$p$, and setting the length of the $k$th pants curve to be $i_{k}$, where the
$i_{k}$ is one of the acceptable lengths. It’s clear that the cardinality of
$\mathcal{Z}$ is at most $J\cdot R^{N}$, which is a polynomial only depending
on the topological type of the surface $S$ and the basepoint $p$.

Suppose now that $y$ is some other point in $B_{R}(p)$. We pick a
$\kappa\in\mathrm{MCG}(S)$ such that the shortest marking on $\kappa y$ is one
of the markings $M_{j}$ for $1\leq j\leq J$. We now need to show that one of
the $z\in\mathcal{Z}$ is close to $\kappa y$. Pick the $z$ such that the
corresponding lengths of the pants curves are closest to the lengths of the
pants curves on $\kappa y$. We can now invoke the combinatorial distance
formula for Teichmüller metric (proved by [Raf07] for the orientable setting,
and Theorem A.6 for the non-orientable case).

$\displaystyle d(z,\kappa y)\mathrel{\mathop{\asymp}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\cdot$}\vss}}}\sum_{Y}\left[d_{Y}(z,\kappa
y)\right]_{k}+\sum_{\alpha\not\in\Gamma}\log\left[d_{\alpha}(z,\kappa)\right]_{k}+\max_{\alpha\in\Gamma}d_{\mathbb{H}_{\alpha}}(z,\kappa
y)$

In the above formula, the first term is the distance between the short
markings when projected to non-annular subsurfaces, the second term is the
distance between the short markings when projected to annular subsurfaces
whose core curves are not the pants curves in the marking, and the third term
corresponds to the length and twist parameters of the short curves.

Since both $z$ and $\kappa y$ have the same short markings, the first two
terms in the above sum become $0$. Also, since we picked $z$ to be the element
of $\mathcal{Z}$ such that the lengths were closest to those on $\kappa y$,
the third term is bounded by some constant, which proves the result. ∎

###  5.3. Using Complexity Length to Count Bad Points

In this section we introduce an alternative to the Teichmüller metric, called
the _complexity length_ (see Definition 6.23). Complexity length (denoted by
$\mathfrak{L}$) was constructed by [DM23], in order to get better estimates on
net points contained in the thin part of Teichmüller space (for orientable
surfaces). We adapt the construction of complexity to the Teichmüller space of
non-orientable surfaces in Section 6. In this section, we state the main
results about complexity length we need in order to prove Theorem 5.1.

For this section, we will state the results with a rescaled version of
complexity length, in order to compare it with Teichmüller length.

######  Definition 5.8 (Rescaled complexity length).

Let $S$ be a surface of finite type. The rescaled complexity length
$d_{\mathrm{comp}}$ on $\mathrm{core}(\mathcal{T}(S))$ is given by the
following formula.

$\displaystyle
d_{\mathrm{comp}}(x,y)=\frac{\mathfrak{L}(x,y)}{h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(S)),\varepsilon_{t})}$

Here, $\mathfrak{L}(x,y)$ is the complexity length between points $x$ and $y$.

The first result we will need is a count of the net points with respect to the
rescaled complexity length.

######  Theorem 5.9 (Theorem 12.1 of [DM23], Theorem 6.24).

There exists a polynomial function $p(R)$ that depends on the net
$\mathfrak{N}$, and a parameter $\varepsilon_{\mathrm{err}}>0$, such that the
following inequality holds for any $\varepsilon_{\mathrm{err}}>0$.

$\displaystyle\\#\left(y\in\mathfrak{N}\mid d_{\mathrm{comp}}(p,x)\leq
R\right)\leq
p(R)\exp((h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(S)),\varepsilon_{t})+\varepsilon_{\mathrm{err}})\cdot
R)$

The next result, which is the main theorem of Section 6, is that if $y$ is a
bad point that is Teichmüller distance $R$ away from $p$, then its rescaled
complexity distance to $p$ is smaller than $R$ by a definite amount. We state
this theorem with an additional hypothesis on the net point entropy of
subsurfaces. We will establish that this hypothesis holds inductively in
Section 5.5.

######  Theorem 5.10 (Linear gap in complexity length, Theorem 6.25).

Suppose that for all proper subsurfaces $V$ of $\mathcal{S}$, the following
inequality holds.

$\displaystyle
h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(V)),\varepsilon_{t})<h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(\mathcal{S}),\varepsilon_{t}))$

Then for any $\varepsilon_{b}>0$, there exists $c>0$, such that for all $R>0$,
and for any bad point $y$, i.e. a point in
$\mathfrak{N}_{b}(p,R,\varepsilon_{b})$, the following upper bound on the
complexity distance between $p$ and $y$ holds.

$\displaystyle d_{\mathrm{comp}}(p,y)\leq R(1-c)$

######  Remark.

We give a brief outline of why the above result should hold. Complexity length
can be thought of as a weighted version of Teichmüller length, where a
specific segment is assigned a weight based on whether it’s traveling in the
thick part of Teichmüller space, or the thin part. In the latter case, it is
assigned a smaller weight that is proportional to the net point entropy of the
product region it is traveling in. The Teichmüller geodesics associated to bad
points spend a significant fraction traveling in the thin part, by the very
definition of bad points. It stands to reason then that the complexity length
assigned to them is smaller by a definite amount, due to the time they spend
in the thin part.

Combining Theorems 5.9 and 5.10, it follows that as $R$ goes to $\infty$, the
proportion of bad points goes to $0$, which is what we need for Theorem 5.1.

###  5.4. Entropy Gap

In the previous section, we saw that the key hypothesis we need for the
complexity length estimate was that for any proper subsurface $V$ of
$\mathcal{S}$, the following strict inequality held.

$\displaystyle
h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(V)),\varepsilon_{t})<h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(\mathcal{S})),\varepsilon_{t})$
(24)

In this section, we will prove that (24) holds by proving a similar inequality
for the lattice point entropy, and using the fact that Theorem 5.1 holds for
all proper subsurfaces $V$, by the inductive hypothesis.

######  Lemma 5.11 (Lattice point entropy gap).

Let $\mathcal{S}$ be a surface, and $\chi(\mathcal{S})\leq-2$. If $V$ is a
proper subsurface and Theorem 5.1 holds for $V$, then we have the following
strict inequality between their lattice point entropy.

$\displaystyle
h_{\mathrm{LP}}(\mathcal{T}(V),\varepsilon_{t})<h_{\mathrm{LP}}(\mathcal{T}(\mathcal{S}),\varepsilon_{t})$

######  Remark.

We do actually need the hypothesis $\chi(\mathcal{S})\leq-2$ in the statement
of the lemma for two reasons. The first reason is that the lemma is actually
false for $\mathcal{S}_{1,0,1}$. Recall that this surface has the torus with
one boundary component as a subsurface, but their mapping class groups are
isomorphic, and have the same lattice point growth entropy. Another reason why
we need the hypothesis is that the proof of the lemma proceeds via a
construction of pseudo-Anosov elements on $\mathcal{S}$, and
$\mathcal{S}_{1,0,1}$ does not admit any pseudo-Anosov mapping classes.

###### Proof of Lemma 5.11.

Observe that $\mathrm{MCG}(V)$ is a subgroup of $\mathrm{MCG}(\mathcal{S})$.
We will first construct an intermediate subgroup
$H=\mathbb{Z}*\mathrm{MCG}(V)$, which is the free product of a pseudo-Anosov
element in $\mathrm{MCG}(\mathcal{S})$ with $\mathrm{MCG}(V)$, and show that
$h_{\mathrm{LP}}(H,\varepsilon_{t})>h_{\mathrm{LP}}(\mathrm{MCG}(V),\varepsilon_{t})$.
This is enough to prove the result, since $H$ is a subgroup of
$\mathrm{MCG}(\mathcal{S})$, we have that
$h_{\mathrm{LP}}(\mathrm{MCG}(\mathcal{S}),\varepsilon_{t})\geq
h_{\mathrm{LP}}(H,\varepsilon_{t})$.

We now need to show that $\mathrm{MCG}(\mathcal{S})$ contains a pseudo-Anosov
element. We can invoke Penner’s construction of pseudo-Anosov mapping classes
([Pen88, Theorem 4.1]), as long as we can construct a filling collection of
_two-sided_ curves in $\mathcal{S}$. This may not be always possible for
$\mathcal{S}$ where $\chi(\mathcal{S})=-1$, but for $\mathcal{S}$ with
$\chi(\mathcal{S})\leq-2$, this is always possible (see [LS18] and [KPW23] for
explicit constructions). Let $\kappa$ denote the pseudo-Anosov mapping class
we construct.

By [Yan18, Proposition 6.6], there exists a large enough $n$ such that
$\kappa^{n}$ and $\mathrm{MCG}(V)$ generate their free product in
$\mathrm{MCG}(S)$: call this subgroup $H$.

We now need to show that the lattice point entropy for $H$ is strictly larger
than the $\mathrm{MCG}(V)$. To see this, we recall an equivalent definition of
the lattice point entropy. The lattice point entropy is the infimum of the set
of exponents $h$ such that the following Poincaré series transitions converges
for any $x\in\mathcal{T}(\mathcal{S})$.

$\displaystyle\sum_{\gamma\in H}\exp\left(-h\cdot d_{\varepsilon_{t}}(x,\gamma
x)\right)$ (25)

Since $H=\mathbb{Z}*\mathrm{MCG}(V)$, we can represent $\gamma\in H$ as
$a_{1}\cdot b_{1}\cdot a_{2}\cdots a_{k}\cdot b_{k}$, where $a_{i}$ belong in
$\mathbb{Z}$ and $b_{i}$ belong in $\mathrm{MCG}(V)$. We use this along with
the triangle inequality to get an upper bound for $d(x,\gamma x)$.

$\displaystyle d_{\varepsilon_{t}}(x,\gamma
x)\leq\sum_{i=1}^{k}d_{\varepsilon_{t}}(x,a_{i}x)+d_{\varepsilon_{t}}(x,b_{i}x)$
(26)

We plug inequality (26) into (25) to get a lower bound.

$\displaystyle\sum_{\gamma\in H}\exp\left(-h\cdot d_{\varepsilon_{t}}(x,\gamma
x)\right)$
$\displaystyle=\sum_{k=1}^{\infty}\left(\sum_{a_{1}}\cdots\sum_{a_{k}}\sum_{b_{1}}\cdots\sum_{b_{k}}\exp(-h\cdot
d_{\varepsilon_{t}}(x,a_{1}\cdot b_{1}\cdots a_{k}\cdot b_{k}x))\right)$ (27)
$\displaystyle\geq\sum_{k=1}^{\infty}\left(\sum_{a\in\mathbb{Z}}\exp(-h\cdot
d_{\varepsilon_{t}}(x,ax))\right)^{k}\left(\sum_{b\in\mathrm{MCG}(V)}\exp(-h\cdot
d_{\varepsilon_{t}}(x,bx))\right)^{k}$ (28)

We have that Theorem 5.1 holds for $V$, which means that
$\mathrm{core}(\mathcal{T}(V))$ is SCC. Corollary 5.4 of [Yan18] states that
group actions that are SCC have Poincaré series that diverge at the critical
exponent. This means there’s small enough $\varepsilon>0$ such that for
$h=h_{\mathrm{LP}}(\mathcal{T}(V),\varepsilon_{t})+\varepsilon$, the series
converges to a value greater than $1$. But that means the Poincaré series for
$H$ diverges at $h_{\mathrm{LP}}(\mathcal{T}(V),\varepsilon_{t})+\varepsilon$,
since we have a lower bound by a geometric series whose ratio is greater than
$1$. This proves that the critical exponent for $H$ is strictly greater than
the critical exponent for $\mathrm{MCG}(V)$. ∎

We can now prove the entropy gap result for $h_{\mathrm{NP}}$.

######  Lemma 5.12 (Net point entropy gap).

Let $\mathcal{S}$ be a surface, and $\chi(\mathcal{S})\leq-2$. If $V$ is a
proper subsurface and Theorem 5.1 holds for $V$, then we have the following
strict inequality between their net point entropy.

$\displaystyle
h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(V)),\varepsilon_{t})<h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(\mathcal{S})),\varepsilon_{t})$

###### Proof.

We have the following inequality, which follows trivially from the definition
of $h_{\mathrm{LP}}$ and $h_{\mathrm{NP}}$.

$\displaystyle h_{\mathrm{LP}}(\mathcal{T}(\mathcal{S}),\varepsilon_{t})\leq
h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(\mathcal{S})),\varepsilon_{t})$ (29)

From Lemma 5.11, we get the following inequality.

$\displaystyle
h_{\mathrm{LP}}(\mathcal{T}(V),\varepsilon_{t})<h_{\mathrm{LP}}(\mathcal{T}(\mathcal{S}),\varepsilon_{t})$
(30)

Finally, since we have Theorem 5.1 for $V$, we have the following equality.

$\displaystyle
h_{\mathrm{LP}}(\mathcal{T}(V),\varepsilon_{t})=h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(V)),\varepsilon_{t})$
(31)

Chaining together (29), (30), and (31) gives us the result. ∎

###  5.5. Proof of Theorem 5.1

We now have all the lemmas we need in order to prove Theorem 5.1.

###### Proof of Theorem 5.1.

We will prove this lemma by inducting on the complexity of the surface
$\mathcal{S}$. Lemma 5.2 proves the result for surfaces with Euler
characteristic equal to $-1$, which serves as the base case of the theorem.

We now assume that Theorem 5.1 already holds for all proper subsurfaces $V$ of
$\mathcal{S}$: it will suffice to show that the result holds for
$\mathcal{S}$.

We will establish that for any $\varepsilon_{b}>0$, there exists a polynomial
$q(R)$, and $R$ large enough, such that the following bound holds.

$\displaystyle\\#\left(B_{R}(p)\cap\mathfrak{N}\right)\leq
q(R)\cdot\exp(h_{\mathrm{LP}}(\mathcal{T}(\mathcal{S}),\varepsilon_{t})\cdot
R\cdot(1+2\varepsilon_{b}))$

We first count the good points in $B_{R}(p)$, by partitioning them according
to the nearest lattice point.

$\displaystyle\mathfrak{N}_{g}(p,R,\varepsilon_{b})=\bigsqcup_{\gamma\in\mathrm{MCG}(\mathcal{S})}\mathfrak{N}_{g}(\gamma,p,R,\varepsilon_{b})$
(32)

Observe that if $y\in\mathfrak{N}_{g}(\gamma,p,R,\varepsilon_{b})$, then
$d(p,\gamma p)\leq R(1+\varepsilon_{b})$, since $d(p,y)\leq R$ and $d(y,\gamma
p)\leq\varepsilon_{b}R$. This observation leads to the following upper bound
on $\\#\left(\mathfrak{N}_{g}(p,R,\varepsilon_{b})\right)$.

