# A conjugacy class counting in Teichmüller space

Pouya Honaryar

(Date: June 2021)

###### Abstract.

Let $\gamma$ be a pseudo-Anosov homeomorphism and $X$ an element of the
Teichmüller space of a genus $g$ surface. In this paper, we find asymptotics
for the number of pseudo-Anosov homeomorphisms that are conjugate to $\gamma$
and the axis of their action on Teichmüller space intersects the ball of
radius $R$ centered at $X$, as $R$ tends to infinity.

##  1\. Introduction

###  1.1. Statement of results

In [Mar04], Margulis obtained asymptotics for the volume growth and orbit
counting for balls of large radius, in the setting of manifolds with negative
curvature. Similar asymptotics were obtained in [ABEM12] for the Teichmüller
space. To state their results, let $\mathcal{T}_{g}$ be the Teichmüller space
of $S$, a surface of genus $g$, and denote the mapping class group of $S$ by
$\Gamma$. Given $X,Y\in\mathcal{T}_{g}$, let $B(X,R)$ be the ball of radius
$R$ centered at $X$, where the distance here is measured with respect to the
Teichmüller metric. Denoting the orbit of $Y$ under the action of $\Gamma$ by
$\Gamma\cdot Y$, Theorem 1.2 of [ABEM12] gives

$|\Gamma\cdot Y\cap
B(X,R)|\sim\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}e^{hR}\,\,\,\,\,\,\text{
as }\,\,\,\,\,\,R\rightarrow\infty.$

Here, $h=6g-6$ is the entropy of the Teichmüller geodesic flow with respect to
Masur-Veech measure and $\Lambda$ is the Hubbard-Masur constant [ABEM12,
Dum15]. The term $\operatorname{Vol}(\mathcal{M}_{g})$ is the normalized
volume of the moduli space $\mathcal{M}_{g}$ as explained at the end of
Section 2.2 of [ABEM12]. The cardinality of a finite set $S$ is denoted by
$|S|$ and $A(R)$ is said to be asymptotic to $B(R)$, written $A(R)\sim B(R)$,
if $A(R)/B(R)\rightarrow 1$ as $R\rightarrow\infty$. Theorem 1.3 of of the
same paper gives the following asymptotics for the volume of $B(X,R)$:

$\operatorname{Vol}(B(X,R))\sim\frac{\Lambda^{2}}{h}e^{hR}\,\,\,\,\,\,\text{
as }\,\,\,\,\,\,R\rightarrow\infty.$

Now fix $\gamma\in\Gamma$ to be a pseudo-Anosov homeomorphism and let
$\mathcal{L}_{\gamma}$ be the axis of its action on Teichmüller space, namely,
the unique geodesic that is kept fixed by this action. The cyclic group
generated by $\gamma$, denoted by $\langle\gamma\rangle$, acts on
$\mathcal{T}_{g}$ properly discontinuously, hence we can form the quotient to
be the cylinder $C_{\gamma}=\langle\gamma\rangle\backslash\mathcal{T}_{g}$.
The elements of $C_{\gamma}$ are of the form $[Y]=\langle\gamma\rangle.Y$ for
$Y\in\mathcal{T}_{g}$. Since the action of $\gamma$ on $\mathcal{L}_{\gamma}$
is by translation, the quotient
$\bar{\mathcal{L}}_{\gamma}=\langle\gamma\rangle\backslash\mathcal{L}$ is a
closed geodesic in $\mathcal{C}_{\gamma}$. Define

$B(\bar{\mathcal{L}}_{\gamma},R)=\\{[Y]\in\mathcal{C}_{\gamma}:d([Y],\bar{\mathcal{L}}_{\gamma})\leq
R\\},$

where the distance in $\mathcal{C}_{\gamma}$ is the one induced by the
Teichmüller distance on its cover $\mathcal{T}_{g}$. Define the $\Gamma$–orbit
of $[X]\in\mathcal{C}_{g}$ to be

$\Gamma\cdot[X]=\\{[gX]:g\in\Gamma\\}.$

The goal of this paper is to establish orbit counting and volume asymptotics,
similar to the ones obtained in [ABEM12], for
$B(\bar{\mathcal{L}}_{\gamma},R)$ instead of $B(X,R)$. To state these results,
we need a few definitions. Define

$B_{\operatorname{Ext}}(\mathcal{L}_{\gamma})=\\{\zeta\in\mathcal{MF}:\inf_{X\in\mathcal{L}_{\gamma}}\operatorname{Ext}(\zeta,X)\leq
1\\}.$

The action of $\langle\gamma\rangle$ on
$B_{\operatorname{Ext}}(\mathcal{L}_{\gamma})$ is proper and discontinuous,
given we remove the two endpoints of ${\mathcal{L}}_{\gamma}$ from the set,
hence we can form the quotient to be

$\mathcal{C}_{\gamma,\operatorname{Ext}}=\langle\gamma\rangle\backslash
B_{\operatorname{Ext}}({\mathcal{L}}_{\gamma}).$

The Thurston measure on $\mathcal{MF}$ induces a measure on
$\mathcal{C}_{\gamma,\operatorname{Ext}}$, which we denote by $\nu$ as well.
The orbit counting asymptotics is given by:

######  Theorem A.

Let $\gamma$ and $\bar{\mathcal{L}}_{\gamma}$ be as above and
$[X]\in\mathcal{C}_{\gamma}$. Then as $R\to\infty$,

$\displaystyle|\Gamma\cdot[X]\cap
B(\bar{\mathcal{L}}_{\gamma},R)|\sim\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}\nu(\mathcal{C}_{\gamma,\operatorname{Ext}})e^{hR}.$
(1)

Note that $\Gamma\cdot[X]\cap B(\bar{\mathcal{L}}_{\gamma},R)$ is in one-to-
one correspondence with the mapping class group translations
$g\mathcal{L}_{\gamma}$ of $\mathcal{L}_{\gamma}$ that intersects $B(X,R)$.
This in turn is in one-to-one correspondence with the conjugates of $\gamma$
whose axis intersects $B(X,R)$. Therefore, we have the following:

######  Corollary B.

Let $\gamma$ be a pseudo-Anosov homeomorphism and $X\in\mathcal{T}_{g}$. Then
as $R\to\infty$,

$|\\{\gamma^{\prime}\in\Gamma:\gamma^{\prime}\text{ is conjugate to $\gamma$
and }\mathcal{L}_{\gamma^{\prime}}\cap
B(X,R)\neq\varnothing\\}|\sim\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}\nu(\mathcal{C}_{\gamma,\operatorname{Ext}})e^{hR}.$

The volume asymptotics is given by:

######  Theorem C.

Let $\gamma$ and $\bar{\mathcal{L}}_{\gamma}$ be as above. Then as
$R\to\infty$,

$\operatorname{Vol}(B(\bar{\mathcal{L}}_{\gamma},R))\sim\frac{\Lambda^{2}}{h}\nu(\mathcal{C}_{\gamma,\operatorname{Ext}})e^{hR}.$

###  1.2. Remarks and the relation to other works

If $\Sigma$ is a surface of constant negative curvature $-1$ and $\Gamma$ its
fundamental group, then Theorem 2.5 of [EM93] gives

$|B(\bar{\mathcal{L}}_{\gamma},R)\cap\Gamma\cdot[x]|\sim\frac{\operatorname{Length}(\bar{\mathcal{L}}_{\gamma})}{\operatorname{Area}(\Sigma)}e^{R}\,\,\,\,\,\,\text{
as }\,\,\,\,\,\,R\to\infty,$

where the terms in the above expression are defined in the same way as before.
A calculation in hyperbolic metric shows Theorem C in this setting, namely, as
$R\rightarrow\infty$,

$\operatorname{Area}(B(\bar{\mathcal{L}}_{\gamma},R))\sim\operatorname{Length}(\bar{\mathcal{L}}_{\gamma})e^{R}.$

For $M$ a compact manifold of (variable) negative curvature, we can define
$\Gamma,\,\gamma,\,\mathcal{L}_{\gamma}$ and $\bar{\mathcal{L}}_{\gamma}$
similarly. The asymptotics for $|\Gamma\cdot[x]\cap
B(\bar{\mathcal{L}}_{\gamma},R)|$ can be obtained as a special case of _common
perpendicular counting_. To explain this, let $\mathcal{L}^{\prime}_{\gamma}$
and $x^{\prime}$ be the images of $\mathcal{L}_{\gamma}$ and $x$ under the
covering map $\Pi\colon\thinspace\widetilde{M}\to M$. Then $\Gamma\cdot[x]\cap
B(\bar{\mathcal{L}}_{\gamma},R)$ is in one-to-one correspondence with
$\operatorname{Perp}(x^{\prime},\mathcal{L}^{\prime}_{\gamma},R)$, the
perpendiculars from $x^{\prime}$ to $\mathcal{L}^{\prime}_{\gamma}$ of length
less than $R$, where such a perpendicular is defined as a locally geodesic
path that starts from $x^{\prime}$ and arrives perpendicularly at
$\mathcal{L}^{\prime}_{\gamma}$. It follwos from Theorem 1 of [PP17] that for
some constant $c_{\gamma}>0$,

$|\Gamma\cdot[x]\cap B(\bar{\mathcal{L}}_{\gamma},R)|\sim c_{\gamma}e^{\delta
R}\,\,\,\,\,\,\text{ as }R\to\infty.$

Here, $\delta$ is the topological entropy of the geodesic flow on $T^{1}M$,
the unit tangent bundle of $M$. Moreover, under some additional conditions, an
exponentially small error term is obtained for the above asymptotics. (See
Theorem 3 of the same paper for the precise statement.)

As a final remark, let us mention that our methods for proving Theorem A are
quite flexible. In particular, Theorem A can be proved for an arbitrary
compact set $\mathcal{K}_{\gamma}\subset\mathcal{C}_{\gamma}$ replacing
$\bar{\mathcal{L}}_{\gamma}$, and the proof is word for word the same, given
we change an $\epsilon$–net of $\bar{\mathcal{L}}_{\gamma}$ by an
$\epsilon$–cover of $\mathcal{K}_{\gamma}$. (see Section 1.3.) Volume
asymptotics in this case can be obtained in the same way as we obtained
Theorem C from Theorem A.

###  1.3. The outline of the proof

Theorem C, proved at the end of Section 5.1, follows easily from Theorem A
using an estimate obtained in Theorem 5.1 of [ABEM12] . Hence the main task is
to prove Theorem A. Let $\gamma$ and $\mathcal{L}$ be as in Section 1.1 and
fix a point $P\in\mathcal{T}_{g}$. By definition, $\Gamma\cdot[P]\cap
B(\bar{\mathcal{L}}_{\gamma},R)$ is in one to one correspondence with

$\displaystyle\langle\gamma\rangle\backslash(\Gamma\cdot P\cap
B(\mathcal{L}_{\gamma},R))=\\{\langle\gamma\rangle.X:X\in\Gamma\cdot P\cap
B(\mathcal{L}_{\gamma},R)\\},$ (2)

where

$B(\mathcal{L}_{\gamma},R)=\\{X\in\mathcal{T}_{g}:d(X,\mathcal{L}_{\gamma})\leq
R\\}.$

Note that taking the quotient by $\langle\gamma\rangle$ in (2) is justified
since $\Gamma\cdot P\cap B(\mathcal{L}_{\gamma},R)$ is
$\langle\gamma\rangle$–invariant.

Fix a point $O\in\mathcal{L}$ and for an $\epsilon>0$, let
$O=X_{0},X_{1},...,X_{N}=\gamma O$ be an $\epsilon$–net in $[O,\gamma O]$,
i.e., the geodesics connecting $X_{i}$ to $X_{i+1}$ are disjoint for $0\leq
i<N$ and $\sup_{0\leq i<N}d(X_{i},X_{i+1})<\epsilon$. Translating this net by
the powers $\gamma$, we get a $\gamma$–invariant $\epsilon$–net
$(...,X_{-1},X_{0},X_{1},...)$ of $\mathcal{L}_{\gamma}$. Define the closest
point map

$\mathcal{P}\colon\thinspace\mathcal{PMF}\to\mathbb{Z}\,\,\,\,\,\,by\,\,\,\,\,\,\mathcal{P}[\zeta]=i\,\text{
if
}\,\operatorname{Ext}(\zeta,X_{i})=\inf_{j\in\mathbb{Z}}\operatorname{Ext}(\zeta,X_{j}).$

Setting $\mathcal{A}_{i}=\mathcal{P}^{-1}(i)$, we obtain a $\gamma$–invariant
partition of $\mathcal{PMF}$. Let $S(X_{i},\mathcal{A}_{i},R)$ be the sector
of radius $R$ centered at $X_{i}$ and observing $\mathcal{A}_{i}$, namely, all
the points $Y\in\mathcal{T}_{g}$ such that $d(X_{i},Y)\leq R$ and the geodesic
connecting $X_{i}$ to $Y$ hits the boundary at an element of
$\mathcal{A}_{i}$. (see 5.1 for a precise definition.)

