# Pseudo-Anosov homeomorphisms of punctured non-orientable surfaces with small
stretch factor

Sayantan Khan  Department of Mathematics, University of Michigan, 530 Church
Street, Ann Arbor, MI 48109<EMAIL_ADDRESS>http://www-
personal.umich.edu/~saykhan/ ,  Caleb Partin  Department of Mathematics,
Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332
<EMAIL_ADDRESS>and  Rebecca R. Winarski  Department of Mathematics and
Computer Science, College of the Holy Cross, 1 College Street, Worcester, MA
01610<EMAIL_ADDRESS>https://sites.google.com/site/rebeccawinarski/

###### Abstract.

We prove that in the non-orientable setting, the minimal stretch factor of a
pseudo-Anosov homeomorphism of a surface of genus $g$ with a fixed number of
punctures is asymptotically on the order of $\frac{1}{g}$. Our result adapts
the work of Yazdi to non-orientable surfaces. We include the details of
Thurston’s theory of fibered faces for non-orientable 3-manifolds.

##  1\. Introduction

Let $S_{g,n}$ be a surface of genus $g$ with $n$ punctures. The mapping class
group of $S_{g,n}$ consists of homotopy classes of orientation preserving
homeomorphisms of $S_{g,n}$. The Nielsen–Thurston classification of mapping
classes (elements of the mapping class group) says that each mapping class is
periodic, preserves some multicurve, or has a representative that is pseudo-
Anosov. For each pseudo-Anosov homeomorphism $\varphi:S_{g,n}\rightarrow
S_{g,n}$, the stretch factor $\lambda(\varphi)$ is an algebraic integer that
describes the amount by which $\varphi$ changes the length of curves.
Arnoux–Yaccoz [4] and Ivanov [15] prove that the set

$\operatorname{Spec}(S_{g,n})=\\{\log(\lambda(\varphi))\mid\varphi\text{ is a
pseudo-Anosov homeomorphism of }S_{g,n}\\}$

is a closed discrete subset of $(0,\infty)$. The minimum of
$\operatorname{Spec}(S_{g,n})$:

$\ell_{g,n}=\min\\{\log(\lambda(\varphi))\mid\varphi\text{ is a pseudo-Anosov
homeomorphism of }S_{g,n}\\}$

quantitatively describes both the dynamics of the mapping class group of
$S_{g,n}$ and the geometry of the moduli space of $S_{g,n}$.

Penner [26] showed that for orientable surfaces,

$\ell_{g,0}\asymp\frac{1}{g}.$

Penner conjectured that $\ell_{g,n}$ will have the same asymptotic behavior
for $n\geq 0$ punctures. Following Penner, substantial attention has been
given to finding bounds for $\ell_{g,n}$ [1, 5, 12, 13, 14, 17, 21, 24],
calculating $\ell_{g,n}$ for specific values of $(g,n)$ [7, 11, 18, 27], and
finding asymptotic behavior of $\ell_{g,n}$ for orientable surfaces with
$n\geq 0$ [17, 30, 31, 33]. We adapt a result of Yazdi [33] to non-orientable
surfaces.

######  Theorem 1.1.

Let $\mathcal{N}_{g,n}$ be a non-orientable surface of genus $g$ with $n$
punctures, and let $\ell_{g,n}^{\prime}$ be the logarithm of the minimum
stretch factor of the pseudo-Anosov mapping classes acting on
$\mathcal{N}_{g,n}$. Then for any fixed $n\in\mathbb{N}$, there is a positive
constant $B^{\prime}_{1}=B^{\prime}_{1}(n)$ and
$B^{\prime}_{2}=B^{\prime}_{2}(n)$ such that for any $g\geq 3$,

the quantity $\ell_{g,n}^{\prime}$ satisfies the following inequalities:

$\displaystyle\frac{B^{\prime}_{1}}{g}\leq\ell^{\prime}_{g,n}\leq\frac{B^{\prime}_{2}}{g}.$

Pseudo-Anosov homeomorphisms. Let $S$ be a (possibly non-orientable) surface
of finite type. A homeomorphism $\varphi:S\rightarrow S$ is said to be pseudo-
Anosov if there exist a pair of transverse measured singular foliations
$\mathcal{F}_{s}$ and $\mathcal{F}_{u}$ and a real number $\lambda$ such that

$\varphi(\mathcal{F}_{s})=\lambda^{-1}\cdot\mathcal{F}_{s}\text{ and
}\varphi(\mathcal{F}_{u})=\lambda\cdot\mathcal{F}_{u}.$

The stretch factor of $\varphi$ is the algebraic integer
$\lambda=\lambda(\varphi)$.

Endow $S$ with a Riemannian metric. The stretch factor $\lambda(\varphi)$
measures the growth rate of the length of geodesic representatives of a simple
closed curve $S$ under iteration of $\varphi$ [8, Proposition 9.21]. Moreover,
$\log(\lambda(\varphi))$ is the minimal topological entropy of any
homeomorphism of $S$ that is istopic to $\varphi$ [8, Exposé 10].

Geometry of moduli space. Let $\mathcal{T}_{g,n}$ denote the Teichmüller space
of $S_{g,n}$, that is: the space of isotopy classes of hyperbolic metrics on
$S_{g,n}$. When endowed with the Teichmüller metric, the mapping class group
of $S_{g,n}$ acts properly discontinuously on $\mathcal{T}_{g,n}$ by
isometries. The quotient of this action is the moduli space of $S_{g,n}$. The
set $\operatorname{Spec}(S_{g,n})$ is the length spectrum of geodesics in the
moduli space of $S_{g,n}$. Therefore the quantity $\ell_{g,n}$ is the length
of the shortest geodesic in the moduli space of $S_{g,n}$.

Explicit bounds. In his foundational work, Penner found $\frac{\log
2}{12g-12+4n}$ to be a lower bound for $\ell_{g,n}$ for orientable surfaces
[26]. He also determined $\frac{\log 11}{g}$ to be an upper bound for
$\ell_{g,0}$. Penner’s work proves that $\ell_{g,0}\asymp\frac{1}{g}$.
McMullen [23] later asked:

######  Question (McMullen).

Does $\displaystyle\lim_{g\rightarrow\infty}g\cdot\ell_{g,0}$ exist, and if
so, what does it converge to?

To this end, Bauer [5] strengthened the upper bound for $g\cdot\ell_{g,0}$ to
$\log 6$, and Minakawa [24] and Hironaka–Kin [13] further sharpened the upper
bounds for $g\cdot\ell_{g,0}$ and $g\cdot\ell_{0,2g+1}$ to $\log(2+\sqrt{3})$.
Later Aaber–Dunfield [1], Hironaka [12], and Kin-Takasawa [16] determined that
$\log\left(\frac{3+\sqrt{5}}{2}\right)$ is an upper bound for
$g\cdot\ell_{g,0}$ and conjectured it is the supremum of $g\cdot\ell_{g,0}$.

Asymptotic behavior of punctured surfaces. Tsai initiated the study of
asymptotic behavior of $\ell_{g,n}$ along lines in the $(g,n)$-plane [30]. In
particular, Tsai determined that for orientable surfaces of fixed genus $g\geq
2$, the asymptotic behavior in $n$ is:

$\ell_{g,n}\asymp\frac{\log n}{n}.$

Further, he showed that $\ell_{0,n}\asymp\frac{1}{n}.$ Later, Yazdi [33]
determined that for an orientable surface with a fixed number of punctures
$n\geq 0$, the asymptotic behavior in $g$ is:

$\ell_{g,n}\asymp\frac{1}{g},$

confirming the conjecture of Penner.

Non-orientable surfaces. Let $\mathcal{N}_{g,n}$ be a non-orientable surface
of genus $g$ with $n$ punctures. As above, let $\ell^{\prime}_{g,n}$ denote
the minimum stretch factor of pseudo-Anosov homeomorphisms of
$\mathcal{N}_{g,n}$. For any $n\geq 0$ and $g\geq 1$, $\ell_{g-1,2n}$ is a
lower bound for $\ell^{\prime}_{g,n}$, which can be seen by passing to the
orientation double cover of $\mathcal{N}_{g,n}$ (note that the definition of
genus is different for orientable and non-orientable surfaces). Because the
upper bounds for $\ell_{g,n}$ are constructed by example, upper bounds for
$\ell^{\prime}_{g,n}$ do not follow immediately from passing to the
orientation double cover. Recently Liechti–Strenner determined
$\ell^{\prime}_{g,0}$ for $g\in\\{4,5,6,7,8,10,12,14,16,18,20\\}$ [20]. Our
work captures the asymptotic behavior for the punctured case.

Techniques. To prove Theorem 1.1, we adapt the strategy of Yazdi [33] to non-
orientable surfaces with punctures. The lower bound of $\ell^{\prime}_{g,n}$
is found by lifting to the orientation double cover of $\mathcal{N}_{g,n}$.
The upper bound (as in all prior work) is constructive. Fix $n\geq 0$: the
desired number of punctures. Yazdi’s construction is as follows. For a
sequence of genera $g_{n,k}$ (where $k$ goes from $3$ to $\infty$, and
$g_{n,k}=(14k-2)n+2$), use the Penner construction [25] to obtain a
homeomorphism $f_{n,k}$ of $S_{g_{n,k},n}$ that is pseudo-Anosov and has low
stretch factor. In order to find pseudo-Anosov homeomorphisms of $S_{g,n}$
with small stretch factor for all $g$ (not just those in the sequence above),
construct a mapping torus for each $f_{n,k}$. To do this Yazdi’s appeals to a
technique involving the use of Thurston’s theory of fibered faces.

Thurston norm for non-orientable 3-manifolds. In Thurston’s development of
what is now called the Thurston norm for 3-manifolds [28], his definitions and
theorems required that all surfaces were orientable. Thurston said that the
theorems should still be true for non-orientable surfaces, but there are some
subtleties that have not been addressed elsewhere in the literature. In this
paper, we write the details of Thurston’s theory of fibered faces to non-
orientable 3-manifolds. In particular, the Thurston norm is a norm on the
second homology of a 3-manifold, that measures the minimum complexity of an
embedded (orientable) surface. However the Thurston norm does not recognize
embedded non-orientable surfaces in the second homology of a non-orientable
3-manifold. To address this limitation, we instead calculate the Thurston norm
on the first cohomology of a non-orientable manifold. We develop a (weak)
version of Poincaré duality in Theorem 2.7 that suffices to define a Thurston
norm on $H^{1}(M;R)$ for a non-orientable 3-manifold $M$.

Fibered faces. A special case of Thurston’s hyperbolization theorem says that
the monodromy of any fibration of a hyperbolic 3-manifold over $S^{1}$ is a
pseudo-Anosov homeomorphism. Therefore by finding other fibrations of the same
3-manifold, one obtains additional pseudo-Anosov homeomorphism. Work of Fried
[9, 10], Matsumoto [22], and Agol–Leininger–Margalit [2] can be used to find a
bound on the stretch factors of certain pseudo-Anosov homeomorphisms obtained
in this way.

In Section 2 we state Thurston’s theory of fibered faces and adapt it to the
non-orientable setting. In Section 3 we show how Thurston’s theory of fibered
faces can be used to construct pseudo-Anosov homeomorphisms of low stretch
factor for non-orientable surfaces. Specifically, we state and prove the
Nielsen–Thurston classification for non-orientable surfaces. Then we adapt the
results of Fried [9, 10], Matsumoto [22], and Agol–Leininger–Margalit [2] used
to construct pseudo-Anosov homeomorphisms with low stretch factor of
orientable surfaces to the non-orientable setting. In Section 4, we prove
Theorem 1.1, following the strategy of Yazdi.

Acknowledgements. This work is the result of an REU at the University of
Michigan in summer 2020. We are grateful to Alex Wright for organizing the REU
and to Livio Liechti for suggesting the project. We are thankful to Dan
Margalit for his comments. The REU was partially funded by NSF Grant No. DMS
185615. We are also grateful to the other co-organizers, mentors, and
participants in the REU: Paul Apisa, Chaya Norton, Christopher Zhang, Bradley
Zykoski, Anne Larsen, and Rafael Saavedra. The third author acknowledges
support of the NSF through grant 2002951.

##  2\. Thurston norm for non-orientable 3-manifolds

Thurston defined a norm on $H_{2}(M;\mathbb{R})$ where $M$ is an orientable
3-manifold [28], and this norm is now called the Thurston norm. In his
manuscript, Thurston wrote “Most of this paper works also for non-orientable
manifolds but for simplicity we only deal with the orientable case.” However,
the details are not explained in Thurston’s work or in subsequent literature.
Therefore the goal of this section is to write the details of the Thurston
norm for non-orientable 3-manifolds. We recall the Thurston norm for
orientable manifolds in Section 2.1 and then adapt it to the non-orientable
setting. In Section 2.2 we describe the challenge of defining the Thurston
norm on $H_{2}(M;\mathbb{R})$ if $M$ is non-orientable and present the
solution of defining the Thurston norm instead on $H^{1}(M;\mathbb{R})$.
However, Poincaré duality does not hold for non-orientable manifolds. We
therefore define a condition – relative orientability – on a pair consisting
of a manifold and an embedded surface. A surface that is relatively orientable
in a non-orientable 3-manifold $M$ will have a corresponding cohomology class
in $H^{1}(M;\mathbb{Z})$, giving a version of Poincaré duality for non-
orientable 3-manifolds as stated in Theorem 2.7. Finally in Section 2.4, we
define the oriented sum for relatively oriented embedded surfaces in non-
orientable manifolds.

