# Counting arcs on hyperbolic surfaces

Nick Bell

###### Abstract

We give the asymptotic growth of the number of (multi-)arcs of bounded length
between boundary components on complete finite-area hyperbolic surfaces with
boundary. Specifically, if $S$ has genus $g$, $n$ boundary components and $p$
punctures, then the number of orthogeodesic arcs in each pure mapping class
group orbit of length at most $L$ is asymptotic to $L^{6g-6+2(n+p)}$ times a
constant. We prove an analogous result for arcs between cusps, where we define
the length of such an arc to be the length of the sub-arc obtained by removing
certain cuspidal regions from the surface.

##  1 Introduction

Let $S$ be an orientable surface of negative Euler characteristic of genus $g$
with $n$ boundary components and $p$ punctures, and we assume
$(g,n+p)\neq(0,3)$. Let $\textup{Mod}(S)$ be the mapping class group

$\textup{Mod}(S)\coloneqq\textup{Mod}(S^{o})=\text{Homeo}^{+}(S^{o})/\text{Homeo}^{+}_{0}(S^{o}),$

where $S^{o}=S\setminus\partial S$ is the interior of $S$,
$\text{Homeo}^{+}(S^{o})$ is the space of orientation-preserving
homeomorphisms of $S^{o}$ and $\text{Homeo}^{+}_{0}(S^{o})$ is the subgroup of
homeomorphisms properly homotopic to the identity. Let $\textup{PMod}(S)$ be
the pure mapping class group: the finite-index subgroup of $\textup{Mod}(S)$
consisting of exactly those elements which fix each boundary component and
each puncture of $S$. See [8] for a thorough treatment of mapping class
groups. Here we will say that two multicurves, by which we mean formal sums of
finitely many weighted curves, are _of the same type_ if they share a
$\textup{PMod}(S)$-orbit.

A celebrated theorem of Mirzakhani ([14],[15]) gives the asymptotic growth of
the number of (homotopy classes of) multicurves of the same type of bounded
hyperbolic length. Letting $Y$ be a complete hyperbolic metric on $S^{o}$ and
$\gamma_{0}$ be a multi-curve on $S$, Mirzakhani showed that

$\lim_{L\to\infty}\frac{|\\{\gamma\ \textup{of type}\
\gamma_{0}\mid\ell_{Y}(\gamma)\leq
L\\}|}{L^{6g-6+2(n+p)}}=\frac{\mathfrak{c}(\gamma_{0})}{\mathfrak{b}_{g,n+p}}\mathfrak{m}_{\textup{Thu}}(\\{\ell_{Y}(\cdot)\leq
1\\})$ (1)

where $\mathfrak{m}_{\textup{Thu}}$ is the Thurston measure on the space
$\mathcal{ML}(S)$ of compactly supported measured laminations on $S^{o}$,
$\mathfrak{c}(\gamma_{0})$ is a constant depending on the type $\gamma_{0}$,
and $\mathfrak{b}_{g,n+p}$ is a constant depending on the topology of $S$. We
refer the reader to [14] and [15] for details of the constants, and to [20],
[21] and [9] for a background on $\mathcal{ML}(S)$. Here, $\ell_{Y}(\gamma)$
denotes the $Y$-length of the geodesic representative of $\gamma$. Mirzakhani
first proved the above result for simple multicurves in [14], and then again
for general multicurves in [15]; also see [6] and [7] for an alternative proof
of this theorem.

In fact, Mirzakhani’s theorem holds if we redefine the type of a multicurve to
correspond to the orbit of _any_ finite-index subgroup of $\textup{Mod}(S)$.

In this paper, we shall show that Mirzakhani’s theorem holds when we replace
multicurves with multi-arcs. By $\partial S$ we shall mean the boundary of
$S$, consisting of the $n$ boundary components. The $p$ punctures correspond
to ends of $S$. We will count both compact arcs, whose endpoints lie on
$\partial S$, and infinite arcs, whose endpoints are punctures of $S$. A
multi-arc is a formal sum of finitely many weighted arcs, and as with
multicurves, two multi-arcs are of the same type if they share a
$\textup{PMod}(S)$-orbit. Detailed definitions can be found in Section 2.

We endow $S$ with a (metrically) complete, finite-area, hyperbolic metric $X$
such that $\partial S$ is geodesic. We note that whilst our metric is
metrically complete, it is not geodesically complete (unless $\partial
S=\emptyset$), and that the punctures correspond to cusps under such a metric.

The length of a compact arc is the length of its orthogeodesic representative,
that is, the unique geodesic compact arc in its homotopy class which meets
$\partial S$ orthogonally. The study of orthogeodesics on hyperbolic surfaces
has a rich history. For example, if one counts all orthogeodesics of length at
most $L$, Basmajian’s Identity [1] gives an upper bound exponential in $L$ for
this number, and the actual asymptotic growth was shown to be exponential by
Parkonnen and Paulin in [16] (see [10] for a generalisation). These results
can be viewed as analogues to Huber’s [11] and Margulis’ [12] Prime Geodesic
theorem, showing asymptotic exponential growth of the number of closed
geodesics on the surface. Our results are instead analogues of Mirzakhani’s
theorem, counting arcs in each (pure) mapping class group orbit and giving
polynomial asymptotic growth. We prove the following.

######  Theorem 1.

Let $X$ be a complete, finite-area, hyperbolic metric on $S$ such that
$\partial S\neq\emptyset$ is geodesic. Let $\alpha_{0}$ be a compact multi-arc
on $S$. Then we have

$\lim_{L\to\infty}\frac{|\\{\alpha\ \textup{of type}\
\alpha_{0}\mid\ell_{X}(\alpha)\leq
L\\}|}{L^{6g-6+2(n+p)}}=\frac{\mathfrak{c}(\alpha_{0})}{\mathfrak{b}_{g,n+p}}\mathfrak{m}_{\textup{Thu}}(\\{\ell_{X}(\cdot)\leq
1\\})$

where $\mathfrak{m}_{\textup{Thu}}$ and $\mathfrak{b}_{g,n+p}$ are as in (1),
and $\mathfrak{c}(\alpha_{0})>0$ is a constant depending only on the type
$\alpha_{0}$.

As implied by the name, infinite arcs have infinite length as they descend
infinitely far down the cusps. Hence we must define a suitable notion of the
length of infinite arcs to allow us to derive an analogue of Theorem 1. A
natural way to do this is to cut off the cusps at some point and consider the
length of the segment of the arc which remains. Given any positive $t\leq 1$,
we define the _$t$ -length_ of an infinite arc $\alpha$ to be
$\ell_{X}^{t}(\alpha)=\ell_{X}(\alpha^{t})$, where $\alpha^{t}$ is the compact
sub-arc of $\alpha$ obtained by removing a cuspidal region of area $t$ at each
cusp (we refer to Section 4 for the precise definition). We prove the
following result.

######  Theorem 2.

Let $X$ be a complete, finite-area, hyperbolic metric on $S$ with (possibly
empty) geodesic boundary. Let $\alpha_{0}$ be an infinite arc on $S$. Then for
any positive $t\leq 1$, we have

$\lim_{L\to\infty}\frac{|\\{\alpha\ \textup{of type}\
\alpha_{0}\mid\ell_{X}^{t}(\alpha)\leq
L\\}|}{L^{6g-6+2(n+p)}}=\frac{\mathfrak{c}(\alpha_{0})}{\mathfrak{b}_{g,n+p}}\mathfrak{m}_{\textup{Thu}}(\\{\ell_{X}(\cdot)\leq
1\\})$

where $\mathfrak{m}_{\textup{Thu}}$ and $\mathfrak{b}_{g,n+p}$ are as in (1),
and $\mathfrak{c}(\alpha_{0})>0$ is a constant depending on the type
$\alpha_{0}$. In particular, the limit does not depend on $t$.

######  Remark.

Theorem 2 also holds for infinite multi-arcs, following the same argument
presented in this paper. Moreover, our arguments can be easily modified to
apply to half-infinite arcs, by which we mean arcs with one endpoint on the
boundary and one at a puncture, and multi-arcs whose components are any
combination of infinite, compact and half-infinite arcs.

There are other natural choices of length to assign to infinite arcs, such as
the truncated length (see [17]) and the closely related $\lambda$-length (see
[18] and [19]). Theorem 2 also holds for the truncated length, as we will
explain in Section 4.

