a sequence of elementary
moves so that

$\left|\tau(f,\Omega)-\frac{n_{T}+2n_{S}}{N}\right|<\epsilon,$

where $n=n_{S}+n_{T}$, as above. Recall that the path metric we defined on
(the components of) $\mathcal{P}(S)$ has edges of length $1$ corresponding to
an elementary move that occurs on a one-holed torus has length $1$, and edges
of length $2$ corresponding to an elementary move that occurs on a four-holed
sphere.

For the path $P_{0},\ldots,P_{n}$ above, let
$P_{\Omega}^{N}\subset\partial\overline{M}_{f^{N}}$ be the pants decomposition
from part (5). Observe that by replacing the sequence $P_{0},\ldots,P_{n}$
with a concatenation of
$P=P_{0}=P_{0}^{\prime},P_{1}^{\prime},\ldots,P_{k}^{\prime}=f^{-1}(P)$ and
its image under $f^{-1},f^{-2},\ldots,f^{-N+1}$, we get the same pants
decomposition $P_{\Omega}^{N}$ on $\partial\overline{M}_{f^{N}}$, by
Proposition 4.3 (5), but this necessarily lifts the pants decomposition
$P_{\Omega}$ of $\partial\overline{M}_{f}$. Since ${\rm
Vol}(\overline{M}_{f^{N}}-P_{\Omega}^{N})$ is the volume of the metric with
totally geodesic boundary on $\overline{M}_{f^{N}}-P_{\Omega}^{N}$, it follows
that ${\rm Vol}(\overline{M}_{f^{N}}-P_{\Omega}^{N})=N{\rm
Vol}(\overline{M}_{f}-P_{\Omega})$. Therefore, if we can prove

(4.1) ${\rm Vol}(\overline{M}_{f^{N}}-P_{\Omega}^{N})\leq V_{\rm
oct}(n_{T}+2n_{S})$

we will have

${\rm Vol}(\overline{M}_{f}-P_{\Omega})=\frac{{\rm
Vol}(\overline{M}_{f^{N}}-P^{N}_{\Omega})}{N}\leq V_{\rm
oct}\frac{n_{T}+2n_{S}}{N}\leq V_{\rm oct}(\tau(f,\Omega)+\epsilon),$

and letting $\epsilon\to 0$ will then complete the proof.

To prove the inequality (4.1), we apply Proposition 4.3 (4) to the path
$P_{0},\ldots,P_{n}$ and $\overline{M}_{f^{N}}$, producing a block
decomposition of $\widehat{M}_{f^{N}}$ relative to $\hat{L}$. By Corollary
4.2, the drilled manifold $\widehat{M}_{f^{N}}-\hat{L}$ also admits a complete
hyperbolic metric $\sigma$ with totally geodesic thrice-punctured sphere
boundary components having volume ${\rm
Vol}(\widehat{M}_{f^{N}}-\hat{L},\sigma)=V_{\rm oct}(n_{T}+2n_{S})$. We now
observe that $\overline{M}_{f^{N}}-P_{\Omega}^{N}$ is homeomorphic to a Dehn
filling on $\widehat{M}_{f^{N}}-\widehat{L}$ of some of the components of
$\hat{L}$. By Proposition 4.3 (6) and Theorem 2.20, there is a convex
hyperbolic metric $\sigma(f^{N},\Omega)$ on
$\overline{M}_{f^{N}}-P_{\Omega}^{N}$ with totally geodesic boundary.
Appealing to Theorem 2.25, we see

${\rm Vol}(\overline{M}_{f^{N}}-P_{\Omega}^{N})\leq{\rm
Vol}(\widehat{M}_{f^{N}}-\hat{L})=V_{\rm oct}(n_{T}+2n_{S}).$

This proves (4.1) and so completes the proof of the theorem. ∎

From Theorem 1.2 we obtain the first theorem.

Theorem 1.1. For any irreducible, end-periodic homeomorphism $f\colon S\to S$
of a surface with finitely many ends, all accumulate by genus, we have
${\rm{V\underline{ol}}}(\overline{M}_{f})\leq V_{\rm oct}\tau(f)$.

###### Proof.

Given $\epsilon>0$, suppose $\Omega\subset\mathcal{P}_{f}(S)$ is a component
such that

$\left|\tau(f)-\tau(f,\Omega)\right|<\epsilon$

and let $N>0$ be such that $f^{N}$ preserves $\Omega$. Then,

$\frac{\tau(f^{N},\Omega)}{N}=\tau(f,\Omega)\leq\tau(f)+\epsilon.$

According to Theorem 1.2 we have

${\rm Vol}(\overline{M}_{f^{N}}-P_{\Omega})\leq V_{\rm
oct}\tau(f^{N},\Omega),$

where $P_{\Omega}$ is the pants decomposition of
$\partial\overline{M}_{f^{N}}$ from Proposition 4.3 (5).

