# simple closed geodesics in cusped hyperbolic $3$-manifolds

Feihuang Xia

###### Abstract.

We show that cusped finite-volume hyperbolic $3$-manifolds contain infinitely
many simple closed geodesics.

##  1\. Introduction

A closed geodesic in a Riemannian manifold is said to be simple if it has no
self-intersections, or nonsimple otherwise. It is an open question which
Riemannian manifolds contain simple closed geodesics or which do not. In this
paper, we focus on simple closed geodesics in hyperbolic manifolds, that is,
complete Riemannian manifolds of constant curvature $-1$.

In any closed hyperbolic $3$-manifold, any homotopy class of a loop admits a
unique geodesic representative, and any shortest closed geodesic must be
simple. On the other hand, some infinite-volume hyperbolic $3$-manifolds have
no simple closed geodesics, such as $\mathbb{H}^{3}$. C. Adams, J. Hass and P.
Scott showed that the Fuchsian group corresponding to the thrice-punctured
sphere generates the only exception of a complete non-elementary orientable
hyperbolic $3$-manifold that does not contain a simple closed geodesic [1].

Heuristically we expect many hyperbolic $3$-manifolds contain infinitely many
simple closed geodesics. For example, S. M. Miller proved that many hyperbolic
$3$-manifolds contain infinitely many simple closed geodesics. Examples
include the figure-eight knot complement [5]. S. M. Kuhlmann showed that if a
closed orientable hyperbolic $3$-manifold satisfies certain geometric and
arithmetic conditions, it contains infinitely many simple closed geodesics
[3]. She actually found many such examples by program searching. In the cusped
manifold case, S. M. Kuhlmann gave a general result. She showed that there are
infinitely many simple closed geodesics in cusped finite-volume orientable
hyperbolic $3$-manifolds [4].

T. Chinburg and A. W. Reid proved that there exist infinitely many mutually
incommensurable closed hyperbolic $3$-manifolds in which all primitive closed
geodesies are simple by some arithmetic hyperbolic $3$-manifold theory [2],
where we say two manifolds $M$ and $N$ are commensurable if there is a
manifold that covers both $M$ and $N$ with ﬁnite degrees and incommensurable
otherwise. They showed that if there is a nonsimple geodesic in a closed
hyperbolic $3$-manifold $M=\mathbb{H}^{3}/\Gamma$, an invariant of the
commensurability class of $\Gamma$ called invariant quaternion algebra must
satify certain condition. They then construct infinitely many nonisomorphic
quaternion algebras that do not meet this condition. However, it is still
unknown whether all closed hyperbolic $3$-manifolds contain infinitely many
simple closed geodesics.

In this paper, we extend Kuhlmann’s result to higher dimensions with toral
ends, and to the non-orientable, $3$-dimensional cusped case:

######  Theorem 1.1.

A cusped nonelementary hyperbolic $n$-manifold contains infinitely many simple
closed geodesics if there is a cusp which is a flat $(n-1)$-manifold
$\mathbb{R}^{n-1}/\Gamma_{c}$ such that $\Gamma_{c}\cong\mathbb{Z}^{n-1}$ for
$n\geq 3$.

######  Theorem 1.2.

Any nonelementary hyperbolic $3$-manifold with a virtually rank-$2$ cusp
contains infinitely many simple closed geodesics. In particular, any cusped
finite-volume hyperbolic $3$-manifold contains infinitely many simple closed
geodesics.

Since it seems true that the self-intersections of a loop must be rare in high
dimensional manifolds, there is a well-known question:

######  Question.

Does every cusped hyperbolic $n$-manifold M contain infinitely many simple
closed geodesics?

Our method requires to analyse the cusp structure for constructing simple
closed geodesics. However, the cusp structure will be complicated as the
dimension increases. Our approach to construct simple closed geodesics depends
on the location of self-tangencies. So we will discuss different situations as
the cusp structure or the self-tangencies change respectively. These
difficulties do not quite show up in the orientable $3$-dimensional case, but
they will always appear when we consider higher dimensional cases. Therefore,
our case-by-case approach to the $3$-dimensional case does not automatically
generalize to higher dimensions.

To prove these theorems, we follow the basic idea of Kuhlmann, and analyze
structure of the cusp. The cusp of a hyperbolic $n$-manifold $M$ admits a flat
metric. Therefore, it can be represented as $\mathbb{R}^{n-1}/\Gamma_{c}$,
where $\mathbb{R}^{n-1}$ contains Euclidean structure and $\Gamma_{c}$ is an
isometric discrete group acting freely.