Although it might be possible to prove Theorems 1 and 2 using methods similar
to the proof of Mirzakhani’s theorem, we instead take a simpler approach here.
The main idea is to associate a multicurve $\gamma_{\alpha}$ to each multi-arc
$\alpha$ in a way which respects length, up to a well-behaved error, and then
use Mirzakhani’s theorem to deduce Theorems 1 and 2. In fact,
$\mathfrak{c}(\alpha_{0})$ will be shown in each case to be closely related to
$\mathfrak{c}(\gamma_{\alpha_{0}})$: we will get that
$\mathfrak{c}(\alpha_{0})=k(\alpha_{0})2^{6g-6+2(n+p)}\mathfrak{c}(\gamma_{\alpha_{0}})$
where $k(\alpha_{0})$ is a combinatorial constant depending on $\alpha_{0}$,
$\mathfrak{c}(\gamma_{\alpha_{0}})$ is as in (1), and $\gamma_{\alpha_{0}}$ is
the curve associated to $\alpha_{0}$ as defined in Sections 3.1 and 4.

In Section 2, we will introduce the necessary tools to formulate our proof,
and deal with a technicality regarding the application of Mirzakhani’s theorem
in our setting. We shall discuss the link between compact arcs and curves in
Section 3.1, before proving Theorem 1 in Section 3.2. Then in Section 4, we
will demonstrate how to apply the same method to infinite arcs and
subsequently prove Theorem 2.

######  Acknowledgements.

The author would like to thank Viveka Erlandsson for suggesting the topic, and
for advising them throughout. Additionally, the author would like to thank
Hugo Parlier and Juan Souto for their invaluable help and feedback, and Lars
Louder for providing Example 1. Finally, the author would like to thank their
family and friends for their constant support.

##  2 Background

As above, let $\partial S$ denote the boundary of $S$, and let
$S^{o}=S\setminus\partial S$ denote the interior of $S$. Let $\mathfrak{C}$
denote the collection of punctures on $S$. When convenient, we may consider
the punctures as marked points on (the closure of) $S$. Throughout the
following, let $X$ denote a complete, finite-area, hyperbolic metric on $S$
such that each component of $\partial S$ is geodesic.

By a _curve_ we mean (the homotopy class of) an immersion of the circle
$\gamma\colon\mathds{S}^{1}\to S$, and we identify curves which differ by an
orientation. We assume curves to be essential, meaning not homotopic to a
point or a puncture, and non-peripheral, meaning not homotopic to a boundary
component. By abuse of notation, we will use $\gamma$ to refer to both a curve
and its homotopy class. If a curve can be realised by an embedding, we call it
simple.

A _compact arc_ is an immersion of the closed interval $\alpha\colon[0,1]\to
S$ such that $\alpha(0),\alpha(1)\in\partial S$ and $\alpha((0,1))\subset
S^{o}$. We consider compact arcs up to homotopy relative to $\partial S$,
where we allow the endpoints to move along $\partial S$, and we assume that
they are not homotopic into the boundary. Similarly, we define an _infinite
arc_ to be an immersion of the open interval $\alpha\colon(0,1)\to S$ such
that the endpoints are in $\mathfrak{C}$, by which we mean that when we
consider the punctures as marked points, the limit of $\alpha$ in each
direction is a marked point. We consider infinite arcs up to homotopy relative
to $\mathfrak{C}$, and we assume that they are not homotopic into
$\mathfrak{C}$. We identify arcs which differ by an orientation, and again by
abuse of notation, we refer to both an arc and its homotopy class by $\alpha$.
If an (infinite or compact) arc can be realised as an embedding, then we call
it simple. We stress that throughout, we allow arcs to have self-
intersections; we do not only count simple arcs.

A _multicurve_ or a _multi-arc_ is a finite formal sum of weighted curves or
(infinite or compact) arcs respectively. Explicitly, if $\omega$ is a
multicurve (resp. multi-arc), then

$\omega=\sum_{i=1}^{m}a^{i}\omega^{i}$

for some $a^{i}\in\mathds{R}_{+}$ and $m\in\mathds{Z}_{+}$, where each
$\omega^{i}$ is a curve (resp. arc). We will refer to the $\omega^{i}$ as the
components of $\omega$.

Each homotopy class of curves has a unique geodesic representative, and each
homotopy class of compact arcs has a unique geodesic representative which
meets the boundary orthogonally, which we refer to as an _orthogeodesic_. We
define the length of (a homotopy class of) a curve or compact arc to be the
length of its geodesic or orthogeodesic representative, which we denote by
$\ell_{X}(\cdot)$. The length of a multicurve or compact multi-arc is defined
to be the weighted sum of the lengths of its components: for
$\omega=\sum_{i=1}^{m}a^{i}\omega^{i}$, we have
$\ell_{X}(\omega)=\sum_{i=1}^{m}a^{i}\ell_{X}(\omega^{i})$. We will discuss
how to assign appropriate finite lengths to infinite arcs in Section 4.

The pure mapping class group $\textup{PMod}(S)$ acts naturally on curves and
arcs in $S$. If $\varphi$ is a mapping class and $\omega$ is either a geodesic
curve, an orthogeodesic compact arc or a geodesic infinite arc, then we define
$\varphi\cdot\omega$ to be the (ortho-)geodesic representative of $f(\omega)$,
where $f$ is any representative of $\varphi$. Let $\omega_{0}$ be a curve or
arc, then for any curve or arc $\omega$, we say that $\omega$ is _of type_
$\omega_{0}$ if they share an orbit in the pure mapping class group, that is,
there exists some $\varphi\in\textup{PMod}(S)$ such that
$\varphi\cdot\omega_{0}=\omega$.

The action of $\textup{PMod}(S)$ on multicurves and multi-arcs is defined
analogously to the above: if $\omega=\sum_{i=1}^{m}a^{i}\omega^{i}$ is a
multicurve or multi-arc, then

$\varphi\cdot\omega=\sum_{i=1}^{m}a^{i}(\varphi\cdot\omega^{i}).$

We say that a multicurve or multi-arc $\omega$ is of type $\omega_{0}$ if
$\omega$ and $\omega_{0}$ share a $\textup{PMod}(S)$-orbit. As a result, we
have that if $\omega=\sum_{i=1}^{m}a^{i}\omega^{i}$ and
$\omega_{0}=\sum_{j=1}^{n}a^{j}_{0}\omega^{j}_{0}$ are of the same type then
$m=n$ and, up to relabelling, for all $i\in\\{1,\dots,m\\}$, $a^{i}=a^{i}_{0}$
and $\omega^{i}$ is of type $\omega^{i}_{0}$.

Since $X$ is complete and finite-area, each puncture corresponds to a _cusp_
under this metric. Recall that a cusp is an end which has a neighbourhood
$H_{t}$ isometric to

$\Big{\\{}z\in\mathds{H}^{2}\Big{|}\textup{Im}(z)>\frac{1}{t}\Big{\\}}/\langle
z\mapsto z+1\rangle$

for some $t>0$, where we have identified the hyperbolic plane $\mathds{H}^{2}$
with the Poincaré upper half-plane. Such a region has volume $t$, and we refer
to $H_{t}$ as a _cuspidal region (of volume $t$)_. The ends of any infinite
arc escape down cusps, and the unique geodesic representative of its homotopy
class eventually intersects the horocyclic foliation of the corresponding
cusps orthogonally.

Let $t>0$. For each $p\in\mathfrak{C}$, let $H_{t}^{p}$ denote the cuspidal
region at $p$ of area $t$. Denote the union of these regions over all $p$ by
$\mathscr{H}_{t}=\cup_{p\in\mathfrak{C}}H_{t}^{p}$. It is well-known that for
any $t<2$, the cuspidal regions $H_{t}^{p}$ are embedded and pairwise
disjoint, as can be seen as a result of the Collar Lemma (see for example
Theorem 4.4.6 of [4]).