The main geometric idea of this paper is that $S(X_{i},\mathcal{A}_{i},R)$’s
are almost disjoint and they almost cover all of $B(\mathcal{L}_{\gamma},R)$.
(See the discussion just before Lemma 5.3.) Since $\gamma
S(X_{i},\mathcal{A}_{i},R)=S(X_{i+N},\mathcal{A}_{i+N},R)$,

$\sum_{i=0}^{N-1}|\Gamma\cdot P\cap S(X_{i},\mathcal{A}_{i},R)|$

gives a good approximation for $|\langle\gamma\rangle\backslash(\Gamma\cdot
P\cap B(\mathcal{L}_{\gamma},R))|$. The asymptotics of $|\Gamma\cdot X\cap
S(X_{i},\mathcal{A}_{i},R)|$ as $R\rightarrow\infty$ is given by [ABEM12].
Summing up these asymptotics as the $\epsilon$–net $(X_{i})$ in
$\mathcal{L}_{\gamma}$ gets finer, namely $\epsilon\to 0$, we obtain the right
hand side of (1).

To make these ideas work, we need to approximate each $\mathcal{A}_{i}$ from
inside and outside by open sets
$\mathcal{U}_{i}\subset\mathcal{A}_{i}\subset\mathcal{V}_{i}$ and squeeze the
above sum between the corresponding sums for $\mathcal{U}_{i}$ and
$\mathcal{V}_{i}$ replacing $\mathcal{A}_{i}$. To prove that these lower and
upper bounds both converge to the right-hand side of (1), we need the boundary
of the partition $\\{\mathcal{A}_{i}\\}$ to have measure zero. This is proved
in Section 3 and the proof uses Theorem D. (see Section 1.4) The only result
from Section 3 that is used in the rest of the paper is Proposition 3.1.
Section 4 is devoted to the statement and proof of Proposition 4.11, which is
the main tool we use to compare extremal and Teichmüller lengths. In Section
5, we carry out the sector approximation scheme that we mentioned earlier.
Both of the facts that $S(X_{i},\mathcal{A}_{i},R)$’s are almost disjoint and
that they almost cover $B(\mathcal{L}_{\gamma},R)$ are applications of
Proposition 4.11.

###  1.4. A formula for the derivative of extremal length

We end this introduction by stating a formula that we obtained in Section 3 in
the course of proving Proposition 3.1. For $X\in\mathcal{T}_{g}$ and
$\zeta\in\mathcal{MF}$, denote the extremal length of $\zeta$ in $X$ by
$\operatorname{Ext}(\zeta,X)$. Fixing $\zeta$, we can consider
$E_{\zeta}=\operatorname{Ext}(\zeta,\centerdot)$ as a function from
$\mathcal{T}_{g}$ to $\mathbb{R}$. This function is differentiable and its
derivative at $X\in\mathcal{T}_{g}$, $d_{X}E_{\zeta}\colon\thinspace
T_{X}(\mathcal{T}_{g})\to\mathbb{R}$, is given by the Gardiner’s formula
[Gar84]

$d_{X}E_{\zeta}(\mu)=2\Re\int_{X}\mu.\mathcal{V}_{X}^{-1}(\zeta),$

where $T_{X}(\mathcal{T}_{g})$ is the tangent space to $\mathcal{T}_{g}$ at
$X$, $\mu\in T_{X}(\mathcal{T}_{g})$ is a Beltrami differential and the
homeomorphism $\mathcal{V}_{X}\colon\thinspace Q(X)\to\mathcal{MF}$ is defined
by sending a quadratic differential to its vertical measured foliation.

If we fix $X\in\mathcal{T}_{g}$ instead, we can define

$E_{X}\colon\thinspace\mathcal{MF}\to\mathbb{R}\,\,\,\,\,\,\text{ by
}\,\,\,\,\,\,E_{X}(\zeta)=\operatorname{Ext}(\zeta,X).$

In order to compute the derivative of $E_{X}$ we need a differential structure
on $\mathcal{MF}$. In general, the manifold $\mathcal{MF}$ equipped with
train-track charts is only piecewise linear. However, if $\zeta$ is generic,
meaning that it does not have a leaf connecting any two singularities and all
the singularities are simple, then $\mathcal{MF}$ is smooth at $\zeta$ (in a
sense to be defined at the beginning of Section 3.2), hence the tangent space
at this point to $\mathcal{MF}$, $T_{\zeta}\mathcal{MF}$, is defined. The
derivative of $E_{X}$ at such a $\zeta$ is given by the following theorem:
(for the precise statement see Theorem 3.4.)

######  Theorem D.

Fix $X\in\mathcal{T}_{g}$ and let $\zeta\in\mathcal{MF}$ be a generic measured
foliation. Then $E_{X}$ is smooth at $\zeta$ and there is an $\eta\in
T_{\zeta}\mathcal{MF}$ such that

$d_{\zeta}E_{X}\colon\thinspace
T_{\zeta}\mathcal{MF}\to\mathbb{R}\,\,\,\,\,\,\text{ is given by
}\,\,\,\,\,\,d_{\zeta}E_{X}(\centerdot)=\omega_{\operatorname{Th}}(\eta,\centerdot),$

where $\omega_{\operatorname{Th}}$ stands for the Thurston symplectic form
(see Section 3.1 for a definition). Moreover, $\eta$ can be completely
described in certain train track coordinates around $\zeta$.

### Notation

For a set $A$ and subsets $A_{\delta}$ indexed by $\delta\in(0,s)$ for some
$s>0$, we say $A_{\delta}\uparrow A$ as $\delta\downarrow 0$ if the follwoing
holds: $A_{\delta_{2}}\supseteq A_{\delta_{1}}$ for $\delta_{2}<\delta_{1}$
and $\bigcup A_{\delta}=A$. Similarly, we say $B_{\delta}\downarrow B$ as
$\delta\downarrow 0$ if $B_{\delta_{2}}\subseteq B_{\delta_{1}}$ for
$\delta_{2}<\delta_{1}$ and $\bigcap B_{\delta}=B$. Finally, for real numbers
$a,b,c$ we write $a\simeq_{c}b$ if $|a-b|<c$.

### Acknowledgements

I would like to thank my advisor, Kasra Rafi, for suggesting the problem and
his constant support during the writing of this paper.

##  2\. Background on Teichmüller space

### Teichmüller space

Let $S$ be a compact surface of genus $g\geq 2$. We denote the Teichmüller
space of $S$ by $\mathcal{T}_{g}$. This is the space of equivalent classes of
orientation preserving homeomorphism $f\colon\thinspace S\to X$, where $X$ is
a Riemann surface and $f\colon\thinspace S\to X$ is said to be equivalent to
$g\colon\thinspace S\to Y$ if there exists a biholomorphism $h\colon\thinspace
X\to Y$ such that $g$ is isotopic to $h\circ f$. We denote an element
$[f\colon\thinspace S\to X]$ of $\mathcal{T}_{g}$ by $X$ and keep the marking
in the back of our mind. The mapping class group (or modular group) of $S$ is
denoted by $\Gamma$. This is the group of orientation preserving
homeomorphisms of $S$ up to isotopy. An element of mapping class group
$[\gamma\colon\thinspace S\to S]$ acts on $[f\colon\thinspace S\to
X]\in\mathcal{T}_{g}$ by change of marking, namely
$[\gamma].[f]=[f\circ\gamma^{-1}]$. Taking the quotient of $\mathcal{T}_{g}$
by $\Gamma$ we obtain the moduli space
$\mathcal{M}_{g}=\Gamma\backslash\mathcal{T}_{g}$.

### Quadratic differentials

For a Riemann surface $X$, let $Q(X)$ be the space of holomorphic quadratic
differentials (or quadratic differentials for short) on $X$. For a $\phi\in
Q(X)$, define the norm of $\phi$ to be

$|\phi|=\int_{X}|\phi(z)||dz|^{2}.$

The union of $Q(X)$ for $X\in\mathcal{T}_{g}$ forms the space of quadratic
differentials, denoted by $\mathcal{QT}_{g}$. More precisely,
$\mathcal{QT}_{g}$ is the space of equivalent classes $[f\colon\thinspace
S\to(X,\phi)]$, where $f$ and $X$ are as before and $\phi\in Q(X)$. We denote
$[f]$ by $(X,\phi)$ or just $\phi$. Sending $(X,\phi)$ to $X$ gives a
projection map $\pi\colon\thinspace\mathcal{QT}_{g}\to\mathcal{T}_{g}$. The
principal domain $\mathcal{P}(1,..,1)\subset\mathcal{QT}_{g}$ is defined to be
quadratic differentials with only simple zeros.

A flat chart for $(X,\phi)$ is a holomorphic chart $\varphi\colon\thinspace
U\subset\mathbb{C}\to X$ on which the pullback of $\phi$ is $dz^{2}$. The
change of coordinates between two flat charts is of the form $z\to\pm z+c$.
For $\phi\in\mathcal{QT}_{g}$ and $A\in\operatorname{SL}_{2}(\mathbb{R})$,
$A.\phi$ is defined as the unique element $\psi\in\mathcal{QT}_{g}$ such that
the change of marking $\phi\rightarrow A.\psi$ is given by multiplication by
$A$ on the corresponding flat charts. With this definition, Teichmüller
geodesic flow is given by

$g_{t}=\begin{pmatrix}e^{t}&0\\\ 0&e^{-t}\end{pmatrix}.$

The cotangent space at $X\in\mathcal{T}_{g}$ is naturally identified with
$Q(X)$, hence the norm on $Q(X)$ induces a Finsler norm on Teichmüller space.
The resulting metric is called the Teichmüller metric, the distance between
two points $X,Y\in\mathcal{T}_{g}$ is denoted by $d(X,Y)$ and the geodesic
connecting $X$ to $Y$ is shown by $[X,Y]$. For $X\in\mathcal{T}_{g}$ and
$\zeta\in\mathcal{MF}$, there exists a unique $\phi\in Q(X)$ such that $\phi$
has $\zeta$ as its vertical measured foliation ([HM79]). For such $X$ and
$\zeta$, define

$[X,\zeta)=\\{\pi(g_{t}.\phi),\,0\leq t\leq\infty\\}.$

### Extremal length

Let $\mathcal{V}\colon\thinspace\mathcal{QT}_{g}\to\mathcal{MF}$ be the
function that sends a quadratic differential to its vertical measured
foliation. For $X\in\mathcal{T}_{g}$, the restriction of $\mathcal{V}$ to
$Q(X)$, $\mathcal{V}_{X}\colon\thinspace Q(X)\to\mathcal{MF}$, is a
homeomorphism, and the extremal length of $\zeta\in\mathcal{MF}$ at such an
$X$ can be defined by

$\operatorname{Ext}(\zeta,X)=|\mathcal{V}_{X}^{-1}(\zeta)|.$

The Busemann functions are defined by

$\displaystyle\beta(\zeta,X)=\frac{1}{2}\log\operatorname{Ext}(\zeta,X);$
$\displaystyle\beta(\zeta,X,Y)=\beta(\zeta,Y)-\beta(\zeta,X).$

Note that $\beta(\zeta,X,Y)$ only depends on $[\zeta]$, hence it can be
denoted by $\beta([\zeta],X,Y)$. Kerschoff inequality states that

$\frac{\operatorname{Ext}(\zeta,Y)}{\operatorname{Ext}(\zeta,X)}\leq
e^{2d(X,Y)},$

hence taking logarithms we obtain

$\beta(\zeta,Y)\leq\beta(\zeta,X)+d(X,Y).$

If we think of $\beta(\zeta,X)$ as the ”length at infinity” of $[X,\zeta)$,
the above can be thought of as the triangle inequality in
$\bigtriangleup(\zeta,X,Y)$.

##  3\. equidistant measured foliations are negligible

###  3.1. Background on train tracks

We define a train track to be an embedded $3$–regular graph in $S$ such that
its vertices are locally modeled on Figure 1. The vertices of this graph are
called _switches_ and the edges are called _branches_ of the train track. In
the same figure, $a$ is called an _incoming_ branch and $b,c$ are called
_outgoing_ branches. A branch is said to be _large_ if it is the outgoing
branch for both of its endpoints. A _splitting_ along the large branch $e$ is
shown in Figure 2. A train track $\tau$ is said to be _complete_ if all the
components of $S-\tau$ are cusped triangles.

Figure 1. A switch

A function $w$ from the set of branches of $\tau$ to $\mathbb{R}$ is called a
_weight_ if we have $w(a)=w(b)+w(c)$ for every switch as in Figure 1. Let
$W(\tau)$ be the set of all weights on $\tau$. A weight $w\in W(\tau)$ is said
to be _positive_ (or a _measure_ on the train track $\tau$), denoted by $w>0$,
if $w(a)>0$ for all the branches $a$ of $\tau$. Denote the set of all measures
on $\tau$ by $W^{+}(\tau)$.

Given $\mu\in W^{+}(\tau)$, we can foliate a rectangular neighborhood of
$\tau$ according to $\mu$. Shrinking the components of the complement of this
neighborhood, we get a measured foliation, denoted by $\mathcal{F}(\tau,\mu)$.
A measured foliation $\zeta$ is said to be _carried_ by $\tau$ if there exists
a measure $\mu\in W^{+}(\tau)$ such that $\zeta=\mathcal{F}(\tau,\mu)$.