$\displaystyle\\#\left(\mathfrak{N}_{g}(p,R,\varepsilon_{b})\right)\leq\sum_{\begin{subarray}{c}\gamma\in\mathrm{MCG}(\mathcal{S})\\\
d(p,\gamma p)\leq
R(1+\varepsilon_{b})\end{subarray}}\\#\left(B_{\varepsilon_{b}R}(\gamma
p)\cap\mathfrak{N}_{g}(\gamma,p,R,\varepsilon_{b})\right)$ (33)

By Lemma 5.6, there exists a polynomial $q(R)$ such that each term in the
above sum is at most $q(R)$.

$\displaystyle\\#\left(\mathfrak{N}_{g}(p,R,\varepsilon_{b})\right)$
$\displaystyle\leq\sum_{\begin{subarray}{c}\gamma\in\mathrm{MCG}(\mathcal{S})\\\
d(p,\gamma p)\leq R(1+\varepsilon_{b})\end{subarray}}q(R)$ (34)
$\displaystyle\leq
q(R)\cdot\exp(h_{\mathrm{LP}}(\mathcal{T}(\mathcal{S}),\varepsilon_{t})\cdot
R\cdot(1+2\varepsilon_{b}))$ (35)

Here, we estimated the cardinality of $\gamma$ such that $d(p,\gamma p)\leq
R(1+\varepsilon_{b})$ as at most
$\exp(h_{\mathrm{LP}}(\mathcal{T}(\mathcal{S}),\varepsilon_{t})\cdot
R\cdot(1+2\varepsilon_{b}))$, for large enough $R$. We have the desired upper
bound on the cardinality for the good points. Now we show that the number of
bad points is much smaller than the total number of points in the ball, which
will then prove the result.

From the inductive hypothesis, we have that Theorem 5.1 holds for all proper
subsurfaces $V$. By Lemma 5.12, we have that
$h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(V)),\varepsilon_{t})<h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(\mathcal{S})),\varepsilon_{t})$:
this is precisely the hypothesis we need to apply Theorem 5.10. Applying the
theorem, we see that if $y$ is a bad point, $d_{\mathrm{comp}}(p,y)\leq
R(1-c)$. We then apply Theorem 5.9 to get an upper bound on the number of bad
points.

$\displaystyle\\#\left(\mathfrak{N}_{b}(p,R,\varepsilon_{b})\right)\leq
kR^{k}\cdot\exp((h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(\mathcal{S})),\varepsilon_{t})+\varepsilon_{\mathrm{err}})\cdot
R\cdot(1-c))$

We pick $\varepsilon_{\mathrm{err}}$ small enough such that the above term
satisfies the following inequality for large enough $R$.

$\displaystyle\exp((h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(\mathcal{S})),\varepsilon_{t})+\varepsilon_{\mathrm{err}})\cdot
R\cdot(1-c))<\exp\left(h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(\mathcal{S})),\varepsilon_{t})\cdot
R\cdot\left(1-\frac{2c}{3}\right)\right)$

On the other hand, we have that for large enough $R$, the total number of net
points is at least
$\exp\left(h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(\mathcal{S})),\varepsilon_{t})\cdot
R\cdot\left(1-\frac{c}{2}\right)\right)$. Combining these two facts, we see
that the proportion of bad points goes to $0$ as $R$ goes to $\infty$, which
proves the result. ∎

##  6\. Linear Gap in Complexity Length

###  6.1. An Example of Counting in Product Regions

Before we define complexity length, we will look at an example that
illustrates why we need complexity length. Theorem 1.3 of [ABEM12] proves an
estimate on the volume of balls in Teichmüller space. From this volume
estimate, we can obtain an estimate on the cardinality of net points of an
$(\varepsilon_{n},2\varepsilon_{n})$-net $\mathfrak{N}$.

######  Theorem 6.1 (Theorem 1.3 of [ABEM12]).

For a point $p$ in $\mathcal{T}(S)$ (where $S$ is a genus $g$ surface with $b$
boundary components), the number of net points in a ball of radius $R$
centered at the origin satisfies the following asymptotic as $R$ goes to
$\infty$.

$\displaystyle\\#\left(\mathfrak{N}\cap
B_{R}(p)\right)\mathrel{\mathop{\asymp}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\cdot$}\vss}}}\exp((6g-6+2b)R)$

Here, the multiplicative and additive constants showing up in
$\mathrel{\mathop{\asymp}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\cdot$}\vss}}}$ only depend on $p$ and
$\varepsilon_{n}$.

Suppose now that we want to use the above theorem to count net points in a
product region. More concretely, let $p$ be a point in $\mathcal{T}(S)$ such
that a non-separating curve $\gamma$ is very short:
$\ell_{\gamma}(p)\leq\delta\cdot\exp(-R_{0})$, for some $\delta>0$, and some
large $R_{0}$, and we want to estimate the cardinality of $\mathfrak{N}\cap
B_{R}(p)$ for $R<R_{0}$. Note that since $R<R_{0}$, the ball $B_{R}(p)$ is
still contained in the product region of Teichmüller space where
$\ell_{\gamma}\leq\delta$.

Since the entire ball $B_{R}(p)$ is in a product region, we have by Minsky’s
product region theorem (see Theorem A.3 for a precise statement) that the ball
decomposes (up to an additive error) as the product of a ball in
$\mathcal{T}(S\setminus\gamma)$ and ball in $\mathbb{H}$ (which corresponds to
the length and twist around $\gamma$). This gives us an alternative estimate
for $\\#\left(\mathfrak{N}\cap B_{R}(p)\right)$.

$\displaystyle\\#\left(\mathfrak{N}\cap B_{R}(p)\right)$ $\displaystyle\leq
C\cdot\left(\mathfrak{N}_{1}\cap
B_{R}(p,S\setminus\gamma)\right)\cdot\left(\mathfrak{N}_{2}\cap
B_{R}(p,\mathbb{H})\right)$ (36)

Here $\mathfrak{N}_{1}$ and $\mathfrak{N}_{2}$ are
$(\varepsilon_{n},2\varepsilon_{n})$ nets for $\mathcal{T}(S\setminus\gamma)$
and $\mathbb{H}$, and $B_{R}(p,S\setminus\gamma)$ and $B_{R}(p,\mathbb{H})$
are projections on the ball $B_{R}(p)$ to the two components. Applying Theorem
6.1 to the right hand side of (36), we get a better estimate than we would
have gotten with a direct application of Theorem 6.1.

$\displaystyle\\#\left(\mathfrak{N}\cap B_{R}(p)\right)$ $\displaystyle\leq
C\cdot\left(\mathfrak{N}_{1}\cap
B_{R}(p,S\setminus\gamma)\right)\cdot\left(\mathfrak{N}_{2}\cap
B_{R}(p,\mathbb{H})\right)$
$\displaystyle\mathrel{\mathop{\asymp}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\cdot$}\vss}}}\exp((6(g-1)-6+2(b+2))R)\cdot\exp(R)$
$\displaystyle=\exp((6g-6+2b-1)R)$

This example illustrates that in order to count net points accurately, it’s
not sufficient to just estimate the distance the between the base point $p$
and the net point $n$: if the geodesic segment $[p,n]$ travels in a product
region, the count will be lower than what Theorem 6.1 predicts. In fact, the
count will also depend on the type of the product region. In the above
example, the product region had just one curve $\gamma$ becoming short, but in
general, a product region can have multiple curves getting short, in which
case, the net point count will be even smaller.

We now consider a geodesic $[x,y]$ (where $x$ and $y$ are net points) that
travels through several product regions $\pi_{i}$, and possibly the thick
part, which we will also consider a product region, albeit a trivial one. Let
$h_{i}$ be the exponent associated to the product region $\pi_{i}$: this is
the exponent that will appear when we invoke Theorem 6.1 to count net points
in the product region $\pi_{i}$. The order in which $[x,y]$ travels through
the product region is specified in Figure 5.

$x$
---
$z_{1}$
---
$z_{2}$
---
$z_{3}$
---
$z_{4}$
---
$z_{5}$
---
$z_{6}$
---
$z_{7}$
---
$z_{8}$
---
$y$
---
$\pi_{1}$
---
$\pi_{2}$
---
$\pi_{3}$
---
$\pi_{4}$
---
$\pi_{5}$
---
Figure 5. A schematic of the geodesic $[x,y]$ traveling through several
product regions.

Let $z_{i}$ denote the points on the geodesic segment that correspond to the
times when the geodesic enters or exits a product region: in Figure 5, we have
labeled $z_{i}$ for $1\leq i\leq 8$. Let $\mathcal{J}_{i}$ denote the interval
$[x,z_{1}]$ for $i=0$, the interval $[z_{i},z_{i+1}]$ for $1\leq i\leq 7$, and
$[z_{8},y]$ for $i=8$. Let $\ell_{i}$ be the length of $\mathcal{J}_{i}$, and
$e_{i}$ be the sum of the $h_{i}$ for each of the product regions that the
interval $\mathcal{J}_{i}$ is in. Also, for each $z_{i}$, let $z_{i}^{\prime}$
denote the nearest net point.

Keeping $x$ fixed, we can try to count the number of net points $y$ that
satisfy the configuration we have described. We have about
$O(\exp(e_{0}\ell_{0}))$ possibilities for $z_{1}^{\prime}$, and then keeping
a $z_{1}^{\prime}$ fixed, $O(\exp(e_{i}\ell_{1}))$ possibilities for
$z_{2}^{\prime}$ and so on. Multiplying all these estimates, we have the
following upper bound for cardinality of $y$.

$\displaystyle\\#\left(y\right)\leq\exp\left(\sum_{i=0}^{8}e_{i}\ell_{i}\right)$

The quantity $\sum e_{i}\ell_{i}$ serves as a re-weighted version of length of
the geodesic in manner that works well with the counting function. This is a
primitive version of the _complexity length_ of $[x,y]$, and motivates the
actual definition.

Before we define complexity length, we note two ways in which the above
estimate overestimates the actual number of net points: that happens when a
point in the geodesic is simultaneously in two or more product regions, which
can happen in two ways.

  1. (i)

The active subsurfaces associated to product regions are disjoint: In this
case, we are accounting the length of the geodesic segment multiple times:
once for each product region we are in. However, this overcounting is still
better than directly invoking Theorem 6.1, since the sums of the exponents
$h_{i}$ associated to each of the disjoint product regions are smaller than
the exponent associated to the entire surface.

  2. (ii)

The active subsurfaces associated toproduct regions are nested: In this case
as well, we are accounting for the length of a geodesic multiple times, once
for each product region we are in. Unlike in the previous case, in this case,
the exponents associated to each product region can add up to a quantity
larger than the exponent associated to the entire surface, which means the
presence of nested product regions can give a worse estimate than Theorem 6.1.
We will get around this problem by looking only at product regions associated
to special subsurfaces which are called _witnesses_.

###  6.2. An Overview of Complexity Length

Now that we have motivated the need for complexity length, as well as
considering special subsurfaces called witnesses, we formally define them in
this section. This section is a summary for Sections 7 through 12 of [DM23],
so we refer the reader to those sections for details we elide. One difference
in our presentation is that we care about these constructions for both
orientable and non-orientable surfaces, while the original authors only work
with orientable surfaces. However, their constructions and proofs go through
for non-orientable surfaces, as long as we provide a proof of the non-
orientable versions of some of the foundational results they use. We list
those theorems here, and link to the proof of the non-orientable version that
appears in Section A.

  1. (i)

Minsky’s product region theorem (see Theorem A.3).

  2. (ii)

Distance formula for Teichmüller space (see Theorem A.6).

  3. (iii)

Active intervals for subsurfaces (see Proposition A.7).

  4. (iv)

Consistency and realization (see Theorem A.9).

Let $\mathcal{S}$ be a surface (not necessarily orientable), and $\mathbf{C}$
some large arbitrary constant, and $\varepsilon_{t}>0$ a small constant we
pick later. We also pick constants $N_{V}$, for each $V\sqsubset\mathcal{S}$,
such that $N_{V}$ only depends on the topological type of $V$. The precise
values of the $N_{V}$’s is specified via Proposition 10.13 of [DM23]. We will
also abuse notation slightly and use $h_{\mathrm{NP}}(V)$ to refer to
$h_{\mathrm{NP}}(\mathrm{core}(\mathcal{T}(V)))$ whenever $V$ is a non-
orientable surface: when $V$ is orientable, $h_{\mathrm{NP}}(V)$ will refer to
$h_{\mathrm{NP}}(\mathcal{T}(V))$.

Let $[x,y]$ be a geodesic segment in $\mathcal{T}(\mathcal{S})$: we describe
the set $\Upsilon(x,y)$ of subsurfaces along which $[x,y]$ has large
projections.

######  Definition 6.2 (Active subsurfaces).

A subsurface $V\sqsubset\mathcal{S}$ is an active subsurface, i.e. in
$\Upsilon(x,y)$, if one of the following two conditions hold.

  1. (i)

The projection to $\mathcal{C}(V)$ has diameter at least $N_{V}$.

  2. (ii)

If $V$ is annular with core curve $\gamma$, then

$\displaystyle\min\left(\ell_{\gamma}(x),\ell_{\gamma}(y)\right)<\varepsilon_{t}$

Associated to each active subsurface $V$, there is a non-empty connected sub-
interval of $[x,y]$, which we call an active interval, and denote
$\mathcal{I}_{V}^{\varepsilon_{t}}$, which we obtain via an application of
Proposition A.7. The active intervals associated to active subsurfaces enjoy
the following properties.

  1. (i)

$\ell_{\alpha}(z)<\varepsilon_{t}$ for $z\in\mathcal{I}_{V}^{\varepsilon_{t}}$
and $\alpha\in\partial V$.

  2. (ii)

For $z\not\in\mathcal{I}_{V}^{\varepsilon_{t}}$,
$\ell_{\alpha}(z)>{\varepsilon_{t}}^{\prime}$ for some $z\in\partial V$, and
some ${\varepsilon_{t}}^{\prime}<{\varepsilon_{t}}$ that only depends on
${\varepsilon_{t}}$.

  3. (iii)

For $[w,z]\subset[x,y]$ with
$[w,z]\cap\mathcal{I}_{V}^{\varepsilon_{t}}=\varnothing$, $d_{V}(w,z)\leq
M_{\varepsilon_{t}}$ for some $M_{\varepsilon_{t}}$ that only depends on
${\varepsilon_{t}}$.

  4. (iv)

For $U\pitchfork V$,
$\mathcal{I}_{U}^{\varepsilon_{t}}\cap\mathcal{I}_{V}^{\varepsilon_{t}}=\varnothing$.

For pairs of transverse subsurfaces $U\pitchfork V$, since
$\mathcal{I}_{U}^{\varepsilon_{t}}\cap\mathcal{I}_{V}^{\varepsilon_{t}}=\varnothing$
we can also determine which of the subsurfaces are active first.