For each boundary curve $\delta$ in $\partial S$, and for any $c>0$, define
the annulus $A_{c}^{\delta}$ to be the set of points at a distance less than
$c$ from $\delta$. That is,

$A_{c}^{\delta}=\\{x\in S\mid d_{X}(x,\delta)<c\\}.$

Denote by $\mathscr{A}_{c}=\cup_{\delta}A_{c}^{\delta}$ the union of these
annuli over all $\delta$ in $\partial S$. It again follows from the Collar
Lemma, applied to the boundary curves, that there exists $c^{\prime}>0$
depending on $X$ such that the annuli $A_{c^{\prime}}^{\delta}$ are embedded
and pairwise disjoint. We can choose $c^{\prime}$ such that for all $t<2$,
$\mathscr{H}_{t}\cap\mathscr{A}_{c^{\prime}}=\emptyset$, and in particular,
$S\setminus\overline{(\mathscr{H}_{t}\cup\mathscr{A}_{c^{\prime}})}$ is
homeomorphic to $S^{o}$.

For any $p$ and $t<2$, any geodesic segment in $H^{p}_{t}$ either never leaves
the cuspidal region and so intersects every horocycle in $H^{p}_{t}$
orthogonally, or it winds around the cusp before leaving the cuspidal region,
and hence, when long enough, creates self-intersections. In the latter case,
we call the segment _returning_. In fact, the deeper into $H_{t}^{p}$ a
returning segment goes the more times it must self-intersect, and there is a
direct relationship between the length of a returning segment and its self-
intersection number which we record below for future reference. We refer to
[2] and [3] for more details about the behaviour of returning segments and for
the proof of the below lemma: in particular, Proposition 3.4 of [3] gives a
much more precise description of the relationship between how far an arc goes
into a cusp and its self-intersection number. Letting $\iota(\cdot,\cdot)$
denote the (geometric) intersection number between curves or arcs (which is
realised by their (ortho-)geodesic representatives), we have:

######  Lemma 3.

Let $d>0$, and suppose $\beta$ is a geodesic segment in $\mathscr{H}_{1}$ with
both endpoints on $\partial\mathscr{H}_{1}$ such that $\iota(\beta,\beta)\leq
d$. Then there exists some positive $B=B(d)$ such that

$\ell_{X}(\beta)\leq B.$

It follows from Lemma 3 that any geodesic curve $\gamma$ with at most $d$
self-intersections never enters $\mathscr{H}_{{\rm e}^{-B(d)}}$. Moreover,
whenever a complete geodesic enters a small annulus around a boundary curve
$\delta$, it spirals towards $\delta$, and unless it is asymptotic to $\delta$
it eventually leaves the annulus, creating self-intersections if long enough.
It follows that if $\gamma$ is a geodesic curve with at most $d$ self-
intersections, there exists some $c<c^{\prime}$ depending on $d$ (and $X$)
such that $\gamma$ never enters $\mathscr{A}_{c}$. Putting this together with
the above gives us that $\gamma$ is contained in the compact subsurface
$S\setminus(\mathscr{H}_{{\rm e}^{-B(d)}}\cup\mathscr{A}_{c})\subset S^{o}$.
Note that as before, $S\setminus\overline{(\mathscr{H}_{{\rm
e}^{-B(d)}}\cup\mathscr{A}_{c})}$ is homeomorphic to $S^{o}$. Furthermore,
since $\textup{Mod}(S)$ preserves the self-intersection number of curves and
arcs, the above is true for any curve of type $\gamma$. We summarise this
well-known fact below for reference (and again refer to [2] and [3] for much
more precise details).

######  Lemma 4.

Let $\gamma_{0}$ be a curve. Then there exists a compact subsurface $K\subset
S^{o}$ with $K^{o}$ homeomorphic to $S^{o}$ such that for any $\gamma$ of type
$\gamma_{0}$, the geodesic representative of $\gamma$ is contained in $K$.

Since multicurves have finitely many components, this lemma holds for
multicurves by taking the union of the compact subsurfaces given for (the
support of) each component.

Let $d$ be some non-negative integer, and let $\alpha$ be an infinite arc such
that $\iota(\alpha,\alpha)=d$. Then similarly to the above,
$\alpha\cap\mathscr{H}_{{\rm e}^{-B(d)}}$ consists of exactly 2 components,
which are simple half-infinite arcs. Equivalently,
$\alpha\cap(S\setminus\mathscr{H}_{{\rm e}^{-B(d)}})$ has exactly one
component. We state this here for reference.

######  Lemma 5.

Let $\alpha$ be an infinite arc. Then there exists some positive
$t_{\alpha}<1$, depending only on $\iota(\alpha,\alpha)$, such that
$\alpha\cap(S\setminus\mathscr{H}_{t_{\alpha}})$ has exactly one component.

We also need the fact that if a geodesic goes far enough into a cusp then it
must intersect itself inside $\mathscr{H}_{2}$. To see this, suppose $\beta$
is a returning geodesic segment in $\mathscr{H}_{2}$ that enters $H^{p}_{t}$
for some $t\leq 1$ and some $p\in\mathfrak{C}$. Consider the cuspidal region
$H^{p}_{2}$ and identify it with

$\Big{\\{}z\in\mathds{H}^{2}\Big{|}\
\textup{Im}(z)>\frac{1}{2}\Big{\\}}/\langle z\mapsto z+1\rangle.$

A fundamental domain for the action of $z\mapsto z+1$ is the region in
$\mathds{H}^{2}$ bounded by $x=0$ and $x=1$. Note that any geodesic in
$\mathds{H}^{2}$ neither of whose endpoints are at $\infty$ which intersects
the line $y=\frac{1}{t}$ also intersects its translate under the map $z\mapsto
z+1$, and this intersection occurs above the line $y=\frac{1}{2}$. Hence
$\beta$ intersects itself inside the embedded cuspidal region $H^{p}_{2}$. In
particular, any simple geodesic not asymptotic to a puncture cannot enter
$\mathscr{H}_{1}$. We record this here for reference, and refer to Section 1.3
of [13] and Proposition 3.2 of [3] for more details.

######  Lemma 6.

Let $0<t\leq 1$. If $\beta$ is a geodesic segment in $\mathscr{H}_{2}$ with
both endpoints on $\partial\mathscr{H}_{2}$, and
$\beta\cap\mathscr{H}_{t}\neq\emptyset$, then $\iota(\beta,\beta)\geq 1$.

Recall that $\mathcal{ML}(S)$ is the space of compactly supported measured
laminations on $S^{o}$. The support of any $\lambda\in\mathcal{ML}(S)$ is a
union of simple geodesics and thus it follows from Lemma 6 that the support of
$\lambda$ is contained in $S^{o}\setminus\mathscr{H}_{1}$. In fact, there
exists a compact subsurface $K\subseteq S^{o}\setminus\mathscr{H}_{1}$ which
contains the support of $\mathcal{ML}(S)$.

As mentioned in the introduction, the idea in this paper is to find a nice way
to associate curves to arcs so that we can use Mirzakhani’s theorem about
counting curves to count arcs. Mirzakhani’s theorem is stated for complete
finite-area hyperbolic metrics on the interior $S^{o}$, and we will need to
use the result for our metric $X$ on $S$ which has geodesic boundary. This
issue is resolved by instead using the following generalisation.

######  Theorem 7 ([5], Corollary 1.3).

Let $Y$ be a complete Riemannian metric on $S^{o}=S\setminus\partial S$. Then
for any multicurve $\gamma_{0}$,

$\lim_{L\to\infty}\frac{|\\{\gamma\ \textup{of type}\
\gamma_{0}\mid\ell_{Y}(\gamma)\leq
L\\}|}{L^{6g-6+2(n+p)}}=\frac{\mathfrak{c}(\gamma_{0})}{\mathfrak{b}_{g,n+p}}\mathfrak{m}_{\textup{Thu}}(\\{\ell_{Y}(\cdot)\leq
1\\})$

where $\mathfrak{m}_{\textup{Thu}}$, $\mathfrak{c}(\gamma_{0})$ and
$\mathfrak{b}_{g,n+p}$ are as in (1) and $\ell_{Y}(\gamma)$ is the length of a
shortest curve homotopic to $\gamma$.

