# A note on an effective characterization of covers with an application to
higher rank representations

Tarik Aougab  Department of Mathematics
Haverford College
Haverford, PA 19041<EMAIL_ADDRESS>,  Max Lahn  Department of
Mathematics
University of Michigan
Ann Arbor, MI 48109<EMAIL_ADDRESS>,  Marissa Loving  Department of
Mathematics
University of Wisconsin
Madison, WI 53706<EMAIL_ADDRESS>and  Nicholas Miller  Department of
Mathematics
University of Oklahoma
Norman, OK 73019<EMAIL_ADDRESS>
###### Abstract.

In this note we prove an effective characterization of when two finite-degree
covers of a connected, orientable surface of negative Euler characteristic are
isomorphic in terms of which curves have simple elevations, weakening the
hypotheses to consider curves with explicitly bounded self-intersection
number. As an application we show that for sufficiently large $\displaystyle
N$, the set of unmarked traces associated to simple closed curves in a
generically chosen representation to
$\displaystyle\mathrm{SL}_{N}\left\lparen\mathbb{R}\right\rparen$
distinguishes between pairs of non-isomorphic covers.

##  1\. Introduction

In the setting of higher Teichmüller theory, marked spectral rigidity results
are known for some of the many class functions on
$\displaystyle\pi_{1}\left\lparen S\right\rparen$ which share roles analogous
to hyperbolic length. For example, Bridgeman–Canary–Labourie [BCL20] show that
a Hitchin representation of $\displaystyle\pi_{1}\left\lparen S\right\rparen$
is determined by either the marked spectral radii of simple curves or the
marked traces associated to simple curves. In this setting, we show that
covers can be distinguished by the set of unmarked traces of simple closed
curves.

theorem+SLnTrace

There is a positive integer $\displaystyle N$, explicitly computable in terms
of $\displaystyle\left\lvert\chi\left\lparen S\right\rparen\right\rvert$,
$\displaystyle\deg\left\lparen p\right\rparen$, and
$\displaystyle\deg\left\lparen q\right\rparen$, so that $\displaystyle p$ and
$\displaystyle q$ are isomorphic if and only if the restrictions of linear
representations $\displaystyle\pi_{1}\left\lparen
S\right\rparen\xrightarrow[]{}\mathrm{SL}_{N}\left\lparen\mathbb{R}\right\rparen$
or $\displaystyle\pi_{1}\left\lparen
S\right\rparen\xrightarrow[]{}\mathrm{SL}_{N}\left\lparen\mathbb{C}\right\rparen$
to $\displaystyle\pi_{1}\left\lparen X\right\rparen$ and
$\displaystyle\pi_{1}\left\lparen Y\right\rparen$ are generically simple trace
isospectral.

We call the reader’s attention to the words _explicitly computable_ in the
above statement of Section 1. We obtain this by first proving the following
effective version of [Aou+23, Theorem 3], with its hypotheses weakened so that
we need only consider the elevations of curves on $\displaystyle S$ with
explicitly bounded self-intersection number.

theorem+effective

There is an integer $\displaystyle M$, explicitly computable in terms of
$\displaystyle\left\lvert\chi\left\lparen S\right\rparen\right\rvert$,
$\displaystyle\deg\left\lparen p\right\rparen$, and
$\displaystyle\deg\left\lparen q\right\rparen$, so that $\displaystyle p$ and
$\displaystyle q$ are isomorphic as covers of $\displaystyle S$ if and only if
for every curve $\displaystyle\gamma$ on $\displaystyle S$ with $\displaystyle
i\left\lparen\gamma,\gamma\right\rparen\leq M$, $\displaystyle\gamma$ has a
simple elevation along $\displaystyle p$ to $\displaystyle X$ if and only if
it has a simple elevation along $\displaystyle q$ to $\displaystyle Y$.

We note that although a non-effective version of Section 1 can be derived
quickly from [Aou+23, Theorem 3] (such an argument is outlined at the
beginning of Section 3), justifying that the integer $\displaystyle M$ is
_explicitly computable_ in terms of $\displaystyle\left\lvert\chi\left\lparen
S\right\rparen\right\rvert$, $\displaystyle\deg\left\lparen p\right\rparen$,
and $\displaystyle\deg\left\lparen q\right\rparen$ requires an entirely
different argument than that used in the proof of [Aou+23, Theorem 3]. In
particular, Section 1 gives a completely distinct proof of [Aou+23, Theorem
3]. We decided to relegate this result to its own note rather than including
it in [Aou+23] because achieving effective control in Section 1 requires
navigating through some computationally heavy lemmas; we use fine properties
of the geometry of the curve complex to facilitate these arguments. Our proof
also requires effective versions of the main theorem of [Riv01] and of [Tan17,
Proposition 6.2], which we prove in detail in Appendix A and Appendix B,
respectively, as they may be of independent interest.

###  1.1. Organization of paper

We provide some background definitions and tools in Section 2. In Section 3,
we prove Section 1. We then prove Section 1 in Section 4. Finally, Appendix A
and Appendix B contain the proofs of effective versions of the main theorem of
[Riv01] and [Tan17, Proposition 6.2], respectively, which are needed for the
proof of Section 1.

###  1.2. Acknowledgements

We gratefully acknowledge support from NSF grants DMS-1939936 (Aougab),
DMS-1906441 (Lahn), DMS-1902729 & DMS-2231286 (Loving), and
DMS-2005438/2300370 (Miller).

##  2\. Preliminaries

The interested reader can refer to [Aou+23] for a more in depth treatment of
much of the relevant background. To keep this note concise, we will only
mention two of the most specialized results that we make use of here which are
both related to the geometry of the curve complex.

###  2.1. Geometry of the curve complex

We will make use of the following work of Aougab-Taylor, which constructs
geodesic rays whose vertices have controlled intersections.

######  Theorem 2.1 ([AT14, Theorem 1.1]).

For any simple curve $\displaystyle\eta\in\mathscr{C}\left\lparen
S\right\rparen$ on a closed surface $\displaystyle S$ of negative Euler
characteristic, there is a sequence $\displaystyle\left\lparen
v_{k}\right\rparen_{k=0}^{\infty}$ in $\displaystyle\mathscr{C}\left\lparen
S\right\rparen$ with $\displaystyle v_{0}=\eta$, $\displaystyle
d_{S}\left\lparen v_{j},v_{k}\right\rparen=\left\lvert j-k\right\rvert$ for
all indices $\displaystyle j,k\in\mathbb{N}$, and

$\displaystyle i\left\lparen v_{j},v_{k}\right\rparen\leq\left\lparen
D+3\right\rparen^{2\max\left\lparen
j,k\right\rparen-5}\left\lparen\frac{3\left\lvert\chi\left\lparen
S\right\rparen\right\rvert}{2}\right\rparen^{\left\lvert
j-k\right\rvert+2}+f_{j,k}\left\lparen\left\lvert\chi\left\lparen
S\right\rparen\right\rvert\right\rparen,$

where $\displaystyle D$ is a fixed, universal constant described in 3.7, and
$\displaystyle f_{j,k}\left\lparen\left\lvert\chi\left\lparen
S\right\rparen\right\rvert\right\rparen$ is $\displaystyle
O\left\lparen\left\lvert\chi\left\lparen
S\right\rparen\right\rvert^{\left\lvert j-k\right\rvert-4}\right\rparen$.

We remark that the above bound on intersection number takes a slightly
different form than the one found in [AT14] since we are only considering
closed surfaces.

As proven by Rafi–Schleimer [RS09], finite-degree covers of surfaces give rise
to quasi-isometric embeddings of their curve complexes. Specifically, given a
finite-degree cover $\displaystyle p\colon X\xrightarrow[]{}S$, we define
$\displaystyle\widetilde{p}\colon\mathscr{C}\left\lparen
S\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen X\right\rparen$ so that
for each simple curve $\displaystyle\gamma\in\mathscr{C}\left\lparen
S\right\rparen$, $\displaystyle\widetilde{p}\left\lparen\gamma\right\rparen$
is a choice of elevation of $\displaystyle\gamma$ along $\displaystyle p$ to
$\displaystyle X$. At first glance, our definition of
$\displaystyle\widetilde{p}$ appears to rely significantly on a choice of
elevation for every simple curve on $\displaystyle S$ and thus its properties
will not be well behaved when defined as described. However, any two
elevations of a simple curve $\displaystyle\gamma$ along $\displaystyle p$ are
disjoint on $\displaystyle X$, since any intersection between distinct
elevations on $\displaystyle X$ would descend to a self-intersection of
$\displaystyle\gamma$ on $\displaystyle S$. Thus different choices of
elevations for $\displaystyle\widetilde{p}$ yield maps which have uniformly
bounded distance at most $\displaystyle 1$, and
$\displaystyle\widetilde{p}\colon\mathscr{C}\left\lparen
S\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen X\right\rparen$ is said
to be _coarsely well-defined_.

######  Theorem 2.2 ([RS09, Theorem 7.1]).

If $\displaystyle p\colon X\xrightarrow[]{}S$ is a finite-degree cover of a
closed, orientable surface $\displaystyle S$ of negative Euler characteristic,
then $\displaystyle\widetilde{p}\colon\mathscr{C}\left\lparen
S\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen X\right\rparen$ is a
quasi-isometric embedding.

Note that in [RS09], the above theorem is formulated in terms of the many-to-
one relation which associates a simple curve on the base surface to its full
pre-image on the cover, which is a multi-curve. Since a multi-curve has
diameter at most $\displaystyle 1$ in the curve complex, this distinction will
not matter for our purposes.

We also note that the quality of the above quasi-isometry induced by such a
finite-degree cover of surfaces can be explicitly bounded in terms of the
Euler characteristic of the base and the degree of the cover. We require such
bounds, as computed by Aougab–Patel–Taylor [APT22, Theorem 7.1], in Section 3.

Since the image $\displaystyle\widetilde{p}\left\lparen\mathscr{C}\left\lparen
S\right\rparen\right\rparen$ of
$\displaystyle\widetilde{p}\colon\mathscr{C}\left\lparen
S\right\rparen\xhookrightarrow{}\mathscr{C}\left\lparen X\right\rparen$ is
quasi-convex in the Gromov hyperbolic space
$\displaystyle\mathscr{C}\left\lparen X\right\rparen$, nearest point
projection $\displaystyle\nu_{S}\colon\mathscr{C}\left\lparen
X\right\rparen\twoheadrightarrow\widetilde{p}\left\lparen\mathscr{C}\left\lparen
S\right\rparen\right\rparen$ is coarsely well-defined and distance non-
increasing. We will make use of the following result of Tang.

[[Tan17, Proposition 6.2]]theorem*Tang

Let $\displaystyle p\colon X\xrightarrow[]{}S$ be a finite-degree regular
cover of a closed surface $\displaystyle S$ of negative Euler characteristic.
There is a uniform upper bound, depending only on
$\displaystyle\left\lvert\chi\left\lparen S\right\rparen\right\rvert$ and
$\displaystyle\deg\left\lparen p\right\rparen$, on the distance between any
circumcenter of the $\displaystyle\mathrm{Deck}\left\lparen
p\right\rparen$-orbit of a simple curve
$\displaystyle\alpha\in\mathscr{C}\left\lparen X\right\rparen$ and its nearest
point projection $\displaystyle\nu_{S}\left\lparen\alpha\right\rparen$.

See Appendix B for an explicit computation of this uniform upper bound.

##  3\. Elevations of curves with bounded self-intersection number

In this section, we prove Section 1, which we restate below for the reader’s
convenience.

*

We stress the importance of the words explicitly computable in the above
theorem statement. Indeed, the fact that some positive integer $\displaystyle
M$ satisfies the conclusions of Section 1 follows immediately from [Aou+23,
Theorem 3] and the following argument:

Let $\displaystyle d=\max\left\lparen\deg\left\lparen
p\right\rparen,\deg\left\lparen q\right\rparen\right\rparen$, and note that
there are only finitely many isomorphism classes of covers of $\displaystyle
S$ of degree at most $\displaystyle d$. In particular, there are only finitely
many _pairs_ of distinct isomorphism classes of covers of $\displaystyle S$ of
degree at most $\displaystyle d$. Choose a representative of each isomorphism
class and note that, for each pair of non-isomorphic covers in this finite
list, there is some curve $\displaystyle\gamma$ on $\displaystyle S$ which has
a simple elevation along one cover but not along the other. Any positive
integer $\displaystyle M$ at least as large as the largest self-intersection
number of these curves $\displaystyle\gamma$ will satisfy the conclusions of
Section 1.

Thus, in the following proof sketch, we take care to justify that
$\displaystyle M$ is, at least in principle, explicitly computable from the
Euler characteristic $\displaystyle\chi\left\lparen S\right\rparen$ and the
degrees $\displaystyle\deg\left\lparen p\right\rparen$ and
$\displaystyle\deg\left\lparen q\right\rparen$. We will use the phrase
_explicitly computable_ to mean precisely this, and we will call a number
_explicitly bounded_ if it can be bounded in terms of an explicitly computable
number.

For example, the quality of the quasi-isometric embedding of curve complexes
induced by a finite-degree cover (for a detailed discussion of this quasi-
isometric embedding, see 2.2) is explicitly bounded in terms of the degree of
the cover and the Euler characteristic of the base surface. This is the
content of the following result of Aougab-Patel-Taylor.

######  Theorem 3.1 ([APT22, Theorem 7.1]).

If $\displaystyle p\colon X\xrightarrow[]{}S$ is a finite-degree cover of a
closed, orientable surface $\displaystyle S$ of negative Euler characteristic,
then

$\displaystyle\frac{d_{S}\left\lparen\alpha,\beta\right\rparen}{80e^{54}\pi\deg\left\lparen
p\right\rparen\cdot\left\lvert\chi\left\lparen
S\right\rparen\right\rvert^{13}}\leq
d_{X}\left\lparen\widetilde{p}\left\lparen\alpha\right\rparen,\widetilde{p}\left\lparen\beta\right\rparen\right\rparen\leq
d_{S}\left\lparen\alpha,\beta\right\rparen,$

for all simple curves $\displaystyle\alpha,\beta\in\mathscr{C}\left\lparen
S\right\rparen$.

We will require this effective statement for the proof of Section 1.

###  3.1. Counting simple curves of bounded hyperbolic length

The following theorem of Rivin gives effective estimates on the growth of the
number of simple geodesics of bounded length on a hyperbolic surface of finite
type as a polynomial in the bound. The degree of this polynomial increases
with the complexity of the surface, so that the number of simple geodesics of
bounded length grows noticeably faster on surfaces of higher genus. By a
combinatorial argument, we show that covers of different degrees can be
distinguished by which curves of explicitly bounded length have simple
elevations.

[[Riv01]]theorem*Rivin

For any hyperbolic surface $\displaystyle\Sigma$ of finite type
$\displaystyle\left\lparen g,b,n\right\rparen$, there are positive real
numbers $\displaystyle c_{1}\left\lparen\Sigma\right\rparen$, $\displaystyle
c_{2}\left\lparen\Sigma\right\rparen$, and $\displaystyle
L_{0}\left\lparen\Sigma\right\rparen$ so that

$\displaystyle
c_{1}\left\lparen\Sigma\right\rparen\leq\frac{\mathcal{N}\left\lparen
L,\Sigma\right\rparen}{L^{6g-6+2b+2c}}\leq
c_{2}\left\lparen\Sigma\right\rparen,$

for all real numbers $\displaystyle L\geq
L_{0}\left\lparen\Sigma\right\rparen$.

Specifically, we will use the following technical corollary, which utilizes
the reliance of the above bounds on the genus $\displaystyle g$ to find curves
of bounded length which have simple elevations to some covers but not to
others.

######  Corollary 3.2.

Consider a finite collection $\displaystyle\left\\{\pi_{k}\right\\}_{k=0}^{n}$
of finite-degree covers
$\displaystyle\pi_{k}\colon\Sigma_{k}\xrightarrow[]{}\Sigma$ of a closed,
hyperbolic surface $\displaystyle\Sigma$. If
$\displaystyle\deg\left\lparen\pi_{0}\right\rparen>\deg\left\lparen\pi_{k}\right\rparen$
for all indices $\displaystyle 1\leq k\leq n$, then there is a simple closed
geodesic $\displaystyle\alpha$ on $\displaystyle\Sigma_{0}$ whose projection
$\displaystyle\pi_{0}\left\lparen\alpha\right\rparen$ has no simple elevations
along $\displaystyle\pi_{k}$ to $\displaystyle\Sigma_{k}$ for all indices
$\displaystyle 1\leq k\leq n$.

Moreover, such a simple closed geodesic $\displaystyle\alpha$ can be chosen so
that $\displaystyle\ell_{\Sigma_{0}}\left\lparen\alpha\right\rparen$ is
explicitly bounded in terms of $\displaystyle n$, the degree
$\displaystyle\deg\left\lparen\pi_{0}\right\rparen$, the Euler characteristic
$\displaystyle\left\lvert\chi\left\lparen\Sigma\right\rparen\right\rvert$, and
the constants guaranteed by Section 3.1.

###### Proof of 3.2.

For each index $\displaystyle 0\leq k\leq n$, let $\displaystyle
c_{1}\left\lparen\Sigma_{k}\right\rparen$, $\displaystyle
c_{2}\left\lparen\Sigma_{k}\right\rparen$, and $\displaystyle
L_{0}\left\lparen\Sigma_{k}\right\rparen$ denote the constants guaranteed by
Section 3.1.

######  Claim 1.

There is a positive real number $\displaystyle L$, explicitly computable in
terms of $\displaystyle n$, the degree
$\displaystyle\deg\left\lparen\pi_{0}\right\rparen$, the Euler characteristic
$\displaystyle\left\lvert\chi\left\lparen\Sigma\right\rparen\right\rvert$, and
the constants guaranteed by Section 3.1, so that

$\displaystyle\frac{\mathcal{N}\left\lparen
L,\Sigma_{0}\right\rparen}{\deg\left\lparen\pi_{0}\right\rparen}>n\max_{1\leq
k\leq n}{\mathcal{N}\left\lparen\deg\left\lparen\pi_{k}\right\rparen
L,\Sigma_{k}\right\rparen}.$

Although we include the following computation for completeness, the key point
is that $\displaystyle L$ is explicitly computable – its exact value plays
little role in the overall argument. As such, we encourage skipping the
following proof of 1 on a first reading.