A measured foliation is called _generic_ if it has only simple singularities
and does not have a leaf connecting any two of its singularities. Let
$\zeta\in\mathcal{MF}$ be generic and assume $\tau$ is a train track carrying
$\zeta$, say, $\zeta=\mathcal{F}(\tau,\mu)$ for some $\mu\in W^{+}(\tau)$.
Since $\zeta$ is generic, $\tau$ should be complete, hence $W^{+}(\tau)$ is of
maximal dimension $6g-6$ and

$\varphi_{\tau}\colon\thinspace W^{+}(\tau)\to\mathcal{MF}\,\,\,\,\,\,\text{
defined by }\,\,\,\,\,\,\mu_{1}\mapsto\mathcal{F}(\tau,\mu_{1})$

gives a chart around $\zeta$. If the train track $\tau^{\prime}$ carries
$\zeta$ as well, then the change of coordinates
$\varphi_{\tau}^{-1}\circ\varphi_{\tau}$ is linear in a neighbourhood of
$\mu$. This gives $\mathcal{MF}$ a linear structure at such a $\zeta$. (see
the explanation after Proposition 3.1.)

For a train track $\tau$, define the antisymmetric pairing
$\omega_{\operatorname{Th}}\colon\thinspace W(\tau)\times
W(\tau)\rightarrow\mathbb{R}$ by

$\displaystyle\omega_{\operatorname{Th}}(w_{1},w_{2})=\frac{1}{2}\sum_{v}\det\begin{pmatrix}w_{1}(b_{v})&w_{1}(c_{v})\\\
w_{2}(b_{v})&w_{2}(c_{v})\end{pmatrix},$ (3)

where the sum is over all the switches $v$ of $\tau$ and at each switch $v$,
the incoming branch and outgoing branches are labeled by $a_{v},b_{v},c_{v}$
respectively, in such a way that $a_{v}b_{v}c_{v}$ is clockwise. Since
$W(\tau)$ is a vector space, $T_{\mu}W^{+}(\tau)$ is naturally identified with
$W(\tau)$ for every $\mu\in W^{+}(\tau)$, hence (3) gives an antisymmetric
form on $W^{+}(\tau)$, denoted by $\omega_{\operatorname{Th}}$ as well. It can
be proved that $\omega_{\operatorname{Th}}$ is invariant under the change of
coordinates, hence it gives rise to an antisymmetric form on $\mathcal{MF}$,
called the _Thurston symplectic form_.

Let $(X,\phi)\in\mathcal{QT}_{g}$ and denote the zeros of $\phi$ on $X$ by
$\Sigma$. A _saddle triangulation_ (or _triangulation_ for short) of $\phi$ is
a triangulation of $X$ whose vertices belong to $\Sigma$ and the edges are
straight lines in the flat metric induced by $\phi$. Fix such a triangulation
$\Delta$ of $\phi$. For a triangle $ABC\in\Delta$, a _comparison triangle_ is
defined as a flat model of $ABC$, namely, this is a Euclidean triangle
$A^{\prime}B^{\prime}C^{\prime}$ together with a flat chart
$\varphi\colon\thinspace A^{\prime}B^{\prime}C^{\prime}\to ABC$ that sends
$A^{\prime}$ to $A$, $B^{\prime}$ to $B$ and $C^{\prime}$ to $C$. (By triangle
here, we mean the union of edges and the interior.) Note that the comparison
triangle is unique up to translation and reflection from the origin.

For a triangulation $\Delta$, we can give the structure of a measured train
track to the dual graph of $\Delta$ by defining the measure of an edge $e$,
dual to the side $BC$ of a triangle $ABC\in\Delta$ to be
$|\Re(\overrightarrow{B^{\prime}C^{\prime}})|$ where
$A^{\prime}B^{\prime}C^{\prime}$ is the corresponding comparison triangle.
Note that if $\mathcal{V}(\phi)$ is generic then
$A^{\prime}B^{\prime}C^{\prime}$ does not have a vertical side, hence the
measure constructed above is indeed positive. The train track obtained in this
way is called the train track _adapted_ to $\Delta$. Observe that, as shown in
Figure 2, a flip in the triangulation $\Delta$ corresponds to a splitting in
the adapted train track and vice versa.

Figure 2. A splitting in the train track corresponds
to a flip in the triangluation

###  3.2. $E(X,Y)$ has measure zero

For given $X,Y\in\mathcal{T}_{g}$, define

$E(X,Y)=\\{\zeta\in\mathcal{MF}:\operatorname{Ext}_{X}(\zeta)=\operatorname{Ext}_{Y}(\zeta)\\}.$

The goal of this section is to prove the following:

######  Proposition 3.1.

Let $X,Y\in\mathcal{T}_{g}$ be distinct. Then $E(X,Y)$ is of Thurston measure
zero.

A manifold $M$ with charts $\varphi_{\alpha}\colon\thinspace U_{\alpha}\to
V_{\alpha}\subset M$ is said to be _smooth_ at $x\in M$ if the transition maps
are smooth near $x$. More precisely, if for all indices $\alpha$ and $\beta$
such that $x\in V_{\alpha}\cap V_{\beta}$,
$\varphi_{\alpha\beta}=\varphi_{\beta}^{-1}\circ\varphi_{\alpha}\colon\thinspace
U_{\alpha\beta}\to U_{\beta\alpha}$ is smooth on a neighbourhood of
$\varphi_{\alpha}^{-1}x$ where $U_{\alpha\beta}$ is the domain of definition
of $\varphi_{\alpha\beta}$. We define a manifold to be _linear_ or _analytic_
at a point in a similar way.

If $M$ is smooth at $x$, the tangent space to $M$ at $x$, denoted by $T_{x}M$,
can be defined in the usual way. A function $f\colon\thinspace M\to\mathbb{R}$
is said to be _smooth_ (_analytic_ , _linear_) at a smooth (analytic, linear)
point $x\in M$, if it is smooth (analytic, linear) in a neighborhood of
$\varphi_{\alpha}^{-1}(x)$ for a chart $\varphi_{\alpha}$ that covers $x$. If
$f$ is smooth at $x$, the differential of $f$ at $x$, $d_{x}f\colon\thinspace
T_{x}M\to\mathbb{R}$, can be defined in the usual way.

For a given $X\in\mathcal{T}_{g}$, define $N\colon\thinspace
Q(X)\to\mathbb{R}$ by $N(\phi)=|\phi|$. Since $Q(X)$ is a vector space, for
every $\phi\in Q(X)$ we can identify the tangent space to $Q(X)$ at $\phi$,
$T_{\phi}Q(X)$, with $Q(X)$. Define the following anti-symmetric pairing on
$T_{\phi}Q(X)$:

$\omega_{\phi}(\psi_{1},\psi_{2})=\frac{1}{4}\Im\left(\int_{X}\frac{\psi_{1}{\overline{\psi}}_{2}}{|\phi|}\right).$

Note that if $\phi\in\mathcal{P}(1,...,1)$, then $\omega_{\phi}$ is defined
for all $\psi_{1},\psi_{2}$, but it’s not necessarily so if $\phi$ has non-
simple zeros.

######  Lemma 3.2.

Given $X\in\mathcal{T}_{g}$, let $\phi\in Q(X)\cap\mathcal{P}(1,..,1)$ and
$\psi\in T_{\phi}Q(X)\simeq Q(X)$, then

$d_{\phi}N(\psi)=4\omega_{\phi}(i\phi,\psi).$

###### Proof.

The fact that $N$ is differentiable around $\phi$ follows from the proof of
Theorem 5.3 of [Dum15]. With the notation introduced in that proof, we have

$N(\phi)=N^{\epsilon}_{0}(\phi)+N^{\epsilon}_{1}(\phi),$

and it is proved that both $N^{\epsilon}_{0}$ and $N^{\epsilon}_{1}$ are
smooth in a neighborhood of $\phi$. The derivative of $N$ is computed in
[Roy71] Lemma 1. ∎

Construction. Given $X\in\mathcal{T}_{g},\,\phi\in Q(X)$ and a triangulation
$\Delta$ of $\phi$, let $(\tau,\mu)$ be the train track adapted to $\Delta$.
For every $\psi\in Q(X)$, define $w(\Delta,\psi)\in W(\tau)$ as follows: If
the branch $e$ of the train track $\tau$ is dual to the side $BC$ of a
triangle $ABC\in\Delta$, set

$w(\Delta,\psi)(e)=\frac{1}{2}\Re\int_{B}^{C}\frac{\psi}{\sqrt{\phi}},$

where the integral is taken over the side $BC$ of the triangle $ABC$, and the
sign for $\sqrt{\phi}$ is chosen so that $\Re\int_{B}^{C}\sqrt{\phi}>0$.

######  Lemma 3.3.

Let $X\in\mathcal{T}_{g}$ and assume $\phi\in Q(X)$ is such that
$\mathcal{V}(\phi)$ is generic. If $\Delta$ is a triangulation of $\phi$ and
$\tau$ is its adapted train track, then $\varphi_{\tau}^{-1}\circ\mathcal{V}$
is defined and smooth (even real analytic) in a neighbourhood of $\phi$ and
its derivative at $\phi$,
$D_{\phi}(\varphi_{\tau}^{-1}\circ\mathcal{V})\colon\thinspace T_{\phi}Q(X)\to
T_{\mu}W^{+}(\tau)$, is given by $\psi\mapsto w(\Delta,\psi)$.

###### Proof.

If $\phi_{1}\in Q(X)$ is near $\phi$, we can choose a triangulation of
$\phi_{1}$, denoted by $\Delta(\phi_{1})$, that is close to
$\Delta=\Delta(\phi)$. Let $(\tau(\phi_{1}),\mu(\phi_{1}))$ be the measured
train track adapted to $\phi_{1}$. Since $\mathcal{V}(\phi)$ is generic,
$\tau(\phi_{1})$ is the same as $\tau=\tau(\phi)$ up to isotopy. This gives us
a map

$W_{\Delta}\colon\thinspace U\to W^{+}(\tau)\,\,\,\,\,\,\text{ defined by
}\,\,\,\,\,\,W_{\Delta}(\phi_{1})=\mu(\phi_{1}),$

where $U$ is a small neighbourhood of $\phi$. Let $A$ and $B$ be two of the
zeros of $\phi$ such that $AB$ is the side of a triangle in $\Delta$ and
assume $e$ is the branch of $\tau$ that is dual to $AB$. The zeros of
$\phi_{1}$ vary as a complex analytic function of $\phi_{1}\in Q(X)$, hence
for $\phi_{1}$ close to $\phi$, $A_{\phi_{1}}$ and $B_{\phi_{1}}$ can be
chosen such that

$W_{\Delta}(\phi_{1})(e)=\int_{A_{\phi_{1}}}^{B_{\phi_{1}}}\Re\sqrt{\phi_{1}}.$

$\int_{A_{\phi_{1}}}^{B_{\phi_{1}}}\Re\sqrt{\phi_{1}}$ is called a period
function and its derivative is given by Douady-Hubbard formula to be
$w(\Delta,\psi)$ ([DH75]). ∎

Recall that for $X\in\mathcal{T}_{g}$,
$E_{X}\colon\thinspace\mathcal{MF}\to\mathbb{R}$ is defined by
$E_{X}(\zeta)=\operatorname{Ext}(\zeta,X)$.

######  Theorem 3.4.

Let $\zeta$ be a generic measured foliation and $X\in\mathcal{T}_{g}$. Then
$E_{X}$ is real analytic at $\zeta$ and its derivative,
$d_{\zeta}E_{X}\colon\thinspace T_{\zeta}\mathcal{MF}\to\mathbb{R}$, is given
by

$\displaystyle
d_{\zeta}E_{X}(\centerdot)=\omega_{\operatorname{Th}}(\eta,\centerdot)$ (4)

for some $\eta\in T_{\zeta}\mathcal{MF}$ that depends on $X$ and $\zeta$.
Moreover, assuming that $\Delta$ is a triangulation of
$\phi=\mathcal{V}_{X}^{-1}\zeta$ and $(\tau,\mu)$ its adapted train track,
then $\eta$ can be computed in the train track chart $\varphi_{\tau}$ to be
$4w(\Delta,i\phi)$, namely,

$\displaystyle\eta=D_{\mu}\varphi_{\tau}\big{(}4w(\Delta,i\phi)\big{)}$ (5)

where $D_{\mu}\varphi_{\tau}\colon\thinspace W(\tau)\cong
T_{\mu}W^{+}(\tau)\to T_{\zeta}\mathcal{MF}$ is the derivative of the chart
$\varphi_{\tau}\colon\thinspace W^{+}(\tau)\to\mathcal{MF}$ at $\mu$.