When $M$ is a finite-volume hyperbolic orientable $3$-manifolds,
$\Gamma_{c}\cong\mathbb{Z}^{2}$, generated by two parabolic transformations.
In [4], S.M. Kuhlmann considered an infinite family of closed geodesics that
surround the cusp and showed that infinitely many of those are embedded.

To find this family, we consider the axis of isometries $t\circ g$, where $g$
is some certain isometry and $t$ is some isometry stabilizing a distinguished
horocusp, where the axis is defined to be the geodesic which is fixed under a
hyperbolic isometry. We actually find out a specific family of such geodesics
by letting $t$ vary along some carefully chosen direction.

To prove these geodesics are simple, we will estimate the locations of these
axis first. Then we claim that the ends of these geodesics are limited in some
certain small open balls when $t$ is ‘large’ enough. Then we can study these
geodesics more conveniently. That are Lemma 2.1 and Lemma 2.2, which we will
prove in the next section.

In this paper, we will use the similar method in [4] to find this family of
closed geodesics and make some extensions. Then we deal with the
$3$-dimensional non-orientable case.

This paper is organized as follows. In section 2, we define some notations and
prove two lemmas which are mentioned above. In section 3, we extend the
conclusion of [4] to $n$-dimension. In section 4, we discuss the case of
nonelementary hyperbolic $3$-manifolds with a virtually rank-$2$ cusp and find
out infinitely simple closed geodesics. In section 5, we discuss some further
work.

######  Acknowledgments.

I would like to thank my supervisor Yi Liu for introducing me to this project,
guiding me through my research and providing me many helpful comments and
remarks along the way.

##  2\. Preparation

In this section, we will introduce some preliminary notations and some
elementary estimates. The following notations will be used throughout this
paper: the horocusp $\mathcal{H}_{0}$ and $\mathcal{H}_{\infty}$, the group of
isometries $\Gamma_{\infty}$, the isometry $g_{0}$, the distance function
$d_{E}$, and the points $A_{0}$, $B_{0}$, $B_{t}$, $C_{t}$ and $D_{t}$. They
will be defined in this section.

We will explain that the ends of axis of $t\circ g$ are limited in some
certain small open balls when $t$ is ‘large’ enough, where $g$ is some certain
isometry and $t$ changes in the set $\Gamma_{\infty}$ which will be defined
later. We estimate the location of the axis of $t\circ g$.

Let $M=\mathbb{H}^{n}/\Gamma$ be a nonelementary hyperbolic $n$-manifold with
at least one cusp, where $\Gamma\cong\pi_{1}(M)$ is a nonelementary discrete
torsion-free subgroup of the isometry group of hyperbolic $n$-space
$\mathrm{Isom}(\mathbb{H}^{n})$. We choose a cusp $\mathcal{C}$ of $M$. Since
$\Gamma$ is a nonelementary, we can always take the maximal embedded horocusp
$\mathcal{T}$ of $\mathcal{C}$ whose boundary contains a finite numbers of
self-tangencies. Pick a distinguished self-tangency $A$.

Let $\widetilde{A}$ be a lift of $A$ under the covering projection
$\pi:\widetilde{M}=\mathbb{H}^{n}\to M$. $\mathcal{H}$ and
$\mathcal{H^{\prime}}$ are two horoballs which are preimages
$\pi^{-1}(\mathcal{T})$ of $\mathcal{T}$ that intersect at $\widetilde{A}$.
Since $\mathcal{H}$ and $\mathcal{H^{\prime}}$ both cover $\mathcal{T}$, there
must be some isometries relating them, namely, there is some isometry
$g\in\Gamma$ such that $g(\mathcal{H})=\mathcal{H^{\prime}}$.

Figure 1. the maximal embedded horoballs

For any isometries $g,g^{\prime}\in\Gamma$ which map $\mathcal{H}$ to
$\mathcal{H^{\prime}}$, $g^{\prime}\circ g^{-1}$ stabilizes
$\mathcal{H^{\prime}}$. It means that $g^{\prime}=t\circ g$ and $t\in\Gamma$
is an isometry stabilizing $\mathcal{H^{\prime}}$. We fix a distinguished $g$
and adjust $t$ to get a family of isometries $t\circ g$ which map
$\mathcal{H}$ to $\mathcal{H^{\prime}}$.