To see that this result implies that we can count curves in our setting, let
$\gamma_{0}$ be a multicurve on $S$ and let $K=K(\gamma_{0})$ be the compact
subsurface of $S^{o}$ given by Lemma 4. By the discussion after Lemma 6, we
may assume that $K$ is such that $\mathcal{ML}(S)\subset K$. Take any complete
Riemannian metric $Y$ on $S^{o}$ which agrees with $X$ on $K$. Since the
geodesic representative of every multicurve $\gamma$ of type $\gamma_{0}$ is
contained in $K$ and $\mathcal{ML}(S)\subset K$, we have that
$\ell_{Y}(\gamma)=\ell_{X}(\gamma)$ for all $\gamma$ of type $\gamma_{0}$ and
$\ell_{Y}(\lambda)=\ell_{X}(\lambda)$ for all $\lambda\in\mathcal{ML}(S)$. We
record this consequence for reference.

######  Corollary 8.

Let $X$ be a complete, finite-area, hyperbolic metric on $S$ such that
$\partial S$ is geodesic. Then for any multicurve $\gamma_{0}$,

$\lim_{L\to\infty}\frac{|\\{\gamma\ \textup{of type}\
\gamma_{0}\mid\ell_{X}(\gamma)\leq
L\\}|}{L^{6g-6+2(n+p)}}=\frac{\mathfrak{c}(\gamma_{0})}{\mathfrak{b}_{g,n+p}}\mathfrak{m}_{\textup{Thu}}(\\{\ell_{X}(\cdot)\leq
1\\})$

where $\mathfrak{m}_{\textup{Thu}}$, $\mathfrak{c}(\gamma_{0})$ and
$\mathfrak{b}_{g,n+p}$ are as in (1).

##  3 Compact arcs

###  3.1 Relating compact arcs and curves

In this section, we will discuss how to associate multicurves to compact
multi-arcs in a way that respects length, up to some well-behaved error. We
will achieve this using the nice geometric properties of pairs of pants.

First, we will discuss how to associate a single curve to a single compact
arc. Fix an orientation on $S$, which induces an orientation on the boundary
components. Let $\alpha$ be some compact arc in $S$, oriented from $\alpha(0)$
to $\alpha(1)$. The endpoints $\alpha(0)$ and $\alpha(1)$ each lie on a
boundary component, which we denote by $\delta_{0}^{\alpha}$ and
$\delta_{1}^{\alpha}$ respectively: note that these are not necessarily
distinct. Pick basepoints $p_{0}$ and $p_{1}$ on $\delta_{0}^{\alpha}$ and
$\delta_{1}^{\alpha}$ respectively, and consider these boundary components as
loops based at their respective basepoints. Apply a homotopy to $\alpha$ so
that $\alpha(0)=p_{0}$ and $\alpha(1)=p_{1}$. Then we define the _curve
associated to_ $\alpha$ to be the geodesic curve $\gamma_{\alpha}$ (freely)
homotopic to the concatenated path

$\alpha^{-1}\cdot\delta_{1}^{\alpha}\cdot\alpha\cdot\delta_{0}^{\alpha}$

which starts and ends at $p_{0}$. In particular, in the case that $\alpha$ is
simple and $\delta_{0}^{\alpha}\neq\delta_{1}^{\alpha}$, $\gamma_{\alpha}$ is
homotopic to the boundary of a small neighbourhood of the union of $\alpha$,
$\delta_{0}^{\alpha}$ and $\delta_{1}^{\alpha}$. See Figure 1. Recall that we
identify arcs and curves that differ by an orientation, and note that the arc
$\alpha^{\prime}$ which differs from $\alpha$ only in orientation gives rise
to exactly the same curve as $\alpha$, even in orientation.

Figure 1: Examples of compact arcs (in red) and their associated curves (in
blue).

Now, let $P$ be a topological surface with $g=0$, $n=3$ and $p=0$ known as a
pair of pants, and fix an orientation on $P$. The boundary components of $P$
are referred to as _cuffs_ , and for each pair of cuffs the unique homotopy
class of simple compact arcs between them is called a _seam_. We label the
cuffs by $\delta_{0}^{P},\delta_{1}^{P}$ and $\delta_{2}^{P}$ and the seam
between $\delta_{0}^{P}$ and $\delta_{1}^{P}$ by $\alpha_{P}$.

For any arc $\alpha$ on $S$ with endpoints on $\delta_{0}^{\alpha}$ and
$\delta_{1}^{\alpha}$, there exists an orientation-preserving immersion
$\iota_{\alpha}\colon P\to S$ such that

$\displaystyle\iota_{\alpha}(\delta_{0}^{P})$
$\displaystyle=\delta_{0}^{\alpha},$
$\displaystyle\iota_{\alpha}(\delta_{1}^{P})$
$\displaystyle=\delta_{1}^{\alpha},$ $\displaystyle\iota_{\alpha}(\alpha_{P})$
$\displaystyle=\alpha.$

Note that the images of the two cuffs and the seam under this map determine
the image of the third cuff up to homotopy, since this is exactly the (free)
homotopy class of
$\alpha^{-1}\cdot\delta_{1}^{\alpha}\cdot\alpha\cdot\delta_{0}^{\alpha}$. That
is,

$\gamma_{\alpha}=\iota_{\alpha}(\delta_{2}^{P})$ (2)

(up to homotopy). Let $\iota_{\alpha}^{\prime}$ be another immersion which
satisfies the above. Then since they agree on $\delta_{0}^{P}$,
$\delta_{1}^{P}$ and $\alpha_{P}$, we have that the images
$\iota_{\alpha}(\delta_{2}^{P})$ and $\iota_{\alpha}^{\prime}(\delta_{2}^{P})$
of the third cuff are homotopic and such a homotopy extends to a homotopy from
$\iota_{\alpha}(P)$ to $\iota_{\alpha}^{\prime}(P)$. Thus any two such
immersions of $P$ are homotopic.

In the case that $\alpha=\sum_{i=1}^{m}a^{i}\alpha^{i}$ is a compact multi-
arc, we define the _multicurve associated to_ $\alpha$ to be the weighted sum
of the the curves associated to its components. That is,

$\gamma_{\alpha}=\sum_{i=1}^{m}a^{i}\gamma_{\alpha^{i}}=\sum_{i=1}^{m}a^{i}\iota_{\alpha^{i}}(\delta_{2}^{P}).$
(3)

In the remainder of the section, we will first prove several statements for
single compact arcs before demonstrating how these also hold for compact
multi-arcs.

Let $\mathcal{A}(S)$ and $\mathcal{C}(S)$ denote the sets of compact arcs and
curves on $S$ respectively. We define the _association map_
$I\colon\mathcal{A}(S)\to\mathcal{C}(S)$ by

$I(\alpha)=\gamma_{\alpha}.$

First, we show that $I$ distorts the length of arcs in a controlled way.

######  Lemma 9.

Let $X$ be a complete, finite-area, hyperbolic metric on $S$ such that
$\partial S$ is geodesic. There exists a constant $C(X)>0$ such that for any
$\alpha\in\mathcal{A}(S)$,

$|\ell_{X}(I(\alpha))-2\ell_{X}(\alpha)|\leq C(X),$

where $I(\alpha)=\gamma_{\alpha}$ is the curve associated to $\alpha$.

###### Proof.

This will follow from basic hyperbolic geometry. Let
$\alpha\in\mathcal{A}(S)$.

Choosing the lengths of 2 cuffs and the seam between them on a pair of pants
fixes the length of the third cuff; that is, the lengths of
$\delta_{0}^{\alpha}$, $\delta_{1}^{\alpha}$ and $\alpha$ determine the length
of $\gamma_{\alpha}$. More precisely,

$\cosh{\frac{\ell_{X}(\gamma_{\alpha})}{2}}=\sinh{\frac{\ell_{X}(\delta_{0}^{\alpha})}{2}}\sinh{\frac{\ell_{X}(\delta_{1}^{\alpha})}{2}}\cosh{\ell_{X}(\alpha)}-\cosh{\frac{\ell_{X}(\delta_{0}^{\alpha})}{2}}\cosh{\frac{\ell_{X}(\delta_{1}^{\alpha})}{2}}$
(4)

(See Theorem 2.4.1 of [4]). Let $\partial
S=\\{\delta_{1},\dots,\delta_{n}\\}$, then for some $i,j$,
$\delta_{0}^{\alpha}=\delta_{i}$ and $\delta_{1}^{\alpha}=\delta_{j}$. To
simplify notation, we will write