###### Proof.

We start by taking derivatives of both sides of the identity

$N(\phi_{1})=E_{X}(\mathcal{V}(\phi_{1}))$

at $\phi$. Lemma 3.2 gives the derivative of the left hand side to be
$\omega_{\phi}(4i\phi,\centerdot)$. Using chain rule for the right hand side
and the fact that $\omega_{\operatorname{Th}}$ is the push-forward of
$\omega_{\phi}$ by $\mathcal{V}_{X}$ ([Dum15] Theorem 5.8), we get

$d_{\zeta}E_{X}(\centerdot)=\omega_{\operatorname{Th}}(4D\mathcal{V}_{\phi}(i\phi),\centerdot).$

This is (4) for $\eta=4D\mathcal{V}_{\phi}(i\phi)$. To obtain
$D\mathcal{V}_{\phi}(i\phi)$ in a train track chart, we should take derivative
of both sides of

$\mathcal{V}_{X}(\phi_{1})=\varphi_{\tau}\circ W_{\Delta}(\phi_{1}),$

where $W_{\Delta}$ is defined in the proof of Lemma 3.3. Chain rule and Lemma
3.3 then gives (5). ∎

######  Lemma 3.5.

Given $X,Y\in\mathcal{T}_{g}$ and a generic measured foliation $\zeta$, let
$\eta_{1},\eta_{2}\in\mathcal{T}_{\zeta}\mathcal{MF}$ be such that
$d_{\zeta}E_{X}=\omega_{\operatorname{Th}}(\eta_{1},\centerdot)$ and
$d_{\zeta}E_{Y}=\omega_{\operatorname{Th}}(\eta_{2},\centerdot)$. Then
$\eta_{1}=\eta_{2}$ implies $X=Y$.

###### Proof.

First let us describe $w(\Delta,i\phi)$ for an arbitrary triangulation
$\Delta$ of a quadratic differential $\phi\in\mathcal{QT}_{g}$. Set
$(\tau,\mu)$ to be the train track adapted to $\Delta$. Then by the definition
of $w(\centerdot,\centerdot)$, if a branch $e$ of $\tau$ is dual to the side
$BC$ of a triangle $ABC\in\Delta$, then

$w(\Delta,i\phi)(e)=-\Im(B^{\prime}C^{\prime}),$

where the comparison triangle $A^{\prime}B^{\prime}C^{\prime}$ is chosen such
that $\Re(\overrightarrow{B^{\prime}C^{\prime}})>0$. Note that $\phi$ here is
uniquely determined by the data $(\tau,\mu,w(\Delta,i\phi))$.

Now to prove the lemma, let $\phi_{1}\in Q(X)$ and $\phi_{2}\in Q(Y)$ be such
that $\mathcal{V}(\phi_{1})=\mathcal{V}(\phi_{2})=\zeta$. Choose an arbitrary
triangulation $\Delta_{j}$ for $\phi_{j}$ and let $(\tau_{j},\mu_{j})$ be its
adapted train track $(j=1,2)$. Since
$\zeta=\mathcal{F}(\tau_{1},\mu_{1})=\mathcal{F}(\tau_{2},\mu_{2})$, the two
train tracks should have a common splitting $(\tau,\mu)$ ([PH92] Theorem
2.3.1). Since every split along an edge corresponds to a flip in the dual
triangulation, we obtain triangulations
$\Delta^{\prime}_{1},\Delta^{\prime}_{2}$ from $\Delta_{1},\Delta_{2}$ such
that they both have $(\tau,\mu)$ as their adapted train track.

By Theorem 3.4, the derivative of $\varphi_{\tau}\colon\thinspace
W^{+}(\tau)\to\mathcal{MF}$ identifies $\eta_{j}$ with
$4w(\Delta^{\prime}_{j},i\phi_{j})\in T_{\mu}W^{+}(\tau)$ for $j=1,2$. Hence,
the assumption $\eta_{1}=\eta_{2}$ implies
$w(\Delta^{\prime}_{1},i\phi_{1})=w(\Delta^{\prime}_{2},i\phi_{2})$. However,
as mentioned before, the data $(\tau,\mu,w(\Delta,i\phi))$ uniquely determines
$\phi$. This implies that $\phi_{1}=\phi_{2}$, hence $X=Y$. ∎

###### Proof of Proposition 3.1.

Let $\mathcal{G}$ be the subset of $\mathcal{MF}$ consisting of generic
measured foliations. Then $\mathcal{G}$ has full measure, i.e.,
$\nu(\mathcal{MF}{\smallsetminus}\mathcal{G})=0$. Define

$f\colon\thinspace\mathcal{MF}\to\mathbb{R}\,\,\,\,\,\,\text{ by
}\,\,\,\,\,\,f(\zeta)=E_{X}(\zeta)-E_{Y}(\zeta),$

hence $E(X,Y)=f^{-1}(0)$. Let $\zeta\in E(X,Y)\cap\mathcal{G}$ be arbitrary,
then according to Theorem 3.4 there exist $\eta_{1},\eta_{2}$ such that

$d_{\zeta}E_{X}(\centerdot)=\omega_{\operatorname{Th}}(\eta_{1},\centerdot)\,\,\,\,\,\,\text{
and
}\,\,\,\,\,\,d_{\zeta}E_{Y}(\centerdot)=\omega_{\operatorname{Th}}(\eta_{2},\centerdot),$

hence the derivative $d_{\zeta}f\colon\thinspace
T_{\zeta}\mathcal{MF}\to\mathbb{R}$ is given by

$d_{\zeta}f(\centerdot)=(\eta_{1}-\eta_{2},\centerdot).$

Since $X\neq Y$, by Lemma 3.5 we have $\eta_{1}-\eta_{2}\neq 0$. Non-
degeneracy of the Thurston form ([PH92] Theorem 3.2.4) then implies that
$d_{\zeta}f\neq 0$. Since $f$ is smooth at $\zeta$, $f(\zeta)=0$ and
$d_{\zeta}f\neq 0$, $f^{-1}(0)$ is locally a submanifold of codimension $1$
around $\zeta$. So there is a neighborhood $U_{\zeta}$ of $\zeta$ such that
$E(X,Y)\cap U_{\zeta}$ is of measure zero. Now, covering
$E(X,Y)\cap\mathcal{G}$ by countably many $U_{\zeta}$’s we obtain
$\nu(E(X,Y)\cap\mathcal{G})=0$. Since $\mathcal{G}$ has full measure,
$\nu(E(X,Y))=0$. ∎

##  4\. Comparing Teichmüller and extremal lengths

###  4.1. Projection to a thick geodesic

In this section, we state a few general facts about the geodesics that lie
completely in the thick part, where by a geodesic we always mean a bi-infinite
geodesic in the Teichmüller space, unless otherwise stated. Denoting the
covering map from Teichmüller space to the moduli space by
$\Pi\colon\thinspace\mathcal{T}_{g}\to\mathcal{M}_{g}$, for a subset
$\mathcal{K}\subset\mathcal{M}_{g}$ we define $\widetilde{\mathcal{K}}$ to be
$\Pi^{-1}(\mathcal{K})$.

######  Definition 4.1.

Let $\mathcal{K}\subset\mathcal{M}_{g}$ be compact. A Teichmüller geodesic
$\mathcal{G}$ is said to be $\mathcal{K}$– _thick_ if
$\mathcal{G}\subset\widetilde{\mathcal{K}}$.

Assume $\mathcal{G}$ is $\mathcal{K}$–thick for some compact
$\mathcal{K}\subset\mathcal{M}_{g}$ and let $X\in\mathcal{T}_{g}$ and
$\zeta\in\mathcal{MF}$ be arbitrary. Define

$\displaystyle\operatorname{proj}_{\mathcal{G}}X$
$\displaystyle=\\{Y\in\mathcal{G}:d(X,Y)=d(X,\mathcal{G})\\};$
$\displaystyle\operatorname{proj}_{\mathcal{G}}[\zeta]$
$\displaystyle=\operatorname{proj}_{\mathcal{G}}\zeta=\\{Y\in\mathcal{G}:\operatorname{Ext}(\zeta,Y)=\operatorname{Ext}(\zeta,\mathcal{G})\\},$

where $d(X,\mathcal{G})=\inf\\{d(X,Y):Y\in\mathcal{G}\\}$ and
$\operatorname{Ext}(\zeta,\mathcal{G})=\inf\\{\operatorname{Ext}(\zeta,Y):Y\in\mathcal{G}\\}$.
Both $\operatorname{diam}(\operatorname{proj}_{\mathcal{G}}X)$ and
$\operatorname{diam}(\operatorname{proj}_{\mathcal{G}}\zeta)$ are bounded by
constants depending only on $\mathcal{K}$, where $\operatorname{diam}$ stands
for the diameter of a set. The boundedness of
$\operatorname{diam}(\operatorname{proj}_{\mathcal{G}}X)$ is a consequence of
the contraction theorem of [Min96] and the boundedness of
$\operatorname{diam}(\operatorname{proj}_{\mathcal{G}}\zeta)$ is also standard
and follows, say, from Proposition 4.5.

There have been many analogies between the Teichmüller space, equipped with
the Teichmüller metric, and a hyperbolic space. More specifically, we expect
the Teichmüller metric to behave like a $\delta$–hyperbolic (Gromov
hyperbolic) metric in the thick part. (see for example [Min96], [MM99],
[Raf14].) The following is an instance of this phenomenon:

######  Theorem 4.2.

([Raf14] Theorem 8.1) Let $\mathcal{K}\subset\mathcal{M}_{g}$ be compact and
$X,Y,Z\in\mathcal{T}_{g}$. Then there are constants $C$ and $D$ only depending
on $\mathcal{K}$ such that the following holds: If $U,V\in[X,Y]$ are such that
$[U,V]\subset\widetilde{\mathcal{K}}$ and $d(U,V)>C$, then for every
$W\in[U,V]$ we have

$\min\\{d(W,[Z,X]),d(W,[Z,Y])\\}<D.$

Note that the above theorem remains true if any number of the vertices of the
triangle $\bigtriangleup(X,Y,Z)$ belongs to the boundary of Teichmüller space.
The following is a consequence of Theorem 4.2:

######  Proposition 4.3.

Let $\mathcal{K}\subset\mathcal{M}_{g}$ be compact and $\mathcal{G}$ be a
$\mathcal{K}$–thick geodesic. Then there exists a constant $C=C(\mathcal{K})$
such that for every $X\in\mathcal{T}_{g}$, $Y\in\mathcal{G}$ and
$H\in\operatorname{proj}_{\mathcal{G}}X$, the geodesic connecting $X$ to $Y$
passes through $B(H,C)$, the ball of radius $C$ centered at $H$.

###### Proof.

Let $X,H,Y$ be as in the proposition. By Theorem 4.2 there exists
$C^{\prime}=C^{\prime}(\mathcal{K})$ such that for every $Z\in[H,Y]$ there is
$W\in[X,H]\cup[X,Y]$ such that $d(Z,W)<C^{\prime}$. We claim that
$C=3C^{\prime}+1$ statisfies the statement. If $d(H,Y)<2C^{\prime}+1$ then we
are done, otherwise let $Z\in[H,Y]$ be such that $d(H,Z)=2C^{\prime}+1$. If
the point $W$ given by Theorem 4.2 lies in $[X,H]$, then by triangle
inequality in $\bigtriangleup(W,H,Z)$ we obtain $d(W,H)>C^{\prime}+1$, hence

$d(X,Z)\leq d(X,W)+d(W,Z)<d(X,H)-1,$

which contradicts the choice of $H$. This contradiction implies $W\in[X,Y]$.
Triangle inequality in $\bigtriangleup(H,Z,W)$ then implies
$d(H,W)<3C^{\prime}+1$. ∎

######  Corollary 4.4.

Let $\mathcal{K}\subset\mathcal{M}_{g}$ be compact and $\mathcal{G}$ be a
$\mathcal{K}$–thick geodesic. Then there is $C=C(\mathcal{K})$ such that for
every $X\in\mathcal{T}_{g}$, $H\in\operatorname{proj}_{\mathcal{G}}X$ and
$Y\in\mathcal{G}$, we have

$d(X,Y)\simeq_{C}d(X,H)+d(H,Y).$

###### Proof.

Let $\mathcal{G}$, $X$ and $H$ be as above and let $C=C(\mathcal{K})$ be the
constant given by Proposition 4.3. If $[X,Y]$ intersects $B(H,C)$ at $Z$ then

$\displaystyle d(X,Z)\simeq_{C}d(X,H)\text{ and }d(Z,Y)\simeq_{C}d(H,Y).$

The statement follows from summing up these two estimates. ∎

Proposition 4.3 can be proved if $X\in\mathcal{T}_{g}$ is replaced by a
measured foliation $\zeta\in\mathcal{MF}$. The proof parallels the one given
above, only instead of the triangle inequality for triangles with a vertex at
infinity, we should use Kerschoff inequality. (see the discussion at the end
of Section 2.) Corollary 4.4 can be proved in this setting as well, hence we
have the following:

######  Proposition 4.5.

Let $\mathcal{K}\subset\mathcal{M}_{g}$ be compact and $\mathcal{G}$ be a
$\mathcal{K}$–thick geodesic. Then there exists a constant $C=C(\mathcal{K})$
such that for every $\zeta\in\mathcal{MF}$,
$H\in\operatorname{proj}_{\mathcal{G}}\zeta$ and $Y\in\mathcal{G}$, we have

$\beta(\zeta,Y)\simeq_{C}\beta(\zeta,H)+d(H,Y).$

Moreover, the geodesic connecting $\zeta$ to $Y$ passes through $B(H,C)$.