###### Proof of 1.

If $\displaystyle L\geq L_{0}\left\lparen\Sigma_{k}\right\rparen$ for all
indices $\displaystyle 0\leq k\leq n$, Section 3.1 implies that

$\displaystyle\displaystyle
n\mathcal{N}\left\lparen\deg\left\lparen\pi_{k}\right\rparen
L,\Sigma_{k}\right\rparen$ $\displaystyle\displaystyle\leq
nc_{2}\left\lparen\Sigma_{k}\right\rparen\left\lparen\deg\left\lparen\pi_{k}\right\rparen
L\right\rparen^{3\left\lvert\chi\left\lparen\Sigma_{k}\right\rparen\right\rvert}$
$\displaystyle\displaystyle\leq
nc_{2}\left\lparen\Sigma_{k}\right\rparen\deg\left\lparen\pi_{0}\right\rparen^{3\deg\left\lparen\pi_{0}\right\rparen\left\lvert\chi\left\lparen\Sigma\right\rparen\right\rvert}L^{3\deg\left\lparen\pi_{k}\right\rparen\left\lvert\chi\left\lparen\Sigma\right\rparen\right\rvert},$

for all indices $\displaystyle 1\leq k\leq n$, and similarly that

$\displaystyle\displaystyle\frac{\mathcal{N}\left\lparen
L,\Sigma_{0}\right\rparen}{\deg\left\lparen\pi_{0}\right\rparen}\geq\frac{c_{1}\left\lparen\Sigma_{0}\right\rparen
L^{3\left\lvert\chi\left\lparen\Sigma_{0}\right\rparen\right\rvert}}{\deg\left\lparen\pi_{0}\right\rparen}=\frac{c_{1}\left\lparen\Sigma_{0}\right\rparen
L^{3\deg\left\lparen\pi_{0}\right\rparen\left\lvert\chi\left\lparen\Sigma\right\rparen\right\rvert}}{\deg\left\lparen\pi_{0}\right\rparen}.$

It therefore suffices to find an explicitly computable solution $\displaystyle
L$ for the inequalities

$\displaystyle\frac{c_{1}\left\lparen\Sigma_{0}\right\rparen
L^{3\deg\left\lparen\pi_{0}\right\rparen\left\lvert\chi\left\lparen\Sigma\right\rparen\right\rvert}}{\deg\left\lparen\pi_{0}\right\rparen}>nc_{2}\left\lparen\Sigma_{k}\right\rparen\deg\left\lparen\pi_{0}\right\rparen^{3\deg\left\lparen\pi_{0}\right\rparen\left\lvert\chi\left\lparen\Sigma\right\rparen\right\rvert}L^{3\deg\left\lparen\pi_{k}\right\rparen\left\lvert\chi\left\lparen\Sigma\right\rparen\right\rvert},$

for all $\displaystyle k\in\\{1,\dots,n\\}$. One can check that

$\displaystyle L\coloneqq\max\left\lparen\max_{0\leq k\leq
n}{L_{0}\left\lparen\Sigma_{k}\right\rparen},1+\max_{1\leq k\leq
n}{\frac{nc_{2}\left\lparen\Sigma_{k}\right\rparen\deg\left\lparen\pi_{0}\right\rparen^{1+3\deg\left\lparen\pi_{0}\right\rparen\left\lvert\chi\left\lparen\Sigma\right\rparen\right\rvert}}{c_{1}\left\lparen\Sigma_{0}\right\rparen}}\right\rparen,$

is one such solution. ∎

Let $\displaystyle U_{0}$ denote the set of simple closed geodesics on
$\displaystyle\Sigma_{0}$ of length at most $\displaystyle L$, so that
$\displaystyle\left\lvert U_{0}\right\rvert=\mathcal{N}\left\lparen
L,\Sigma_{0}\right\rparen$. Moreover, denote by $\displaystyle
U\coloneqq\pi_{0}\left\lparen U_{0}\right\rparen$ the set of closed geodesics
on $\displaystyle\Sigma$ which have elevations along $\displaystyle\pi_{0}$ to
$\displaystyle\Sigma_{0}$ lying in $\displaystyle U_{0}$, and note that every
curve in $\displaystyle U$ has length at most $\displaystyle L$, although such
a curve need not be simple. Finally, for each index $\displaystyle 1\leq k\leq
n$, denote by $\displaystyle U_{k}\coloneqq\pi_{k}^{-1}\left\lparen
U\right\rparen$ the set of (not necessarily simple) elevations along
$\displaystyle\pi_{k}$ to $\displaystyle\Sigma_{k}$ of curves in
$\displaystyle U$, each of which has length at most
$\displaystyle\deg\left\lparen\pi_{k}\right\rparen L$.

For each index $\displaystyle 0\leq k\leq n$, the cover
$\displaystyle\pi_{k}\colon\Sigma_{k}\xrightarrow[]{}\Sigma$ gives a
surjective map $\displaystyle U_{k}\twoheadrightarrow U$ which is at most
$\displaystyle\deg\left\lparen\pi_{k}\right\rparen$-to-one. Thus

$\displaystyle\displaystyle\left\lvert U\right\rvert$
$\displaystyle\displaystyle\geq\frac{\left\lvert
U_{0}\right\rvert}{\deg\left\lparen\pi_{0}\right\rparen}=\frac{\mathcal{N}\left\lparen
L,\Sigma_{0}\right\rparen}{\deg\left\lparen\pi_{0}\right\rparen}>n\max_{1\leq
k\leq n}{\mathcal{N}\left\lparen\deg\left\lparen\pi_{k}\right\rparen
L,\Sigma_{k}\right\rparen}\geq\sum_{k=1}^{n}{\mathcal{N}\left\lparen\deg\left\lparen\pi_{k}\right\rparen
L,\Sigma_{k}\right\rparen},$

which implies that there is a closed geodesic in $\displaystyle U$ for which
every corresponding curve in $\displaystyle\bigcup_{k=1}^{n}{U_{k}}$ is non-
simple; that is, there is a simple closed geodesic $\displaystyle\alpha$ on
$\displaystyle\Sigma_{0}$ of length
$\displaystyle\ell_{\Sigma_{0}}\left\lparen\alpha\right\rparen\leq L$ so that
for all indices $\displaystyle 1\leq k\leq n$,
$\displaystyle\pi_{0}\left\lparen\alpha\right\rparen$ has no simple elevations
along $\displaystyle\pi_{k}$ to $\displaystyle\Sigma_{k}$. ∎

See Appendix A for a complete and self-contained proof of an effective version
of Section 3.1. Specifically, we will exhibit a dependence of the constants in
question on the injectivity radius to choose a metric in the proof of Section
1 so that the constants are explicitly bounded.

###  3.2. The collar lemma

The collar lemma is a classical result relating the geometry of certain
regular neighborhoods of a simple closed geodesic on a hyperbolic surface to
its length.

######  Lemma 3.3 (Collar lemma).

If $\displaystyle\alpha$ is a simple closed geodesic on a hyperbolic surface
$\displaystyle S$,

$\displaystyle\ell_{S}\left\lparen\beta\right\rparen>i\left\lparen\alpha,\beta\right\rparen\cdot
2\mathrm{sinh}^{-1}\left\lparen\mathrm{csch}\left\lparen\frac{\ell_{S}\left\lparen\alpha\right\rparen}{2}\right\rparen\right\rparen$

for any closed curve $\displaystyle\beta$ on $\displaystyle S$.

See [Bus10, Corollary 4.1.1] for a proof of 3.3. When all the elevations of a
(not necessarily simple) closed geodesic along a finite-degree cover are
simple, we will use the above result to bound the self-intersection number in
terms of its length.

######  Corollary 3.4.

Let $\displaystyle\pi\colon Z\xrightarrow[]{}S$ be a finite-degree cover of a
hyperbolic surface $\displaystyle S$. If every elevation of some curve
$\displaystyle\gamma$ on $\displaystyle S$ along $\displaystyle\pi$ to
$\displaystyle Z$ is simple, then its self intersection number $\displaystyle
i\left\lparen\gamma,\gamma\right\rparen$ is explicitly bounded in terms of its
length $\displaystyle\ell_{S}\left\lparen\gamma\right\rparen$ and the degree
$\displaystyle\deg\left\lparen\pi\right\rparen$.

###### Proof.

Fix a parametrization of $\displaystyle\gamma$ without triple (or higher
order) intersections, and let
$\displaystyle\widetilde{\gamma}_{1},\dots,\widetilde{\gamma}_{n}$ be the
elevations of $\displaystyle\gamma$ along $\displaystyle\pi$ to $\displaystyle
Z$, all of which are simple by assumption and have infimal length

$\displaystyle\ell_{Z}\left\lparen\widetilde{\gamma}_{k}\right\rparen\leq\deg\left\lparen\pi\right\rparen\cdot\ell_{S}\left\lparen\gamma\right\rparen.$

Note that $\displaystyle n\leq\deg\left\lparen\pi\right\rparen$. Moreover,
every intersection point between elevations of $\displaystyle\gamma$ lies
above a self-intersection point of $\displaystyle\gamma$, and conversely, the
fiber of $\displaystyle\pi$ over any self-intersection point of
$\displaystyle\gamma$ consists of exactly
$\displaystyle\deg\left\lparen\pi\right\rparen$ intersection points between
elevations. Hence

$\displaystyle\displaystyle i\left\lparen\gamma,\gamma\right\rparen$
$\displaystyle\displaystyle=\frac{1}{2\deg\left\lparen\pi\right\rparen}\sum_{j,k=1}^{n}{i\left\lparen\widetilde{\gamma}_{j},\widetilde{\gamma}_{k}\right\rparen}<\frac{1}{2\deg\left\lparen\pi\right\rparen}\sum_{j,k=1}^{n}{\frac{\ell_{Z}\left\lparen\widetilde{\gamma_{j}}\right\rparen}{2\mathrm{sinh}^{-1}\left\lparen\mathrm{csch}\left\lparen\frac{\ell_{Z}\left\lparen\widetilde{\gamma_{k}}\right\rparen}{2}\right\rparen\right\rparen}}$
$\displaystyle\displaystyle\leq\frac{1}{2\deg\left\lparen\pi\right\rparen}\sum_{j,k=1}^{n}{\frac{\deg\left\lparen\pi\right\rparen\ell_{S}\left\lparen\gamma\right\rparen}{2\mathrm{sinh}^{-1}\left\lparen\mathrm{csch}\left\lparen\frac{\deg\left\lparen\pi\right\rparen\ell_{S}\left\lparen\gamma\right\rparen}{2}\right\rparen\right\rparen}}=\frac{n^{2}\ell_{S}\left\lparen\gamma\right\rparen}{4\mathrm{sinh}^{-1}\left\lparen\mathrm{csch}\left\lparen\frac{\deg\left\lparen\pi\right\rparen\ell_{S}\left\lparen\gamma\right\rparen}{2}\right\rparen\right\rparen}$
$\displaystyle\displaystyle\leq\frac{\deg\left\lparen\pi\right\rparen^{2}\ell_{S}\left\lparen\gamma\right\rparen}{4\mathrm{sinh}^{-1}\left\lparen\mathrm{csch}\left\lparen\frac{\deg\left\lparen\pi\right\rparen\ell_{S}\left\lparen\gamma\right\rparen}{2}\right\rparen\right\rparen}.$

That is, the self-intersection number $\displaystyle
i\left\lparen\gamma,\gamma\right\rparen$ of $\displaystyle\gamma$ is
explicitly bounded in terms of its length
$\displaystyle\ell_{S}\left\lparen\gamma\right\rparen$ and the degree
$\displaystyle\deg\left\lparen\pi\right\rparen$. ∎

###  3.3. Subsurface projection and the bounded geodesic image theorem

For this section, fix a basepoint $\displaystyle s\in S$ on a closed surface
$\displaystyle S$ of negative Euler characteristic. Any curve
$\displaystyle\gamma$ on $\displaystyle S$ has an associated _annular complex_
$\displaystyle\mathscr{C}\left\lparen\gamma\right\rparen$, which we describe
below.

A curve $\displaystyle\gamma$ on $\displaystyle S$ determines a conjugacy
class of cyclic subgroups of $\displaystyle\pi_{1}\left\lparen
S,s\right\rparen$, and so the Galois correspondence yields a based cover
$\displaystyle\Pi_{\gamma}\colon\left\lparen
S_{\gamma},s_{\gamma}\right\rparen\xrightarrow[]{}\left\lparen
S,s\right\rparen$. The induced homomorphism
$\displaystyle\left\lparen\Pi_{\gamma}\right\rparen_{*}\colon\pi_{1}\left\lparen
S_{\gamma},s_{\gamma}\right\rparen\xhookrightarrow{}\pi_{1}\left\lparen
S,s\right\rparen$ is injective with image the cyclic subgroup
$\displaystyle\left\langle\gamma\right\rangle$ generated by
$\displaystyle\gamma$, and so $\displaystyle S_{\gamma}$ is a surface of
finite type with infinite cyclic fundamental group
$\displaystyle\pi_{1}\left\lparen
S_{\gamma},s_{\gamma}\right\rparen\cong\left\langle\gamma\right\rangle$. In
particular, $\displaystyle S_{\gamma}$ is homeomorphic to an open annulus.

Fix a hyperbolic metric on $\displaystyle S$, which pulls back along
$\displaystyle\Pi_{\gamma}$ to a complete hyperbolic metric on $\displaystyle
S_{\gamma}$, and denote by $\displaystyle\overline{S_{\gamma}}$ the Gromov
compactification of $\displaystyle S_{\gamma}$, which is homeomorphic to a
closed annulus. $\displaystyle\mathscr{C}\left\lparen\gamma\right\rparen$ is a
flag simplicial complex whose vertices are path-homotopy classes of properly
embedded, non-peripheral arcs in $\displaystyle\overline{S_{\gamma}}$, any
pairwise disjoint finite collection of which spans a simplex. As in the case
of the curve complex of a closed surface, we equip
$\displaystyle\mathscr{C}\left\lparen\gamma\right\rparen$ with a metric
$\displaystyle d_{\gamma}$ so that every simplex is a regular Euclidean
simplex of side length $\displaystyle 1$.

Let $\displaystyle\mathrm{B}_{2}\left\lparen\gamma\right\rparen$ denote the
open ball of radius $\displaystyle 2$ in $\displaystyle\mathscr{C}\left\lparen
S\right\rparen$, so that $\displaystyle\mathscr{C}\left\lparen
S\right\rparen\setminus\mathrm{B}_{2}\left\lparen\gamma\right\rparen$ is the
sub-complex of $\displaystyle\mathscr{C}\left\lparen S\right\rparen$
consisting of simplices all of whose vertices intersect $\displaystyle\gamma$.
The pre-image in the cover $\displaystyle S_{\gamma}$ of the geodesic
representative of such a simplex
$\displaystyle\alpha\in\mathscr{C}\left\lparen
S\right\rparen\setminus\mathrm{B}_{2}\left\lparen\gamma\right\rparen$ contains
exactly $\displaystyle i\left\lparen\alpha,\gamma\right\rparen$ properly
embedded, non-peripheral geodesic arcs, all of which are disjoint. By choosing
one of these arcs, we obtain a coarsely well-defined map
$\displaystyle\mathrm{proj}_{\gamma}\colon\mathscr{C}\left\lparen
S\right\rparen\setminus\mathrm{B}_{2}\left\lparen\gamma\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen\gamma\right\rparen$,
called the _subsurface projection_ associated to $\displaystyle\gamma$.

The following result of Minsky–Taylor gives an explicit bound on the diameter
of the subsurface projection onto the annular complex of a non-simple curve.

######  Theorem 3.5 ([MT22, Theorem 1.3]).

$\displaystyle\mathrm{diam}_{\mathscr{C}\left\lparen\gamma\right\rparen}\left\lparen\mathrm{proj}_{\gamma}\left\lparen\mathscr{C}\left\lparen
S\right\rparen\setminus\mathrm{B}_{2}\left\lparen\gamma\right\rparen\right\rparen\right\rparen\leq
38$ for any curve $\displaystyle\gamma$ on a closed surface $\displaystyle S$
of negative Euler characteristic which is not a power of a simple curve.

The following corollary follows immediately from the fact that the quasi-
isometric embedding of curve complexes induced by a finite-degree cover is
distance non-increasing.

######  Corollary 3.6.

Let $\displaystyle p\colon X\xrightarrow[]{}S$ be a finite-degree cover of a
closed surface $\displaystyle S$ of negative Euler characteristic. Then

$\displaystyle\mathrm{diam}_{\mathscr{C}\left\lparen\widetilde{\alpha}\right\rparen}\Big{\lparen}\mathrm{proj}_{\widetilde{\alpha}}\big{\lparen}\widetilde{p}\left\lparen\mathscr{C}\left\lparen
S\right\rparen\right\rparen\setminus\mathrm{B}_{2}\left\lparen\widetilde{\alpha}\right\rparen\big{\rparen}\Big{\rparen}\leq
38$

for all simple curves
$\displaystyle\widetilde{\alpha}\in\mathscr{C}\left\lparen X\right\rparen$ for
which $\displaystyle\alpha\coloneqq
p\left\lparen\widetilde{\alpha}\right\rparen$ is not a power of a simple
curve.

###### Proof.

Fix basepoints $\displaystyle x\in X$ and $\displaystyle s\in S$ so that
$\displaystyle p\left\lparen x\right\rparen=s$. There are choices of cyclic
subgroups of $\displaystyle\pi_{1}\left\lparen X,x\right\rparen$ and
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ representing
$\displaystyle\widetilde{\alpha}$ and $\displaystyle\alpha$, respectively, so
that the induced homomorphism $\displaystyle p_{*}\colon\pi_{1}\left\lparen
X,x\right\rparen\xhookrightarrow{}\pi_{1}\left\lparen S,s\right\rparen$
restricts to an isomorphism of these cyclic subgroups. This implies that there
are choices of the corresponding covers
$\displaystyle\Pi_{\widetilde{\alpha}}\colon
X_{\widetilde{\alpha}}\xrightarrow[]{}X$ and $\displaystyle\Pi_{\alpha}\colon
S_{\alpha}\xrightarrow[]{}S$ so that $\displaystyle
X_{\widetilde{\alpha}}=S_{\alpha}$ and the following diagram commutes.

$\displaystyle{X_{\widetilde{\alpha}}=S_{\alpha}}$$\displaystyle{X}$$\displaystyle{S}$$\displaystyle\scriptstyle{\Pi_{\widetilde{\alpha}}}$$\displaystyle\scriptstyle{\Pi_{\alpha}}$$\displaystyle\scriptstyle{p}$

Recall that $\displaystyle\widetilde{p}$ depends on a choice of elevation of
every simple curve on $\displaystyle S$. If we choose the elevations of curves
intersecting $\displaystyle\alpha$ to intersect
$\displaystyle\widetilde{\alpha}$, then $\displaystyle\widetilde{p}$ restricts
to a map $\displaystyle\mathscr{C}\left\lparen
S\right\rparen\setminus\mathrm{B}_{2}\left\lparen\alpha\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen
X\right\rparen\setminus\mathrm{B}_{2}\left\lparen\widetilde{\alpha}\right\rparen$
so that the following diagram commutes.

$\displaystyle{\mathscr{C}\left\lparen
S\right\rparen\setminus\mathrm{B}_{2}\left\lparen\alpha\right\rparen}$$\displaystyle{\mathscr{C}\left\lparen
X\right\rparen\setminus\mathrm{B}_{2}\left\lparen\widetilde{\alpha}\right\rparen}$$\displaystyle{\mathscr{C}\left\lparen\alpha\right\rparen=\mathscr{C}\left\lparen\widetilde{\alpha}\right\rparen}$$\displaystyle\scriptstyle{\mathrm{proj}_{\alpha}}$$\displaystyle\scriptstyle{\widetilde{p}}$$\displaystyle\scriptstyle{\mathrm{proj}_{\widetilde{\alpha}}}$

In particular, 3.5 implies that

$\displaystyle\displaystyle\mathrm{diam}_{\mathscr{C}\left\lparen\widetilde{\alpha}\right\rparen}\left\lparen\mathrm{proj}_{\widetilde{\alpha}}\left\lparen\widetilde{p}\left\lparen\mathscr{C}\left\lparen
S\right\rparen\right\rparen\setminus\mathrm{B}_{2}\left\lparen\widetilde{\alpha}\right\rparen\right\rparen\right\rparen$
$\displaystyle\displaystyle=\mathrm{diam}_{\mathscr{C}\left\lparen\alpha\right\rparen}\left\lparen\mathrm{proj}_{\alpha}\left\lparen\mathscr{C}\left\lparen
S\right\rparen\setminus\mathrm{B}_{2}\left\lparen\alpha\right\rparen\right\rparen\right\rparen\leq
38.\qed$

We will also make use of the Bounded Geodesic Image theorem, which describes
the subsurface projection of a geodesic in the curve complex. Originally
proved by Masur–Minsky [MM00, Theorem 3.1], we will use the following
strengthening due to Webb.