$\displaystyle A_{i,j}$
$\displaystyle=\sinh{\frac{\ell_{X}(\delta_{i})}{2}}\sinh{\frac{\ell_{X}(\delta_{j})}{2}},$
$\displaystyle B_{i,j}$
$\displaystyle=\cosh{\frac{\ell_{X}(\delta_{i})}{2}}\cosh{\frac{\ell_{X}(\delta_{j})}{2}}.$

Then equation (4) gives the length of $\gamma_{\alpha}$ as

$\ell_{X}(\gamma_{\alpha})=2\cosh^{-1}(A_{i,j}\cosh{\ell_{X}(\alpha)}-B_{i,j}),$

and we want to show that this length is close to $2\ell_{X}(\alpha)$. To this
end, we define the error function $E_{i,j}\colon[m_{i,j},\infty)\to\mathds{R}$
by

$E_{i,j}(\ell)=2\cosh^{-1}(A_{i,j}\cosh{\ell}-B_{i,j})-2\ell,$

where $m_{i,j}$ is a lower bound on the lengths of arcs between $\delta_{i}$
and $\delta_{j}$ which can be taken as
${m_{i,j}\coloneqq\cosh^{-1}\Big{(}\frac{B_{i,j}+1}{A_{i,j}}\Big{)}}.$ This
function is continuous, and the limit

$\lim_{\ell\to\infty}E_{i,j}(\ell)=2\ln(A_{i,j})$

exists. Hence $|E_{i,j}(\ell)|$ is bounded for all $\ell\in[m_{i,j},\infty)$,
thus there exists $C(i,j)>0$ such that for any arc $\alpha$ between
$\delta_{i}$ and $\delta_{j}$, we have
$|\ell_{X}(\gamma_{\alpha})-2\ell_{X}(\alpha)|\leq C(i,j)$. Therefore as S has
finitely many boundary components, there exists $C(X)>0$ such that for any
$\alpha\in\mathcal{A}(S)$,

$|\ell_{X}(\gamma_{\alpha})-2\ell_{X}(\alpha)|\leq C(X).$

∎

This association map $I$ is not one-to-one: for example, suppose $S$ is a
four-holed sphere with boundary components
$\delta_{1},\delta_{2},\delta_{3},\delta_{4}$. Let $\alpha$ be a simple arc
connecting $\delta_{1}$ and $\delta_{2}$, and $\beta$ a simple arc connecting
$\delta_{3}$ and $\delta_{4}$. Then $\gamma_{\alpha}$ and $\gamma_{\beta}$ are
the same (homotopy class of) curve, up to orientation.

There are also less trivial examples, such as where $\alpha$ and $\beta$ are
arcs between the same boundary components, as the following example
illustrates.

######  Example 1.

Let $P$ be the pair of pants from above. Fix a base point $\star\in P$ and
choose generators $A$ and $B$ for $\pi_{1}(P,\star)\simeq F_{2}$ such that the
loops $A$ and $B$ are freely homotopic (as oriented curves) to
$\delta_{0}^{P}$ and $\delta_{1}^{P}$ respectively. Note that the third
boundary component $\delta_{2}^{P}$ corresponds to the conjugacy class of
$B^{-1}A^{-1}$. Let $S=S_{0,3,0}$ be another oriented pair of pants and let
$a$, $b$ be generators of $\pi_{1}(S,\ast)$ (for some basepoint $\ast\in S$)
such that $a$ and $b$ are freely homotopic to two of the boundary components
of $S$, say $\delta_{0}^{S}$ and $\delta_{1}^{S}$ respectively.

Now consider the homomorphism $h_{1}\colon\pi_{1}(P,\star)\to\pi_{1}(S,\ast)$
defined by

$A\mapsto a\qquad\textup{and}\qquad B\mapsto b^{-1}aba^{-1}b.$

As $h_{1}(A)$ and $h_{1}(B)$ are conjugate to $a$ and $b$, $h_{1}(A)$ and
$h_{1}(B)$ are (freely) homotopic to $\delta_{0}^{S}$ and $\delta_{1}^{S}$,
preserving orientation. Thus $h_{1}$ induces an immersion $\iota_{1}\colon
P\to S$.

Similarly, the homomorphism $h_{2}\colon\pi_{1}(P,\star)\to\pi_{1}(S,\ast)$
defined by

$A\mapsto ab^{-1}aba^{-1}\qquad\textup{and}\qquad B\mapsto b$

also induces an immersion $\iota_{2}\colon P\to S$.

Note that (the homotopy classes of) all three boundary components of
$\iota_{1}(P)$ and $\iota_{2}(P)$ agree; they correspond to the conjugacy
classes of $a$, $b$ and $b^{-1}ab^{-1}a^{-1}ba^{-1}$. Further note that

$\langle a,b^{-1}aba^{-1}b\rangle\neq\langle ab^{-1}aba^{-1},b\rangle,$

and hence $\iota_{1}(P)$ is not homotopic to $\iota_{2}(P)$. In particular,
the arcs $\iota_{1}(\alpha_{P})$ and $\iota_{2}(\alpha_{P})$ are not homotopic
whilst $\gamma_{\iota_{1}(\alpha_{P})}$ and $\gamma_{\iota_{2}(\alpha_{P})}$
determine the same curve (namely, the conjugacy class of
$b^{-1}ab^{-1}a^{-1}ba^{-1}$).

Despite this complication, if we restrict $I$ to a (pure) mapping class group
orbit of an arc, it is uniformly bounded-to-one. This will be enough for our
purposes, as $I$ is equivariant with respect to $\textup{PMod}(S)$, which we
now demonstrate.

######  Lemma 10.

Let $\varphi\in\textup{PMod}(S)$ and $\alpha\in\mathcal{A}(S)$. Then

$I(\varphi\cdot\alpha)=\varphi\cdot I(\alpha).$

###### Proof.

To see this, write

$\gamma_{\varphi\cdot\alpha}=\iota_{\varphi\cdot\alpha}(\delta_{2}^{P})$

using (2). We have that $\iota_{\varphi\cdot\alpha}(P)\subset S$ is an
immersed pair of pants with boundary components $\delta_{0}^{\alpha}$ and
$\delta_{1}^{\alpha}$, and the seam between them is $\varphi\cdot\alpha$.
Similarly, $\varphi\cdot\iota_{\alpha}(P)\subset S$ is an immersed pair of
pants with boundary components $\delta_{0}^{\alpha}$ and $\delta_{1}^{\alpha}$
and seam between them $\varphi\cdot\alpha$, since $\varphi$ fixes the boundary
components of $S$. Therefore,
$\iota_{\varphi\cdot\alpha}(P)=\varphi\cdot\iota_{\alpha}(P)$, and in
particular,
$\iota_{\varphi\cdot\alpha}(\delta_{2}^{P})=\varphi\cdot\iota_{\alpha}(\delta_{2}^{P})$.
Since $\gamma_{\alpha}=\iota_{\alpha}(\delta^{P}_{2})$ by (2), we have

$\gamma_{\varphi\cdot\alpha}=\varphi\cdot\gamma_{\alpha}$

and so the lemma holds. ∎

Let $\alpha_{0}$ be a compact arc, and let

$I_{\alpha_{0}}\colon\textup{PMod}(S)\cdot\alpha_{0}\to\textup{PMod}(S)\cdot\gamma_{\alpha_{0}}$

be the restriction of $I$ to arcs of type $\alpha_{0}$. By Lemma 10, this map
is well-defined. Moreover, as mentioned above, it is uniformly bounded-to-1,
which we prove in the following proposition.

######  Proposition 11.

Let $\alpha_{0}$ be a compact arc. Then there exits $k=k(\alpha_{0})$ such
that $I_{\alpha_{0}}$ is surjective and $k$-to-1.

###### Proof.

Let $\gamma\in\textup{PMod}(S)\cdot\gamma_{\alpha_{0}}$, so
$\gamma=\varphi\cdot\gamma_{\alpha_{0}}$ for some
$\varphi\in\textup{PMod}(S)$. Let $\alpha=\varphi\cdot\alpha_{0}$. Then by
Lemma 10,

$\gamma=\varphi\cdot\gamma_{\alpha_{0}}=\gamma_{\varphi\cdot\alpha_{0}}=I_{\alpha_{0}}(\varphi\cdot\alpha_{0})=I_{\alpha_{0}}(\alpha),$

thus $I_{\alpha_{0}}$ is surjective.