On the other hand, by Theorem 2.22 and Theorem 2.24,

${\rm{V\underline{ol}}}(\overline{M}_{f^{N}})\leq{\rm{V\underline{ol}}}(\overline{M}_{f^{N}}-P_{\Omega})={\rm
Vol}(\overline{M}_{f^{N}}-P_{\Omega}),$

and so by Theorem 2.23 and the inequalities above we have

${\rm{V\underline{ol}}}(\overline{M}_{f})=\frac{{\rm{V\underline{ol}}}(\overline{M}_{f^{N}})}{N}\leq\frac{V_{\rm
oct}\tau(f^{N},\Omega)}{N}\leq V_{\rm oct}(\tau(f)+\epsilon).$

Letting $\epsilon\to 0$ completes the proof. ∎

Next, we prove the lower bound on translation length stated in the
introduction. Recall that $V_{\rm tet}$ is the volume of a regular ideal
tetrahedron in $\mathbb{H}^{3}$.

Corollary 1.4 Given an irreducible, end-periodic homeomorphism $f$, we have

$\tau(f)\geq\frac{V_{\rm tet}\xi(\partial\overline{M}_{f})}{2V_{\rm oct}}.$

###### Proof.

First observe that $\tau(f)=\inf\tau(f,\Omega)$, where the infimum is taken
over all components $\Omega\subset\mathcal{P}(S)$, so it suffices to prove
that for any component $\Omega$, $\tau(f,\Omega)$ is bounded below by the
quantity on the right in the corollary. We need only consider components
$\Omega\subset\mathcal{P}_{f}(S)$, so we fix such a component. Let $N>0$ be
such that $f^{N}(\Omega)=\Omega$, and let
$P_{\Omega}\subset\partial\overline{M}_{f^{N}}$ be the pants decomposition
given by Proposition 4.3 (5). By Proposition 4.3 (6) together with Theorem
2.20, $\overline{M}_{f^{N}}-P_{\Omega}$ admits a convex hyperbolic structure
$\sigma_{min}$ with totally geodesic boundary so that

${\rm{V\underline{ol}}}(\overline{M}_{f^{N}}-P_{\Omega})={\rm
Vol}(\overline{M}_{f^{N}}-P_{\Omega}).$

The number of pants curves in $P_{\Omega}$ is
$\xi(\partial\overline{M}_{f^{N}})=N\xi(\partial\overline{M}_{f})$, and so the
double of $\overline{M}_{f^{N}}-P_{\Omega}$ admits a complete hyperbolic
structure, obtained by doubling $\sigma_{min}$, having

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

cusps. According to Adams [Ada88], the volume of this hyperbolic structure on
the double of $\overline{M}_{f^{N}}$ is at least $V_{\rm tet}$ times the
number of cusps, and thus

$\tau(f,\Omega)=\frac{\tau(f^{N},\Omega)}{N}\geq\frac{{\rm{V\underline{ol}}}(\overline{M}_{f^{N}}-P_{\Omega})}{N\,V_{\rm
oct}}\geq\frac{V_{\rm tet}\,N\xi(\partial\overline{M}_{f})}{2N\,V_{\rm
oct}}=\frac{V_{\rm tet}\,\xi(\partial\overline{M}_{f})}{2V_{\rm oct}}.$

∎

The following provides an alternate description of the lower bound in
Corollary 1.4 in terms of the coarse end behavior $\Phi^{*}(f)$ of $f$ as
discussed in Section 2.3.

######  Corollary 4.4.

Given an irreducible, end-periodic homeomorphism $f$ with coarse end behavior
$\Phi^{*}(f)$, we have

$\tau(f)\geq\frac{3V_{\rm tet}|\Phi^{*}(f)|}{2V_{\rm oct}}.$

###### Proof.

Since $\partial\overline{M}_{f}=S_{+}\cup S_{-}$, the result is immediate from
Corollary 2.8 and Corollary 1.4. ∎

##  5\. Proof of Proposition 4.3

The bulk of the proof of Proposition 4.3 involves defining the quotient map
$h:\overline{M}_{f}\to\widehat{M}_{f}$, analyzing the structure of the sets of
the associated decomposition, and then appealing to Theorem 2.16. Since the
hypotheses of that theorem require a closed manifold, we will double
$\overline{M}_{f}$ over $\partial\overline{M}_{f}$.

We now define a quotient map $h:\overline{M}_{f}\to\widehat{M}_{f}$ in terms
of a decomposition of $\overline{M}_{f}$. The elements in the decomposition
are either:

  1. (1)

singletons in $W$; or

  2. (2)

maximal components of a flow line of the local flow $(\bar{\psi}_{s})$
(defined at the end of Section 3) intersected with $\overline{M}_{f}-W$.

We note that there may, a priori, be a flow line that entirely misses $W$. We
will see that this does not happen, and as Corollary 5.5 below shows, every
component of intersection of a flow line with $\overline{M}_{f}-W$ is a
compact arc. We first verify this for points in $L$ and then for points
outside. Along the way, we verify that the image of $L$ under $h$ is the
appropriate union of circles from Proposition 4.3 (3).