######  Theorem 3.7 ([Web15, Corollary 1.3]).

There is an explicit positive real number $\displaystyle C>0$ so that

$\displaystyle\mathrm{diam}_{\mathscr{C}\left\lparen\gamma\right\rparen}\left\lparen\mathrm{proj}_{\gamma}\left\lparen
R\right\rparen\right\rparen\leq C$

for any simple curve $\displaystyle\gamma$ on a surface $\displaystyle S$ of
negative Euler characteristic and any geodesic ray $\displaystyle R$ in
$\displaystyle\mathscr{C}\left\lparen S\right\rparen$ so that $\displaystyle
R\subseteq\mathrm{B}\left\lparen\gamma\right\rparen$.

It follows from Webb’s argument that $\displaystyle C$ can be taken to be at
most $\displaystyle 1000$.

###  3.4. An explicit computation of M

We are now ready to prove Section 1 by computing the integer $\displaystyle M$
explicitly as a function of the Euler characteristic
$\displaystyle\left\lvert\chi\left\lparen S\right\rparen\right\rvert$ and the
degrees $\displaystyle\deg\left\lparen p\right\rparen$ and
$\displaystyle\deg\left\lparen q\right\rparen$.

Before proceeding, we give a brief sketch of the proof for the reader’s
convenience. Without loss of generality, throughout we will assume that
$\displaystyle\deg\left\lparen p\right\rparen\leq\deg\left\lparen
q\right\rparen$. First, in 1 and 2, we reduce to the case where
$\displaystyle\deg\left\lparen p\right\rparen$ and
$\displaystyle\deg\left\lparen q\right\rparen$ are equal by using well-known
asymptotics of the growth of simple closed curves with respect to length. In
particular, we show that if $\displaystyle\deg\left\lparen
p\right\rparen<\deg\left\lparen q\right\rparen$, then there is always a simple
closed curve on $\displaystyle Y$ whose image in $\displaystyle S$ has no
simple elevation to $\displaystyle X$, and that the self-intersection number
of such a curve can be explicitly bounded. This comprises the simplest part of
the argument, and the primary difficulty is confirming that all of the
relevant constants are explicitly computable.

In the case that $\displaystyle\deg\left\lparen
p\right\rparen=\deg\left\lparen q\right\rparen$, for each $\displaystyle g\in
G=\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$, we form an intermediate
cover, $\displaystyle W_{g}$, corresponding to the subgroup $\displaystyle
H_{g}$ generated by $\displaystyle
g\mathrm{Deck}\left\lparen\pi_{X}\right\rparen g^{-1}$ and
$\displaystyle\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen$ (see Figure 1 and
the surrounding text). Under the assumption that $\displaystyle p$ and
$\displaystyle q$ are non-isomorphic covers, it follows that $\displaystyle Y$
covers each $\displaystyle W_{g}$ non-trivially. In 3, we then find a simple
curve $\displaystyle\alpha$ on $\displaystyle X$ and a simple curve
$\displaystyle\eta$ on $\displaystyle S$ for which we can explicitly compute
bounds on the intersection number of elevations
$\displaystyle\widetilde{\alpha}$ and $\displaystyle\widetilde{\eta}$ of
$\displaystyle\alpha$ and $\displaystyle\eta$ to the cover $\displaystyle Z$,
and for which $\displaystyle p\left\lparen\alpha\right\rparen$ has no simple
elevations to $\displaystyle W_{g}$ for any $\displaystyle g\in G$.

Using work of Aougab-Taylor, we then construct a quasi-geodesic ray
$\displaystyle\widetilde{R}=\left\lparen\widetilde{v_{n}}\right\rparen_{n=0}^{\infty}$
in $\displaystyle\mathscr{C}\left\lparen Z\right\rparen$ emanating from
$\displaystyle\widetilde{\eta}$ where $\displaystyle\widetilde{v_{i}}$ and
$\displaystyle\widetilde{v_{j}}$ have explicitly bounded intersection number
for each $\displaystyle i,j\in\mathbb{N}$. The content of 4 is then to find
some computable universal constant $\displaystyle E\in\mathbb{N}$ so that
twisting $\displaystyle\widetilde{R}$ $\displaystyle E$ times around
$\displaystyle\widetilde{\alpha}$ makes positive progress away from the curve
complexes $\displaystyle\mathscr{C}\left\lparen W_{g}\right\rparen$; that is,
the distance from
$\displaystyle\tau_{\widetilde{\alpha}}^{E}\left\lparen\widetilde{v_{n}}\right\rparen$
to the quasi-isometrically embedded image of each
$\displaystyle\mathscr{C}\left\lparen W_{g}\right\rparen$ in
$\displaystyle\mathscr{C}\left\lparen Z\right\rparen$ is bounded below
linearly in $\displaystyle n$.

Combined with work of Tang, we then use this to show in 5 that if there is
some $\displaystyle g\in G$ for which $\displaystyle
g\cdot\tau_{\widetilde{\alpha}}^{E}\left\lparen\widetilde{v_{n}}\right\rparen$
lies in the quasi-isometrically embedded image of
$\displaystyle\mathscr{C}\left\lparen Y\right\rparen$ in
$\displaystyle\mathscr{C}\left\lparen Z\right\rparen$, then the index
$\displaystyle n$ is explicitly bounded. In particular, this claim implies
that if $\displaystyle n$ is large enough, $\displaystyle
g\cdot\tau_{\widetilde{\alpha}}^{E}\left\lparen\widetilde{v_{n}}\right\rparen$
cannot lie in this image and so for any such $\displaystyle n$,
$\displaystyle\gamma_{n}=\pi_{S}\left\lparen\tau_{\widetilde{\alpha}}^{E}\left\lparen\widetilde{v_{n}}\right\rparen\right\rparen$
will be a curve on $\displaystyle S$ with a simple elevation to $\displaystyle
X$ but no simple elevation to $\displaystyle Y$. Finally in 6, we use the work
of Aougab-Taylor and Webb to effectively compute an upper bound for the
intersection number of one such $\displaystyle\gamma_{n}$, completing the
proof of Section 1.

As in the proof of 3.2, we emphasize the importance of the existence of
certain explicitly computable numbers over that of their exact form computed
below. As such, we encourage the reader to skim the proof of each claim on a
first reading.

###### Proof of Section 1.

For ease of reference, let $\displaystyle\chi\coloneqq\chi\left\lparen
S\right\rparen$ and $\displaystyle d\coloneqq\max\left\lparen\deg\left\lparen
p\right\rparen,\deg\left\lparen q\right\rparen\right\rparen$. Fix basepoints
$\displaystyle x\in X$, $\displaystyle y\in Y$, and $\displaystyle s\in S$ so
that $\displaystyle p\left\lparen x\right\rparen=q\left\lparen
y\right\rparen=s$, and let

$\displaystyle K\coloneqq\bigcap_{\gamma\in\pi_{1}\left\lparen
S,s\right\rparen}{\gamma\left\lparen p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen\cap q_{*}\left\lparen\pi_{1}\left\lparen
Y,y\right\rparen\right\rparen\right\rparen\gamma^{-1}},$

be the normal core of $\displaystyle p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen\cap q_{*}\left\lparen\pi_{1}\left\lparen
Y,y\right\rparen\right\rparen$ in $\displaystyle\pi_{1}\left\lparen
S,s\right\rparen$. Then $\displaystyle K$ is a normal subgroup of
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ of explicitly bounded
index

$\displaystyle\displaystyle\left[{\pi_{1}\left\lparen
S,s\right\rparen}:{K}\right]$
$\displaystyle\displaystyle\leq\left[{\pi_{1}\left\lparen
S,s\right\rparen}:{p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen\cap q_{*}\left\lparen\pi_{1}\left\lparen
Y,y\right\rparen\right\rparen}\right]!$
$\displaystyle\displaystyle\leq\left\lparen\left[{\pi_{1}\left\lparen
S,s\right\rparen}:{p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen}\right]\cdot\left[{\pi_{1}\left\lparen
S,s\right\rparen}:{q_{*}\left\lparen\pi_{1}\left\lparen
Y,y\right\rparen\right\rparen}\right]\right\rparen!$
$\displaystyle\displaystyle=\left\lparen\deg\left\lparen
p\right\rparen\cdot\deg\left\lparen q\right\rparen\right\rparen!\leq d^{2}!$

which is contained in both $\displaystyle p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen$ and $\displaystyle
q_{*}\left\lparen\pi_{1}\left\lparen Y,y\right\rparen\right\rparen$. As in the
proof of [Aou+23, Theorem 3], there is a surjective homomorphism
$\displaystyle\rho\colon\pi_{1}\left\lparen S,s\right\rparen\twoheadrightarrow
G$ onto a finite group $\displaystyle G$ with kernel
$\displaystyle\mathrm{ker}\left\lparen\rho\right\rparen=K$. Let $\displaystyle
A\coloneqq\rho\left\lparen p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen\right\rparen$ and $\displaystyle
B\coloneqq\rho\left\lparen q_{*}\left\lparen\pi_{1}\left\lparen
Y,y\right\rparen\right\rparen\right\rparen$ be the corresponding subgroups of
$\displaystyle G$.

Up to re-labeling, we may assume without loss of generality that
$\displaystyle\left\lvert A\right\rvert\geq\left\lvert B\right\rvert$. For
each element $\displaystyle g\in G$, let $\displaystyle
H_{g}\coloneqq\left\langle gAg^{-1},B\right\rangle$. The Galois correspondence
between finite-index subgroups of $\displaystyle\pi_{1}\left\lparen
S,s\right\rparen$ and finite-degree based covers of $\displaystyle\left\lparen
S,s\right\rparen$ then yields for each element $\displaystyle g\in G$ a based
surface $\displaystyle\left\lparen W_{g},w_{g}\right\rparen$ and a commutative
diagram of based covers of the form found in Figure 1.

$\displaystyle{\pi_{1}\left\lparen
S,s\right\rparen}$$\displaystyle{\left\lparen
Z,z\right\rparen}$$\displaystyle{\rho^{-1}\left\lparen
H_{g}\right\rparen}$$\displaystyle{\left\lparen
X,x\right\rparen}$$\displaystyle{\left\lparen
Y,y\right\rparen}$$\displaystyle{p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen}$$\displaystyle{q_{*}\left\lparen\pi_{1}\left\lparen
Y,y\right\rparen\right\rparen}$$\displaystyle{\left\lparen
W_{g},w_{g}\right\rparen}$$\displaystyle{K}$$\displaystyle{\left\lparen
S,s\right\rparen}$$\displaystyle\scriptstyle{\pi_{S}}$$\displaystyle\scriptstyle{\pi_{X}}$$\displaystyle\scriptstyle{\pi_{g}}$$\displaystyle\scriptstyle{\pi_{Y}}$$\displaystyle\scriptstyle{p}$$\displaystyle\scriptstyle{q_{g}}$$\displaystyle\scriptstyle{q}$$\displaystyle\scriptstyle{r_{g}}$
Figure 1. Covers from the Galois correspondence

Note that the regular cover $\displaystyle\pi_{S}$ has explicitly bounded
degree

$\displaystyle\deg\left\lparen\pi_{S}\right\rparen=\left[{\pi_{1}\left\lparen
S,s\right\rparen}:{K}\right]\leq d^{2}!$

Moreover, since $\displaystyle K$ is normal in
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$, this cover is regular
with deck group

$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cong\frac{N_{\pi_{1}\left\lparen
S,s\right\rparen}\left\lparen K\right\rparen}{K}=\frac{\pi_{1}\left\lparen
S,s\right\rparen}{K}\cong\mathrm{im}\left\lparen\rho\right\rparen=G.$

Similarly, the covers $\displaystyle\pi_{X}\colon Z\xrightarrow[]{}X$,
$\displaystyle\pi_{Y}\colon Z\xrightarrow[]{}Y$, and
$\displaystyle\pi_{g}\colon Z\xrightarrow[]{}W_{g}$ are regular, with deck
groups $\displaystyle\mathrm{Deck}\left\lparen\pi_{X}\right\rparen\cong A$,
$\displaystyle\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen\cong B$,
$\displaystyle\mathrm{Deck}\left\lparen\pi_{g}\right\rparen\cong H_{g}$,
respectively. By [Aou+23, Proposition 3.5], we may identify the deck groups of
these covers by their isomorphic images in
$\displaystyle\mathrm{Mod}\left\lparen Z\right\rparen$.

For each element $\displaystyle g\in G$, the covering relations described in
2.2 yield from the above commutative diagram of finite-degree covering spaces
a commutative diagram of coarsely well-defined quasi-isometric embeddings of
corresponding curve complexes of the form found in Figure 2.

$\displaystyle{\mathscr{C}\left\lparen
Z\right\rparen}$$\displaystyle{\mathscr{C}\left\lparen
X\right\rparen}$$\displaystyle{\mathscr{C}\left\lparen
Y\right\rparen}$$\displaystyle{\mathscr{C}\left\lparen
W_{g}\right\rparen}$$\displaystyle{\mathscr{C}\left\lparen
S\right\rparen}$$\displaystyle\scriptstyle{\widetilde{\pi_{X}}}$$\displaystyle\scriptstyle{\widetilde{\pi_{Y}}}$$\displaystyle\scriptstyle{\widetilde{\pi_{g}}}$$\displaystyle\scriptstyle{\widetilde{q_{g}}}$$\displaystyle\scriptstyle{\widetilde{\pi_{S}}}$$\displaystyle\scriptstyle{\widetilde{p}}$$\displaystyle\scriptstyle{\widetilde{r_{g}}}$$\displaystyle\scriptstyle{\widetilde{q}}$
Figure 2. Quasi-isometric embeddings induced by finite-degree covering maps

We will later use the fact that an explicitly computable bound on the degree
of each cover yields explicit bounds on the quality of the above quasi-
isometric embeddings.

######  Claim 1.

$\displaystyle p$ and $\displaystyle q$ are isomorphic covers if and only if
$\displaystyle B=H_{g}$ for some element $\displaystyle g\in G$.

###### Proof of 1.

The Galois correspondence implies that $\displaystyle p$ and $\displaystyle q$
are isomorphic as covers of $\displaystyle S$ precisely when the subgroups
$\displaystyle p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen$ and $\displaystyle
q_{*}\left\lparen\pi_{1}\left\lparen Y,y\right\rparen\right\rparen$ are
conjugate in $\displaystyle\pi_{1}\left\lparen S,s\right\rparen$. Since
$\displaystyle G=\mathrm{Deck}(\pi_{S})$, this occurs precisely when the
images $\displaystyle\rho\left\lparen p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen\right\rparen=A$ and
$\displaystyle\rho\left\lparen q_{*}\left\lparen\pi_{1}\left\lparen
Y,y\right\rparen\right\rparen\right\rparen=B$ are conjugate in $\displaystyle
G$. Note that if $\displaystyle gAg^{-1}=B$ for some element $\displaystyle
g\in G$, then

$\displaystyle H_{g}=\left\langle gAg^{-1},B\right\rangle=\left\langle
B,B\right\rangle=B.$

Conversely, if $\displaystyle H_{g}=B$ for some element $\displaystyle g\in
G$, then $\displaystyle gAg^{-1}$ is a subgroup of $\displaystyle H_{g}=B$ of
size $\displaystyle\left\lvert gAg^{-1}\right\rvert=\left\lvert
A\right\rvert\geq\left\lvert B\right\rvert$, and so $\displaystyle
gAg^{-1}=B$. ∎

Thus we may assume that $\displaystyle B\subsetneq H_{g}$ for every element
$\displaystyle g\in G$, so that $\displaystyle p$ and $\displaystyle q$ are
not isomorphic as covers of $\displaystyle S$. In particular, this implies
that $\displaystyle\deg\left\lparen
q_{g}\right\rparen=\left[{H_{g}}:{B}\right]>1$ for each element $\displaystyle
g\in G$. It remains to find an explicitly computable positive integer
$\displaystyle M$ and a curve $\displaystyle\gamma$ on $\displaystyle S$ with
$\displaystyle i\left\lparen\gamma,\gamma\right\rparen\leq M$ which has a
simple elevation to one of $\displaystyle X$ and $\displaystyle Y$ but not to
both.

To that end, fix an arbitrary hyperbolic metric on $\displaystyle S$, which
pulls back along the covers described above to a hyperbolic metric on each
surface in question.

######  Claim 2.

If $\displaystyle\left\lvert A\right\rvert>\left\lvert B\right\rvert$, then
there is an explicitly computable positive integer $\displaystyle M_{1}$
satisfying the conclusions of Section 1; that is, $\displaystyle
i\left\lparen\gamma,\gamma\right\rparen\leq M_{1}$ for some curve
$\displaystyle\gamma$ on $\displaystyle S$ which has a simple elevation along
$\displaystyle q$ to $\displaystyle Y$ but no simple elevations along
$\displaystyle p$ to $\displaystyle X$.

###### Proof of 2.

Under these hypotheses, note that

$\displaystyle\displaystyle\deg\left\lparen q\right\rparen$
$\displaystyle\displaystyle=\left[{\pi_{1}\left\lparen
S\right\rparen}:{q_{*}\left\lparen\pi_{1}\left\lparen
Y,y\right\rparen\right\rparen}\right]=\left[{G}:{B}\right]>\left[{G}:{A}\right]=\left[{\pi_{1}\left\lparen
S\right\rparen}:{p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen}\right]=\deg\left\lparen p\right\rparen,$

and so 3.2 implies that there is an explicitly computable positive number
$\displaystyle L_{1}$ and a simple closed geodesic $\displaystyle\beta$ on
$\displaystyle Y$ of hyperbolic length at most $\displaystyle L_{1}$ so that
$\displaystyle\gamma\coloneqq q\left\lparen\beta\right\rparen$ has no simple
elevations along $\displaystyle p$ to $\displaystyle X$. In particular,

$\displaystyle L_{1}\coloneqq\max\left\lparen L_{0}\left\lparen
S\right\rparen,L_{0}\left\lparen X\right\rparen,L_{0}\left\lparen
Y\right\rparen,1+\frac{c_{2}\left\lparen X\right\rparen
d^{1+3d\left\lvert\chi\right\rvert}}{c_{1}\left\lparen
Y\right\rparen}\right\rparen.$

It suffices to find an explicitly computable bound on the self-intersection
number $\displaystyle i\left\lparen\gamma,\gamma\right\rparen$. To that end,
note that while $\displaystyle\gamma$ has no simple elevations along
$\displaystyle p$ to $\displaystyle X$, all of its elevations along
$\displaystyle\pi_{S}$ to $\displaystyle Z$ are simple, since
$\displaystyle\beta$ is simple on $\displaystyle Y$ and $\displaystyle\pi_{S}$
is regular. Thus 3.4 implies that $\displaystyle
i\left\lparen\gamma,\gamma\right\rparen$ is explicitly bounded; specifically,

$\displaystyle
i\left\lparen\gamma,\gamma\right\rparen\leq\frac{\deg\left\lparen\pi_{S}\right\rparen^{2}\ell_{S}\left\lparen\gamma\right\rparen}{4\mathrm{sinh}^{-1}\left\lparen\mathrm{csch}\left\lparen\frac{\deg\left\lparen\pi_{S}\right\rparen\ell_{S}\left\lparen\gamma\right\rparen}{2}\right\rparen\right\rparen}\leq\frac{\left\lparen
d^{2}!\right\rparen^{2}L_{1}}{4\mathrm{sinh}^{-1}\left\lparen\mathrm{csch}\left\lparen\frac{d^{2}!L_{1}}{2}\right\rparen\right\rparen}\eqqcolon
M_{1},$

as required. ∎

Thus we may assume for the remainder of this proof that
$\displaystyle\left\lvert A\right\rvert=\left\lvert B\right\rvert<\left\lvert
H_{g}\right\rvert$ for each element $\displaystyle g\in G$. Under these
assumptions, we will use the full power of 3.2 to show that
$\displaystyle\widetilde{\pi_{X}}\left\lparen\mathscr{C}\left\lparen
X\right\rparen\right\rparen$ is not contained in $\displaystyle\bigcup_{g\in
G}{\widetilde{\pi_{g}}\left\lparen\mathscr{C}\left\lparen
W_{g}\right\rparen\right\rparen}$, and in fact that the relative complement
contains a curve with explicitly bounded length.