######  Definition 6.3 (Behrstock partial order).

If $U$ and $V$ are a pair of transverse subsurfaces in $\Upsilon(x,y)$, we say
$U\lessdot V$ if $\mathcal{I}_{U}^{\varepsilon_{t}}$ appears to the left of
$\mathcal{I}_{V}^{\varepsilon_{t}}$ in $[x,y]$.

Observe that when restricted to $\mathcal{I}_{V}^{\varepsilon_{t}}$, the
geodesic is traveling in a product region, one of whose components is
$\mathcal{T}(V)$, but trying to apply the technique from the previous
subsection leads to the problem of overcounting, namely overcounting arising
from subsurfaces either nested in $V$, or subsurfaces $V$ is nested in.

To deal with this issue, we will consider a subset of $\Upsilon(x,y)$, called
a _witness family_. However, to avoid overcounting, some additional properties
are required of the witness families. Rather than defining all of those
properties without context, we introduce them one at a time, after motivating
the need for the property.

######  Definition 6.4 (Witness family).

A witness family $\Omega(x,y)$ associated to the geodesic $[x,y]$ is a subset
of $\Upsilon(x,y)$ satisfying the following properties.

  1. (i)

For any $Z\in\Upsilon(x,y)$, $Z\sqsubset W$ for some $W\in\Omega(x,y)$.

  2. (ii)

If $Z\sqsubset W$, and $Z\in\Omega(x,y)$ and $W\in\Upsilon(x,y)$, then $W$
must also either be a witness, or must be transverse to a witness
$V\in\Omega(x,y)$ such that $Z\sqsubset V$.

The first condition of the definition ensures that when we restrict our
attention from all active subsurfaces to witnesses, we do not lose
information, i.e. every active subsurface contributes to whichever witness it
is contained in. The second condition is a more technical requirement that is
required to ensure that the other properties we define later work nicely.

We now make the notion of an active subsurface _contributing to a witness_
more precise.

######  Definition 6.5 (Complete witness family).

For an active subsurface $V$, a witness $W$ is said to be the
$\Omega$-completion of $V$, denoted $\overline{V}^{\Omega}$ if $W$ is the
minimal (by inclusion) witness containing $V$. If $W=\overline{V}^{\Omega}$,
we say $V$ _contributes_ to $W$. Furthermore, a witness family is _complete_
if every active subsurface has a unique $\Omega$-completion.

By partitioning off the collection active subsurfaces into classes, where each
class is represented by a witness, and only considering the product regions
associated to the witnesses, rather than all the active subsurfaces, we can
cut down on the overcount we obtain by considering all active subsurfaces.

We now look at an extreme example of a complete witness families to motivate
further properties that we will need from the witness families in order to
count well.

######  Example 6.6 (Trivial witness family).

Let $\gamma$ be a pseudo-Anosov mapping class on $\mathcal{S}$, such that
$\gamma$ has large translation distance on $\mathcal{C}(\mathcal{S})$, and
$\delta$ a reducible mapping class, acting on a subsurface $V$ such that the
action of $\delta$ on $\mathcal{C}(V)$ has large translation distance as well.

Let $x$ be a point in $\mathcal{T}(\mathcal{S})$, and
$y=\delta\gamma\delta^{-1}x$. The active subsurfaces for $[x,y]$ contain the
surfaces $\mathcal{S}$, $V$, and $\gamma V$: however, we can pick
$\Omega(x,y)=\left\\{\mathcal{S}\right\\}$, and check that this is a complete
witness family.

In the above example, since we only have one subsurface in our witness family,
we certainly do not overcount via overlapping product regions, but we do end
up ignoring the fact that the geodesic $[x,y]$ travels in a smaller product
region near the beginning of the segment, as well as the end. For the initial
and the final segment of the geodesic, the witness $\mathcal{S}$ is too big
for the subsurface the geodesic is actually traveling in. This suggest that a
better choice of a witness family would be to include both $V$ and $\gamma V$
as witnesses too. We can take this approach further, and include every
subsurface in $\Upsilon(x,y)$ as a witness: this will still form a complete
witness family. However, this approach also leads to multiple witnesses nested
within one another, which is something we want to avoid as much as possible.

The drawback of Example 6.6 motivates the next property we will require from
witness families, which is the notion of being _insulated_. Informally, a
witness family $\Omega(x,y)$ is insulated if all the maximal active
subsurfaces that are active near the beginning or end of $[x,y]$ are also
witnesses.

######  Definition 6.7 (Insulated witness family).

A witness family $\Omega(x,y)$ is insulated if for every $E\in\Omega(x,y)$,
all subsurfaces $V\sqsubset E$ satisfying the following properties are also
witnesses.

  1. (i)

$V\in\Upsilon(x,y)$.

  2. (ii)

$d_{E}(\mathcal{C}(V),x)\leq 9\mathbf{C}$, or $d_{E}(\mathcal{C}(V),y)\leq
9\mathbf{C}$, where we consider $\mathcal{C}(V)$ to be a subset of
$\mathcal{C}(E)$.

  3. (iii)

$V$ is topologically maximal among the subsurfaces that satisfy (i) and (ii).

Once we have an insulated witness family, we can order a nested pair of
witnesses $W\sqsubset V$ based on whether $W$ is active near the beginning or
the end of the geodesic $[x,y]$ projected to $\mathcal{C}(V)$.

######  Definition 6.8 (Subordering).

Let $[x,y]$ be a geodesic in $\mathcal{T}(\mathcal{S})$ and $\Omega(x,y)$ a
complete insulated witness family associated to $[x,y]$. Then for each nested
pair of witnesses $W\sqsubset V$, a subordering is an assignment of exactly
one of the following two possibilities:

  1. (i)

$W\swarrow V$

  2. (ii)

$V\searrow W$

The orderings $\swarrow$ and $\searrow$ satisfy the following properties.

  1. (i)

If $Z$, $V$, and $W$ are witnesses such that $Z\sqsubset V\sqsubset W$, then
$Z\swarrow W$ iff $V\swarrow W$ (equivalently, $W\searrow Z$ iff $W\searrow
V$).

  2. (ii)

If $Z$, $V$ and $W$ are witnesses such that $Z\swarrow V\searrow W$, then
$Z\pitchfork_{V}W$ and $Z\lessdot W$.

  3. (iii)

If $Z$ and $V$ are witnesses, and $W$ an active subsurface such that
$Z\swarrow V\lessdot W$, or $W\lessdot V\searrow Z$, then $Z\pitchfork_{V}W$.

  4. (iv)

If $Z$ and $V$ are witnesses, such that $Z\swarrow V$ (or $Z\searrow V$), then
there does not exist any active subsurface $W$ such that the $\Omega$-closure
of $W$ is $V$ and $W\lessdot Z$ (or $Z\lessdot W$).

Here $Z\pitchfork_{V}W$ refers to notion of two subsurfaces cutting each other
relative to $V$.

######  Definition 6.9 (Relative cutting).

Given a subsurface $V$ of $S$, we say two subsurfaces $Z$ and $W$ of $S$ _cut
relative_ to $V$ if for any subsurfaces $Z^{\prime}\sqsubset Z$ and
$W^{\prime}\sqsubset W$ that intersect $V$, $Z^{\prime}\pitchfork W^{\prime}$.

We now provide some motivation for the various conditions that appear in the
above definition. First of all, when we see $Z\swarrow W$, we are to read that
as _the geodesic $[x,y]$ makes progress in the nested subsurface $Z$, before
making progress in the supersurface $W$_. Similarly, when we see $W\searrow
Z$, we are to read that as _the geodesic $[x,y]$ make progress in the
supersurface $W$ before making progress in the nested subsurface $Z$_. With
this description of the subordering, conditions (i) and (iv) of the definition
are easy to understand. The conditions (ii) and (iii) let us upgrade
$\swarrow$ and $\searrow$ to transversality and time-ordering. A more
intuitive reading of condition (ii) for instance would be, if $Z\swarrow
V\searrow W$, that means the geodesic makes progress in $Z$ before $V$, and
then makes progress in $W$. That means if we just look at $V$ and $W$, it
makes progress in $V$ and then $W$. And since neither of them are nested in
the other, the only way they can be time-ordered is by cutting relative to
$V$.

We can also see how the subordering on a witness family interacts with the
witness family being insulated: recall the pair of witnesses $V\sqsubset E$
from Definition 6.7.

  * •

If $d_{E}(\mathcal{C}(V),x)\leq 9\mathbf{C}$, then $V\swarrow E$, since the
geodesic makes progress in $V$ before $E$.

  * •

If $d_{E}(\mathcal{C}(V),Y)\leq 9\mathbf{C}$, then $E\searrow V$, since the
geodesic makes progress in $E$ before $V$.

However, the above example does not capture all the ways in which we can have
$V\swarrow E$ or $E\searrow V$. Consider a decomposition of a Teichmüller
geodesic by the active intervals corresponding to witnesses illustrated in
Figure 6.

$x$
---
$y$
---
$I_{E}$
---
$I_{V_{1}}$
---
$I_{V_{2}}$
---
$I_{V_{3}}$
---
$I_{W_{1}}$
---
$I_{W_{2}}$
---
$I_{W_{3}}$
---
Figure 6. Examples of $V_{i}\swarrow E$ and $E\searrow W_{i}$.

In this example, all of the witnesses $V_{i}$ satisfy $V_{i}\swarrow E$, and
all of the witnesses $W_{i}$ satisfy $E\searrow W_{i}$, but
$d_{E}(\mathcal{C}(V_{i}),x)$ need not be less than $9\mathbf{C}$ for $i=2$ or
$i=3$, and similarly, $d_{E}(\mathcal{C}(W_{i}),y)$ need not be less than
$9\mathbf{C}$ for $i=2$ or $i=3$.

In fact, the above example illustrates that for witnesses $V$ that are not
within distance $9\mathbf{C}$ from one of the endpoints, the choice between
assigning $V\swarrow E$ and $E\searrow V$ is ambiguous, which is what the
property of being _wide_ tries to fix. The property of being _wide_ also tells
when an active subsurface $V$ nested in a witness $E$ contributes to $E$: this
happens when $V$ appears in the “middle” of the segment $[x,y]$.

######  Definition 6.10 (Wide witness families).

An insulated complete subordered witness family is wide if for each $V$ in the
witness family, both of the following quantities are at most
$\frac{N_{V}}{3}$.

  * •

For $W$ a witness such that $W\swarrow V$, the quantity
$\mathrm{diam}_{V}(x,\mathcal{C}(W))$.

  * •

For $W$ a witness such that $V\searrow W$, the quantity
$\mathrm{diam}_{V}(y,\mathcal{C}(W))$.

The idea behind a wide witness family is to create a buffer zone of length at
least $\frac{N_{V}}{3}$ in the middle of the projection of the geodesic to
$\mathcal{C}(V)$ for any witness $V$ such that:

  * •

If any subsurface $W$ is active to the left of the buffer zone, it contributes
to a subsurface $Z$ such that $Z\swarrow V$.

  * •

If any subsurface $W$ is active to the right of the buffer zone, it
contributes to a subsurface $Z$ such that $V\searrow Z$.

  * •

If a subsurface $W$ is active within the buffer zone, it contributes to $V$.

The upshot of defining wide, insulated, subordered, and complete witness
families (which will be abbreviated to WISC witness families) is that it gives
us a better idea the order in which progress is made in various active
subsurfaces. If we were working with just the collection of active
subsurfaces, the only time we can tell if a geodesic makes progress in a
subsurface $V$ followed by the subsurface $W$ is when $V\pitchfork W$. When
working with a WISC witness family, we can do that, but we can also make
similar statements about pairs of nested witnesses $V\sqsubset W$, namely we
can have either $V\swarrow W$ or $W\searrow V$.

The following lemma asserts that WISC witness families exist, and their
cardinality can be uniformly bounded.

######  Lemma 6.11 (Lemmas 7.29 and 7.30 from [DM23]).

Let $\mathcal{S}$ be surface, and $[x,y]$ a geodesic segment in
$\mathcal{T}(\mathcal{S})$. Then there exists a WISC witness family
$\Omega(x,y)$ for $[x,y]$. Furthermore, the cardinality of $\Omega(x,y)$
depends only on $\mathcal{S}$, and not the points $x$ and $y$.

######  Remark.

While the statements of Lemmas 7.29 and 7.30 in [DM23] are for orientable
surfaces, they go through without any changes for non-orientable surfaces as
well.

We now get to the raison d’être of witness families: turning points on the
geodesic segment $[x,y]$ in $\mathcal{T}(\mathcal{S})$ into points in
$\mathcal{T}(V)$, where $V$ is a witness in $\Omega(x,y)$. We will do so by
assigning to each point $w$ in a neighbourhood of $[x,y]$ a point
$\widetilde{w}_{Z}$ in $\mathcal{C}(Z)$ for all subsurfaces $Z$ contained in
$V$, and then showing this assignment is consistent. Then the realization
theorem (Theorem A.9) will give us a point $\widehat{w}_{V}^{\Omega}$ in
$\mathcal{T}(V)$ which has the same projections in $\mathcal{C}(Z)$ as the
original point $W$.

######  Definition 6.12 (Projection tuple).

Let $\Omega(x,y)$ be a WISC witness family for a Teichmüller geodesic $[x,y]$
in $\mathcal{T}(\mathcal{S})$. Let $w$ be a point in
$\mathcal{T}(\mathcal{S})$ satisfying the following bound for every subsurface
$V$.

$\displaystyle d_{V}(x,w)+d_{V}(w,y)\leq d_{V}(x,y)+9\mathbf{C}$

Then for any $U\in\Omega(x,y)$, the projection tuple $\widetilde{w}$ of $w$ is
the point in $\prod_{Z\sqsubseteq U}\mathcal{C}(Z)$ given by the following
formula (where $\pi_{Z}$ is the usual projection map from
$\mathcal{T}(\mathcal{S})$ to $\mathcal{C}(Z)$).

$\displaystyle\widetilde{w}_{Z}=\begin{cases}\pi_{Z}(y),&\text{if
$Z\in\Upsilon(x,y)$ and $\overline{Z}^{\Omega}\swarrow U$}\\\
\pi_{Z}(x),&\text{if $Z\in\Upsilon(x,y)$ and
$U\searrow\overline{Z}^{\Omega}$}\\\ \pi_{Z}(w),&\text{otherwise}\end{cases}$

Observe that this is different from the usual projection map from
$\mathcal{T}(\mathcal{S})$ to $\mathcal{C}(Z)$: for subsurfaces $Z$ that
contribute to a witness nested in $U$, and consequently,
$\overline{Z}^{\Omega}\swarrow U$ or $U\searrow\overline{Z}^{\Omega}$, we
change the projection from $\pi_{Z}(w)$ to $\pi_{Z}(y)$ or $\pi_{Z}(x)$
respectively.

This new projection map, despite being a modification of the usual projection
map, is still consistent.

######  Proposition 6.13 (Proposition 8.4 of [DM23]).

The projection tuple $\widetilde{w}_{Z}$ is $k$-consistent for some $k$
depending only on $\mathbf{C}$.