For each component $\alpha\subset L$, we say that $\alpha$ flips forward if
$\psi_{1/n}(\alpha)\not\subset L$. Note that $\alpha$ flips forward if and
only if it is equal to $\alpha_{k}^{-}\subset Z_{k}\subset
S\times\left\\{\tfrac{k}{n}\right\\}$ for some $0\leq k<n$. Thus, there are
precisely $n$ components of $L$ that flip forward. We say that $\alpha$ flips
forward after $k\geq 0$ steps if either $k=0$ and $\alpha$ flips forward, or
if $k>0$ and $\psi_{j/n}(\alpha)$ does not flip forward for any $0\leq j<k$,
but $\psi_{k/n}(\alpha)$ flips forward. We similarly say that a component
$\alpha\subset L$ flips backward if $\psi_{-1/n}(\alpha)\not\subset L$ and
note that there are exactly $n$ components of $L$ that flip backward. Flipping
backward after $k\geq 0$ steps is defined in the analogous manner. Briefly, we
will also say that $\alpha$ eventually flips forward/backward, if it does so
after $k$ steps, for some $k\geq 0$.

For any component $\alpha\subset L$ and (possibly infinite) interval
$I\subseteq\mathbb{R}$ containing $0$, we let

$A(\alpha,I)=\bigcup_{t\in I}\psi_{t}(\alpha).$

######  Lemma 5.1.

Every component $\alpha\subset L$ eventually flips forward or backward.

###### Proof.

We suppose $\alpha$ does not eventually flip forward or backward and arrive at
a contradiction. Indeed, under this assumption we have $A(\alpha,\mathbb{R})$
is a bi-infinite annulus that meets $S=S\times\\{0\\}$ in an $f$–invariant
subset of $P_{0}$. This means that there is either a periodic simple closed
curve in $P_{0}$ or else the $f$–orbit of a curve in $P_{0}$ contains
components in both $U_{+}$ and $U_{-}$. Either conclusion contradicts the
irreducibility of $f$. ∎

For the next lemma, we let $n=n_{S}+n_{T}$, as in Section 4.2.

######  Lemma 5.2.

The intersection of $\overline{M}_{f}-W$ with the union of $(\bar{\psi}_{s})$
flow lines through $L$ consists of $n+\tfrac{3}{2}|\Phi^{*}(f)|$ (possibly
degenerate) compact annuli. There are $3|\Phi^{*}(f)|$ boundary components of
these annuli on $\partial\overline{M}_{f}$ forming a pants decomposition of
$\partial\overline{M}_{f}$, and $n$ boundary components which are curves of
the form $\alpha_{k}^{\pm}$ on some $\overline{W}_{k}$.

By a “degenerate annulus”, we mean an annulus $S^{1}\times[a,b]$, where $a=b$,
which is therefore just a circle. In this case, the two boundary components of
the annulus consist of the circle, and this has the form
$\alpha_{k}^{+}=\alpha_{k+1}^{-}$.

###### Proof.

Fix a component $\alpha\subset L$. By Lemma 5.1, $\alpha$ either flips forward
after $k\geq 0$ steps, backward after $k^{\prime}\geq 0$ steps, or both (for
some $k$ and $k^{\prime}$). If it flips forward after $k$ steps and does not
eventually flip backward, then $A(\alpha,(-\infty,\tfrac{k}{n}])$ is a half-
open annulus with its boundary component given by
$\alpha_{j}^{-}=\psi_{k/n}(\alpha)\subset\overline{W}_{j}$, for some $j$. The
open end limits to a curve in $S_{-}\subset\partial\overline{M}_{f}$.

Similarly, if $\alpha$ flips backward after $k^{\prime}\geq 0$ steps, but does
not eventually flip forward, then $A(\alpha,[\tfrac{-k^{\prime}}{n},\infty))$
is a half-open annulus with boundary component
$\alpha_{j}^{+}=\psi_{-k^{\prime}/n}(\alpha)\subset\overline{W}_{j}$, for some
$j$. The open end limits to a curve in $S_{+}$.

If $\alpha$ both flips forward after $k\geq 0$ steps and backward after
$k^{\prime}\geq 0$ steps, then
$A(\alpha,[\tfrac{-k^{\prime}}{n},\tfrac{k}{n}])$ is a compact annulus with
one boundary component
$\alpha_{j}^{-}=\psi_{k/n}(\alpha)\subset\overline{W}_{j}$ and the other
$\alpha_{j^{\prime}}^{+}=\psi_{-k^{\prime}/n}(\alpha)\subset\overline{W}_{j^{\prime}}$,
for some $j,j^{\prime}$.

The annuli of the first two types compactify in $\overline{M}_{f}$ to have
boundary components on $\partial\overline{M}_{f}$. Since $(\bar{\psi}_{s})$ is
the local flow extending the reparameterization of $(\psi_{s})$, the union of
all resulting compact annuli (from all three cases) are exactly the union of
the flow lines through $L$ intersected with $\overline{M}_{f}-W$.