######  Claim 3.

There are simple curves $\displaystyle\alpha\in\mathscr{C}\left\lparen
X\right\rparen$ and $\displaystyle\eta\in\mathscr{C}\left\lparen
S\right\rparen$ so that  (i) the intersection number $\displaystyle
i\left\lparen\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen,\widetilde{\pi_{S}}\left\lparen\eta\right\rparen\right\rparen$
is explicitly bounded; and  (ii) for all elements $\displaystyle g\in G$,
$\displaystyle p\left\lparen\alpha\right\rparen$ has no simple elevations
along $\displaystyle r_{g}$ to $\displaystyle W_{g}$.

###### Proof of 3.

Since $\displaystyle\left\lvert A\right\rvert<\left\lvert H_{g}\right\rvert$
for each element $\displaystyle g\in G$,

$\displaystyle\displaystyle\deg\left\lparen p\right\rparen$
$\displaystyle\displaystyle=\left[{\pi_{1}\left\lparen
S,s\right\rparen}:{p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen}\right]=\left[{G}:{A}\right]$
$\displaystyle\displaystyle>\left[{G}:{H_{g}}\right]=\left[{\pi_{1}\left\lparen
S,s\right\rparen}:{\left\lparen
r_{g}\right\rparen_{*}\left\lparen\pi_{1}\left\lparen
W_{g},w_{g}\right\rparen\right\rparen}\right]=\deg\left\lparen
r_{g}\right\rparen,$

and so 3.2 implies that there is a simple closed geodesic
$\displaystyle\alpha$ on $\displaystyle X$ so that for all elements
$\displaystyle g\in G$, $\displaystyle p\left\lparen\alpha\right\rparen$ has
no simple elevations along $\displaystyle r_{g}$ to $\displaystyle W_{g}$.
Moreover, $\displaystyle\alpha$ can be chosen to be of explicitly bounded
length $\displaystyle\ell_{X}\left\lparen\alpha\right\rparen\leq L_{2}$;
specifically,

$\displaystyle\displaystyle\ell_{X}\left\lparen\alpha\right\rparen$
$\displaystyle\displaystyle\leq\max\left\lparen\max_{g\in G}{L_{0}\left\lparen
W_{g}\right\rparen},L_{0}\left\lparen X\right\rparen,1+\max_{g\in
G}{\frac{\left\lvert G\right\rvert c_{2}\left\lparen
W_{g}\right\rparen\deg\left\lparen p\right\rparen^{1+3\deg\left\lparen
p\right\rparen\left\lvert\chi\right\rvert}}{c_{1}\left\lparen
X\right\rparen}}\right\rparen$
$\displaystyle\displaystyle\leq\max\left\lparen\max_{g\in G}{L_{0}\left\lparen
W_{g}\right\rparen},L_{0}\left\lparen
X\right\rparen,1+\frac{d^{2}!\left\lparen\max_{g\in G}{c_{2}\left\lparen
W_{g}\right\rparen}\right\rparen
d^{1+3d\left\lvert\chi\right\rvert}}{c_{1}\left\lparen
X\right\rparen}\right\rparen\eqqcolon L_{2}.$

Explicit bounds on $\displaystyle\deg\left\lparen\pi_{X}\right\rparen$ and
$\displaystyle\ell_{X}\left\lparen\alpha\right\rparen$ now give explicit
bounds on the elevation
$\displaystyle\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen$;
specifically,

(1)
$\ell_{Z}\left\lparen\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen\right\rparen\leq\deg\left\lparen\pi_{X}\right\rparen\cdot\ell_{X}\left\lparen\alpha\right\rparen\leq\deg\left\lparen\pi_{S}\right\rparen\cdot\ell_{X}\left\lparen\alpha\right\rparen\leq
d^{2}!\cdot L_{2}.$

By Bers’ theorem (see [Bus10, Theorem 5.1.2]), there is a simple closed
geodesic $\displaystyle\eta\in S$ of length at most $\displaystyle
4\mathrm{log}\left\lparen 4\pi\left\lvert\chi\right\rvert\right\rparen$. A
similar argument now gives an explicit bound on the length of the elevation
$\displaystyle\widetilde{\pi_{S}}\left\lparen\eta\right\rparen$. Explicitly,

$\displaystyle\ell_{Z}\left\lparen\widetilde{\pi_{S}}\left\lparen\eta\right\rparen\right\rparen\leq\deg\left\lparen\pi_{S}\right\rparen\cdot\ell_{S}\left\lparen\eta\right\rparen\leq
d^{2}!\cdot 4\mathrm{log}\left\lparen
4\pi\left\lvert\chi\right\rvert\right\rparen.$

3.3 now gives the following explicit bound on the intersection number
$\displaystyle
i\left\lparen\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen,\widetilde{\pi_{S}}\left\lparen\eta\right\rparen\right\rparen$:

$\displaystyle
i\left\lparen\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen,\widetilde{\pi_{S}}\left\lparen\eta\right\rparen\right\rparen\leq\frac{d^{2}!\cdot
4\mathrm{log}\left\lparen
4\left\lvert\chi\right\rvert\right\rparen}{2\mathrm{sinh}^{-1}\left\lparen\mathrm{csch}\left\lparen\frac{d^{2}!L_{2}}{2}\right\rparen\right\rparen}\coloneqq
K_{1},$

as required. ∎

Fix curves $\displaystyle\alpha$ and $\displaystyle\eta$ satisfying the
conclusions of 3. For ease of notation, we will denote by
$\displaystyle\widetilde{\alpha}$ and $\displaystyle\widetilde{\eta}$ the
elevations $\displaystyle\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen$
and $\displaystyle\widetilde{\pi_{S}}\left\lparen\eta\right\rparen$,
respectively. Note that Hempel’s bound of [Hem01, Lemma 2.1] gives an explicit
bound

(2) $d_{Z}\left\lparen\widetilde{\alpha},\widetilde{\eta}\right\rparen\leq
2\mathrm{log}_{2}\left\lparen
i\left\lparen\widetilde{\alpha},\widetilde{\eta}\right\rparen\right\rparen+2\leq
2\mathrm{log}_{2}\left\lparen K_{1}\right\rparen+2,$

on the distance $\displaystyle
d_{Z}\left\lparen\widetilde{\alpha},\widetilde{\eta}\right\rparen$. Moreover,
for each element $\displaystyle g\in G$, since
$\displaystyle\pi_{g}\left\lparen\widetilde{\alpha}\right\rparen$ is an
elevation of $\displaystyle p\left\lparen\alpha\right\rparen$ along
$\displaystyle r_{g}$ to $\displaystyle W_{g}$, it is non-simple. 3.6
therefore implies that

$\displaystyle\mathrm{diam}_{\mathscr{C}\left\lparen\widetilde{\alpha}\right\rparen}\left\lparen\mathrm{proj}_{\widetilde{\alpha}}\left\lparen\widetilde{\pi_{g}}\left\lparen\mathscr{C}\left\lparen
W_{g}\right\rparen\right\rparen\setminus\mathrm{B}_{2}\left\lparen\widetilde{\alpha}\right\rparen\right\rparen\right\rparen\leq
38.$

Let $\displaystyle\left\lparen v_{k}\right\rparen_{k=0}^{\infty}$ be a
sequence in $\displaystyle\mathscr{C}\left\lparen S\right\rparen$ with
$\displaystyle v_{0}=\eta$ and

(3) $d_{S}\left\lparen v_{j},v_{k}\right\rparen=\left\lvert
j-k\right\rvert,\quad i\left\lparen v_{j},v_{k}\right\rparen\leq\left\lparen
C+3\right\rparen^{2\max\left\lparen
j,k\right\rparen-5}\left\lparen\frac{3\left\lvert\chi\right\rvert}{2}\right\rparen^{\left\lvert
j-k\right\rvert+2}+f_{j,k}\left\lparen\left\lvert\chi\right\rvert\right\rparen,$

for all indices $\displaystyle j,k\in\mathbb{N}$, whose existence is
guaranteed by 2.1. We recall that the constant $\displaystyle C$ appearing
above is the fixed, computable universal constant appearing in 3.7.

For ease of reference, we will denote

$\displaystyle\displaystyle R$
$\displaystyle\displaystyle\coloneqq\left\lparen
v_{n}\right\rparen_{n=0}^{\infty}$
$\displaystyle\displaystyle\widetilde{v_{n}}$
$\displaystyle\displaystyle\coloneqq\widetilde{\pi_{S}}\left\lparen
v_{n}\right\rparen$ $\displaystyle\displaystyle\widetilde{R}$
$\displaystyle\displaystyle\coloneqq\left\lparen\widetilde{v_{n}}\right\rparen_{n=0}^{\infty}=\left\lparen\widetilde{\pi_{S}}\left\lparen
v_{n}\right\rparen\right\rparen_{n=0}^{\infty},$

and we will denote by
$\displaystyle\tau_{\widetilde{\alpha}}\in\mathrm{Mod}\left\lparen
Z\right\rparen$ a Dehn twist around $\displaystyle\widetilde{\alpha}$.

######  Claim 4.

There is a computable universal constant $\displaystyle E\in\mathbb{N}$ and an
explicitly computable constant $\displaystyle K_{2}\geq 1$ so that

$\displaystyle
d_{Z}\left\lparen\tau_{\widetilde{\alpha}}^{E}\left\lparen\widetilde{v_{n}}\right\rparen,\widetilde{\pi_{g}}\left\lparen\mathscr{C}\left\lparen
W_{g}\right\rparen\right\rparen\right\rparen\geq\frac{n}{K_{2}}-K_{2},$

for all indices $\displaystyle n\in\mathbb{N}$ and elements $\displaystyle
g\in G$.

###### Proof of 4.

We will find some $\displaystyle E$ so that for each $\displaystyle n$ and for
any $\displaystyle z\in\mathscr{C}\left\lparen W_{g}\right\rparen$, any
geodesic segment
$\displaystyle[\tau^{E}_{\widetilde{\alpha}}(\widetilde{v_{n}}),\widetilde{\pi_{g}}(z)]$
passes within distance $\displaystyle 1$ of $\displaystyle\widetilde{\alpha}$
in $\displaystyle\mathscr{C}(Z)$. The claim will follow since the confluence
of Equation 2 and Equation 3 show that $\displaystyle\widetilde{\alpha}$ is at
distance at least $\displaystyle|n-2\mathrm{log}_{2}(K_{1})-2|$ from
$\displaystyle\tau^{E}_{\widetilde{\alpha}}(\widetilde{v_{n}})$, where the
multiplicative constant comes from the quasi-geodesic constants in 3.1 which
depend explicitly on the degrees of the covers.

Let $\displaystyle E\coloneqq C+41$ and for ease of notation let
$\displaystyle\widetilde{z}=\widetilde{\pi_{g}}(z)$. For contradiction, we
will assume that a geodesic segment connecting
$\displaystyle\tau_{\widetilde{\alpha}}^{E}(\widetilde{v_{n}})$ and
$\displaystyle\widetilde{z}$ does not pass within distance $\displaystyle 1$
of $\displaystyle\widetilde{\alpha}$ from which we will violate the bound in
3.7. Note that 3.6 implies that

$\displaystyle\mathrm{diam}_{\mathscr{C}\left\lparen\widetilde{\alpha}\right\rparen}\left\lparen\mathrm{proj}_{\widetilde{\alpha}}\left\lparen\widetilde{\pi_{g}}\left\lparen\mathscr{C}\left\lparen
W_{g}\right\rparen\right\rparen\right\rparen\right\rparen=\mathrm{diam}_{\mathscr{C}\left\lparen\widetilde{\alpha}\right\rparen}\left\lparen\mathrm{proj}_{\widetilde{\alpha}}\left\lparen\widetilde{\pi_{g}}\left\lparen\mathscr{C}\left\lparen
W_{g}\right\rparen\right\rparen\setminus\mathrm{B}_{2}\left\lparen\widetilde{\alpha}\right\rparen\right\rparen\right\rparen\leq
38.$

Since given any $\displaystyle\kappa\in\mathscr{C}\left\lparen Z\right\rparen$
which lies outside of the $\displaystyle 1$-neighborhood of
$\displaystyle\widetilde{\alpha}$, the distance in the annular complex of
$\displaystyle\widetilde{\alpha}$ between $\displaystyle\kappa$ and
$\displaystyle\tau^{E}_{\widetilde{\alpha}}(\kappa)$ is at least
$\displaystyle E-2$, one has that

$\displaystyle\displaystyle
d_{\mathscr{C}\left\lparen\widetilde{\alpha}\right\rparen}(\mathrm{proj}_{\widetilde{\alpha}}(\tau^{E}_{\widetilde{\alpha}}(\widetilde{v_{n}})),\mathrm{proj}_{\widetilde{\alpha}}(\widetilde{z}))$
$\displaystyle\displaystyle\geq
d_{\mathscr{C}\left\lparen\widetilde{\alpha}\right\rparen}(\mathrm{proj}_{\widetilde{\alpha}}(\tau^{E}_{\widetilde{\alpha}}(\widetilde{v_{n}})),\mathrm{proj}_{\widetilde{\alpha}}(\widetilde{v_{n}}))-d_{\mathscr{C}\left\lparen\widetilde{\alpha}\right\rparen}(\mathrm{proj}_{\widetilde{\alpha}}(\widetilde{v_{n}}),\mathrm{proj}_{\widetilde{\alpha}}(\widetilde{z}))$
$\displaystyle\displaystyle\geq E-40,$

where the final line follows since
$\displaystyle\widetilde{v}_{n},\widetilde{z}\in\widetilde{\pi_{g}}(\mathscr{C}(W_{g}))$.
By choice of $\displaystyle E$, this lower bound is at least $\displaystyle
C+1$, contradicting 3.7. Therefore any geodesic connecting
$\displaystyle\tau^{E}_{\widetilde{\alpha}}(\widetilde{v_{n}})$ to
$\displaystyle\widetilde{z}$ passes through the $\displaystyle 1$-neighborhood
of $\displaystyle\widetilde{\alpha}$, as desired. ∎

For each index $\displaystyle n\in\mathbb{N}$, let
$\displaystyle\tau_{n}\coloneqq\tau_{\widetilde{\alpha}}^{E}\left\lparen\widetilde{v_{n}}\right\rparen$.
Note that
$\displaystyle\widetilde{\alpha},\widetilde{v_{n}}\in\widetilde{\pi_{X}}(\mathscr{C}(X))$
and therefore $\displaystyle\tau_{n}\in\widetilde{\pi_{X}}(\mathscr{C}(X))$.
Moreover, for each element $\displaystyle g\in G$, consider the nearest point
projection map $\displaystyle\nu_{g}\colon\mathscr{C}\left\lparen
Z\right\rparen\twoheadrightarrow\widetilde{\pi_{g}}\left\lparen\mathscr{C}\left\lparen
W_{g}\right\rparen\right\rparen$, which is coarsely well-defined, and let
$\displaystyle\beta_{g}\left\lparen\tau_{n}\right\rparen$ denote a
circumcenter of the $\displaystyle H_{g}$-orbit of $\displaystyle\tau_{n}$.
2.2 implies that

$\displaystyle
d_{Z}\left\lparen\beta_{g}\left\lparen\tau_{n}\right\rparen,\nu_{g}\left\lparen\tau_{n}\right\rparen\right\rparen\leq
D,$

for some positive real number $\displaystyle D$ which, as we will show in
Appendix B, can be explicitly bounded in terms of
$\displaystyle\left\lvert\chi\right\rvert$ and $\displaystyle d$. In
particular, note that

$\displaystyle\displaystyle\frac{n}{K_{2}}-K_{2}$
$\displaystyle\displaystyle\leq
d_{Z}\left\lparen\tau_{n},\widetilde{\pi_{g}}\left\lparen\mathscr{C}\left\lparen
W_{g}\right\rparen\right\rparen\right\rparen=d_{Z}\left\lparen\tau_{n},\nu_{g}\left\lparen\tau_{n}\right\rparen\right\rparen$
$\displaystyle\displaystyle\leq
d_{Z}\left\lparen\tau_{n},\beta_{g}\left\lparen\tau_{n}\right\rparen\right\rparen+d_{Z}\left\lparen\beta_{g}\left\lparen\tau_{n}\right\rparen,\nu_{g}\left\lparen\tau_{n}\right\rparen\right\rparen\leq\mathrm{diam}_{\mathscr{C}\left\lparen
Z\right\rparen}\left\lparen H_{g}\cdot\tau_{n}\right\rparen+D,$

so that $\displaystyle\mathrm{diam}_{\mathscr{C}\left\lparen
Z\right\rparen}\left\lparen
H_{g}\cdot\tau_{n}\right\rparen\geq\frac{n}{K_{2}}-K_{2}-D$.

######  Claim 5.

If $\displaystyle
g\cdot\tau_{n}\in\widetilde{\pi_{Y}}\left\lparen\mathscr{C}\left\lparen
Y\right\rparen\right\rparen$ for some element $\displaystyle g\in G$, then
$\displaystyle n$ is explicitly bounded; specifically, $\displaystyle n\leq
K_{2}\left\lparen d^{2}!+K_{2}+D\right\rparen$.