Using the above proposition, and the realization theorem for non-orientable
surfaces (Theorem A.9), we can turn a projection tuple into a point in
$\mathcal{T}(U)$, which [DM23] refer to as _resolving a point $w$ in
$\mathcal{T}(U)$_.

######  Definition 6.14 (Resolution point).

Let $[x,y]$ be a geodesic segment in $\mathcal{T}(\mathcal{S})$, and
$\Omega(x,y)$ an associated WISC witness family. For
$w\in\left\\{x,y\right\\}$, and $U\in\Omega(x,y)$, we define
$\widehat{w}_{U}^{\Omega}$ as follows.

  * •

If $U$ is non-annular, then $\widehat{w}_{U}^{\Omega}\in\mathcal{T}(U)$ is the
thick point whose projections to $\mathcal{C}(V)$ for $V\sqsubseteq U$ are
coarsely equal to the projection tuple $\widehat{w}_{U}$ (which exists due to
the Realization theorem (Theorem A.9)).

  * •

If $U$ is annular, then $\widetilde{w}_{U}$ is an element of $\mathbb{Z}$, and
we set $\widehat{w}_{U}^{\Omega}$ to be the point in $\mathbb{H}$ whose twist
coordinate is $\widetilde{w}_{U}$, and whose length coordinate is
$\displaystyle\frac{1}{\min\left({\varepsilon_{t}},\ell_{\partial
U}(w)\right)}$.

We can now define the complexity length associated to a witness family
$\Omega$.

######  Definition 6.15 (Complexity of witness family).

Let $[x,y]$ be a geodesic segment in $\mathcal{T}(\mathcal{S})$, and $\Omega$
an associated WISC witness family. The complexity $\mathfrak{L}_{\Omega}(x,y)$
of $\Omega$ is the following quantity.

$\displaystyle\mathfrak{L}_{\Omega}(x,y)\coloneqq\sum_{U\in\Omega}h_{\mathrm{NP}}^{\ast}(U)\cdot
d_{\mathcal{T}(U)}(\widehat{x}_{U}^{\Omega},\widehat{y}_{U}^{\Omega})$

Here, $h_{\mathrm{NP}}^{\ast}(U)$ is the net point growth entropy for
$\mathcal{T}(U)$ when $U$ is non-annular, and when $U$ is annular,
$h_{\mathrm{NP}}^{\ast}(U)$ is $1$ when both $\widehat{x}_{U}^{\Omega}$ and
$\widehat{y}_{U}^{\Omega}$ are ${\varepsilon_{t}}$-thick, and $2$ if not.

We now address why we used a modified version of the projection map in
Definition 6.12 instead of the usual projection map to curve complexes, by
revisiting Example 6.6.

######  Example 6.16.

Let $\gamma$ be a pseudo-Anosov mapping class on $\mathcal{S}$, with large
translation distance on $\mathcal{C}(\mathcal{S})$ and small projections
elsewhere. Let $\delta$ be a reducible mapping class, which is psuedo-Anosov
on a subsurface $V$, with large translation distance on $\mathcal{C}(V)$ and
small translation distance everywhere. Let $x$ be a point in
$\mathcal{T}(\mathcal{S})$ such that $\partial V$ is a component of the short
marking on $x$, and $y=\delta\gamma\delta^{-1}x$. We first verify that
$\Upsilon(x,y)=\left\\{\mathcal{S},V,\delta\gamma V\right\\}$. To see this, we
consider the following tuple of points $(x,\delta x,\delta\gamma
x,\delta\gamma\delta^{-1}x)$. We claim that each of the points in the tuple
lies coarsely on the geodesic $[x,y]$. To see this, we compute the projections
of adjacent pairs of points in tuple to various curve complexes.

  * -

$[x,\delta x]$ has large projections on $\mathcal{C}(V)$ and small projections
on other curve complexes.

  * -

$[\delta x,\delta\gamma x]$ has large projections on the
$\mathcal{C}(\delta\mathcal{S})$, which is the same as
$\mathcal{C}(\mathcal{S})$, and small projections elsewhere.

  * -

$[\delta\gamma x,\delta\gamma\delta^{-1}x]$ has large projections on
$\mathcal{C}(\delta\gamma V)$, and small projections elsewhere.

Consider the path $\kappa$ obtained by concatenating Teichmüller geodesics
between $x$ and $\delta x$, $\delta x$ and $\delta\gamma x$, and $\delta\gamma
x$ and $\delta\gamma\delta^{-1}x$. The curve complex calculations above show
that the projection of this path to any curve complex is a quasi-geodesic.
That indicates this path is a _hierarchy path_ , in the language of
hierarchically hyperbolic spaces. Furthermore, $\kappa$ does not have large
projections on disjoint subsurfaces: this indicates the hierarchy path
$\kappa$ fellow travels the geodesic $[x,y]$.

Let $\Omega(x,y)=\Upsilon(x,y)=\left\\{\mathcal{S},V,\delta\gamma V\right\\}$.
One can verify that this is a WISC witness family for $[x,y]$. Furthermore, we
have that $V\swarrow\mathcal{S}$ and $\mathcal{S}\searrow\gamma V$.

We now compute the resolution of the points $x$ and $y$ in $\mathcal{T}(V)$,
$\mathcal{T}(\gamma V)$, and $\mathcal{T}(\mathcal{S})$. Observe that the
geodesic $[x,y]$ almost immediately moves into a product region associated to
$V$ at the beginning, leaves that product region at some point $w$ along the
geodesic, and then enters the product region associated to $\gamma V$ at some
point $z$, and then stays in that product region almost all the way up to the
end. We will abuse notation slightly, and refer to $x$ and $w$ as points in
$\mathcal{T}(V)$, when we mean their projection via the product region map,
and $z$ and $y$ will refer to points in $\mathcal{T}(\gamma V)$ Resolving
points in $\mathcal{T}(V)$ and $\mathcal{T}(\gamma V)$ is easy, since there’s
no other witnesses nested in them, which means the projection tuple for those
subsurfaces is the usual projection map.

$\displaystyle\widehat{x}_{V}^{\Omega}$ $\displaystyle=x$
$\displaystyle\widehat{y}_{V}^{\Omega}$ $\displaystyle=w$
$\displaystyle\widehat{x}_{\gamma V}^{\Omega}$ $\displaystyle=z$
$\displaystyle\widehat{y}_{\gamma V}^{\Omega}$ $\displaystyle=y$

To resolve points in $\mathcal{T}(\mathcal{S})$, we have to use our modified
projection map, instead of the usual one. Doing so, the points $x$ and $y$
resolve in the following manner.

$\displaystyle\widehat{x}_{\mathcal{S}}^{\Omega}=w$
$\displaystyle\widehat{y}_{\mathcal{S}}^{\Omega}=z$

With these resolutions, we get the following estimate for complexity in terms
of Teichmüller distance.

$\displaystyle\mathfrak{L}_{\Omega}(x,y)$
$\displaystyle=h_{\mathrm{NP}}^{\ast}(V)\cdot
d_{\mathcal{T}(V)}(x,w)+h_{\mathrm{NP}}^{\ast}(\mathcal{S})\cdot
d_{\mathcal{T}(\mathcal{S})}(w,z)+h_{\mathrm{NP}}^{\ast}(\gamma V)\cdot
d_{\mathcal{T}(\gamma V)}(z,y)$
$\displaystyle<h_{\mathrm{NP}}^{\ast}(\mathcal{S})\cdot
d_{\mathcal{T}(\mathcal{S})}(x,y)$

Compare this to the complexity $\mathfrak{L}_{\Omega}^{\prime}$ we would have
gotten if we used the usual projection map instead of the modified projection
map.

$\displaystyle\mathfrak{L}_{\Omega}^{\prime}(x,y)$
$\displaystyle=h_{\mathrm{NP}}^{\ast}(V)\cdot
d_{\mathcal{T}(V)}(x,w)+h_{\mathrm{NP}}^{\ast}(\mathcal{S})\cdot
d_{\mathcal{T}(\mathcal{S})}(x,y)+h_{\mathrm{NP}}^{\ast}(\gamma V)\cdot
d_{\mathcal{T}(\gamma V)}(z,y)$
$\displaystyle>h_{\mathrm{NP}}^{\ast}(\mathcal{S})\cdot
d_{\mathcal{T}(\mathcal{S})}(x,y)$

If we use Rafi’s distance formula to estimate the distance between the
resolution points $\widehat{x}_{V}^{\Omega}$ and $\widehat{x}_{V}^{\Omega}$ we
do not get very good bounds for
$d_{\mathcal{T}(V)}(\widehat{x}_{V}^{\Omega},\widehat{x}_{V}^{\Omega})$: at
best, we accrue multiplicative and additive errors. For our applications
however, the most we can tolerate is additive error. To do this, we will need
to refine to notion of active interval for a subsurface to something more
useful for the estimate: the _contribution set_ $\mathcal{A}_{V}^{\Omega}$ of
a witness $V$. We first define two intermediate collections of subintervals of
a geodesic segment $[x,y]$.

$\displaystyle M(V)$
$\displaystyle\coloneqq\bigcup\left\\{\mathcal{I}_{W}^{{\varepsilon_{t}}}\mid\text{$W\in\Omega$
with $W\sqsubset V$ }\right\\}$ $\displaystyle C(V)$
$\displaystyle\coloneqq\bigcup\left\\{\mathcal{I}_{Z}^{{\varepsilon_{t}}}\mid\text{$Z$
contributes to $V$}\right\\}$

######  Definition 6.17 (Contribution set).

For a witness $V\in\Omega$, the contribution set $\mathcal{A}_{V}^{\Omega}$ is
a subset of the geodesic segment $[x,y]$ defined in the following manner.

$\displaystyle\mathcal{A}_{V}^{\Omega}\coloneqq\left(\mathcal{I}_{V}^{\varepsilon_{t}}\setminus
M(V)\right)\cup C(V)$

######  Remark.

The reason we remove $M(V)$ and then later add $C(V)$ again is because it is
possible for several different subsurfaces to be active at the same time:
orthogonal subsurfaces for instance. One can have $V$ and $W$ as witnesses,
with $W\sqsubset V$, and $Z\sqsubset V$ an active subsurface but not a
witness, such that $W\perp Z$ with $\mathcal{I}_{W}^{\varepsilon_{t}}$ and
$\mathcal{I}_{Z}^{\varepsilon_{t}}$ overlapping. In that case, removing $M(V)$
would also remove part of $\mathcal{I}_{Z}^{\varepsilon_{t}}$, and adding back
$C(V)$ would add back the deleted portion.

The following theorem estimates
$d_{\mathcal{T}(V)}(\widehat{x}_{V}^{\Omega},\widehat{y}_{V}^{\Omega})$ using
$\mathcal{A}_{V}^{\Omega}$.

######  Theorem 6.18 (Theorem 9.4 of [DM23]).

There exists a uniform constant $C$ such that the following bound holds for
any $x$, $y$, and witness $V$.

$\displaystyle
d_{\mathcal{T}(V)}(\widehat{x}_{V}^{\Omega},\widehat{y}_{V}^{\Omega})\leq\int_{x}^{y}\mathbbm{1}_{\mathcal{A}_{V}^{\Omega}}+C$

Contribution sets help us make precise the notion of “overcounting” when
multiple product regions are active at the same time. More precisely, when a
segment of $[x,y]$ is a part of two or more contribution sets, that segment
shows up multiple times when computing $\mathfrak{L}_{\Omega}(x,y)$, thanks to
Theorem 6.18. If the overlapping segment is sufficiently long, one could even
end up having
$\mathfrak{L}_{\Omega}(x,y)>h_{\mathrm{NP}}^{\ast}(\mathcal{S})\cdot
d_{\mathcal{T}(\mathcal{S})}(x,y)$, which as we will see, leads to a worse
count for net points than the usual methods. This phenomenon of contribution
sets overlapping is called _badness_ , and while we will not be able to
eliminate it entirely, we will be able to minimize it.

######  Definition 6.19 (Bad set).

We say a point $p$ in $\mathcal{A}_{V}^{\Omega}$ is bad if there exists some
other witness $W$ such that $p$ also belongs in $\mathcal{A}_{W}^{\Omega}$.
The bad set $\mathcal{B}_{V}^{\Omega}$ denotes the set of all bad points in
$\mathcal{A}_{V}^{\Omega}$, and $\left|\mathcal{B}_{V}^{\Omega}\right|$
denotes the total length of this set, when we think of
$\mathcal{B}_{V}^{\Omega}$ as a subset of the geodesic segment $[x,y]$.

For our applications, we won’t need to eliminate badness entirely, or even
bound the length of the bad set uniformly: it will suffice to show that the
length of the bad set is a very small multiple of
$d_{\mathcal{T}(\mathcal{S})}(x,y)$.

######  Definition 6.20 (Admissible and limited).

A witness family $\Omega$ associated to a geodesic segment $[x,y]$ is said to
be:

  * •

_admissible_ if
$\displaystyle\left|\mathcal{B}_{V}^{\Omega}\right|\leq\frac{d_{\mathcal{T}(\mathcal{S})}(x,y)}{K_{V}\mathbf{C}}$,
for all $V\in\Omega$, and some constants $K_{V}$ that only depend on the
topological type of $V$.

  * •

_limited_ if $\left|\Omega\right|$ is uniformly bounded, independent of $x$
and $y$.

######  Remark.

Our definition of limited is a weaker version of Definition 10.7 from [DM23],
but since we don’t need the stronger version, we present this version instead.

[DM23] prove that WISC witness families that are admissible and limited exist.
They call these witness families WISCAL witness families.

######  Proposition 6.21 (Section 10.3 of [DM23]).

For all $[x,y]$, there exists an associated WISC witness family that is also
admissible and limited.

For WISCAL witness families, the following result relating complexity and
Teichmüller distance follows easily from Theorem 6.18 and the definition of
admissible.

######  Proposition 6.22.

If $\Omega$ is a WISCAL witness family associated to $[x,y]$, then the
following inequality holds.

$\displaystyle\mathfrak{L}_{\Omega}(x,y)\leq\left(h_{\mathrm{NP}}(\mathcal{S})+\frac{K}{\mathbf{C}}\right)d_{\mathcal{T}(\mathcal{S})}(x,y)+K\mathbf{C}$

Here, $K$ is some uniform constant depending only on $\mathcal{S}$.

We now define complexity length, which follows from the definition of the
complexity of a witness family.

######  Definition 6.23 (Complexity length).

For a pair of points $x$ and $y$ in $\mathcal{T}(\mathcal{S})$, the complexity
length $\mathfrak{L}(x,y)$ is defined to be the following.

$\displaystyle\mathfrak{L}(x,y)\coloneqq\inf_{\Omega}\mathfrak{L}_{\Omega}(x,y)$

Here, we take the infimum over all WISCAL witness families for $[x,y]$.