We can count the number of annuli by counting the number of boundary
components of the annuli and dividing by $2$. There are exactly two boundary
components on each $\overline{W}_{k}$, and this accounts for $2n$ boundary
components. The remaining boundary components form a multicurve in
$\partial\overline{M}_{f}$. If this is a pants decomposition, then we have
$\xi(\partial\overline{M}_{f})=3|\Phi^{*}(f)|$ such boundary components, and
hence the number of annuli is $n+\tfrac{3}{2}|\Phi^{*}(f)|$, as required.

To see that the boundary components on $S_{+}$ form a pants decomposition, we
argue as follows (the case of $S_{-}$ is similar). Choose a good nesting
neighborhood $U_{+}$ of the attracting end with the property that $P_{j}\cap
U_{+}=P_{k}\cap U_{+}$, for all $j,k$, and so that $f(U_{+})\subset U_{+}$.
Note that this means that no component of $P_{k}\cap U_{+}\subset
S\times\\{\tfrac{k}{n}\\}$ eventually flips forward. Every point of $U_{+}$
flows forward to $S_{+}$ (since $U_{+}\subset\mathcal{U}_{+}$) and this flow
defines a surjective local diffeomorphism $U_{+}\to S_{+}$; more precisely,
this is one end of the infinite cyclic covering $\mathcal{U}_{+}\to S_{+}$.
Because $P_{k}$ defines a pants decomposition of the end $U_{+}$, this
projects by the flow to a pants decomposition of $S_{+}$, completing the
proof. ∎

Let $A_{1},\ldots,A_{m}$ be the annuli from Lemma 5.2, with
$m=n+\tfrac{3}{2}|\Phi^{*}(f)|$, and let $\mathcal{A}$ be the union of these
annuli. Note that the product structure on $A_{j}\cong S^{1}\times I$ obtained
from the flow has the property that $\\{\ast\\}\times I$ is an element of the
decomposition defining the quotient $\widehat{M}_{f}$. Furthermore, every
component of $L$ has the form $S^{1}\times\\{\ast\\}\subset A_{j}$, for some
$j$. Therefore, we have the following.

######  Corollary 5.3.

We have $\hat{L}=h(L)=h(\mathcal{A})$ and the restriction
$h|_{\mathcal{A}}\colon\mathcal{A}\to\widehat{M}_{f}$ factors as

$\mathcal{A}\to\bigsqcup_{j=1}^{m}S^{1}\to\widehat{M}_{f},$

where the first map projects out the interval directions of the annuli, and
the second map is injective.

Thus far we have described the restriction of $h$ to both $W$ (on which $h$ is
injective) and $\mathcal{A}$. The next lemma describes the structure of the
rest of $\overline{M}_{f}$.

######  Lemma 5.4.

Every component $X$ of $\overline{M}_{f}-(\mathcal{A}\cup W)$ is homeomorphic
to a product of a $3$-punctured sphere with a (possibly degenerate) interval
$I\subset[-\infty,\infty]$ with $0\in\partial I$. The $3$-punctured sphere $Y$
is a component of $\bigcup\partial_{2}\overline{W}_{j}$, and the homeomorphism
$Y\times I\to X$ is given by $(x,t)\mapsto\psi_{t}(x)$. Furthermore,
$Y\times\partial I$ consists of two (possibly equal) $3$-punctured sphere
components of

$\bigcup_{j=0}^{n-1}\partial_{2}\overline{W}_{j}\cup(\partial\overline{M}_{f}-\mathcal{A}).$

Note that in the statement of this lemma, we may have to make sense of
$\psi_{\infty}(x)$ for some $x$, which is defined to be the point
$\displaystyle{\lim_{s\to\infty}\psi_{s}(x)}$ in
$S_{+}\subset\partial\overline{M}_{f}$, when this limit exists. We make a
similar convention for $\psi_{-\infty}(x)$. It might seem more natural to use
the local flow, $(\bar{\psi}_{s})$, but the parameterization of
$(\bar{\psi}_{s})$ is not well-behaved with respect to the $S$–fibers. In
particular, we cannot produce the nice homeomorphism described in the
statement.

###### Proof.

Consider any $3$-punctured sphere component
$Y\subset\partial_{2}^{+}\overline{W}_{j}$ for any $j$. This is the interior
of a (compact) pair of pants we denote
$\overline{Y}\subset\overline{Z}_{j}\times\\{\tfrac{j+1}{n}\\}$, and the
boundary consists of three (not necessarily distinct) pants curves
$\alpha_{1},\alpha_{2},\alpha_{3}$ in $P_{j+1}$ (with $j+1$ taken modulo $n$).
If one of these pants curves flips forward after $k$ steps, then taking
$I=[0,\tfrac{k}{n}]$, the map $Y\times I\to M_{f}$, given by the formula in
the lemma maps to $\overline{M}_{f}-(\mathcal{A}\cup W)$, with
$Y\times\\{\tfrac{k}{n}\\}$ mapping to a $3$-punctured sphere component of
$\partial_{2}^{-}W_{i}$, for some $i$. If none of these curves eventually flip
forward, then we take $I=[0,\infty]$ and the formula provides the
homeomorphism, this time sending $Y\times\\{\infty\\}$ to a $3$-punctured
sphere component of $S_{+}-\mathcal{A}$.