Consider the upper half-space model of $\mathbb{H}^{n}$, which is the space
$\mathbb{H}^{n}=\\{(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}|x_{n}>0\\}$ with the
Riemannian metric $(dx_{1}^{2}+\cdots+dx_{n}^{2})/x_{n}^{2}$. Without loss of
generality, we assume that $\mathcal{H}$ is a Euclidean ball of radius $1$
centered at $(0,\cdots,0,\frac{1}{2})$, and $\mathcal{H^{\prime}}$ is the
domain $\\{(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}|x_{n}\geq 1\\}$ above the
hyperplane $x_{n}=1$. $\mathcal{H^{\prime}}$ intersects $\mathcal{H}$ at the
point $A_{0}=(0,\cdots,0,1)$. We rewrite $\mathcal{H}$ as $\mathcal{H}_{0}$,
and $\mathcal{H^{\prime}}$ as $\mathcal{H}_{\infty}$ below.

Let $g_{0}$ be our fixed isometry that sends $\mathcal{H}_{0}$ to
$\mathcal{H}_{\infty}$ and $\Gamma_{\infty}\subset\Gamma$ consist of all the
isometries stabilizing $\mathcal{H}_{\infty}$. We fix a fundamental domain
$\mathcal{D}\subset\partial H_{\infty}$ contains $A_{0}$ with respect to the
action of $\Gamma_{\infty}$, for example, a suitable parallelogram. Possibly
after modifying $g_{0}$ with a translation in $\Gamma_{\infty}$, we can assume
that the point $B_{0}=g_{0}(A_{0})$ lies in $\mathcal{D}$. Denote $t\circ
g_{0}$ as $g_{t}$ where $t\in\Gamma_{\infty}$ is an isometry stabilizing
$\mathcal{H}_{\infty}$ and $B_{t}=g_{t}(A_{0})=t(B_{0})$. we use $\|t\|$ to
denote $d_{E}(A_{0},B_{t})$, where $d_{E}$ stands for the Euclidean distance
with respect to the induced Riemannian metric on
$\partial\mathcal{H}_{\infty}$.

Figure 2. the upper half-space model

Suppose $g_{t}$ is hyperbolic and denote the axis of $g_{t}$ as $\gamma_{t}$
which intersects
$\partial\mathcal{H}_{\infty}=\\{(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}|x_{n}=1\\}$
at the points $C_{t}$ and $D_{t}$. Let $\gamma_{\infty}$ be the line which is
perpendicular to $\partial{\mathbb{H}}^{n}$ through the point $A_{0}$. We will
see that $C_{t}$, $D_{t}$ tend to $A_{0}$, $B_{t}$ respectively, then we can
study the geodesic $\gamma_{t}$ via $A_{0}$ and $B_{t}$ when $\|t\|$ is large
enough.

In [4], S. M. Kuhlmann use fractional linear transformations to get the
similar result in $3$-dimensional case. For general dimension $n$, we switch
to vector calculations in the upper half space model.

######  Lemma 2.1.

For $t\in\Gamma_{\infty}$, $C_{t}\to A_{0},D_{t}\to B_{t}$, when
$\|t\|\to\infty$.

Here $D_{t}\to B_{t}$ means $d_{E}(D_{t},B_{t})$ will be arbitrarily small
with $\|t\|$ increasing.

###### Proof.

We estimate the endpoints of $\gamma_{t}$. First we describe $g_{t}$ by some
simple isometries.

Denote the inversion about the Euclidean $(n-1)$-sphere of radius $1$ centered
at the origin as $h_{1}$:

$h_{1}(\overrightarrow{X})=\frac{\overrightarrow{X}}{\|\overrightarrow{X}\|^{2}},$

where $\|\cdot\|$ means the Euclidean norm
$\sqrt{(\overrightarrow{x},\overrightarrow{x})}$ in vector space as usual.

Denote the reflection about a Euclidean $(n-1)$-plane orthogonal to
$\partial\mathbb{H}^{n}$ which sends $A_{0}$ to $B_{0}$ as $h_{2}$:

$h_{2}(\overrightarrow{X})=\overrightarrow{X}-\frac{2(\overrightarrow{X},\overrightarrow{b})}{\|\overrightarrow{b}\|^{2}}\overrightarrow{b}+\overrightarrow{b},where\overrightarrow{b}=\overrightarrow{A_{0}B_{0}}.$

It is obvious that $h_{1}$ sends $\mathcal{H}_{0}$ to $\mathcal{H}_{\infty}$
fixing $A_{0}$, and $h_{2}$ keeps $\mathcal{H}_{\infty}$ sending $A_{0}$ to
$B_{0}$. Let $h=h_{2}\circ h_{1}$. Then we see that
$h(\mathcal{H}_{0})=\mathcal{H}_{\infty}$ and $h(A_{0})=B_{0}$.