Consider the collection of compact arcs $\alpha_{i}$ such that
$\gamma_{\alpha_{i}}=\gamma$. This set is finite: by Lemma 9, the maximum
length of such an arc is $\frac{1}{2}\ell_{X}(\gamma)+\frac{1}{2}C(X)$, and
thus there are only finitely many. Suppose that there are exactly $k$ such
arcs, and suppose further that for some other curve $\gamma^{\prime}$ of type
$\gamma_{\alpha_{0}}$, there are exactly $k^{\prime}$ arcs
$\alpha^{\prime}_{i}$ such that
$\gamma_{\alpha^{\prime}_{i}}=\gamma^{\prime}$. Since $\gamma$ and
$\gamma^{\prime}$ are of the same type, there exists some
$\psi\in\textup{PMod}(S)$ such that $\gamma^{\prime}=\psi\cdot\gamma$. Again
by Lemma 10, we have that for each $i\in\\{1,\dots,k\\}$,

$\gamma_{\psi\cdot\alpha_{i}}=\psi\cdot\gamma_{\alpha_{i}}=\psi\cdot\gamma=\gamma^{\prime}.$

Thus we have constructed a set of $k$ arcs which are associated to
$\gamma^{\prime}$, and so $k\leq k^{\prime}$. Analogously, we can construct a
set of $k^{\prime}$ arcs which are associated to $\gamma$, hence
$k^{\prime}\leq k$. Therefore $k=k^{\prime}$, and so $k$ is uniform across all
curves of type $\gamma_{\alpha_{0}}$. ∎

We will need to use these results for compact multi-arcs. Let
$\mathcal{A}_{\text{multi}}(S)$ and $\mathcal{C}_{\text{multi}}(S)$ be the
sets of weighted compact multi-arcs and weighted multicurves respectively. By
abuse of notation, define the association map on multi-arcs
$I\colon\mathcal{A}_{\text{multi}}(S)\to\mathcal{C}_{\text{multi}}(S)$ by

$I(\alpha)=I\Big{(}\sum_{i=1}^{m}a^{i}\alpha^{i}\Big{)}=\sum_{i=1}^{m}a^{i}I(\alpha^{i}).$

As multi-arcs have finitely many components, the following is a short
corollary of Lemma 9.

######  Corollary 12.

Let $X$ be a complete, finite-area, hyperbolic metric on $S$ such that
$\partial S$ is geodesic. There exists a constant $C(X)>0$ such that for any
$\alpha\in\mathcal{A}_{\text{multi}}(S)$,

$|\ell_{X}(I(\alpha))-2\ell_{X}(\alpha)|\leq C(X),$

where $I(\alpha)=\gamma_{\alpha}$ is the multicurve associated to $\alpha$.

Furthermore, as an immediate corollary to Lemma 10, $I$ remains
$\textup{PMod}(S)$-equivariant when defined on compact multi-arcs. Following
the proof of Proposition 11, we can see that the restriction of the
association map to multi-arcs of a particular type is surjective and $k$-to-1,
for some $k$ depending only on the type. We record this here for reference.

######  Corollary 13.

Let $\alpha_{0}$ be a compact multi-arc and $\gamma_{\alpha_{0}}$ be as in
(3). Let
$I_{\alpha_{0}}\colon\textup{PMod}(S)\cdot\alpha_{0}\to\textup{PMod}(S)\cdot\gamma_{\alpha_{0}}$
be the restriction of $I$ to multi-arcs of type $\alpha_{0}$. Then there
exists $k=k(\alpha_{0})$ such that $I_{\alpha_{0}}$ is surjective and
$k$-to-1.

######  Remark.

It would be interesting to understand the value of $k$. For example, if
$\alpha_{0}$ is simple, is $k(\alpha_{0})=1$? Is it always 1? Future work
could study this association more closely to give us an idea of how it varies
with the type of arc, and if it is not identically 1, derive some explicit
examples of arcs of the same type which are associated to the same curve.

###  3.2 Counting compact arcs

We can now prove Theorem 1.

###### Proof of Theorem 1.

Let $\alpha_{0}$ be a compact multi-arc. Consider the set

$\\{\alpha\ \text{of type}\ \alpha_{0}\mid\ell_{X}(\alpha)\leq L\\}$

for some $L>0$. By Corollary 13, there exists $k$ such that the association
map $I_{\alpha_{0}}$ is $k$-to-1, and by Corollary 12, the maximum length of a
multicurve associated to a multi-arc in this set is $2L+C(X)$. Hence we can
write

$|\\{\alpha\ \text{of type}\ \alpha_{0}\mid\ell_{X}(\alpha)\leq L\\}|\leq
k|\\{\gamma\ \text{of type}\ \gamma_{\alpha_{0}}\mid\ell_{X}(\gamma)\leq
2L+C(X)\\}|.$

Then we have

$\displaystyle\limsup_{L\to\infty}$ $\displaystyle\frac{|\\{\alpha\ \text{of
type}\ \alpha_{0}\mid\ell_{X}(\alpha)\leq
L\\}|}{L^{6g-6+2(n+p)}}\leq\limsup_{L\to\infty}\frac{k|\\{\gamma\ \text{of
type}\ \gamma_{\alpha_{0}}\mid\ell_{X}(\gamma)\leq
2L+C(X)\\}|}{L^{6g-6+2(n+p)}}$
$\displaystyle=k\cdot\limsup_{L\to\infty}\frac{|\\{\gamma\ \text{of type}\
\gamma_{\alpha_{0}}\mid\ell_{X}(\gamma)\leq
2L+C(X)\\}|}{(2L+C(X))^{6g-6+2(n+p)}}\frac{(2L+C(X))^{6g-6+2(n+p)}}{L^{6g-6+2(n+p)}}$
$\displaystyle=k\cdot
2^{6g-6+2(n+p)}\frac{\mathfrak{c}(\gamma_{\alpha_{0}})}{\mathfrak{b}_{g,n+p}}\mathfrak{m}_{\textup{Thu}}(\\{\ell_{X}(\cdot)\leq
1\\})$

using Corollary 8. Using a similar argument, we have

$|\\{\alpha\ \text{of type}\ \alpha_{0}\mid\ell_{X}(\alpha)\leq L\\}|\geq
k|\\{\gamma\ \text{of type}\ \gamma_{\alpha_{0}}\mid\ell_{X}(\gamma)\leq
2L-C(X)\\}|$

and therefore

$\liminf_{L\to\infty}\frac{|\\{\alpha\ \text{of type}\
\alpha_{0}\mid\ell_{X}(\alpha)\leq L\\}|}{L^{6g-6+2(n+p)}}\geq k\cdot
2^{6g-6+2(n+p)}\frac{\mathfrak{c}(\gamma_{\alpha_{0}})}{\mathfrak{b}_{g,n+p}}\mathfrak{m}_{\textup{Thu}}(\\{\ell_{X}(\cdot)\leq
1\\}).$

Hence, since the limit superior and inferior both exist and agree, we have
that the limit exists and equals the same value. In other words,

$\lim_{L\to\infty}\frac{|\\{\alpha\ \text{of type}\
\alpha_{0}\mid\ell_{X}(\alpha)\leq
L\\}|}{L^{6g-6+2(n+p)}}=\frac{\mathfrak{c}(\alpha_{0})}{\mathfrak{b}_{g,n+p}}\mathfrak{m}_{\textup{Thu}}(\\{\ell_{X}(\cdot)\leq
1\\}),$

where $\mathfrak{c}(\alpha_{0})\coloneqq k\cdot
2^{6g-6+2(n+p)}\mathfrak{c}(\gamma_{\alpha_{0}})$, $k$ is as in Corollary 13,
and $\mathfrak{c}(\gamma_{\alpha_{0}})$, $\mathfrak{b}_{g,n+p}$ and
$\mathfrak{m}_{\textup{Thu}}$ are as in (1). ∎

##  4 Counting infinite arcs

The main work in this section is to prove the lemmas from Section 3.1 for
infinite arcs, with modifications to account for the range of values the
$t$-length of an infinite arc can take. The proof of Theorem 2 will then be
analogous to that of Theorem 1.