We can argue similarly for a $3$-punctured sphere component
$Y\subset\partial_{2}^{-}\overline{W}_{j}$ for any $j$, this time producing an
interval $I$ of the form $[-\tfrac{k}{n},0]$ if one of the boundary pants
curves flips backward after $k$ steps or $[-\infty,0]$ if it does not
eventually flip backward.

Each of the products produced above is a component of
$\overline{M}_{f}-(\mathcal{A}\cup W)$, and thus it suffices to show that this
accounts for all components of this complement. To see this, we simply note
that any point $x$ in such a component $X$ flows forward (remaining inside
$X$) for time less than $\tfrac{1}{n}$ until it reaches some fiber
$S\times\\{\tfrac{j}{n}\\}$. It meets this in a $3$-punctured sphere component
of the complement of $P_{j}$, and (each of) the boundary pants curves must
eventually flip forward or backward. The first time one of these flips forward
or backward specifies a length of time we need to flow $x$ (inside $X$) until
it hits $\bigcup\partial_{2}\overline{W}_{i}$. Therefore, $X$ is one of the
components already discussed, completing the proof. ∎

######  Corollary 5.5.

Every flow line of $(\bar{\psi}_{s})$ has nontrivial intersection with $W$.
Consequently, every set in the decomposition is either a point or compact arc
of a flow line with at most one endpoint on $\partial\overline{M}_{f}$.

We are now ready for the proof of Proposition 4.3. Most of the proof involves
analyzing the decomposition defining the quotient
$h\colon\overline{M}_{f}\to\widehat{M}_{f}$ using the description of the
decomposition elements from the lemmas/corollary above. From this analysis, we
will show that $\widehat{M}_{f}$ is a $3$-manifold with boundary and that $h$
restricts to a homeomorphism
$\partial\overline{M}_{f}\to\partial\widehat{M}_{f}$. Proving that $h$ is
homotopic to a homeomorphism involves analyzing the associated quotient of the
double $D\overline{M}_{f}$ and applying Theorem 2.16. The remaining items
follow fairly quickly after this is done.

###### Proof of Proposition 4.3..

We begin by proving that $\widehat{M}_{f}$ is a $3$-manifold with boundary.
The first step is to find a neighborhood of each point in $\widehat{M}_{f}$
homeomorphic to a ball or half-ball in $\mathbb{R}^{3}$. For this, observe
that the restriction of $h$ to $W$ is a homeomorphism onto its image, since
$W$ is an open set on which $h$ is injective. So, all points in $h(W)$ have
neighborhoods homeomorphic to a ball in $\mathbb{R}^{3}$. Next, observe that
by Lemma 5.4, each component $X$ of $\overline{M}_{f}-(\mathcal{A}\cup W)$ is
homeomorphic to a product $X\cong Y\times I$, with $\\{y\\}\times
I=\bigcup_{s\in I}\psi_{s}(y)$ an arc of a flow line. The restriction of $h$
to $X$ factors as the composition of the projection $X\to Y$ and an inclusion
$Y\to\widehat{M}_{f}$,

$X\to Y\hookrightarrow\widehat{M}_{f}.$

If the interval $I$ does not contain either $\infty$ or $-\infty$, then any
point in the image of $Y$ in $\widehat{M}_{f}$ has a neighborhood homeomorphic
to a ball in $\mathbb{R}^{3}$, constructed from half-balls in
$\overline{W}_{j}$ and $\overline{W}_{i}$ where the bottom and top
$3$-punctured spheres $Y\times\partial I$ of $X$ are components of
$\partial_{2}^{+}\overline{W}_{j}$ and $\partial_{2}^{-}\overline{W}_{i}$,
respectively. If $I$ contains $\infty$, then it cannot contain $-\infty$, and
$Y\times\\{0\\}\subset X$ is a component of some
$\partial_{2}^{+}\overline{W}_{j}$ while $Y\times\\{\infty\\}$ is a component
of $S_{+}-\mathcal{A}$, and in this case every point in the image of $Y$ in
$\widehat{M}_{f}$ has a neighborhood homeomorphic to a half-ball in
$\mathbb{R}^{3}$, coming from a half-ball in $\overline{W}_{j}$. If $I$
contains $-\infty$, we similarly find a half-ball neighborhood.

The only remaining points of $\widehat{M}_{f}$ are those in the image of
$\mathcal{A}$. For these, we consider each component annulus
$A_{j}\subset\mathcal{A}$ and build neighborhoods from all the
$\overline{W}_{k}$ it meets. There are different cases depending on whether
the annulus $A_{j}$ meets $\partial\overline{M}_{f}$ or not.