###  2.1. Thurston norm and mapping tori

In this section we recall the Thurston norm for orientable surfaces and how it
detects when a 3-manifold fibers over a circle.

Mapping tori. Let $S$ be a surface, and let $\varphi:S\to S$ be a
homeomorphism. A mapping torus of $S$ by $\varphi$ is a $3$-manifold
$M_{\varphi}$ given by the identification:

$\displaystyle
M_{\varphi}\coloneqq\frac{S\times[0,1]}{(x,1)\sim(\varphi(x),0)}.$

A mapping torus is _fibration over $S^{1}$_, denoted $S\rightarrow
M_{\varphi}\rightarrow S^{1}$. A fibration defines a flow on $M$, called the
_suspension flow_ , where for any $x_{0}\in S$ and $t_{0}\in S^{1}$ the pair
$(x_{0},t_{0})$ is sent to $(x_{0},t_{0}+t)$. The fiber of a fibration is the
preimage of any point $\theta\in S^{1}$ under the projection map from
$M_{\varphi}\to S^{1}$. If we do not specify $\theta$, the fiber as a subset
of $M_{\varphi}$ is only well defined up to isotopy. The homology class of the
fiber in $H_{2}(M_{\varphi};\mathbb{R})$ is well defined.

A natural inverse question is to determine when a 3-manifold fibers over a
circle, and the possible fibers. To this end, Thurston established a
correspondence between second cohomology of 3-manifolds and surfaces embedded
in 3-manifolds.

Complexity of an embedded surface. Let $M$ be a compact orientable closed
$3$-manifold. Let $S$ be a connected surface embedded in $M$. The complexity
of $S$ is $\chi_{-}(S)=\max\left\\{-\chi(S),0\right\\}$. If the surface $S$
has multiple components $S_{1},\ldots,S_{m}$ then
$\chi_{-}(S)=\displaystyle\sum_{i=1}^{m}\chi_{-}(S_{i})$. The elements in
$H_{2}(M;\mathbb{Z})$ can be represented by embedded surfaces inside of $M$
[28, Lemma 1].

Thurston norm. Let $a$ be a homology class in $H_{2}(M;\mathbb{Z})$. Define
the integer valued norm $x:H_{2}(M;\mathbb{Z})\rightarrow\mathbb{Z}$ as the
following:

$\displaystyle x(a)=\min\\{\chi_{-}(S)\mid[S]=a\text{ and $S$ is compact,
properly embedded and oriented}\\}.$

We then linearly extend $x$ to $H_{2}(M;\mathbb{Q})$. The _Thurston norm_ is
the unique continuous $\mathbb{R}$ valued function that is an extension of $x$
to $H_{2}(M;\mathbb{R})$. The unit ball for the Thurston norm is a convex
polyhedron in $H_{2}(M;\mathbb{R})$.

The following remarkable theorem of Thurston [28] determines all possible
fibrations of an oriented 3-manifold over the circle. We use the restatement
of Yazdi [33].

######  Theorem 2.1 (Thurston).

Let $M$ be an orientable 3-manifold. Let $\mathcal{F}$ be the set of homology
classes in $H_{2}(M;\mathbb{R})$ that are representable by fibers of
fibrations of $M$ over the circle.

  1. (i)

Elements of $\mathcal{F}$ are in one-to-one correspondence with (non-zero)
lattice points inside some union of cones over open faces of the unit ball in
the Thurston norm.

  2. (ii)

If a surface $F$ is transverse to the suspension flow associated to some
fibration of $M\xrightarrow[]{}S^{1}$ then $[F]$ lies in the closure of the
corresponding cone in $H_{2}(M;\mathbb{R})$.

The class $[F]$ has orientation such that the positive flow direction is
pointing outwards relative to the surface. An open face of the unit ball is
said to be a _fibered face_ if the cone over the face contains the fibers of
the corresponding fibration.

The goal for the rest of this section is to prove a version of Theorem 2.1 for
compact non-orientable $3$-manifolds. Most of the work in the proof will
involve reducing the version for non-orientable $3$-manifolds to the
orientable version by passing to the double cover.

###  2.2. Thurston norm on cohomology of non-orientable mapping tori

Let $\mathcal{N}$ be a compact non-orientable surface. A naïve first attempt
at defining the Thurston norm would be to define it on the
$H_{2}(\mathcal{N};\mathbb{R})$, like in the orientable case. However, if the
norm is defined on $H_{2}(\mathcal{N};\mathbb{R})$, the non-orientable version
of Theorem 2.1 will not be true. Let $\varphi:\mathcal{N}\to\mathcal{N}$ be a
homeomorphism and let $N_{\varphi}$ be associated mapping torus. Clearly,
$N_{\varphi}$ fibers over $S^{1}$, and $\mathcal{N}$ is the fiber of this
fibration. However, the homology class associated to $\mathcal{N}$ is the zero
homology class, since the top-dimensional homology of non-orientable compact
surfaces is $0$-dimensional.

Our workaround for this problem will be to define a norm on the first
cohomology $H^{1}(N_{\varphi})$ rather than the second homology
$H_{2}(N_{\varphi})$. By Poincaré duality they are isomorphic for orientable
3-manifolds, but that is not true for non-orientable $3$-manifolds.

Poincaré Duality. To see why Poincaré duality fails for non-orientable
3-manifolds, we will work through the construction of the isomorphism between
first cohomology and second homology for orientable 3-manifolds. Let $M$ be a
3-manifold. To define the Poincaré dual of $H^{1}(M;\mathbb{Z})$, we first
define a homotopy class of maps $M\rightarrow S^{1}$. Then we construct an
element of $H_{2}(M;\mathbb{Z})$. Let $\alpha$ be a 1-form on $M$ and
$[\alpha]$ its class in $H^{1}(M;\mathbb{Z})$. Fix a basepoint $y_{0}\in M$.
The associated map $f_{\alpha}:M\rightarrow S^{1}$ is given by:

(1) $\displaystyle
f_{\alpha}(y)\coloneqq\int_{y_{0}}^{y}\alpha\mod\mathbb{Z}.$

The choice of basepoint does not affect the homotopy class of $f_{\alpha}$
(see Section 5.1 of [6] for the details).

Now let $\theta\in S^{1}$ be a regular value so that $S=f^{-1}(\theta)$ is a
surface. To construct the associated element of $H_{2}(M;\mathbb{Z})$, we
choose an orientation on $S$ by assigning positive values of $\alpha$ to the
outward pointing normal vectors on $S$. Then $S$ inherits an orientation from
the orientation on $M$, and we have defined a fundamental class $[S]\in
H_{2}(M;\mathbb{Z})$. We claim that $[S]$ is the Poincaré dual to $\alpha$.

######  Lemma 2.2.

Let $\theta$ and $\theta^{\prime}$ be two regular values of the function
$f_{\alpha}$ and let $S=f_{\alpha}^{-1}(\theta)$ and
$S^{\prime}=f_{\alpha}^{-1}(\theta^{\prime})$. Then for any closed $2$-form
$\omega$ on $M$, the following identities hold:

  1. (i)

$\displaystyle\int_{S}\omega=\displaystyle\int_{S^{\prime}}\omega$ and

  2. (ii)

$\displaystyle\int_{S}\omega=\displaystyle\int_{M}\alpha\wedge\omega.$

In particular, the homology class of $S$ is Poincaré dual to $\alpha$.

###### Proof.

To see (i), observe that $S$ and $S^{\prime}$ are homologous, i.e.
$f^{-1}_{\alpha}([\theta,\theta^{\prime}])$ is a singular $3$-chain that has
$S$ and $S^{\prime}$ as boundaries. From Stokes’ theorem, we have the
following:

$\displaystyle\int_{S-S^{\prime}}\omega$
$\displaystyle=\int_{f_{\alpha}^{-1}([\theta,\theta^{\prime}])}d\omega=0.$

To prove (ii), observe that because $\alpha$ is the pullback of $d\xi$ along
the map $f_{\alpha}$ we can write the right hand side:

$\displaystyle\int_{M}\alpha\wedge\omega$
$\displaystyle=\int_{S^{1}}\left(\int_{f_{\alpha}^{-1}(\xi)}\omega\right)d\xi.$

By Sard’s theorem, almost every $\xi\in[0,1]$ is a regular value. Therefore
the right hand side is well-defined. By (i), the inner integral is a constant
function, as we vary over the $\xi$ which are regular values of $f_{\alpha}$.
Then the integral of $d\xi$ over $S^{1}$ is $1$, giving us the identity we
want:

$\displaystyle\int_{M}\alpha\wedge\omega=\int_{S}\omega.$

∎

What we have here is an explicit formula for the Poincaré duality map from
$H^{1}(M;\mathbb{R})$ to $H_{2}(M;\mathbb{R})$. For orientable $3$-manifolds,
this is an isomorphism.

######  Theorem 2.3 (Poincaré duality for orientable $3$-manifolds).

Let $M$ be an orientable $3$-manifold, and let $S$ be an orientable embedded
surface. Then there exists a $1$-form $\alpha$ and a regular value $\theta\in
S^{1}$ such that $S$ and $f_{\alpha}^{-1}(\theta)$ are homologous surfaces.

Let $N$ be a non-orientable 3-manifold. The map above from
$H^{1}(N;\mathbb{R})$ to $H_{2}(N;\mathbb{R})$ is still well-defined. However
the map from $H^{1}(N;\mathbb{Z})$ to $H_{2}(N;\mathbb{Z})$ has a nontrivial
kernel. For example, when $N_{\varphi}$ is the mapping torus of a non-
orientable surface $\mathcal{N}$, as above, the fiber is trivial in
$H_{2}(N;\mathbb{Z})$.

Non-orientable manifolds. Let $N$ be a non-orientable $3$-manifold. Let
$\widetilde{N}$ and the covering map $p:\widetilde{N}\rightarrow N$ be the
orientation double covering space of $N$. Let $\iota$ be the orientation
reversing deck transformation of $\widetilde{N}$. Let $N=N_{\varphi}$ be the
mapping torus of the non-orientable surface $\mathcal{N}$ by a homeomorphism
$\varphi:\mathcal{N}\to\mathcal{N}$. Then $\widetilde{N}$ is the mapping torus
of $(\mathcal{S},\widetilde{\varphi})$, where $\mathcal{S}$ is the orientation
double cover of $\mathcal{N}$, and $\widetilde{\varphi}$ is the orientation
preserving lift of $\varphi$.

Defining the Thurston norm on cohomology. In order to define the Thurston norm
on $H^{1}(N;\mathbb{Z})$, we first need to relate $H^{1}(N;\mathbb{R})$ and
$H^{1}(\widetilde{N};\mathbb{R})$. We do so by pulling back
$H^{1}(N;\mathbb{R})$ to $H^{1}(\widetilde{N};\mathbb{R})$ via $p$. We also
state the following lemma without proof (the proof is elementary).

######  Lemma 2.4.

The pullback $p^{\ast}:H^{1}(N;\mathbb{R})\rightarrow
H^{1}(\widetilde{N};\mathbb{R})$ maps $H^{1}(N;\mathbb{R})$ bijectively to the
$\iota^{\ast}$-invariant subspace of $H^{1}(\widetilde{N};\mathbb{R})$.

Next we use Lemma 2.4 to define the Thurston norm on $H^{1}(N;\mathbb{R})$.

Thurston norm for non-orientable $3$-manifolds. Let $\alpha\in
H^{1}(N;\mathbb{R})$ and let $\widetilde{x}$ be the Thurston norm on
$H^{1}(\widetilde{N};\mathbb{R})\cong H_{2}(\widetilde{N};\mathbb{R})$. The
_Thurston norm on $H^{1}(N;\mathbb{R})$_, is the norm
$x:H^{1}(N;\mathbb{R})\rightarrow\mathbb{R}$ defined:

$\displaystyle x(\alpha)\coloneqq\widetilde{x}(p^{\ast}\alpha).$

Note that defining the Thurston norm on $H^{1}(N;\mathbb{R})$ rather than
$H_{2}(N;\mathbb{R})$ is not quite satisfactory. In particular, fibers of
fibrations are embedded surfaces in $N$. In the orientable case, the embedded
surfaces define the Thurston norm. In Section 2.3, we develop a (weak) version
of Poincaré duality for non-orientable 3-manifolds.

###  2.3. Weak inverse to the Poincaré duality map

Let $M$ be a 3-manifold and $[\alpha]$ an element of $H^{1}(M;\mathbb{R})$.
Regardless of whether $M$ is orientable or not, we can construct a dual map
$f_{\alpha}:M\rightarrow S^{1}$, using equation (1). The preimage of a regular
value $\theta$ is an embedded surface $f_{\alpha}^{-1}(\theta)$ in $M$. If we
assume that $M$ is orientable, Poincaré duality allows us to reverse this
correspondence. Indeed, let $S$ is an embedded surface in (the orientable
manifold) $M$. Poincaré duality determines a cohomology class in
$H^{1}(M;\mathbb{Z})$ corresponding to $S$. Now let $\alpha$ be a $1$-form
representative of an element of $H^{1}(M;\mathbb{R})$ and let $\theta$ be a
regular value of $f_{\alpha}$. The surfaces $f_{\alpha}^{-1}(\theta)$ and $S$
are homologous. In this section, we state a weak version of Poincaré duality
for non-orientable surfaces in Theorem 2.7. This version of Poincaré duality
will allow us to associate 1-forms to a certain subset of surfaces embedded in
non-orientable 3-manifolds. We will require that the embedded surfaces are
_relative oriented_ in the (non-orientable) 3-manifold.