Notice that $g_{0}$ sends $\mathcal{H}_{0}$ to $\mathcal{H}_{\infty}$ and
$g_{0}(A_{0})=B_{0}$, so $h^{-1}g_{0}$ is an isometry keeping $A_{0}$ and
$\mathcal{H}_{\infty}$. Therefore, $h^{-1}g_{0}$ fixes all plane that parallel
with $\partial\mathbb{H}^{n}$. So $h^{-1}g_{0}\in O(\mathbb{R}^{n})$, where
$O(\mathbb{R}^{n})$ consists of orthogonal $n\times n$ matrices. We denote
$h^{-1}g_{0}$ as $V$.

Now we have $g_{0}=h\circ V,g_{t}=t\circ h\circ V$ where $t$ is a Euclidean
isometry stabilizing $\mathcal{H}_{\infty}$. Assume
$t(\overrightarrow{X})=A\overrightarrow{X}+\overrightarrow{c}$ where $A\in
O(\mathbb{R}^{n})$.

To solve the endpoints of $\gamma_{\infty}$, let
$g_{t}(\overrightarrow{X})=\overrightarrow{X}$. Then we obtain

$A(\frac{V\overrightarrow{X}}{\|V\overrightarrow{X}\|^{2}}-\frac{2(V\overrightarrow{X},\overrightarrow{b})}{\|V\overrightarrow{X}\|^{2}\|\overrightarrow{b}\|^{2}}\overrightarrow{b}+\overrightarrow{b})+\overrightarrow{c}=\overrightarrow{X}$
$\overrightarrow{X}-A\frac{V\overrightarrow{X}}{\|V\overrightarrow{X}\|^{2}}+\frac{2(V\overrightarrow{X},\overrightarrow{b})}{\|V\overrightarrow{X}\|^{2}\|\overrightarrow{b}\|^{2}}A\overrightarrow{b}=A\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{A_{0}B_{t}}$

Notice that $A,V\in O(\mathbb{R}^{n})$, so:

(1) $\displaystyle\|\overrightarrow{A_{0}B_{t}}\|$
$\displaystyle\leq\|\overrightarrow{X}\|+\left\|\frac{AV\overrightarrow{X}}{\|V\overrightarrow{X}\|^{2}}\right\|+\left\|\frac{2(V\overrightarrow{X},\overrightarrow{b})}{\|V\overrightarrow{X}\|^{2}\|\overrightarrow{b}\|^{2}}V\overrightarrow{b}\right\|$
$\displaystyle\leq\|\overrightarrow{X}\|+\frac{3}{\|\overrightarrow{X}\|}$

and

(2) $\displaystyle\|\overrightarrow{A_{0}B_{t}}-\overrightarrow{X}\|$
$\displaystyle\leq\left\|\frac{AV\overrightarrow{X}}{\|V\overrightarrow{X}\|^{2}}\right\|+\left\|\frac{2(V\overrightarrow{X},\overrightarrow{b})}{\|V\overrightarrow{X}\|^{2}\|\overrightarrow{b}\|^{2}}V\overrightarrow{b}\right\|$
$\displaystyle\leq\frac{3}{\|\overrightarrow{X}\|}$

When $\|t\|=\|\overrightarrow{A_{0}B_{t}}\|\to\infty$, we get
$\|\overrightarrow{X}\|\to 0$ or $\infty$ by $(1)$ and
$\|\overrightarrow{A_{0}B_{t}}-\overrightarrow{X}\|\to 0$ as
$\|\overrightarrow{X}\|\to\infty$ by $(2)$.

Denote the two roots as $x_{1}$ and $x_{2}$, then $x_{1}\to a_{0}$, $x_{2}\to
b_{t}$ as $\|t\|\to\infty$, where $a_{0}$, $b_{t}$ are the images of $A_{0}$
,$B_{t}$ respectively under the orthogonal projection from
$\partial\mathcal{H}_{\infty}$ to $\partial\mathbb{H}^{n}$.

