# Left orderability, foliations, and transverse $(\pi_{1},\mathbb{R})$
structures for $3$-manifolds with sphere boundary

Bojun Zhao  Department of Mathematics, University at Buffalo, Buffalo, NY
14260, USA<EMAIL_ADDRESS>
###### Abstract.

Let $M$ be a closed orientable irreducible $3$-manifold with a left orderable
fundamental group, and $M_{0}=M-Int(B^{3})$. We show that there exists a
Reebless co-orientable foliation $\mathcal{F}$ in $M_{0}$, whose leaves may be
transverse to $\partial M_{0}$ or tangent to $\partial M_{0}$ at their
intersections with $\partial M_{0}$, such that $\mathcal{F}$ has a transverse
$(\pi_{1}(M_{0}),\mathbb{R})$ structure, and $\mathcal{F}$ is analogue to taut
foliations (in closed $3$-manifolds) in the following sense: there exists a
compact $1$-manifold (i.e. a finite union of properly embedded arcs and/or
simple closed curves) transverse to $\mathcal{F}$ that intersects every leaf
of $\mathcal{F}$. We conjecture that $\mathcal{F}$ is obtained from removing a
$3$-ball foliated with horizontal disks from a taut foliation in $M$.

##  1\. Introduction

Throughout this paper, all $3$-manifolds are assumed to be orientable, and all
foliations and laminations are assumed to be co-orientable.

The L-space conjecture was proposed by Boyer-Gordon-Watson in [BGW] and by
Juhaz in [J]:

######  Conjecture 1 (L-space conjecture).

Let $M$ be an irreducible rational homology $3$-sphere. Then the following
statements are equivalent:

(1) $M$ is a non-L-space.

(2) $\pi_{1}(M)$ is left orderable.

(3) $M$ admits a co-orientable taut foliation.

(3) $\Rightarrow$ (1) is known by [B], [OS], [KR]. In [Ga1], Gabai proved that
all closed orientable irreducible $3$-manifolds with $b_{1}>0$ admit taut
foliations. In [BRW], Boyer, Rolfsen and Wiest proved that for every compact
orientable irreducible $3$-manifold $M$ with $b_{1}>0$, $\pi_{1}(M)$ is left
orderable. Hence (3) and (2) hold for all closed orientable irreducible
$3$-manifolds that have positive first Betti number.

Furthermore, connections between foliations and laminations, and orders and
group actions (on $1$-manifolds) were developed in many works. Morgan and
Shalen proved the duality of incompressible measured laminations and
$\pi_{1}$-actions on $\mathbb{R}$-trees ([MS1], [MS2], [MS3]). Through his
universal circle ([T]), Thurston showed that: for every atoroidal $3$-manifold
$M$ that admits a taut foliation, there exists an effective action of
$\pi_{1}(M)$ on $S^{1}$ induced by the taut foliation. In [CD], Calegari and
Dunfield showed that tight essential laminations with solid tori guts (in
atoroidal $3$-manifolds) also induce effective actions of $\pi_{1}$ on
$S^{1}$.

The following question was proposed by Thurston and summarized by Calegari in
[Cal1, Question 8.1]:

######  Question 1.1 (Thurston).

Let $M$ be a closed orientable irreducible $3$-manifold such that $\pi_{1}(M)$
is left orderable. When does $M$ admits a taut folation with a transverse
$(\pi_{1}(M),\mathbb{R})$ structure?

Here, the transverse $(\pi_{1}(M),\mathbb{R})$ structure means that: let $L$
denote the leaf space of the pull-back foliation in the universal cover of
$M$, then there exists an immersion $i:L\to\mathbb{R}$ descending the action
of $\pi_{1}(M)$ on $L$ (induced by deck transformations) to a nontrivial
action on $\mathbb{R}$. Also, see the remarks in [Cal1, Question 8.1] for
Question 1.1 with some flexibility.

In this paper, we consider the related topics for compact $3$-manifolds whose
boundary is a $2$-sphere. Taut foliations are not generically defined in
compact $3$-manifolds with nonempty boundary (but are defined in some
situations, notably, defined for sutured manifolds in [Ga1, Definition 2.11]).
We consider its analogues (in some weak sense) as follows: leaves of the
foliation may be transverse to the boundary component or tangent to the
boundary component at their intersections with the boundary component, and
there exists a compact $1$-manifold transverse to the foliation which
intersects all leaves. This setting is weaker than the setting of taut
foliations in closed $3$-manifolds.

Our main result is:

######  Theorem 1.2.

Let $M$ be a closed orientable irreducible $3$-manifold such that $\pi_{1}(M)$
is left orderable. Let $M_{0}=M-Int(B^{3})$. Then there exists a Reebless
foliation $\mathcal{F}$ in $M_{0}$, where the leaves of $\mathcal{F}$ may be
transverse to $\partial M_{0}$ or tangent to $\partial M_{0}$ at their
intersections with $\partial M_{0}$, such that:

(1) $\mathcal{F}$ is analogue to “taut” in the following sense:

$\bullet$ There exists a compact $1$-manifold (i.e. a finite union of properly
embedded arcs and/or simple closed curves) in $M_{0}$ that is transverse to
$\mathcal{F}$ and has nonempty intersection with every leaf of $\mathcal{F}$.

(2) $\mathcal{F}$ has a transverse $(\pi_{1}(M_{0}),\mathbb{R})$ structure:

$\bullet$ Let $L$ denote the leaf space of the pull-back foliation of
$\mathcal{F}$ in the universal cover of $M_{0}$, then there exists an
immersion $i:L\to\mathbb{R}$ descending the action of $\pi_{1}(M_{0})$ on $L$
to an effective action on $\mathbb{R}$.

######  Remark 1.3.

There are many equivalent definitions for taut foliations in closed
$3$-manifolds (e.g. [Cal2, Chapter 4] and [Cal3]). We don’t know if the
generalizations of these definitions in our setting are equivalent. In Theorem
1.2 (1), our condition is only analogue to the one of these definitions for
taut foliations in closed $3$-manifolds: there exists a compact $1$-manifold
(i.e. a finite union of simple closed curves) transverse to the foliation and
intersects every leaf.

Suppose that $M$ is a closed orientable irreducible $3$-manifold, and
$M_{0}=M-Int(B^{3})$. If $M_{0}$ admits the foliation $\mathcal{F}$ as given
in Theorem 1.2, then $\pi_{1}(M_{0})$ acts on $\mathbb{R}$ effectively, which
implies that $\pi(M_{0})$ is left orderable ([BRW, Theorem 2.4]). Therefore,
the existence of the foliation $\mathcal{F}$ as given in Theorem 1.2 is a
necessary and sufficient condition for $\pi_{1}(M)$ is left orderable.

######  Corollary 1.4.

Let $M$ be a closed orientable irreducible $3$-manifold. Let
$M_{0}=M-Int(B^{3})$. Then $\pi_{1}(M)$ is left orderable if and only if there
exists a foliation $\mathcal{F}$ in $M_{0}$ as given in Theorem 1.2.

Notice that $\pi_{1}(M)$ is left orderable if it acts on $\mathbb{R}$
nontrivially ([BRW, Theorem 3.2]). So Corollary 1.4 would still hold if we
only request $i:L\to\mathbb{R}$ to descend the $\pi_{1}$-action on $L$ to a
nontrivial action.

The foliation $\mathcal{F}$ in Theorem 1.2 is very similar to usual taut
foliations in closed $3$-manifolds. In the spirit of this, apparently it’s
interesting to think about its connection with removing a $3$-ball foliated
with horizontal disks from a taut foliation in the closed $3$-manifold.

At first, we note that: if Conjecture 1 (2) $\Rightarrow$ (3) holds and
Question 1.1 is true for all closed orientable irreducible $3$-manifolds, then
Theorem 1.2 can be implied by them directly. Let $M$ be a closed orientable
irreducible $3$-manifold such that $\pi_{1}(M)$ is left orderable. Suppose
that $M$ admits a taut foliation $\mathcal{F}_{1}$. If $B\subseteq M$ is a
$3$-ball such that $\mathcal{F}_{1}\mid_{B}\cong D^{2}\times I$, then we say

$(M-Int(B),\mathcal{F}_{1}\mid_{M-Int(B)})$

is obtained from _removing a $3$-ball_ from $(M,\mathcal{F}_{1})$. Obviously,
if $(M_{0},\mathcal{F})$ is obtained from removing a $3$-ball from
$(M,\mathcal{F}_{1})$, then $(M_{0},\mathcal{F})$ satisfies Theorem 1.2 (1),
and whenever $\mathcal{F}_{1}$ has a transverse $(\pi_{1}(M),\mathbb{R})$
structure, $(M_{0},\mathcal{F})$ satisfies Theorem 1.2 (2). Therefore, Theorem
1.2 is involved in Conjecture 1 (2) $\Rightarrow$ (3) and Question 1.1 for
closed orientable irreducible $3$-manifolds. We can view Theorem 1.2 as a weak
version of them.

Furthermore, we conjecture the following holds:

######  Conjecture 2.

Let $M$ be a closed orientable irreducible $3$-manifold such that $\pi_{1}(M)$
is left orderable. Let $M_{0}=M-Int(B^{3})$. If $\mathcal{F}$ is a foliation
of $M_{0}$ that is constructed in Theorem 1.2, then there exists a taut
foliation $\mathcal{F}_{1}$ in $M$ such that $(M_{0},\mathcal{F})$ is obtained
from removing a $3$-ball from $(M,\mathcal{F}_{1})$ (this also implies that
$\mathcal{F}_{1}$ has a transverse $(\pi_{1}(M),\mathbb{R})$ structure).

We give some equivalent forms of Conjecture 2 in Remark 5.1. Conjecture 2
would be true if part of the Novikov’s Theorem ([N], [R]) can be generalized
to the setting of Theorem 1.2. In Theorem 1.2, we say $\mathcal{F}$ satisfies
the _Novikov’s condition_ if the following always holds: if $\alpha$ is an
embedded loop in a leaf $\lambda$ of $\mathcal{F}$ and $\alpha$ is
contractible in $M_{0}$, then $\alpha$ either bounds a disk in $\lambda$ or
co-bounds a surface in $\lambda$ with some components of $\lambda\cap\partial
M_{0}$.

######  Proposition 1.5.

With the conditions of Theorem 1.2, if $\mathcal{F}$ satisfies the Novikov’s
condition, then Conjecture 2 is true.

At last, we give the following summary:

######  Remark 1.6.

(a) We explain the relevance between $\mathcal{F}$ and the left orderability
of $\pi_{1}(M_{0})$. Before constructing $\mathcal{F}$, we will choose an
arbitrary left-invariant order “$>$” of $\pi_{1}(M_{0})$. The order “$>$”
affects the construction of $\mathcal{F}$ in many places, especially on the
transverse $(\pi_{1}(M_{0}),\mathbb{R})$ structure of $\mathcal{F}$ (see
Subsection 4.6).

(b) Let $\widetilde{\mathcal{F}}$ be the pull-back foliation of $\mathcal{F}$
in the universal cover of $M_{0}$. Then $\widetilde{\mathcal{F}}$ contains a
leaf that has trivial stabilizer under deck transformations. Especially, by
the Novikov’s Theorem ([N], [R]), if Conjecture 2 is true for
$(M_{0},\mathcal{F})$, then $\mathcal{F}$ has a planar leaf.

(c) In Proposition 4.1, we give a more comprehensive version of Theorem 1.2 in
terms of a lamination. This statement needs some definitions in Section 3, so
we give it in Section 4.

###  1.1. The plan of the construction of $\mathcal{F}$ in Theorem 1.2

Let $M$ be a closed, orientable, irreducible $3$-manifold such that
$\pi_{1}(M)$ is left orderable. To construct the foliation $\mathcal{F}$ in
$M-Int(B^{3})$ as given in Theorem 1.2, we will construct a lamination
$\Lambda$ in $M$ with a single $B^{3}$ gut region (see Proposition 4.1). Then
$\mathcal{F}$ can be obtained from filling in the interstitial regions of
$\Lambda$. Notice that we will not consider whether or not $\Lambda$ is an
essential lamination in the construction. Actually, whenever $\Lambda$ is an
essential lamination, Conjecture 2 is true for $(M-Int(B^{3}),\mathcal{F})$.

At first, we give the following conventions:

$\bullet$ Let $p:\widetilde{M}\to M$ be the universal cover of $M$, and we
choose a base point $x\in\widetilde{M}$. Let $G=\pi_{1}(M,p(x))$. For each
$h\in G$, we denote by

$t_{h}\in Aut(\widetilde{M})$

the deck transformation such that: if $\eta$ is a path in $\widetilde{M}$ that
starts at $x$ and ends at $t_{h}(x)$, then $p(\eta)$ is a loop in $M$ such
that $[p(\eta)]=h$.

$\bullet$ We fix an arbitrary left-invariant order “$<$” of $G$.

$\bullet$ Our setting of _fundamental domains_ will be given in Subsection
2.1. Since $M$ is irreducible and $\pi_{1}(M)$ is left orderable (then
$\pi_{1}(M)$ is torsion-free, see [BRW, Proposition 2.1]), $M$ is covered by
$\mathbb{R}^{3}$, and the fundamental domains of $M$ in $\widetilde{M}$ are
homeomorphic to $B^{3}$. In the following, we will fix a fundamental domain
$\Gamma$ given by Lemma 3.3. Then $p(\partial\Gamma)$ is a standard spine
([Cas], see Definition 3.2 and Figure 1, which is topologically homeomorphic
to a branched surface but has no cusp directions). And we fix an orientation
of $\Gamma$. Moreover, without loss of generality, we assume that the base
point $x\in Int(\Gamma)$.

We will do the following steps:

######  Step 1.

We will determine a branched surface $B$ in $M$ in this step. Recall that we
have fixed a fundamental domain $\Gamma$ (with an orientation on it) in the
above such that $p(\partial\Gamma)$ is a standard spine. We say $F$ is a
_face_ of $\Gamma$ if $F$ is a component of a nonempty intersection
$\Gamma\cap t_{h}(\Gamma)$ for some $h\in G-\\{1\\}$, and there is no $g\in
G-\\{1,h\\}$ such that $F\subseteq\Gamma\cap t_{g}(\Gamma)$. Then every face
of $\Gamma$ is a subsurface of $\partial\Gamma$ since $p(\partial\Gamma)$ is a
standard spine.

At first, we orient all faces of $\Gamma$. For each face $F$ of $\Gamma$,
there exists a unique $h\in G-\\{1\\}$ such that $F$ is a common face of
$\Gamma$ and $t_{h}(\Gamma)$. We assign $F$ the positive orientation with
respect to the orientation of $\Gamma$ if $h>1$, and we assign $F$ the
negative orientation with respect to the orientation of $\Gamma$ if $h<1$.

If $F_{1}$ is a face of $\Gamma$, then there exists a unique different face
$F_{2}$ of $\Gamma$ such that $p(F_{1})=p(F_{2})$, and there exists a unique
$h\in G-\\{1\\}$ such that $F_{1}\subseteq\Gamma\cap t_{h}(\Gamma)$,
$F_{2}\subseteq\Gamma\cap t_{h^{-1}}(\Gamma)$. Then $F_{1},F_{2}$ are assigned
with opposite orientations (with respect to the orientation of $\Gamma$),
since exactly one of $h,h^{-1}$ is greater than $1$. Therefore, $p$ takes the
associated orientations on $F_{1},F_{2}$ to the same orientation on
$p(F_{1})=p(F_{2})$.