Relative oriented surfaces. Let $M$ be a $3$-manifold, and $S$ an embedded
surface in $M$. The surface $S$ is said to be _relatively oriented with
respect to $M$_ if there is a nowhere vanishing vector field on $S$ that is
transverse to the tangent plane of $S$. Two such vector fields are said to
induce the same orientation if they induce the same local orientation after
choosing a local frame for $S$. A surface $S$ is _relatively oriented_ in $M$
if both $S$ and the choice of positive normal vector field are specified.

If $S$ and $M$ are orientable, then $S$ is relatively oriented with respect to
$M$. But even if $M$ is non-orientable, a non-orientable embedded surface $S$
may be relatively oriented in $M$. In particular, we have the following Lemma.

######  Lemma 2.5.

Let $\mathcal{N}$ be the fiber of a fibration $f:N\to S^{1}$. Then
$\mathcal{N}$ is relatively oriented in $N$.

###### Proof.

Consider a non-zero tangent vector $v$ pointing in the positive direction at a
point $\theta\in S^{1}$. One can pull back the tangent vector $v$ to a nowhere
vanishing vector field over $f^{-1}(\theta)=\mathcal{N}$ because $f$ is a
fibration, i.e. a submersion. The pulled back vector field defines a relative
orientation for $\mathcal{N}$ in $N$. ∎

Orientable manifolds. Now let $M$ be an orientable 3-manifold, and let $S$ be
an orientable embedded surface. If $S$ is relatively oriented with respect to
$M$, then a choice of orientation on $S$ determines an orientation on $M$ and
vice versa.

We also need to define the notion of _incompressible surfaces_ to state our
version of Poincaré duality.

Incompressible surfaces. Let $S$ be a surface with positive genus embedded in
a $3$-manifold $M$. The surface $S$ is said to be _incompressible_ if there
does not exist an embedded disc $D$ in $M$ such that $D$ intersects $S$
transversely and $D\cap S=\partial D$. The following result of Thurston
demonstrates the link between incompressible surfaces and fibers of
fibrations.

######  Theorem 2.6 (Theorem 4 of [28]).

Let $M$ be an oriented 3-manifold that fibers over $S^{1}$. Let $S$ be an
incompressible surface embedded in $M$. If $S$ is homologous to a fiber, then
$S$ is isotopic to the fiber.

In the remainder of the section, we will be working with a non-orientable
3-manifold $N$ and an embedded non-orientable surface $\mathcal{N}$. Let
$\widetilde{N}$ and the covering map $p:\widetilde{N}\rightarrow N$ be the
orientation double covering space of $N$. Let $\widetilde{\mathcal{N}}$ be the
preimage of $\mathcal{N}$ under $p$. Let
$\iota:\widetilde{N}\rightarrow\widetilde{N}$ be the orientation-reversing
deck transformation of $p$. We will initiate $N$ and $\mathcal{N}$ in each
result below, but we surpress the initiation of the orientation double cover.

######  Theorem 2.7 (Poincaré Duality for non-orientable 3-manifolds).

Let $N$ be a compact non-orientable $3$-manifold, and let $\mathcal{N}$ be a
relatively oriented incompressible surface embedded in $N$. Then there exists
$[\alpha]\in H^{1}(N;\mathbb{Z})$ such that the pullback of $[\alpha]$ to
$\widetilde{N}$ is the Poincaré dual of $\widetilde{\mathcal{N}}$ in
$\widetilde{N}$. If $[\alpha]$ has a $1$-form representative $\alpha$ that
vanishes nowhere on $N$, then $\mathcal{N}$ is homeomorphic to
$f_{\alpha}^{-1}(\theta)$ for all $\theta\in S^{1}$.

We will refer to the 1-form $\alpha$ given in Theorem 2.7 as the Poincaré dual
of the non-orientable surface $\mathcal{N}$. Before proving Theorem 2.7, we
need three lemmas.

######  Lemma 2.8.

Let $N$ be a non-orientable 3-manifold. Let $\mathcal{N}$ be a relatively
oriented embedded surface in $N$, and let
$\widetilde{\mathcal{N}}=p^{-1}(\mathcal{N})$ in $\widetilde{N}$. Then the
Poincaré dual to $[\widetilde{\mathcal{N}}]$ is $\iota^{\ast}$-invariant.

###### Proof.

The positive vector space on $\mathcal{N}$ that is transverse to its tangent
plane in $N$ lifts to a relative orientation of $\widetilde{\mathcal{N}}$ in
$\widetilde{N}$. Since $\widetilde{\mathcal{N}}$ and $\widetilde{N}$ are
orientable, the relative orientation of $\widetilde{\mathcal{N}}$ defines an
orientation of $\widetilde{\mathcal{N}}$, and thus the homology class
$[\widetilde{\mathcal{N}}]$ in $H_{2}(\widetilde{N};\mathbb{R})$ is well-
defined.

Next we show that $\iota$ reverses the orientation of
$\widetilde{\mathcal{N}}$. To do so, we first observe that because
$\mathcal{N}$ is relatively oriented in $N$, the outward pointing transverse
vector field on ${\mathcal{N}}$ must lift to an outward pointing transverse
vector field on $\widetilde{\mathcal{N}}$. In particular, for any outward
pointing vector $\widetilde{v}$ on $\mathcal{N}$, the vector
$\iota(\widetilde{v})$ is also outward pointing.

Lift an outward pointing transverse vector field on $\mathcal{N}$ to an
outward pointing transverse vector field $\widetilde{V}$ on
$\widetilde{\mathcal{N}}$. Let $(v_{1},v_{2},v_{3})$ be a local frame for some
point in $\widetilde{\mathcal{N}}$ such that $v_{3}$ is in $\widetilde{V}$.
Since $\iota$ reverses the orientation of $\widetilde{N}$ but preserves the
direction of $v_{3}$, $\iota$ must reverse the orientation of the pair
$(v_{1},v_{2})$. In particular, that means $\iota$ reverses the orientation of
$\widetilde{\mathcal{N}}$.

Therefore $[\widetilde{\mathcal{N}}]$ is in the $(-1)$-eigenspace of the
$\iota_{\ast}$ action on $H_{2}(\widetilde{N};\mathbb{R})$. Let the cohomology
class $[\widetilde{\alpha}]$ be the the Poincaré dual to
$[\widetilde{\mathcal{N}}]$. Let $\widetilde{\alpha}$ be a representative
$1$-form $\widetilde{\alpha}$ of $[\widetilde{\alpha}]$ (that need not be
$\iota^{\ast}$-invariant). We use the fact that $\iota^{2}=\mathrm{id}$ in the
first and third equalities:

$\displaystyle\int_{\iota_{\ast}\widetilde{\mathcal{N}}}\omega$
$\displaystyle=\int_{\widetilde{\mathcal{N}}}\iota^{\ast}\omega$ (By a change
of variables)
$\displaystyle=\int_{\widetilde{N}}\widetilde{\alpha}\wedge\iota^{\ast}\omega$
(Poincaré duality)
$\displaystyle=\int_{\widetilde{N}}\iota^{\ast}\left(\iota^{\ast}\widetilde{\alpha}\wedge\omega\right)$
$\displaystyle=\int_{\widetilde{N}}-\left(\iota^{\ast}\widetilde{\alpha}\wedge\omega\right)$
$\displaystyle\text{($\iota$ is orientation reversing)}.$

Because $\iota_{\ast}[\widetilde{\mathcal{N}}]=-[\widetilde{\mathcal{N}}]$, we
have the following:

$\displaystyle\int_{\iota_{\ast}\widetilde{\mathcal{N}}}\omega$
$\displaystyle=-\int_{\widetilde{\mathcal{N}}}\omega$
$\displaystyle=-\int_{\widetilde{N}}\widetilde{\alpha}\wedge\omega.$

Since

$\int_{\widetilde{N}}\widetilde{\alpha}\wedge\omega=\int_{\widetilde{N}}\iota^{\ast}\widetilde{\alpha}\wedge\omega$

for all $\omega$, it follows that $\widetilde{\alpha}$ and
$\iota^{\ast}\widetilde{\alpha}$ differ by an exact form, and therefore the
cohomology class $[\widetilde{\alpha}]$ is $\iota^{\ast}$-invariant. ∎

As above, we will denote the Poincaré dual to $[\widetilde{\mathcal{N}}]$ by
$[\widetilde{\alpha}]$. The class $[\widetilde{\alpha}]$ is an
$\iota^{\ast}$-invariant element of $H^{1}(\widetilde{N};\mathbb{Z})$, but it
is not clear that $[\widetilde{\alpha}]$ is the pullback of an element of
$H^{1}(N;\mathbb{Z})$ under $p$. In the next lemma, we show that is indeed the
case, i.e. $[\widetilde{\alpha}]$ is the pullback of an element in
$H^{1}(N;\mathbb{Z})$.

######  Lemma 2.9.

Let $N$ be a non-orientable 3-manifold. Let $[\widetilde{\alpha}]\in
H^{1}(\widetilde{N},Z)$ and let $\widetilde{S}$ be the Poincaré dual of
$[\widetilde{\alpha}]$ in $\widetilde{N}$. There exists $[\alpha]\in
H^{1}(N;\mathbb{Z})$ such that $\widetilde{\alpha}=p^{\ast}\alpha$.

###### Proof.

It will suffice to show that for any simple closed curve $\gamma$ in $N$, the
integral of $\widetilde{\alpha}$ along any path lift of $\gamma$ is an
integer. Let $x_{0}\in N$ be a base point of $\gamma$. Note that $\gamma$ has
two (path) lifts $\widetilde{\gamma}_{1},\widetilde{\gamma}_{2}$ under $p$ in
$\widetilde{N}$, one based at each element of $p^{-1}(x_{0})$. Either
$\widetilde{\gamma}_{1},\widetilde{\gamma}_{2}$ are both simple closed curves
based at the each of the two preimages $p^{-1}(x_{0})$ or
$\widetilde{\gamma}_{1},\widetilde{\gamma}_{2}$ are both arcs between the two
points of $p^{-1}(x_{0})$. If each lift
$\widetilde{\gamma}_{1},\widetilde{\gamma}_{2}$ of $\gamma$ is a closed curve
in $\widetilde{N}$, the integral
$\displaystyle\int_{\widetilde{\gamma}_{i}}\widetilde{\alpha}$ will be an
integer since $[\widetilde{\alpha}]\in H^{1}(\widetilde{N};\mathbb{Z})$.

If each lift $\widetilde{\gamma}_{1},\widetilde{\gamma}_{2}$ of $\gamma$ is an
arc between the two preimages of $p^{-1}(x_{0})$, we consider the simple
closed curve
$\widetilde{\gamma}=\widetilde{\gamma}_{1}\cup\widetilde{\gamma}_{2}$. We note
that $\iota(\widetilde{\gamma})=\widetilde{\gamma}$. By Lemma 2.8,
$\widetilde{\alpha}$ is $\iota^{\ast}$-invariant. Therefore we have that
$\displaystyle\int_{\widetilde{\gamma}_{1}}\widetilde{\alpha}=\displaystyle\int_{\widetilde{\gamma}_{2}}\widetilde{\alpha}.$
Therefore

$\int_{\widetilde{\gamma}}\widetilde{\alpha}=2\int_{\widetilde{\gamma}_{1}}\widetilde{\alpha}.$

It will suffice to show that
$\displaystyle\int_{\widetilde{\gamma}}\widetilde{\alpha}$ is an even integer.
Without loss of generality, we can assume all intersections of the simple
closed curve $\widetilde{\gamma}$ with the surface $\widetilde{\mathcal{S}}$
are transverse. Since $\widetilde{\alpha}$ is a representative of the Poincaré
dual to $[\widetilde{\mathcal{S}}]$, the integral of $\widetilde{\alpha}$
along $\widetilde{\gamma}$ is the signed intersection number of
$\widetilde{\gamma}$ with $\widetilde{\mathcal{S}}$. The intersection number
must be even, for if $\widetilde{\gamma}$ and $\widetilde{\mathcal{S}}$
intersect at a point $y$, then they also intersect at $\iota(y)$. This proves
the lemma. ∎

The last lemma we need is the claim that lifts of incompressible surfaces are
incompressible.

######  Lemma 2.10.

Let $N$ be a non-orientable 3-manifold. If $\mathcal{N}$ is a relatively
oriented incompressible surface in $N$, then
$\widetilde{\mathcal{N}}=p^{-1}(\mathcal{N})$ is also incompressible in
$\widetilde{N}$.

###### Proof.

Because $\mathcal{N}$ is incompressible in $N$, the map on fundamental groups
induced by inclusion $\mathcal{N}\rightarrow N$ is injective. Since
$p_{\ast}:\pi_{1}(\widetilde{N})\rightarrow\pi_{1}(N)$ is injective, the
induced map $\pi_{1}(\widetilde{\mathcal{N}})\to\pi_{1}(\widetilde{N})$ must
also be injective. An injective induced map on fundamental groups is
equivalent to the orientable surface $\widetilde{\mathcal{N}}$ being
incompressible. ∎

We now have everything we need to finish proving Theorem 2.7.