Recall that $C_{t},D_{t}$ are the intersection of $\gamma_{t}$ and
$\partial\mathcal{H}_{\infty}$. We get $C_{t}\to A_{0}$ and $D_{t}\to B_{t}$.
∎

Consider the projection image $\pi(\gamma_{t})$ of $\gamma_{t}$ in $M$. We
will show that it contains two distinct arcs. Denote the arc
$g_{t}^{-1}(D_{t})C_{t}$ as $s_{t}$ and arc $C_{t}D_{t}$ as $l_{t}$. Then
$\pi(s_{t})$ is the part of $\pi(\gamma_{t})$ outside $\mathcal{T}$ and
$\pi(l_{t})$ is the part inside $\mathcal{T}$, where $\mathcal{T}$ is the
maximal embedded horocusp and $\pi$ is the covering map of $\mathbb{H}^{n}\to
M$ mentioned before.

Intuitively, when $\|t\|$ is sufficiently large, $\pi(s_{t})$ will be a short
geodesic segment very close to the self-tangency of the maximal horocusp, so
$\pi(s_{t})$ has no self-intersection. The following Lemma 2.2 gives a
sufficient condition for $\pi(l_{t})$ to have no self-intersection. Recall
that $\Gamma_{\infty}\subset\Gamma$ consists of all the isometries stabilizing
$\mathcal{H}_{\infty}$. Then $\Gamma_{\infty}$ stabilizes every Euclidean
hyperplane parallel to $\partial H_{\infty}$.

######  Lemma 2.2.

When $\|t\|$ is large enough and if there is no points $M,N$ on the arc
$l_{t}=C_{t}D_{t}$ such that there is an isometry $\tau\in\Gamma_{\infty}$
satisfying $\tau(M)=N$, then $\pi(\gamma_{t})$ is simple closed geodesic.
Moreover, the lemma is also true if we replace the arc $l_{t}=C_{t}D_{t}$ by
the projected line segment $C_{t}D_{t}$ on $\partial\mathcal{H}_{\infty}$.

###### Proof.

By Lemma 2.1, $C_{t}$ and $g_{t}^{-1}(D_{t})$ approach $A_{0}$ as
$\|t\|\to\infty$. $\pi(s_{t})$ is embedding because $\Gamma$ is discrete.

If $\pi(l_{t})$ is not embedding, there must be some points $M,N$ such that
$\tau(M)=N$ where $\tau\in\Gamma_{\infty}$. This is the first claim. Then we
consider the case of the projected line segment $C_{t}D_{t}$. Denote
$M^{\prime},N^{\prime}$ to be the images of $M$ and $N$ under projection.
According to the given conditions, there are no maps from $M^{\prime}$ to
$N^{\prime}$, hence there are no maps from $M$ to $N$ neither. ∎

##  3\. The $n$-dimensional case with toral cusps

We are now ready to prove the first theorem in the paper which is Theorem 1.1
in Introduction:

######  Theorem 3.1.

A cusped nonelementary hyperbolic $n$-manifold contains infinitely many simple
closed geodesics if there is a cusp which is a flat $(n-1)$-manifold
$\mathbb{R}^{n-1}/\Gamma_{c}$ such that $\Gamma_{c}\cong\mathbb{Z}^{n-1}$ for
$n\geq 3$.

In this case, $\Gamma_{\infty}$ corresponding to this cusp is generated by
$(n-1)$ translations when we restrict isometries in $\Gamma_{\infty}$ to the
boundary of the horoball $\mathcal{H}_{\infty}$.

###### Proof.

It suffices to show that there are infinitely many $\gamma_{t}$ whose $l_{t}$
is embedded by Lemma 2.2.

Let $\Gamma_{\infty}$ is generated by $t_{1},t_{2},\cdots,t_{n-1}$ which are
all linearly independent translations. Consider
$\partial\mathcal{H}_{\infty}=\mathbb{R}^{n-1}$ and take the translation
vectors of $t_{1},t_{2},\cdots,t_{n-1}$ as a basis in it.

Let $A_{0}=(0,\cdots,0),B_{0}=(b_{1},\cdots,b_{n-1})$ where $0\leq b_{i}<1$
because we set $B_{0}$ in tha same fundamental domain of
$\mathcal{H}_{\infty}$ with $A_{0}$. It is also easy to see that not all
$b_{i}$ equal to $0$. We assume $b_{2}\neq 0$ without loss of generality.