First, suppose $A_{j}\cap\partial\overline{M}_{f}=\emptyset$. See Figure 7 for
an illustration of the construction of the neighborhood of a point in the
image of $A_{j}$. Then we can write

$A_{j}=\bigcup_{s\in[0,t]}\psi_{s}(\alpha_{k}^{+}),$

for some some $t\geq 0$ and integer $k$ so that
$\psi_{t}(\alpha_{k}^{+})=\alpha_{\ell}^{-}$, for some integer $\ell$. Observe
that $k$ is such that
$\alpha_{k}^{+}\subset\overline{Z}_{k}\times\\{\tfrac{k+1}{n}\\}$ and does not
flip backward (c.f. proof of Lemma 5.2). The annulus may also meet various
pieces $\overline{W}_{k_{1}},\ldots,\overline{W}_{k_{j}}$ along the vertical
annuli
$\partial^{v}\overline{W}_{k_{1}},\ldots,\partial^{v}\overline{W}_{k_{j}}$.
Any point in the image of $A_{j}$ is given by the decomposition element of the
form $\\{\ast\\}\times I\subset S^{1}\times I\cong A_{j}$, and a neighborhood
of this point is obtained from neighborhoods in $\overline{W}_{k}$ and
$\overline{W}_{\ell}$ on the “top and bottom”, together with neighborhoods
“along the sides” from the images of
$\overline{W}_{k_{1}},\ldots,\overline{W}_{k_{j}}$.

When the annulus $A_{j}$ meets $S_{+}$ or $S_{-}$, we have a similar picture,
except in the former case the top reaches $S_{+}$ instead of some
$\overline{W}_{\ell}$, and in the latter, the bottom reaches $S_{-}$ instead
of “starting” on some $\overline{W}_{k}$. In this setting, we get half-ball
neighborhoods (since the top/bottom are missing).

$W_{\ell}$$W_{k}$$W_{k_{1}}$$W_{k_{5}}$$W_{k_{2}}$$W_{k_{3}}$$W_{k_{4}}$$\alpha_{k}^{+}$$\alpha_{\ell}^{-}$$h$$A_{j}$
Figure 7. The left-hand side of the figure shows a slice of a neighborhood of
some annulus $A_{j}\cong S^{1}\times I$ (specifically, the slice
$\\{\ast\\}\times I$). The figure illustrates the situation that the boundary
components $\alpha_{k}^{+}$ and $\alpha_{\ell}^{-}$ of $A_{j}$ are on a pair
of regions $\overline{W}_{k}$ and $\overline{W}_{\ell}$. The annulus meets
some (possibly zero) number of other regions
$\overline{W}_{k_{1}},\ldots,\overline{W}_{k_{j}}$ along the “vertical”
boundary annuli. The right-hand side of the figure shows the result of
collapsing. Neighborhoods in the $\overline{W}_{i}$ pieces are glued together
in the quotient to produce a neighborhood of points in $h(A_{j})$.

We have found neighborhoods of every point in $\widehat{M}_{f}$ homeomorphic
to either an open ball or half-ball in $\mathbb{R}^{3}$. From the construction
of these neighborhoods, it is straightforward to show that any two distinct
points in $\widehat{M}_{f}$ have disjoint neighborhoods, and that there is a
countable basis for the topology. Since $\widehat{M}_{f}$ is the image of the
compact space $\overline{M}_{f}$, it is also compact. Thus, $\widehat{M}_{f}$
is a compact $3$-manifold with boundary.

Since the points with neighborhoods homeomorphic to half-balls are precisely
those in the image of $\partial\overline{M}_{f}$, we have
$\partial\widehat{M}_{f}=h(\partial\overline{M}_{f})$. To see that the
restriction of $h$ to $\partial\overline{M}_{f}$ is injective, observe that by
Corollary 5.5, any element of the decomposition meets
$\partial\overline{M}_{f}$ in at most one point. This proves item (1).
According to Corollary 5.3, $\hat{L}=h(L)=h(\mathcal{A})$ is the continuous
image of a disjoint union of $n+\tfrac{3}{2}|\Phi^{*}(f)|$ circles. Since
$\widehat{M}_{f}$ is Hausdorff, the injection is an embedding, proving item
(3). Moreover, since $\mathcal{A}$ meets $\partial\overline{M}_{f}$ in a pants
decomposition, $\hat{L}$ meets $\partial\widehat{M}_{f}$ in a pants
decomposition which we denote by $P_{\Omega}$, where $\Omega$ is the component
of $\mathcal{P}(S)$ containing $P$.

Next, we want to see that
$h|_{\overline{W}_{k}}\colon\overline{W}_{k}\to\widehat{M}_{f}$ factors
through a block map relative to $\hat{L}=h(L)$, and that the maps
$h|_{\overline{W}_{k}}$ define a block decomposition. This follows from (a)
the fact that $h|_{W_{k}}$ is injective on the interior of product (by
definition of the decomposition), (b) $h$ collapses
$\partial^{v}\overline{W}_{k}$ to circles contained in $\hat{L}$ since these
annuli are sub-annuli of $\mathcal{A}$, (c) $h$ is injective on each pair of
pants $Y\subset\partial_{2}\overline{W}_{k}$ by Lemma 5.4, and (d) $h$ is
injective on each curve $\alpha_{k}^{\pm}$ since this is either the bottom or
top boundary component of an annulus in $\mathcal{A}$.