First, we discuss the assignment of appropriate finite lengths to infinite
arcs. For any $t\in(0,1]$, let
$\mathscr{H}_{t}=\cup_{p\in\mathfrak{C}}H_{t}^{p}$ be the union of the
cuspidal regions of volume $t$ as before, and define the $t$_-length_
$\ell^{t}_{X}(\alpha)$ of any arc $\alpha$ to be the length of
$\alpha^{t}=\alpha\cap(S\setminus\mathscr{H}_{t})$. That is,

$\ell^{t}_{X}(\alpha)=\ell_{X}(\alpha^{t}).$

Note that in general, $\alpha^{t}$ could consist of multiple connected
components and in this case, $\ell_{X}(\alpha^{t})$ is the sum of the lengths
of its components. Fix an infinite arc $\alpha$ and let $t_{\alpha}$ be given
by Lemma 5. Then $\alpha\cap(S\setminus\mathscr{H}_{t_{\alpha}})$ is
connected, and moreover for any $t\leq t_{\alpha}$, $\alpha^{t}$ has exactly
one component.

As mentioned in the introduction, the $t$-length of an arc is closely related
to the _truncated length_ as defined by Parlier in [17]. Choose a standard
collection of cuspidal regions, which we may take as $\mathscr{H}_{1}$. For
any infinite arc $\alpha$, the (doubly) truncated length of $\alpha$ is the
length of the segment of $\alpha$ between the first and last times $\alpha$
crosses $\partial\mathscr{H}_{1}$. We denote this length by
$\ell_{X}^{Tr}(\alpha)$.

This length is also closely related to the _$\lambda$ -length_ introduced by
Penner in [18] and [19]. In our setting, and choosing the appropriate cuspidal
regions, we have that

$\lambda(\alpha)={\rm e}^{\frac{1}{2}\ell_{X}^{Tr}(\alpha)}.$

Note that for any $\alpha\in\mathcal{A}(S)$, $\ell_{X}^{t_{\alpha}}(\alpha)$
and the truncated length $\ell_{X}^{Tr}(\alpha)$ differ by a constant, and
this constant depends only on $\iota(\alpha,\alpha)$. This is because the
$t_{\alpha}$-length of $\alpha$ is exactly the truncated length plus the
lengths of the two geodesic segments of $\alpha$ between
$\partial\mathscr{H}_{1}$ and $\partial\mathscr{H}_{t_{\alpha}}$, which each
have length $\ln(\frac{1}{t_{\alpha}})$. Thus we can write

$|\ell_{X}^{t_{\alpha}}(\alpha)-\ell_{X}^{Tr}(\alpha)|\leq
2\ln\Big{(}\frac{1}{t_{\alpha}}\Big{)}.$ (5)

Recall that by Lemma 5, $t_{\alpha}$ depends only on $\iota(\alpha,\alpha)$.
Using this fact, one can show that Theorem 2 also holds when we replace
$\ell_{X}^{t}$ by $\ell_{X}^{Tr}$.

The curve $\gamma_{\alpha}$ associated to an infinite arc $\alpha$ is defined
analogously to the compact case. Denote by $p^{\alpha}_{0}$ and
$p^{\alpha}_{1}$ the cusps at each end of $\alpha$, where $\alpha$ is oriented
from $p^{\alpha}_{0}$ to $p^{\alpha}_{1}$. With $t_{\alpha}$ as above, define
$\gamma_{\alpha}$ to be the geodesic curve (freely) homotopic to the loop
given by the concatenation

$(\alpha^{t_{\alpha}})^{-1}\cdot h^{\alpha}_{1}\cdot\alpha^{t_{\alpha}}\cdot
h^{\alpha}_{0},$

where $h^{\alpha}_{0}=\partial
H_{t_{\alpha}}^{p_{0}^{\alpha}},h^{\alpha}_{1}=\partial
H_{t_{\alpha}}^{p_{1}^{\alpha}}$ are the horocycles at $p^{\alpha}_{0}$ and
$p^{\alpha}_{1}$ of length $t_{\alpha}$, viewed as loops with appropriate
basepoints and orientations. Note that if we replaced $t_{\alpha}$ with any
$t<t_{\alpha}$, we would get the same curve $\gamma_{\alpha}$. Let $P$ be a
(generalised) pair of pants with one boundary component and two cusps,
labelled $\delta$, $p_{0}$ and $p_{1}$ respectively. There is an orientation-
preserving immersion $\iota_{\alpha}\colon P\to S$ which sends $p_{0}$ and
$p_{1}$ to $p^{\alpha}_{0}$ and $p^{\alpha}_{1}$ respectively, and such that
(the homotopy class of) the simple infinite arc between them is mapped to
$\alpha$. Then equivalently, $\gamma_{\alpha}$ is the geodesic representative
of $\iota_{\alpha}(\delta)$.

We define $J\colon\mathcal{A}_{\infty}(S)\to\mathcal{C}(S)$ to be the
association map from infinite arcs to curves, where $\mathcal{A}_{\infty}(S)$
is the set of all infinite arcs on $S$. That is, for any
$\alpha\in\mathcal{A}_{\infty}(S)$,

$J(\alpha)=\gamma_{\alpha}.$

We will now prove an analogue of Lemma 9 for infinite arcs. As $t$ can be
taken arbitrarily close to 0, the $t$-length of an arc can be arbitrarily
long, and so any bound on the difference between the $t$-lengths of infinite
arcs and the lengths of their associated curves must depend on $t$.
Furthermore, arcs which self-intersect arbitrarily often will go arbitrarily
deep into the cusps, and therefore so will their curves. Thus for a fixed
value of $t$, this difference can become arbitrarily large. Hence, any such
bound must also depend on self-intersection number.

######  Lemma 14.

Let $\alpha$ be an infinite arc. Then for any positive $t<1$, there exists
$C\big{(}\iota(\alpha,\alpha),t\big{)}>0$ such that

$|\ell_{X}(J(\alpha))-2\ell_{X}({\alpha}^{t})|\leq
C\big{(}\iota(\alpha,\alpha),t\big{)}$

where $J(\alpha)=\gamma_{\alpha}$ is the curve associated to $\alpha$.

Figure 2: The pre-image of an infinite arc $\alpha$ in the generalised pair of
pants $P$, with the perpendiculars which we cut along.

###### Proof.

Let $\alpha$ be an infinite arc, and let $t_{\alpha}$ be given by Lemma 5. We
will start by proving the lemma in the case that $t\leq t_{\alpha}$. Then we
will demonstrate that for $t>t_{\alpha}$, the difference between
$\ell_{X}^{t}(\alpha)$ and $\ell_{X}^{t_{\alpha}}(\alpha)$ is uniformly
bounded across all arcs with the same self-intersection number, and so
complete the proof.

Suppose that $t\leq t_{\alpha}$. Equip the generalised pair of pants $P$ with
a metric using the pullback of $X$ through $\iota_{\alpha}$. Cut $P$ along
four geodesic arcs: the pre-image of $\alpha$, the perpendicular compact
simple geodesic arc from the boundary component to itself, and the two simple
half-infinite geodesic arcs between the boundary component and the cusps. See
Figure 2. We are left with 4 isometric copies of a quadrilateral with three
right angles and one ideal vertex, which we label as in Figure 3.

Since $t\leq t_{\alpha}$, we have that the length of the edge $qw$ is
$\frac{1}{2}\ell_{X}(\alpha^{t})$, and the length of the edge $uv$ is
$\frac{1}{4}\ell_{X}(\gamma_{\alpha})$. Consider this quadrilateral in the
upper-half space model for $\mathds{H}^{2}$ and normalise it such that the
ideal vertex is at $\infty$ and the edges incident to it are on the lines
$x=0$ and $x=1$. As the length of the boundary of the cuspidal region of area
$t$ is $t$, the length of the segment which lives in this quadrilateral is
$\frac{t}{2}$. Therefore it lies on the line $y=\frac{2}{t}$.

Figure 3: One of the quadrilaterals acquired from cutting $P$ (left), and the
same quadrilateral in the upper-half plane model after normalising (right).