###  4.2. Busemann approximation

We make the following definition:

######  Definition 4.6.

Let $\mathcal{K}\subset\mathcal{M}_{g}$ be compact. The geodesic $[X,Y]$
connecting two points $X,Y\in\mathcal{T}_{g}$ is called $\mathcal{K}$–
_typical_ if it spends at least half of its time in $\widetilde{\mathcal{K}}$

We say that two geodesics
$\mathcal{G}_{1}\colon\thinspace[0,a]\to\mathcal{T}_{g}$ and
$\mathcal{G}_{2}\colon\thinspace[0,b]\to\mathcal{T}_{g}$, parametrized with
respect to arc length, $D$– _fellow travel_ , if $|a-b|<D$ and for all $0\leq
t\leq\min\\{a,b\\}$ we have

$d(\mathcal{G}_{1}(t),\mathcal{G}_{2}(t))<D.$

######  Theorem 4.7.

([Raf14] Theorem 7.1) Let $\mathcal{K}\subset\mathcal{M}_{g}$ be compact and
$C>0$. Then there exists a constant $D=D(\mathcal{K},C)$ such that the
following holds: for every $X,Y\in\widetilde{\mathcal{K}}$ and
$\bar{X},\bar{Y}\in\mathcal{T}_{g}$ such that $d(X,\bar{X})$ and
$d(Y,\bar{Y})$ are both less than $C$, the geodesics $[X,Y]$ and
$[\bar{X},\bar{Y}]$ $D$– _fellow travel_.

Note that the conclusion of this theorem remains valid if $X=\bar{X}$ belongs
to the boundary of the Teichmüller space ([Raf14] Remark 7.2). In that case,
we should allow $a=b=\infty$ in the definition of fellow traveling.

######  Remark 4.8.

It is a consequence of this theorem that for every compact set
$\mathcal{K}\subset\mathcal{M}_{g}$ and real number $C>0$, there exists an
enlargement $\mathcal{K}^{\prime}\supset\mathcal{K}$, depending only
$\mathcal{K}$ and $C$, such that if $X,Y,\bar{X},\bar{Y}$ are as in the
theorem and $[X,Y]$ is $\mathcal{K}$–typical, then $[\bar{X},\bar{Y}]$ is
$\mathcal{K}^{\prime}$–typical.

Generically, two geodesic rays going to the same point in the boundary of
Teichmüller space become exponentially close to each other. The precise
statement is as follows:

######  Theorem 4.9.

([EMR21] Corollary ??) Let $\mathcal{K}\subset\mathcal{M}_{g}$ be compact and
$C>0$. Then there are positive numbers $\alpha=\alpha(\mathcal{K})$ and
$D=D(\mathcal{K},C)$ such that the following holds: If
$X,Y\in\widetilde{\mathcal{K}}$ and $\zeta\in\mathcal{MF}$ are such that
$\operatorname{Ext}(\zeta,X)=\operatorname{Ext}(\zeta,Y)$; $d(X,Y)<C$; and
$Z_{1}\in[X,\zeta)$ and $Z_{2}\in[Y,\zeta)$ are such that
$d(X,Z_{1})=d(Y,Z_{2})=T$, then

$d(Z_{1},Z_{2})<De^{-\alpha T}.$

We also need the following ([Min96] Corollary 4.1):

######  Lemma 4.10.

Let $\mathcal{K}\subset\mathcal{M}_{g}$ be compact and $\mathcal{G}$ be a
$\mathcal{K}$–thick geodesic. Then there exists a constant $C=C(\mathcal{K})$
such that for every $X,Y\in\mathcal{T}_{g}$ we have

$\operatorname{diam}(\operatorname{proj}_{\mathcal{G}}(X)\cup\operatorname{proj}_{\mathcal{G}}(Y))<d(X,Y)+C.$

The next proposition is the main tool that we use to relate the extremal and
Teichmüller lengths:

######  Proposition 4.11.

Let $\mathcal{K}\subset\mathcal{M}_{g}$ be compact and $\mathcal{G}$ be a
$\mathcal{K}$–thick geodesic. Then for every $\epsilon>0$ there exists
$C=C(\mathcal{K},\epsilon)$ such that the following holds: if
$\zeta\in\mathcal{MF}$; $X,Y\in\mathcal{G}$;
$Z\in[X,\zeta)\cap\widetilde{\mathcal{K}}$ and
$H\in\operatorname{proj}(Z,\mathcal{G})$ are such that the goedesic $[Z,H]$ is
$\mathcal{K}$–typical and of length greater than $C$, then

$d(Z,Y)-d(Z,X)\simeq_{\epsilon}\beta(\zeta,Y)-\beta(\zeta,X)=\beta([\zeta],X,Y).$

###### Proof.

Let $H_{\zeta}\in\operatorname{proj}_{\mathcal{G}}\zeta$ and
$C=C(\mathcal{K})$ be the constant given by Proposition 4.5, so there is
$X^{\prime}\in(\zeta,X]$ such that $d(X^{\prime},H_{\zeta})<C$. By Theorem 4.7
there is a constant $D$ depending on $C$ and $\mathcal{K}$ such that
$(\zeta,X^{\prime}]$ and $(\zeta,H_{\zeta}]$ $D$–fellow travel; as a result,
we can find $Z_{\zeta}\in(\zeta,H_{\zeta}$] with $d(Z,Z_{\zeta})<D$. Since
$H_{\zeta}\in\operatorname{proj}_{\mathcal{G}}Z_{\zeta}$, by Lemma 4.10, there
exists $D^{\prime}$ depending on $D$ and $\mathcal{K}$ such that
$d(H,H_{\zeta})<D^{\prime}$. Triangle inequality then implies
$d(X^{\prime},H)<D^{\prime}+C$, hence $[Z,H]$ and $[Z,X^{\prime}]$ fellow
travel, so $[Z,X^{\prime}]$ is $\mathcal{K}^{\prime}$–typical for some
enlargement $\mathcal{K}^{\prime}$ of $\mathcal{K}$ (Remark 4.8) and since the
constants $C,D,D^{\prime}$ depend on $\mathcal{K}$, the enlargement
$\mathcal{K}^{\prime}$ only depends on $\mathcal{K}$ as well. Applying
Proposition 4.5 once again gives $Y^{\prime}\in(\zeta,Y]$ with
$d(Y^{\prime},H_{\zeta})<C$, hence $d(X^{\prime},Y^{\prime})<2C$. Using
Kerckhoff inequality we get

$|\beta(\zeta,X^{\prime},Y^{\prime})|\leq d(X^{\prime},Y^{\prime})<2C,$

so by moving $Y^{\prime}$ along $(\zeta,Y]$ by at most $2C$ we may obtain
$Y^{\prime\prime}$ such that
$\beta(\zeta,X^{\prime})=\beta(\zeta,Y^{\prime\prime})$. Let $Z^{\prime}$ be
the point obtained by flowing $Y^{\prime\prime}$ along
$[Y^{\prime\prime},\zeta)$ by time $T=d(X^{\prime},Z)$. By Theorem 4.9 there
exists $T_{0}=T_{0}(\mathcal{K}^{\prime},C,\epsilon)$ such that if $T>T_{0}$
we have $d(Z,Z^{\prime})<\epsilon$. Hence if $d(Z,H)>T_{0}+D^{\prime}+C$, we
have

$\displaystyle d(Z,Y)-d(Z,X)$
$\displaystyle\simeq_{\epsilon}d(Z^{\prime},Y)-d(Z,X)=d(Y^{\prime\prime},Y)-d(X^{\prime},X)$
$\displaystyle=(\beta(\zeta,Y)-\beta(\zeta,Y^{\prime\prime}))-(\beta(\zeta,X)-\beta(\zeta,X^{\prime}))$
$\displaystyle=\beta(\zeta,Y)-\beta(\zeta,X).$

∎

##  5\. Proofs

###  5.1. Preliminary discussion

Fix a pseudo-Anosov homeomorphism $\gamma$ throughout this section and let
$\mathcal{L}_{\gamma}$ be the axis of its action on Teichmüller space and
$L=\tau(\gamma)$ its translation length. Define $\mathcal{C}_{\gamma}$ and
$\bar{\mathcal{L}}_{\gamma}$ as in the introduction and denote the covering
map from $\mathcal{T}_{g}$ to $\mathcal{C}_{\gamma}$ by $\Pi_{\gamma}$. To
lighten the notation, we denote $\mathcal{L}_{\gamma}$ and
$\bar{\mathcal{L}}_{\gamma}$ by $\mathcal{L}$ and $\bar{\mathcal{L}}$
respectively. Recall the following definitions:

$\displaystyle B(X,R)$ $\displaystyle=\\{Y\in\mathcal{T}_{g}:d(X,Y)\leq R\\};$
$\displaystyle B(\mathcal{L},R)$
$\displaystyle=\\{Y\in\mathcal{T}_{g}:d(Y,\mathcal{L})\leq R\\};$
$\displaystyle B_{\operatorname{Ext}}(\mathcal{L})$
$\displaystyle=\\{\zeta\in\mathcal{MF}:\operatorname{Ext}(\zeta,\mathcal{L})\leq
1\\};$

and define the following:

$\displaystyle\operatorname{Typ}(\mathcal{L},\mathcal{K})$
$\displaystyle=\\{Y\in\mathcal{T}_{g}:[Y,H]\text{ is
$\mathcal{K}$--typical}\text{ for some
}H\in\operatorname{proj}_{\mathcal{L}}Y\\};$
$\displaystyle\operatorname{Typ}(X,\mathcal{K})$
$\displaystyle=\\{Y\in\mathcal{T}_{g}:[X,Y]\text{ is
$\mathcal{K}$--typical}\\}.$

Given an open subset $\mathcal{U}$ of $\mathcal{PMF}$, let

$\displaystyle S(X,\mathcal{U},R)$ $\displaystyle=\\{\pi(g_{t}\phi):\phi\in
Q(X),[\mathcal{V}(\phi)]\in\mathcal{U}\text{ and }0\leq t\leq R\\};$
$\displaystyle S_{\operatorname{Ext}}(X,\mathcal{U})$
$\displaystyle=\\{\zeta\in\mathcal{MF}:[\zeta]\in\mathcal{U}\text{ and
}\operatorname{Ext}(\zeta,X)\leq 1\\}.$

Let $X,P\in\mathcal{T}_{g}$ and assume $\mathcal{U}$ is an open subset of
$\mathcal{PMF}$ . we need the following two facts from [ABEM12]:

  * •

As $R\rightarrow\infty$,

$\displaystyle|\Gamma\cdot P\cap
S(X,\mathcal{U},R)|\sim\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}\nu(S_{\operatorname{Ext}}(X,\mathcal{U}))e^{hR}.$
(6)

  * •

For every $\epsilon>0$, there exists a compact set
$\mathcal{K}\subset\mathcal{M}_{g}$ depending on $X,P,\epsilon$ such that

$\displaystyle\limsup_{R\rightarrow\infty}e^{-hR}|\Gamma\cdot P\cap
S(X,\mathcal{U},R){\smallsetminus}\operatorname{Typ}(X,\mathcal{K})|<\epsilon.$
(7)

The first fact follows from [ABEM12] Theorem 2.9, since, with the notation of
Proposition 2.1 of the same paper, we have

$\displaystyle\int_{\mathcal{U}}\lambda^{-}(q)ds_{X}(q)=\int_{\mathcal{U}}\frac{d(\delta^{+}_{X})_{*}\bar{\nu}}{ds_{X}}ds_{X}=(\delta_{X}^{+})_{*}\bar{\nu}(\mathcal{U})=\nu\big{(}S_{\operatorname{Ext}}(X,\mathcal{U})\big{)},$

where the first equality uses part (i) of the same proposition.

The second fact follows from [ABEM12] Theorem 2.7 since (again, with the
paper’s notation) if
$K^{\prime}\subset\mathcal{P}(1,...,1)\subset\mathcal{QT}^{1}_{g}$ is a
compact subset of the principal domain of quadratic differentials with norm
$1$, then $\mathcal{K}=\pi(K^{\prime})$ is compact as well.

Let $[\gamma^{\pm}]$ be the set containing the two elements of $\mathcal{PMF}$
that are fixed by $\gamma$. By Theorem 6.9 of [MP89], the action of
$\langle\gamma\rangle$ on $\mathcal{PMF}{\smallsetminus}[\gamma^{\pm}]$ is a
covering space action. Define

$\mathcal{C}_{\operatorname{Ext},\gamma}=\langle\gamma\rangle\backslash
B_{\operatorname{Ext}}(\mathcal{L})\,\,\,\,\,\,\text{ and let
}\,\,\,\,\,\,\Pi_{\operatorname{Ext},\gamma}\colon\thinspace
B_{\operatorname{Ext}}(\mathcal{L})\to\mathcal{C}_{\operatorname{Ext},\gamma}$

be the corresponding covering map. Since $\Gamma\cdot[P]\cap
B(\bar{\mathcal{L}},R)$ is in one-to-one correspondence with
$\langle\gamma\rangle\backslash\big{(}\Gamma\cdot P\cap
B(\mathcal{L},R)\big{)}$, Theorem A is equivalent to the following:

######  Theorem 5.1.

Let $\gamma\in\mathcal{MCG}$ be pseudo-Anosov and
$\mathcal{L}=\mathcal{L}_{\gamma}$ its axis. Then for a given
$P\in\mathcal{T}_{g}$ we have

$|\langle\gamma\rangle\backslash\big{(}\Gamma\cdot P\cap
B(\mathcal{L},R)\big{)}|\sim\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}\nu(\mathcal{C}_{\operatorname{Ext},\gamma})e^{hR},$

as $R\rightarrow\infty$.