###### Proof of Theorem 2.7.

Let $\widetilde{\mathcal{N}}=p^{-1}(\mathcal{N}).$ The relative orientation of
$\widetilde{\mathcal{N}}$ determines a homology class
$[\widetilde{\mathcal{N}}]\in H_{2}(N;\mathbb{Z})$. Let the $1$-form
$\widetilde{\alpha}$ be the Poincaré dual to $[\widetilde{\mathcal{N}}]$ in
$\widetilde{N}$. By Lemma 2.9, there exists a 1-form $\alpha\in
H^{1}(N;\mathbb{Z})$ such that $\widetilde{\alpha}=p^{\ast}\alpha$.

We define the map $f_{\alpha}:N\rightarrow S^{1}$ according to equation (1).
Because $\alpha$ is non-vanishing, $f_{\alpha}$ has full rank everywhere.
Therefore $f_{\alpha}$ is a fibration. The map $f_{\alpha}\circ p$ is a lift
of $f_{\alpha}$ to $\widetilde{N}$ under $p$, and is therefore also a
fibration. By Lemma 2.10, $\widetilde{\mathcal{N}}$ is incompressible. It
follows from the orientable version of Poincaré duality that
$\widetilde{\mathcal{N}}$ and $p^{-1}(f_{\alpha}^{-1}(\theta))$ are homologous
surfaces in $\widetilde{N}$. Theorem 2.6 then tells us
$\widetilde{\mathcal{N}}$ must be isotopic to a fiber of $f_{\alpha}\circ p$.
The restriction of $p$ to the homeomorphic surfaces $\widetilde{\mathcal{N}}$
and $p^{-1}(f_{\alpha}^{-1})(\theta)$ determines two equivalent 2-fold
covering maps $p^{-1}(f^{-1}_{\alpha}(\theta))\rightarrow
f^{-1}_{\alpha}(\theta)$. Therefore the image surfaces $\mathcal{N}$ and
$f_{\alpha}^{-1}(\theta)$ must also be homeomorphic. ∎

Note that the above proof does not tell us that $\mathcal{N}$ and
$f_{\alpha}^{-1}(\theta)$ are isotopic. Isotopy of the fibers of $N$ requires
the isotopy between $\widetilde{\mathcal{N}}$ and
$p^{-1}(f^{-1}_{\alpha}(\theta))$ to be $\iota^{\ast}$-equivariant. However,
the theorem is sufficient for our application.

We conclude the section with a non-orientable version of Theorem 2.1.

######  Theorem 2.11.

Let $N$ be a compact non-orientable $3$-manifold, and let $\mathcal{F}$ be the
elements of $H^{1}(N;\mathbb{Z})$ corresponding to fibrations of $N$ over
$S^{1}$.

  1. (i)

Elements of $\mathcal{F}$ are in one-to-one correspondence with (non-zero)
lattice points (i.e. points of $H^{1}(N;\mathbb{Z})$) inside some union of
cones over open faces of the unit ball in the Thurston norm.

  2. (ii)

Let $\mathcal{N}$ be relatively oriented surface in $N$ that transverse to the
suspension flow associated to some fibration $f:N\to S^{1}$. Let $[\alpha]$ be
the Poincaré dual $[\alpha]$ to $\mathcal{N}$. Then $[\alpha]$ lies in the
closure of the cone in $H^{1}(N;\mathbb{R})$ containing the $1$-form
corresponding to $f$.

###### Proof.

For (i), we observe that by Theorem 2.1 the fibrations of $\widetilde{N}$ are
in one-to-one correspondence with lattice points inside a union of cones over
open faces of the unit ball in $H_{2}(\widetilde{N};\mathbb{R})$. Let
$\widetilde{\mathcal{K}}$ be the union of cones in
$H_{2}(\widetilde{N};\mathbb{R})$. By Poincaré duality,
$\widetilde{\mathcal{K}}$ is in one-to-one correspondence to a union of cones
in $H^{1}(\widetilde{N};\mathbb{R})$, which we will call
$\widetilde{\mathcal{K}}^{\ast}$

Because $H^{1}(N;\mathbb{R})$ is isomorphic to a subspace of
$H^{1}(\widetilde{N};\mathbb{R})$, we can construct a union of cones in
$H^{1}(N;\mathbb{R})$ that map to the intersection of
$p^{\ast}(H^{1}(N;\mathbb{R}))$ with $\widetilde{\mathcal{K}}^{\ast}$. Indeed,
every lattice point in $\widetilde{\mathcal{K}}^{\ast}$ corresponds to a
fibration $f:N\to S^{1}$, since the pullback of $f$ to
$H^{1}(\widetilde{N};\mathbb{Z})$ corresponds to a fibration of
$\widetilde{N}$. Conversely, every fibration of $f:N\rightarrow S^{1}$ must
correspond to an element of $\widetilde{\mathcal{K}}^{\ast}$, since the
composition $f\circ p$ is a fibration of $\widetilde{N}\rightarrow S^{1}$.

For (ii), assume that the surface $\mathcal{N}$ is transverse to the
suspension flow of a fibration $f:N\rightarrow S^{1}$. Then
$\widetilde{\mathcal{N}}$ is transverse to the suspension flow $p\circ
f:\widetilde{N}\rightarrow S^{1}$. Let $\widetilde{\alpha}$ be the pullback of
$\alpha$ under $p$. Then $\widetilde{\alpha}$ is the Poincaré dual of
$\widetilde{\mathcal{N}}$. By Theorem 2.1, the 1-form $\widetilde{\alpha}$
lies in the closure of a component of $\widetilde{\mathcal{K}}^{\ast}$ that
contains the 1-form corresponding to $f\circ p$. Let $\widetilde{K}$ be this
component. Let $K\subset H^{1}(N;\mathbb{R})$ be the preimage of
$\widetilde{K}$ under $p^{\ast}$. The cone $K$ contains both $\alpha$ and the
1-form corresponding to $f$, as desired. ∎

###  2.4. Oriented sums

The next step in studying embedded non-orientable surfaces will be to describe
_oriented sums_. Let $M$ be a 3-manifold. The oriented sum of two embedded
surfaces in $M$ is additive in both the Euler characteristic and
$H^{1}(M;\mathbb{R})$. This operation is well-known in the case of orientable
$3$-manifolds (along with orientable embedded surfaces), but we will sketch
the relevant details. We then extend the construction to relatively oriented
embedded surfaces.

#### Oriented sum for oriented manifolds

Let $M$ be an orientable manifold. Let $S$ and $S^{\prime}$ be orientable
embedded surfaces in $M$. Assume that $S$ and $S^{\prime}$ intersect
transversally. Thus $S\cap S^{\prime}$ is a disjoint union of copies of
$S^{1}$. For each component of $S\cap S^{\prime}$, take a tubular neighborhood
that has cross section as in Figure 1.

$S$
---
$S$
---
$S^{\prime}$
---
$S^{\prime}$
---
Figure 1. Cross section of intersection of $S$ and $S^{\prime}$.

We then perform a surgery on the leaves of $S$ and $S^{\prime}$ so that the
outward pointing normal vector fields match as in Figure 2.

$S$
---
$S$
---
$S^{\prime}$
---
$S^{\prime}$
---
$S^{\prime}$
---
$S^{\prime}$
---
$S$
---
$S$
---
Figure 2. On the left, the normal vectors on $S$ and $S^{\prime}$ are
consistent. On the right, they are not.

By performing this surgery at all the intersections, we get a new submanifold
$S^{\prime\prime}$ of $M$ (which may have multiple components). This new
submanifold $S^{\prime\prime}$ is called the oriented sum of $S$ and
$S^{\prime}$. The operation of taking oriented sums is additive on Euler
characteristic, as well as the homology classes (and thus the cohomology
classes of their Poincaré duals):

$\displaystyle\chi(S^{\prime\prime})$ $\displaystyle=\chi(S)+\chi(S^{\prime})$
$\displaystyle[S^{\prime\prime}]$ $\displaystyle=[S]+[S^{\prime}].$

#### Oriented sum for non-orientable manifolds

Let $N$ be a non-orientable manifold and let $\mathcal{N}$ and
$\mathcal{N}^{\prime}$ be embedded surfaces in $N$ that are relatively
oriented. We define the oriented sum on $\mathcal{N}$ and
$\mathcal{N}^{\prime}$ as follows. As above, let $p:\widetilde{N}\rightarrow
N$ be the orientation double cover and let $\iota$ be the orientation
reversing deck transformation of $\widetilde{N}$. Let
$\widetilde{\mathcal{N}}=p^{-1}(\mathcal{N})$ and
$\widetilde{\mathcal{N}}^{\prime}=p^{-1}(\mathcal{N}^{\prime})$, which are
embedded oriented surfaces in $\widetilde{N}$. The oriented sum of
$\mathcal{N}$ and $\mathcal{N}^{\prime}$ is the image under $p$ of the
oriented sum of $\widetilde{\mathcal{N}}$ and
$\widetilde{\mathcal{N}}^{\prime}$ (as defined above for oriented surfaces in
oriented manifolds).

To see that the operation is well defined, we recall that $\iota$ preserves
the relative orientation of $\widetilde{\mathcal{N}}$ and
$\widetilde{\mathcal{N}}^{\prime}$. Therefore $\iota$ leaves the outward
normal vector fields on $\widetilde{\mathcal{N}}$ and
$\widetilde{\mathcal{N}}^{\prime}$ invariant (see the proof of Lemma 2.8).
Thus a leaf $L$ of $\widetilde{\mathcal{N}}$ is surgered with a leaf of
$L^{\prime}$ of $\widetilde{\mathcal{N}}^{\prime}$ if and only if $\iota(L)$
and $\iota(L^{\prime})$ are surgered. Therefore surgery factors through $p$
and $[\mathcal{N}]+[\mathcal{N}^{\prime}]$ is well-defined for non-orientable
surfaces.

Example. Let $\gamma$ be a component of $\mathcal{N}\cap\mathcal{N}^{\prime}$
and $\widetilde{\gamma}_{1}$ and $\widetilde{\gamma}_{2}$ be the path lifts of
$\gamma$. One possible orientation of $\widetilde{S}$ and
$\widetilde{S}^{\prime}$ is given in Figure 3. The outward pointing normal
vectors to $\widetilde{\mathcal{N}}$ and $\widetilde{\mathcal{N}^{\prime}}$
determine which leaves are glued together along $\widetilde{\gamma}_{1}$ and
$\widetilde{\gamma}_{2}$.

$\widetilde{\gamma}_{1}$
---
$\widetilde{S^{\prime}}$
---
$\widetilde{S^{\prime}}$
---
$\widetilde{S}$
---
$\widetilde{S}$
---
$\widetilde{\gamma}_{2}$
---
$\widetilde{S^{\prime}}$
---
$\widetilde{S^{\prime}}$
---
$\widetilde{S}$
---
$\widetilde{S}$
---
Figure 3. Neighborhoods of $\widetilde{\gamma}_{1}$ and
$\widetilde{\gamma}_{2}$, with the outward pointing normal vector field.

To preserve the normal vector field, glue the left $\widetilde{\mathcal{N}}$
leaf to the bottom $\widetilde{\mathcal{N}^{\prime}}$ leaf near
$\widetilde{\gamma}_{1}$ and $\widetilde{\gamma}_{2}$. Since
$\iota(\widetilde{\gamma}_{1})=\widetilde{\gamma_{2}}$, the outward pointing
normal vector fields point the same (relative) directions.

#### Additivity

By the consistency of the oriented sum in $N$ and $\widetilde{N}$, it easily
follows that the oriented sum is additive in Euler characteristic, as well as
in terms of Poincaré dual, since the Poincaré dual was also defined by passing
to the orientation double cover.

##  3\. Mapping classes with small stretch factors

In this section, we provide a strategy to compute pseudo-Anosov homeomorphisms
with small stretch factors.

###  3.1. Mapping class groups of non-orientable surfaces

Let $\mathcal{N}$ be a non-orientable surface and let
$\widetilde{\mathcal{N}}$ and the covering map
$p:\widetilde{\mathcal{N}}\rightarrow\mathcal{N}$ be its orientation double
covering space. Every homeomorphism $\varphi:\mathcal{N}\to\mathcal{N}$, has a
unique orientation-preserving lift
$\widetilde{\varphi}:\widetilde{\mathcal{N}}\to\widetilde{\mathcal{N}}$.

A consequence is that lifting homeomorphisms induces a monomorphism between
homeomorphisms of $\mathcal{N}$ and orientation-preserving homeomorphisms of
$\widetilde{\mathcal{N}}$. Every homotopy of $\mathcal{N}$ lifts to a homotopy
of $\widetilde{\mathcal{N}}$. Therefore there is an inclusion from the mapping
class group of $\mathcal{N}$ to the (orientation-preserving) mapping class
group of $\widetilde{\mathcal{N}}$. This inclusion also respects the Nielsen-
Thurston classification of mapping classes, both qualitatively, and
quantitatively, as the following proposition shows.

######  Proposition 3.1.

Let $\varphi:\mathcal{N}\rightarrow\mathcal{N}$ be a homeomorphism and let
$\widetilde{\varphi}:\widetilde{\mathcal{N}}\rightarrow\widetilde{\mathcal{N}}$
be the orientation-preserving lift of $\varphi$. Then:

  1. (i)

$\varphi$ is periodic if and only if $\widetilde{\varphi}$ is periodic,

  2. (ii)

$\varphi$ is reducible if and only if $\widetilde{\varphi}$ is reducible, and

  3. (iii)

$\varphi$ is pseudo-Anosov if and only if $\widetilde{\varphi}$ is pseudo-
Anosov. Moreover if $\varphi$ has stretch factor $\lambda$, then
$\widetilde{\varphi}$ also has stretch factor $\lambda$.