With the machinery of complexity length set up, it is now possible to count
net points with respect to complexity length.

######  Theorem 6.24 (Theorem 12.1 of [DM23]).

For any large enough $\mathbf{C}>0$, and any $\varepsilon_{\mathrm{err}}>0$,
there exists an polynomial function $p(r)$, and $r>0$ large enough such that
the following bound holds for net points in $\mathcal{T}(\mathcal{S})$.

$\displaystyle\\#\left(y\in\mathfrak{N}\mid\mathfrak{L}(x,y)\leq r\right)\leq
p(r)\cdot\exp\left((1+\varepsilon_{\mathrm{err}})r\right)$

######  Remark.

The above theorem is a weaker version of the theorem that appears in [DM23]:
their version does not have the $\varepsilon_{\mathrm{err}}$. The reason we
have the weaker version is that in the proof of their theorem, they count the
number of net points in $\mathcal{T}(V)$ in a ball of radius $R$, where $V$ is
a witness, using Theorem 6.1, which gives them that the number of net points
is equal, up to multiplicative error, to $\exp(h_{\mathrm{NP}}(U)R)$. Since
Theorem 6.1 only holds orientable surfaces, and we want to state our results
for non-orientable surfaces as well, we will need to use a weaker counting
result to count net points in $\mathcal{T}(V)$, namely the following bound,
which holds for any $\varepsilon_{\mathrm{ent}}>0$ and large enough $R$.

$\displaystyle\\#\left(y\in\mathfrak{N}\mid d_{\mathcal{T}(V)}(x,y)\leq
R\right)\leq\exp\left((h_{\mathrm{NP}}(V)+\varepsilon_{\mathrm{ent}})R\right)$

We sketch out a proof of Theorem 6.24 below: the proof proceeds identically to
the proof in [DM23], except at one point, where we plug in our weaker bound
for net points in $\mathcal{T}(V)$ for witnesses $V$.

###### Sketch of proof for Theorem 6.24.

For each $y\in\mathfrak{N}$ such that $\mathfrak{L}(x,y)\leq r$, we have a
WISCAL witness family $\Omega$ such that $\mathfrak{L}_{\Omega}(x,y)\leq r$.
We can turn that witness family into a graph in the following manner.

  * •

Add a vertex for every witness $V\in\Omega$.

  * •

Label the vertex associated with $V$ with the tuple
$(h_{\mathrm{NP}}^{\ast}(V),\lfloor
d_{\mathcal{T}(V)}(\widehat{x}_{V}^{\Omega},\widehat{y}_{V}^{\Omega})\rfloor)$.

  * •

If we have a pair of witnesses $V\swarrow W$, we join the vertices associated
to them with a directed edge labeled “SW”: $V\xrightarrow{SW}W$.

  * •

If we have a pair of witnesses $W\searrow V$, we join the vertices associated
to them with a directed edge labeled “SE”: $W\xrightarrow{SE}V$.

  * •

If we have a pair of witnesses $W\pitchfork V$, with $W\lessdot V$, we join
the vertices associated to them with a directed edge labelled “P”:
$W\xrightarrow{P}V$.

We first count how many distinct possibilities are there for such labeled
graphs that correspond to $y$ for which $\mathfrak{L}(x,y)\leq r$. Since the
cardinality of a WISCAL family is uniformly bounded, there are at most $k$
many vertices, for some constant $k$. As for the labels on the vertices, there
are at most $\frac{r}{h_{\mathrm{NP}}^{\ast}(V)}$ possibilities for a label on
vertex which corresponds to a subsurface which is homeomorphic to $V$. From
this, we conclude that there are at most $p(r)$ possibilities for the
combinatorial type of the graph, where $p(r)$ is a polynomial in $r$.

It will suffice to compute how many distinct net points give rise to witness
families whose graph is of a given type. To do so, we consider _initial
subsets_ of the graph, i.e. a subset $\mathcal{W}$ of the vertices
$\mathcal{V}$ of the graph such that there is no directed edge from
$\mathcal{V}\setminus\mathcal{W}$ to $\mathcal{W}$.

Given an initial subset $\mathcal{W}$ of the graph, we construct points $y$
such that the witness family associated to $[x,y]$ has the combinatorial type
$\mathcal{W}$. We then consider an enlargement of $\mathcal{W}$ by one-
additional vertex $v$, such that the enlargement is still an initial subset.

######  Claim.

The entire graph $\mathcal{V}$ can be built up from such one-step
enlargements.

We then count the number of net points whose associated witness families have
the combinatorial type $\mathcal{W}\cup\left\\{v\right\\}$, after we fix one
witness family associated to $\mathcal{W}$. More concretely, let $w$ be a
point such that the combinatorial type of the witness family associated to
$[x,w]$ is $\mathcal{W}$. Suppose now that we add a vertex $(h,r_{0})$ to the
graph $\mathcal{W}$. To extend $\Omega(x,w)$ so that its combinatorial type is
$\mathcal{W}\cup\left\\{(h,r_{0})\right\\}$, we need to add a witness $U$
whose net point entropy is $h$, and a point $y\in\mathcal{T}(U)$ such that the
following holds.

$\displaystyle d_{\mathcal{T}(U)}(\widehat{x}_{U}^{\Omega},y)\leq r_{0}$

There are only finitely many choices for such subsurfaces $U$ (because their
boundary curves must get short near $w$), and once we’ve made a choice of $U$,
we have a choice
$\exp\left((h_{\mathrm{NP}}(V)+\varepsilon_{\mathrm{ent}})r_{0}\right)$ points
for $y$.

Multiplying out the counts for each vertex added, we get the following
estimate for the cardinality associated to each combinatorial type.

$\displaystyle\\#\left(y\mid\text{$\Omega(x,y)$ has combinatorial type
$\mathcal{V}$}\right)$
$\displaystyle=\sum_{(h,s)\in\mathcal{V}}\exp\left((h+\varepsilon_{\mathrm{ent}})s\right)$
$\displaystyle\leq\exp\left((1+\varepsilon_{\mathrm{err}})r\right)$

We get the second inequality by picking $\varepsilon_{\mathrm{ent}}$ small
enough, and observing that $\sum hs\leq r$. ∎

###  6.3. Linear Gap for Bad Points

In this subsection, we will prove our main result involving complexity length:
on the complexity length of bad points.

######  Theorem 6.25.

Suppose that for all proper subsurfaces $V$ of $\mathcal{S}$, the following
inequality holds.

$\displaystyle h_{\mathrm{NP}}(V)<h_{\mathrm{NP}}(\mathcal{S})$

Then for any $\varepsilon_{b}>0$, there exists $c>0$, and $R$ large enough,
such that for any bad point $y$, i.e. a point in
$\mathfrak{N}_{b}(p,R,\varepsilon_{b})$, the following upper bound holds for
the complexity length between $p$ and $y$.

$\displaystyle\mathfrak{L}(p,y)\leq h_{\mathrm{NP}}(\mathcal{S})(1-c)R$

###### Proof.

Let $\Omega$ be a WISCAL witness family for $[p,y]$: the proof of Theorem 6.25
splits into two cases depending on whether the surface $\mathcal{S}$ is a
witness in $\Omega$ or not.

The case where $\mathcal{S}$ is a witness is harder, so we deal with that
first.

We consider the triple of points $(p,\widehat{y}_{\mathcal{S}}^{\Omega},y)$,
and first estimate $\mathfrak{L}(p,\widehat{y}_{\mathcal{S}}^{\Omega})$.

By applying Proposition 6.22, we get a bound for
$\mathfrak{L}(p,\widehat{y}_{\mathcal{S}}^{\Omega})$.

$\displaystyle\mathfrak{L}(p,\widehat{y}_{\mathcal{S}}^{\Omega})\leq\left(h_{\mathrm{NP}}(\mathcal{S})+\frac{K}{\mathbf{C}}\right)(d_{\mathcal{T}(\mathcal{S})}(p,\widehat{y}_{\mathcal{S}}^{\Omega}))+K\mathbf{C}$
(37)

We next estimate $\mathfrak{L}(\widehat{y}_{\mathcal{S}}^{\Omega},y)$: we
claim that there exists a WISCAL witness family $\Omega^{\prime}$ for
$[\widehat{y}_{\mathcal{S}}^{\Omega},y]$ that does not have $\mathcal{S}$ as a
witness. The first thing we need in order to get such a witness family is
verify that on the Teichmüller geodesic
$[\widehat{y}_{\mathcal{S}}^{\Omega},y]$, $\mathcal{S}$ is not an active
subsurface. This follows from the fact that the $\mathcal{S}$-coordinate in
the projection tuple of $y$ is equal to the $\mathcal{S}$-coordinate of the
projection of $\widehat{y}_{\mathcal{S}}^{\Omega}$ to
$\mathcal{C}(\mathcal{S})$, by construction of
$\widehat{y}_{\mathcal{S}}^{\Omega}$. We now use Proposition 6.21 to get a
witness family which does not contain $\mathcal{S}$, since $\mathcal{S}$ is
not an active subsurface.

Since the witness family $\Omega^{\prime}$ does not have $\mathcal{S}$ as a
witness, we can do better than Proposition 6.22 when estimating
$\mathfrak{L}(\widehat{y}_{\mathcal{S}}^{\Omega},y)$. We have from our
hypothesis that $h_{\mathrm{NP}}(\mathcal{S})>h_{\mathrm{NP}}(V)$, so there
exists a constant $h$ such that $h<h_{\mathrm{NP}}(\mathcal{S})$ but
$h>h_{\mathrm{NP}}(V)$. Using Theorem 6.18, we get the following estimate for
$\mathfrak{L}(\widehat{y}_{\mathcal{S}}^{\Omega},y)$.

$\displaystyle\mathfrak{L}(\widehat{y}_{\mathcal{S}}^{\Omega},y)\leq\left(h+\frac{K}{\mathbf{C}}\right)d_{\mathcal{T}(\mathcal{S})}(\widehat{y}_{\mathcal{S}}^{\Omega},y)+K\mathbf{C}$
(38)

From the triangle inequality for complexity length, we also have the following

$\displaystyle\mathfrak{L}(p,y)\leq\mathfrak{L}(p,\widehat{y}_{\mathcal{S}}^{\Omega})+\mathfrak{L}(\widehat{y}_{\mathcal{S}}^{\Omega},y)$
(39)

We plug in (37) and (38) into (39).

$\displaystyle\mathfrak{L}(p,y)$
$\displaystyle\leq\left(h_{\mathrm{NP}}(\mathcal{S})+\frac{K}{\mathbf{C}}\right)(d_{\mathcal{T}(\mathcal{S})}(p,\widehat{y}_{\mathcal{S}}^{\Omega}))+\left(h+\frac{K}{\mathbf{C}}\right)d_{\mathcal{T}(\mathcal{S})}(\widehat{y}_{\mathcal{S}}^{\Omega},y)+2K\mathbf{C}$
$\displaystyle=\left(h_{\mathrm{NP}}(\mathcal{S})+\frac{K}{\mathbf{C}}\right)(d_{\mathcal{T}(\mathcal{S})}(p,\widehat{y}_{\mathcal{S}}^{\Omega})+d_{\mathcal{T}(\mathcal{S})}(\widehat{y}_{\mathcal{S}}^{\Omega},y))$
$\displaystyle-\left(h_{\mathrm{NP}}(\mathcal{S})-h\right)d_{\mathcal{T}(\mathcal{S})}(\widehat{y}_{\mathcal{S}}^{\Omega},y)$
$\displaystyle+2K\mathbf{C}$

We now claim that $\widehat{y}_{\mathcal{S}}^{\Omega}$ is within a bounded
distance of a point $q$ on $[p,y]$. Consider the triple
$(x,\widehat{y}_{\mathcal{S}}^{\Omega},y)$: this is a strongly aligned tuple,
i.e. its projections onto all curve complexes satisfy a coarse reverse
triangle inequality (see Definition 3.21 of [DM23]). Lemma 9.10 of [DM23]
asserts the existence of $q$ on $[x,y]$ that satisfies the following
properties.

  * -

For any subsurface $V$ such that $\overline{V}^{\Omega}=\mathcal{S}$,
$d_{V}(\widehat{y}_{\mathcal{S}}^{\Omega},q)\leq M$ for some fixed constant
$M$.

  * -

For any subsurfaces $V$ such that $\overline{V}^{\Omega}\swarrow\mathcal{S}$
or $\mathcal{S}\searrow\overline{V}^{\Omega}$, $q$ lies outside the active
interval for $V$.

In the case where $\overline{V}^{\Omega}\swarrow\mathcal{S}$, we have that
$d_{V}(q,y)\leq M$, since $q$ lies outside the active interval for $V$. But
note that by construction of the projection tuple,
$\pi_{V}(\widehat{y}_{\mathcal{S}}^{\Omega})=y$, thus
$d_{V}(\widehat{y}_{\mathcal{S}}^{\Omega},q)\leq M$. Similarly, we have
$d_{V}(\widehat{y}_{\mathcal{S}}^{\Omega},q)\leq M$ for
$\mathcal{S}\searrow\overline{V}^{\Omega}$ as well. Thus, by Rafi’s distance
formula, we have that $\widehat{y}_{\mathcal{S}}^{\Omega}$ is within a bounded
distance of $q$ on $[p,y]$: we have
$d_{\mathcal{T}(\mathcal{S})}(p,\widehat{y}_{\mathcal{S}}^{\Omega})+d_{\mathcal{T}(\mathcal{S})}(\widehat{y}_{\mathcal{S}}^{\Omega},y)\leq
R+J$, for some constant $J$. Furthermore,
$d_{\mathcal{T}(\mathcal{S})}(\widehat{y}_{\mathcal{S}}^{\Omega},y)\geq\varepsilon_{b}R$,
by the hypothesis of $y$ being a bad point, since
$\widehat{y}_{\mathcal{S}}^{\Omega}$ is in the thick part of Teichmüller
space. This simplifies the expression for $\mathfrak{L}(x,y)$.

$\displaystyle\mathfrak{L}(x,y)$
$\displaystyle\leq\left(h_{\mathrm{NP}}(\mathcal{S})+\frac{K}{\mathbf{C}}\right)R-(h_{\mathrm{NP}}(\mathcal{S})-h)(\varepsilon_{b})R+2K\mathbf{C}$
$\displaystyle=R\left(h_{\mathrm{NP}}(\mathcal{S})-d+\frac{K}{\mathbf{C}}\right)+2K\mathbf{C}$

Here, $d=\varepsilon_{b}\left(h_{\mathrm{NP}}(\mathcal{S})-h\right)$, which is
a positive constant, since $h_{\mathrm{NP}}(\mathcal{S})>h$. By picking
$\mathbf{C}$ and $R$ large enough, we get the statement of the theorem, which
proves the result in the first case of $\mathcal{S}$ being in the witness
family.