Take the isometry $\tau_{n}=t_{1}^{n}$, so
$B_{\tau_{n}}=(b_{1}+n,b_{2},\cdots,b_{n-1})$. We assume the $l_{\tau_{n}}$ is
not embedded to get a contradiction. Assume there are points $M,N$ in the line
segment $C_{\tau_{n}}D_{\tau_{n}}$ on $\partial\mathcal{H}_{\infty}$ and there
is a $t=\sum_{i=0}^{n-1}w_{i}t_{i}\in\Gamma_{\infty}$ such that $t(M)=N$ where
$w_{i}\in\mathbb{Z}$. Then
$(N_{1},\cdots,N_{n-1})=(M_{1}+w_{1},\cdots,M_{n-1}+w_{n-1})$ where
$M_{i},N_{i}$ is the $i$-th coordinate of $M,N$.

Due to Lemma 2.1, we take $n$ large enough such that
$\|C_{\tau_{n}}-A_{0}\|,\|D_{\tau_{n}}-B_{\tau_{n}}\|<\varepsilon$ for some
$\varepsilon$ small suffiently. Then
$\|M_{i}-N_{i}\|\leq\|(C_{\tau_{n}})_{i}-(D_{\tau_{n}})_{i}\|\leq
b_{i}+2\varepsilon<1$, we know that $w_{i}=0$, $2\leq i\leq n-1$.

Then $t=w_{1}t_{1}$. Hence $\overrightarrow{MN}\parallel t_{1}$, here we
denote the translation vector of $t_{1}$ as $\overrightarrow{t_{1}}$. Since
$M$ and $N$ are on the line segment $C_{\tau_{n}}D_{\tau_{n}}$,
$C_{\tau_{n}}D_{\tau_{n}}\parallel\overrightarrow{t_{1}}$, too. However,
$\|(C_{\tau_{n}})_{2}-(D_{\tau_{n}})_{2}\|\geq b_{2}-2\varepsilon>0$ since
$b_{2}\neq 0$. It means that
$\overrightarrow{MN}\nparallel\overrightarrow{t_{1}}$. This is a contraction.

So $\gamma_{\tau_{n}}$ is a simple closed geodesic when $n$ is large enough.
This leads to the conclusion. ∎

######  Remark.

The method follows [4]. In [4], S.M. Kuhlmann first use fractional linear
transformation to get the result of Lemma 2.1 in $3$-dimensional case. Then
she proposes a certain family of translations $B_{\tau_{n}}$ and use this
method to finish her proof, since the cusp of cusped orientable hyperbolic
$3$-manifold of finite-volume is a torus, which means
$\Gamma_{c}\cong\mathbb{Z}^{2}$.

The main theorem of [4] is a special case of Theorem 3.1:

######  Corollary 3.1.

Any cusped finite-volume orientable hyperbolic $3$-manifold contains
infinitely many simple closed geodesics.

##  4\. The non-orientable $3$-dimensional case

In this section, we consider the case of nonelementary hyperbolic
$3$-manifolds with a virtually rank-$2$ cusp. First, we discuss all the
elements in $\Gamma_{\infty}$ if there are some non-orientable transformations
in it.

If there is an isometry $s\in\Gamma_{\infty}$ and $s$ is orientation-
reversing, then $s$ is a glide reflection. Since $\Gamma_{\infty}$ act freely,
all the translation vectors of glide reflections in $\Gamma_{\infty}$ are
parallel to the translation vector of $s$. Suppose $s$ contains the shortest
translation distance without loss of generality. Take an appropriate
coordinate system, we can set $s(\vec{X})=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\vec{X}+\begin{pmatrix}\alpha\\\ 0\end{pmatrix}$, where
$\alpha>0$.

Since the hyperbolic manifolds we consider have a virtually rank-$2$ cusp,
there exists some translation whose vector is not parallel to the vector of
$s$. For any translation $t^{\prime}(\vec{X})=\vec{X}+\begin{pmatrix}a\\\
b\end{pmatrix}\in\Gamma_{\infty},(b\neq 0)$,
$st^{\prime}s^{-1}(\vec{X})=\vec{X}+\begin{pmatrix}a\\\ -b\end{pmatrix}$ and
$\vec{X}+\begin{pmatrix}0\\\
2b\end{pmatrix}=(t^{\prime}-sts^{-1})(\vec{X})\in\Gamma_{\infty}$. Then the
translation vector of $t^{\prime}-sts^{-1}$ is perpendicular to
$\begin{pmatrix}\alpha\\\ 0\end{pmatrix}$.