Then we assign all sectors of $p(\partial\Gamma)$ a co-orientation induced
from the orientations on the faces of $\Gamma$. We can verify that this
assignment is compatible, and this co-orientation gives $p(\partial\Gamma)$
well-defined cusp-directions. Then $p(\partial\Gamma)$ is oriented to be a
branched surface. We denote by $B=p(\partial\Gamma)$. After the process, the
covering map takes every face of $\Gamma$ to a branch sector of $B$.

The precise process will be given in Subsection 3.2. Notice that $B$ is an
essential branched surface only if the vertical boundary of a fibered
neighborhood of $B$ is connected.

######  Step 2.

Let $N(B)$ be a fibered neighborhood of $B$, then $N(B)$ can be regarded as an
interval bundle over $B$. Let $\widetilde{B}=p^{-1}(B)$ be the pull-back of
$B$ in $\widetilde{M}$. Let $\widetilde{N(B)}=p^{-1}(N(B))$ be the pull-back
of $N(B)$ in $\widetilde{M}$. Let
$\widetilde{\pi}:\widetilde{N(B)}\to\widetilde{B}$ denote the map to collapse
each interval fiber of $\widetilde{N(B)}$ into a single point.

We will introduce the _abstract lamination_ in Definition 3.1. In this step,
we will determine an abstract lamination $\mathcal{A}$. $\mathcal{A}$ will be
constructed by the following way: define

$\mathcal{C}=\\{C\varsubsetneqq G\mid$ if $a\in C,b<a,$ then $b\in C\\}$,

then

$\mathcal{A}=\\{\partial(\bigcup_{h\in C}t_{h}(\Gamma))\mid
C\in\mathcal{C}\\}$.

Notice that the leaves of $\mathcal{A}$ (we say each element of $\mathcal{A}$
is a _leaf_) may be disconnected. $\mathcal{A}$ will be used to construct the
lamination $\Lambda$ in $M$ (which is carried by $B$) in the next step. The
components of leaves of $\mathcal{A}$ turn out to be the classes

$\\{\widetilde{\pi}(\widetilde{\lambda})\mid\widetilde{\lambda}$ is a leaf of
$\widetilde{\Lambda}\\},$

where $\widetilde{\Lambda}$ is the pull-back of $\Lambda$ in $\widetilde{M}$.

The precise statements will be given in Subsection 3.3 and Subsection 3.4.

######  Step 3.

Let F be the set of faces of $\Gamma$ that are oriented positively with
respect to $\Gamma$. Then $p(\textbf{F})$ is the set of branch sectors of $B$.
Then $N(B)$ can be decomposed to

$\coprod_{F\in\textbf{F}}(p(F)\times I)$

by cutting along the interval fibers at the branch locus of $B$. We will
construct the leaves of $\Lambda$ by gluing some horizontal pieces in
$\\{p(F)\times\\{t\\}\\}_{F\in\textbf{F},t\in I}$ through their boundaries. A
requirement of the gluing is that it only produce Euclidean neighborhoods in
the boundary points of these horizontal pieces. We will give the precise
process in Section 4.

###  1.2. Acknowledgements

We wish to thank Professor Xingru Zhang for his help.

##  2\. Preliminaries

###  2.1. Conventions

For metric spaces $A$ and $B$, we denote by $A\setminus\setminus B$ the
closure of $A-B$ under the path metric.

When we use the notation “$<$” for an order, $a\leqslant b$ will always mean
either $a<b$ or $a=b$.

Let $M$ be a closed $3$-manifold. Let $p:\widetilde{M}\to M$ be the universal
cover of $M$. We say $\Gamma$ is a _fundamental domain_ (or a _fundamental
polyhedron_ , see [Gr]) of $M$ in $\widetilde{M}$ if
$\Gamma\subseteq\widetilde{M}$ is a connected, compact, contractible
polyhedron with topologically locally flat boundary, $p\mid_{Int(\Gamma)}$ is
injective, $p\mid_{\Gamma}$ is surjective, and $\partial\Gamma$ is a finite
union of polygons that are identified pairwise by deck transformations. Notice
that every compact set in $\widetilde{M}$ only meets finitely many images of
$\Gamma$ under deck transformations (see [Gr, Lemma 1]). We say
$F\subseteq\partial\Gamma$ is a _face_ (or is a _common face_ of $\Gamma$ and
$h(\Gamma)$) if there exists $h$ a non-identity deck transformation of
$\widetilde{M}$ such that $F$ is a component of $\Gamma\cap h(\Gamma)$, and
there is no deck transformation $g$ such that $g\neq 1,h$ and
$F\subseteq\Gamma\cap g(\Gamma)$.

Throught this paper, we will always discuss closed irreducible $3$-manifolds
with left orderable fundamental groups. Since left orderable groups are
torsion-free ([BRW, Proposition 2.1]), these $3$-manifolds are always covered
by $\mathbb{R}^{3}$, and their fundamental domains are always homeomorphic to
$B^{3}$.

###  2.2. Branched surfaces

Our notations and definitions of branched surfaces follow from [FO], [O],
[Ga2], and [L]:

######  Definition 2.1.

A _branched surface_ $B$ in a closed $3$-manifold $M$ is a subspace obtained
from gluing a finite union of compact surfaces and orient them to be locally
modeled as Figure 2.

Figure 1. The local models of standard spines ([Cas]).

Figure 2. The local models of branched surfaces. Figure 3. The fibered
neighborhood $N(B)$.

######  Notation 2.2.

Let $B$ be a branched surface in a closed $3$-manifold $M$.

(a) Let $N(B)$ be a regular neighborhood of the branched surface $B$ as shown
in Figrue 3. Then $N(B)$ can be regarded as an interval bundle $\mathcal{I}$
over $B$. $\partial N(B)$ can be decomposed to $\partial_{h}N(B)$ (_horizontal
boundary_) and $\partial_{v}N(B)$ (_vertical boundary_) such that:
$\partial_{h}N(B)$ is transverse to $\mathcal{I}$, and $\partial_{v}N(B)$ is
tangent to $\mathcal{I}$. We call $N(B)$ the _fibered neighborhood_ of $B$,
and call $\mathcal{I}$ the _$I$ -fibers_ (or _interval fibers_) of $B$.

(b) Let $\pi:N(B)\to B$ be the projection that takes each $I$-fiber to a
single point. We call $\pi$ the _collapsing map_.

(c) A lamination $\mathcal{L}$ is _carried_ by $B$ if there exists a fibered
neighborhood $N(B)$ such that $\mathcal{L}\subseteq N(B)$ and each leaf of
$\mathcal{L}$ is transverse to the $I$-fibers of $N(B)$. $\mathcal{L}$ is
_fully carried_ by $B$ if it is carried by $B$ and $\mathcal{L}$ intersects
every $I$-fiber of $N(B)$.

(d) The _branch locus_ of $B$ is the set of points in $B$ that have no
Euclidean neighborhood in $B$. We denote it by $L(B)$. Each component of
$B\setminus\setminus L(B)$ is called a _branch sector_. We say each point in
$L(B)$ that has no $\mathbb{R}$-neighborhood in $L(B)$ is a _double point_ of
$L(B)$. Let $V$ denote the set of double points of $L(B)$. We say each
component of $L(B)\setminus\setminus V$ is an _edge_ of $L(B)$.

######  Remark 2.3.

Suppose that $B$ is a branched surface in a closed $3$-manifold $M$. Let
$p:\widetilde{M}\to M$ be the universal cover of $M$. Let
$\widetilde{B}=p^{-1}(B)$. Because $\widetilde{M}$ is not a closed
$3$-manifold and $\widetilde{B}$ can not be obtained from gluing a finite
union of compact surfaces, $\widetilde{B}$ is not a branched surface in
$\widetilde{M}$ in terms of Definition 2.1. For our convenience, we also say
$\widetilde{B}$ is a branched surface in $\widetilde{M}$.

Moreover, for a fibered neighborhood $N(B)$ of $B$, we say
$\widetilde{N(B)}=p^{-1}(B)$ is a fibered neighborhood of $\widetilde{B}$, and
we will always assume that the $I$-fibers of $\widetilde{N(B)}$ are the pull-
back of $I$-fibers of $N(B)$. And we also say the map
$\widetilde{\pi}:\widetilde{N(B)}\to\widetilde{B}$ to collapse the $I$-fibers
of $\widetilde{N(B)}$ into $\widetilde{B}$ is the collapsing map.

Branched surface is an important tool to describe sutured manifolds (see
[Ga2]) and essential laminations (see [GO]). A method to construct essential
laminations is given in [L] by showing an essential branched surface is
laminar.

In this paper, we attempt a different approach. We will choose a branched
surface with a single $B^{3}$ complementary region, and we will construct a
lamination fully carried by this branched surface directly.

##  3\. Construction of the abstract laminations

Let $M$ be a closed, orientable, irreducible $3$-manifold such that
$\pi_{1}(M)$ is left orderable. Let $p:\widetilde{M}\to M$ be the universal
cover of $M$. Fix $x\in\widetilde{M}$ a base point. Let $G=\pi_{1}(M,p(x))$.
For each $h\in G$, there exists a unique deck transformation $t_{h}\in
Aut(\widetilde{M})$ such that: if $\eta\subseteq\widetilde{M}$ is a path that
starts at $x$ and ends at $t_{h}(x)$, then $[p(\eta)]=h$. In the following, we
will always adopt the notations $\\{t_{h}\\}_{h\in G}$ for the deck
transformations in $\widetilde{M}$. And we fix “$<$” a left-invariant order of
the group $G$.

###  3.1. Abstract laminations

Assume that $\mathcal{L}$ is a co-orientable lamination in $M$ carried by a
branched surface $B$. Let $N(B)$ be a fibered neighborhood of $B$ (such that
$\mathcal{L}\subseteq N(B)$) and $\pi:N(B)\to B$ the collapsing map. Let
$\widetilde{B}=p^{-1}(B)$, $\widetilde{N(B)}=p^{-1}(N(B))$, and
$\widetilde{\pi}:\widetilde{N(B)}\to\widetilde{B}$ the collapsing map. Let
$\widetilde{\mathcal{L}}$ be the pull-back lamination of $\mathcal{L}$ in
$\widetilde{M}$. Assume further that the restriction of $\widetilde{\pi}$ on
every leaf of $\widetilde{\mathcal{L}}$ is injective. Then we can describe
$\widetilde{\mathcal{L}}$ by the collection

$\\{\widetilde{\pi}(\widetilde{l})\mid\widetilde{l}$ is a leaf in
$\widetilde{\mathcal{L}}\\}$.

For every leaf $\widetilde{l}$ in $\widetilde{\mathcal{L}}$,
$\widetilde{\pi}(\widetilde{l})$ is a $2$-manifold subspace of $\widetilde{B}$
since $\widetilde{\pi}\mid_{\widetilde{l}}$ is injective, and it is transverse
to the interval fibers of $\widetilde{N(B)}$. Moreover, the elements in
$\\{\widetilde{\pi}(\widetilde{l})\mid\widetilde{l}$ is a leaf in
$\widetilde{\mathcal{L}}\\}$ are equivariant under deck transformations, and
for arbitrary
$s_{1},s_{2}\in\\{\widetilde{\pi}(\widetilde{l})\mid\widetilde{l}$ is a leaf
in $\widetilde{\mathcal{L}}\\}$, $s_{2}$ only intersects at most one component
of $\widetilde{M}-s_{1}$.

To describe this collection, we give the following definition:

######  Definition 3.1 (Abstract lamination).

Let $B$ be a branched surface in $M$. Let $\widetilde{B}=p^{-1}(B)$. We say
$\mathcal{A}$ is an _abstract lamination_ in $\widetilde{M}$ (_carried by
$\widetilde{B}$_) if the following holds:

(1) Let S be the set of branch sectors of $B$. Each $\sigma\in\mathcal{A}$ is
a subspace of $\widetilde{B}$ such that: there exists $S_{0}\subseteq
p^{-1}(\textbf{S})$ such that

$\sigma=\bigcup_{X\in S_{0}}X,$

$\sigma$ is a $2$-manifold (possibly disconnected) with
$\partial\sigma=\emptyset$, and each component of $\sigma$ separates
$\widetilde{M}$. We say $\sigma$ is a _leaf_ of $\mathcal{A}$.

(2) Let $N(B)$ be a fibered neighborhood of $B$ and
$\widetilde{N(B)}=p^{-1}(N(B))$. Then $\widetilde{N(B)}$ is a fibered
neighborhood of $\widetilde{B}$. For each $\sigma\in\mathcal{A}$, $\sigma$ is
transverse to the interval fibers of $\widetilde{N(B)}$.

(3) For each $\sigma\in\mathcal{A}$ and $h\in G$,
$t_{h}(\sigma)\in\mathcal{A}$.

(4) If $\sigma_{1},\sigma_{2}\in\mathcal{A}$, $s_{1}$ is a component of
$\sigma_{1}$, $s_{2}$ is a component of $\sigma_{2}$, then one of the $2$
components of $\widetilde{M}-s_{1}$ is disjoint from $s_{2}$. This also
implies that one of the $2$ components of $\widetilde{M}-s_{2}$ is disjoint
from $s_{1}$.

Notice that a leaf of an abstract lamination may be disconnected.

In the remainder of this section, our plan is: firstly, choose a branched
surface $B$ in $M$, and secondly, determine an abstract lamination
$\mathcal{A}$ in $\widetilde{M}$ carried by $\widetilde{B}=p^{-1}(B)$. The
choice of this branched surface $B$ will be shown in next subsection.

###  3.2. The choice of the branched surface

######  Definition 3.2 (Standard spine).

(a) A standard spine ([Cas]) is a topological space locally modeled as Figure
1, which is topologically homeomorphic to a branched surface but does not have
well-defined cusp directions.

(b) Suppose that $S$ is a standard spine and $L$ is the set of points in $S$
that have no Euclidean neighborhood in $S$. Then we say each component of
$S\setminus\setminus L$ is a _sector_ of $S$.

Recall our settings of fundamental domains from Subsection 2.1. A fundamental
domain of $M$ in $\widetilde{M}$ is a compact polyhedron with topologically
locally flat boundary, and it is homeomorphic to $B^{3}$.

######  Lemma 3.3.

There exists a fundamental domain $\Gamma$ of $M$ in $\widetilde{M}$ such that
$p(\partial\Gamma)$ is a standard spine.

(a)  (b)

Figure 4. Thicken an edge.

(a)  (b)

Figure 5. Thicken a vertex.

###### Proof.

For an arbitrary fundamental domain $\Gamma$ of $M$ in $\widetilde{M}$, we
will always denote by $G(\Gamma)$ the set of points in $p(\partial\Gamma)$
that has no $\mathbb{R}^{2}$-neighborhood in $p(\partial\Gamma)$.