###### Proof.

The fact that the map from $\operatorname{Mod}(\mathcal{N})$ to
$\operatorname{Mod}(\widetilde{\mathcal{N}})$ is type-preserving follows from
Aramayona–Leininger–Souto [3, Lemma 10] (while the statement of the Lemma is
for orientable surfaces, the argument, which we will skip, is identical for
non-orientable surfaces).

Suppose now that $\varphi:\mathcal{N}\rightarrow\mathcal{N}$ is a psuedo-
Anosov homeomorphism with stretch factor $\lambda$ and stable and unstable
foliations $\mathcal{F}_{s}$ and $\mathcal{F}_{u}$ respectively. Let
$\widetilde{\mathcal{F}_{s}}$ and $\widetilde{\mathcal{F}_{u}}$ denote the
lifts of the stable and unstable foliations to the orientation double cover.
We need to show that the following identities hold for all simple closed
curves $\gamma$ in $\widetilde{\mathcal{N}}$ (see [8, Exposé 5] for the
definition of intersection number with measured foliations; the fact that
these identities suffice follows from [8, Lemma 9.15]):

(2) $\displaystyle i(\gamma,\widetilde{\varphi}(\widetilde{\mathcal{F}_{u}}))$
$\displaystyle=\lambda\cdot i(\gamma,\widetilde{\mathcal{F}_{u}})$ (3)
$\displaystyle i(\gamma,\widetilde{\varphi}(\widetilde{\mathcal{F}_{s}}))$
$\displaystyle=\frac{1}{\lambda}\cdot i(\gamma,\widetilde{\mathcal{F}_{s}}).$

To see that these identities hold, we partition $\gamma$ into short arcs
$\\{\gamma_{i}\\}$ such that the restriction of the covering map $p$ to a
neighborhood of each arc is a homeomorphism. Then we have:

(4) $\displaystyle i(\gamma_{i},\widetilde{\mathcal{F}_{u}})$
$\displaystyle=i(p(\gamma_{i}),\mathcal{F}_{u})$ (5) $\displaystyle
i(\gamma_{i},\widetilde{\varphi}(\widetilde{\mathcal{F}_{u}}))$
$\displaystyle=i(p(\gamma_{i}),\varphi(\mathcal{F}_{u})).$

Since we know that $\mathcal{F}_{u}$ is the unstable foliation for $\varphi$
with stretch factor $\lambda$, we can compute the ratio of the right hand side
of (4) and (5):

(6) $\displaystyle i(p(\gamma_{i}),\varphi(\mathcal{F}_{u}))=\lambda\cdot
i(p(\gamma_{i}),\mathcal{F}_{u}).$

Combining (4), (5), and (6), and summing over all $\gamma_{i}$ gives us that
(2) holds. A similar argument also verifiproves that (3) holds. ∎

###  3.2. Constructing pseudo-Anosov maps on nearby surfaces using oriented
sums

The goal of this section is to prove that the stretch factor of any pseudo-
Anosov homeomorphism provides an asymptotic upper bound for the minimum
stretch factor. We do this in Proposition 3.2.

######  Proposition 3.2.

Let $\mathcal{N}_{g}$ be a non-orientable surface of genus $g$ and let
$\varphi:\mathcal{N}_{g}\rightarrow\mathcal{N}_{g}$ be a pseudo-Anosov
homeomorphism with stretch factor $\lambda$. Let $N_{\varphi}$ be the mapping
torus of $\mathcal{N}_{g}$ by $\varphi$. Let $\mathcal{N}_{g^{\prime}}$ be an
incompressible surface embedded in $N_{\varphi}$ that is transverse to the
supension flow associated to $\varphi$. Then for all $k\in\mathbb{Z}^{+}$,
there is a pseudo-Anosov homeomorphism of the oriented sum
$\mathcal{N}_{g}+k\mathcal{N}_{g^{\prime}}$ with stretch factor at most
$\lambda$.

Our strategy for proving Proposition 3.2 is to find fiber bundles of
$N_{\varphi}$ over $S^{1}$ that have fiber
$\mathcal{N}_{g}+k\mathcal{N}_{g^{\prime}}$. We then apply a special case of
Thurston’s hyperbolization theorem, which says that the mapping torus of an
orientable surface $S$ by a homeomorphism $\varphi$ is hyperbolic if and only
if $\varphi$ is pseudo-Anosov [29, Theorem 0.1]. In particular, Thurston’s
theorem implies that if $M=M_{\varphi}$ fibers over $S^{1}$ in two ways,
either both mondromies are pseudo-Anosov or neither monodromy is pseudo-
Anosov. Finally, we adapt theorems of Fried and Matsumoto (Theorem 3.4) and
Agol–Leininger–Margalit (Theorem 3.5) to work for mapping tori with non-
orientable fibers.

We will repeatedly use the following two facts for orientable surfaces and
3-manifolds:

  1. (1)

An orientable surface $S$ minimizes the Thurston norm in its homology class if
and only if $S$ is incompressible.

  2. (2)

If an orientable 3-manifold $M$ fibers over $S^{1}$, then the fiber is
incompressible.

######  Proposition 3.3.

Let $\mathcal{N}^{\prime}$ be an incompressible surface embedded in $N$ that
is transverse to the suspension flow associated to $\varphi$. Let $\alpha$ be
the Poincaré dual of $\mathcal{N}$ and $\alpha^{\prime}$ the Poincaré dual of
$\mathcal{N}^{\prime}$. If the oriented sum of $\mathcal{N}$ and
$\mathcal{N}^{\prime}$ is connected, then $\mathcal{N}+\mathcal{N}^{\prime}$
is homeomorphic to the fiber of the fibration given by
$\alpha+\alpha^{\prime}$.

###### Proof.

Let $p:\widetilde{N}\rightarrow N$ be the orientation double cover of $N$. The
surface $\mathcal{N}$ is incompressible because it is a fiber of $f$;
therefore $p^{-1}(\mathcal{N})$ is also incompressible. Then the Thurston norm
of $\mathcal{N}$ of $\alpha$ is $2\chi_{-}(\mathcal{N})$. Likewise, the
Thurston norm of $\alpha^{\prime}$ is $2\chi_{-}(\mathcal{N}^{\prime})$.

By Theorem 2.11 (ii), $\alpha^{\prime}$ lies in the same cone in
$H^{1}(N;\mathbb{Z})$ as $\alpha$. The Thurston norm $x$ on
$H^{1}(N;\mathbb{Z})$ is linear function on that cone. Since the Thurston norm
is also linear on oriented sums of $\mathcal{N}$ and $\mathcal{N}^{\prime}$,
we have:

$\displaystyle x(\alpha+\alpha^{\prime})$
$\displaystyle=x(\alpha)+x(\alpha^{\prime})$
$\displaystyle=2\chi_{-}(\mathcal{N})+2\chi_{-}(\mathcal{N}^{\prime})$
$\displaystyle=2\chi_{-}(\mathcal{N}+\mathcal{N}^{\prime}).$

Because $2\chi_{-}(\mathcal{N}+\mathcal{N}^{\prime})$ achieves the Thurston
norm of $\alpha+\alpha^{\prime}$, the preimage
$p^{-1}(\mathcal{N}+\mathcal{N}^{\prime})$ achieves the Thurston norm of the
pullback of $\alpha+\alpha^{\prime}$ under $p$. Therefore
$p^{-1}(\mathcal{N}+\mathcal{N}^{\prime})$ is incompressible. Then
$\mathcal{N}+\mathcal{N}^{\prime}$ is also incompressible.

By Theorem 2.11 (i), we have that $\alpha+\alpha^{\prime}$ corresponds to some
other fibration $f^{\prime\prime}:N\rightarrow S^{1}$. By Theorem 2.7, the
fiber of $f^{\prime\prime}$ must be homeomorphic to
$\mathcal{N}+\mathcal{N}^{\prime}$. ∎

In the proof of Proposition 3.2, we will use Proposition 3.3 along with a
theorem of Thurston to obtain a pseudo-Anosov homemorphism $\varphi_{k}$ of
the surface of genus of genus $g+kg^{\prime}$. We the use Theorems 3.4 and 3.5
to obtain a upper bound on the stretch factor of $\varphi_{k}$.

######  Theorem 3.4 (Fried [9, 10], Matsumoto[22]).

Let $M$ be an orientable hyperbolic $3$-manifold and let $\mathcal{K}$ be the
union of cones in $H^{1}(M;\mathbb{R})$ whose lattice points correspond to
fibrations over $S^{1}$. There exists a strictly convex function
$h:\mathcal{K}\to\mathbb{R}$ satisfying the following properties:

  1. (i)

For all $c>0$ and $u\in\mathcal{K}$, $h(cu)=\frac{1}{c}h(u)$,

  2. (ii)

For every primitive lattice point $u\in\mathcal{K}$, $h(u)=\log(\lambda)$,
where $\lambda$ is the stretch factor of the pseudo-Anosov map associated to
this lattice point, and

  3. (iii)

$h(u)$ goes to $\infty$ as $u$ approaches $\partial\mathcal{K}$.

######  Theorem 3.5 (Agol-Leininger-Margalit).

Let $\mathcal{K}$ be a fibered cone for a mapping torus $M$ and let
$\overline{\mathcal{K}}$ be its closure in $H^{1}(M;\mathbb{R})$. If
$u\in\mathcal{K}$ and $v\in\overline{\mathcal{K}}$, then $h(u+v)<h(u)$.

###### Proof of Proposition 3.2.

The oriented sum $\mathcal{S}=\mathcal{N}_{g}+k\mathcal{N}_{g^{\prime}}$
constructed in Proposition 3.3 is a surface of genus $g+kg^{\prime}$, and is
$\mathcal{S}$ is homeomorphic to a fiber of $N_{\varphi}$ given by
$\alpha+k\alpha^{\prime}$. Let $\varphi_{k}:\mathcal{S}\rightarrow\mathcal{S}$
be the monodromy of $N_{\varphi}$ over $\mathcal{S}$. By Thurston’s theorem,
$\varphi_{k}$ is pseudo-Anosov. We claim that $\varphi_{k}$ has stretch factor
at most $\lambda$.

Let $p:\widetilde{N}\rightarrow N_{\varphi}$ be the orientation double cover
of $N_{\varphi}$. Let $h\mid_{N}$ be the restriction of $h$ to the pullback
$p^{\ast}(H^{1}(N_{\varphi};\mathbb{R}))$ in
$H^{1}(\widetilde{N};\mathbb{R})$. The restriction $h\mid_{N}$ satisfies all
the properties of Theorems 3.4 and 3.5.

Let $\widetilde{\varphi}$ be the orientation preserving lift of $\varphi$ to
$p^{-1}(\mathcal{N})$. Since $\widetilde{\alpha}$ is the pullback of $\alpha$,
the $\widetilde{\varphi}$ is the pseudo-Anosov homeomorphism associated to
$\widetilde{\alpha}$. By Proposition 3.1, the stretch factor of
$\widetilde{\varphi}$ is $\lambda$.

Let $\mathcal{K}$ be the cone in $H^{1}(N_{\varphi};\mathbb{R})$ containing
$\alpha$. Since $\mathcal{N}_{g^{\prime}}$ is transverse to the suspension
flow in the direction of $\varphi$, we have that $\alpha^{\prime}$ is in the
closure of $\mathcal{K}$ in $H^{1}(N;\mathbb{R})$. Let $\widetilde{\alpha}$ be
the pullback of $\alpha$ under $p$ and let $\widetilde{\alpha}^{\prime}$ be
the pullback of $\alpha^{\prime}$ under $p$. Then
$h\mid_{N}(\widetilde{\alpha}+\widetilde{\alpha}^{\prime})<h\mid_{N}(\widetilde{\alpha})$.
By Theorem 3.4, $h(\widetilde{\alpha})$ is equal to the stretch factor of the
pseudo-Anosov homeomorphism associated to $\widetilde{\alpha}$. Therefore we
have
$h\mid_{N}(\widetilde{\alpha}+\widetilde{\alpha}^{\prime})<\log(\lambda)$. It
follows that the stretch factor of $\varphi_{k}$ is less than $\lambda$. ∎

##  4\. Minimal stretch factors for non-orientable surfaces with marked points

In this section we will use Theorem 2.11 and Proposition 3.2 to adapt the
methods of Yazdi [33] to non-orientable surfaces. We recall the statement of
the main theorem:

######  Theorem 1.1.

Let $\mathcal{N}_{g,n}$ be a non-orientable surface of genus $g$ with $n$
punctures, and let $\ell_{g,n}^{\prime}$ be the logarithm of the minimum
stretch factor of the pseudo-Anosov mapping classes acting on
$\mathcal{N}_{g,n}$. Then for any fixed $n\in\mathbb{N}$, there is a positive
constant $B^{\prime}_{1}=B^{\prime}_{1}(n)$ and
$B^{\prime}_{2}=B^{\prime}_{2}(n)$ such that for any $g\geq 3$, the quantity
$\ell_{g,n}^{\prime}$ satisfies the following inequalities:

$\displaystyle\frac{B^{\prime}_{1}}{g}\leq\ell^{\prime}_{g,n}\leq\frac{B^{\prime}_{2}}{g}.$

Observe that the lower bound for the non-orientable case follows easily from
the lower bound for the orientable case. Indeed, let $\varphi$ be a pseudo-
Anosov map with the minimal stretch factor on $\mathcal{N}_{g,n}$. The
orientation double cover of $\mathcal{N}_{g,n}$ is $\mathcal{S}_{G-1,2n}$
where $G=2g$. Note that in the non-orientable case we measure genus as the
number of copies of the projective plane attached to $S^{2}$ via a connect sum
and in the orientable case we measure genus as the number of copies of the
torus attached to $S^{2}$ via a connected sum. Let
$\widetilde{\varphi}:\mathcal{S}_{G-1,2n}\to\mathcal{S}_{G-1,2n}$ be the
orientation preserving lift of $\varphi$. By Proposition 3.1,
$\widetilde{\varphi}$ has the same stretch factor as $\varphi$. The former is
bounded below by $\frac{B_{1}}{G-1}$, and thus the stretch factor of $\varphi$
is bounded below as well. The more challenging part of the proof is showing
that the upper bound holds.