In fact, Corollary 5.3 and Lemma 5.4 tell us that $\widehat{M}_{f}$ is built
by gluing the blocks together along pairs of pants and unions of circle
components of $1$–boundaries $\partial_{1}\mathcal{B}_{k}$ contained in
$\hat{L}$ (by definition of the decomposition space, the interiors of the
blocks are disjoint). This gives a block decomposition relative to $\hat{L}$,
proving item (4).

Next we prove item (2), namely that $h$ is homotopic to a homeomorphism
$H\colon\overline{M}_{f}\to\widehat{M}_{f}$. For this, we consider the doubles
$D\overline{M}_{f}$ and $D\widehat{M}_{f}$, and let $Dh\colon
D\overline{M}_{f}\to D\widehat{M}_{f}$ be the associated quotient map, which,
restricted to each copy of $\overline{M}_{f}$, maps via $h$ to one of the
copies of $\widehat{M}_{f}$. Note that because $D\widehat{M}_{f}$ is a
$3$-manifold, hence Hausdorff, and $Dh$ is a quotient map from the compact
space $D\overline{M}_{f}$, it follows that $Dh$ is a closed map. By
Proposition 2.14, the decomposition is upper semicontinuous. Furthermore, by
Corollary 5.5, each element of the decomposition defining the quotient map $h$
is an arc (or point) meeting $\partial\overline{M}_{f}$ in at most one point,
hence each element of the decomposition defining the quotient map $Dh$ is also
an arc or point. These are in fact arcs of flow lines of the extension of
$(\bar{\psi}_{s})$ to $D\overline{M}_{f}$ and so it is easy to find a sequence
of neighborhoods, each homeomorphic to a cell, whose intersection is the
arc/point. Therefore, the decomposition defining $Dh$ is cellular, hence $Dh$
is cellular, and so by Theorem 2.16, $Dh$ is homotopic to a homeomorphism.

Now we note that the inclusion of $\overline{M}_{f}$ to $D\overline{M}_{f}$
induces an injection $\pi_{1}\overline{M}_{f}$, and since $Dh_{*}$ is an
isomorphism, it follows that
$h_{*}\colon\pi_{1}\overline{M}_{f}\to\pi_{1}\widehat{M}_{f}$ is also
injective. By a result of Waldhausen [Wal68, Theorem 6.1], $h$ is homotopic to
a homeomorphism; in fact, this is through a homotopy $h_{t}$ with
$h_{t}|_{\partial\overline{M}_{f}}=h|_{\partial\overline{M}_{f}}$ for all $t$.
This completes the proof of item (2).

By construction, $\hat{L}\cap\partial\widehat{M}_{f}=h(P_{\Omega})$ and this
is equal to $H(P_{\Omega})$, since
$h|_{\partial\overline{M}_{f}}=H|_{\partial\overline{M}_{f}}$. Suppose we
replace the path $P=P_{0},\ldots,P_{n}=f^{-1}(P)$ in the construction with a
path between $P^{\prime}$ and $f^{-1}(P^{\prime})$ in the same component
$\Omega\subset\mathcal{P}(S)$. Since $P=P^{\prime}$ outside a compact set of
$S$ (and the same is true for all pants in the path from $P^{\prime}$ to
$f^{-1}(P^{\prime})$), one sees that the annuli $\mathcal{A}^{\prime}$
constructed analogously to $\mathcal{A}$ meet $\partial\overline{M}_{f}$ in
the same pants decomposition. Therefore, $P_{\Omega}$ does indeed depend only
on the component $\Omega$. This proves item (5).

All that remains is to prove item (6), stating that
$(\overline{M}_{f},P_{\Omega})$ is a pared acylindrical manifold. All the
conditions in the definition of pared manifold are immediate from the
preceding results, except condition (4), stating that an incompressible
annulus $(A,\partial A)\subset(\overline{M}_{f},N(P_{\Omega}))$ with boundary
in a neighborhood $N(P_{\Omega})$ of $P_{\Omega}$ must be homotopic into
$N(P_{\Omega})$, via a homotopy of pairs. Consider such an annulus
$(A,\partial A)\subset(\overline{M}_{f},N(P_{\Omega}))$.

Since the preimage $\widetilde{P}_{\Omega}$ of $P_{\Omega}$ in
$\widetilde{M}_{\infty}\subset S\times[-\infty,\infty]$ is a pants
decomposition of
$\partial\widetilde{M}_{\infty}=\mathcal{U}_{\pm}\times\\{\pm\infty\\}$, $A$
lifts to an annulus $\widetilde{M}_{\infty}$ with boundary in
$N(\widetilde{P}_{\Omega})$. By Lemma 3.5, $A$ is not essential, and hence
must be boundary parallel. Since $N(P_{\Omega})$ is a neighborhood of a pants
decomposition, being boundary parallel gives a homotopy of pairs taking $A$
into $N(P_{\Omega})$, proving that $(\overline{M}_{f},P_{\Omega})$ is a pared
manifold.