Let $\Gamma_{0}$ be a fundamental domain of $M$ in $\widetilde{M}$. Then
$G(\Gamma_{0})$ is a graph, and we can assume that it has no isolated point.
Let $V(\Gamma_{0})\subseteq G(\Gamma_{0})$ be a finite set of points such
that: for each component $e$ of $G(\Gamma_{0})-V(\Gamma_{0})$, $\overline{e}$
is homeomorphic to a closed interval. We define each element of
$V(\Gamma_{0})$ to be a vertex of $G(\Gamma_{0})$ and define the closure of
each component of $G(\Gamma_{0})-V(\Gamma_{0})$ to be an edge of
$G(\Gamma_{0})$. For an edge $e$ of $G(\Gamma_{0})$, we say $e$ is _$k$
-valent_ if $p^{-1}(e)\cap\Gamma_{0}$ has $k$ components.

Starting from $\Gamma_{0}$, we repeat the following operation to obtain
$\Gamma_{i+1}$ from $\Gamma_{i}$ ($i\geqslant 0$):

$\bullet$ (Thicken an edge) Assume that $\Gamma_{i}$ is a fundamental domain
of $M$ in $\widetilde{M}$, and there is an edge $e$ of $G(\Gamma_{i})$ that
does not have valency $3$. Let $N_{e}=e\times B^{2}$ be a $B^{2}$ bundle over
$e$. Let $\widetilde{N_{e}}$ be a component of $p^{-1}(N_{e})$ such that
$\widetilde{N_{e}}\cap\Gamma_{i}\neq\emptyset$. Let

$\Gamma_{i+1}=\overline{\Gamma_{i}-p^{-1}(N_{e})}\cup\widetilde{N_{e}}$

(see Figure 4). Then $\Gamma_{i+1}$ is still a fundamental domain of $M$ in
$\widetilde{M}$. We define each vertex of $G(\Gamma_{i})$ to be a vertex of
$G(\Gamma_{i+1})$, and define each component of
$G(\Gamma_{i+1})\setminus\setminus\\{$vertices of $G(\Gamma_{i+1})\\}$ to be
an edge of $G(\Gamma_{i+1})$. Then

$\\#\\{s$ is an edge of $G(\Gamma_{i+1})$, $s$ is not $3$-valent$\\}<\\#\\{s$
is an edge of $G(\Gamma_{i})$, $s$ is not $3$-valent$\\}$.

We can repeat the above operation continuously, until we obtain $\Gamma_{k}$
($k\in\mathbb{N}$) such that all edges of $G(\Gamma_{k})$ have valency $3$.
Starting from $\Gamma_{k}$, we do the following operation to obtain
$\Gamma_{i+1}$ from $\Gamma_{i}$ ($i\geqslant k$):

$\bullet$ (Thicken a vertex) Assume that $\Gamma_{i}$ is a fundamental domain
of $M$ in $\widetilde{M}$ such that every edge $e$ of $G(\Gamma_{i})$ is
$3$-valent, and there is a vertex $v$ of $G(\Gamma_{i})$ that is not
$2$-valent or $4$-valent. Let $N_{v}$ be a regular neighborhood of $v$. Let
$\widetilde{N_{v}}$ be a component of $p^{-1}(N_{v})$ such that
$\widetilde{N_{v}}\cap\Gamma_{i}\neq\emptyset$. Let

$\Gamma_{i+1}=\overline{\Gamma_{i}-p^{-1}(N_{v})}\cup\widetilde{N_{v}}$

(see Figure 5). Then $\Gamma_{i+1}$ is still a fundamental domain of $M$ in
$\widetilde{M}$. We define every point in $(\\{$vertices of
$G(\Gamma_{i})\\}-\\{v\\})\cup(\partial N_{v}\cap G(\Gamma_{i}))$ to be a
vertex of $G(\Gamma_{i+1})$, and define each component of
$G(\Gamma_{i+1})\setminus\setminus\\{$vertices of $G(\Gamma_{i+1})\\}$ to be
an edge of $G(\Gamma_{i+1})$. Then

$\\#\\{$vertices of $G(\Gamma_{i+1})$ have not valency $2$ or
$4\\}<\\#\\{$vertices of $G(\Gamma_{i})$ have not valency $2$ or $4\\}$.

Moreover, all new edges produced in this process still have valency $3$.

We can repeat this operation continuously. At last, we can obtain a
fundamental domain $\Gamma_{n}$ such that all vertices of $G(\Gamma_{n})$ are
$2$-valent or $4$-valent, and all edges of $G(\Gamma_{n})$ are still
$3$-valent. Then $p(\partial\Gamma_{n})$ is a standard spine. We can
accomplish the proof of Lemma 3.3 by choosing $\Gamma=\Gamma_{n}$. ∎

Guaranteed by Lemma 3.3, we can fix $\Gamma$ a fundamental domain of $M$ in
$\widetilde{M}$ such that $p(\partial\Gamma)$ is a standard spine. Without
loss of generality, we assume the base point $x\in Int(\Gamma)$. And we fix an
orientation of $\Gamma$.

Recall from Subsection 2.1, a face $F$ of $\Gamma$ is a component of a
nonempty intersection $\Gamma\cap t_{h}(\Gamma)$ ($h\in G$, $h\neq 1$) such
that there is no $g\in G-\\{1,h\\}$ such that $F\subseteq\Gamma\cap
t_{g}(\Gamma)$. Since $p(\partial\Gamma)$ is a standard spine, every face of
$\Gamma$ is a subsurface of $\partial\Gamma$. We orient the faces of $\Gamma$
as follows:

$\bullet$ Let $F$ be a face of $\Gamma$. Then there exists a unique $h\in
G-\\{1\\}$ such that $F$ is a common face of $\Gamma$ and $t_{h}(\Gamma)$. If
$h>1$, then we assign $F$ the positive orientation (with respect to the
orientation of $\Gamma$). Otherwise, $h<1$, we assign $F$ the negative
orientation (with respect to the orientation of $\Gamma$).

Note that for every face $F$ of $\Gamma$, there exists a unique different face
$F^{{}^{\prime}}$ of $\Gamma$ such that $p(F)=p(F^{{}^{\prime}})$. There
exists a unique $h\in G-\\{1\\}$ such that $F$ is a common face of $\Gamma$
and $t_{h}(\Gamma)$, then $F^{{}^{\prime}}$ is a common face of $\Gamma$ and
$t_{h^{-1}}(\Gamma)$. Since exactly one of $h,h^{-1}$ is greater than $1$,
$F,F^{{}^{\prime}}$ are assigned with opposite orientations with respect to
$\Gamma$. Then $p$ descends their orientations to the same orientation on
$p(F)=p(F^{{}^{\prime}})$. Therefore, the orientations of all faces of
$\Gamma$ induce a well-defined co-orientation on their images in
$p(\partial\Gamma)$ (i.e. the sectors of $p(\partial\Gamma)$). We orient
$p(\partial\Gamma)$ in $M$ by this co-orientation. Moreover, we orient
$\bigcup_{h\in G}t_{h}(\partial\Gamma)$ in $\widetilde{M}$ in a consistent way
with the pull-back of this co-orientation on $p(\partial\Gamma)$.

The following lemma shows that the orientations on $\bigcup_{h\in
G}t_{h}(\partial\Gamma)$ are decided by the order “$<$” of $G$ in a uniform
way:

######  Lemma 3.4.

Let $S$ be a sector of $p(\partial\Gamma)$. Let $\widetilde{S}$ be a component
of $p^{-1}(S)$. Then there exists $h,g\in G$ such that $\widetilde{S}$ is a
common face of $t_{h}(\Gamma)$ and $t_{g}(\Gamma)$. Without loss of
generality, we assume that $h>g$. We assume that the orientations of
$t_{h}(\Gamma),t_{g}(\Gamma)$ are induced from the orientation of $\Gamma$
(under deck transformations). Recall that the orientation of $\widetilde{S}$
is induced from the orientation of $S$. Then the orientation of
$\widetilde{S}$ is positive with respect to $t_{g}(\Gamma)$ and is negative
with respect to $t_{h}(\Gamma)$.

###### Proof.

There exists a component $\widetilde{S_{0}}$ of $p^{-1}(S)$ such that
$\widetilde{S_{0}}\subseteq\Gamma\cap t_{g^{-1}h}(\Gamma)$. Then
$\widetilde{S}=t_{g}(\widetilde{S_{0}})$.

Since $h>g$, $g^{-1}h>1$. So $\widetilde{S_{0}}$ is oriented positively with
respect to $\Gamma$ and is oriented negatively with respect to
$t_{g^{-1}h}(\Gamma)$. This implies that $\widetilde{S}$ is oriented
positively with respect to $t_{g}(\Gamma)$, and is oriented negatively with
respect to $t_{h}(\Gamma)$. ∎

(a)  (b)

Figure 6. Assume that the $4$ points $a_{1},a_{2},a_{3},a_{4}\in\widetilde{M}$
in (a) lies in $t_{h_{1}}(\Gamma)$, $t_{h_{2}}(\Gamma)$, $t_{h_{3}}(\Gamma)$,
$t_{h_{4}}(\Gamma)$ respectively, and $h_{1},h_{2},h_{3},h_{4}$ has the
relation $h_{1}<h_{2}<h_{3}<h_{4}$. Then (a) can be oriented to be (b).

From now on, we assign $p(\partial\Gamma)$ the cusp directions given by the
assignment of this co-orientation on its sectors. And the fundamental domain
$\Gamma$ is assumed to be deformed in $\widetilde{M}$ correspondently. Next,
we will verify that $p(\partial\Gamma)$ is a branched surface.

######  Lemma 3.5.

$p(\partial\Gamma)$ is a branched surface.

###### Proof.

By Lemma 3.4, the orientations on $\bigcup_{h\in G}t_{h}(\partial\Gamma)$ can
give well-defined cusp directions on the set of points that possess no
Euclidean neighborhood, as shown in Figure 6. Then $\bigcup_{h\in
G}t_{h}(\partial\Gamma)$ is locally modeled as Figure 2.

Correspondently, $p(\partial\Gamma)$ is locally modeled as Figure 2.
Therefore, $p(\partial\Gamma)$ is oriented to be a branched surface. ∎

We denote by $B=p(\partial\Gamma)$ this branched surface.

######  Notation 3.6.

In the remainder of this section (and together with Section 4), we will always
assume $B=p(\partial\Gamma)$. And we fix the following notations associated to
$B$:

$\bullet$ Let $N(B)$ be a fibered neighborhood of $B$. Let $\pi:N(B)\to B$ be
the collapsing map.

$\bullet$ Let $\widetilde{B}=p^{-1}(B)$ be the pull-back branched surface in
$\widetilde{M}$. Then $\widetilde{B}=\bigcup_{h\in G}t_{h}(\partial\Gamma)$.
And we assume that the orientations on the branch sectors of $\widetilde{B}$
are induced from $B$ (by the pull-back).

$\bullet$ Let $\widetilde{N(B)}=p^{-1}(N(B))$, and we denote by
$\widetilde{\pi}:\widetilde{N(B)}\to B$ the collapsing map.

In the rest of this section, we will prove:

######  Proposition 3.7.

There exists an abstract lamination $\mathcal{A}$ carried by $\widetilde{B}$.

We will introduce order divisions of the left orderable group $G$ in next
subsection. They will be used to construct the abstract lamination
$\mathcal{A}$ in Subsection 3.4.

###  3.3. Order divisions of $G$

We say a non-empty set $C\varsubsetneqq G$ is an _order division_ of $G$ if it
safitfies that: whenever $a\in C$ and $b<a$, then $b\in C$. Let

$\mathcal{C}=\\{C\mid C$ is an order division of $G\\}$.

######  Fact 3.8.

For each $C\in\mathcal{C}$ and $g\in G$, $gC=\\{gt\mid t\in
C\\}\in\mathcal{C}$.

###### Proof.

For each $y\in gC$ and $z<y$, we have

$g^{-1}z<g^{-1}y\in C.$

Then $z=g(g^{-1}z)\in gC$. Therefore, $gC$ is also an order division. ∎

######  Fact 3.9.

If $C_{1},C_{2}\in\mathcal{C}$, then one of the following holds:

$\bullet$ $C_{1}=C_{2}$.

$\bullet$ $C_{1}\varsubsetneqq C_{2}$.

$\bullet$ $C_{2}\varsubsetneqq C_{1}$.

###### Proof.

If $C_{1}\neq C_{2}$, without loss of generality, we assume that there exists
$g\in C_{1}$ and $g\notin C_{2}$.

Then for each $t>g$, $t\notin C_{2}$. Hence

$C_{2}\subseteq G-\bigcup_{t\geqslant g}t=(\bigcup_{t<g}t)\varsubsetneqq
C_{1}.$

Therefore, $C_{2}\varsubsetneqq C_{1}$. ∎

The following two facts will be used in the next section, for explaining the
“continuity” of $\mathcal{C}$.

######  Fact 3.10.

Suppose that $\mathcal{X}\subseteq\mathcal{C}$, and there exists
$C_{0}\in\mathcal{C}$ such that for every $C\in\mathcal{X}$, $C\varsubsetneqq
C_{0}$. Then $\bigcup_{C\in\mathcal{X}}C\in\mathcal{C}$, and
$\bigcup_{C\in\mathcal{X}}C\subseteq C_{0}$.

###### Proof.

$\bigcup_{C\in\mathcal{X}}C\subseteq C_{0}$ since every element of
$\bigcup_{C\in\mathcal{X}}C$ is in some $C^{{}^{\prime}}\in\mathcal{X}$, and
$C^{{}^{\prime}}\subseteq C_{0}$.

For every $a\in\bigcup_{C\in\mathcal{X}}C$, there exists $C_{1}\in\mathcal{X}$
such that $a\in C_{1}$. Then for every $b<a$,

$b\in C_{1}\subseteq\bigcup_{C\in\mathcal{X}}C.$

Moreover, $\bigcup_{C\in\mathcal{X}}C\subseteq C_{0}\varsubsetneqq G$, and
$\bigcup_{C\in\mathcal{X}}C$ is non-empty obviously. Hence
$\bigcup_{C\in\mathcal{X}}C\in\mathcal{C}$. ∎

######  Fact 3.11.

Suppose that $\mathcal{X}\subseteq\mathcal{C}$, and there exists
$C_{0}\in\mathcal{C}$ such that for every $C\in\mathcal{X}$,
$C_{0}\varsubsetneqq C$. Then $\bigcap_{C\in\mathcal{X}}C\in\mathcal{C}$, and
$C_{0}\subseteq\bigcap_{C\in\mathcal{X}}C$.

###### Proof.

$C_{0}\subseteq\bigcap_{C\in\mathcal{X}}C$ since for every $C\in\mathcal{X}$,
$C_{0}\subseteq C$.

Let $a\in\bigcap_{C\in\mathcal{X}}C$. Then $a\in C$ for every
$C\in\mathcal{X}$. If $b<a$, then $b\in C$ for every $C\in\mathcal{X}$. Then

$b\in\bigcap_{C\in\mathcal{X}}C.$

We choose $C_{1}\in\mathcal{X}$, then $\bigcap_{C\in\mathcal{X}}C\subseteq
C_{1}\varsubsetneqq G$. And $C_{0}\subseteq\bigcap_{C\in\mathcal{X}}C$ ensures
that $\bigcap_{C\in\mathcal{X}}C$ is non-empty. Therefore,
$\bigcap_{C\in\mathcal{X}}C\in\mathcal{C}$. ∎

###  3.4. Abstract laminations constructed by order divisions

Next, we will construct an abstract lamination $\mathcal{A}$ carried by the
pull-back branched surface $\widetilde{B}=p^{-1}(B)$.