We will closely follow Yazdi’s construction, which proceeds in five steps that
we will reorder for clarity. In steps 1 and 2, we construct a family of
psuedo-Anosov homeomorphisms of $\mathcal{N}_{g_{i},n}$, where $\\{g_{i}\\}$
is an unbounded increasing sequence. However the sequence $\\{g_{i}\\}$ does
not contain all natural numbers. In step 3 we gives an upper bound to the
stretch factor of the previously constructed homeomorphisms. In steps 4 and 5,
we construct pseudo-Anosov maps on surfaces of genera that do not belong to
the sequence $\\{g_{i}\\}$. It is in steps 4 and 5 that we uses Thurston’s
fibered face theory. We have adapted each of Yazdi’s five steps to work for
non-orientable surfaces.

Step 1: Constructing the surfaces.

We begin by defining a family of surfaces $P_{n,k}$. Let $S$ be an orientable
surface of genus 5 with 3 boundary components. Call the boundary components
$c,d$ and $e$. Choose an orientation of $S$ and let $c,d$ and $e$ inherit the
induced orientations. We obtain a non-orientable surface $T$ from $S$ by
adding two cross caps to $S$ (retaining the orientation of the boundary
components of $S$). Let $p$ and $q$ be marked points in the boundary component
$e$. In Step 5 we will remove $p$ and all copies. Let $r$ and $s$ be the
components of $e\setminus\\{p,q\\}$. The resulting surface $T$ is shown in
Figure 4.

Figure 4. The surface $T$, which will be the building block of the
construction.

Let $T_{i,j}$ be copies of the surface $T$, where $i,j\in\mathbb{Z}$. Let
$c_{i,j},d_{i,j}$ and $e_{i,j}$ be the (oriented) boundary components of
$T_{i,j}$ and let $r_{i,j}$ and $s_{i,j}$ be the copies of the arcs $r$ and
$s$ in $T_{i,j}$. Define a connected infinite surface $T_{\infty}$ as the
quotient:

$\displaystyle T_{\infty}\coloneqq\left.\left(\bigcup
T_{i,j}\right)\right/\sim$

for all integers $i$ and $j$. The gluing $\sim$ is given by orientation-
reversing identifications:

(7) $\displaystyle c_{i,j}$ $\displaystyle\sim d_{i+1,j}$ (8) $\displaystyle
r_{i,j}$ $\displaystyle\sim s_{i,j+1}.$

We have two natural shift maps
$\overline{\rho_{1}},\overline{\rho_{2}}:T_{\infty}\to T_{\infty}$ that act in
the following manner:

$\displaystyle\overline{\rho_{1}}:T_{i,j}$ $\displaystyle\mapsto T_{i+1,j}$
$\displaystyle\overline{\rho_{2}}:T_{i,j}$ $\displaystyle\mapsto T_{i,j+1}.$

Note that $\overline{\rho_{1}}$ and $\overline{\rho_{2}}$ commute. Define the
surface $P_{n,k}$ as the quotient of the surface $T_{\infty}$ by the covering
action of the group generated by $(\overline{\rho_{1}})^{n}$ and
$(\overline{\rho_{2}})^{k}$. Then $\overline{\rho_{1}}$ and
$\overline{\rho_{2}}$ are equivariant with respect to the covering map. We
denote the induced homeomorphisms of the quotient $P_{n,k}$ by $\rho_{1}$ and
$\rho_{2}$. Note that later we will require that $k\geq 3$ and $n$ is the
number of punctures, given in Theorem 1.1.

######  Lemma 4.1.

Let

$\displaystyle g_{n,k}$ $\displaystyle=(14k-2)n+2$

for $n\geq 1$ and $k\geq 1$. The genus of $P_{n,k}$ is $g_{n,k}$.

###### Proof.

Let $U\subset P_{n,k}$ be the subsurface

$\displaystyle
U=\left.\left(\bigcup_{j=0}^{k-1}T_{0,j}\right)\right/\sim^{\prime}$

where $\sim^{\prime}$ is given by (7) and by identifying $r_{i,k-1}$ and
$s_{i,0}$. Then $U$ is a compact, non-orientable surface of genus $12k$ with
$2k$ boundary components. The surface $P_{n,k}$ consists of $n$ copies of $U$
identified along the $2k$ boundary components. Therefore the Euler
characteristic of $P_{n,k}$ is:

$\displaystyle\chi(P_{n,k})$ $\displaystyle=n\cdot\chi(U)$
$\displaystyle=n\cdot(2-12k-2k)$ $\displaystyle=-n(14k-2).$

Since $P_{n,k}$ is a non-orientable surface with empty boundary, we have that:

$\displaystyle g_{n,k}=n(14k-2)+2.$

∎

Step 2: Constructing the maps. In what is now a classical paper, Penner gives
a construction of pseudo-Anosov homeomorphisms on both orientable and non-
orientable surfaces [25]. Below we outline the Penner construction for non-
orientable surfaces following the details of Liechti–Strenner [20, Section 2].

Inconsistent markings. Let $\mathcal{N}$ be a non-orientable surface and let
$c$ be a two-sided curve in $\mathcal{N}$. There exists a neighborhood of $c$
that is homeomorphic to an annulus. Let $\mathcal{A}_{c}$ be an annulus and
let $\zeta_{c}:\mathcal{A}_{c}\xrightarrow{}\mathcal{N}$ be the homeomorphism
that maps to a neighborhood of $c$. The homeomorphism $\zeta_{c}$ is called a
marking of $c$. A pair consisting of a curve $c$ and $\zeta_{c}$ is called a
marked curve. If we fix an orientation of $\mathcal{A}_{c}$, then we can
pushforward this orientation to $\mathcal{N}$. Let $(c,\zeta_{c})$ and let
$(d,\zeta_{d})$ be two marked curves that intersect at one point $p$. We say
that $(c,\zeta_{c})$ and $(d,\zeta_{d})$ are marked inconsistently if the
pushforward of the orientation of $\mathcal{A}_{c}$ and disagrees with the
pushforward of the orientation of $\mathcal{A}_{d}$ in a neighborhood of $p$.
We emphasize that we can also say that two disjoint curves are inconsistently
marked.

Dehn twists. We define the Dehn twist $\phi_{c,\zeta_{c}}(x)$ around a marked
curve $(c,\zeta_{c})$ as:

$\displaystyle\phi_{c,\zeta_{c}}(x)=\begin{cases}\zeta_{c}\circ\tau_{c}\circ\zeta_{c}^{-1}(x)&\text{for
}x\in\zeta_{c}(\mathcal{A}_{c})\\\ x&\text{for
}x\in\mathcal{N}-\zeta_{c}(\mathcal{A}_{c})\end{cases}.$

Here $\tau_{c}$ is the left-handed Dehn twist on $\mathcal{A}_{c}$, i.e.
$\tau_{c}(\theta,t)=(\theta+2\pi t,t)$.

The Penner construction for non-orientable surfaces. Let $\mathcal{C}$ be a
set of marked essential simple closed curves in $\mathcal{N}$ such that no two
curves in $\mathcal{C}$ are homotopic. A Penner construction on $\mathcal{N}$
is a composition of Dehn twists about the marked curves in $\mathcal{C}$ such
that:

  1. (1)

the complement of curves in $\mathcal{C}$ in $\mathcal{N}$ consists of disks
with at most one puncture or marked point,

  2. (2)

the marked curves $(c_{i},\zeta_{i}),(c_{j},\zeta_{j})\in\mathcal{C}$ with
$i\neq j$ are marked inconsistently,

  3. (3)

a Dehn twist about each marked curve in $\mathcal{C}$ is included in the
composition, and

  4. (4)

all powers of Dehn twists are positive (alternatively, all powers are
negative).

Construction of $f_{n,k}$. We now construct homeomorphisms $f_{n,k}:P_{n,k}\to
P_{n,k}$ that are defined as a composition of specific Dehn twists followed by
a finite order mapping class. The key insight is that a power of this map will
be a composition of Dehn twists that satisfy the criteria to be a Penner
construction. Therefore $f_{n,k}$ is pseudo-Anosov. Here we are using the
rotational symmetry of the $P_{n,k}$.

Figure 5. Part of surface $P_{n,k}$ that includes the subsurface $T_{0,0}$ and
the curves $\alpha_{i}$, $\beta_{j}$, and $\gamma$.

Let $\\{\alpha_{1},\cdots,\alpha_{8}\\}$ be the multi-curve in $T_{0,0}$ as
shown in Figure 5. Let $\\{\beta_{1},\cdots,\beta_{7}\\}$ be the multi-curve
in $T_{0,0}\cup T_{0,1}\cup T_{1,0}$ shown in Figure 5.

For any $\alpha_{i}$, we choose a marking $\zeta_{\alpha_{i}}$ to be
orientation-preserving. For any $\beta_{j}$ let $\zeta_{\beta_{j}}$ be
orientation reversing. From here forward, we will think of $\alpha_{i}$ and
$\beta_{j}$ as (inconsistently) marked curves but we will suppress the marking
maps. These choices give an inconsistent marking of
$\\{\alpha_{1},\cdots,\alpha_{8}\\}\cup\\{\beta_{1},\cdots,\beta_{7}\\}$.

Let

$\mathcal{R}=\displaystyle\bigcup_{i=2}^{8}\alpha_{i}.$

Then $\mathcal{R}$ is a marked multi-curve that is disjoint from $\gamma$. Let

$\overline{\mathcal{R}}=\mathcal{R}\cup\rho_{1}(\mathcal{R})\cup\dots\cup\rho_{1}^{n-1}(\mathcal{R}).$

Let $\Phi_{r}$ be the composition of Dehn twists about the marked curves in
$\overline{\mathcal{R}}$. Because the curves in $\overline{\mathcal{R}}$ are
disjoint, the Dehn twists about the curves commute.

Let

$\mathcal{B}=\displaystyle\bigcup_{j=2}^{7}\beta_{j}$

in $T_{0,0}\cup T_{0,1}\cup T_{1,0}$. As above, $\mathcal{B}$ is a marked
multi-curve that is disjoint from $\gamma$. Let

$\overline{\mathcal{B}}=\mathcal{B}\cup\rho_{1}(\mathcal{B})\cup\dots\cup\rho_{1}^{n-1}(\mathcal{B}).$

Let $\Phi_{b}$ as the composition of Dehn twists about all of the marked
curves in $\overline{\mathcal{B}}$. As with $\overline{\mathcal{R}}$, the Dehn
twists about curves in $\overline{\mathcal{B}}$ commute.

Let $\alpha_{1},\beta_{1}\subset T_{0,0}$ be the (marked) curves in Figure 5.
Let $\Phi$ be the composition of Dehn twists along all the curves
$\alpha_{1},\rho_{1}(\alpha_{1}),\dots,\rho_{1}^{n-1}(\alpha_{1})$ followed by
Dehn twists along all the curves
$\beta_{1},\rho_{1}(\beta_{1}),\dots,\rho_{1}^{n-1}(\beta_{1})$. Define the
map $f_{n,k}$ as:

$\displaystyle f_{n,k}$
$\displaystyle\coloneqq\rho_{2}\circ\Phi\circ\Phi_{b}\circ\Phi_{r}$

Since the curves about which we twist to construct $f_{n,k}$ satisfy the
conditions of Penner’s construction, $f_{n,k}$ is a pseudo-Anosov
homeomorphism.

Step 3: Bounding the Stretch Factor. Following Yazdi, our next goal is to find
an upper bound for the stretch factor of the pseudo-Anosov homeomorphisms
$f_{n,k}$.

Train tracks. Let $S$ be a surface. A train track in $S$ is graph embedded in
$S$ with that property that for every vertex $v$ of valence three or greater,
all edges adjacent to $v$ have the same tangent vector at $v$. Let
$\varphi:S\rightarrow S$ be a pseudo-Anosov homeomorphism. The map $\varphi$
is equipped with a train track whose image under $\varphi$ is homotopic to
itself. Such a train track is an invariant train track associated to
$\varphi$. Invariant train tracks have an associated matrix whose Perron-
Frobenius eigenvalue is the stretch factor of $\varphi$.

Yazdi uses the Lemma 4.2 to bound the spectral radius of the associated
matrices.