###### Proof of 5.

Since $\displaystyle
g\cdot\tau_{n}\in\widetilde{\pi_{Y}}\left\lparen\mathscr{C}\left\lparen
Y\right\rparen\right\rparen$, its orbit under $\displaystyle
B=\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen$ is a multi-curve on
$\displaystyle Z$; that is, $\displaystyle d_{Z}\left\lparen
b\cdot\left\lparen g\cdot\tau_{n}\right\rparen,g\cdot\tau_{n}\right\rparen\leq
1$ for all elements $\displaystyle b\in B$. Similarly, since
$\displaystyle\tau_{n}\in\widetilde{\pi_{X}}\left\lparen\mathscr{C}\left\lparen
X\right\rparen\right\rparen$, its orbit under $\displaystyle
A=\mathrm{Deck}\left\lparen\pi_{X}\right\rparen$ is a multi-curve on
$\displaystyle Z$, which implies that

$\displaystyle d_{Z}\left\lparen\left\lparen
gag^{-1}\right\rparen\cdot\left\lparen
g\cdot\tau_{n}\right\rparen,g\cdot\tau_{n}\right\rparen=d_{Z}\left\lparen
ga\cdot\tau_{n},g\cdot\tau_{n}\right\rparen=d_{Z}\left\lparen
a\cdot\tau_{n},\tau_{n}\right\rparen\leq 1.$

Since $\displaystyle\left\lparen gAg^{-1}\right\rparen\cup B$ generates
$\displaystyle H_{g}$, every element $\displaystyle h\in H_{g}$ can be written
as a word in $\displaystyle\left\lparen gAg^{-1}\right\rparen\cup B$ of length
at most $\displaystyle\left\lvert H_{g}\right\rvert$. In particular,

$\displaystyle d_{Z}\left\lparen h\cdot\left\lparen
g\cdot\tau_{n}\right\rparen,g\cdot\tau_{n}\right\rparen\leq d_{\left\lparen
gAg^{-1}\right\rparen\cup B}\left\lparen 1,h\right\rparen\leq\left\lvert
H_{g}\right\rvert\leq\left\lvert G\right\rvert\leq d^{2}!$

for all elements $\displaystyle h\in H_{g}$, and so

$\displaystyle n\leq K_{2}\cdot\mathrm{diam}_{\mathscr{C}\left\lparen
Z\right\rparen}\left\lparen
H_{g}\cdot\tau_{n}\right\rparen+K_{2}^{2}+DK_{2}\leq K_{2}\cdot\left\lvert
G\right\rvert+K_{2}^{2}+DK_{2}\leq K_{2}\left\lparen
d^{2}!+K_{2}+D\right\rparen,$

as required. ∎

Fix an explicitly computable natural number $\displaystyle K_{3}\in\mathbb{N}$
so that $\displaystyle K_{2}\cdot\left\lparen
d^{2}!+K_{2}+D\right\rparen+1<2^{K_{3}}+2$. Let $\displaystyle N\coloneqq
2^{K_{3}}+2$, and note that 5 implies that
$\displaystyle\tau_{N}\in\widetilde{\pi_{X}}\left\lparen\mathscr{C}\left\lparen
X\right\rparen\right\rparen$ but $\displaystyle
g\cdot\tau_{N}\notin\widetilde{\pi_{Y}}\left\lparen\mathscr{C}\left\lparen
Y\right\rparen\right\rparen$ for all elements $\displaystyle g\in G$. In
particular,
$\displaystyle\gamma\coloneqq\pi_{S}\left\lparen\tau_{N}\right\rparen$ has a
simple elevation to $\displaystyle X$ but no simple elevations to
$\displaystyle Y$. It therefore suffices to show that the self-intersection
number $\displaystyle i\left\lparen\gamma,\gamma\right\rparen$ is explicitly
bounded. In fact, we will compute an explicit bound on the length
$\displaystyle\ell_{S}\left\lparen\gamma\right\rparen$ and again argue via the
collar lemma.

######  Claim 6.

The self-intersection number $\displaystyle
i\left\lparen\gamma,\gamma\right\rparen$ is explicitly bounded; that is,
$\displaystyle i\left\lparen\gamma,\gamma\right\rparen\leq M_{2}$ for some
explicitly computable positive integer $\displaystyle M_{2}$.

###### Proof of 6.

Continuing to let $\displaystyle(v_{k})_{k=0}^{\infty}$ denote the sequence
from Theorem 2.2, we point out that the construction of the $\displaystyle
v_{k}$ is via an iterated Dehn twisting argument. In particular, following
[AT14, Section 5], the $\displaystyle v_{k}$ are defined so that
$\displaystyle v_{4}=T^{C+3}_{v_{3}}(v_{0})$ and for each $\displaystyle k\geq
2$, $\displaystyle v_{2^{k}+2}=T^{C+3}_{v^{2^{k-1}+2}}(v_{0})$ where
$\displaystyle C$ is the constant coming from the bounded geodesic image
theorem. The constant $\displaystyle C$ can be taken to be less than
$\displaystyle 938$ by work of Webb [Web15]. One can then compute inductively
that

$\displaystyle\displaystyle\ell_{S}(v_{4})$
$\displaystyle\displaystyle=\ell_{S}(T^{941}_{v_{3}}(v_{0}))\leq
941~{}\\!i(v_{3},v_{0})\ell_{S}(v_{3})+\ell_{S}(v_{0}),$
$\displaystyle\displaystyle\ell_{S}(v_{6})$
$\displaystyle\displaystyle=\ell_{S}(T^{941}_{v_{4}}(v_{0}))\leq
941~{}\\!i(v_{4},v_{0})\ell_{S}(v_{4})+\ell_{S}(v_{0}),$
$\displaystyle\displaystyle\vdots$
$\displaystyle\displaystyle\ell_{S}(v_{N_{K}})=\ell_{S}(v_{2^{K}+2})$
$\displaystyle\displaystyle=\ell_{S}(T^{941}_{v_{2^{K-1}+2}}(v_{0}))\leq
941~{}\\!i(v_{2^{K-1}+2},v_{0})\ell_{S}(v_{2^{K-1}+2})+\ell_{S}(v_{0}).$

Using the fact that $\displaystyle i(v_{k},v_{j})$ can be explicitly bounded
using Equation 3 we conclude inductively from the above equation that there is
an effectively computable upper bound on $\displaystyle\ell_{S}(v_{N_{K}})$
with leading term of the form

$\displaystyle\displaystyle
941~{}\\!i(v_{3},v_{0})\ell_{S}(v_{3})\prod_{k=1}^{K-1}941i(v_{2^{k}+2},v_{0})$
$\displaystyle\displaystyle\leq
941^{K+1}\left(\frac{3|\chi|}{2}\right)^{5}\ell_{S}(v_{3})\prod_{k=1}^{K-1}\left(941^{2^{k+1}-1}\left(\frac{3|\chi|}{2}\right)^{2^{k}+4}\right),$
$\displaystyle\displaystyle=941^{2^{K+1}-2}\left(\frac{3|\chi|}{2}\right)^{2^{K}+4K-1},$

where we assume that the product on the left side of the first equation is
empty whenever $\displaystyle K<2$. Consequently the same holds of
$\displaystyle\ell_{N_{K}}=\ell_{Z}(\widetilde{\pi_{S}}(v_{N_{K}}))$ since
$\displaystyle\ell_{Z}(\widetilde{\pi_{S}}(v_{N_{K}}))\leq|G|~{}\\!\ell_{S}(v_{N_{K}})$.
It then follows that

(4)
$\ell_{Z}(\tau^{E}_{\widetilde{\alpha}}(\widetilde{\pi_{S}}(v_{N_{K}})))\leq
E~{}\\!i(\widetilde{\pi_{S}}(v_{N_{K}}),\widetilde{\alpha})\ell_{Z}(\widetilde{\alpha})+\ell_{N_{K}}.$

We have an effective bound on the length of $\displaystyle\widetilde{\alpha}$
via Equation 1. Using 3.3 it therefore follows that we can effectively compute
an upper bound on $\displaystyle
i(\widetilde{\pi_{S}}(v_{N_{K}}),\widetilde{\alpha})$ and hence all of the
terms on the righthand side of Equation 4 are effectively computable and
therefore so is a bound on
$\displaystyle\ell_{Z}(\tau^{E}_{\widetilde{\alpha}}(\widetilde{\pi_{S}}(v_{N_{K}})))$.
Since
$\displaystyle\gamma=\pi_{S}(\tau^{E}_{\widetilde{\alpha}}(\widetilde{\pi_{S}}(v_{N_{K}})))$
it follows that

$\displaystyle\ell_{S}(\gamma)\leq\ell_{Z}(\tau^{E}_{\widetilde{\alpha}}(\widetilde{\pi_{S}}(v_{N_{K}}))).$

We are therefore done since the collar lemma again provides an effectively
computable upper bound for $\displaystyle i(\gamma,\gamma)$. ∎

Finally, let us summarize what we have shown. If $\displaystyle p$ and
$\displaystyle q$ are not isomorphic as covers of $\displaystyle S$, then
there are two cases; either $\displaystyle\left\lvert
A\right\rvert=\left\lvert B\right\rvert$, or $\displaystyle\left\lvert
A\right\rvert>\left\lvert B\right\rvert$. If $\displaystyle\left\lvert
A\right\rvert=\left\lvert B\right\rvert$, we showed that $\displaystyle
i\left\lparen\gamma,\gamma\right\rparen\leq M_{1}$ for some curve
$\displaystyle\gamma$ on $\displaystyle S$ which has a simple elevation along
$\displaystyle q$ to $\displaystyle Y$ but no simple elevations along
$\displaystyle p$ to $\displaystyle X$. On the other hand, if
$\displaystyle\left\lvert A\right\rvert>\left\lvert B\right\rvert$, we showed
that $\displaystyle i\left\lparen\gamma,\gamma\right\rparen\leq M_{2}$ for
some curve $\displaystyle\gamma$ on $\displaystyle S$ which has a simple
elevation along $\displaystyle p$ to $\displaystyle X$ but no simple
elevations along $\displaystyle q$ to $\displaystyle Y$. Therefore,
$\displaystyle M\coloneqq\max\left\lparen M_{1},M_{2}\right\rparen$ satisfies
the conclusions of Section 1. ∎

We remark that the above proof of Section 1 shows that so long as
$\displaystyle\alpha\in\mathscr{C}\left\lparen X\right\rparen$ is sufficiently
far from the pullback of each $\displaystyle\mathscr{C}\left\lparen
W_{g}\right\rparen$ in $\displaystyle\mathscr{C}\left\lparen Z\right\rparen$,
it cannot lie over a curve in $\displaystyle S$ that lifts simply to
$\displaystyle Y$. Indeed, being far away from each
$\displaystyle\mathscr{C}\left\lparen W_{g}\right\rparen$ guarantees that the
$\displaystyle\mathrm{Deck}\left\lparen\pi_{g}\right\rparen$-orbits of
$\displaystyle\widetilde{\alpha}$ are of large diameter. On the other hand, we
show that some of these orbits are relatively small when $\displaystyle
p\left\lparen\alpha\right\rparen$ has simple elevations along $\displaystyle
q$ to $\displaystyle Y$. Thus, in the case that the covers in question are not
isomorphic, our proof of Section 1 exhibits not only one curve on
$\displaystyle S$ with bounded self-intersection and which lifts simply to one
cover but not the other, but in fact shows that _“almost all”_ curves on
$\displaystyle S$ that lift simply to one cover do not lift simply to the
other.

We state an example of this phenomenon below, using slightly stronger
hypotheses on the covers $\displaystyle p$ and $\displaystyle q$ to simplify
the statement. Note that the hypothesis below on the subgroups of
$\displaystyle\pi_{1}\left\lparen S\right\rparen$ corresponding to the covers
$\displaystyle p$ and $\displaystyle q$ is satisfied precisely when there is
no intermediate cover $\displaystyle W\xrightarrow[]{}S$ that is covered by
both $\displaystyle X$ and $\displaystyle Y$. When there are no intermediate
covers, $\displaystyle S$ itself plays the role of each $\displaystyle W_{g}$.
The constant $\displaystyle D$ stated therein is explicitly computable using
the details in Appendix B and the main theorems of [APT22]. See Figure 3 for a
visual description of this result.

######  Theorem 3.8.

Suppose that (conjugates of) the subgroups of
$\displaystyle\pi_{1}\left\lparen S\right\rparen$ corresponding to the covers
$\displaystyle p$ and $\displaystyle q$ jointly generate
$\displaystyle\pi_{1}\left\lparen S\right\rparen$. Then there is an explicitly
computable $\displaystyle D$, depending only on
$\displaystyle\left\lvert\chi\left\lparen S\right\rparen\right\rvert$,
$\displaystyle\deg\left\lparen p\right\rparen$, and
$\displaystyle\deg\left\lparen q\right\rparen$, so that if
$\displaystyle\gamma$ is any simple closed curve on $\displaystyle X$
satisfying $\displaystyle
d_{X}\left\lparen\gamma,\widetilde{p}\left\lparen\mathscr{C}\left\lparen
S\right\rparen\right\rparen\right\rparen\geq D$, then $\displaystyle
p\left\lparen\gamma\right\rparen$ does not admit a simple elevation to
$\displaystyle Y$.

$\displaystyle\cdots$$\displaystyle\cdots$$\displaystyle\cdots$$\displaystyle\omega$$\displaystyle
p\left\lparen\omega\right\rparen$$\displaystyle\bullet$$\displaystyle\omega$$\displaystyle>D$$\displaystyle
X$$\displaystyle Y$$\displaystyle
S$$\displaystyle\widetilde{p}\left\lparen\mathscr{C}\left\lparen
S\right\rparen\right\rparen$$\displaystyle\mathscr{C}\left\lparen
X\right\rparen$ Figure 3. The only simple curves on $\displaystyle X$ whose
projections to $\displaystyle S$ admit simple elevations to $\displaystyle Y$
are those in the $\displaystyle D$-neighborhood of
$\displaystyle\widetilde{p}\left\lparen\mathscr{C}\left\lparen
S\right\rparen\right\rparen$ in $\displaystyle\mathscr{C}\left\lparen
X\right\rparen$, which is drawn above in green. For example, the simple curve
$\displaystyle\omega$ drawn on $\displaystyle X$ has distance more than
$\displaystyle D$ to
$\displaystyle\widetilde{p}\left\lparen\mathscr{C}\left\lparen
S\right\rparen\right\rparen$, and so $\displaystyle
p\left\lparen\omega\right\rparen$ has no simple elevations to $\displaystyle
Y$.

##  4\. Unmarked simple trace spectral rigidity

The goal of this section is to apply the proof of Section 1 to prove Section
1, which we restate below for the reader’s convenience.

*

We will prove an even stronger statement: if $\displaystyle p$ and
$\displaystyle q$ are not isomorphic, then for
$\displaystyle\mathbb{K}=\mathbb{R}$ and $\displaystyle\mathbb{C}$, the set of
linear representations $\displaystyle\pi_{1}\left\lparen
S\right\rparen\xrightarrow[]{}\mathrm{SL}_{N}\left\lparen\mathbb{K}\right\rparen$
whose restrictions to $\displaystyle\pi_{1}\left\lparen X\right\rparen$ and
$\displaystyle\pi_{1}\left\lparen Y\right\rparen$ are simple trace isospectral
has null Lebesgue measure in
$\displaystyle\mathrm{hom}\left\lparen\pi_{1}\left\lparen
S\right\rparen,\mathrm{SL}_{N}\left\lparen\mathbb{K}\right\rparen\right\rparen$,
and its complement is dense. Thus, the term “generic” in the above statement
of Section 1 can be interpreted both measure-theoretically and topologically.

In what follows, we will denote by $\displaystyle R_{\mathbb{K}}\left\lparen
S,N\right\rparen$ the _representation variety_
$\displaystyle\mathrm{hom}\left\lparen\pi_{1}\left\lparen
S\right\rparen,\mathrm{SL}_{N}\left\lparen\mathbb{K}\right\rparen\right\rparen$.
We will require the following theorem of Rapinchuk–Benyash-Krivetz–Chernousov.

######  Theorem 4.1 ([RBC96, Theorem 3]).

The representation variety $\displaystyle R_{\mathbb{K}}\left\lparen
S,N\right\rparen$ is an irreducible variety.

We will also require the following basic arithmetic lemma.

######  Lemma 4.2.

Given complex numbers $\displaystyle x,y,z\in\mathbb{C}$, if $\displaystyle
x\neq y$, $\displaystyle z\neq 0$, and $\displaystyle x+\frac{1}{z}=y+z$, then
$\displaystyle 2x+\frac{1}{z^{2}}\neq 2y+z^{2}$.

###### Proof.

If $\displaystyle z=\pm 1$, then the equation $\displaystyle
x+\frac{1}{z}=y+z$ implies that $\displaystyle x=y$, and so this contradiction
implies that $\displaystyle z\neq\pm 1$. Note that

$\displaystyle\displaystyle 2x+\frac{1}{z^{2}}$
$\displaystyle\displaystyle=2\left\lparen
x+\frac{1}{z}\right\rparen-\frac{2}{z}+\frac{1}{z^{2}}=2\left\lparen
y+z\right\rparen-\frac{2}{z}+\frac{1}{z^{2}}=2y+z^{2}-z^{2}+2z-\frac{2}{z}+\frac{1}{z^{2}}$
$\displaystyle\displaystyle=2y+z^{2}-\frac{z^{4}-2z^{3}+2z-1}{z^{2}}=2y+z^{2}-\frac{\left\lparen
z-1\right\rparen^{3}\left\lparen z+1\right\rparen}{z^{2}}.$

Since $\displaystyle z\neq\pm 1$, $\displaystyle\frac{\left\lparen
z-1\right\rparen^{3}\left\lparen z+1\right\rparen}{z^{2}}$ is non-zero, and so
$\displaystyle 2x+\frac{1}{z^{2}}\neq 2y+z^{2}$, as desired. ∎

###### Proof of Section 1.

Fix basepoints $\displaystyle s\in S$, $\displaystyle x\in X$, and
$\displaystyle y\in Y$ so that $\displaystyle s=p\left\lparen
x\right\rparen=q\left\lparen y\right\rparen$, and suppose that $\displaystyle
p\colon X\xrightarrow[]{}S$ and $\displaystyle q\colon Y\xrightarrow[]{}S$ are
non-isomorphic. The above proof of Section 1 implies that there is a curve
$\displaystyle\left[\gamma\right]$ whose word length in some chosen finite
generating set for $\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ is
explicitly bounded in terms of $\displaystyle\deg\left\lparen p\right\rparen$,
$\displaystyle\deg\left\lparen q\right\rparen$, and
$\displaystyle\left\lvert\chi\left\lparen S\right\rparen\right\rvert$, and
with a simple elevation along one of these covers but no simple elevations
along the other.

Without loss of generality, assume that $\displaystyle\left[\gamma\right]$ has
a simple elevation $\displaystyle\left[\alpha\right]$ along $\displaystyle p$
to $\displaystyle X$ but no simple elevations along $\displaystyle q$ to
$\displaystyle Y$; that is, there is some simple element
$\displaystyle\alpha\in\pi_{1}\left\lparen X,x\right\rparen$ so that
$\displaystyle p_{*}\left\lparen\alpha\right\rparen$ is conjugate to
$\displaystyle\gamma^{m}$ in $\displaystyle\pi_{1}\left\lparen
S,s\right\rparen$ for some integer $\displaystyle 1\leq m\leq d$. On the other
hand, for any simple element $\displaystyle\beta\in\pi_{1}\left\lparen
Y,y\right\rparen$, $\displaystyle q_{*}\left\lparen\beta\right\rparen$ is not
conjugate to $\displaystyle\gamma^{m}$ or $\displaystyle\gamma^{-m}$.

Let
$\displaystyle\Omega\coloneqq\left\\{\omega_{1},\dotsc,\omega_{r}\right\\}\subseteq\pi_{1}\left\lparen
S,s\right\rparen$ be a set of trace twins of $\displaystyle\gamma$ for which
the commensurability constants are $\displaystyle m$ and $\displaystyle 1$;
that is,

$\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen^{2}=\mathrm{tr}\left\lparen\rho\left\lparen\omega_{i}\right\rparen\right\rparen^{2},$

for all linear representations $\displaystyle\rho\in
R_{\mathbb{C}}\left\lparen S,2\right\rparen$ and indices $\displaystyle 1\leq
i\leq r$. In particular, [Lei03, Theorem 1.4] implies that $\displaystyle
m\ell_{\mu}\left\lparen\gamma\right\rparen=\ell_{\mu}\left\lparen\omega_{i}\right\rparen$
for all $\displaystyle\mu\in\mathscr{T}\left\lparen S\right\rparen$ and
indices $\displaystyle 1\leq i\leq r$.

By [Lei03, Corollary 3.4], we may assume that $\displaystyle\gamma^{m}$ and
$\displaystyle\omega_{i}$ are homologous (possibly replacing
$\displaystyle\omega_{i}$ with $\displaystyle\omega_{i}^{-1}$ in order to do
so), and we may furthermore choose $\displaystyle\Omega$ to be a maximal such
set subject to the additional condition that no two distinct elements are
conjugate in $\displaystyle\pi_{1}\left\lparen S,s\right\rparen$. Such a
maximal set is finite, since any closed hyperbolic surface has finitely many
distinct curves of length at most any finite bound.