Take a translation of smallest distance among those translations whose vector
is perpendicular to $\begin{pmatrix}\alpha\\\ 0\end{pmatrix}$. Denote it as
$t(\vec{X})=\vec{X}+\begin{pmatrix}0\\\ \beta\end{pmatrix}$. Then $s$, $t$ are
the generators of $\Gamma_{\infty}$. For any $w\in\Gamma_{\infty}$, we can
write it as $w=s^{n}t^{m}$ since $tst=s$. Then we can write down all the
elements in $\Gamma_{\infty}$:

If $n=2k$, $w(\vec{X})=\vec{X}+\begin{pmatrix}n\alpha\\\ m\beta\end{pmatrix}$
is a translation.

If $n=2k+1$, $w(\vec{X})=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\vec{X}+\begin{pmatrix}n\alpha\\\ -m\beta\end{pmatrix}$ is a
glide reflection whose reflection axis is $y=-m\beta/2$ and translation
distance is $n\alpha$ along $x$-axis. So all the reflection axis of glide
reflection in $\Gamma_{\infty}$ is $m\beta/2$ for some $m\in\mathbb{Z}$.

With the above description of a nonorientable cusp in mind, we prove the
following theorem.

######  Theorem 4.1.

Any nonelementary hyperbolic $3$-manifold with a virtually rank-$2$ cusp
contains infinitely many simple closed geodesics. In particular, any cusped
finite-volume hyperbolic $3$-manifold contains infinitely many simple closed
geodesics.

###### Proof.

It is suffices to show the case if there are some orientation-reversing
transformations in $\Gamma_{\infty}$.

Due to the discussion above, we assume that $\Gamma_{\infty}$ is generated by
the glide reflection $s$ and the translation $t$. Since we have showed that
the translation vector of $s$ and $t$ are perpendicular, consider the
fundamental region $\mathcal{D}=\\{(x,y)~{}|~{}0\leq x<\alpha,-\beta/2\leq
y<\beta/2\\}$ in $\partial\mathcal{H}_{\infty}=\mathbb{R}^{2}$ with respect to
the action of $\Gamma_{\infty}$. We have points $A_{0},B_{0}\in\mathcal{D}$
and $A_{0}\neq B_{0}$. Below we consider different cases according to the
position of $A_{0}$ and $B_{0}$.

Case 1. Suppose that the x-coordinates of $A_{0}$ and $B_{0}$ are different.
Consider $\tau_{n}(\vec{X})=\vec{X}+\begin{pmatrix}0\\\ n\beta\end{pmatrix}$
for any integer $n$. Then $\pi(\gamma_{\tau_{n}})$ is a simple closed geodesic
when $n$ is large enough as we proved in Theorem 3.1.

Figure 3. Case 1

Case 2. Suppose that the x-coordinates of $A_{0}$ and $B_{0}$ are the same.
Hence their y-coordinates are different. Without loss of generality, we can
assume that $A_{0}$ and $B_{0}$ are both in the y-axis.

Case 2.1. We consider the case that their y-coordinates are symmetric about
$y=0$ first.

Take $\tau_{n}=t^{n}\cdot s$ and $n$ large enough such that
$d_{E}(C_{\tau_{n}},A_{0}),d_{E}(D_{\tau_{n}},B_{\tau_{n}})<\varepsilon$ for
$\varepsilon$ small sufficiently where $d_{E}$ is the Euclidean distance on
$\partial\mathcal{H}_{\infty}$.

For all $M,N$ on the line segment $C_{\tau_{n}}D_{\tau_{n}}$,

$\displaystyle 0<\|(M)_{1}-(N)_{1}\|$
$\displaystyle\leq\|(C_{\tau_{n}})_{1}-(D_{\tau_{n}})_{1}\|$
$\displaystyle\leq\|(A_{0})_{1}-(B_{\tau_{n}})_{1}\|+2\varepsilon$
$\displaystyle=\alpha+2\varepsilon<2\alpha$

So if there is a $w\in\Gamma_{\infty}$ such that $w(M)=N$, $w$ will not be a
translation since all the distance of translations along x-axis in
$\Gamma_{\infty}$ are even number multiples of $\alpha$. Then we assume $w$ is
a glide reflection whose axis is $m\beta/2$ for some $m\in\mathbb{Z}$

Figure 4. Case 2.1

Let $A_{0}=(0,-a),B_{0}=(0,a),0<a<\frac{\beta}{2}$, then
$B_{\tau_{n}}=(\alpha,-a+n\beta)$. The y-coordinate of the middle point of
$A_{0}$ and $B_{\tau_{n}}$ is $(-a+\frac{n\beta}{2})$. Then we should discuss
the points on the geodesic segment $l_{t}$ whose endpoints are $C_{0}$ and
$D_{\tau_{n}}$ in place of the line segment $C_{0}D_{\tau_{n}}$. We will proof
that if we take $n$ such that $0<|-a\pm\varepsilon|<\frac{\beta}{2}$, $l_{t}$
will not have self-intersections under any transformations in
$\Gamma_{\infty}$.