Theorem 5.1 is proved in Section 5.2. Assuming this theorem, we now give a
proof of Theorem C.

###### Proof of Theorem C.

Fix a point $O\in\mathcal{L}$ and let $Y\in\mathcal{T}_{g}$ be arbitrary. Pick
a point $H_{Y}\in\operatorname{proj}_{\mathcal{L}}(Y)$ and choose
$k=k(Y)\in\mathbb{Z}$ in such a way that $d(O,\gamma^{k}H_{Y})<L$. Define

$h\colon\thinspace\langle\gamma\rangle\backslash\big{(}\Gamma\cdot X\cap
B(\mathcal{L},R)\big{)}\to\Gamma\cdot X\cap B(O,R+L)$

by sending $\langle\gamma\rangle.(gX)$ to $\gamma^{k{(gX)}}(gX)$ (there might
be more than one option for $H_{gX}$ and $k{(gX)}$, then choose one.) Note
that $h$ is an injection, hence

$|\langle\gamma\rangle\backslash\big{(}\Gamma\cdot X\cap
B(\mathcal{L},R)\big{)}|\leq|\Gamma\cdot X\cap B(O,R+L)|.$ (8)

By [ABEM12] Theorem 5.1, there exists a constant $C$, only depending on the
base point $O$, such that for all $X\in\mathcal{T}_{g}$ and $R>0$,

$\displaystyle|\Gamma\cdot X\cap B(O,R)|<Ce^{hR}.$

This, combined with (8), implies that for $C^{\prime}=Ce^{hL}$ and every
$X\in\mathcal{T}_{g}$ we have

$\displaystyle|\langle\gamma\rangle\backslash\big{(}\Gamma\cdot X\cap
B(\mathcal{L},R)\big{)}|<C^{\prime}e^{hR}.$ (9)

Define the covering map

$\Pi_{\gamma,\Gamma}\colon\thinspace\mathcal{C}_{\gamma}\to\mathcal{M}_{g}\,\,\,\,\,\,\text{
by }\,\,\,\,\,\,\langle\gamma\rangle.X\mapsto\Gamma\cdot X.$

Since $\Pi_{\gamma}$ is a local diffeomorphism, we have

$\displaystyle\operatorname{Vol}(B(\bar{\mathcal{L}},R))$
$\displaystyle=\int_{\mathcal{M}_{g}}|\Pi_{\gamma,\Gamma}^{-1}(X)\cap
B(\bar{\mathcal{L}},R)|d\operatorname{Vol}(X)$
$\displaystyle=\int_{\mathcal{M}_{g}}|\langle\gamma\rangle\backslash\big{(}\Gamma\cdot
X\cap B(\mathcal{L},R)\big{)}|d\operatorname{Vol}(X).$

We multiply the left and right-hand side of this equation by $e^{-hR}$ and
take the limit as $R\rightarrow\infty$. Since $C^{\prime}$ in (9) does not
depend on $X$, we can apply Lebesgue’s dominated convergence theorem to take
the limit inside the integral. Theorem 5.1 then concludes the proof. ∎

###  5.2. Concluding the proof

The goal of this section is to give a proof of Theorem 5.1. Fix two points
$P\in\mathcal{T}_{g}$ and $O\in\mathcal{L}$ for the rest of this section. To
lighten the notation, we do not show the dependance of constants on
$\mathcal{L},O,P$. So, for example, we write $C=C(\mathcal{K})$ instead of
$C=C(\mathcal{K},\mathcal{L},O,P)$. For $n,m\in\mathbb{Z}\cup\\{\pm\infty\\}$,
a sequence of points $\mathcal{X}=(X_{i})_{n\leq i\leq m}\subset\mathcal{L}$
is called a _net_ if $(X_{i},X_{i+1})$’s are disjoint for $n\leq i<m$. For
such a net $\mathcal{X}$, $\sup_{n\leq i<m}d(X_{i},X_{i+1})$ is called the
_mesh_ of $\mathcal{X}$ and for an $\epsilon>0$, a net with mesh less than
$\epsilon$ is called an $\epsilon$– _net_.

Let $\epsilon>0$ and take an $\epsilon$–net $O=X_{0},...,X_{N}=\gamma O$ in
$[O,\gamma O]$. Letting $X_{i+N}=\gamma X_{i}$, we obtain a $\gamma$–invariant
$\epsilon$–net $\mathcal{X}=(...,X_{-1},X_{0},X_{1},...)$ in $\mathcal{L}$,
and every $\gamma$–invariant $\epsilon$–net containing $O$ is obtained in this
way. From now on, by an $\epsilon$–net $\mathcal{X}=(X_{i})_{i\in\mathbb{Z}}$
we always mean a $\gamma$–invariant $\epsilon$–net in $\mathcal{L}$ with
$X_{0}=O$, and we show the number $N$ such that $X_{N}=\gamma O$ by
$N=N(\mathcal{X})$. For such an $\epsilon$–net $\mathcal{X}$, and for all
$i\in\mathbb{Z}$, define

$\displaystyle\mathcal{A}_{i}(\mathcal{X})=\\{\zeta\in\mathcal{MF}:\operatorname{Ext}(\zeta,X_{i})\leq
1\text{ and }\beta(\zeta,X_{i})=\inf_{j\in\mathbb{Z}}\beta(\zeta,X_{j})\\}.$

######  Lemma 5.2.

Let $\mathcal{X}$ be an $\epsilon$–net, then

$\frac{1}{e^{\epsilon(6g-6)}}\nu(\mathcal{C}_{\operatorname{Ext},\gamma})\leq\sum_{i=0}^{N(\mathcal{X})-1}\nu(\mathcal{A}_{i}(\mathcal{X}))\leq\nu(\mathcal{C}_{\operatorname{Ext},\gamma}).$

###### Proof.

Let $N=N(\mathcal{X})$. It follows from the definition that the interior of
$\mathcal{A}_{i}(\mathcal{X})$’s, denoted by
$\mathring{\mathcal{A}}_{i}(\mathcal{X})$’s, are disjoint and
$\mathcal{A}_{i}(\mathcal{X})$’s are $\gamma$– _equivariant_ , meaning that
$\gamma\mathcal{A}_{i}(\mathcal{X})=\mathcal{A}_{i+N}(\mathcal{X})$. These
facts imply that
$\Pi_{\operatorname{Ext},\gamma}\colon\thinspace\mathring{\mathcal{A}}_{i}(\mathcal{X})\to\mathcal{C}_{\operatorname{Ext},\gamma}$
is an injection for $i\in\mathbb{Z}$, and
$\Pi_{\operatorname{Ext},\gamma}(\mathring{\mathcal{A}}_{i}(\mathcal{X}))\subset\mathcal{C}_{\operatorname{Ext},\gamma}$
are disjoint for $0\leq i<N$. Thus we have

$\sum_{i=0}^{N-1}\nu(\mathring{\mathcal{A}}_{i}(\mathcal{X}))\leq\nu(\mathcal{C}_{\operatorname{Ext},\gamma}).$

By the definition of $E(\cdot,\cdot)$, given at the beginning of Section 3.2,
$\mathcal{A}_{i}(\mathcal{X})\cap\mathcal{A}_{j}(\mathcal{X})\subset
E(X_{i},X_{j})$, hence by proposition 3.1

$\partial\mathcal{A}_{i}(\mathcal{X})=\bigcup_{j\neq
i}\mathcal{A}_{i}(\mathcal{X})\cap\mathcal{A}_{j}(\mathcal{X})\subset\bigcup_{j\neq
i}E(X_{i},X_{j})\implies\nu(\partial\mathcal{A}_{i}(\mathcal{X}))=0.$

From this, we get

$\displaystyle\sum_{i=0}^{N-1}\nu(\mathring{\mathcal{A}}_{i}(\mathcal{X}))=\sum_{i=0}^{N-1}\nu(\mathcal{A}_{i}(\mathcal{X}))\leq\nu(\mathcal{C}_{\operatorname{Ext},\gamma}).$

This proves the right hand inequality in the statement of the lemma.

To prove the left hand inequality, let $\zeta\in
B_{\operatorname{Ext}}(\mathcal{L})$ be arbitrary and assume
$H_{\zeta}\in\operatorname{proj}_{\mathcal{L}}\zeta$, so we have
$\operatorname{Ext}(\zeta,H_{\zeta})\leq 1$. Since the mesh of $\mathcal{X}$
is less than $\epsilon$, if $X_{i_{0}}$ is the element of the net that is
closest to $H_{\zeta}$, by triangle inequality (Kerckhoff inequality) in
$\bigtriangleup(\zeta,H_{\zeta},X_{i_{0}})$ we have

$\beta(\zeta,X_{i_{0}})\leq\beta(\zeta,H_{\zeta})+\beta(H_{\zeta},X_{i_{0}})<\epsilon\implies\inf_{i\in\mathbb{Z}}\beta(\zeta,X_{i})<\epsilon\implies\inf_{i\in\mathbb{Z}}\operatorname{Ext}(\zeta,X_{i})<e^{2\epsilon}.$

If the latter infimum is attained at $i=i_{1}$ then we have
$\zeta/e^{\epsilon}\in\mathcal{A}_{i_{1}}$, hence $\zeta\in
e^{\epsilon}\mathcal{A}_{i_{1}}$. As a result,

$B_{\operatorname{Ext}}(\mathcal{L})\subset\bigcup
e^{\epsilon}\mathcal{A}_{i}(\mathcal{X})\implies\mathcal{C}_{\operatorname{Ext},\gamma}\subset\bigcup\Pi_{\operatorname{Ext},\gamma}(e^{\epsilon}\mathcal{A}_{i}(\mathcal{X})),$

which implies

$\displaystyle\frac{1}{e^{\epsilon(6g-6)}}\nu(\mathcal{C}_{\operatorname{Ext},\gamma})\leq\sum_{i=0}^{N-1}\nu(\mathcal{A}_{i}(\mathcal{X})).$

∎

Let $\mathcal{X}$ be an $\epsilon$–net and for $\delta>0$, define the
following subsets of $\mathcal{MF}$:

$\displaystyle\mathcal{U}^{\delta}_{i}(\mathcal{X})$
$\displaystyle=\\{\zeta\in\mathcal{MF}:\operatorname{Ext}(\zeta,X_{i})\leq
1\text{ and }\beta(\zeta,X_{i})<\inf_{j\neq i}\beta(\zeta,X_{j})-\delta\\};$
$\displaystyle\mathcal{V}^{\delta}_{i}(\mathcal{X})$
$\displaystyle=\\{\zeta\in\mathcal{MF}:\operatorname{Ext}(\zeta,X_{i})\leq
1\text{ and }\beta(\zeta,X_{i})<\inf_{j\neq i}\beta(\zeta,X_{j})+\delta\\}.$

Note that $\mathcal{U}^{\delta}_{i}(\mathcal{X})$’s are open and
$\gamma$–eqivariant, and the same is true for $\mathcal{V}^{\delta}_{i}$’s.
Moreover,
$\mathcal{U}^{\delta}_{i}(\mathcal{X})\subset\mathcal{A}_{i}(\mathcal{X})\subset\mathcal{V}^{\delta}_{i}$
and
$\mathcal{U}^{\delta}_{i}(\mathcal{X})\uparrow\mathring{\mathcal{A}}_{i}(\mathcal{X})$
as $\delta\downarrow 0$; also,
$\mathcal{V}^{\delta}_{i}(\mathcal{X})\downarrow\mathcal{A}_{i}(\mathcal{X})$
as $\delta\downarrow 0$. For a compact set
$\mathcal{K}\subset\mathcal{M}_{g}$, define the following subsets of
$\Gamma\cdot P$ associated to $\mathcal{U}^{\delta}_{i}(\mathcal{X})$ and
$\mathcal{V}^{\delta}_{i}(\mathcal{X})$:

$\displaystyle\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K})$
$\displaystyle=\Gamma\cdot P\cap
S\left(X_{i},[\mathcal{U}^{\delta}_{i}(\mathcal{X})]\right)\cap\operatorname{Typ}(X_{i},\mathcal{K});$
$\displaystyle\mathsf{V}^{\delta}_{i}(\mathcal{X})$ $\displaystyle=\Gamma\cdot
P\cap S\left(X_{i},[\mathcal{V}^{\delta}_{i}(\mathcal{X})]\right),$

where for a subset $\mathcal{U}\subset\mathcal{MF}$, we denote
$\\{[\zeta]:\zeta\in\mathcal{U}\\}\subset\mathcal{PMF}$ by $[\mathcal{U}]$. As
before, both $\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K})$’s and
$\mathsf{V}^{\delta}_{i}(\mathcal{X})$’s are $\gamma$–equivariant. For the
moment, let $N=N(\mathcal{X})$,
$\mathsf{U}_{i}(R)=\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K})\cap
B(X_{i},R)$ and $\mathsf{V}_{i}(R)=\mathsf{V}^{\delta}_{i}(\mathcal{X})\cap
B(X_{i},R)$. As mentioned in 1.3, the idea of the proof is to show that for
$\mathcal{K}$ large enough:

  * •

For every $i\in\mathbb{Z}$,
$\Pi_{\gamma}\colon\thinspace\mathsf{U}_{i}(R)\to\Gamma\cdot[P]\cap
B(\bar{\mathcal{L}},R)$ is almost an injection and
$\Pi_{\gamma}(\mathsf{U}_{i}(R))$’s are more or less disjoint for $0\leq
i<N$(lemma 5.3). Hence $\sum_{i=0}^{N-1}|\mathsf{U}_{i}(R)|$ gives a lower
bound for $|\Gamma\cdot[P]\cap B(\bar{\mathcal{L}},R)|$

  * •

The union of $\Pi_{\gamma}(\mathsf{V}_{i}(R+\epsilon))$’s for $0\leq i<N$
cover almost all of $C(\bar{\mathcal{L}},R)$ (lemma 5.4 \+ lemma 5.5), hence
$\sum_{i=0}^{N-1}|\mathsf{V}_{i}(R+\epsilon)|$ gives an upper bound for
$|\Gamma\cdot[P]\cap B(\bar{\mathcal{L}},R)|$.