Now observe that $\overline{M}_{f}$ has incompressible boundary, and hence so
does $(\overline{M}_{f},P_{\Omega})$. Since the paring locus $P_{\Omega}$ is a
pants decomposition of the boundary, $(\overline{M}_{f},P_{\Omega})$ is
acylindrical, as explained in the discussion after the definition of pared
manifolds. This completes the proof of item (6) of the proposition, and we are
done. ∎

##  6\. Sharpness of the Upper Bound

In this section, we describe a construction of end-periodic homeomorphisms and
provide sufficient conditions which ensure that they are either strongly
irreducible or irreducible (but not strongly irreducible). The construction
allows us to prescribe the end behavior (Theorem 1.5), and also produce
sequences of examples showing that the bound on volume given in Theorem 1.1 is
asymptotically sharp (Theorem 1.6). The proof of Theorem 1.5 will follow
immediately from Proposition 6.3 together with Corollary 2.9, while the proof
of Theorem 1.1 will appear at the end of Section 6.3.

###  6.1. Examples of irreducible, end-periodic homeomorphisms

Recall that $S$ is a boundaryless infinite-type surface with $n$ ends, each
accumulated by genus, with $2\leq n<\infty$. We will construct a general
family of examples of irreducible and strongly irreducible end-periodic
homeomorphisms of $S$ with arbitrary coarse end behavior. All examples will be
of the form $f=\rho h$, where $\rho$ is as in 2.10 and $h$ is supported on a
compact subsurface $C$. Whether the resulting homeomorphism $f$ is irreducible
or strongly irreducible will depend on the choice of subsurface $C$.

######  Definition 6.1 (Fully separating/partially separating).

We say a subsurface $C$ is fully separating if $C$ separates every end of $S$,
in the sense that the end space of each component of $S-C$ is a singleton. We
say a subsurface $C$ is partially separating (with respect to a given end-
periodic homeomorphism) if $C$ separates the collection of attracting ends
from the collection of repelling ends.

Note that a fully separating subsurface is necessarily partially separating
with respect to any end-periodic homeomorphism, but the converse is not
necessarily true. For any partially separating subsurface $C$, the boundary
$\partial C$ decomposes as a disjoint union $\partial
C=\partial_{+}C\cup\partial_{-}C$ according whether the ends cut off are
attracting or repelling. Examples and a non-example are illustrated in Figure
8.

Figure 8. Consider an end-periodic homeomorphism of the $5$-ended surface
shown here with $3$-repelling ends on the left and $2$-attracted ends on the
right. Both the finite-type subsurface bounded by the red curves and the
finite-type subsurface bounded by the green curves are partially separating
(upper left). The finite-type subsurface bounded by the blue curves is fully
separating (upper right). The finite-type surface bounded by the pink curves
is neither fully nor partially separating (bottom).

######  Convention 6.2.

As in 2.10, we fix any coarse end behavior of $S$ and realize it by an end-
periodic homeomorphism (isotopic to) $\rho=\Pi_{1}^{m}\rho_{i}$, where
$\\{\rho_{i}\\}_{1}^{m}$ are pairwise-commuting handle shifts. We continue
with the notation and assumptions from that construction, letting
$v_{1},\ldots,v_{n}$ be separating curves cutting off good nesting
neighborhoods of the ends, and let $U_{+}$ and $U_{-}$ denote the resulting
union of good nesting neighborhoods of the attracting and repelling ends,
respectively. In addition, we now let $C\subset S$ be a compact subsurface
which is partially separating with respect to $\rho$ and disjoint from $U_{+}$
and $U_{-}$. We further assume, as we may, that

  1. (1)

$S-(U_{+}\sqcup U_{-}\sqcup C)$ is either empty or a union of planar
subsurfaces;

  2. (2)

$\partial C$ meets each of the handle strips in arcs isotopic to the arcs of
intersection with the curves $v_{1},\ldots,v_{n}$; and

  3. (3)

$C$ intersects each handle strip in a subsurface with one boundary component
and genus at least $2$.

See Figure 9 for more explanation. When $C$ is fully separating, we observe
that item (1) implies that $S-C=U_{+}\sqcup U_{-}$. Fix a component $\eta$ of
$\partial_{-}C$ and fix a component $\alpha$ of $\partial_{+}C$. By items (2)
and (3), we have $i(\rho(\eta),\rho^{-1}(\alpha))=0$. For this fixed $\eta$,
we choose $h\in{\rm Map}(S)$ which is supported on $C$ and for which
$d_{C}(\rho(\eta),h(\rho(\eta)))\geq 9$ (see Section 2.4 for discussion of
subsurface distance). We also let $B\subset B^{\prime}\subset\mathcal{AC}(C)$
denote the balls of radius two and three about $\rho(\eta)$, respectively.
Since $i(\rho^{-1}(\alpha),\rho(\eta))=0$, we have
$d_{C}(\rho^{-1}(\alpha),\rho(\eta))=1$ and so $\rho^{-1}(\alpha)\in B$. We
will consider the end-periodic homeomorphism $f=\rho h$. By construction,
$U_{+}$ and $U_{-}$ are still good nesting neighborhoods of the attracting and
repelling ends of $f$, respectively.

$v_{1}$
---
$v_{2}$
---
$v_{3}$
---
$v_{4}$
---
$v_{5}$
---