When $\mathcal{S}$ is not in the witness family, we set
$\widehat{y}_{\mathcal{S}}^{\Omega}=p$, and the rest of the proof follows
identically. ∎

##  Appendix A Geometry of $\mathcal{T}(\mathcal{N}_{g})$

In this section, we prove some standard results about the geometry of
Teichmüller spaces of non-orientable surfaces that we use in Section 6. We do
so by lifting the hyperbolic structures and markings on the non-orientable
surfaces to their double covers, which give us points in the Teichmüller space
and curve complex of the double cover.

The fact that these lifts are well-defined and respect the metric properties
are encapsulated in the following two theorems.

######  Theorem A.1 (Isometric embedding of Teichmüller spaces (Theorem 2.1 of
[Kha23])).

The map $i:\mathcal{T}(\mathcal{N}_{g})\to\mathcal{T}(\mathcal{S}_{g-1})$
given by lifting the hyperbolic structure and marking from $\mathcal{N}_{g}$
to $\mathcal{S}_{g-1}$ is an isometric embedding. Furthermore, the image of
$\mathcal{T}(\mathcal{N}_{g})$ in $\mathcal{T}(\mathcal{S}_{g-1})$ is the
subset of $\mathcal{T}(\mathcal{S}_{g-1})$ is fixed by $\iota^{\ast}$, where
$\iota^{\ast}$ is the map induced by the orientation reversing deck
transformation $\iota$ on $\mathcal{S}_{g-1}$.

######  Theorem A.2 (Quasi isometric embedding of curve complexes (Lemma 6.3
from [MS13])).

The map $\mathcal{C}(\mathcal{N}_{g})\to\mathcal{C}(\mathcal{S}_{g-1})$
obtained by lifting curves in $\mathcal{N}_{g}$ to $\mathcal{S}_{g-1}$ is a
quasi-isometric embedding.

We will use the above two theorems, along with Lemma A.5, to reduce statements
about the geometry of $\mathcal{T}(\mathcal{N}_{g})$ to statements about the
geometry of $\mathcal{T}(\mathcal{S}_{g-1})$. However, we postpone the
statement and the proof of Lemma A.5 until Section A.3, since it’s not
required for Section A.1.

We set up some notation for this section.

  * -

$d(x,y)$ and $d(\widetilde{x},\widetilde{y})$: Given points $x$ and $y$ in
$\mathcal{T}(\mathcal{N}_{g})$, $d(x,y)$ is the distance in Teichmüller metric
between them, and $d(\widetilde{x},\widetilde{y})$ is the distance in
$\mathcal{T}(\mathcal{S}_{g-1})$ between their images, $\widetilde{x}$ and
$\widetilde{y}$.

  * -

$\pi_{V}(\mu_{x})$ and $\pi_{V}(x)$: If $\mu_{x}$ is a marking/curve on a
surface, the $\pi_{V}(\mu_{x})$ denotes the subsurface projection to the
subsurface $V$. If $x$ is a point in the Teichmüller space, the
$\pi_{V}(x)=\pi_{V}(\mu_{x})$, where $\mu_{x}$ is the Bers marking on $x$.

  * -

$d_{V}(\mu_{x},\mu_{y})$ and $d_{V}(x,y)$: If $\mu_{x}$ and $\mu_{y}$ are
markings/curves on a surface, and $V$ is a subsurface, then
$d_{V}(\mu_{x},\mu_{y})$ refers to the curve complex distance between the
subsurface projections of $\mu_{x}$ and $\mu_{y}$ in $\mathcal{C}(V)$. When
$x$ and $y$ are points in Teichmüller space, $d_{V}(x,y)$ refers to
$d_{V}(\mu_{x},\mu_{y})$, where $\mu_{x}$ and $\mu_{y}$ are the Bers marking
on $x$ and $y$.

###  A.1. Minsky’s Product Region Theorem

In this section, we prove a version of Minsky’s product region theorem
[Min96a, Theorem 6.1] for non-orientable surfaces.

We recall the following objects that were defined in Section 3.3.

  1. (i)

The multicurve $\gamma$ on $\mathcal{N}_{g}$.

  2. (ii)

The metric space $X_{\gamma}$, and the projection map $\Pi$.

  3. (iii)

The thin region $\mathcal{T}_{\gamma\leq\varepsilon_{t}}(\mathcal{N}_{g})$.

######  Theorem A.3 (Product region theorem for non-orientable surfaces).

For any $c>0$, there exists a small enough $\varepsilon_{t}>0$, such that the
restriction of $\Pi$ to
$\mathcal{T}_{\gamma\leq\varepsilon_{t}}(\mathcal{N}_{g})$ is an isometry with
additive error at most $c$, i.e. the following holds for any $x$ and $y$ in
$\mathcal{T}_{\gamma\leq\varepsilon_{t}}(\mathcal{N}_{g})$.

$\displaystyle\left|d(x,y)-d_{X_{\gamma}}(\Pi(x),\Pi(y))\right|\leq c$

###### Proof.

We will prove this result by reducing the distance calculation in
$\mathcal{T}(\mathcal{N}_{g})$ to a distance calculation in
$\mathcal{T}(\mathcal{S}_{g-1})$, where $\mathcal{S}_{g-1}$ is the orientation
double cover, and invoking the classical product region theorem in that
setting.

We begin the proof by constructing some points in
$\mathcal{T}(\mathcal{S}_{g-1})$ and a multicurve on $\mathcal{S}_{g-1}$.
Recall that $\mathcal{T}(\mathcal{N}_{g})$ isometrically embeds inside
$\mathcal{T}(\mathcal{S}_{g-1})$: let $\widetilde{x}$ and $\widetilde{y}$
denote the points in $\mathcal{T}(\mathcal{S}_{g-1})$ that are the images of
$x$ and $y$ under the embedding. Let $\widetilde{\gamma}$ denote the lift of
the multicurve $\gamma$: if $\gamma_{i}$ is a two-sided curve, it will have
two disjoint lifts in the cover, and if $\gamma_{i}$ is a one-sided curve, it
will have single lift in the double cover. We have that the region
$\mathcal{T}_{\widetilde{\gamma}\leq\varepsilon_{t}}(\mathcal{S}_{g-1})\subset\mathcal{T}(\mathcal{S}_{g-1})$
intersects the image of $\mathcal{T}(\mathcal{N}_{g})$ at the image of
$\mathcal{T}_{\gamma\leq\varepsilon_{t}}(\mathcal{N}_{g})\subset\mathcal{T}(\mathcal{N}_{g})$.
Let $\iota$ denote the orientation reversing deck transformation on
$\mathcal{S}_{g-1}$ which corresponds to the covering map.

######  Claim.

Let $\Pi_{k}$ denote the projection map from $\mathcal{T}(\mathcal{N}_{g})$ to
the $k$th component of $X_{\gamma}$, and $\widetilde{\Pi_{k}}$ denote the
projection map from $\mathcal{T}(\mathcal{S}_{g-1})$ to the lift of the $k$th
component of $\gamma$ to $\mathcal{S}_{g-1}$. This map is an isometric
embedding.

$\displaystyle
d(\Pi_{k}(x),\Pi_{k}(y))=d(\widetilde{\Pi_{k}}(\widetilde{x}),\widetilde{\Pi_{k}}(\widetilde{y}))$

###### Proof of claim.

We need to verify the claim on the three kinds of factors of $X_{\gamma}$.

  1. (i)

$\mathcal{N}_{g}\setminus\gamma$: The lift of $\mathcal{N}_{g}\setminus\gamma$
to $\mathcal{S}_{g-1}$ will have two components if
$\mathcal{N}_{g}\setminus\gamma$ is orientable, which we call $S_{1}$ and
$S_{2}$. Both $S_{1}$ and $S_{2}$ are homeomorphic to
$\mathcal{N}_{g}\setminus\gamma$. If $\mathcal{N}_{g}\setminus\gamma$ is non-
orientable, then its lift in $\mathcal{S}_{g-1}$ is the orientation double
cover.

In the first case, $\mathcal{T}(\mathcal{N}_{g}\setminus\gamma)$ maps to the
diagonal subspace in $\mathcal{T}(S_{1})\times\mathcal{T}(S_{2})$, and the
metric on $\mathcal{T}(S_{1})\times\mathcal{T}(S_{2})$ is the $\sup$ metric.
The space $\mathcal{T}(\mathcal{N}_{g})$ maps to the diagonal subspace because
its image must be invariant under the map $\iota$, which isometrically swaps
$S_{1}$ and $S_{2}$. This map is an isometric embedding, and thus for any
points $x$ and $y$ in $\mathcal{T}(\mathcal{N}_{g}\setminus\gamma)$, the
distance between their images in $\mathcal{T}(S_{1})\times\mathcal{T}(S_{2})$
is the same as the distance in $\mathcal{T}(\mathcal{N}_{g}\setminus\gamma)$.

In the second case, we have that $\mathcal{T}(\mathcal{N}_{g}\setminus\gamma)$
also isometrically embeds inside the Teichmüller space of its double cover, by
Theorem A.1, so the claim follows.

  2. (ii)

$\gamma_{i}$ (for $\gamma_{i}$ two-sided): The lift of $\gamma_{i}$ in this
case are two disjoint curves on $\mathcal{S}_{g-1}$, which are swapped by the
deck transformation $\iota$. This means the $\mathbb{H}$-coordinate given by
length and twist of $\gamma_{i}$ maps to the diagonal in
$\mathbb{H}\times\mathbb{H}$, which correspond the length and twist around the
two lifts. Since $\mathbb{H}$ mapped to the diagonal in
$\mathbb{H}\times\mathbb{H}$ is an isometric embedding with $\sup$ metric, the
claim follows in this case.

  3. (iii)

$\gamma_{i}$ (for $\gamma_{i}$ one-sided): The lift of $\gamma_{i}$ in this
case is a single curve $\widetilde{\gamma_{i}}$ on $\mathcal{S}_{g-1}$ which
is left invariant by the deck transformation $\iota$. We will show that the
twist coordinate around $\widetilde{\gamma_{i}}$ cannot be changed without
leaving the image of $\mathcal{T}(\mathcal{N}_{g})$ in
$\mathcal{T}(\mathcal{S}_{g-1})$, i.e. any $\widetilde{x}$ and $\widetilde{y}$
have the same twist coordinate around $\widetilde{\gamma_{i}}$. Once we have
established that, the claim will follow, since only the length coordinate of
$\gamma_{i}$ can be changed, which corresponds to $\mathbb{R}_{>0}$.

Suppose now that $x$ is a point in $\mathcal{T}(\mathcal{N}_{g})$ and
$\widetilde{x}$ the corresponding point in $\mathcal{T}(\mathcal{S}_{g-1})$.
Consider a pants decomposition on $\mathcal{N}_{g}$ that contains $\gamma_{i}$
as one of the curves. There is a unique one-sided curve $\kappa$ that
intersects $\gamma_{i}$ and does not intersect any of the other pants curves.
Let $\widetilde{\kappa}$ be the lift of $\kappa$ to $\mathcal{S}_{g-1}$: we
will use this curve to measure twisting around $\widetilde{\gamma_{i}}$. Let
$x^{\prime}$ be another point in $\mathcal{T}(\mathcal{S}_{g-1})$ obtained by
taking $\widetilde{x}$, and twisting by some amount around
$\widetilde{\gamma_{i}}$, without changing the length of
$\widetilde{\gamma_{i}}$. On $x^{\prime}$, the length of $\widetilde{\kappa}$
will be different from the length on $\widetilde{x}$. However, this means that
$x^{\prime}$ is not contained in the image of $\mathcal{T}(\mathcal{N}_{g})$,
since if it were, the length of $\widetilde{\kappa}$ would have to be the
same, since that’s the lift of the curve $\kappa$, whose length only depends
on the length of $\gamma_{i}$.

∎

The following equality follows from the claim.

$\displaystyle
d_{X_{\widetilde{\gamma}}}(\Pi(\widetilde{x}),\Pi(\widetilde{y}))=d_{X_{{\gamma}}}(\Pi({x}),\Pi({y}))$

We also have that $\mathcal{T}(\mathcal{N}_{g})$ isometrically embeds into
$\mathcal{T}(\mathcal{S}_{g-1})$.

$\displaystyle d(\widetilde{x},\widetilde{y})=d(x,y)$

And finally, have that the region
$\mathcal{T}_{\widetilde{\gamma}\leq\varepsilon_{t}}(\mathcal{S}_{g-1})\subset\mathcal{T}(\mathcal{S}_{g-1})$
intersects the image of $\mathcal{T}(\mathcal{N}_{g})$ at the image of
$\mathcal{T}_{\gamma\leq\varepsilon_{t}}(\mathcal{N}_{g})\subset\mathcal{T}(\mathcal{N}_{g})$.
Combining these three facts, and applying Minsky’s product region theorem for
orientable surfaces, the result follows. ∎

###  A.2. Uniform Bounds for the Volume of a Ball

In this section, , we show that the for balls of fixed radius in
$\mathrm{core}(\mathcal{T}(\mathcal{N}_{g}))$, the $\nu_{N}$ volume of the
ball is bounded above and below by constants that are independent of the
center of the ball.

Let $\mathcal{P}$ be a pants decomposition for $\mathcal{N}_{g}$: recall the
formula for $\nu_{N}$.

$\displaystyle\nu_{N}=\left(\bigwedge_{\text{$\gamma_{i}$ one-
sided}}\coth(\ell(\gamma_{i}))d\ell(\gamma_{i})\right)\wedge\left(\bigwedge_{\text{$\gamma_{i}$
two-sided}}d\tau(\gamma_{i})\wedge d\ell(\gamma_{i})\right)$

Here $\ell(\gamma_{i})$ denotes the length of the curve $\gamma_{i}$, and
$\tau(\gamma_{i})$ denotes the twist, when $\gamma_{i}$ is two-sided.

######  Proposition A.4.

For any $\kappa>0$, and $\varepsilon_{t}>0$ small enough, there exist positive
constants222We will also consider $c_{1}$ and $c_{2}$ as functions of $\kappa$
elsewhere in the paper. $c_{1}$ and $c_{2}$ (depending only on $\kappa$ and
$\varepsilon_{t}$) such the $\nu_{N}$ volume of a ball
$B_{\kappa}^{\varepsilon_{t}}(x)$ of radius $\kappa$ centered at
$x\in\mathcal{T}_{\varepsilon_{t}}^{-}(\mathcal{N}_{g})$ are bounded below and
above by $c_{1}$ and $c_{2}$.

$\displaystyle c_{1}\leq\nu_{N}(B_{\kappa}^{\varepsilon_{t}}(x))\leq c_{2}$

###### Proof.

Note that since the points we are considering lie in
$\mathcal{T}_{\varepsilon_{t}}^{-}(\mathcal{N}_{g})$, we have the following
upper bound and lower bound for $\coth(\ell(\gamma_{i}))$, where $\gamma_{i}$
is a one sided curve.