As $R\rightarrow\infty$, we can use (6) to count the points in each
$\mathsf{U}_{i}(R)$ and $\mathsf{V}_{i}(R)$, and if $\epsilon,\delta$ are
small enough ($\delta$ moves to $0$ much faster than $\epsilon$), the upper
and lower bounds obtained in this way are close to each other, and they
approximate the right hand side of 5.1 from above and below.

######  Lemma 5.3.

Let $\mathcal{X}=(X_{i})_{i\in\mathbb{Z}}$ be an $\epsilon$–net. Then, for
every compact $\mathcal{K}\subset\mathcal{M}_{g}$ and $\delta>0$, there exists
a constant $C=C(\epsilon,\mathcal{K},\delta)$ such that for $i\neq j$,

$\left(\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K}){\smallsetminus}B(X_{i},C)\right)\cap\left(\mathsf{U}^{\delta}_{j}(\mathcal{X},\mathcal{K}){\smallsetminus}B(X_{j},C)\right)=\varnothing$

###### Proof.

Fix the net $\mathcal{X}$ throughout the proof. Let
$Y\in\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K})$ and
$H\in\operatorname{proj}_{\mathcal{L}}Y$. We claim that there is a compact set
$\mathcal{K}^{\prime}$, only depending on $\mathcal{K}$, such that $[Y,H]$ is
$\mathcal{K}^{\prime}$–typical. Let $\zeta\in\mathcal{U}_{i}^{\delta}$ be such
that $Y$ lies on $[X_{i},\zeta)$. Note that, since the net’s mesh is less than
$\epsilon$,

$\beta(\zeta,X_{i})=\inf_{k\in\mathbb{Z}}\beta(\zeta,X_{k})\simeq_{\epsilon}\inf_{Y\in\mathcal{L}}\beta(\zeta,Y).$

Because of the shape of $\beta(\zeta,X)$ as a function of $X\in\mathcal{L}$,
described in Proposition 4.5, the latter infimum should be attained near
$X_{i}$, namely, there exists $C_{1}=C_{1}(\mathcal{K})$ such that if
$H_{\zeta}\in\operatorname{proj}_{\mathcal{L}}\zeta$, then
$d(X_{i},H_{\zeta})<C_{1}$. Thus, by Theorem 4.7, $(\zeta,X_{i}]$ and
$(\zeta,H_{\zeta}]$ $D$–fellow travel for some $D=D(\mathcal{K},C_{1})$. As a
result, if $Z_{\zeta}\in(\zeta,H_{\zeta}]$ is such that
$d(Y,X_{i})=d(Z_{\zeta},H_{\zeta})$ then $d(Y,Z_{\zeta})<D$. Since
$H_{\zeta}=\operatorname{proj}_{\mathcal{L}}Z_{\zeta}$, Lemma 4.10 implies
$d(H_{\zeta},H)<D_{1}$ for some $D_{1}=D_{1}(\mathcal{K},D)$. Triangle
inequality then implies $d(X_{i},H)<C_{2}=C_{1}+D_{1}$ and since $[Y,X_{i}]$
is $\mathcal{K}$–typical, Remark 4.8 implies that $[Y,H]$ is
$\mathcal{K}^{\prime}$–typical for an enlargement
$\mathcal{K}^{\prime}\supset\mathcal{K}$ that only depends on
$\mathcal{K},C_{2}$.

This proves the claim.

If we apply Proposition 4.11 to $\mathcal{K}^{\prime}$ and $\delta/2$, we
obtain $C_{3}=C_{3}(\mathcal{K}^{\prime},\delta)$ such that if $Y$ and $H$ are
as above, $d(Y,\mathcal{L})=d(Y,H)>C_{3}$ and $[Y,H]$ is
$\mathcal{K}^{\prime}$–typical, then

$d(Y,X_{i})-d(Y,X_{j})\simeq_{\delta/2}\beta(\zeta,X_{i})-\beta(\zeta,X_{j}).$

Note that by triangle inequality in $\bigtriangleup(Y,H,X_{i})$,

$d(Y,H)\geq d(Y,X_{i})-d(X_{i},H)>d(Y,X_{i})-C_{2},$

so for $C=C_{2}+C_{3}$,

$d(Y,X_{i})>C\implies d(Y,H)>C_{3}.$

As a result, if
$Y\in\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K}){\smallsetminus}B(X_{i},C)$
we have

$\displaystyle
d(Y,X_{i})-d(Y,X_{j})\simeq_{\delta/2}d(\zeta,X_{i})-d(\zeta,X_{j})$
$\displaystyle\implies d(Y,X_{i})-d(Y,X_{j})<-\delta/2<0$
$\displaystyle\implies d(Y,X_{i})<d(Y,X_{j}).$

The second impication is because
$\zeta\in\mathcal{U}^{\delta}_{i}(\mathcal{X})$ implies
$d(\zeta,X_{i})-d(\zeta,X_{j})<-\delta$. If
$Y\in\mathsf{U}^{\delta}_{j}(\mathcal{K}){\smallsetminus}B(X_{j},C)$ as well,
then by changing the role of $X_{i}$ and $X_{j}$ in the argument above we
obtain $d(Y,X_{j})<d(Y,X_{i})$. This contradiction proves that the
intersection mentioned in the lemma is empty.

∎

###### Proof of the lower bound for Theorem 5.1.

For an $\epsilon$–net $\mathcal{X}$, a compact set
$\mathcal{K}\subset\mathcal{M}_{g}$ and $\delta>0$, let
$C=C(\epsilon,\mathcal{K},\delta)$ be the constant given by Lemma 5.3. For
every $i\in\mathbb{Z}$, define

$\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K};C,R)=\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K})\cap
B(X_{i},R){\smallsetminus}B(X_{i},C).$

We claim that

$\Pi_{\gamma}\colon\thinspace\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K};C,R)\to\Gamma\cdot[P]\cap
B(\bar{\mathcal{L}},R)$

Is an injection. To prove this claim, assume $\Pi_{\gamma}(X)=\Pi_{\gamma}(Y)$
for $X,Y\in\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K};C,R)$. This imlies
that there exists $k\in\mathbb{Z}$ such that $\gamma^{k}X=Y$, so $Y$ belongs
to the intersection of $\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K};C,R)$
with
$\gamma^{k}\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K};C,R)=\mathsf{U}^{\delta}_{i+kN}(\mathcal{X},\mathcal{K};C,R)$.
By the definition of $C$, we have $k=0$, hence $X=Y$. This proves the claim.

A similar argument implies that
$\Pi_{\gamma}\left(\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K};C,R)\right)$
are disjoint for $0\leq i<N(\mathcal{X})$, so

$\displaystyle\sum_{i=0}^{N(\mathcal{X})-1}|\mathsf{U}^{\delta}_{i}(\mathcal{X},\mathcal{K};C,R)|\leq|\Gamma\cdot[P]\cap
B(\bar{\mathcal{L}},R)|.$

Multiplying both sides by $e^{-hR}$ and taking $\liminf$ as
$R\rightarrow\infty$, we get

$\displaystyle\sum_{i=0}^{N(\mathcal{X})-1}s^{\delta}_{i}(\mathcal{X},\mathcal{K})$
$\displaystyle\leq\liminf_{R\rightarrow\infty}e^{-hR}|\Gamma\cdot[P]\cap
B(\bar{\mathcal{L}},R)|,$ (10)

where $s^{\delta}_{i}(\mathcal{X},\mathcal{K})$ is defined by

$\displaystyle s^{\delta}_{i}(\mathcal{X},\mathcal{K})$
$\displaystyle=\liminf_{R\rightarrow\infty}e^{-hR}|\mathsf{U}^{\delta}_{i}(\mathcal{K})\cap
B(X_{i},R)|$

Note that (10) is valid for every $\epsilon$–net $\mathcal{X}$, compact set
$\mathcal{K}\subset\mathcal{M}_{g}$ and $\delta>0$. Now, fixing $\mathcal{X}$
and $\delta$, we let the compact sets
$(\mathcal{K}_{n})_{n\in\mathbb{N}}\subset\mathcal{M}_{g}$ form an exhaustion
of $\mathcal{M}_{g}$. This means that
$\mathcal{K}_{n}\subset\mathring{\mathcal{K}}_{n+1}$ and
$\mathcal{M}_{g}=\bigcup_{n\in\mathbb{N}}\mathcal{K}_{n}$. Then, by (6) and
(7),

$s^{\delta}_{i}(\mathcal{X},\mathcal{K}_{n})\uparrow\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}\nu(\mathcal{U}^{\delta}_{i}(\mathcal{X}))\,\,\,\,\,\,\text{
as }\,\,\,\,\,\,n\to\infty,$

so we get

$\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}\sum_{i=0}^{N(\mathcal{X})-1}\nu(\mathcal{U}^{\delta}_{i}(\mathcal{X}))\leq\liminf_{R\rightarrow\infty}e^{-hR}|\Gamma\cdot[P]\cap
B(\bar{\mathcal{L}},R)|.$

Now, keeping the net $\mathcal{X}$ fixed in the above expression and letting
$\delta\downarrow 0$, we have
$\mathcal{U}^{\delta}_{i}(\mathcal{X})\uparrow\mathring{\mathcal{A}}_{i}(\mathcal{X})$,
hence
$\nu(\mathcal{U}^{\delta}_{i}(\mathcal{X}))\uparrow\nu(\mathring{\mathcal{A}}_{i}(\mathcal{X}))=\nu(\mathcal{A}_{i}(\mathcal{X}))$.
Thus,

$\displaystyle\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}\sum_{i=0}^{N(\mathcal{X})-1}\nu(\mathcal{A}_{i}(\mathcal{X}))\leq\liminf_{R\rightarrow\infty}e^{-hR}|\Gamma\cdot[P]\cap
B(\bar{\mathcal{L}},R)|.$

Finally, using Lemma 5.2 and making the $\epsilon$–net $\mathcal{X}$ finer,
i.e., letting $\epsilon\rightarrow 0$, proves the lower bound

$\displaystyle\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}\nu(\mathcal{C}_{\operatorname{Ext},\gamma})\leq\liminf_{R\rightarrow\infty}e^{-hR}|\Gamma\cdot[P]\cap
B(\bar{\mathcal{L}},R)|.$

∎

For the upper bound we first prove:

######  Lemma 5.4.

Let $\gamma$ and $\mathcal{L}$ be as before. Then, for every $\kappa>0$, there
exists a compact set $\mathcal{K}\subset\mathcal{M}_{g}$, only depending on
$\kappa$, such that

$\displaystyle\limsup_{R\rightarrow\infty}e^{-hR}|\langle\gamma\rangle\backslash\big{(}\Gamma.P\cap
B(\mathcal{L},R){\smallsetminus}\operatorname{Typ}(\mathcal{L},\mathcal{K})\big{)}|<\kappa.$

###### Proof.

Let

$h\colon\thinspace\langle\gamma\rangle\backslash\big{(}\Gamma\cdot P\cap
B(\mathcal{L},R)\big{)}\to\Gamma\cdot P\cap B(O,R+L)$

be the map defined in the proof of Theorem C, given after the statement of
Theorem 5.1. By (7), there exists
$\mathcal{K}^{\prime}\subset\mathcal{M}_{g}$, only depending on $O$ and
$\kappa$, such that

$\displaystyle e^{-hR}|\Gamma\cdot P\cap
B(O,R+L){\smallsetminus}\operatorname{Typ}(O,\mathcal{K}^{\prime})|<\kappa$
(11)

for $R$ large enough. Let $[Y]\in\langle\gamma\rangle\backslash(\Gamma\cdot
P\cap B(\mathcal{L},R)$ and recall that $h$ sends $[Y]$ to $\gamma^{k}Y$ such
that $d(O,\gamma^{k}H)<L$ for some $H\in\operatorname{proj}_{\mathcal{L}}Y$.
Since $d(O,\gamma^{k}H)<L$, by Remark 4.8, there exists an enlargement
$\mathcal{K}\supset\mathcal{K}^{\prime}$, only depending on
$\mathcal{K}^{\prime}$ and $L$, such that $[\gamma^{k}Y,O]$
$\mathcal{K}^{\prime}$–typical implies $[\gamma^{k}Y,\gamma^{k}H]$ is
$\mathcal{K}$–typical. Thus, $h$ sends

$\langle\gamma\rangle\backslash\big{(}\Gamma\cdot P\cap
B(\mathcal{L},R){\smallsetminus}\operatorname{Typ}(\mathcal{L},\mathcal{K})\big{)}\,\,\,\,\,\,\text{
to }\,\,\,\,\,\,\Gamma\cdot P\cap
B(O,R+L){\smallsetminus}\operatorname{Typ}(O,\mathcal{K}^{\prime}).$

The injectivity of $h$ and (11) then concludes the proof.