######  Claim 1.

$\displaystyle\mathrm{det}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen=\mathrm{det}\left\lparen\rho\left\lparen\omega_{i}\right\rparen\right\rparen$
for any linear representation $\displaystyle\rho\colon\pi_{1}\left\lparen
S,s\right\rparen\xrightarrow[]{}\mathrm{GL}_{n}\left\lparen\mathbb{C}\right\rparen$
and any index $\displaystyle 1\leq i\leq r$.

###### Proof of 1.

The map $\displaystyle\mathrm{det}\circ\rho\colon\pi_{1}\left\lparen
S,s\right\rparen\xrightarrow[]{}\mathbb{C}^{\times}$ is a homomorphism to an
abelian group $\displaystyle\mathbb{C}^{\times}$, and so it factors through
the abelianization $\displaystyle\mathrm{H}_{1}\left\lparen
S;\mathbb{Z}\right\rparen$. The desired claim now follows immediately from our
choice of orientation of $\displaystyle\omega_{i}$ so that
$\displaystyle\gamma^{m}$ and $\displaystyle\omega_{i}$ are homologous. ∎

Since surface groups and free groups are conjugacy-separable (see [Mar07],
[Dye79]), there is for each index $\displaystyle 1\leq i\leq r$ a homomorphism
$\displaystyle\phi_{i}\colon\pi_{1}\left\lparen
S\right\rparen\xrightarrow[]{}G_{i}$ onto a finite group $\displaystyle G_{i}$
so that $\displaystyle\phi_{i}\left\lparen\gamma\right\rparen$ and
$\displaystyle\phi_{i}\left\lparen\omega_{i}\right\rparen$ are not conjugate
in $\displaystyle G_{i}$. Note that $\displaystyle\left\lvert
G_{i}\right\rvert$ may, in principle, be explicitly bounded in terms of the
word lengths of $\displaystyle\gamma$ and $\displaystyle\omega_{i}$, and that
the Milnor-Švarc lemma implies that the word length of
$\displaystyle\omega_{i}$ is explicitly bounded in terms of the word length of
$\displaystyle\gamma$.

Moreover, basic representation theory implies that there is for each index
$\displaystyle 1\leq i\leq r$ an irreducible representation
$\displaystyle\psi_{i}\colon
G_{i}\xrightarrow[]{}\mathrm{GL}_{N_{i}}\left\lparen\mathbb{C}\right\rparen$
such that

$\displaystyle\mathrm{tr}\left\lparen\psi_{i}\left\lparen\phi_{i}\left\lparen\gamma^{m}\right\rparen\right\rparen\right\rparen\neq\mathrm{tr}\left\lparen\psi_{i}\left\lparen\phi_{i}\left\lparen\omega_{i}\right\rparen\right\rparen\right\rparen,$

and whose dimension $\displaystyle N_{i}$ is explicitly bounded in terms of
the character table for $\displaystyle G_{i}$.

Fix $\displaystyle N\coloneqq\max\left\\{2,2N_{1}+1,\dotsc,2N_{r}+1\right\\}$.

######  Claim 2.

For any element $\displaystyle\delta\in\pi_{1}\left\lparen S,s\right\rparen$,
either $\displaystyle\delta$ is conjugate to $\displaystyle\gamma^{m}$, or
there is some linear representation $\displaystyle\rho\in
R_{\mathbb{C}}\left\lparen S,N\right\rparen$ such that
$\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen\neq\mathrm{tr}\left\lparen\rho\left\lparen\delta\right\rparen\right\rparen$.

###### Proof of 2.

Suppose that $\displaystyle\delta$ is not conjugate to
$\displaystyle\gamma^{m}$. There are three cases; either $\displaystyle\gamma$
and $\displaystyle\delta$ are trace twins with commensurability constants
$\displaystyle m$ and $\displaystyle 1$ and $\displaystyle\gamma^{m}$ and
$\displaystyle\delta$ are homologous, or they are trace twins with these
commensurability constants and $\displaystyle\gamma^{m}$ and
$\displaystyle\delta^{-1}$ are homologous, or they are not trace twins with
these commensurability constants. Suppose first that $\displaystyle\gamma$ and
$\displaystyle\delta$ are trace twins with commensurability constants
$\displaystyle m$ and $\displaystyle 1$; that is,

$\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen^{2}=\mathrm{tr}\left\lparen\rho\left\lparen\delta\right\rparen\right\rparen^{2}$

for all linear representations $\displaystyle\rho\in
R_{\mathbb{C}}\left\lparen S,2\right\rparen$, and that
$\displaystyle\gamma^{m}$ and $\displaystyle\delta$ are homologous. The
maximality of $\displaystyle\Omega$ implies that $\displaystyle\delta$ is
conjugate to $\displaystyle\omega_{i}$ for some index $\displaystyle 1\leq
i\leq r$. Consider the representation $\displaystyle\rho\in
R_{\mathbb{C}}\left\lparen S,N\right\rparen$ defined by the formula

(5)
$\rho\left\lparen\eta\right\rparen\coloneqq\psi_{i}\left\lparen\phi_{i}\left\lparen\eta\right\rparen\right\rparen\oplus\begin{pmatrix}\frac{1}{\mathrm{det}\left\lparen\psi_{i}\left\lparen\phi_{i}\left\lparen\eta\right\rparen\right\rparen\right\rparen}\end{pmatrix}\oplus\mathbf{I}_{N-N_{i}-1};$

that is, $\displaystyle\rho\left\lparen\eta\right\rparen$ is block diagonal,
with an $\displaystyle N_{i}\times N_{i}$ block given by
$\displaystyle\psi_{i}\left\lparen\phi_{i}\left\lparen\eta\right\rparen\right\rparen$,
a $\displaystyle 1\times 1$ block with entry
$\displaystyle\frac{1}{\mathrm{det}\left\lparen\psi_{i}\left\lparen\phi_{i}\left\lparen\eta\right\rparen\right\rparen\right\rparen}$,
and an $\displaystyle\left\lparen N-N_{i}-1\right\rparen\times\left\lparen
N-N_{i}-1\right\rparen$ identity block. Note that

$\displaystyle\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen$
$\displaystyle\displaystyle=\mathrm{tr}\left\lparen\psi_{i}\left\lparen\phi_{i}\left\lparen\gamma^{m}\right\rparen\right\rparen\right\rparen+\frac{1}{\mathrm{det}\left\lparen\psi_{i}\left\lparen\phi_{i}\left\lparen\eta\right\rparen\right\rparen\right\rparen}+N-N_{i}-1$
$\displaystyle\displaystyle\neq\mathrm{tr}\left\lparen\psi_{i}\left\lparen\phi_{i}\left\lparen\delta\right\rparen\right\rparen\right\rparen+\frac{1}{\mathrm{det}\left\lparen\psi_{i}\left\lparen\phi_{i}\left\lparen\delta\right\rparen\right\rparen\right\rparen}+N-N_{i}-1=\mathrm{tr}\left\lparen\rho\left\lparen\delta\right\rparen\right\rparen.$

Now suppose that $\displaystyle\gamma$ and $\displaystyle\delta$ are trace
twins with commensurability constants $\displaystyle m$ and $\displaystyle 1$,
but that $\displaystyle\gamma^{m}$ and $\displaystyle\delta^{-1}$ are
homologous. The maximality of $\displaystyle\Omega$ implies that
$\displaystyle\delta$ is conjugate to $\displaystyle\omega_{i}^{-1}$ for some
index $\displaystyle 1\leq i\leq r$. Consider the representation
$\displaystyle\rho,\rho^{\prime}\in R_{\mathbb{C}}\left\lparen
S,N\right\rparen$, where $\displaystyle\rho$ is defined by Equation 5, and
$\displaystyle\rho^{\prime}$ is defined by the formula

$\displaystyle\rho^{\prime}\left\lparen\eta\right\rparen\coloneqq\psi_{i}\left\lparen\phi_{i}\left\lparen\eta\right\rparen\right\rparen\oplus\psi_{i}\left\lparen\phi_{i}\left\lparen\eta\right\rparen\right\rparen\oplus\begin{pmatrix}\frac{1}{\mathrm{det}\left\lparen\psi_{i}\left\lparen\phi_{i}\left\lparen\eta\right\rparen\right\rparen\right\rparen^{2}}\end{pmatrix}\oplus\mathbf{I}_{N-2N_{i}-1}.$

Therefore either
$\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen\neq\mathrm{tr}\left\lparen\rho\left\lparen\delta\right\rparen\right\rparen$
or
$\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen=\mathrm{tr}\left\lparen\rho\left\lparen\delta\right\rparen\right\rparen$,
in which case 4.2 implies that
$\displaystyle\mathrm{tr}\left\lparen\rho^{\prime}\left\lparen\gamma^{m}\right\rparen\right\rparen\neq\mathrm{tr}\left\lparen\rho^{\prime}\left\lparen\delta\right\rparen\right\rparen$.
Either way we have a representation with the desired property.

Finally, suppose that $\displaystyle\gamma$ and $\displaystyle\delta$ are not
trace twins with commensurability constants $\displaystyle m$ and
$\displaystyle 1$. Then [Lei03, Theorem 1.4] implies that
$\displaystyle\left[\gamma\right]$ and $\displaystyle\left[\delta\right]$ are
not length twins with commensurability constants $\displaystyle m$ and
$\displaystyle 1$, and so

$\displaystyle\mathrm{tr}\left\lparen\rho^{\prime}\left\lparen\gamma^{m}\right\rparen\right\rparen^{2}\neq\mathrm{tr}\left\lparen\rho^{\prime}\left\lparen\delta\right\rparen\right\rparen^{2}$

for some linear representation
$\displaystyle\rho^{\prime}\colon\pi_{1}\left\lparen
S,s\right\rparen\xhookrightarrow{}\mathrm{SL}_{2}\left\lparen\mathbb{R}\right\rparen$.
Such a linear representation $\displaystyle\rho^{\prime}$ can be taken to be a
lift to $\displaystyle\mathrm{SL}_{2}\left\lparen\mathbb{R}\right\rparen$ of
any monodromy representation of a metric in which $\displaystyle\gamma^{m}$
and $\displaystyle\delta$ have different lengths. In particular,
$\displaystyle\mathrm{tr}\left\lparen\rho^{\prime}\left\lparen\gamma^{m}\right\rparen\right\rparen\neq\mathrm{tr}\left\lparen\rho^{\prime}\left\lparen\delta\right\rparen\right\rparen$.

Let
$\displaystyle\iota\colon\mathrm{SL}_{2}\left\lparen\mathbb{R}\right\rparen\xhookrightarrow{}\mathrm{SL}_{N}\left\lparen\mathbb{C}\right\rparen$
denote the reducible embedding defined by the formula
$\displaystyle\iota\left\lparen\mathbf{A}\right\rparen=\mathbf{A}\oplus\mathbf{I}_{N-2}$,
where $\displaystyle\mathbf{I}_{N-2}$ is the $\displaystyle\left\lparen
n-2\right\rparen\times\left\lparen n-2\right\rparen$ identity matrix. Note
that

$\displaystyle\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen$
$\displaystyle\displaystyle=\mathrm{tr}\left\lparen\iota\left\lparen\rho^{\prime}\left\lparen\gamma^{m}\right\rparen\right\rparen\right\rparen=\mathrm{tr}\left\lparen\rho^{\prime}\left\lparen\gamma^{m}\right\rparen\oplus\mathbf{I}_{N-2}\right\rparen=\mathrm{tr}\left\lparen\rho^{\prime}\left\lparen\gamma^{m}\right\rparen\right\rparen+n-2$
$\displaystyle\displaystyle\neq\mathrm{tr}\left\lparen\rho^{\prime}\left\lparen\delta\right\rparen\right\rparen+n-2=\mathrm{tr}\left\lparen\rho^{\prime}\left\lparen\delta\right\rparen\oplus\mathbf{I}_{n-2}\right\rparen=\mathrm{tr}\left\lparen\iota\left\lparen\rho^{\prime}\left\lparen\delta\right\rparen\right\rparen\right\rparen=\mathrm{tr}\left\lparen\rho\left\lparen\delta\right\rparen\right\rparen,$

where
$\displaystyle\rho\coloneqq\iota\circ\rho^{\prime}\colon\pi_{1}\left\lparen
S,s\right\rparen\xhookrightarrow{}\mathrm{SL}_{N}\left\lparen\mathbb{C}\right\rparen$.
∎

We are now ready to show that $\displaystyle X$ and $\displaystyle Y$ are
generically simple trace non-isospectral over $\displaystyle S$ for linear
representations $\displaystyle\pi_{1}\left\lparen
S,s\right\rparen\xrightarrow[]{}\mathrm{SL}_{N}\left\lparen\mathbb{K}\right\rparen$,
where $\displaystyle\mathbb{K}=\mathbb{R}$ or $\displaystyle\mathbb{C}$.

Let $\displaystyle\mathscr{U}_{\mathbb{K}}\subseteq R_{\mathbb{K}}\left\lparen
S,N\right\rparen$ be the set of linear representations
$\displaystyle\pi_{1}\left\lparen
S,s\right\rparen\xrightarrow[]{}\mathrm{SL}_{N}\left\lparen\mathbb{C}\right\rparen$
whose restrictions to $\displaystyle\pi_{1}\left\lparen X,x\right\rparen$ and
$\displaystyle\pi_{1}\left\lparen Y,y\right\rparen$ along $\displaystyle
p_{*}$ and $\displaystyle q_{*}$, respectively, are simple trace isospectral.
We will show that $\displaystyle\mathscr{U}_{\mathbb{K}}$ has null Lebesgue
measure in $\displaystyle R_{\mathbb{K}}\left\lparen S,N\right\rparen$. To
that end, for each element $\displaystyle\delta\in\pi_{1}\left\lparen
S,s\right\rparen$ not conjugate to $\displaystyle\gamma^{m}$, let
$\displaystyle Z_{\mathbb{K}}\left\lparen\delta\right\rparen$ be the zero set
of the function
$\displaystyle\mathrm{tr}\left\lparen\gamma^{m}\right\rparen-\mathrm{tr}\left\lparen\delta\right\rparen$
on $\displaystyle R_{\mathbb{K}}\left\lparen S,N\right\rparen$; that is,

$\displaystyle
Z_{\mathbb{K}}\left\lparen\delta\right\rparen\coloneqq\left\\{\rho\in
R_{\mathbb{K}}\left\lparen
S,N\right\rparen:\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen=\mathrm{tr}\left\lparen\rho\left\lparen\delta\right\rparen\right\rparen\right\\}.$

2 above shows that the regular function
$\displaystyle\mathrm{tr}\left\lparen\gamma^{m}\right\rparen-\mathrm{tr}\left\lparen\delta\right\rparen$
on the irreducible variety $\displaystyle R_{\mathbb{C}}\left\lparen
S,N\right\rparen$ is not identically zero. The identity theorem for
holomorphic functions implies that it is also not identically zero on the real
points $\displaystyle R_{\mathbb{R}}\left\lparen S,N\right\rparen$.

######  Claim 3.

For any element $\displaystyle\delta\in\pi_{1}\left\lparen S,s\right\rparen$
not conjugate to $\displaystyle\gamma^{m}$, $\displaystyle
Z_{\mathbb{K}}\left\lparen\delta\right\rparen$ is of positive codimension in
$\displaystyle R_{\mathbb{K}}\left\lparen S,N\right\rparen$.

###### Proof of 3.

The codimension of $\displaystyle
Z_{\mathbb{K}}\left\lparen\delta\right\rparen$ in $\displaystyle
R_{\mathbb{K}}\left\lparen S,N\right\rparen$ is the height of the principal
ideal generated by
$\displaystyle\mathrm{tr}\left\lparen\gamma^{m}\right\rparen-\mathrm{tr}\left\lparen\delta\right\rparen$.
Since this function is not nilpotent and $\displaystyle
R_{\mathbb{K}}\left\lparen S,N\right\rparen$ is irreducible by 4.1, this
height is positive. ∎

An immediate consequence of 3 is that for each element
$\displaystyle\delta\in\pi_{1}\left\lparen S,s\right\rparen$ not conjugate to
$\displaystyle\gamma^{m}$, $\displaystyle
Z_{\mathbb{K}}\left\lparen\delta\right\rparen$ is closed and of null Lebesgue
measure in $\displaystyle R_{\mathbb{K}}\left\lparen S,N\right\rparen$, and
its complement $\displaystyle R_{\mathbb{K}}\left\lparen
S,N\right\rparen\setminus Z_{\mathbb{K}}\left\lparen\delta\right\rparen$ is
open and dense. In particular, as a countable union of null measure sets,

$\displaystyle
Z_{\mathbb{K}}\coloneqq\bigcup_{\delta}{Z_{\mathbb{K}}\left\lparen\delta\right\rparen},$

has null Lebesgue measure, where in the above union, $\displaystyle\delta$
varies over all elements of $\displaystyle\pi_{1}\left\lparen
S,s\right\rparen$ not conjugate to $\displaystyle\gamma^{m}$. Moreover, its
complement

$\displaystyle R_{\mathbb{K}}\left\lparen S,N\right\rparen\setminus
Z_{\mathbb{K}}=R_{\mathbb{K}}\left\lparen
S,N\right\rparen\setminus\bigcup_{\delta}{Z_{\mathbb{K}}\left\lparen\delta\right\rparen}=\bigcap_{\delta}{R_{\mathbb{K}}\left\lparen
S,N\right\rparen\setminus Z_{\mathbb{K}}\left\lparen\delta\right\rparen},$

is dense in $\displaystyle R_{\mathbb{K}}\left\lparen S,N\right\rparen$, since
$\displaystyle R_{\mathbb{K}}\left\lparen S,N\right\rparen$ is a Baire space.
It therefore suffices to show that
$\displaystyle\mathscr{U}_{\mathbb{K}}\subseteq Z_{\mathbb{K}}$.

To that end, let $\displaystyle\rho\in\mathscr{U}_{\mathbb{K}}$ and
$\displaystyle
T\coloneqq\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen\in\mathbb{K}$.
Since

$\displaystyle\mathrm{tr}\left\lparen\left\lparen\rho\circ
p_{*}\right\rparen\left\lparen\alpha\right\rparen\right\rparen=\mathrm{tr}\left\lparen\rho\left\lparen
p_{*}\left\lparen\alpha\right\rparen\right\rparen\right\rparen=\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen=T,$

and $\displaystyle\rho\circ p_{*}$ and $\displaystyle\rho\circ q_{*}$ are
simple trace isospectral, there is a simple element
$\displaystyle\beta\in\pi_{1}\left\lparen S,s\right\rparen$ so that
$\displaystyle\mathrm{tr}\left\lparen\left\lparen\rho\circ
q_{*}\right\rparen\left\lparen\beta\right\rparen\right\rparen=T$. Let
$\displaystyle\delta\coloneqq
q_{*}\left\lparen\beta\right\rparen\in\pi_{1}\left\lparen S,s\right\rparen$,
and note that

$\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\gamma^{m}\right\rparen\right\rparen=\mathrm{tr}\left\lparen\rho\left\lparen
p_{*}\left\lparen\alpha\right\rparen\right\rparen\right\rparen=T=\mathrm{tr}\left\lparen\rho\left\lparen
q_{*}\left\lparen\beta\right\rparen\right\rparen\right\rparen=\mathrm{tr}\left\lparen\rho\left\lparen\delta\right\rparen\right\rparen.$

On the other hand, $\displaystyle\delta$ cannot be conjugate to
$\displaystyle\gamma^{m}$, since there are no simple elevations of
$\displaystyle\gamma$ along $\displaystyle q$ to $\displaystyle Y$. Thus
$\displaystyle\rho\in Z_{\mathbb{K}}\left\lparen\delta\right\rparen\subseteq
Z_{\mathbb{K}}$. In particular,
$\displaystyle\mathscr{U}_{\mathbb{K}}\subseteq Z_{\mathbb{K}}$, and so
$\displaystyle\mathscr{U}_{\mathbb{K}}$ has null Lebesgue measure in
$\displaystyle R_{\mathbb{K}}\left\lparen S,N\right\rparen$, and its
complement $\displaystyle R_{\mathbb{K}}\left\lparen
S,N\right\rparen\setminus\mathscr{U}_{\mathbb{K}}$ is dense. ∎

##  Appendix A Effectivizing an argument of Rivin

In this section, we effectivize the result of Rivin [Riv01] to show that the
constants $\displaystyle c_{1}(\Sigma)$, $\displaystyle c_{2}(\Sigma)$ are
effectively computable. We first restate the theorem for the reader’s
convenience.