We consider the points $\bar{M}$ and $\bar{N}$ on the geodesic segment $l_{t}$
The images of projection on $\partial\mathcal{H}_{\infty}$ of them are $M$ and
$N$. If there is some isometry $w\in\Gamma_{\infty}$ such that $w(M)=N$,
$\bar{M}$ and $\bar{N}$ must be on the same plane parallel with
$\partial\mathcal{H}_{\infty}$. It means they are equidistant from the middle
point $\bar{P}$ of $C_{0}$ and $D_{\tau_{n}}$ on the geodesic segment $l_{t}$.

Figure 5. $M$ and $N$ are symmetric

Then the projected image $P$ of $\bar{P}$ on the
$\partial\mathcal{H}_{\infty}$ is also the middle point of $C_{0}$ and
$D_{\tau_{n}}$ on $\partial\mathcal{H}_{\infty}$, then
$-a+\frac{n\beta}{2}-\varepsilon\leq(P)_{2}\leq-a+\frac{n\beta}{2}+\varepsilon$
because
$d_{E}(C_{\tau_{n}},A_{0}),d_{E}(D_{\tau_{n}},B_{\tau_{n}})<\varepsilon$.
However, $w$ is a glide reflection whose axis must be $m\beta/2$ for some
$m\in\mathbb{Z}$. So $(P)_{2}=m\beta/2$.

Now we take $n$ large enough such that $0<|-a\pm\varepsilon|<\frac{\beta}{2}$
to construct a contradiction. Hence $\pi(\gamma_{\tau_{n}})$ is a simple
closed geodesic by Lemma 2.2.

Case 2.2. Now we consider the situation that $A_{0}$ and $B_{0}$ are not
symmetric about $y=0$.

Figure 6. Case 2.2

We eliminate glide reflections first. According to the prove above, we only
need to show that there are no reflection axis such that the $y$-coordinate of
$A_{0}$ and $B_{t}$ are symmetric about it. Take
$\tau_{2n}(\vec{X})=s^{2n}(\vec{X})=\vec{X}+\begin{pmatrix}2n\alpha\\\
0\end{pmatrix}$. Then the $y$-coordinate of $B_{\tau_{2n}}=\tau_{2n}(B_{0})$
and $A_{0}$ are not symmetric about any reflection axis since we have assumed
that $A_{0}$ and $B_{0}$ are not symmetric about $y=0$. So there is no glide
reflection $w$ such that $l_{t}$ has self-intersections.

On the other hand, the y-coordinate of $A_{0}$ and $B_{\tau_{2n}}$ are
different, as well as $C_{\tau_{2n}}$ and $D_{\tau_{2n}}$ when we take $n$
large enough, so there is no translation $w^{\prime}$ and points $M,N$ on the
line segment $C_{\tau_{2n}}D_{\tau_{2n}}$ such that $w^{\prime}(M)=N$ as the
argment in Theorem 3.1.

With Corollary 3.1 and discussions in Case 1, Case 2.1 and Case 2.2 above, we
complete the proof of Theorem 4.1. ∎

##  5\. Conclusions

The method is based on the picture of the structure of cusp. It is known that
the cusp of a hyperbolic $n$-manifold admits a flat metric. It means that when
we consider $n$-dimension cusped hyperbolic case, we should know the
classification of $(n-1)$-dimension flat manifolds. It seems that the method
of this paper will likely work in $4$-dimensional case due to our knowledge
about $3$-dimension flat manifolds. We will continue to consider if any cusped
finite-volume hyperbolic $4$-manifolds contain infinitely many simple closed
geodesics.

However, the complexity of the structure of flat manifolds increases rapidly
as the number of dimensions increases. So when we consider the location of
self-tangencies to construct simple closed geodesics, it will appear a lot of
complications. It seems that there will always be some special cases such that
we have very few options to construct simple closed geodesics. This will cause
a lot of difficulties for our case-by-case approach to extend our conclusions
to higher dimensions directly.

## References

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