∎

######  Lemma 5.5.

Let the axis $\mathcal{L}$ and the point $P\in\mathcal{T}_{g}$ be as before,
and assume $\mathcal{X}=(X_{i})_{i\in\mathbb{Z}}$ is an $\epsilon$–net. Then,
for every compact set $\mathcal{K}\subset\mathcal{M}_{g}$ and $\delta>0$,
there exists $C=C(\epsilon,\mathcal{K},\delta)$ such that the following holds:
if

$Y\in\Gamma\cdot P\cap
B(\mathcal{L},R)\cap\operatorname{Typ}(\mathcal{L},\mathcal{K}),\,\,\,\,\,\,d(Y,\mathcal{L})>C,$

and $i_{0}\in\mathbb{Z}$ is such that
$d(Y,X_{i_{0}})=\inf_{i\in\mathbb{Z}}d(Y,X_{i}),$ then

$Y\in\mathsf{V}^{\delta}_{i_{0}}(\mathcal{X})\cap B(X_{i_{0}},R+\epsilon).$

###### Proof.

Let

$Y\in\Gamma\cdot P\cap
B(\mathcal{L},R)\cap\operatorname{Typ}(\mathcal{L},\mathcal{K}).$

Note that $\inf_{i\in\mathbb{Z}}d(Y,X_{i})$ is attained for some
$i\in\mathbb{Z}$ because of the shape of $d(Y,X)$ as a function of
$X\in\mathcal{L}$, described in Corollary 4.4. Let the geodesic from
$X_{i_{0}}$ to $Y$ hit the boundary at $[\zeta]$, i.e.,
$Y\in[X_{i_{0}},\zeta)$. By Proposition 4.11, there exists
$C=C(\mathcal{K},\delta)$ such that if $d(Y,\mathcal{L})>C$, then for all
$i\in\mathbb{Z}$ we have

$\displaystyle\beta(\zeta,X_{i})-\beta(\zeta,X_{i_{0}})\simeq_{\delta}d(Y,X_{i})-d(Y,X_{i_{0}})$
$\displaystyle\implies$
$\displaystyle\beta(\zeta,X_{i})-\beta(\zeta,X_{i_{0}})>d(Y,X_{i})-d(Y,X_{i_{0}})-\delta\geq-\delta$
$\displaystyle\implies$
$\displaystyle\beta(\zeta,X_{i_{0}})<\beta(\zeta,X_{i})+\delta$
$\displaystyle\implies$
$\displaystyle\zeta\in\mathcal{V}^{\delta}_{i_{0}}(\mathcal{X})$

Let $H\in\operatorname{proj}_{\mathcal{L}}Y$ and note that since $Y\in
B(\mathcal{L},R)$, we have $d(Y,H)\leq R$. Choose $X_{i_{1}}$ to be the point
of $\mathcal{X}$ that is closest to $H$. Since the mesh of $\mathcal{X}$ is
less than $\epsilon$, we have $d(H,X_{i_{1}})<\epsilon$, hence by triangle
inequality $d(Y,X_{i_{1}})<R+\epsilon$. So

$d(Y,X_{i_{0}})=\inf_{j\in\mathbb{Z}}d(Y,X_{i})<R+\epsilon.$

This proves the lemma.

∎

###### Proof of the upper bound for Theorem 5.1.

For an $\epsilon$–net $\mathcal{X}$, a compact set
$\mathcal{K}\subset\mathcal{M}_{g}$ and $\delta>0$, let
$C=C(\epsilon,\mathcal{K},\delta)$ be the constant given by Lemma 5.5, and for
$R>C$, write

$\Gamma\cdot P\cap
B(\mathcal{L},R)=\mathsf{B}_{1}(\mathcal{K};C,R)\cup\mathsf{B}_{2}(\mathcal{K},R)\cup\mathsf{B}_{3}(C),$

where

$\displaystyle\mathsf{B}_{1}(\mathcal{K};C,R)$ $\displaystyle=\Gamma\cdot
P\cap
B(\mathcal{L},R)\cap\operatorname{Typ}(\mathcal{L},\mathcal{K}){\smallsetminus}B(\mathcal{L},C);$
$\displaystyle\mathsf{B}_{2}(\mathcal{K},R)$ $\displaystyle=\Gamma\cdot P\cap
B(\mathcal{L},R){\smallsetminus}\operatorname{Typ}(\mathcal{L},\mathcal{K});$
$\displaystyle\mathsf{B}_{3}(C)$ $\displaystyle=\Gamma\cdot P\cap
B(\mathcal{L},C).$

Note that $\mathsf{B}_{1}(\mathcal{K},C,R)$ (
$\mathsf{B}_{2}(\mathcal{K},R),\,\,\mathsf{B}_{3}(C)$ ) is $\gamma$–invariant,
hence we can form the quotient
$\overline{\mathsf{B}}_{1}(\mathcal{K},C,R)=\langle\gamma\rangle\backslash\mathsf{B}_{1}(\mathcal{K},C,R)$
(
$\overline{\mathsf{B}}_{2}(\mathcal{K},R)=\langle\gamma\rangle\backslash\mathsf{B}_{2}(\mathcal{K},R),\,\,\overline{\mathsf{B}}_{3}(C)=\langle\gamma\rangle\backslash\mathsf{B}_{3}(C)$
) and obtain

$\displaystyle|\langle\gamma\rangle\backslash\big{(}\Gamma\cdot P\cap
B(\mathcal{L},R)\big{)}|\leq\overline{\mathsf{B}}_{1}(\mathcal{K},C,R)+\overline{\mathsf{B}}_{2}(\mathcal{K},R)+\overline{\mathsf{B}}_{3}(C).$
(12)

Since $|\overline{\mathsf{B}}_{3}(C)|$ is a constant only depending on $C$,

$\limsup_{R\rightarrow\infty}e^{-hR}|\overline{\mathsf{B}}_{3}(C)|\rightarrow
0.$

To control $|\overline{\mathsf{B}}_{2}(\mathcal{K},R)|$, we define

$\kappa(\mathcal{K})=\limsup_{R\rightarrow\infty}e^{-hR}|\overline{\mathsf{B}}_{2}(\mathcal{K},R)|.$

To find an upper bound for $\overline{\mathsf{B}}_{1}(\mathcal{K},C,R)$, note
that by Lemma 5.5,

$\displaystyle\mathsf{B}_{1}(\mathcal{K},C,R)$
$\displaystyle\subset\bigcup_{i\in\mathbb{Z}}\big{(}\mathsf{V}^{\delta}_{i}(\mathcal{X})\cap
B(X_{i},R+\epsilon)\big{)}$
$\displaystyle\implies|\overline{\mathsf{B}}_{1}(\mathcal{K},C,R)|$
$\displaystyle\leq\sum_{i=0}^{N(\mathcal{X})-1}|\mathsf{V}^{\delta}_{i}(\mathcal{X})\cap
B(X_{i},R+\epsilon)|.$

Multiplying both sides of (12) by $e^{-hR}$ and taking the $\limsup$ as
$R\rightarrow\infty$ (while keeping $\mathcal{X}$, $\mathcal{K}$ and $\delta$
fixed), we obtain

$\displaystyle\limsup_{R\rightarrow\infty}e^{-hR}|\langle\gamma\rangle\backslash\Gamma\cdot
P\cap B(\mathcal{L},R)|$
$\displaystyle\leq\sum_{i=0}^{N(\mathcal{X})-1}\lim_{R\rightarrow\infty}e^{-hR}|\mathsf{V}^{\delta}_{i}\cap
B(X_{i},R+\epsilon)|+\kappa(\mathcal{K})$
$\displaystyle=\sum_{i=0}^{N(\mathcal{X})-1}\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}e^{h\epsilon}\nu(\mathcal{V}^{\delta}_{i}(\mathcal{X}))+\kappa(\mathcal{K}),$

where the equality is by (6). Note that the above is valid for every
$\epsilon$–net $\mathcal{X}$, compact set $\mathcal{K}\subset\mathcal{M}_{g}$
and $\delta>0$. Keeping $\mathcal{X}$ and $\delta$ fixed, we let
$\mathcal{K}_{n}\uparrow\mathcal{M}_{g}$ to be an exhaustion of
$\mathcal{M}_{g}$. By Lemma 5.4, $\kappa(\mathcal{K}_{n})\downarrow 0$ as
$n\to\infty$, thus

$\displaystyle\limsup_{R\rightarrow\infty}{e^{-hR}|\langle\gamma\rangle\backslash\Gamma\cdot
P\cap
B(\mathcal{L},R)|}\leq\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}e^{h\epsilon}\sum_{i=0}^{N(\mathcal{X})-1}\nu(\mathcal{V}^{\delta}_{i}(\mathcal{X})).$

As in the proof of the lower bound, we first let $\delta\downarrow 0$ and use
$\bigcap\mathcal{V}^{\delta}_{i}(\mathcal{X})=\mathcal{A}_{i}(\mathcal{X})$,
then let $\epsilon\downarrow 0$ and use Lemma 5.2 to obtain

$\displaystyle\limsup_{R\rightarrow\infty}{e^{-hR}|\langle\gamma\rangle\backslash\Gamma\cdot
P\cap
B(\mathcal{L},R)|}\leq\frac{\Lambda^{2}}{h\operatorname{Vol}(\mathcal{M}_{g})}\nu(\mathcal{C}_{\operatorname{Ext},\gamma}).$

∎

## References

  * [ABEM12] Jayadev Athreya, Alexander Bufetov, Alex Eskin, and Maryam Mirzakhani.  Lattice point asymptotics and volume growth on Teichmüller space.  Duke Math. J., 161(6):1055–1111, 2012.
  * [DH75] A. Douady and J. Hubbard.  On the density of Strebel differentials.  Invent. Math., 30(2):175–179, 1975.
  * [Dum15] David Dumas.  Skinning maps are finite-to-one.  Acta Math., 215(1):55–126, 2015.
  * [EM93] Alex Eskin and Curt McMullen.  Mixing, counting, and equidistribution in Lie groups.  Duke Math. J., 71(1):181–209, 1993.
  * [EMR21] Alex Eskin, Maryam Mirzakhani, and Kasra Rafi.  Absolutely continuous stationary measures for the mapping class group.  preprint, 2021.
  * [Gar84] Frederick P. Gardiner.  Measured foliations and the minimal norm property for quadratic differentials.  Acta Math., 152(1-2):57–76, 1984.
  * [HM79] John Hubbard and Howard Masur.  Quadratic differentials and foliations.  Acta Math., 142(3-4):221–274, 1979.
  * [Mar04] Grigoriy A. Margulis.  On some aspects of the theory of Anosov systems.  Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2004.  With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska.
  * [Min96] Yair N. Minsky.  Quasi-projections in Teichmüller space.  J. Reine Angew. Math., 473:121–136, 1996.
  * [MM99] Howard A. Masur and Yair N. Minsky.  Geometry of the complex of curves. I. Hyperbolicity.  Invent. Math., 138(1):103–149, 1999.
  * [MP89] John McCarthy and Athanase Papadopoulos.  Dynamics on Thurston’s sphere of projective measured foliations.  Comment. Math. Helv., 64(1):133–166, 1989.
  * [PH92] R. C. Penner and J. L. Harer.  Combinatorics of train tracks, volume 125 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 1992.
  * [PP17] Jouni Parkkonen and Frédéric Paulin.  Counting common perpendicular arcs in negative curvature.  Ergodic Theory Dynam. Systems, 37(3):900–938, 2017.
  * [Raf14] Kasra Rafi.  Hyperbolicity in Teichmüller space.  Geom. Topol., 18(5):3025–3053, 2014.
  * [Roy71] H. L. Royden.  Automorphisms and isometries of Teichmüller space.  In Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969), pages 369–383. Ann. of Math. Studies, No. 66. Princeton Univ. Press, Princeton, N.J., 1971.