For each $C\in\mathcal{C}$, let

$R_{C}=\bigcup_{h\in C}t_{h}(\Gamma),$ $l_{C}=\partial R_{C}.$

$\\{l_{C}\\}_{C\in\mathcal{C}}$ is our intentional abstract lamination
$\mathcal{A}$ carried by $\widetilde{B}$ in Proposition 3.7. In the following,
we will verify Condition (1) $\sim$ (4) of Definition 3.1 for
$\\{l_{C}\\}_{C\in\mathcal{C}}$.

For each $C\in\mathcal{C}$, $\emptyset\varsubsetneqq C\varsubsetneqq G$. So
$\emptyset\varsubsetneqq R_{C}\varsubsetneqq\widetilde{M}$. This implies
$l_{C}=\partial R_{C}\neq\emptyset$. And obviously, $\partial l_{C}=\emptyset$
for each $C\in\mathcal{C}$.

######  Lemma 3.12.

For each $C\in\mathcal{C}$, $l_{C}$ is a topologically locally flat surface
that has no component which is a closed surface.

###### Proof.

Notice that $\widetilde{B}$ only has edges of valency $3$ and only has
vertices of valency $4$. So $R_{C}$ must have topologically locally flat
boundary. Then $l_{C}$ must be a topologically locally flat surface.

Recall that $\Gamma$ has a face $F$ oriented with the negative orientation
(with respect to the orientation of $\Gamma$). We denote by $s\in G-\\{1\\}$
such that $F=\Gamma\cap t_{s}(\Gamma)$. Then $s<1$. For each $g\in C$, we have

$g>gs>gs^{2}>\ldots,$

which implies $\\{g,gs,gs^{2},\ldots\\}\subseteq C$, and $t_{g}(\Gamma)$,
$t_{gs}(\Gamma)$, $t_{gs^{2}}(\Gamma)$, $\ldots$ lie in $R_{C}$. Note that
$t_{g}(\Gamma)$, $t_{gs}(\Gamma)$, $t_{gs^{2}}(\Gamma)$, $\ldots$ are adjacent
one-by-one (since $\Gamma\cap t_{s}(\Gamma)\neq\emptyset$) and different from
each other (since $G$ is torsion-free). So every component of $R_{C}$ is
composed of infinitely many members of $\\{t_{h}(\Gamma)\\}_{h\in G}$, and
similarly, every component of $\widetilde{M}\setminus\setminus R_{C}$ is
composed of infinitely many members of $\\{t_{h}(\Gamma)\\}_{h\in G}$.
Therefore, no component of $l_{C}=\partial R_{C}$ is a closed surface
(otherwise, a closed surface component of $l_{C}$ should separate
$\widetilde{M}$ into $2$ components, and one of them is composed of finitely
many members of $\\{t_{h}(\Gamma)\\}_{h\in G}$). ∎

Since $l_{C}$ separates $\widetilde{M}$ into the $2$ parts $R_{C}$ and
$\widetilde{M}\setminus\setminus R_{C}$, we can ensure that each component of
$l_{C}$ separates $\widetilde{M}$. Therefore, $\\{l_{C}\\}_{C\in\mathcal{C}}$
satisfies Condition (1) of Definition 3.1.

######  Lemma 3.13.

For each $C\in\mathcal{C}$, $l_{C}$ is transverse to the interval fibers of
$\widetilde{N(B)}$.

###### Proof.

Suppose that $S$ is a branch sector of $\widetilde{B}$ such that $S\subseteq
l_{C}$. There exist $g,h\in G$ such that $S$ is a common face of
$t_{h}(\Gamma)$ and $t_{g}(\Gamma)$. Assume without loss of generality, $h>g$.
Then $t_{g}(\Gamma)$ lies inside $R_{C}$ and $t_{h}(\Gamma)$ lies outside
$R_{C}$. By Lemma 3.4, $S$ has the positive orientation with respect to the
orientation of $t_{g}(\Gamma)$.

Hence $l_{C}$ can be oriented so that its orientation is consistent with the
orientations of all branch sectors of $\widetilde{B}$ that lie on $l_{C}$.
Therefore, $l_{C}$ is transverse to the interval fibers of $\widetilde{N(B)}$.
∎

Therefore, $\\{l_{C}\\}_{C\in\mathcal{C}}$ satisfies Condition (2) of
Definition 3.1.

By Fact 3.8, $\\{l_{C}\\}_{C\in\mathcal{C}}$ is $G$-equivariant. Then
Condition (3) of Definition 3.1 is true for $\\{l_{C}\\}_{C\in\mathcal{C}}$.

By Fact 3.9, whenever $C_{1},C_{2}\in\mathcal{C}$ and $C_{1}\neq C_{2}$, then
either $R_{C_{1}}\varsubsetneqq R_{C_{2}}$, or $R_{C_{2}}\varsubsetneqq
R_{C_{1}}$. Assume without loss of generality, $R_{C_{1}}\varsubsetneqq
R_{C_{2}}$. We choose $l_{1}$ an arbitrary component of $l_{C_{1}}$ and
$l_{2}$ an arbitrary component of $l_{C_{2}}$. $l_{2}\subseteq\partial
R_{C_{2}}$ is embedded in the closure of $\widetilde{M}-R_{C_{1}}$, then it
must has empty intersection with one of the two components of
$\widetilde{M}-l_{1}$. Hence we have verified Condition (4) of Definition 3.1
for $\\{l_{C}\\}_{C\in\mathcal{C}}$. The proof of Proposition 3.7 is now
completed.

Now we know that $\\{l_{C}\\}_{C\in\mathcal{C}}$ is an abstract lamination
carried by $\widetilde{B}$. We denote this abstract lamination by

$\mathcal{A}=\\{l_{C}\\}_{C\in\mathcal{C}}.$

Recall that each $l_{C}$ ($C\in\mathcal{C}$) is called a _leaf_ , which may be
disconnected.

(a)  (b)

Figure 7. (a) A face $F$ of $\Gamma$ in the pull-back branched surface
$\widetilde{B}$. The blue region denotes the fibered neighborhood
$\widetilde{N(B)}=p^{-1}(N(B))$. (b) The leaves of $\mathcal{A}$ that contain
$F$ are subspaces of $\widetilde{B}$ and are overlapped together at $F$, but
we can draw them separately in $\widetilde{N(B)}$. This separated view
reflects their sequence under the order “$\prec$” (Definition 4.2).

##  4\. The construction in the main theorem

In this section, we follow the settings and notations of the last section,
including the branched surface $B$ (see Subsection 3.2), the set $\mathcal{C}$
of order divisions (see Subsection 3.3), and the abstract lamination
$\mathcal{A}$ (see Subsection 3.4). We will prove the following proposition in
this section:

######  Proposition 4.1.

There exists a co-orientable lamination $\Lambda$ in $M$, $\Lambda$ is fully
carried by $B$, and the leaves of its pull-back lamination
$\widetilde{\Lambda}$ are mapped to the components of leaves of $\mathcal{A}$
by the collapsing map $\widetilde{\pi}:\widetilde{N(B)}\to\widetilde{B}$.
Moreover, $\Lambda$ satisfies the following:

$\bullet$ $\Lambda$ has no torus leaf that bounds a solid torus (hence has no
Reeb component).

$\bullet$ $M-\Lambda$ is connected and simply connected.

$\bullet$ There exists a simple closed curve in $M$ which is co-orientably
transverse to $\Lambda$, and it intersects every leaf of $\Lambda$.

$\bullet$ $\Lambda$ has a single gut region, which is homeomorphic to $B^{3}$.

$\bullet$ $\widetilde{\Lambda}$ has a leaf that has trivial stabilizer under
deck transformations.

###  4.1. The $G$-equivariant linear order on $\mathcal{A}$

######  Definition 4.2.

Let “$\prec$” be the order on $\\{l_{C}\\}_{C\in\mathcal{C}}$ such that for
arbitrary $C_{1},C_{2}\in\mathcal{C}$, $l_{C_{1}}\prec l_{C_{2}}$ if
$C_{1}\varsubsetneqq C_{2}$.

By Fact 3.9, for $C_{1},C_{2}\in\mathcal{C}$, whenever $C_{1}\neq C_{2}$,
either $C_{1}\varsubsetneqq C_{2}$ or $C_{2}\varsubsetneqq C_{1}$. So
“$\varsubsetneqq$” is a linear order on $\mathcal{C}$, and “$\prec$” is a
linear order on $\mathcal{A}$.

Moreover, $\prec$ is $G$-equivariant:

######  Fact 4.3.

Let $C_{1},C_{2}\in\mathcal{C}$. If $l_{C_{1}}\prec l_{C_{2}}$, then

$t_{h}(l_{C_{1}})=l_{hC_{1}}\prec l_{hC_{2}}=t_{h}(l_{C_{2}}).$

###### Proof.

We have $C_{1}\varsubsetneqq C_{2}$ since $l_{C_{1}}\prec l_{C_{2}}$. Then

$hC_{1}\varsubsetneqq hC_{2}.$

Hence $l_{hC_{1}}\prec l_{hC_{2}}$. ∎

###  4.2. The map descending $\mathcal{C}$ into $\mathbb{R}$

######  Lemma 4.4.

There exists an order-preserving injective map

$e:\mathcal{C}\to\mathbb{R}$

with respect to the inclusion order “$\varsubsetneqq$” of $\mathcal{C}$ and
the increasing order of $\mathbb{R}$ (i.e. for $X,Y\in\mathcal{C}$,
$e(X)<e(Y)$ if and only if $X\varsubsetneqq Y$), and for each
$N\in\mathbb{R}_{+}$, there exists $C_{1},C_{2}\in\mathcal{C}$ such that
$e(C_{1})>N$, $e(C_{2})<-N$.

###### Proof.

Let

$H=\\{h\in G\mid h<1\\}\in\mathcal{C},$
$\overline{H}=H\cup\\{1\\}\in\mathcal{C}.$

Let

$\mathcal{C}_{0}=(\bigcup_{g\in G}gH)\cup(\bigcup_{g\in
G}g\overline{H})\subseteq\mathcal{C}.$

Since $G$ is a countable group, $\mathcal{C}_{0}$ is a countable set. So there
exists

$e_{0}:\mathcal{C}_{0}\to\mathbb{R}$

an order-preserving injective map (with respect to the inclusion order
“$\varsubsetneqq$” of $\mathcal{C}_{0}$ and the increasing order of
$\mathbb{R}$). Then we can extend the map $e_{0}:\mathcal{C}_{0}\to\mathbb{R}$
to an order-preserving injective map $e:\mathcal{C}\to\mathbb{R}$ as follows:

$\bullet$ Suppose that $C\in\mathcal{C}-\mathcal{C}_{0}$, then

$\sup_{g\in C}e_{0}(g\overline{H})\leqslant\inf_{g\in G-C}e_{0}(gH),$

since for every $a\in C$ and $b\in G-C$, $a\overline{H}\subseteq bH$. We set

$\sup_{g\in C}e_{0}(g\overline{H})\leqslant e(C)\leqslant\inf_{g\in
G-C}e_{0}(gH).$

Next, we verify the injectivity and order-preserving property of
$e:\mathcal{C}\to\mathbb{R}$:

$\bullet$ Suppose that $C_{1},C_{2}\in\mathcal{C}$ and $C_{1}\neq C_{2}$.
Assume without loss of generality, $C_{1}\varsubsetneqq C_{2}$. Then there
exists $g\in G$ such that $g\in C_{2}$ but $g\notin C_{1}$. Then

$e(C_{1})\leqslant e_{0}(gH)<e_{0}(g\overline{H})\leqslant e(C_{2}).$

Hence we have verified that $e:\mathcal{C}\to\mathbb{R}$ is an order-
preserving injective map.

Notice that $\mathcal{C}$ has no maximum element and has no minimum element
(with respect to the inclusion order “$\varsubsetneqq$”). So we can request
$e:\mathcal{C}\to\mathbb{R}$ to satisfy that: for each $N\in\mathbb{R}_{+}$,
there exists $C_{1},C_{2}\in\mathcal{C}$ such that $e(C_{1})>N$,
$e(C_{2})<-N$. ∎

Next, we define a few terms related to $\mathcal{C}$:

######  Definition 4.5.

Fix an order-preserving injective map $e:\mathcal{C}\to\mathbb{R}$ as given in
Lemma 4.4.

(a) For each $C\in\mathcal{C}$, we define

$e^{-}(C)=\sup\\{e(X)\mid X\in\mathcal{C},X\varsubsetneqq C\\},$

and

$e^{+}(C)=\inf\\{e(X)\mid X\in\mathcal{C},C\varsubsetneqq X\\}.$

(b) Let $C\in\mathcal{C}$. We say $C$ is an _upward gap place_ (with respect
to $e$) if

$e(C)>e^{-}(C),$

and say $C$ is a _downward gap place_ (with respect to $e$) if

$e(C)<e^{+}(C).$

Notice that for each $C\in\mathcal{C}$, $e^{-}(C)\leqslant e(C)\leqslant
e^{+}(C)$. If $C$ is an upward gap place (resp. downward gap place) with
respect to $e$, then there exists an interval component $J$ of
$\mathbb{R}-e(\mathcal{C})$ such that $e(C)=\sup J$ (resp. $e(C)=\inf J$).

In the following, we will show that we can make the map
$e:\mathcal{C}\to\mathbb{R}$ has an additional property such that
$e(\mathcal{C})$ has $G$-equivariant gaps in $\mathbb{R}$.

######  Lemma 4.6.

There exists an order-preserving injective map $e:\mathcal{C}\to\mathbb{R}$ as
given in Lemma 4.4 that has the following additional properties:

$\bullet$ If $C_{1}\in\mathcal{C}$ is an upward gap place with respect to $e$,
then for each $h\in G$, $hC_{1}$ is an upward gap place with respect to $e$.

$\bullet$ If $C_{2}\in\mathcal{C}$ is a downpward gap place with respect to
$e$, then for each $h\in G$, $hC_{2}$ is a downward gap place with respect to
$e$.

###### Proof.

We choose an arbitrary order-preserving injective map
$\overline{e}:\mathcal{C}\to\mathbb{R}$ as given in Lemma 4.4. Let $E_{+}$
(resp. $E_{-}$) be the set of upward gap places (resp. downward gap places) in
$\mathcal{C}$ with respect to $\overline{e}$.

Since $\mathbb{R}$ contains at most countably many disjoint intervals,
$\mathbb{R}-\overline{e}(\mathcal{C})$ has countably many interval components.
So $E_{+},E_{-}$ are both countable sets. Since $G$ is a countable group,
$\\{hE_{+}\\}_{h\in G},\\{hE_{-}\\}_{h\in G}$ are both countable sets.

We can blow up countably many places in $\mathbb{R}-\overline{e}(\mathcal{C})$
to obtain a map $e:\mathcal{C}\to\mathbb{R}$ such that $\\{hE_{+}\\}_{h\in G}$
is exactly the set of upward gap places with respect to $e$, and
$\\{hE_{-}\\}_{h\in G}$ is exactly the set of downward gap places with respect
to $e$. Then $e:\mathcal{C}\to\mathbb{R}$ has the above addtional properties.
∎

From now on, we fix a map $e:\mathcal{C}\to\mathbb{R}$ as given in Lemma 4.6.
In the remainder of this section, we assume all upward gap places and downward
gap places in $\mathcal{C}$ are with respect to $e$, and we will not emphasize
that they are with respect to $e$.