*

In what follows, we use the strategy of [Riv01] to build the relevant
constants recursively from the base case of the sphere with four boundary
components $\displaystyle\Sigma_{0,4}$. We outline an effective computation
for the constants in the case of the four-holed sphere, and then we note that
Rivin’s proof for general surfaces allows for a recursive computation of
constants from surface to surface. It follows that given explicit constants in
the four-holed sphere case as input, $\displaystyle c_{1}(\Sigma)$ and
$\displaystyle c_{2}(\Sigma)$ are recursively computable, as desired.

For the upper bound $\displaystyle c_{2}(\Sigma)$ where $\displaystyle\Sigma$
is some finite type hyperbolic surface, fix a Bers pants decomposition
$\displaystyle\mathcal{P}$ on $\displaystyle\Sigma$, and let
$\displaystyle\epsilon$ denote the injectivity radius of
$\displaystyle\Sigma$. Recall that $\displaystyle\mathcal{P}$ has total
geodesic length at most some $\displaystyle B$, where $\displaystyle B$
depends only on the topology of $\displaystyle\Sigma$. Then by basic
hyperbolic geometry, there is some constant
$\displaystyle\tau=\tau(\epsilon,B)$ so that the minimum length of a geodesic
arc connecting two pants curves (or, one pants curve to itself) is at least
$\displaystyle\tau$. Then given a simple closed geodesic $\displaystyle\alpha$
with length at most $\displaystyle L$, it follows that

$\displaystyle i(\alpha,\mathcal{P})\leq L/\tau,$

and moreover, $\displaystyle\alpha$ can twist about a given pants curve at
most $\displaystyle L/B$ times. It therefore follows that every Dehn-Thurston
coordinate of $\displaystyle\alpha$ has absolute value bounded above by
$\displaystyle L/B+L/\tau$. Thus, $\displaystyle\mathcal{N}(L,\Sigma)$ is at
most the number of integer points in a cube in
$\displaystyle\mathbb{R}^{\mathrm{dim}(\mathscr{T}(\Sigma))}$, which is

$\displaystyle\left(L/\tau+L/B\right)^{6g-6+b+2c}.$

One thus obtains a bound on $\displaystyle c_{2}$ of the form

$\displaystyle c_{2}(\Sigma)\leq\left(1/\tau+1/B\right)^{6g-6+b+2c}.$

We next turn to the lower bound $\displaystyle c_{1}$. The reader familiar
with [Riv01] may notice some subtle differences between Rivin’s work and the
numerology that follows. Since we could not fully reconcile the proof in
[Riv01], we instead opted to describe our understanding of how to implement
the basic strategy.

$\displaystyle E$$\displaystyle E$$\displaystyle E$$\displaystyle
E$$\displaystyle+$$\displaystyle=$$\displaystyle P_{1}$$\displaystyle
P_{1}$$\displaystyle P_{2}$$\displaystyle
P_{2}$$\displaystyle\gamma_{1}$$\displaystyle\gamma_{2}$$\displaystyle\gamma_{p/2k}$
Figure 4. The construction of $\displaystyle\gamma_{p/2k}$ (with
$\displaystyle p=3$ and $\displaystyle k=4$)

In what follows, we use $\displaystyle E$ to denote a simple separating curve
cutting $\displaystyle\Sigma_{0,4}$ into two pair of pants
$\displaystyle\mathcal{P}_{1}$, $\displaystyle\mathcal{P}_{2}$ glued along
$\displaystyle E$. Note that any simple closed curve on
$\displaystyle\Sigma_{0,4}$ is either a power of a boundary component or is
obtained by the following procedure (an example of which is drawn in Figure
4):

  * •

Fix $\displaystyle k\in\mathbb{N}$ and $\displaystyle p\in\mathbb{Z}$,

  * •

For $\displaystyle i\in\\{1,2\\}$, let $\displaystyle\gamma_{i}$ be
$\displaystyle k$ copies of the seam $\displaystyle\eta$ connecting
$\displaystyle E$ to $\displaystyle E$ and separating the other boundary
components in $\displaystyle\mathcal{P}_{i}$, where we ask that each of the
$\displaystyle k$ seams has a distinct endpoint on $\displaystyle E$ and that
the seams are pairwise disjoint,

  * •

Glue $\displaystyle\gamma_{1}$ to $\displaystyle\gamma_{2}$ using a
$\displaystyle p/2k$-twist along $\displaystyle E$, where this twist is
clockwise if $\displaystyle p/2k>0$ and this twist is counter-clockwise if
$\displaystyle p/2k<0$.

We will denote a curve obtained by this procedure by
$\displaystyle\gamma_{p/2k}$ and note that this procedure gives a simple
closed curve precisely when $\displaystyle\mathrm{gcd}(p,2k)=1$. Recall that
for any fixed $\displaystyle k\in\mathbb{N}$, $\displaystyle\phi(2k)$ is the
number of $\displaystyle 1\leq p\leq 2k$ such that
$\displaystyle\mathrm{gcd}(p,2k)=1$ and hence for any $\displaystyle
N\in\mathbb{N}$

$\displaystyle\left|\left\\{\gamma_{p/2k}\mid\left|\frac{p}{2k}\right|<N\right\\}\right|=\begin{cases}4N-1,&k=1,\\\
2\phi(2k)N,&k\neq 1.\end{cases}$

Indeed the $\displaystyle k\neq 1$ case follows because there are precisely
$\displaystyle 2\phi(2k)$ possible fractional twists (twists with
$\displaystyle-1<p/2k<1$), $\displaystyle\phi(2k)$ positive and
$\displaystyle\phi(2k)$ negative, and therefore there are $\displaystyle
2\phi(2k)N$ curves $\displaystyle\gamma_{p/2k}$ such that
$\displaystyle-N<p/2k<N$. The $\displaystyle k=1$ case differs because only
half integer twists are allowed and so there are $\displaystyle 4N-1$ half
integers with absolute value less than $\displaystyle N$. Note that in either
case, the size of this set if bounded below by $\displaystyle 2\phi(2k)N$.

Now let $\displaystyle m$ by a hyperbolic metric on
$\displaystyle\Sigma_{0,4}$ and note that the length of
$\displaystyle\gamma_{p/2k}$ is bounded above by

(6) $\ell(\gamma_{p/2k})\leq 2k\ell(\eta)+|p|\ell(E),$

where $\displaystyle\eta$ is the geodesic representative of the aforementioned
seam. Let $\displaystyle\mathcal{N}_{0,4}(L,m)$ denote the size of the set of
isotopy classes of non-boundary parallel simple closed geodesics of length at
most $\displaystyle L$. Suppose moreover that we require that $\displaystyle
m$ is such that $\displaystyle\ell(\eta)\leq\ell(E)$. Then it follows from
Equation 6 that

$\displaystyle\ell(\gamma_{p/2k})\leq(2k+|p|)\ell(E).$

Fix some $\displaystyle L\in\mathbb{R}$, then if $\displaystyle N$ is any
natural number for which $\displaystyle N+1\leq\frac{L}{2k\ell(E)}$, it
follows that if $\displaystyle|p/2k|=|p|/2k<N$ then

$\displaystyle\ell(\gamma_{p/2k})\leq(2k+|p|)\ell(E)<2k(N+1)\ell(E)\leq L.$

Moreover, by the calculation from the previous paragraph, there are at least

$\displaystyle\left\lfloor\frac{L}{2k\ell(E)}-1\right\rfloor\cdot 2\phi(2k),$

such curves. Note that the maximum $\displaystyle k$ for which
$\displaystyle\ell(\gamma_{p/2k})\leq\ell$ for all $\displaystyle-1<p/2k<1$ is
$\displaystyle\left\lfloor\frac{L}{4\ell(E)}\right\rfloor$. Combining the
above, it follows that

$\displaystyle\mathcal{N}_{0,4}(L,m)=\\{\gamma\mid\gamma\text{ is a simple
closed curve on }\Sigma_{0,4}\\}\geq
A(L)=\sum_{k=1}^{\left\lfloor\frac{L}{4\ell(E)}\right\rfloor}\left\lfloor\frac{L}{2k\ell(E)}-1\right\rfloor\cdot
2\phi(2k).$

We now assume that $\displaystyle L$ is a multiple of $\displaystyle 4\ell(E)$
so that

$\displaystyle\mathcal{N}_{0,4}(L,m)\geq
A(L)=\sum_{k=1}^{\frac{L}{4\ell(E)}}\left(\frac{L}{2k\ell(E)}-1\right)\cdot
2\phi(2k),$

then importantly $\displaystyle A(L)$ is asymptotic to
$\displaystyle\frac{3\pi^{2}}{2\ell(E)^{2}}L^{2}$. One can show this
rigorously using some basic number theoretic calculations however, because we
want an effective theorem, we will not prove this asymptotic directly but
rather content ourselves to find a lower bound on $\displaystyle A(L)$ which
is quadratic in $\displaystyle L$ for large enough $\displaystyle L$.

For ease of notation define $\displaystyle x=\frac{L}{4\ell(E)}$, then
decomposing $\displaystyle A(L)$ as

$\displaystyle
A(L)=A^{+}(L)-A^{-}(L)=4x\sum_{k=1}^{x}\frac{\phi(2k)}{k}-2\sum_{k=1}^{x}\phi(2k),$

then it is straightforward that

$\displaystyle A^{-}(L)=2\sum_{k=1}^{x}\phi(2k)\leq
4\sum_{k=1}^{x}\phi(k)=4\Phi(x).$

By the Möbius inversion formula,

$\displaystyle\Phi(x)=\sum_{k=1}^{x}\phi(k)=\frac{1}{2}\sum_{k=1}^{x}\mu(k)\left\lfloor\frac{x}{k}\right\rfloor\left(\left\lfloor\frac{x}{k}\right\rfloor+1\right)=\frac{1}{2}\sum_{k=1}^{x}\mu(k)\left\lfloor\frac{x}{k}\right\rfloor^{2}+\frac{1}{2}\sum_{k=1}^{x}\mu(k)\left\lfloor\frac{x}{k}\right\rfloor.$

Note that

$\displaystyle\frac{1}{2}\sum_{k=1}^{x}\mu(k)\left\lfloor\frac{x}{k}\right\rfloor\leq\frac{1}{2}\sum_{k=1}^{x}\frac{x}{k}\leq\frac{1}{2}x(\mathrm{ln}(x)+1).$

Similarly,

$\displaystyle\displaystyle\frac{1}{2}\sum_{k=1}^{x}\mu(k)\left\lfloor\frac{x}{k}\right\rfloor^{2}$
$\displaystyle\displaystyle=\frac{1}{2}\sum_{k=1}^{x}\mu(k)\left(\left\lfloor\frac{x}{k}\right\rfloor^{2}-\left(\frac{x}{k}\right)^{2}\right)+\frac{1}{2}\sum_{k=1}^{x}\mu(k)\left(\frac{x}{k}\right)^{2},$
$\displaystyle\displaystyle=\frac{1}{2}\sum_{k=1}^{x}\mu(k)\left(\left\lfloor\frac{x}{k}\right\rfloor-\frac{x}{k}\right)\left(\left\lfloor\frac{x}{k}\right\rfloor+\frac{x}{k}\right)+\frac{x^{2}}{2}\sum_{k=1}^{x}\frac{\mu(k)}{k^{2}},$
$\displaystyle\displaystyle\leq\sum_{k=1}^{x}\frac{x}{k}+\frac{x^{2}}{2}\left(\frac{6}{\pi^{2}}-\sum_{k=x+1}^{\infty}\frac{\mu(k)}{k^{2}}\right),$
$\displaystyle\displaystyle\leq
x(\mathrm{ln}(x)+1)+\frac{3}{\pi^{2}}x^{2}+\frac{x^{2}}{2}\sum_{k=x+1}^{\infty}\frac{1}{k^{2}},$
$\displaystyle\displaystyle\leq
x(\mathrm{ln}(x)+1)+\frac{3}{\pi^{2}}x^{2}+\frac{x^{2}}{2}\frac{1}{x-1}.$

Therefore, provided $\displaystyle x\geq 2$ to simplify the last term in the
previous inequality, we find that

$\displaystyle A^{-}(L)\leq\frac{12}{\pi^{2}}x^{2}+6x\mathrm{ln}(x)+8x+4.$

For $\displaystyle A^{+}(L)$, we similarly see that

$\displaystyle A^{+}(L)=4x\sum_{k=1}^{x}\frac{\phi(2k)}{k}\geq
4x\sum_{k=1}^{x}\frac{\phi(k)}{k}=4x\Psi(x).$

Again by the Möbius inversion formula

$\displaystyle\Psi(x)=\sum_{k=1}^{x}\frac{\phi(k)}{k}=\sum_{k=1}^{x}\left\lfloor\frac{x}{k}\right\rfloor\frac{\mu(k)}{k}.$

In particular, similar to the case of $\displaystyle\Phi(x)$ we conclude that

$\displaystyle\displaystyle\Psi(x)$
$\displaystyle\displaystyle=\sum_{k=1}^{x}\left(\left\lfloor\frac{x}{k}\right\rfloor-\frac{x}{k}\right)\frac{\mu(k)}{k}+\sum_{k=1}^{x}\frac{x}{k}\frac{\mu(k)}{k},$
$\displaystyle\displaystyle\geq-\sum_{k=1}^{x}\frac{1}{k}+x\sum_{k=1}^{x}\frac{\mu(k)}{k^{2}},$
$\displaystyle\displaystyle\geq-\mathrm{ln}(x)+x\left(\frac{6}{\pi^{2}}-\sum_{k=x+1}^{\infty}\frac{\mu(k)}{k^{2}}\right),$
$\displaystyle\displaystyle\geq-\mathrm{ln}(x)+x\left(\frac{6}{\pi^{2}}-\sum_{k=x+1}^{\infty}\frac{1}{k^{2}}\right),$
$\displaystyle\displaystyle\geq-\mathrm{ln}(x)+x\left(\frac{6}{\pi^{2}}-\frac{1}{x-1}\right),$
$\displaystyle\displaystyle\geq-\mathrm{ln}(x)+\frac{6}{\pi^{2}}x-1.$

Therefore we find that

$\displaystyle
A^{+}(L)=4x\Psi(x)\geq\frac{24}{\pi^{2}}x^{2}-4x\mathrm{ln}(x)-4x,$

and consequently

$\displaystyle A(L)\geq
A^{+}(L)-A^{-}(L)\geq\frac{12}{\pi^{2}}x^{2}-10x\mathrm{ln}(x)-12x-4,$

which, when remembering that $\displaystyle x=\frac{L}{4\ell(E)}$, has leading
term of the form $\displaystyle\frac{3\pi^{2}}{4\ell(E)}L^{2}.$ If
$\displaystyle x\geq 30$, then
$\displaystyle-2x\mathrm{ln}(x)-12x-4\geq-\frac{6}{\pi^{2}}x^{2}$ and
consequently if we assume that $\displaystyle L\geq L_{0}=120\ell(E)$, then we
find that

$\displaystyle
A(L)\geq\frac{3}{8\pi^{2}\ell(E)^{2}}L^{2}=c_{1}(\Sigma_{0,4})L^{2},$

as required.

##  Appendix B Effectivizing an argument of Tang

The goal of this appendix is to make clear that the bound in 2.2 of Tang,
which we restate below for the reader’s convenience, can be made explicitly
computable in the sense of Section 3.

*

The _radius_ $\displaystyle\mathrm{rad}\left\lparen U\right\rparen$ of a non-
empty finite subset $\displaystyle U\subseteq X$ of a Gromov hyperbolic metric
space $\displaystyle X$ is defined by

$\displaystyle\mathrm{rad}\left\lparen
U\right\rparen\coloneqq\min\left\\{r\in\left[{0},{\infty}\right\rparen:U\subseteq\mathrm{D}_{r}\left\lparen
x\right\rparen\textrm{ for some }x\in X\right\\}.$

We say that a point $\displaystyle x\in X$ is a _circumcenter_ of
$\displaystyle U$ if $\displaystyle U$ is contained in a closed disk of radius
$\displaystyle\mathrm{rad}\left\lparen U\right\rparen$ and center
$\displaystyle x$.

In the remainder of this section, we will briefly explain how to make 2.2
effective.

###  B.1. Definitions

In [Tan17], Tang defines the _hull_ of a tuple of simple curves
$\displaystyle\alpha=\left\lparen\alpha_{1},\dotsc,\alpha_{n}\right\rparen$ on
$\displaystyle X$ to be the set of all simple curves on $\displaystyle X$
lying on some geodesic segment connecting two curves in $\displaystyle\alpha$.
Given such a tuple $\displaystyle\alpha$, fix
$\displaystyle\mathbf{t}=\left\lparen
t_{1},\dotsc,t_{n}\right\rparen\in\mathbb{R}_{\geq 0}^{n}$, and let
$\displaystyle\mathbf{t}\cdot\alpha$ denote the formal sum

$\displaystyle\mathbf{t}\cdot\alpha\coloneqq\sum_{k=1}^{n}{t_{k}\alpha_{k}}.$

Tang also defines $\displaystyle\left\lVert\mathbf{t}\right\rVert_{\alpha}$ by
the formula

$\displaystyle\left\lVert\mathbf{t}\right\rVert_{\alpha}\coloneqq\sqrt{\sum_{j,k=1}^{n}{t_{j}t_{k}i\left\lparen\alpha_{j},\alpha_{k}\right\rparen}},$

and the _$\displaystyle L$ -short set_ of $\displaystyle\mathbf{t}\cdot\alpha$
by

$\displaystyle\mathrm{short}\left\lparen\mathbf{t}\cdot\alpha,L\right\rparen\coloneqq\left\\{\gamma\in\mathscr{C}\left\lparen
X\right\rparen:i\left\lparen\mathbf{t}\cdot\alpha,\gamma\right\rparen\leq
L\left\lVert\mathbf{t}\right\rVert_{\alpha}\right\\}.$

###  B.2. The constants in [Tan17, Lemma 6.4] and [Tan17, Lemma 5.1]

In this subsection, we discuss a few lemmas from [Tan17] and the dependencies
of various constants that arise therein. We recall that $\displaystyle p\colon
X\xrightarrow[]{}S$ is a finite-degree cover of a closed surface
$\displaystyle S$ of negative Euler characteristic, and we begin with the
following lemma.

######  Lemma B.1 ([Tan17, Lemma 6.4]).

There is a positive number $\displaystyle L_{0}>0$, depending only on
$\displaystyle\left\lvert\chi\left\lparen X\right\rparen\right\rvert$, so that

$\displaystyle
p\left\lparen\alpha\right\rparen\in\mathrm{short}\left\lparen\sum_{g\in\mathrm{Deck}\left\lparen
p\right\rparen}{g\left\lparen\alpha\right\rparen},L_{0}\left\lvert\mathrm{Deck}\left\lparen
p\right\rparen\right\rvert\right\rparen$

for all simple curves $\displaystyle\alpha$ on $\displaystyle X$.