######  Lemma 4.2 (Lemma 2.3 of [33]).

Let $A$ be a non-negative integral matrix, $\Gamma$ be the adjacency graph of
$A$, and $V(\Gamma)$ the set of vertices of $\Gamma$. For each $v\in
V(\Gamma)$, define $v^{+}$ to be the set of vertices $u\in V(\Gamma)$ such
that there is an oriented edge from $v$ to $u$. Let $D$ and $k$ be fixed
natural numbers. Assume the following conditions hold for $\Gamma$:

  1. (i)

For each $v\in V(\Gamma)$ we have $\deg_{\text{out}}(v)\leq D$,

  2. (ii)

There is a partition $V(\Gamma)=V_{1}\cup\dots\cup V_{\ell}$ such that for
each $v\in V_{i}$ we have $v^{+}\subset V_{i+1}$, for any $1\leq i\leq\ell$
except possibly when $i=1$ or 3 (indices are mod $\ell$),

  3. (iii)

For each $v\in V_{1}$, we have $v^{+}\subset V_{2}\cup V_{3}$,

  4. (iv)

For each $v\in V_{3}$ we have $v^{+}\subset V_{3}\cup V_{4}$, and for $u\in
v^{+}\cap V_{3}$ we have $u^{+}\subset V_{4}$, and

  5. (v)

For all $3<j\leq k$ and each $v\in V_{j}$, the set $v^{+}$ consists of a
single element.

Then the spectral radius of $A^{\ell-1}$ is at most $4D^{4}$.

With this result in hand, we can find an upper bound for the stretch factor of
$f_{n,k}$

######  Lemma 4.3.

Let $\lambda_{n,k}$ be the stretch factor of $f_{n,k}$. Then there exists a
universal positive constant $C^{\prime}$ such that for every $n\geq 1$ and
$k\geq 3$, we have the following upper bound on $\log(\lambda_{n,k})$.

$\displaystyle\log(\lambda_{n,k})\leq C^{\prime}\frac{n}{g_{n,k}}$

###### Proof.

We deliberately constructed our curves so that all intersections of the multi-
curve $\\{\alpha_{1},\dots,\alpha_{8}\\}$ and
$\\{\beta_{1},\dots,\beta_{7}\\}$ occur in the subsurface $T_{0,0}$. The curve
$\beta_{3}$ intersects $\rho_{2}(\alpha_{3})$ at one point in $T_{0,1}$ and
$\beta_{7}$ intersects $\rho_{1}(\alpha_{8})$ at one point in $T_{1,0}$.

We define the following unions of marked curves:

$\displaystyle\mathcal{A}$
$\displaystyle\coloneqq\mathcal{B}\cup\mathcal{R}\cup\\{\alpha_{1},\beta_{1}\\}=\bigcup_{i=1}^{8}\alpha_{i}\cup\bigcup_{j=1}^{7}\beta_{j}$
$\displaystyle\vskip 3.0pt plus 1.0pt minus 1.0pt\overline{\mathcal{A}}$
$\displaystyle\coloneqq\mathcal{A}\cup\rho_{1}(\mathcal{A})\cup\dots\cup\rho_{1}^{n-1}(\mathcal{A})$
$\displaystyle\widehat{\mathcal{A}}$
$\displaystyle\coloneqq\overline{\mathcal{A}}\cup\rho_{2}\left(\overline{\mathcal{A}}\right)\cup\dots\cup\rho_{2}^{k-1}\left(\overline{\mathcal{A}}\right).$

Because $f_{n,k}$ is pseudo-Anosov, it has a corresponding invariant train
track $\tau$. Let $V_{\tau}$ be the space of all measured foliations that can
be obtained by varying the weights on the tracks of $\tau$. This forms a
finite dimensional cone of measures, all of which can be carried by the
combinatorial train track $\tau$. Furthermore, $f_{n,k}$ acts linearly on this
cone, and leaves the cone invariant, since $\tau$ is an invariant track for
$f_{n,k}$. Consider now the transverse measure $\mu_{\delta}$ for any curve
$\delta$ in $\widehat{\mathcal{A}}$. This transverse measure is carried by
$\tau$, and thus $\mu_{\delta}$ belongs in the cone of measures $V_{\tau}$.
Let $H$ be the subspace spanned by
$\\{\mu_{\delta}\mid\delta\subset\widehat{\mathcal{A}}\\}$. This linear
subspace is also left invariant by $f_{n,k}$, so we can restrict our attention
to the matrix of $f_{n,k}$ with respect to the basis given by the
$\\{\mu_{\delta}\mid\delta\subset\widehat{\mathcal{A}}\\}$. Let $M$ be the
matrix representing this linear action on $H$. Let $\Gamma$ be the adjacency
graph for $M$. Work of Penner [25] tells us that the Perron–Frobenius
eigenvalue of $M$ is the stretch factor of $f_{n,k}$.

To bound the spectral radius of $M$, we need to show that $\Gamma$ satisfies
the criteria of Lemma 4.2.

  1. (i)

There exists a constant $D^{\prime}$, independent of $n$ and $k$, such that
for every curve $\delta\in\widehat{\mathcal{A}}$, the geometric intersection
number between $\delta$ and every curve in $\overline{\mathcal{A}}$ is at most
$D^{\prime}$. Recall that
$f_{n,k}=\rho_{2}\circ\Phi\circ\Phi_{b}\circ\Phi_{r}$. Let $M_{1},M_{2},M_{3}$
and $M_{4}$ be the matrices describing the linear action of
$\Phi_{r},\,\Phi_{b},\,\Phi$ and $\rho_{2}$ on $H$, respectively. The matrix
$M$ can then be written as a product:

$\displaystyle M=M_{4}M_{3}M_{2}M_{1}.$

For a curve $\delta\in\widehat{\mathcal{A}}$, the $L^{1}$-norm of
$M_{i}(\mu_{\delta})$ is bounded above by the geometric intersection of
$f_{n,k}(\delta)$ with the curves in $\overline{\mathcal{A}}$. Thus each of
$M_{1}$, $M_{2}$ and $M_{3}$ will change the norm by a factor of at most
$(1+D^{\prime})$. Since $\rho_{2}$ will not change intersection numbers,
$M_{4}$ will preserve the $L^{1}$-norm. If we let $D=(1+D^{\prime})^{3}$, then
the outward degree of each vertex in $\Gamma$ is at most $D$.

For the remaining conditions, we partition the vertices of $\Gamma$. Observe

$\widehat{\mathcal{A}}=\rho_{2}^{-1}(\overline{\mathcal{A}})\cup\overline{\mathcal{A}}\cup\bigcup_{i=3}^{k}\rho_{2}^{i-2}(\overline{\mathcal{A}}).$

Then define $V_{1}$ as the vertices of $\Gamma$ corresponding to
$\rho_{2}^{-1}(\overline{\mathcal{A}})$, the set $V_{2}$ as the vertices of
$\Gamma$ corresponding to $\overline{\mathcal{A}}$, and $V_{i}$ for $3\leq
i\leq k$ as the vertices of $\Gamma$ corresponding to elements in
$\rho_{2}^{i-2}(\overline{\mathcal{A}})$.

  2. (ii)

Suppose that $v\in V_{i}$ for $i\neq 1,3$, is a vertex that corresponds to
$\mu_{\delta}$ for a curve $\delta\in\widehat{\mathcal{A}}$. Then $\delta$ is
disjoint from all curves in $\overline{\mathcal{A}}$. The action of
$\Phi\circ\Phi_{b}\circ\Phi_{r}$ on $\widehat{\mathcal{A}}$ will preserve the
set $\rho_{2}^{(i-2)\mod k}(\overline{\mathcal{A}})$ (and therefore $\delta$)
for each $i\neq 1,3$. Then $\rho_{2}$ will rotate the curve
$\Phi\circ\Phi_{b}\circ\Phi_{r}(\delta)$ to $\rho_{2}^{(i-1)\mod
k}(\overline{\mathcal{A}})$. That is:
$f_{n,k}=\rho_{2}\circ\Phi\circ\Phi_{b}\circ\Phi_{r}$ maps $\mu_{\delta}\in H$
to

$\sum_{\zeta\in\mathcal{Z}}\mu_{\zeta}$

where $\mathcal{Z}$ is a subset of $\rho_{2}^{(i-1)\mod
k}(\overline{\mathcal{A}})$. Therefore $f_{n,k}$ maps $v$ to a subset of
$V_{i+1}$.

  3. (iii)

To verify the third condition, we first look at the vertices $v\in V_{1}$ such
that $v^{+}\not\subset V_{2}$. Such vertices will correspond to the curves in
$\rho_{2}^{-1}(\overline{\mathcal{A}})$ that $\Phi\circ\Phi_{b}\circ\Phi_{r}$
maps to curves that are not in $\rho_{2}(\overline{\mathcal{A}})$. Because
$\rho_{1}$ and $\rho_{2}$ commute, we can write the curves of
$\rho_{2}^{-1}(\overline{\mathcal{A}})$ as:

$\rho_{2}^{-1}(\overline{\mathcal{A}})=\rho_{2}^{-1}(\mathcal{A})\cup\rho_{1}(\rho_{2}^{-1}(\mathcal{A}))\cup\cdots\cup\rho_{1}^{n-1}(\rho_{2}^{-1}(\mathcal{A})).$

The elements of $v^{+}$ that are not in $V_{2}$ correspond to the images of
curves in $\rho_{2}^{-1}(\overline{\mathcal{A}})$ under $f_{n,k}$ that are not
in $\overline{\mathcal{A}}$. As in Yazdi, the only curves in
$\rho_{2}^{-1}(\overline{\mathcal{A}})$ that intersect curves in
$\overline{\mathcal{A}}$ are those in the set:

$\displaystyle\mathcal{X}=\\{\rho_{1}^{i}(\rho_{2}^{-1}(\beta_{7}))\,\mid\,0\leq
i\leq n-1\\}.$

Therefore $\Phi\circ\Phi_{b}\circ\Phi_{r}$ maps curves in $\mathcal{X}$ to
curves in $\rho_{2}^{-1}(\overline{\mathcal{A}})\cup\overline{\mathcal{A}}$.
Then $f_{n,k}=\rho_{2}\circ\Phi\circ\Phi_{b}\circ\Phi_{r}$ maps curves in
$\mathcal{X}$ to curves in
$\overline{\mathcal{A}}\cup\rho_{2}(\overline{\mathcal{A}}).$ For any curve in
$\mathcal{X}$, the corresponding vertex $v\in V_{1}$ will have $v^{+}\subset
V_{2}\cup V_{3}.$ Moreover, $f_{n,k}$ maps the curves
$\rho_{2}^{-1}(\overline{\mathcal{A}})\setminus\mathcal{X}$ to curves in
$\overline{\mathcal{A}}$. Thus for any vertex $v\in V_{1}$ that does not
correspond to an element of $\mathcal{X}$, the set $v^{+}$ is contained in
$V_{2}$.

  4. (iv)

Similarly, we look for the $v\in V_{3}$ such that $v^{+}\not\subset V_{4}$.
Such vertices will correspond to the curves in
$\rho_{2}(\overline{\mathcal{A}})$ that $\Phi\circ\Phi_{b}\circ\Phi_{r}$ maps
to curves that are not in $\rho_{2}^{2}(\overline{\mathcal{A}})$. As above, we
have:

$\rho_{2}(\overline{\mathcal{A}})=\rho_{2}(\mathcal{A})\cup\rho_{1}(\rho_{2}(\mathcal{A}))\cup\cdots\cup\rho_{1}^{n-1}(\rho_{2}(\mathcal{A})).$

The elements of $v^{+}$ that are not in $V_{4}$ correspond to the the images
of $\rho_{2}(\overline{\mathcal{A}})$ that intersect the curves in
$\overline{\mathcal{A}}$. The only vertices of $V_{4}$ that correspond to such
curves are those in the set:

$\mathcal{Y}=\\{\rho_{1}^{i}(\rho_{2}(\alpha_{8}))\,\mid\,0\leq i\leq n-1\\}.$

For any element $v\in V_{3}$ corresponding to a curve in $\mathcal{Y}$ and any
$u\in v^{+}\cap V_{3}$, the vertex $u$ does not correspond to an element of
$\mathcal{Y}$. Therefore $u^{+}\subset V_{4}$.

  5. (v)

All the curves corresponding to an element of $V_{j}$, $3<j\leq k$ are
disjoint from all the curves in $\overline{\mathcal{A}}$. Thus, $f_{n,k}$ just
acts by rotation.

Let $\lambda=\lambda_{n,k}$ be the stretch factor of $f_{n,k}$. By Lemma 4.2,
we have:

$\displaystyle\lambda^{k-1}=\rho(M)^{k-1}=\rho(M^{k-1})\leq 4D^{4}.$

Then the logarithm of $\lambda$ satisfies:

$\log(\lambda^{k-1})=(k-1)\cdot\log(\lambda)\leq\log(4D^{4}).$

Then for $k\geq 2$

$\frac{k}{2}\log(\lambda)\leq(k-1)\log(\lambda)\leq\log(4D^{4}).$

On the other hand, we know $g_{n,k}=(14k-2)n+2\leq 14kn$ by Lemma 4.1.
Therefore

$\displaystyle\log(\lambda)\leq 2\log(4D^{4})\cdot\frac{1}{k}\leq
2\log(4D^{4})\cdot\frac{14n}{g_{n,k}}.$

Let $C^{\prime}\coloneqq 28\log(4D^{4})$ to complete the result. ∎

Step 4: The Mapping Torus.

We have now constructed an infinite family of non-orientable surfaces
$P_{n,k}$ and pseudo-Anosov homeomorphisms $f_{n,k}:P_{n,k}\to P_{n,k}$, but
this is not enough. In Lemma 4.1, we show that $\\{P_{n,k}\\}$ does not
include surfaces of infinitely many genera. We use the strategy of McMullen
[23] and our extension of the Thurston’s fibered face theory to fill in the
gaps.

Next we follow the strategy of Leininger–Margalit [19] to find a surface
embedded in the mapping torus of minimal genus. In our situation, this means
that we will construct an embedded surface homeomorphic to $\mathcal{N}_{3}$.