######  Lemma 4.7.

For each $t\in\mathbb{R}-e(\mathcal{C})$, we denote by $J$ the comonent of
$\mathbb{R}-e(\mathcal{C})$ that contains $t$, and denote by

$a=\sup(e(\mathcal{C})\cap(-\infty,t])\leqslant t,$
$b=\inf(e(\mathcal{C})\cap[t,+\infty))\geqslant t.$

Then $a\neq b$, and one of the following holds:

$\bullet$ $J=(a,b)$.

$\bullet$ $J=[a,b)$.

$\bullet$ $J=(a,b]$.

###### Proof.

Recall that the map $e$ is unbounded on the both sides, so $a,b$ exist. We
consider the following $3$ cases:

######  Case 1.

Assume that $a\notin e(\mathcal{C})$. We denote by

$C_{1}=\bigcup_{C\in\mathcal{C},e(C)\leqslant t}C.$

By Fact 3.10, $C_{1}\in\mathcal{C}$. Then $e(C_{1})\geqslant b$ (since
$e(C_{1})\geqslant a$ but $a\notin e(\mathcal{C})$). If $e(C_{1})>b$, then
there exists $C^{{}^{\prime}}_{1}\in\mathcal{C}$ such that $b\leqslant
e(C^{{}^{\prime}}_{1})<e(C_{1})$, which implies
$C^{{}^{\prime}}_{1}\varsubsetneqq C_{1}$. By Fact 3.10,

$C_{1}=\bigcup_{C\in\mathcal{C},e(C)\leqslant t}C\subseteq
C^{{}^{\prime}}_{1}$

since $C\subseteq C^{{}^{\prime}}_{1}$ for every $C\in\mathcal{C}$ such that
$e(C)\leqslant t$ (because $e(C^{{}^{\prime}}_{1})\geqslant b\geqslant
t\geqslant e(C)$), which is contradict to $C^{{}^{\prime}}_{1}\varsubsetneqq
C_{1}$. Then $b=e(C_{1})\in e(\mathcal{C})$. So $J=[a,b)$ since $a\notin
e(\mathcal{C})$ and $b\in e(\mathcal{C})$.

######  Case 2.

Assume that $b\notin e(\mathcal{C})$. We denote by

$C_{2}=\bigcap_{C\in\mathcal{C},e(C)\geqslant t}C.$

By Fact 3.11, $C_{2}\in\mathcal{C}$. Then $e(C_{1})\leqslant a$ since
$e(C_{1})\leqslant b$ but $b\notin e(\mathcal{C})$. If $e(C_{2})<a$, then
there exists $C^{{}^{\prime}}_{2}\in\mathcal{C}$ such that
$e(C_{2})<e(C^{{}^{\prime}}_{2})\leqslant a$, which implies
$C_{2}\varsubsetneqq C^{{}^{\prime}}_{2}$. By Fact 3.11,

$C^{{}^{\prime}}_{2}\subseteq\bigcap_{C\in\mathcal{C},e(C)\geqslant t}C=C_{2}$

since $C^{{}^{\prime}}_{2}\subseteq C$ for every $C\in\mathcal{C}$ such that
$e(C)\geqslant t$ (because $e(C^{{}^{\prime}}_{2})\leqslant a\leqslant
t\leqslant e(C)$), which is contradict to $C_{2}\varsubsetneqq
C^{{}^{\prime}}_{2}$. Hence $a=e(C_{2})\in e(\mathcal{C})$. And $J=(a,b]$
since $b\notin e(\mathcal{C})$ and $a\in e(\mathcal{C})$.

######  Case 3.

If $a,b\in e(\mathcal{C})$, then $b\neq a$ (otherwise, $t=a=b\in
e(\mathcal{C})$, which is contradictory). So $J=(a,b)$.

Therefore, $a\neq b$, and $J$ is one of $(a,b),[a,b),(a,b]$. ∎

######  Definition 4.8.

Let $J$ be a component of $\mathbb{R}-e(\mathcal{C})$.

(a) By Lemma 4.7, $J$ is an interval that has the form of $(a,b)$ or $[a,b)$
or $(a,b]$. We say $J$ has $(a,b)$-type (resp. $[a,b)$-type, $(a,b]$-type) if
$J\cong(a,b)$ (resp. $J\cong[a,b)$, $J\cong(a,b]$).

(b) If $J$ has $(a,b)$-type, then there exists $C_{1}\in\mathcal{C}$ a
downward gap place, $C_{2}\in\mathcal{C}$ an upward gap place, such that
$a=e(C_{1})=e^{-}(C_{2})$, $b=e(C_{2})=e^{+}(C_{1}))$. We say $C_{1}$ is an
_absolute downward gap place_ , and say $C_{2}$ is an _absolute upward gap
place_.

(c) If $J$ has $[a,b)$-type, then there exists $C_{0}\in\mathcal{C}$ an upward
gap place such that $b=e(C_{0})$, $a=e^{-}(C_{0})$. We say $C_{0}$ is a
_relative upward gap place_.

(d) If $J$ has $(a,b]$-type, then there exists $C_{0}\in\mathcal{C}$ a
downward gap place such that $a=e(C_{0})$, $b=e^{+}(C_{0})$. We say $C_{0}$ is
a _relative downward gap place_.

######  Lemma 4.9.

Suppose that $C_{0}\in\mathcal{C}$ is an absolute upward gap place (resp.
absolute downward gap place, relative upward gap place, relative downward gap
place). Then for each $h\in G$, $hC_{0}$ is an absolute upward gap place
(resp. absolute downward gap place, relative upward gap place, relative
downward gap place).

###### Proof.

This follows directly from the following properties, and each $hC_{0}$ ($h\in
G$) preserves these properties:

(1) If $C_{0}\in\mathcal{C}$ is an absolute upward gap place, then there
exists $C_{1}\in\mathcal{C}$ such that $C_{1}\varsubsetneqq C_{0}$, and for
each $C\in\mathcal{C}$ such that $C\varsubsetneqq C_{0}$ and $C\neq C_{1}$, we
have $C\varsubsetneqq C_{1}$.

(2) If $C_{0}\in\mathcal{C}$ is an absolute downward gap place, then there
exists $C_{1}\in\mathcal{C}$ such that $C_{0}\varsubsetneqq C_{1}$, and for
each $C\in\mathcal{C}$ such that $C_{0}\varsubsetneqq C$ and $C\neq C_{1}$, we
have $C_{1}\varsubsetneqq C$.

(3) If $C_{0}\in\mathcal{C}$ is a relative upward gap place, then
$\bigcup_{C\in\mathcal{C},C\varsubsetneqq C_{0}}C=C_{0}$.

(4) If $C_{0}\in\mathcal{C}$ is a relative downward gap place, then
$\bigcap_{C\in\mathcal{C},C_{0}\varsubsetneqq C}C=C_{0}$. ∎

###  4.3. Leaves of $\mathcal{A}$ passing a fixed face

Let

$\textbf{F}=\\{F$ is a face of $\Gamma$, $F$ is positively oriented with
respect to $\Gamma\\}.$

Notice that $p(\textbf{F})$ is exactly the set of the branch sectors of $B$.

######  Definition 4.10.

Let $F\in\textbf{F}$. We denote by $g\in G$ for which $F$ is a common face of
$\Gamma$ and $t_{g}(\Gamma)$ (then $g>1$, since $F$ is positively oriented
with respect to $\Gamma$).

(a) We denote by

$H=\\{h\in G\mid h<1\\}$

and

$\overline{H}=\\{h\in G\mid h\leqslant 1\\}.$

Let

$\Psi_{F}=\\{l_{C}\mid C\in\mathcal{C},F\subseteq l_{C}\\}=\\{l_{C}\mid
C\in\mathcal{C},\overline{H}\subseteq C\subseteq gH\\}.$

(b) Suppose that $|\Psi_{F}|>1$. Then $\overline{H}\varsubsetneqq gH$. Let
$j:\mathcal{A}\to\mathcal{C}$ be the map that takes every $l_{C}$ to $C$ (for
each $C\in\mathcal{C}$). And we choose a homeomorphism

$h_{F}:[e(\overline{H}),e(gH)]\to I.$

Then we define $e_{F}:\Psi_{F}\to I$ to be the map

$h_{F}\circ e\circ(j\mid_{\Psi_{F}}).$

From now on, we assume that $|\Psi_{F}|>1$ for each $F\in\textbf{F}$. The case
of $|\Psi_{F}|=1$ for some $F\in\textbf{F}$ will be considered in Remark 4.16.

Then we obtain a collection of injective maps $\\{e_{F}:\Psi_{F}\to
I\\}_{F\in\textbf{F}}$ such that: for each $F\in\textbf{F}$, $e_{F}$ is order-
preserving with respect to the order “$\prec$” of $\Psi_{F}$ (see Definition
4.2) and the increasing order of $\mathbb{R}$, and
$e_{F}(l_{\overline{H}})=0$, $e_{F}(l_{gH})=1$.

Similar to the map $e:\mathcal{C}\to\mathbb{R}$, we give the following
definitions for $\\{e_{F}:\Psi_{F}\to I\\}_{F\in\textbf{F}}$:

######  Definition 4.11.

(a) Let $C\in\mathcal{C}$. Suppose that $l_{C}\in\Psi_{F}$ for some
$F\in\textbf{F}$. If $e_{F}(l_{C})\neq 0$, then we denote by

$e^{-}_{F}(l_{C})=\sup\\{e_{F}(l_{X})\mid l_{X}\in\Psi_{F},l_{X}\prec
l_{C}\\}.$

If $e_{F}(l_{C})\neq 1$, then we denote by

$e^{+}_{F}(l_{C})=\inf\\{e_{F}(l_{X})\mid l_{X}\in\Psi_{F},l_{X}\succ
l_{C}\\}.$

Then $e^{-}_{F}(l_{C})\leqslant e_{F}(l_{C})$ (if $e_{F}(l_{C})\neq 0$) and
$e_{F}(l_{C})\leqslant e^{+}_{F}(l_{C})$ (if $e_{F}(l_{C})\neq 1$).

(b) Let $F\in\textbf{F}$. Let $J$ be a component of $I-e_{F}(\Psi_{F})$. Then
$J$ is an interval that has the form of $(a,b)$ or $[a,b)$ or $(a,b]$ (see
Lemma 4.7). We say $J$ has $(a,b)$-type (resp. $[a,b)$-type, $(a,b]$-type) if
$J\cong(a,b)$ (resp. $J\cong[a,b)$, $J\cong(a,b]$).

######  Lemma 4.12.

Suppose that $C\in\mathcal{C}$ and $l_{C}\in\Psi_{F}$ for some
$F\in\textbf{F}$. Then:

(a) Assume that $e_{F}(l_{C})\neq 0$. Then $e_{F}(l_{C})>e^{-}_{F}(l_{C})$ if
and only if $C$ is an upward gap place in $\mathcal{C}$. Moreover, if $C$ is
an absulote upward gap place (resp. relative upward gap place), then
$e^{-}_{F}(l_{C})\in e_{F}(\Psi_{C})$ (resp. $e^{-}_{F}(l_{C})\notin
e_{F}(\Psi_{C})$).

(b) Assume that $e_{F}(l_{C})\neq 1$. Then $e_{F}(l_{C})<e^{+}_{F}(l_{C})$ if
and only if $C$ is a downward gap place in $\mathcal{C}$. Moreover, if $C$ is
an absulote downward gap place (resp. relative downward gap place), then
$e^{+}_{F}(l_{C})\in e_{F}(\Psi_{C})$ (resp. $e^{+}_{F}(l_{C})\notin
e_{F}(\Psi_{C})$).

###### Proof.

Follows from Definition 4.5 and Definition 4.8 directly. ∎

######  Definition 4.13.

For each $F\in\textbf{F}$, let $\mu_{F}:I-e_{F}(\Psi_{F})\to I$ be a map such
that:

$\bullet$ If $J$ is a component of $I-e_{F}(\Psi_{F})$ that has $(a,b)$-type,
then $\mu_{F}\mid_{J}$ is a homeomorphism that takes $J$ to $(0,1)$.

$\bullet$ If $J$ is a component of $I-e_{F}(\Psi_{F})$ that has $[a,b)$-type,
then $\mu_{F}\mid_{J}$ is a homeomorphism that takes $J$ to $[0,1)$.

$\bullet$ If $J$ is a component of $I-e_{F}(\Psi_{F})$ that has $(a,b]$-type,
then $\mu_{F}\mid_{J}$ is a homeomorphism that takes $J$ to $(0,1]$.

######  Lemma 4.14.

Let s be an edge of the branch locus of $B$. Then there exists
$F,F_{1},F_{2}\in\textbf{F}$ such that $p(F),p(F_{1}),p(F_{2})$ are the branch
sectors glued along s in $B$, and the cusp direction at s points out of
$p(F_{1}),p(F_{2})$ and points in $p(F)$. Then there exist $h_{1},h_{2}\in G$
such that $F,t_{h_{1}}(F_{1}),t_{h_{2}}(F_{2})$ have a common edge in
$\widetilde{M}$ (and the common edge is a component of $p^{-1}(\textbf{s})$).
Notice that one of $t_{h_{1}}(F_{1}),t_{h_{2}}(F_{2})$ lies on
$\partial\Gamma$, then exactly one of $h_{1},h_{2}$ is $1$. We assume that
$h_{2}=1$ and $h_{1}\neq 1$. For our convenience, we will not do the
substitution $h_{2}=1$ in the following (we will deal with $i=1,2$ in a
similar way).

(a) For $i=1,2$, there is an injective order-preserving map (with respect to
the order “$\prec$”)

$d_{(\textbf{s},i)}:e_{F_{i}}(\Psi_{F_{i}})\to e_{F}(\Psi_{F})$

that takes every $e_{F_{i}}(l_{C})$ ($l_{C}\in\Psi_{F_{i}}$) to
$e_{F}(l_{h_{i}C})$. Moreover,

$e_{F}(\Psi_{F})=d_{(\textbf{s},1)}(e_{F_{1}}(\Psi_{F_{1}}))\cup
d_{(\textbf{s},2)}(e_{F_{2}}(\Psi_{F_{2}})).$

(b) For $i=1,2$, the map $d_{(\textbf{s},i)}:e_{F_{i}}(\Psi_{F_{i}})\to
e_{F}(\Psi_{F})$ can extend to an embedding $\delta_{(\textbf{s},i)}:I\to I$
such that for each $t\in I-e_{F_{i}}(\Psi_{F_{i}})$,
$\mu_{F_{i}}(t)=\mu_{F}(\delta_{(\textbf{s},i)}(t))$.