$\displaystyle 1\leq\coth(\ell(\gamma_{i}))\leq\coth(\varepsilon_{t})$ (40)

In particular, the $\nu_{N}$ volume of a ball can be bounded above and below
by $\coth(\varepsilon_{t})\nu_{N}^{\prime}$ and $\nu_{N}^{\prime}$, where
$\nu_{N}^{\prime}=\left(\bigwedge_{\text{$\gamma_{i}$ one-
sided}}d\ell(\gamma_{i})\right)\wedge\left(\bigwedge_{\text{$\gamma_{i}$ two-
sided}}d\kappa(\gamma_{i})\wedge d\ell(\gamma_{i})\right)$.

We now split up $\mathcal{T}_{\varepsilon_{t}}^{-}(\mathcal{N}_{g})$ into two
regions: $\mathcal{T}_{\varepsilon_{t}}^{\pm}(\mathcal{N}_{g})$, and the
complementary region. Since $\mathrm{MCG}(\mathcal{N}_{g})$ acts cocompactly
on $\mathcal{T}_{\varepsilon_{t}}^{\pm}(\mathcal{N}_{g})$, and
$\nu_{N}(B_{\kappa}^{\varepsilon_{t}}(x))$ is continuous in $x$, the desired
bounds hold in this region. It will therefore suffice to prove the bounds in
the complementary region.

Note that for any $x$ in the complementary region, there is some two-sided
curve $\gamma$ that is short. By Theorem A.3, the ball
$B_{\kappa}^{\varepsilon_{t}}(x)$ is contained in a product of balls, one in
$\mathbb{H}$, and one in
$\mathcal{T}_{\varepsilon_{t}}^{-}(\mathcal{N}_{g}\setminus\gamma)$. We pick
$\gamma$ to be part of a pants decomposition $\mathcal{P}$, and write
$\nu_{N}$ as follows.

$\displaystyle\nu_{N}=\left(d\kappa(\gamma)\wedge
d\ell(\gamma)\right)\wedge\nu_{N}^{\mathcal{N}_{g}\setminus\gamma}$ (41)

Here, $\nu_{N}^{\mathcal{N}_{g}\setminus\gamma}$ denotes the volume form on
$\mathcal{T}(\mathcal{N}_{g}\setminus\gamma)$. As a result, we have that the
$\nu_{N}$ measure of a product of the two balls is the product of the
corresponding measures of those balls.

The measure of any ball of a fixed radius in $\mathbb{H}$ is constant, since
$\mathbb{H}$ is homogeneous. The $\nu_{N}^{\mathcal{N}_{g}\setminus\gamma}$
measure of a ball in
$\mathcal{T}_{\varepsilon_{t}}^{-}(\mathcal{N}_{g}\setminus\gamma)$ is again
bounded above and below by fixed constants, by inducting on a surface of lower
complexity.

Since we have uniform bounds for both the terms in the product, we get uniform
bounds for the measure of a ball in
$\mathcal{T}_{\varepsilon_{t}}^{-}(\mathcal{N}_{g})$. ∎

###  A.3. Teichmüller Geodesics and Geodesics in the Curve Complex

In this section, we will deduce some standard results about Teichmüller
geodesics and the corresponding curve complex geodesics for non-orientable
surfaces by reducing to the orientable case. The following lemma will be the
main tool for the reduction to the orientable case.

######  Lemma A.5.

Let $[x,y]$ be a Teichmüller geodesic segment in $\mathcal{T}(\mathcal{N})$,
where $\mathcal{N}$ is a non-orientable surface, and
$[\widetilde{x},\widetilde{y}]$ be its image in $\mathcal{T}(\mathcal{S})$,
where $\mathcal{S}$ is the orientable double cover of $\mathcal{N}$. Let $V$
be a subsurface of $S$: then the following statements hold for
$d_{V}(\widetilde{x},\widetilde{y})$.

  1. (i)

If $V$ is the lift of an orientable subsurface $W$ in $\mathcal{N}$, then
$d_{V}(\widetilde{x},\widetilde{y})=d_{\iota(V)}(\widetilde{x},\widetilde{y})=d_{W}(x,y)$.

  2. (ii)

If $V$ is the lift of a non-orientable subsurface $W$ in $\mathcal{N}$, then
$d_{V}(\widetilde{x},\widetilde{y})\mathrel{\mathop{\asymp}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\cdot$}\vss}}}d_{W}(x,y)$.

  3. (iii)

If $V$ is not a lift of a subsurface in $\mathcal{N}$, then there exists a
uniform constant $k_{0}$, independent of $x$, $y$, and $V$, such that
$d_{V}(\widetilde{x},\widetilde{y})\leq k_{0}$.

###### Proof.

We deal with the proof in cases.

  1. (i)

If $V$ is the lift of an orientable surface, we have that the covering map
restricted to $V$ is a homeomorphism, and the same holds for $\iota(V)$, so
the result follows in this case as well.

  2. (ii)

If $V$ is the lift of a non-orientable subsurface $W$, then by Theorem A.2, we
have that $\mathcal{C}(W)$ quasi-isometrically embeds into $\mathcal{C}(V)$,
and the result follows.

  3. (iii)

If $V$ is not a lift at all, that means $V$ and $\iota(V)$ are transverse
subsurfaces. By the Behrstock inequality, there exists a $k_{0}$ such that the
following holds.

$\displaystyle\min\left(d_{V}(\widetilde{x},\partial\iota(V)),d_{\iota(V)}(\widetilde{x},\partial
V)\right)\leq\frac{k_{0}}{2}$ (42)

But we also have that $\widetilde{x}$ is fixed by $\iota$, which gives us the
following equality of curve complex distances.

$\displaystyle
d_{V}(\widetilde{x},\partial\iota(V))=d_{\iota(V)}(\widetilde{x},\partial V)$
(43)

Combining (42) and (43), we get the following bound on the $\mathcal{C}(V)$
distance between $\widetilde{x}$ and $\partial\iota(V)$.

$\displaystyle d_{V}(\widetilde{x},\partial\iota(V))\leq\frac{k_{0}}{2}$

We have that the same bounds also hold for $\widetilde{y}$, so the result
follows from the above inequality and the triangle inequality.

This shows the result for all the cases and concludes the proof. ∎

We begin by proving the distance formula for points in Teichmüller space. Let
$x$ and $y$ be a pair of points in $\mathcal{T}(\mathcal{N}_{g})$, and let
$\Gamma$ be the set of curves that are short on both $x$ and $y$, $\Gamma_{x}$
the set of curves that are only short on $x$, and $\Gamma_{y}$ the set of
curves that are only short on $y$. Let $\mu_{x}$ and $\mu_{y}$ be short
markings on $x$ and $y$ respectively. Let $\mathcal{C}^{+}$ and
$\mathcal{C}^{-}$ denote the set of two-sided and one-sided curves on
$\mathcal{N}_{g}$. Finally, let $\left[x\right]_{k}$ be the function which is
$0$ for $x\leq k$, and identity for $x>k$.

######  Theorem A.6 (Distance formula).

The distance between $x$ and $y$ in $\mathcal{T}(\mathcal{N}_{g})$ is given by
the following formula.

$\displaystyle d(x,y)$ $\displaystyle\mathrel{\mathop{\asymp}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\cdot$}\vss}}}\sum_{Y}\left[d_{Y}(\mu_{x},\mu_{y})\right]_{k}+\sum_{\alpha\in\Gamma^{c}\cap\mathcal{C}^{+}}\left[\log(d_{\alpha}(\mu_{x},\mu_{y}))\right]_{k}$
(44)
$\displaystyle+\max_{\alpha\in\Gamma\cap\mathcal{C}^{+}}d_{\mathbb{H}_{\alpha}}(x,y)+\max_{\alpha\in\Gamma\cap\mathcal{C}^{-}}d_{(\mathbb{R}_{>0})_{\alpha}}(x,y)$
$\displaystyle+\max_{\alpha\in\Gamma_{x}}\log\frac{1}{\ell_{x}(\alpha)}+\max_{\alpha\in\Gamma_{y}}\log\frac{1}{\ell_{y}(\alpha)}$

###### Proof.

Let $\widetilde{x}$ and $\widetilde{y}$ be the images of $x$ and $y$ in
$\mathcal{T}(\mathcal{S}_{g-1})$ under the isometric embedding map. Since
$d(x,y)=d(\widetilde{x},\widetilde{y})$, it will suffice to estimate
$d(\widetilde{x},\widetilde{y})$ using distances in the curve complexes. Let
$\widetilde{\mu_{x}}$ and $\widetilde{\mu_{y}}$ be the lifts of $\mu_{x}$ and
$\mu_{y}$. Both $\widetilde{\mu_{x}}$ and $\widetilde{\mu_{y}}$ are short
markings on $\widetilde{x}$ and $\widetilde{y}$ respectively. We have by
Rafi’s distance formula [Raf07, Theorem 6.1], the following estimate on
$d(\widetilde{x},\widetilde{y})$.

$\displaystyle d(\widetilde{x},\widetilde{y})$
$\displaystyle\mathrel{\mathop{\asymp}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\cdot$}\vss}}}\sum_{Y}\left[d_{Y}(\widetilde{\mu_{x}},\widetilde{\mu_{y}})\right]_{k}+\sum_{\alpha\in\widetilde{\Gamma}^{c}}\left[\log(d_{\alpha}(\widetilde{\mu_{x}},\widetilde{\mu_{y}}))\right]_{k}$
(45)
$\displaystyle+\max_{\alpha\in\widetilde{\Gamma}}d_{\mathbb{H}_{\alpha}}(\widetilde{x},\widetilde{y})$
$\displaystyle+\max_{\alpha\in\widetilde{\Gamma_{\widetilde{x}}}}\log\frac{1}{\ell_{\widetilde{x}}(\alpha)}+\max_{\alpha\in\Gamma_{\widetilde{y}}}\log\frac{1}{\ell_{\widetilde{y}}(\alpha)}$

Here, $\widetilde{\Gamma}$, $\widetilde{\Gamma_{\widetilde{x}}}$, and
$\widetilde{\Gamma_{\widetilde{y}}}$ are curves on $\widetilde{x}$ and
$\widetilde{y}$ that are simultaneously short, short on $\widetilde{x}$ and
not on $\widetilde{y}$, and short on $\widetilde{y}$ and not on
$\widetilde{x}$ respectively.

It will suffice to show that for a large enough choice of $k$, the right hand
side of (44) is equal to the right hand side of (45), up to an additive and
multiplicative constant. We consider the first term in the right hand side of
(45), namely the sum over the non-annular subsurfaces $Y$. There are three
possibilities for $Y$ in $\mathcal{S}_{g-1}$, which we deal with using Lemma
A.5.

  1. (i)

$Y$ is one component of a lift of an orientable subsurface $Z$ of
$\mathcal{N}_{g}$: In this case we have
$d_{Y}(\widetilde{\mu_{x}},\widetilde{\mu_{y}})=d_{Z}(\mu_{x},\mu_{y})$ (and
the same equality with $Y$ replaced with $\iota(Y)$). Thus, for every term
associated to an orientable non-annular subsurface $Z$ in (44), we get two
corresponding equal terms in (45).

  2. (ii)

$Y$ is the lift of a non-orientable subsurface $Z$ of $\mathcal{N}_{g}$: In
this case, we have
$d_{Y}(\widetilde{\mu_{x}},\widetilde{\mu_{y}})\mathrel{\mathop{\asymp}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\cdot$}\vss}}}d_{Z}(\mu_{x},\mu_{y})$.

  3. (iii)

$Y$ is not a lift of a subsurface of $\mathcal{N}_{g}$: In this case, we have
the following for some $k_{0}$.

$\displaystyle d_{Y}(\widetilde{\mu_{x}},\widetilde{\mu_{y}})\leq k_{0}$

If we pick a threshold $k>k_{0}$, the subsurfaces $Y$ that do not arise from
lifts will not contribute to the right hand side of (45).

We now do the same case analysis for annular subsurfaces: consider a curve
$\alpha$ on $\mathcal{S}_{g-1}$ that is contained in $\Gamma^{c}$, i.e. it is
not simultaneously short on $\widetilde{x}$ and $\widetilde{y}$. There are
three possibilities for $\alpha$.

  1. (i)

$\alpha$ is one component of a lift of a two-sided curve $\gamma$ on
$\mathcal{N}_{g}$: In this case, $\alpha$ and $\iota(\alpha)$ are disjoint,
and the restriction of the covering map to these curves is a homeomorphism. We
have
$d_{\alpha}(\widetilde{\mu_{x}},\widetilde{\mu_{y}})=d_{\gamma}(\mu_{x},\mu_{y})$:
consequently, for every term in $\Gamma^{c}\cap\mathcal{C}^{+}$ in (44), we
have two equal terms in (45).

  2. (ii)

$\alpha$ is the lift of a one-sided curve on $\mathcal{N}_{g}$: In this case
$\alpha=\iota(\alpha)$, but the transformation $\iota$ reverses orientation on
the surface $\mathcal{S}_{g-1}$. That means $\widetilde{(}\mu_{x})$ and
$\widetilde{\mu_{y}}$ cannot have a relative twist between them along
$\alpha$, because if they did, $\iota(\widetilde{(}\mu_{x}))$ and
$\iota(\widetilde{(}\mu_{y}))$ would have the opposite twist. On the other
hand $\widetilde{\mu_{i}}=\iota(\widetilde{\mu_{i}})$ for $i=x$ and $i=y$,
which means the relative twist must be $0$. This proves that the $\alpha$
which are lifts of one-sided curves do not contribute to the second term of
(45).

  3. (iii)

$\alpha$ is not a lift of a curve on $\mathcal{N}_{g}$: In this case $\alpha$
and $\iota(\alpha)$ intersect each other, and are not equal, which means they
are transverse. We deal with this the same way we dealt with transverse non-
annular subsurfaces, i.e. via the Behrstock inequality.

This case analysis proves that the second terms on the right hand side of (44)
and (45) are equal, up to an additive and multiplicative constant.

We now deal with the last three terms of (45). These terms deal with short
curves on $x$ or $y$: we claim that the short curves must be lifts of either
one-sided or two-sided curves in $\mathcal{N}_{g}$. Suppose a curve $\alpha$
is short and not a lift. Then $\alpha$ has positive intersection number with
$\iota(\alpha)$, but since $\iota$ is an isometry, $\iota(\alpha)$ must also
be short. For a sufficiently small threshold for what we call short, we can’t
have a short curve intersecting another short curve, which proves the claim
that all the short curves arise as lifts.

Since the curves in $\widetilde{\Gamma}$ are all lifts, the third term of (45)
can be split up into two terms: the lifts of the two-sided and one-sided
curves. For the two-sided curves, the distance calculation involves both the
length and twist coordinate, and for the one-sided curves, only the length
coordinate is involved. This follows from Theorem A.3.

Finally, the last two terms in (45) are the same as the last two terms of
(44), up to an additive error of $(6g)\cdot\log(2)$, since the lift of a short
curve can double its length, and there are no more than $6g$ short curves.