Figure 6. The curves $\gamma$ and $\widehat{\gamma}$ bound an a non-orientable
surface of genus 1.

######  Proposition 4.4.

Let $M_{n,k}$ be the mapping torus of $f_{n,k}$. Let $\mathcal{K}_{n,k}$
denote the fibered cone of $H^{1}(M_{n,k};\mathbb{R})$ corresponding to the
map $f_{n,k}$. There is a relatively orientable incompressible surface
$F_{n,k}$ embedded in $M_{n,k}$ that is homeomorphic to $\mathcal{N}_{3}$.
Moreover $F_{n,k}$ is transverse to the suspension flow direction given by
$f_{n,k}$ and the Poincaré dual of $F_{n,k}$ is in the closure
$\overline{\mathcal{K}_{n,k}}$.

###### Proof.

Let $\gamma\subset T_{0,0}$ be the curve shown in Figure 6. Note that $\gamma$
and $\Phi(\gamma)$ bound a non-orientable surface $\widehat{F}$ of genus 1
with boundary. For convenience, we will denote $\Phi(\gamma)$ by
$\widehat{\gamma}$. We are going to follow the image of $\gamma$ under powers
of $f_{n,k}$. Then we attach annuli to the boundary of $\widehat{F}$ to obtain
$\mathcal{N}_{3}$. Since $\gamma$ is disjoint from all curves in
$\overline{\mathcal{R}}$ and $\overline{\mathcal{B}}$ (as seen in Figure 5),
the maps $\Phi_{r}$ and $\Phi_{b}$ act trivially on $\gamma$. Recalling that
$f_{n,k}=\rho_{2}\circ\Phi\circ\Phi_{b}\circ\Phi_{r}$, we have the following:

$\displaystyle f_{n,k}(\gamma)$
$\displaystyle=\rho_{2}\circ\Phi\circ\Phi_{b}\circ\Phi_{r}(\gamma)$
$\displaystyle=\rho_{2}\circ\Phi(\gamma)$
$\displaystyle=\rho_{2}(\widehat{\gamma})$

It follows that for all $1\leq i\leq k$, the curve $f_{n,k}^{i}(\gamma)$ is
$\rho_{2}^{i}(\widehat{\gamma})$. For $1\leq i\leq k$, let $A_{i}$ be an
annulus in $M_{n,k}$ that connects $f_{n,k}^{i-1}(\gamma)$ to
$f_{n,k}^{i}(\gamma)$ obtained by following the suspension flow of $f_{n,k}$
around $M_{n,k}$. Let $A$ be the union of all of the $A_{i}$, which is also an
annulus. We can now construct the embedded surface $F_{n,k}$ by taking the
union of $A$ and $\widehat{F}$. The union of $\widehat{F}$ with $A$ has empty
boundary, so $F_{n,k}$ is homeomorphic to $\mathcal{N}_{3}$.

We now need to show that $F_{n,k}$ is relatively orientable. We construct a
outwards pointing normal vector field by combining the outwards pointing
vector fields on $\widehat{F}$ and $A$ given by following $\gamma$ along the
suspension flow. Let $v_{1}$ be a vector field on $\widehat{F}$ pointing in
the flow direction. Define $v_{2}$ to be a vector field on $A$ as follows: on
$\gamma$ define $v_{2}$ to be the vector field pointing in to $\widehat{F}$.
Flow the vector field along the suspension flow so $v_{2}$ is pointing away
from $\widehat{F}$ on $\widehat{\gamma}$.

Let $U$ be a neighborhood of $\gamma$ in $F_{n,k}=\widehat{F}\cup A$. Define
two bump functions $c_{1}$ and $c_{2}$ supported in $U$. Let $c_{1}$ be $1$ on
$\partial U\cap A$ and $0$ on $\widehat{F}$. Let $c_{2}$ be $1$ on $\partial
U\cap\widehat{F}$ and $0$ on $A$. We add the vector fields $v_{1}$ on
$\widehat{F}$ and $v_{2}$ on $A$ using these bump functions: the resulting
vector field is $c_{1}v_{1}+c_{2}v_{2}$. Observe that since $v_{1}$ points in
the flow direction, and $v_{2}$ points into the surface, the new vector field
$c_{1}v_{1}+c_{2}v_{2}$ is transverse, in the neighborhood of $\gamma$, to
$F_{n,k}$ (see Figure 7 for a picture of the resulting transverse vector
field).

We perform a similar construction in a small neighborhood of
$\widehat{\gamma}$: in this case, the fact that the vector field on
$\widehat{F}$ points in the flow direction, and the vector field on $A$ points
away from the surface $\widehat{F}$ ensures that the new vector field is
transverse, in a neighborhood of $\widehat{\gamma}$, to the surface $F_{n,k}$.

A key fact we use in this construction is that the vector field along $A$ that
starts pointing into $\widehat{F}$ at $\gamma$ comes back pointing away from
the surface at $\widehat{\gamma}$: this is because the homeomorphism
$f_{n,kj}$ maps the inner tubular neighborhood of $\gamma$ to the outer
tubular neighborhood of $\widehat{\gamma}$, where inner and outer tubular
neighborhood are the half tubular neighborhoods contained in $\widehat{F}$ and
the complement respectively. This fact about $f_{n,k}$ follows from its
definition, i.e. following the four homeomorphisms whose composition is
$f_{n,k}$.

$\gamma$
---
$\gamma$
---
$\widehat{\gamma}$
---
$\widehat{\gamma}$
---
$\widehat{F}$
---
$\widehat{F}$
---
$\widehat{F}$
---
$\widehat{F}$
---
Figure 7. The left side shows the vector fields $v_{1}$ on $\widehat{F}$ and
$v_{2}$ on $A$. The upper picture is a neighborhood of $\gamma$ and the lower
picture is a neighborhood of $\widehat{\gamma}$. The right side shows the
vector fields $c_{1}v_{1}+c_{2}v_{2}$ on neighborhoods of $\gamma$ and
$\widehat{\gamma}$.

The proof that $F_{n,k}$ can be isotoped to be transverse to the suspension
flow is the same as the proof Yazdi uses [33], which is a restatement of that
of Leininger–Margalit [19]. We include it here for completeness.

Let $N(\gamma)$ be a tubular neighborhood of $\gamma$ in $\widehat{F}$. Let
$\eta:\widehat{F}\xrightarrow{}[0,1]$ be a smooth function supported on
$N(\gamma)$ with the following properties:

  * •

$\eta^{-1}(1)=\gamma$ and

  * •

the derivative of $\eta$ vanishes on $\gamma$.

Let $\pi:M_{n,k}\rightarrow S^{1}$ be the projection map and let $t_{0}$ be
such that $\widehat{F}\subset\pi^{-1}(t_{0})$. Let
$g:\widehat{F}\xrightarrow{}M_{n,k}$ be the suspension flow of $f_{n,k}$
defined as $g(x)=(x,t_{0}+k\cdot\eta(x))$. Then the restriction of $g$ to the
interior of $\widehat{F}$ is an embedding into $M_{n,k}$ and
$g(\gamma)=\widehat{\gamma}$. Therefore the image of $\widehat{F}$ under $g$
is an embedded non-orientable surface of genus three. Moreover,
$g(\widehat{F})$ is isotopic to the natural embedding of $F_{n,k}$ in
$M_{n,k}$, and is transverse to the suspension flow. Therefore, the Poincaré
dual of $F_{n,k}$ is in $\overline{\mathcal{K}_{n,k}}$ by Theorem 2.11.

Finally, $F_{n,k}$ is incompressible in $M_{n,k}$ because $M_{n,k}$ is
hyperbolic, and $F_{n,k}$ is genus $3$, the lowest possible genus for a
hyperbolic non-orientable surface. ∎

Step 5: Filling in the Gaps. Recall that the family of surfaces $P_{n,k}$ that
we have constructed have genera in the set $\\{(14k-2)n+2\\}$. We now want to
construct surfaces of genera not in the set $\\{(14k-2)n+2\\}$ and pseudo-
Anosov homeomorphisms of those surfaces that have small stretch factors. To do
this we use the mapping torus $M_{n,k}=(P_{n,k},f_{n,k})$. Recall from
Proposition 4.4 that there exists a relatively incompressible surface
$F_{n,k}$ in $M_{n,k}$ that is homeomorphic to $\mathcal{N}_{3}$. Let
$P_{n,k}^{r}$ be the oriented sum of $P_{n,k}$ and $rF_{n,k}$, as defined in
Proposition 3.3. The surfaces $P_{n,k}^{r}$ will be surfaces of the remaining
genera.

######  Lemma 4.5.

The surface $P^{r}_{n,k}$ is of genus $g^{r}_{n,k}=g_{n,k}+r$. In particular,
as $r$ varies between $0$ and $14n$, the genera of $\\{P^{r}_{n,k}\\}$ spans
the range between $g_{n,k}$ and $g_{n,k+1}$. Moreover, $P^{r}_{n,k}$ is
isotopic to a fiber of a fibration of $M_{n,k}$ with pseudo-Anosov monodromy
that fixes $2n$ of the singularities of its invariant foliation.

###### Proof.

The Euler characteristic of an oriented sum is the sum of the Euler
characteristics of the summands:

$\displaystyle\chi(P^{r}_{n,k})$
$\displaystyle=\chi(P_{n,k})+r\cdot\chi(F_{n,k})$
$\displaystyle=(-2g_{n,k}+2)-2r$ $\displaystyle=-2(g_{n,k}+r)+2.$

Since $P_{n,k}^{r}$ has no boundary or punctures, we have that the genus of
$P_{n,k}^{r}$ is $g_{n,k}+r$.

By Lemma 4.4 we know that $F_{n,k}$ is incompressible and transverse to the
suspension flow of given by $f_{n,k}$. Therefore by Proposition 3.2, there is
a pseudo-Anosov homeomorphism of $P_{n,k}^{r}=P_{n,k}+rF_{n,k}$.

As in Yazdi [32, Lemma 3.5], $f_{n,k}$ fixes the $2n$ singularities of the
stable foliation that are the intersection points of the axis of $\rho_{1}$
with $P_{n,k}$. By Lemma 4.4, the surface $F_{n,k}$ can be isotoped to be
transverse to the suspension flow and disjoint from the orbit of the $2n$
singularities of $f_{n,k}$. Hence the monodromy $f^{r}_{n,k}$ still fixes the
corresponding $2n$ singularities on $P^{r}_{n,k}$. ∎

We now prove the non-orientable version of the final piece of Yazdi’s proof
[33, Lemma 3.6].

######  Lemma 4.6.

Let $\lambda_{n,k}^{r}$ be the stretch factor of
$f_{n,k}^{r}:P_{n,k}^{r}\rightarrow P_{n,k}^{r}$. Then there exists a constant
$C>0$ such that for every $n\geq 1$, $k\geq 3$, and $0\leq r\leq 14n$ we have
the following upper bound on $\log(\lambda_{n,k}^{r})$:

$\displaystyle\log(\lambda^{r}_{n,k})\leq C\frac{n}{g^{r}_{n,k}}.$

###### Proof.

Let $\mathcal{K}=\mathcal{K}_{n,k}$ be the fibered cone in
$H^{1}(M_{n,k};\mathbb{R})$ corresponding to $f_{n,k}$ and
$h:\mathcal{K}\xrightarrow[]{}\mathbb{R}$ the function described in Theorem
3.4. Note that $g_{n,k}\geq 42$, therefore we have the following bounds on
$g_{n,k}^{r}$:

$\displaystyle g^{r}_{n,k}$ $\displaystyle=g_{n,k}+r$ $\displaystyle\leq
g_{n,k}+14n$ $\displaystyle<2g_{n,k}.$

Let $\omega$ be the Poincaré dual of $P^{r}_{n,k}$ and $\alpha$ the Poincaré
dual of $P_{n,k}$. Then the following string of inequalities holds:

$\displaystyle h([\omega])$ $\displaystyle<h([\alpha])$ (Convexity of $h$)
$\displaystyle\leq C^{\prime}\frac{n}{g_{n,k}}$ (Lemma 4.3) $\displaystyle\leq
2C^{\prime}\frac{n}{g^{r}_{n,k}}$ $\displaystyle\text{(upper bound for
$g_{n,k}^{r}$)}.$

∎

In the initial construction of $P_{n,k}$, there were $2n$ marked points, which
were singularities of the map $f_{n,k}$. By the construction of $P_{n,k}^{r}$,
these marked points are also singularities of $f^{r}_{n,k}$. Now we puncture
$P_{n,k}^{r}$ at $n$ of these marked points. We could think of this as
removing all copies of the point $p$ in the construction of $P_{n,k}$ in step
1.

We can now give a proof of Theorem 1.1.

###### Proof of Theorem 1.1.

As above, the lower bound follows easily from the lower bound in the
orientable setting.

To find the upper bound, let $C^{\prime}=\frac{C}{2}$ be the value given in
Lemma 4.6. Let $B^{\prime}_{2}(n)$ be the following quantity:

$\displaystyle
B^{\prime}_{2}(n)=\max\\{2C^{\prime}n,\ell^{\prime}_{1,n},2\ell^{\prime}_{2,n},\dots,(40n+1)\ell^{\prime}_{40n+1,n}\\}.$

By Proposition 3.2 and Lemma 4.6, $B^{\prime}_{2}(n)$ is an upper bound for
$g\cdot\ell^{\prime}_{g,n}$. ∎

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