###### Proof.

If the leaves $l_{C_{1}},l_{C_{2}}\in\mathcal{A}$ pass $F_{1}$ and
$l_{C_{1}}\prec l_{C_{2}}$, then

$l_{h_{1}C_{1}}=t_{h_{1}}(l_{C_{1}})\in
t_{h_{1}}(\Psi_{F_{1}})\subseteq\Psi_{F}$

and

$l_{h_{1}C_{2}}=t_{h_{1}}(l_{C_{2}})\in
t_{h_{1}}(\Psi_{F_{1}})\subseteq\Psi_{F}$

must pass $F$, and $l_{h_{1}C_{1}}\prec l_{h_{1}C_{2}}$. So the map
$d_{(\textbf{s},1)}:e_{F_{1}}(\Psi_{F_{1}})\to e_{F}(\Psi_{F})$ in (a) exists
naturally, and it is order-preserving. Samely, the map
$d_{(\textbf{s},2)}:e_{F_{2}}(\Psi_{F_{2}})\to e_{F}(\Psi_{F})$ exists, and it
is order-preserving. Moreover, for every $l_{C}\in\Psi_{F}$, $l_{C}$ either
passes $t_{h_{1}}(F_{1})$ or passes $t_{h_{2}}(F_{2})$. Hence
$e_{F}(\Psi_{F})=d_{(\textbf{s},1)}(e_{F_{1}}(\Psi_{F_{1}}))\cup
d_{(\textbf{s},2)}(e_{F_{2}}(\Psi_{F_{2}}))$.

We will prove (b) in the following. For our convenience, we only give the
proof in the case of $i=1$.

Let $J$ be a component of $I-e_{F_{1}}(\Psi_{F_{1}})$. We denote by $u=\min J$
and $v=\max J$. Then $J$ is one of $(u,v)$, $(u,v]$, $[u,v)$, and at least one
of $u,v$ is in $e_{F_{1}}(\Psi_{F_{1}})$ (see Lemma 4.7). Without loss of
generality, assume $u\in e_{F_{1}}(\Psi_{F_{1}})$. Then there exists
$C_{1}\in\mathcal{C}$ a downward gap place such that $u=e_{F_{1}}(C_{1})$.
Then $v=e^{+}_{F_{1}}(C_{1})$.

We denote by

$u^{{}^{\prime}}=e_{F}(l_{h_{1}C_{1}})=d_{(\textbf{s},1)}(u).$

Since $h_{1}C_{1}$ is a downward gap place (Lemma 4.6) and
$u^{{}^{\prime}}=d_{(\textbf{s},1)}(u)<1$ (because $u<1$), we denote by

$v^{{}^{\prime}}=e^{+}_{F}(l_{h_{1}C_{1}})>u^{{}^{\prime}}.$

We consider the following cases:

######  Case 1.

If $v\in e_{F_{1}}(\Psi_{F_{1}})$, then $C_{1}$ is an absolute downward gap
place, and so is $h_{1}C_{1}$ (by Lemma 4.9). Hence $v^{{}^{\prime}}\in
e_{F}(\Psi_{F})$, and $(u^{{}^{\prime}},v^{{}^{\prime}})$ is a component of
$I-e_{F}(\Psi_{F})$. Let $J^{{}^{\prime}}=(u^{{}^{\prime}},v^{{}^{\prime}})$.
We define

$\delta_{(\textbf{s},1)}\mid_{(u,v)}=(\mu_{F}\mid_{J^{{}^{\prime}}})^{-1}\circ(\mu_{F_{1}}\mid_{J}).$

Then $\mu_{F_{1}}(t)=\mu_{F}(\delta_{(\textbf{s},1)}(t))$ for each $t\in J$.

######  Case 2.

If $v\notin e_{F_{1}}(\Psi_{F_{1}})$, then $C_{1}$ is a relative downward gap
place, and so is $h_{1}C_{1}$ (by Lemma 4.9). Hence $v^{{}^{\prime}}\notin
e_{F}(\Psi_{F})$, and $(u^{{}^{\prime}},v^{{}^{\prime}}]$ is a component of
$I-e_{F}(\Psi_{F})$. Let $J^{{}^{\prime}}=(u^{{}^{\prime}},v^{{}^{\prime}}]$.
We define

$\delta_{(\textbf{s},1)}\mid_{(u,v]}=(\mu_{F}\mid_{J^{{}^{\prime}}})^{-1}\circ(\mu_{F_{1}}\mid_{J}).$

Then $\mu_{F_{1}}(t)=\mu_{F}(\delta_{(\textbf{s},1)}(t))$ for each $t\in J$.

Hence $d_{(\textbf{s},1)}$ can extend to an injective order-preserving map
$\delta_{(\textbf{s},1)}:I\to I$. By Lemma 4.9 and Lemma 4.12, the components
of $I-e_{F_{1}}(\Psi_{F_{1}})$ and the components of
$[d_{(\textbf{s},1)}(0),d_{(\textbf{s},1)}(1)]-e_{F}(\Psi_{F})$ are in one-to-
one correspondence under the map $\delta_{(\textbf{s},1)}$. So
$\delta_{(\textbf{s},1)}$ is an embedding. And

$\mu_{F_{1}}(t)=\mu_{F}(\delta_{(\textbf{s},1)}(t))$

for each $t\in I-e_{F_{1}}(\Psi_{F_{1}})$, since this holds for every $t$ in
every component of $I-e_{F_{1}}(\Psi_{F_{1}})$ (see Case 1, Case 2). ∎

Lemma 4.14 gives the relations at edges of the branch locus of $B$. Such
relations also can be given at the double points of the branch locus of $B$
(although there are more than $3$ branch sectors at a double point of the
branch locus). We will generalize Lemma 4.14 as follows:

######  Remark 4.15.

(a) We can generalize Lemma 4.14 in the following way:

Let $q$ be a double point of $B$. Suppose that $F_{1},F_{2}\in\textbf{F}$,
$h\in G$ such that there exists $\widetilde{q}\in p^{-1}(q)$,
$\widetilde{q}\in t_{h}(F_{1})\cap F_{2}$. Then $q\in p(F_{1})\cap p(F_{2})$.
We denote by

$\Psi=\\{l_{C}\mid C\in\mathcal{C},t_{h}(F_{1})\subseteq l_{C},F_{2}\subseteq
l_{C}\\}$.

Assume that $\Psi\neq\emptyset$. Similar to Lemma 4.14 (a), there exists an
order-preserving bijection

$d_{(t_{h}(F_{1}),F_{2})}:e_{F_{1}}(t^{-1}_{h}(\Psi))\to e_{F_{2}}(\Psi)$

that takes every $e_{F_{1}}(l_{C})$ ($l_{C}\in
t^{-1}_{h}(\Psi)\subseteq\Psi_{F_{1}}$) to $e_{F_{2}}(l_{hC})$. Similar to
Lemma 4.14 (b), $d_{(t_{h}(F_{1}),F_{2})}$ can extend to a homeomorphism

$\delta_{(t_{h}(F_{1}),F_{2})}:[\inf e_{F_{1}}(t^{-1}_{h}(\Psi)),\sup
e_{F_{1}}(t^{-1}_{h}(\Psi))]\to[\inf e_{F_{2}}(\Psi),\sup e_{F_{2}}(\Psi)]$

such that for each $t\in[\inf e_{F_{1}}(t^{-1}_{h}(\Psi)),\sup
e_{F_{1}}(t^{-1}_{h}(\Psi))]-e_{F_{1}}(\Psi)$,
$\mu_{F_{1}}(t)=\mu_{F_{2}}(\delta_{(t_{h}(F_{1}),F_{2})}(t))$.

(b) Let $q$ be a double point of $B$. Suppose that
$F_{1},F_{2},F_{3}\in\textbf{F}$, $h_{1},h_{2}\in G$ such that there exists
$\widetilde{q}\in p^{-1}(q)$, $\widetilde{q}\in t_{h_{1}}(F_{1})\cap
t_{h_{2}}(F_{2})\cap F_{3}$. Then $q\in p(F_{1})\cap p(F_{2})\cap p(F_{3})$.
We denote by

$\Psi=\\{l_{C}\mid C\in\mathcal{C}$ and
$t_{h_{1}}(F_{1}),t_{h_{2}}(F_{2}),F_{3}\subseteq l_{C}\\}$.

Assume that $\Psi\neq\emptyset$. Then

$d_{(t_{h_{1}}(F_{1}),F_{3})}\mid_{e_{F_{1}}(t^{-1}_{h_{1}}(\Psi))}=(d_{(t_{h_{2}}(F_{2}),F_{3})}\mid_{e_{F_{2}}(t^{-1}_{h_{2}}(\Psi))})\circ(d_{(t_{h^{-1}_{2}h_{1}}(F_{1}),F_{2})}\mid_{e_{F_{1}}(t^{-1}_{h_{1}}(\Psi))}).$

And we denote by $J_{i}=[\inf e_{F_{i}}(t^{-1}_{h_{i}}(\Psi)),\sup
e_{F_{i}}(t^{-1}_{h_{i}}(\Psi))]$ ($i=1,2$), then

$\delta_{(t_{h_{1}}(F_{1}),F_{3})}\mid_{J_{1}}=(\delta_{(t_{h_{2}}(F_{2}),F_{3})}\mid_{J_{2}})\circ(\delta_{(t_{h^{-1}_{2}h_{1}}(F_{1}),F_{2})}\mid_{J_{1}}).$

In Remark 4.15, we only consider the case that branch sectors intersect at a
double point. Actually, for arbitrary $F_{1},F_{2}\in\textbf{F}$ and $h\in G$
such that $\\{l_{C}\mid C\in\mathcal{C},t_{h}(F_{1})\subseteq
l_{C},F_{2}\subseteq l_{C}\\}\neq\emptyset$, the maps
$d_{(t_{h}(F_{1}),F_{2})},\delta_{(t_{h}(F_{1}),F_{2})}$ can be defined as
Remark 4.15 (a), and we have similar arguments as Remark 4.15 (b).

We have completed the work on “numbering” the leaves of $\mathcal{A}$ that
pass every face $F\in\textbf{F}$. In our discussions, we always assume that
$|\Psi_{F}|>1$ for each $F\in\textbf{F}$. We explain the case of
$|\Psi_{F}|=1$ for some $F\in\textbf{F}$ in the following.

######  Remark 4.16.

Assume that there exists $F\in\textbf{F}$ such that $\Psi_{F}=\\{l_{C}\\}$ for
a $C\in\mathcal{C}$. We can replace $\\{l_{gC}\\}_{g\in
G}\subseteq\mathcal{A}$ by $\\{l^{+}_{gC}\\}_{g\in
G}\cup\\{l^{-}_{gC}\\}_{g\in G}$ in $\mathcal{A}$ such that
$l^{+}_{gC},l^{-}_{gC},l_{gC}$ represent the same topological space in
$\widetilde{M}$, but $l_{C_{1}}\prec l^{-}_{gC}\prec l^{+}_{gC}\prec
l_{C_{2}}$ for every $g\in G$ and $C_{1}\varsubsetneqq gC\varsubsetneqq
C_{2}$. Then $\Psi_{F}=\\{l^{-}_{C},l^{+}_{C}\\}$, and $e_{F}:\Psi_{F}\to I$
takes $l^{-}_{C},l^{+}_{C}$ to $0,1$ respectively. For every $g\in G$ and
$F^{{}^{\prime}}\in\textbf{F}$ such that $l_{gC}\in\Psi_{F^{{}^{\prime}}}$,
$l_{gC}$ is replaced by $l^{+}_{gC},l^{-}_{gC}$, and $e_{F^{{}^{\prime}}}$
takes them into two different points in $I$.

In the following, we will always state with assuming $|\Psi_{F}|>1$ for all
$F\in\textbf{F}$. The case of $|\Psi_{F}|=1$ for some $F\in\textbf{F}$ can be
dealt with the above process and be transformed to the case of $|\Psi_{F}|>1$.

Next, we will construct the lamination $\Lambda$ carried by $B$ in $M$.

###  4.4. Construction of the leaves

Figure 8. With the conditions of Lemma 4.14, for $i=1,2$, the pieces of
$\coprod_{t\in I}p(F_{i})\times\\{t\\}$ are glued to pieces in $\coprod_{t\in
I}p(F)\times\\{t\\}$ along their boundaries. The gluing is given by the
embedding $\delta_{(\textbf{s},i)}:I\to I$. Especially, $p(F_{i})\times
e_{F_{i}}(\Psi_{F_{i}})$ is glued to $p(F)\times e_{F}(\Psi_{F})$, induced by
the map $d_{(\textbf{s},i)}:e_{F_{i}}(\Psi_{F_{i}})\to e_{F}(\Psi_{F})$.

Recall that $\\{p(F)\\}_{F\in\textbf{F}}$ is the set of branch sectors of $B$.
We can decompose $N(B)$ along the union of interval fibers at the branch locus
of $B$, and obtain a decomposition

$N(B)=\bigcup_{F\in\textbf{F}}(p(F)\times I).$

Let

$\mathcal{E}=\coprod_{F\in\textbf{F},t\in I}(p(F)\times\\{t\\}).$

At first, we will construct leaves by gluing the pieces in $\mathcal{E}$
through their boundaries. Note that this process requests the boundary points
of the pieces in $\mathcal{E}$ be glued horizontally and only produce
Euclidean neighborhoods.

We glue the pieces in $\mathcal{E}$ as follows:

$\bullet$ Let s be an edge of the branch locus of $B$. We denote by
$F,F_{1},F_{2}\in\textbf{F}$ the faces given by the conditions of Lemma 4.14
(then $p(F),p(F_{1}),p(F_{2})$ are the branch sectors glued along s in $B$,
the cusp direction at s points out of $p(F_{1}),p(F_{2})$ and points in
$p(F)$, and there exists $h_{1}\in G-\\{1\\}$ such that
$F,t_{h_{1}}(F_{1}),F_{2}$ has a common edge). Recall from Lemma 4.14, we have
defined the maps $\delta_{(\textbf{s},i)}:I\to I$ ($i=1,2$). For $i=1,2$ and
each $t\in I$, we homeomorphically glue

$\textbf{s}\times\\{t\\}\subseteq p(F_{i})\times\\{t\\}$

with

$\textbf{s}\times\\{\delta_{(\textbf{s},i)}(t)\\}\subseteq
p(F)\times\\{\delta_{(\textbf{s},i)}(t)\\}$

(for arbitrary $r\in\textbf{s}$, $(r,t)\in p(F_{i})\times\\{t\\}$ is
identified with $(r,\delta_{(\textbf{s},i)}(t))\in
p(F)\times\\{\delta_{(\textbf{s},i)}(t)\\}$).

Notice that $\textbf{s}\times I\subseteq p(F_{i})\times I$ is locally
homeomorphically glued into $\textbf{s}\times I\subseteq p(F)\times I$ (by the
relation
$\textbf{s}\times\\{t\\}\sim\textbf{s}\times\\{\delta_{(\textbf{s},i)}(t)\\}$).
Moreover, by Remark 4.15, we can verify that in the interval fibers at the
double points of the branch locus of $B$, the pieces of $\mathcal{E}$ are
glued horizontally and only produce Euclidean neighborhoods in the
identification space. Hence the identification space forms a division of
$N(B)$, and each component of the identification space is a $2$-manifold
(which may have nonempty boundary that lies on $\partial_{v}N(B)$).

Let $M_{0}=N(B)\cong M-Int(B^{3})$. Then the components of the identification
space form a foliation of $M_{0}$. We denote by $\mathcal{F}$ this foliation.
Notice that $\mathcal{F}$ may be tangent to $\partial M_{0}$ or transverse to
$\partial M_{0}$ at its intersections with $\partial M_{0}$ (the intersections
of $\mathcal{F}$ and $\partial M_{0}$ lie on $\partial_{v}N(B)$).
$\mathcal{F}$ is our intentional foliation in Theorem 1.2. We still need to
show some information about $\mathcal{F}$ in the following.