This means that $w=\frac{2}{t}i{\rm e}^{-\frac{1}{2}\ell_{X}(\alpha^{t})}$,
and a computation shows that
$\ell_{X}(\gamma_{\alpha})=4\cosh^{-1}(\frac{t}{2}{\rm
e}^{\frac{1}{2}\ell_{X}(\alpha^{t})})$. Hence, the difference
$\ell_{X}(\gamma_{\alpha})-2\ell_{X}(\alpha^{t})$ can be written as

$4\cosh^{-1}\Big{(}\frac{t}{2}{\rm
e}^{\frac{1}{2}\ell_{X}(\alpha^{t})}\Big{)}-2\ell_{X}(\alpha^{t}).$

The function $E_{t}(\ell)=4\cosh^{-1}(\frac{t}{2}{\rm
e}^{\frac{1}{2}\ell})-2\ell$ is continuous on $[m_{t},\infty)$, where
$m_{t}\coloneqq 2\ln(\frac{2}{t})$ is a lower bound on the length of
$\alpha^{t}$, and $\lim_{\ell\to\infty}E_{t}(\ell)=4\ln(t)$. It follows that
there exists $C_{1}(t)>0$ such that

$|\ell_{X}(\gamma_{\alpha})-2\ell_{X}(\alpha^{t})|\leq C_{1}(t).$ (6)

Now suppose that $t>t_{\alpha}$. Note that $\alpha^{t}$ is contained in
$\alpha^{t_{\alpha}}$, and so
$\ell^{t}_{X}(\alpha)<\ell^{t_{\alpha}}_{X}(\alpha)$. Consider
$\alpha^{t_{\alpha}}\setminus\alpha^{t}$, which lies in
$\mathscr{H}_{t}\setminus\mathscr{H}_{t_{\alpha}}$. Exactly two of the
components of $\alpha^{t_{\alpha}}\setminus\alpha^{t}$ are simple geodesic
arcs from $\partial\mathscr{H}_{t}$ to $\partial\mathscr{H}_{t_{\alpha}}$
which meet each boundary orthogonally. Hence, these two components each have
length $\ln(\frac{t}{t_{\alpha}})\leq\ln(\frac{1}{t_{\alpha}})$. The other
components, if any, are returning segments in $\mathscr{H}_{t}$ with both
endpoints on $\partial\mathscr{H}_{t}$. Let $\beta$ be some such segment of
$\alpha$, and let $d=\iota(\alpha,\alpha)$. Then we must have
$\iota(\beta,\beta)\leq d$. Thus, as $t\leq 1$, $\ell_{X}(\beta)\leq B(d)$
where $B$ is given by Lemma 3. As $B$ only depends on $d$, this holds for any
such segment. Now we need to show that there are only finitely many segments
of $\alpha$ in $\mathscr{H}_{t}\setminus\mathscr{H}_{t_{\alpha}}$. From Lemma
6, we have that the self-intersection number of each segment is at least 1,
and indeed the sum of the self-intersection numbers of these segments is at
most $d$. Hence,
$\alpha\cap(\mathscr{H}_{t}\setminus\mathscr{H}_{t_{\alpha}})$ has at most
$d+2$ components and so

$\ell^{t_{\alpha}}_{X}(\alpha)-\ell^{t}_{X}(\alpha)\leq
dB(d)+2\ln\Big{(}\frac{t}{t_{\alpha}}\Big{)}\leq
dB(d)+2\ln\Big{(}\frac{1}{t_{\alpha}}\Big{)}.$

Note that by Lemma 5, $t_{\alpha}$ depends only on $d=\iota(\alpha,\alpha)$.
Thus there exists some $C_{2}\big{(}\iota(\alpha,\alpha)\big{)}>0$ such that

$|\ell_{X}^{t_{\alpha}}(\alpha)-\ell_{X}^{t}(\alpha)|\leq
C_{2}\big{(}\iota(\alpha,\alpha)\big{)}.$

Now by applying (6) to $t_{\alpha}$, we have that
$|\ell_{X}(\gamma_{\alpha})-2\ell_{X}(\alpha^{t_{\alpha}})|\leq
C_{1}(t_{\alpha}),$ and thus we can write

$|\ell_{X}(\gamma_{\alpha})-2\ell_{X}(\alpha^{t})|\leq
C_{1}(t_{\alpha})+2C_{2}\big{(}\iota(\alpha,\alpha)\big{)}.$

Therefore, for any $t\leq 1$,

$|\ell_{X}(\gamma_{\alpha})-2\ell_{X}(\alpha^{t})|\leq
C\big{(}\iota(\alpha,\alpha),t\big{)}$

where

$C\big{(}\iota(\alpha,\alpha),t\big{)}=\begin{cases}C_{1}(t)&\text{if}\ t\leq
t_{\alpha},\\\
C_{1}(t_{\alpha})+2C_{2}\big{(}\iota(\alpha,\alpha)\big{)}&\text{if}\
t>t_{\alpha}.\end{cases}$

∎

The fact that $J$ is $\textup{PMod}(S)$-equivariant holds by an argument
analogous to the proof of Lemma 10. That is, for any infinite arc $\alpha$ and
any $\varphi\in\textup{PMod}(S)$,

$J(\varphi\cdot\alpha)=\varphi\cdot J(\alpha).$ (7)

For any infinite arc $\alpha_{0}$,
$J_{\alpha_{0}}\colon\textup{PMod}(S)\cdot\alpha_{0}\to\textup{PMod}(S)\cdot\gamma_{\alpha_{0}}$
is the restriction of $J$ to $\textup{PMod}(S)\cdot\alpha_{0}$. Given any
curve $\gamma$ of type $\gamma_{\alpha_{0}}$ and a fixed value of $t$, there
are only finitely many arcs $\alpha$ of type $\alpha_{0}$ such that
$\ell(\alpha^{t})\leq\frac{1}{2}\ell_{X}(\gamma)+\frac{1}{2}C\big{(}\iota(\alpha_{0},\alpha_{0}),t\big{)}$.
By Lemma 14, this means that there are only finitely many arcs $\alpha$ of
type $\alpha_{0}$ such that $\gamma_{\alpha}=\gamma$. Using this together with
(7), Proposition 11 holds for $J_{\alpha_{0}}$ by an analogous argument.

######  Proposition 15.

Let $\alpha_{0}$ be an infinite arc. Then there exists some $k=k(\alpha_{0})$
such that $J_{\alpha_{0}}$ is surjective and $k$-to-1.

Armed with this, we can follow the argument from the proof of Theorem 1 to
prove Theorem 2.

###### Proof of Theorem 2.

Let $\alpha_{0}$ be an infinite arc, and fix some positive $t\leq 1$. Let
$\gamma_{\alpha_{0}}$ be the curve associated to $\alpha_{0}$, as defined
above. Using the same argument as in the proof of Theorem 1, replacing
Corollary 12 and Corollary 13 with Lemma 14 and Proposition 15, we have

$\displaystyle\lim_{L\to\infty}\frac{|\\{\alpha\ \textup{of type}\
\alpha_{0}\mid\ell_{X}^{t}(\alpha)\leq L\\}|}{L^{6g-6+2(n+p)}}$
$\displaystyle=k\cdot 2^{6g-6+2(n+p)}\lim_{L\to\infty}\frac{|\\{\gamma\
\textup{of type}\ \gamma_{\alpha_{0}}\mid\ell_{X}(\gamma)\leq
L\\}|}{L^{6g-6+2(n+p)}}$ $\displaystyle=k\cdot
2^{6g-6+2(n+p)}\frac{\mathfrak{c}(\gamma_{\alpha_{0}})}{\mathfrak{b}_{g,n+p}}\mathfrak{m}_{\textup{Thu}}(\\{\ell_{X}(\cdot)\leq
1\\})$
$\displaystyle=\frac{\mathfrak{c}(\alpha_{0})}{\mathfrak{b}_{g,n+p}}\mathfrak{m}_{\textup{Thu}}(\\{\ell_{X}(\cdot)\leq
1\\})$

where $\mathfrak{c}(\alpha_{0})=k\cdot
2^{6g-6+2(n+p)}\mathfrak{c}(\gamma_{\alpha_{0}})$, $k$ is as in Proposition
15, and $\mathfrak{c}(\gamma_{\alpha_{0}})$ is as in (1). ∎

######  Remark.

As previously mentioned, Theorem 2 holds when we replace the $t$-length by the
truncated length $\ell_{X}^{Tr}$. This can be seen by applying Theorem 2 in
the case that $t=t_{\alpha_{0}}$, and using the bound on the difference in the
$t_{\alpha_{0}}$-length and the truncated length from (5).

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