By following the proof of [Tan17, Lemma 6.4], one may observe that this
constant $\displaystyle L_{0}$ is the same as that which appears in [Tan17,
Lemma 5.1].

######  Lemma B.2 ([Tan17, Lemma 5.1]).

There are some positive numbers $\displaystyle L_{0}$ and $\displaystyle
k_{0}$, depending only on $\displaystyle\left\lvert\chi\left\lparen
X\right\rparen\right\rvert$, so that for all $\displaystyle L\geq L_{0}$, the
$\displaystyle L$-short set
$\displaystyle\mathrm{short}\left\lparen\mathbf{t}\cdot\alpha,L\right\rparen$
is non-empty and has diameter

$\displaystyle\mathrm{diam}_{\mathscr{C}\left\lparen
X\right\rparen}\left\lparen\mathrm{short}\left\lparen\mathbf{t}\cdot\alpha,L\right\rparen\right\rparen\leq
4\mathrm{log}_{2}\left\lparen L\right\rparen+k_{0}.$

This is proved in [Tan17, Section 8]. In that section, Tang shows that it
suffices to choose $\displaystyle L_{0}\geq\frac{\sqrt{2}}{W_{0}}$, where
$\displaystyle W_{0}>0$ depends only on
$\displaystyle\left\lvert\chi\left\lparen S\right\rparen\right\rvert$ and
satisfies the following condition:

######  Proposition B.3 ([Tan17, Proposition 4.1(3)]).

$\displaystyle S\left\lparen\mathbf{t}\cdot\alpha\right\rparen$ contains an
essential annulus of width at least $\displaystyle
W_{0}\left\lVert\mathbf{t}\right\rVert_{\alpha}$.

Above, $\displaystyle S\left\lparen\mathbf{t}\cdot\alpha\right\rparen$ is the
singular flat surface dual to the union of curves in $\displaystyle\alpha$,
where the rectangle corresponding to an intersection point between
$\displaystyle\alpha_{j}$ and $\displaystyle\alpha_{k}$ has side lengths
$\displaystyle t_{j}$ and $\displaystyle t_{k}$. This proposition is proved
using the following lemma of Bowditch.

######  Lemma B.4 ([Bow06, Lemma 5.1]).

Let $\displaystyle\rho$ be a singular Riemannian metric on an orientable
closed surface $\displaystyle\Sigma$ with area $\displaystyle 1$, and let
$\displaystyle A\subsetneq\Sigma$ be a finite set. If there is a homeomorphism
$\displaystyle
f\colon\left[{0},{\infty}\right\rparen\xrightarrow[]{}\left[{0},{\infty}\right\rparen$
so that

$\displaystyle\mathrm{area}\left\lparen D\right\rparen\leq
f\left\lparen\ell\left\lparen\partial D\right\rparen\right\rparen$

for any region $\displaystyle D$ of $\displaystyle\Sigma$ containing at most
one point in $\displaystyle A$, then $\displaystyle\Sigma\setminus A$ contains
an essential annulus of width at least $\displaystyle\eta>0$, where
$\displaystyle\eta$ depends only on
$\displaystyle\left\lvert\chi\left\lparen\Sigma\right\rparen\right\rvert$ and
$\displaystyle\left\lvert A\right\rvert$.

In our desired context, the set $\displaystyle A$ of marked points is empty,
and so B.3 follows from B.4 once one has shown that
$\displaystyle\mathrm{area}\left\lparen D\right\rparen\leq
4\ell\left\lparen\partial D\right\rparen^{2}$ for all appropriate regions
$\displaystyle D$ in $\displaystyle
S\left\lparen\mathbf{t}\cdot\alpha\right\rparen$.

###  B.3. A weaker version of [Tan17, Lemma 5.1]

The second clause to [Tan17, Lemma 5.1] states that
$\displaystyle\mbox{short}(\textbf{t}\cdot\alpha,L)$ has uniformly bounded
diameter in $\displaystyle\mathcal{C}(\Sigma)$ in terms only of $\displaystyle
L$ and $\displaystyle\Sigma$ and not on t. We will need this second clause;
however, Tang’s proof is complicated by the possibility that
$\displaystyle\alpha$ does not fill the surface. Since in our context,
$\displaystyle\alpha$ will always be the $\displaystyle G$-orbit of some curve
in $\displaystyle\mathcal{C}(\Sigma)$, we can simplify matters by proving a
basic lemma which guarantees that $\displaystyle G\cdot\gamma$ fills
$\displaystyle\Sigma$ once $\displaystyle\gamma$ is at least distance
$\displaystyle 2$ from $\displaystyle\pi(\gamma)$:

######  Lemma B.5.

Suppose $\displaystyle d_{\Sigma}(\gamma,\pi(\gamma))\geq 2$, then
$\displaystyle G\cdot\gamma$ fills $\displaystyle\Sigma$.

###### Proof.

If $\displaystyle G\cdot\gamma$ does not fill $\displaystyle\Sigma$, then
there is some essential subsurface $\displaystyle Y$ in its complement.
Restricting the covering map $\displaystyle P$ to $\displaystyle Y$ yields a
cover $\displaystyle P|_{Y}:Y\rightarrow Z$ where $\displaystyle Z$ is some
non-simply connected subsurface of $\displaystyle S$. Thus, $\displaystyle Z$
supports a (potentially boundary parallel) simple closed curve
$\displaystyle\eta$ which is necessarily disjoint from $\displaystyle
P(\gamma)$. It follows that $\displaystyle\widetilde{P}(\eta)$ (which is
contained in $\displaystyle\widetilde{P}(\mathcal{C}(S))$) is distance
$\displaystyle 1$ from some elevation of $\displaystyle\gamma$ to
$\displaystyle\Sigma$. ∎

###  B.4. Assembling the proof modulo $\displaystyle W_{0}$ and Bowditch’s
lemma

In this subsection, we will obtain an effective version of [Tan17, Proposition
6.2], save for the constant $\displaystyle W_{0}$. First, we effectivize the
proof of [Tan17, Lemma 5.1] in the event that $\displaystyle\alpha$ fills.
There, Tang shows that any two curves in
$\displaystyle\mbox{short}(\textbf{t}\cdot\alpha,L)$ (for $\displaystyle
L>L_{0}$ with $\displaystyle L_{0}$ as above) intersect at most $\displaystyle
L/W_{0}$ times.

We can then use an effective version of Hempel’s bound on curve complex
distance in terms of intersection number, due to Bowditch [Bow14] (this
particular version can be found in [MT22])

$\displaystyle
d_{\mathscr{C}\left\lparen\Sigma\right\rparen}(\alpha,\beta)<6+2\cdot\frac{\mathrm{log}(2\cdot
i(\alpha,\beta)+1)}{\mathrm{log}((2g+p-4)/2)},$

where $\displaystyle g,p$ denotes the genus and number of punctures of
$\displaystyle\Sigma$, respectively. Tang points out that $\displaystyle L$
need only be bigger than $\displaystyle\sqrt{2}/W_{0}$, and Bowditch obtains
an effective bound on $\displaystyle W_{0}$ in [Bow14] which is the following
bound

$\displaystyle W_{0}\geq\frac{1}{8\cdot(2g+p-1)\cdot(2g+p+6)}.$

We note that this also uses Tang’s isoperimetric inequality mentioned above;
Bowditch states the result in terms of some constant $\displaystyle h$ so that
the area of a trivial region is at most $\displaystyle h$ times the length of
its boundary squared and moreover shows that a factor of $\displaystyle
4\sqrt{h}$ should appear in the bound. Thus, the appearance of $\displaystyle
8$ in the denominator above is actually $\displaystyle 4\sqrt{4}$, where the
$\displaystyle\sqrt{4}$ corresponds to the $\displaystyle 4$ in Tang’s
isoperimetric bound.

Lemma 6.4 of [Tan17] above therefore says that $\displaystyle\pi(\alpha)$ lies
in $\displaystyle\mbox{short}({\bf{1}}\cdot G\alpha,L_{0}|G|)$, which by the
above, has diameter at most

$\displaystyle
12+4\cdot\frac{\mathrm{log}(2\cdot(8\sqrt{2}\cdot(2g+p-1)\cdot(2g+p+6))^{2}\cdot|G|+1)}{\mathrm{log}((2g+p-4)/2)},$

which is bounded uniformly above independent of $\displaystyle g,p$.

We would be done if the circumcentre of $\displaystyle G\cdot\alpha$ were also
in this short curve set. It need not be, but [Tan17, Proposition 5.2] asserts
the existence of some $\displaystyle k_{1}$ so that for all $\displaystyle
L>L_{0}$, $\displaystyle\mbox{short}(\rho,L)$ (which is defined to be the
union, taken over all t, of
$\displaystyle\mbox{short}(\textbf{t}\cdot\rho,L)$) is Hausdorff distance at
most $\displaystyle k_{1}$ from $\displaystyle\mbox{Hull}(\rho)$, for any
finite collection of curves $\displaystyle\rho$. Here $\displaystyle k_{1}$
depends only on $\displaystyle L$, the topology of the surface, and the number
of curves in $\displaystyle\rho$. In particular when
$\displaystyle\rho=G\cdot\alpha$, $\displaystyle k_{1}$ will depend only on
$\displaystyle|G|$, the topology of $\displaystyle\Sigma$ (or alternatively,
the topology of $\displaystyle S$ and the degree of $\displaystyle P$), and
$\displaystyle L_{0}$.

In [Tan17, Proposition 5.2], Tang shows that $\displaystyle k_{1}$ can be
taken to be a constant $\displaystyle r=r(\Sigma,|G|,L_{0})$, where
$\displaystyle r$ is defined to be the distance between

$\displaystyle\mbox{short}(\mbox{{t}}^{\prime}\cdot\alpha^{\prime},|G|L_{0}/\sqrt{2}),$

and $\displaystyle[\alpha_{1},\alpha_{2}]$, where

  1. (1)

$\displaystyle\alpha^{\prime}=\left\\{\alpha_{1},\alpha_{2}\right\\}$, the
points in the orbit of $\displaystyle G\cdot\alpha$ maximizing $\displaystyle
t_{i}t_{j}i(\alpha_{i},\alpha_{j})$ over all pairs
$\displaystyle(i,j)\in\left\\{1,...,|G|\right\\}^{2}$, and

  2. (2)

$\displaystyle[\alpha_{1},\alpha_{2}]$ is any geodesic in the curve graph
between $\displaystyle\alpha_{1},\alpha_{2}$.

Since curve graphs are $\displaystyle 17$-hyperbolic, any two geodesics are at
most $\displaystyle 92\cdot 17=1564$ apart (see for instance [GS19, Theorem
1.1]), so up to addition of this universal constant, the choice of geodesic
will not matter.

###  B.5. Effectivizing the Hausdorff distance statement from Bowditch

It remains to determine how close the short curve set
$\displaystyle\mbox{short}(\left\\{\alpha_{1},\alpha_{2}\right\\},|G|W_{0}^{-1})$
is to a geodesic segment $\displaystyle[\alpha_{1},\alpha_{2}]$. For this, we
use [Bow06, Proposition 6.2], which asserts that there is some $\displaystyle
q$ and some $\displaystyle h$ so that for all $\displaystyle Q>q$, the set

$\displaystyle
G_{Q}(\alpha,\beta)=\left\\{\gamma\in\mathcal{C}:i(\gamma,\alpha)\cdot
i(\gamma,\beta)<Q\cdot i(\alpha,\beta)\right\\},$

is within $\displaystyle h$ of $\displaystyle[\alpha,\beta]$. To understand
the relevance of this, first consider some
$\displaystyle\gamma\in\mbox{short}(\left\\{\alpha,\beta\right\\},Q)$. Thus,

$\displaystyle
i(\gamma,t_{1}\alpha)+i(\gamma,t_{2}\beta)<Q\cdot\sqrt{t_{1}t_{2}i(\alpha,\beta)},$

for some nonnegative $\displaystyle t_{1},t_{2}$. Squaring both sides yields

$\displaystyle t_{1}t_{2}i(\gamma,\alpha)\cdot
i(\gamma,\beta)+T<Q^{2}t_{1}t_{2}i(\alpha,\beta),$

where $\displaystyle T$ is non-negative. Therefore, it follows that
$\displaystyle\gamma\in G_{Q^{2}}(\alpha,\beta)$.

Bowditch proves [Bow06, Proposition 6.2] using the “lines” he constructs in
the proof of hyperbolicity, which he denotes by
$\displaystyle\Lambda_{\alpha\beta}$. In [Bow06, Section 3], he outlines a
series of criteria involving a family of lines such that if any graph admits
such a family, it is hyperbolic, and the lines are uniformly quasigeodesic.
The criteria appear directly before [Bow06, Proposition 3.1].

The proof of [Bow06, Proposition 3.1] uses an auxiliary set of paths denoted
$\displaystyle\pi_{\alpha\beta}$. In [Bow06, Lemma 3.3], Bowditch shows that
these paths are $\displaystyle(K,2K)$-quasigeodesic, where $\displaystyle K$
is as in the statement of the three axioms he uses as a criteria for
hyperbolicity. He also shows, directly before this lemma, that
$\displaystyle\Lambda_{\alpha\beta}$ is Hausdorff distance at most
$\displaystyle K$ from $\displaystyle\pi_{\alpha\beta}$. Putting all of this
together and using the Morse Lemma, it follows that
$\displaystyle\Lambda_{\alpha,\beta}$ is within a distance of $\displaystyle
92K^{2}(2K+2+17)$ of a geodesic $\displaystyle[\alpha,\beta]$ (again using
[GS19]). In [Bow06, Section 4], Bowditch shows that the curve graph satisfies
the three required axioms, for $\displaystyle K=18D+2$.

Furthermore, Bowditch shows in [Bow14] that it suffices to choose
$\displaystyle D=20$ for all surfaces. Putting all of this together, we get
that $\displaystyle\Lambda_{\alpha\beta}$ is within a neighborhood of
$\displaystyle[\alpha,\beta]$ of radius at most

$\displaystyle\displaystyle 92K^{2}(2K+19)$
$\displaystyle\displaystyle=92\cdot(18D+2)^{2}\cdot(36D+23),$
$\displaystyle\displaystyle=8,957,643,664.$

In the direction of the proof of [Bow06, Proposition 6.2] that is relevant to
us, Bowditch only needs to establish that $\displaystyle G_{Q}(\alpha,\beta)$
is nearby $\displaystyle\Lambda_{\alpha\beta}$, since as mentioned above, we
can estimate the distance between $\displaystyle\Lambda_{\alpha\beta}$ and
$\displaystyle[\alpha,\beta]$.

Bowditch’s proof of [Bow06, Proposition 6.2] shows that one may set
$\displaystyle q=(W_{0}^{-1})^{2}$, and then that

$\displaystyle G_{Q}(\alpha,\beta)\subset L_{\alpha\beta}(t,\sqrt{Q}),$

for all $\displaystyle Q>q$, where $\displaystyle L_{\alpha\beta}(t,\sqrt{Q})$
is as defined in [Bow06, Section 4]. Thus we are interested in the diameter of
$\displaystyle L_{\alpha\beta}(t,(W_{0}^{-1})^{2})$, which, again by
Bowditch’s proof of uniform hyperbolicity [Bow14], is uniformly bounded above,
for instance by $\displaystyle 20$.

###  B.6. Putting it all together.

Now, assume we begin with some
$\displaystyle\gamma\in\mbox{short}(\left\\{\alpha_{1},\alpha_{2}\right\\},|G|W_{0}^{-1})$.
Then

$\displaystyle\gamma\in G_{|G|^{2}(W_{0}^{-1})^{2}}(\alpha,\beta).$

It follows from the very end of the previous section, and [Bow06, Lemma 4.2],
that $\displaystyle\gamma$ is at most

$\displaystyle 2(W_{0}^{-1})^{3}|G|+2,$

away from $\displaystyle\Lambda_{\alpha\beta}$. Therefore,
$\displaystyle\gamma$ is at most

$\displaystyle 2(W_{0}^{-1})^{3}|G|+2+8,957,643,664,$

away from $\displaystyle[\alpha,\beta]$.

It finally follows, assuming $\displaystyle G\cdot\alpha$ fills
$\displaystyle\Sigma$, that $\displaystyle\pi(\alpha)$ is within

$\displaystyle
2(W_{0}^{-1})^{3}|G|+2+8,957,643,664+12+4\cdot\frac{\mathrm{log}(2\cdot(8\sqrt{2}\cdot(2g+p-1)\cdot(2g+p+6))^{2}\cdot|G|+1)}{\mathrm{log}((2g+p-4)/2)},$

of the circumcentre of $\displaystyle G\cdot\alpha$.

If $\displaystyle G\cdot\alpha$ does not fill $\displaystyle\Sigma$, B.5
implies that the radius of $\displaystyle G\cdot\alpha$ is at most
$\displaystyle 2$ and that $\displaystyle\pi(\alpha)$ is distance at most
$\displaystyle 2$ away, and so the circumcentre of $\displaystyle
G\cdot\alpha$ is at most distance $\displaystyle 4$ from
$\displaystyle\pi(\alpha)$.

Since the log term above is bounded above independently of $\displaystyle
g,p$, the entire bound is $\displaystyle O(|G|)$ and on the order of a
polynomial of degree $\displaystyle 6$ in $\displaystyle g$ and $\displaystyle
p$. For a less explicit upper bound that is stronger asymptotically, note that
$\displaystyle|G|$ is necessarily bounded above by a linear function of
$\displaystyle g$ and $\displaystyle p$. Therefore
$\displaystyle|G|^{2}\cdot(W_{0}^{-1})^{2}$ grows at most polynomially in
$\displaystyle g,p$ of degree at most $\displaystyle 6$. The techniques in
[Aou13] show how to obtain a bound that is uniform in $\displaystyle g,p$ on
the diameter of $\displaystyle L_{\alpha\beta}(t,R)$ whenever $\displaystyle
R$ grows slower than some polynomial in $\displaystyle g,p$.

## References

  * [Aou+23] Tarik Aougab, Max Lahn, Marissa Loving and Nicholas Miller  “Unmarked simple length spectral rigidity for covers”, 2023
  * [Aou13] Tarik Aougab  “Uniform hyperbolicity of the graphs of curves”  In _Geometry and Topology_ 17, 2013, pp. 2855–2875  DOI: 10.2140/gt.2013.17.2855
  * [APT22] Tarik Aougab, Priyam Patel and Samuel J. Taylor  “Covers of surfaces, Kleinian groups, and the curve complex”  In _Journal of Topology_ 15.4, 2022, pp. 1833–1863  DOI: 10.1112/topo.12261
  * [AT14] Tarik Aougab and Samuel Taylor  “Small intersection numbers in the curve graph”  In _Bulletin of the London Mathematical Society_ 46.5, 2014, pp. 989–1002  DOI: 10.1112/blms/bdu057
  * [BCL20] Martin Bridgeman, Richard Canary and Francois Labourie  “Simple length rigidity for Hitchin representations”  In _Advances in Mathematics_ 360, 2020  DOI: 10.1016/j.aim.2019.106901
  * [Bow06] Brian H. Bowditch  “Intersection numbers and the hyperbolicity of the curve complex”  In _Journal für die reine und angewandte Mathematik_ 2006.598, 2006, pp. 105–129  DOI: 10.1515/CRELLE.2006.070