From now on, when we mention a piece $p(F)\times\\{t\\}$ ($F\in\textbf{F}$,
$t\in I$), $p(F)\times\\{t\\}$ is assumed to lie on the identification space
of the gluing. Then it is contained in a leaf of $\mathcal{F}$.

######  Definition 4.17.

Let

$\Lambda_{0}=\bigcup_{F\in\textbf{F}}(p(F)\times e_{F}(\Psi_{F})).$

######  Lemma 4.18.

$\Lambda_{0}$ is a union of leaves in $\mathcal{F}$.

###### Proof.

Suppose that $\lambda$ is a leaf of $\mathcal{F}$ such that
$\lambda\cap\Lambda_{0}\neq\emptyset$. Then there exists
$F\in\textbf{F},l_{C}\in\Psi_{F}$ such that
$p(F)\times\\{e_{F}(l_{C})\\}\subseteq\lambda$. By Lemma 4.14 (a), if
$p(F^{{}^{\prime}})\times\\{t\\}\subseteq\lambda$
($F^{{}^{\prime}}\in\textbf{F}$, $t\in I$), then there exists $h\in G$ such
that $l_{hC}\in\Psi_{F^{{}^{\prime}}}$ and $t=e_{F^{{}^{\prime}}}(l_{hC})$. So
$p(F^{{}^{\prime}})\times\\{t\\}\subseteq\Lambda_{0}$.

This implies $\lambda\subseteq\Lambda_{0}$ for each leaf $\lambda$ of
$\mathcal{F}$ such that $\lambda\cap\Lambda_{0}\neq\emptyset$. Therefore,
$\Lambda_{0}$ is a union of leaves in $\mathcal{F}$. ∎

######  Remark 4.19.

$\Lambda_{0}$ may not be a lamination, since the leaves in $\Lambda_{0}$ may
not compose a closed subspace of $M$.

In $\widetilde{M}$, we have the decomposition

$\widetilde{N(B)}=\bigcup_{F\in\textbf{F},h\in G}(t_{h}(F)\times I)$

given by the pull-back. Then $p(t_{h}(F)\times\\{t\\})=p(F)\times\\{t\\}$
($\forall F\in\textbf{F},h\in G,t\in I$). We denote by
$\widetilde{\Lambda_{0}}=p^{-1}(\Lambda_{0})$ the pull-back of $\Lambda_{0}$
in $\widetilde{M}$.

######  Definition 4.20.

For each $C\in\mathcal{C}$, we denote by

$\widetilde{\lambda_{C}}=\bigcup_{F\in\textbf{F},h\in
G,l_{hC}\in\Psi_{F}}(t^{-1}_{h}(F)\times\\{e_{F}(l_{hC})\\}).$

######  Lemma 4.21.

(a) For each $C\in\mathcal{C}$, $\widetilde{\lambda_{C}}$ is a union of leaves
of $\widetilde{\Lambda_{0}}$, and
$\widetilde{\pi}(\widetilde{\lambda_{C}})=l_{C}$.

(b) $\\{\widetilde{\lambda_{C}}\\}_{C\in\mathcal{C}}$ is a division of
$\widetilde{\Lambda_{0}}$.

###### Proof.

We only prove Lemma 4.21 (a).

Let $C\in\mathcal{C}$. Suppose that $\widetilde{\lambda}$ is a leaf of
$\widetilde{\Lambda_{0}}$ such that
$\widetilde{\lambda}\cap\widetilde{\lambda_{C}}\neq\emptyset$. Then there
exists

$t^{-1}_{h}(F)\times\\{e_{F}(l_{hC})\\}\subseteq\widetilde{\lambda}$

for some $F\in\textbf{F}$ and $h\in G$ with $l_{hC}\in\Psi_{F}$. Then
$t^{-1}_{h}(F)\subseteq l_{C}$ (since $F\subseteq l_{hC}=t_{h}(l_{C})$). If
$t^{-1}_{g}(F^{{}^{\prime}})\times\\{t\\}\subseteq\widetilde{\lambda}$ for
some $F^{{}^{\prime}}\in\textbf{F},g\in G,t\in I$, then
$t^{-1}_{g}(F^{{}^{\prime}})\subseteq l_{C}$ (which implies
$l_{gC}\in\Psi_{F^{{}^{\prime}}}$) and $t=e_{F^{{}^{\prime}}}(l_{gC})$. Hence

$t^{-1}_{g}(F^{{}^{\prime}})\times\\{t\\}\subseteq\widetilde{\lambda_{C}}.$

And we can ensure that $\widetilde{\lambda}\subseteq\widetilde{\lambda_{C}}$.
Therefore, $\widetilde{\lambda_{C}}$ is a union of leaves of
$\widetilde{\Lambda_{0}}$.

$\widetilde{\pi}(\widetilde{\lambda_{C}})=l_{C}$ since for each
$F\in\textbf{F},h\in G$ such that $t^{-1}_{h}(F)\subseteq l_{C}$,

$\widetilde{\lambda_{C}}\cap(t^{-1}_{h}(F)\times
I)=t^{-1}_{h}(F)\times\\{e_{F}(l_{hC})\\},$

which implies
$t^{-1}_{h}(F)\subseteq\widetilde{\pi}(\widetilde{\lambda_{C}})$. ∎

Hence $\widetilde{\pi}$ induces a bijection

$\\{\widetilde{\lambda}\mid\widetilde{\lambda}$ is a component of
$\widetilde{\Lambda_{0}}\\}\to\\{l\mid l$ is a component of a leaf of
$\mathcal{A}\\}$,

which maps the components of $\widetilde{\lambda_{C}}$ to the components of
$l_{C}$ (for each $C\in\mathcal{C}$).

######  Lemma 4.22.

Suppose that $X$ is a component of
$N(B)-\bigcup_{\lambda\in\Lambda_{0}}\lambda$. Then:

(a) $\mathcal{F}\mid_{X}$ is a surface bundle over $(0,1)$ or $(0,1]$ or
$[0,1)$.

(b) Suppose that $\mathcal{F}\mid_{X}$ is a surface bundle over $(0,1)$ (resp.
$(0,1]$, $[0,1)$). For each $q\in(0,1)$ (resp. $[0,1),(0,1]$),
$X\cap(\bigcup_{F\in\textbf{F},t\in\mu^{-1}_{F}(q)}(p(F)\times\\{t\\}))$ is a
leaf in $X$.

###### Proof.

Recall from Lemma 4.14, for $i=1,2$, the map $\delta_{(\textbf{s},i)}:I\to I$
homeomorphically takes each component of $I-e_{F_{i}}(\Psi_{F_{i}})$ (which is
an interval) to a component of $I-e_{F}(\Psi_{F})$. For the component $X$ of
$N(B)-\bigcup_{\lambda\in\Lambda_{0}}\lambda$, the intersection of $X$ and
interval fibers of $N(B)$ are pairwise homeomorphic intervals (may have the
form in one of $(a,b),[a,b),(a,b]$). Assume without loss of generality, these
intervals are homeomorphic to $(a,b)$.

By Lemma 4.14 (b), all pieces $p(F)\times\\{t\\}$ ($F\in\textbf{F}$, $t\in
I-e_{F}(\Psi_{F})$) contained in $X$ that satisfy $\mu_{F}(t)=q$ ($q\in(0,1)$)
are glued into a leaf. Therefore, we can confirm Lemma 4.22 (a) and (b). ∎

Next, we will define a projection $\mu:\mathcal{F}-\Lambda_{0}\to I$ (which
maps $\mathcal{F}\mid_{X}$ to $[0,1]$ or $(0,1]$ or $[0,1)$, for each
component $X$ of $N(B)-\bigcup_{\lambda\in\Lambda_{0}}\lambda$).

######  Definition 4.23.

Let

$\mu:\mathcal{F}-\Lambda_{0}\to I$

be the map such that: for every leaf $\lambda\in\mathcal{F}-\Lambda_{0}$, we
choose $p(F)\times\\{t\\}\subseteq\lambda$ ($F\in\textbf{F}$, $t\in
I-e_{F}(\Psi_{F})$), and we define $\mu(\lambda)=\mu_{F}(t)$.

By Lemma 4.22 (b), $\mu$ is well-defined.

In the next subsection, we will choose a lamination
$\Lambda\subseteq\mathcal{F}$ (composed of all leaves that are not transverse
to $\partial M_{0}=\partial N(B)$) to accomplish Proposition 4.1.

###  4.5. The lamination $\Lambda$

The complementary region of a lamination can decompose to _guts_ and
_interstitial regions_ , along a union of properly embedded annuli. Each
interstitial region is an $I$-bundle over a (typically non-compact) surface,
and each gut is compact. A comprehensive introduction can be found in [CD,
Section 2.5].

Recall that $M_{0}=N(B)\cong M-Int(B^{3})$. We denote by $\Lambda$ the
collection of leaves of $\mathcal{F}$ that are not transverse to $\partial
M_{0}$. Then $\Lambda$ is a lamination and $\Lambda_{0}\subseteq\Lambda$,
which implies that $\Lambda$ is fully carried by $B$. Obviously, $\Lambda$ has
a single gut region $M-Int(N(B))\cong B^{3}$. Then $\mathcal{F}$ can be viewed
as extending $\Lambda$ by filling the interstitial regions (which is
$M_{0}\setminus\setminus\Lambda$) with surface bundles over intervals.

We can ensure that $\Lambda$ has no torus leaf that bounds a solid torus,
since $B$ does not separate $M$.

Let $\widetilde{\Lambda_{0}}=p^{-1}(\Lambda_{0})$ be the pull-back of
$\Lambda_{0}$ in $\widetilde{M}$. By Lemma 4.21, $\widetilde{\pi}$ maps each
leaf of $\widetilde{\Lambda_{0}}$ to a component of a leaf of $\mathcal{A}$.
If $\lambda$ is a leaf of $\Lambda-\Lambda_{0}$, then $\lambda$ is not
transverse to $\partial M_{0}$, and by Lemma 4.22, the component of
$N(B)-\Lambda_{0}$ that contains $\lambda$ is foliated by a surface bundle
over an interval that has the form in one of $(a,b)$, $[a,b)$, $(a,b]$. Then
there exists $\lambda_{0}$ a leaf of $\Lambda_{0}$ and $S\times
I\subseteq\Lambda$ a surface bundle over $I$ such that
$\\{\lambda,\lambda_{0}\\}=\\{S\times\\{0\\},S\times\\{1\\}\\}$. For each
component $\widetilde{\lambda}$ of $p^{-1}(\lambda)$, there exists a component
$\widetilde{\lambda_{0}}$ of $p^{-1}(\lambda_{0})$ such that
$\widetilde{\pi}(\widetilde{\lambda})=\widetilde{\pi}(\widetilde{\lambda_{0}})$.
Therefore, $\widetilde{\pi}$ takes all leaves of $\widetilde{\Lambda}$ to the
components of the leaves of $\mathcal{A}$.

Let $\Upsilon$ be the complementary region of the lamination $\Lambda$ in $M$.
Then $\Upsilon$ is connected, since $M-Int(N(B))\subseteq\Upsilon$ and each
component of $\Upsilon\cap N(B)$ has nonempty intersection with $M-Int(N(B))$
(they intersect at some components of $\partial_{v}N(B)$). Choose

$z\in\Gamma-\widetilde{N(B)}\subseteq p^{-1}(\Upsilon).$

Let $\widetilde{\Upsilon}$ be the component of $p^{-1}(\Upsilon)$ such that
$z\in\widetilde{\Upsilon}$. In the following, we will explain that $\Upsilon$
is simply connected by showing $\widetilde{\Upsilon}$ has trivial stabilizer
under deck transformations in $\widetilde{M}$:

$\bullet$ Let $H=\\{h\in G\mid h<1\\}\in\mathcal{C}$. Choose $g\in G$ such
that $g<1$. Then $g\in H$ but $1\notin H$. So $l_{H}$ separates $z$ and
$t_{g}(z)$. Recall from Definition 4.20, there exists
$\widetilde{\lambda_{H}}\subseteq p^{-1}(\Lambda_{0})$ such that
$\widetilde{\pi}(\widetilde{\lambda_{H}})=l_{H}$. So $\widetilde{\lambda_{H}}$
separates $z$ and $t_{g}(z)$. This implies
$t_{g}(\widetilde{\Upsilon})\neq\widetilde{\Upsilon}$. Similarly, for each
$g\in G$ such that $g>1$,
$t_{g}(\widetilde{\Upsilon})\neq\widetilde{\Upsilon}$.

Therefore, $\Upsilon$ is simply connected. Moreover, since $H=\\{h\in G\mid
h<1\\}$ has trivial stabilizer under the left action of $G$,
$\widetilde{\lambda_{H}}\subseteq\widetilde{\Lambda}$ has trivial stabilizer
under deck transformations. It only remains to choose a simple closed curve
that is co-orientably transverse to $\Lambda$ and intersects all leaves of
$\Lambda$.

We can choose a path $\eta:I\to\widetilde{M}$ such that
$p(\eta(0))=p(\eta(1))$, and $\eta$ meets all branch sectors of
$\widetilde{B}$ nontrivially and co-orientably. We can isotope $\eta(I)$ so
that $\eta(I)$ is co-orientably transverse to $\widetilde{\Lambda}$. Then
$p(\eta(I))$ is co-orientably transverse to $\Lambda$, and it intersects all
leaves of $\Lambda$.

We have completed the proof of Propostion 4.1. Then Theorem 1.2 (1) can be
deduced directly. Next, we will prove Theorem 1.2 (2).

###  4.6. The transverse $(\pi_{1}(M_{0}),\mathbb{R})$ structure of
$\mathcal{F}$

Recall from Subsection 4.4, $M_{0}=N(B)\cong M-Int(B^{3})$ and $\mathcal{F}$
is a foliation in $M_{0}$. We will show that $\mathcal{F}$ has a transverse
$(\pi_{1}(M_{0}),\mathbb{R})$ structure in the following.

Let $p_{0}:\widetilde{M_{0}}\to M_{0}$ be the universal cover of $M_{0}$. Let
$\widetilde{\mathcal{F}}$ be the pull-back foliation of $\mathcal{F}$ in
$\widetilde{M_{0}}$. Let $\widetilde{\Lambda}$ be the pull-back lamination of
$\Lambda$ in $\widetilde{M_{0}}$. Throught this subsection, we view $\Lambda$
as a subset of $\mathcal{F}$, view $\widetilde{\Lambda}$ as a subset of
$\widetilde{\mathcal{F}}$, and view $\widetilde{M_{0}}$ as a subspace of
$\widetilde{M}$.

We denote by $L$ the leaf space of $\widetilde{\mathcal{F}}$. Then $L$ is a
simply-connected $1$-manifold that may not be Hausdorff. For our convenience,
we will not distinguish the leaves of $\widetilde{\mathcal{F}}$ and the points
in $L$. Then $\widetilde{\mathcal{F}}$ can be thought as the set $L$
regardless of its topology. For $\widetilde{\mathcal{F}_{0}}$ a subset of
$\widetilde{\mathcal{F}}$, we use the notation
$L(\widetilde{\mathcal{F}_{0}})$ to denote the subset of $L$ that is exactly
composed of every leaf of $\widetilde{\mathcal{F}_{0}}$.