We have shown that the right hand sides of (44) and (45) are equal, up to a
multiplicative and additive constant, which proves the result. ∎

We now verify that Teichmüller geodesics can be broken up into _active
intervals associated to subsurfaces_ , which are subintervals of the geodesic
associated to each subsurface $V$, along which the projection to $V$ is large,
and outside of which, the projection is bounded. The following lemma of [DM23,
Lemma 3.26] (which itself is a generalization of [Raf07, Proposition 3.7])
describes the subsegments of $[x,y]$ along which the geodesic makes progress
in the curve complex of a subsurface.

######  Proposition A.7.

For each sufficiently small $\varepsilon_{t}>0$, there exists
$0<{\varepsilon_{t}}^{\prime}<\varepsilon_{t}$ and $M_{\varepsilon_{t}}\geq 0$
such that for any subsurface $V\sqsubset S$, there’s a (possibly empty)
connected interval $\mathcal{I}_{V}^{\varepsilon_{t}}\subset[x,y]$ such that
the following five conditions hold.

  1. (i)

If $d_{V}(x,y)\geq M_{\varepsilon_{t}}$, then
$\mathcal{I}_{V}^{{\varepsilon_{t}}}$ is a non-empty subinterval of $[x,y]$.

  2. (ii)

$\ell_{\alpha}(z)<{\varepsilon_{t}}$ for all
$z\in\mathcal{I}_{V}^{{\varepsilon_{t}}}$ and $\alpha\in\partial V$.

  3. (iii)

For all $z\in[x,y]\setminus\mathcal{I}_{V}^{{\varepsilon_{t}}}$, some
component $\alpha$ of $\partial V$ has
$\ell_{\alpha}(z)>{\varepsilon_{t}}^{\prime}$.

  4. (iv)

$d_{V}(w,z)\leq M_{\varepsilon_{t}}$ for every subinterval $[w,z]\subset[x,y]$
if $[w,z]\cap\mathcal{I}_{V}^{\varepsilon_{t}}=\varnothing$.

  5. (v)

For a pair of traverse subsurfaces $U$ and $V$,
$\mathcal{I}_{U}^{\varepsilon_{t}}\cap\mathcal{I}_{V}^{\varepsilon_{t}}=\varnothing$.

###### Proof of Theorem A.7 for non-orientable surfaces.

Let $\mathcal{N}$ be the non-orientable surface, and $\mathcal{S}$ its double
cover. We consider the image $[\widetilde{x},\widetilde{y}]$ of the geodesic
$[x,y]$ in $\mathcal{T}(\mathcal{S})$. We know that the result holds for
$[\widetilde{x},\widetilde{y}]$, although with ${\varepsilon_{t}}^{\prime}$
replaced with $\frac{{\varepsilon_{t}}^{\prime}}{2}$, since lifting can double
the lengths of some curves.

The main fact we need to verify is that the only subsurfaces $V$ that have
non-empty $\mathcal{I}_{V}^{\varepsilon_{t}}$ come from lifts. If $V$ is a
subsurface of $\mathcal{S}$ that is not a lift, we use case (iii) of Lemma A.5
to conclude that $d_{V}(x,y)\leq k_{0}$ for some fixed constant $k_{0}$.
Picking $M_{\varepsilon_{t}}>k_{0}$ guarantees that the only subsurfaces for
which $\mathcal{I}_{V}^{\varepsilon_{t}}$ is non-empty arise from lifts, which
proves the result for non-orientable surfaces. ∎

Finally, we show that the consistency and the realization results ([BKMM12])
hold for Teichmüller spaces of non-orientable surfaces as well. We begin by
recalling the definition of consistency.

######  Definition A.8 (Consistency).

For a connected surface $S$, and a parameter $\theta\geq 1$, we say a tuple
$(z_{V})\in\prod_{V\sqsubset S}\mathcal{C}(V)$ is $\theta$-consistent if the
following two conditions holds for all pairs of subsurfaces $U$ and $V$.

  1. (i)

If $U\pitchfork V$, then

$\displaystyle\min(d_{U}(z_{U},\partial V),d_{V}(z_{V},\partial U))\leq\theta$

  2. (ii)

If $U\sqsubset V$, then

$\displaystyle\min(d_{U}(z_{U},\pi_{U}(z_{V})),d_{V}(z_{V},\partial
U))\leq\theta$

The following theorem ([BKMM12, Theorem 4.3]) states that the projection from
Teichmüller space to the curve complexes of all the subsurfaces is coarsely
surjective onto the set of consistent tuples.

######  Theorem A.9 (Consistency and realization).

There is a constant $K\geq 1$, and function
$\mathfrak{C}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that the following holds
for any surface $S$.

  * •

(Consistency) For every $x\in\mathcal{T}(S)$, the projection tuple
$(\pi_{V}(x))_{V\sqsubset S}$ is $K$-consistent.

  * •

(Realization) For every $\theta$-consistent tuple $(z_{V})_{V\sqsubset S}$,
there exists a point $z\in\mathcal{T}(S)$ such that
$d_{V}(\pi_{V}(z),z_{V})\leq\mathfrak{C}(\theta)$ for all $V$.

###### Proof sketch of Theorem A.9 for non-orientable surfaces.

We first show that the projection map is consistent, and then show consistent
tuples lie coarsely in the image of the projection map.

  * •

(Consistency) We map $x$ to $\widetilde{x}$ in the Teichmüller space of the
double cover $\widetilde{S}$. By applying the theorem for orientable surfaces,
we have the $(\pi_{W}(\widetilde{x}))_{W\sqsubset\widetilde{S}}$, and we
restrict to the subsurfaces in the tuple which arise as lifts. These points
lie in the image of the quasi-isometric embedding map from Theorem A.2, which
means consistency also holds for the tuples in $S$.

  * •

(Realization) Given a $\theta$-consistent tuple $(z_{V})_{V\sqsubset S}$, we
construct a $\theta^{\prime}$-consistent tuple in the double cover
$\widetilde{S}$. For subsurfaces of $\widetilde{S}$ that arise as lifts, we
use the map from Theorem A.2. For the subsurfaces $W$ that are not lifts, we
set $z_{W}=\pi_{W}(\partial(\iota(W)))$. The fact that this is a
$\theta^{\prime}$-consistent tuple follows from the Behrstock inequality (for
some $\theta^{\prime}>\theta$)333A longer but a more thorough way of seeing
this would be to verify that the Teichmüller space of a non-orientable surface
satisfies the $9$ axioms for hierarchical hyperbolicity that are enumerated in
[BHS19].. We now use this point to construct $y\in\mathcal{T}(\widetilde{S})$,
and deduce that $y$ is coarsely fixed by $\iota^{\ast}$, i.e. the distance
between $y$ and $\iota^{\ast}(y)$ is bounded above by a uniform constant $C$.
Consider the midpoint $\widetilde{x}$ of the Teichmüller geodesic segment
joining $y$ and $\iota^{\ast}(y)$: by Teichmüller’s uniqueness theorem,
$\widetilde{x}$ is fixed under the $\iota^{\ast}$-action. This means there is
some $x\in\mathcal{T}(S)$ whose image is $\widetilde{x}$, and therefore the
projection maps are coarsely $(z_{V})$.

∎

## References

  * [ABEM12] Jayadev Athreya, Alexander Bufetov, Alex Eskin and Maryam Mirzakhani  “Lattice point asymptotics and volume growth on Teichmüller space”  In _Duke Mathematical Journal_ 161.6 Duke University Press, 2012, pp. 1055–1111  DOI: 10.1215/00127094-1548443
  * [Ara22] Francisco Arana-Herrera  “Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves”  In _arXiv e-prints_ , 2022, pp. arXiv:2202.04156  DOI: 10.48550/arXiv.2202.04156
  * [Bab02] Martine Babillot  “On the mixing property for hyperbolic systems”  In _Israel journal of mathematics_ 129 Springer, 2002, pp. 61–76
  * [BHS19] Jason Behrstock, Mark Hagen and Alessandro Sisto  “Hierarchically hyperbolic spaces II: Combination theorems and the distance formula”  In _Pacific Journal of Mathematics_ 299.2 Mathematical Sciences Publishers, 2019, pp. 257–338
  * [BKMM12] Jason Behrstock, Bruce Kleiner, Yair Minsky and Lee Mosher  “Geometry and rigidity of mapping class groups”  In _Geometry & Topology_ 16.2 Mathematical Sciences Publishers, 2012, pp. 781–888
  * [CGTY] Remi Coulon, Ilya Gekhtman, Samuel Tapie and Wenyuan Yang  “Mixing of the geodesic flow for a group with a contracting element”
  * [Cou22] Rémi Coulon  “Patterson-Sullivan theory for groups with a strongly contracting element”  In _arXiv preprint arXiv:2206.07361_ , 2022
  * [Cou24] Rémi Coulon  “Ergodicity of the geodesic flow for groups with a contracting element”, 2024  arXiv:2303.01390 [math.DS]
  * [DM23] Spencer Dowdall and Howard Masur  “Lattice point counting for finite-order mapping classes”, 2023  arXiv:2305.03721 [math.GT]
  * [EGPS23] Viveka Erlandsson, Matthieu Gendulphe, Irene Pasquinelli and Juan Souto  “Mapping class group orbit closures for non-orientable surfaces”  In _Geometric and Functional Analysis_ Springer, 2023, pp. 1–57
  * [EM11] Alex Eskin and Maryam Mirzakhani  “Counting closed geodesics in moduli space”  In _Journal of Modern Dynamics_ 5.1, 2011, pp. 71–105
  * [EM22] Alex Eskin and Shahar Mozes  “10. Margulis functions and their applications”  In _The Impact of Margulis on Modern Mathematics_ Chicago: University of Chicago Press, 2022, pp. 342–361  DOI: doi:10.7208/chicago/9780226804163-011
  * [FM02] Benson Farb and Lee Mosher  “Convex cocompact subgroups of mapping class groups”  In _Geometry & Topology_ 6.1 Mathematical Sciences Publishers, 2002, pp. 91–152
  * [Gen17] Matthieu Gendulphe  “What’s wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces”  In _arXiv e-prints_ , 2017, pp. arXiv:1706.08798  arXiv:1706.08798 [math.GT]
  * [GM23] Ilya Gekhtman and Biao Ma  “Dynamics of subgroups of mapping class groups”, 2023  arXiv:2311.03779 [math.GT]
  * [Kha23] Sayantan Khan  “The limit set of non-orientable mapping class groups”  In _Journal of Modern Dynamics_ 19.0, 2023  DOI: 10.3934/jmd.2023007
  * [KPW23] Sayantan Khan, Caleb Partin and Rebecca R Winarski  “Pseudo-Anosov homeomorphisms of punctured nonorientable surfaces with small stretch factor”  In _Algebraic & Geometric Topology_ 23.6 Mathematical Sciences Publishers, 2023, pp. 2823–2856
  * [LR06] Christopher J Leininger and Alan W Reid  “A combination theorem for Veech subgroups of the mapping class group”  In _Geometric & Functional Analysis GAFA_ 16 Springer, 2006, pp. 403–436
  * [LS18] Livio Liechti and Bal’azs Strenner  “Minimal pseudo-Anosov stretch factors on nonoriented surfaces”  In _Algebraic & Geometric Topology_, 2018  URL: https://api.semanticscholar.org/CorpusID:56112676
  * [Mag18] Michael Magee  “Counting One-Sided Simple Closed Geodesics on Fuchsian Thrice Punctured Projective Planes”  In _International Mathematics Research Notices_ 2020.13, 2018, pp. 3886–3901  DOI: 10.1093/imrn/rny112
  * [Mar04] GA Margulis  “On some aspects of the theory of Anosov systems”  Springer, 2004
  * [Mas82] Howard Masur  “Interval exchange transformations and measured foliations”  In _Annals of Mathematics_ 115.1 JSTOR, 1982, pp. 169–200
  * [Min96] Yair N Minsky  “Quasi-projections in Teichmüller space.”  In _Journal für die reine und angewandte Mathematik (Crelles Journal)_ 1996.473 De Gruyter Berlin, New York, 1996, pp. 121–136
  * [Min96a] Yair N. Minsky  “Extremal length estimates and product regions in Teichmüller space”  In _Duke Math. J._ 85.1, 1996, pp. 249–286  URL: http://dml.mathdoc.fr/item/1077244446
  * [Mir08] Maryam Mirzakhani  “Growth of the number of simple closed geodesics on hyperbolic surfaces”  In _Annals of Mathematics_ JSTOR, 2008, pp. 97–125
  * [Miy08] Hideki Miyachi  “Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space”  In _Geometriae Dedicata_ 137 Springer, 2008, pp. 113–141
  * [MS13] Howard Masur and Saul Schleimer  “The geometry of the disk complex”  In _Journal of the American Mathematical Society_ 26.1, 2013, pp. 1–62
  * [Nor08] Paul Norbury  “Lengths of geodesics on non-orientable hyperbolic surfaces”  In _Geometriae Dedicata_ 134 Springer, 2008, pp. 153–176
  * [Pen88] Robert C Penner  “A construction of pseudo-Anosov homeomorphisms”  In _Transactions of the American Mathematical Society_ 310.1, 1988, pp. 179–197
  * [Qui06] Jean-François Quint  “An overview of Patterson-Sullivan theory”  In _Workshop The barycenter method, FIM, Zurich_ , 2006
  * [Raf07] Kasra Rafi  “A combinatorial model for the Teichmüller metric”  In _GAFA Geometric And Functional Analysis_ 17 Springer, 2007, pp. 936–959
  * [Rob03] Thomas Roblin  “Ergodicité et équidistribution en courbure négative”  Société mathématique de France, 2003
  * [Sch82] Martin Scharlemann  “The Complex of Curves on Non-Orientable Surfaces”  In _Journal of the London Mathematical Society_ 2.1 Citeseer, 1982, pp. 171–184
  * [Sis18] Alessandro Sisto  “Contracting elements and random walks”  In _Journal für die reine und angewandte Mathematik (Crelles Journal)_ 2018.742, 2018, pp. 79–114  DOI: doi:10.1515/crelle-2015-0093
  * [Sta23] Douglas Stanford  “A Mirzakhani recursion for non-orientable surfaces”, 2023  arXiv:2303.04049 [hep-th]
  * [Sul79] Dennis Sullivan  “The density at infinity of a discrete group of hyperbolic motions”  In _Publications Mathématiques de l’IHÉS_ 50, 1979, pp. 171–202
  * [Yan18] Wen-yuan Yang  “Statistically Convex-Cocompact Actions of Groups with Contracting Elements”  In _International Mathematics Research Notices_ 2019.23, 2018, pp. 7259–7323  DOI: 10.1093/imrn/rny001
  * [Yan20] Wen-yuan Yang  “Genericity of contracting elements in groups”  In _Mathematische Annalen_ 376.3 Springer, 2020, pp. 823–861