Recall that $\Lambda_{0}\subseteq\Lambda$ is a collection of leaves
(Definition 4.17 and Lemma 4.18). Let $\widetilde{\Lambda_{0}}$ denote the
pull-back of $\Lambda_{0}$ in $\widetilde{M}$. Recall from Definition 4.20,
for each $C\in\mathcal{C}$, $\widetilde{\lambda_{C}}$ is a union of leaves in
$\widetilde{\Lambda_{0}}$ such that
$\widetilde{\pi}(\widetilde{\lambda_{C}})=l_{C}$, and
$\\{\widetilde{\lambda_{C}}\\}_{C\in\mathcal{C}}$ is a division of
$\widetilde{\Lambda_{0}}$.

######  Step 1.

We will define an immersion

$\widetilde{\mu}:L-L(\widetilde{\Lambda_{0}})\to I,$

which is $G$-equivariant under the $\pi_{1}$-action on $L$.

Recall from Definition 4.23, we have defined a map

$\mu:\mathcal{F}-\Lambda_{0}\to I.$

Let $\widetilde{\mu}:L-L(\widetilde{\Lambda_{0}})\to I$ be the map such that
for each $q\in L-L(\widetilde{\Lambda_{0}})$,

$\widetilde{\mu}(q)=\mu(p_{0}(q)).$

Then $\widetilde{\mu}$ is $\pi_{1}$-equivariant, since the value of
$\widetilde{\mu}(q)$ ($q\in L-L(\widetilde{\Lambda_{0}})$) deponds only on the
leaf $p_{0}(q)$ in $\mathcal{F}-\Lambda_{0}$.

Notice that for each component $V$ of $L-L(\widetilde{\Lambda_{0}})$,
$\widetilde{\mu}$ takes $V$ homeomorphically to one of $(0,1)$, $[0,1)$,
$(0,1]$. Then $\overline{V}$ is a closed interval. We denote by
$\min\overline{V}$, $\max\overline{V}$ the $2$ endpoints of $\overline{V}$
such that there exists a positively oriented transversal that starts at
$\min\overline{V}$ and ends at $\max\overline{V}$ (if $\min\overline{V}\in V$,
then $\widetilde{\mu}(\min\overline{V})=0$, and if $\max\overline{V}\in V$,
then $\widetilde{\mu}(\max\overline{V})=1$).

######  Step 2.

We will define an immersion

$i:L\to\mathbb{R},$

which is $G$-equivariant under the $\pi_{1}$-action on $L$.

Recall from Subsection 4.2, we have fixed an injective map

$e:\mathcal{C}\to\mathbb{R},$

which is order-preserving with respect to the inclusion order
“$\varsubsetneqq$” of $\mathcal{C}$ and the increasing order of $\mathbb{R}$,
and $\mathbb{R}-e(\mathcal{C})$ has $G$-equivariant gaps (see Lemma 4.6).
Notice that the map $e$ is unbounded (see Lemme 4.4).

For each leaf $q\in L(\widetilde{\Lambda_{0}})$, we denote by
$C_{q}\in\mathcal{C}$ such that $q$ is a component of
$\widetilde{\lambda_{C_{q}}}$. Let

$i_{0}:L(\widetilde{\Lambda_{0}})\to\mathbb{R}$

be the map such that $i_{0}(q)=e(C_{q})$ for each $q\in
L(\widetilde{\Lambda_{0}})$. Notice that $i_{0}$ is $G$-equivariant under the
$\pi_{1}$-action on $L(\widetilde{\Lambda_{0}})$: for every $q_{1},q_{2}\in
L(\widetilde{\Lambda_{0}})$, $i_{0}(q_{1})=i_{0}(q_{2})$ if and only if
$C_{q_{1}}=C_{q_{2}}$.

Let

$\nu:\mathbb{R}-e(\mathcal{C})\to I$

be a map such that: if $J$ is a component of $\mathbb{R}-e(\mathcal{C})$ that
has the form of $(a,b)$ (resp. $[a,b)$, $(a,b]$), then $\nu$ maps $J$
homeomorphically to $(0,1)$ (resp. $[0,1)$, $(0,1]$). Next, we show the way to
extend $i_{0}$ to an immersion $i:L\to\mathbb{R}$. We consider the following
$3$ cases:

######  Case 1.

Suppose that $V$ is a component of $L-L(\widetilde{\Lambda_{0}})$ such that
$\widetilde{\mu}(V)=(0,1)$. Then $\min\overline{V},\max\overline{V}\in
L(\widetilde{\Lambda_{0}})$. There exists $C_{1},C_{2}\in\mathcal{C}$ such
that $\min\overline{V}$ is a component of $\widetilde{\lambda_{C_{1}}}$ and
$\max\overline{V}$ is a component of $\widetilde{\lambda_{C_{2}}}$. Then
$C_{1}$ is an absolute downward gap place of $\mathcal{C}$, $C_{2}$ is an
absolute upward gap place of $\mathcal{C}$, and $(e(C_{1}),e(C_{2}))$ is a
component of $\mathbb{R}-e(\mathcal{C})$. We define $i_{V}:V\to\mathbb{R}$ to
be the map $(\nu\mid_{(e(C_{1}),e(C_{2}))})^{-1}\circ\widetilde{\mu}$.

######  Case 2.

Suppose that $V$ is a component of $L-L(\widetilde{\Lambda_{0}})$ such that
$\widetilde{\mu}(V)=[0,1)$. Then $\max\overline{V}\in
L(\widetilde{\Lambda_{0}})$, but $\min\overline{V}\notin
L(\widetilde{\Lambda_{0}})$. There exists $C_{0}\in\mathcal{C}$ such that
$\max\overline{V}$ is a component of $\widetilde{\lambda_{C_{0}}}$, and
$C_{0}$ is a relative upward gap place of $\mathcal{C}$. Then
$[e^{-}(C_{0}),e(C_{0}))$ is a component of $\mathbb{R}-e(\mathcal{C})$
(recall from Definition 4.5, $e^{-}(C_{0})=\sup\\{e(C)\mid
C\in\mathcal{C},C\varsubsetneqq C_{0}\\}$). We define $i_{V}:V\to\mathbb{R}$
to be the map $(\nu\mid_{[e^{-}(C_{0}),e(C_{0}))})^{-1}\circ\widetilde{\mu}$.

######  Case 3.

Suppose that $V$ is a component of $L-L(\widetilde{\Lambda_{0}})$ such that
$\widetilde{\mu}(V)=(0,1]$. Then $\min\overline{V}\in
L(\widetilde{\Lambda_{0}})$, but $\max\overline{V}\notin
L(\widetilde{\Lambda_{0}})$. There exists $C_{0}\in\mathcal{C}$ such that
$\min\overline{V}$ is a component of $\widetilde{\lambda_{C_{0}}}$, and
$C_{0}$ is a relative downward gap place of $\mathcal{C}$. Then
$(e(C_{0}),e^{+}(C_{0})]$ is a component of $\mathbb{R}-e(\mathcal{C})$
(recall from Definition 4.5, $e^{+}(C_{0})=\inf\\{e(C)\mid
C\in\mathcal{C},C_{0}\varsubsetneqq C\\}$). We define $i_{V}:V\to\mathbb{R}$
to be the map $(\nu\mid_{(e(C_{0}),e^{+}(C_{0})]})^{-1}\circ\widetilde{\mu}$.

Let $i:L\to\mathbb{R}$ be the map such that:
$i\mid_{L(\widetilde{\Lambda})}=i_{0}$, and for each component $V$ of
$L-L(\widetilde{\Lambda})$, $i\mid_{V}=i_{V}$ (see Case 1, Case 2, and Case
3). Then $i$ is an immersion. Moreover, $i$ is a surjective map since $i_{0}$
is unbounded.

Next, we will show that $i:L\to\mathbb{R}$ is $G$-equivariant under the
$\pi_{1}$-action on $L$. For our convenience, we still use the notation
$\\{t_{h}\\}_{h\in G}$ for the $\pi_{1}$-action on $L$ (induced by deck
transformations). Recall that $i_{0}=i\mid_{L(\widetilde{\Lambda_{0}})}$ is
$\pi_{1}$-equivariant. In the following, we will show that
$i\mid_{L-L(\widetilde{\Lambda_{0}})}$ is $\pi_{1}$-equivariant.

Suppose that $q,s\in L-L(\widetilde{\Lambda_{0}})$ such that $i(q)=i(s)$. Then
$\widetilde{\mu}(q)=\widetilde{\mu}(s)$. We denote by $J$ the component of
$\mathbb{R}-i_{0}(L(\widetilde{\Lambda_{0}}))$ that contains $i(q)$. Then at
least one of $\sup J$ and $\inf J$ is in
$i_{0}(L(\widetilde{\Lambda_{0}}))=e(\mathcal{C})$. Without loss of
generality, we assume that $\sup J\in e(\mathcal{C})$. Then there exists
$C\in\mathcal{C}$ such that $e(C)=\sup J$. We denote by $V_{1},V_{2}$ the
components of $L-L(\widetilde{\Lambda_{0}})$ that contain $q,s$. Then $\max
V_{1},\max V_{2}$ are components of $\widetilde{\lambda_{C}}$.

Choose $h\in G$. Then $t_{h}(\max V_{1}),t_{h}(\max V_{2})$ are components of
$\widetilde{\lambda_{hC}}$. This implies that $t_{h}(q),t_{h}(s)$ are both lie
in a component $J_{0}$ of $\mathbb{R}-e(\mathcal{C})$ such that $e(hC)=\sup
J_{0}$. Since $\widetilde{\mu}(q)=\widetilde{\mu}(s)$ and $\widetilde{\mu}$ is
a $\pi_{1}$-equivariant map, we have
$\widetilde{\mu}(t_{h}(q))=\widetilde{\mu}(t_{h}(s))$. Therefore, $i$ is
$G$-equivariant under the $\pi_{1}$-action on $L$.

Let $H=\\{h\in G\mid h<1\\}$. Then $\widetilde{\lambda_{H}}$ has trivial
stabilizer under the deck transformations of $\widetilde{M_{0}}$. Therefore,
$i$ descends the $\pi_{1}$-action on $L$ to an effective action on
$\mathbb{R}$. We have completed the proof of Theorem 1.2 (2).

##  5\. Relations with taut foliations in closed $3$-manifolds

###  5.1. Extending conditions of $\mathcal{F}$

With the conditions of Section 3 and Section 4, we consider when does
$\mathcal{F}$ extend to a taut foliation in $M$:

######  Remark 5.1.

With the conditions in Section 4, $\mathcal{F}$ can extend to a taut foliation
in $M$ if either of the following pairwise equivalent conditions holds:

(1) The vertical boundary of $N(B)$ is connected (then $\partial_{v}N(B)$ is
homeomorphic to an annulus). Equivalently, both of the union of faces of
$\Gamma$ oriented positively with respect to $\Gamma$ and the union of faces
of $\Gamma$ oriented negatively with respect to $\Gamma$ are connected (then
each of them is homeomorphic to a disk).

(2) The foliation $\mathcal{F}$ satisfies the following generalization of the
Novikov’s Theorem ([N], [R]): for every embedded loop $\alpha$ in a leaf
$\lambda$ of $\mathcal{F}$ such that $\alpha$ is non-essential in $M_{0}$,
$\alpha$ either bounds a disk in $\lambda$ or co-bounds a surface with some
components of $\lambda\cap\partial M_{0}$.

Notice that if the above conditions are not satisfied, then $\Gamma-
Int(N(B))\cong B^{3}$ contains at least $2$ components of $\partial_{v}N(B)$,
which implies that the lamination $\Lambda$ is not essential.

###  5.2. Examples

(a)  (b)

Figure 9. (a) The pull-back train track $\widetilde{\tau}=p^{-1}(\tau)$ in the
universal cover of a torus $S$. The complementary region of $\tau$ in $S$ has
$2$ cusps, so $\tau$ fully carries a lamination that can extend to a foliation
in $S$. (b) In a closed orientable surface $S$ of genus $2$, the complementary
region of the train track $\tau$ has $6$ cusps. $\tau$ fully carries a
lamination, but it can not extend to a foliation in $S$. Figure 10. $M=S\times
I/\stackrel{{\scriptstyle id}}{{\sim}}$ is a mapping torus over a closed
orientable surface $S$ of genus $2$ with identity monodromy. We can get a
branched surface $B$ such that $\partial_{v}N(B)$ is connected (the picture is
a fundamental domain of $M$ in $\widetilde{M}$, which is homeomorphic to
$M\setminus\setminus B$, and the red curve is the cusp curve on it). Then $B$
fully carries a lamination that can extend to a taut foliation in $M$.

Notice that surface groups are left orderable. We can do the processes similar
to Section 3 and Section 4 for closed orientable surfaces.

######  Example 5.2.

Let $S$ be a closed orientable surface (of genus $\geqslant 1$). Let
$p:\widetilde{S}\to S$ be the universal cover of $S$. We can choose a
fundamental domain $\Gamma$ of $S$ such that $p(\partial\Gamma)$ is a
trivalent graph, and we can orient $p(\partial\Gamma)$ to be a train track
$\tau$ through a left-invariant order of $\pi_{1}(S)$, as given in Subsection
3.2. Then $\tau$ fully carries a lamination $\Lambda$ as given in Section 4.

(a) Let $S$ be a torus. Figure 9 (a) shows an example of the pull-back train
track $\widetilde{\tau}=p^{-1}(\tau)\subseteq\widetilde{S}$ for a choice of
the fundamental domain $\Gamma$ and the left-invariant order of
$\pi_{1}(S)=\mathbb{Z}\oplus\mathbb{Z}$. Then $S\setminus\setminus\tau$ has
$2$ cusps. So $\tau$ fully carries a lamination $\Lambda$ that can extend to a
foliation in $S$.

(b) Let $S$ be a closed orientable surface of genus $g>1$. We can also get a
train track $\tau$ that fully carries a lamination $\Lambda$, but
$S\setminus\setminus\tau$ has more than $2$ cusps. So $\Lambda$ can not extend
to a foliation similar to (a). On the other hand, we know that $S$ can not
admit a foliation since $g>1$.

######  Example 5.3.

Let $M=S\times I/\stackrel{{\scriptstyle id}}{{\sim}}$ be a mapping torus over
a closed orientable surface (of genus $\geqslant 1$) whose monodromy is
identity. Fix a left-invariant order $\stackrel{{\scriptstyle S}}{{<}}$ of
$\pi_{1}(S)$. Then $\pi_{1}(M)=\pi_{1}(S)\oplus\mathbb{Z}$ has a left-
invariant order “$\stackrel{{\scriptstyle M}}{{<}}$” such that: for
$(a,b),(c,d)\in\pi_{1}(S)\oplus\mathbb{Z}$, $(a,b)\stackrel{{\scriptstyle
M}}{{<}}(c,d)$ if either $b<d$ ($b<d$ under the increasing order of
$\mathbb{Z}$), or $a\stackrel{{\scriptstyle S}}{{<}}c$ and $b=d$. Follows
Example 5.2, our process can obtain a branched surface $B$ in $M$ such that
$\partial_{v}N(B)$ is connected (see Figure 10 for an example). By Remark 5.1
(a), $B$ fully carries a lamination $\Lambda$ that can extend to a taut
foliation $\mathcal{F}_{1}$ in $M$. Notice that $\mathcal{F}_{1}$ is different
from the foliation of horizontal surfaces in $M$, since it contains a planar
leaf (Remark 1.6 (b)).

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