# Unmarked simple length spectral rigidity for covers

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.

We prove that every closed orientable surface $\displaystyle S$ of negative
Euler characteristic admits a pair of finite-degree covers which are length
isospectral over $\displaystyle S$ but generically not simple length
isospectral over $\displaystyle S$. To do this, we first characterize when two
finite-degree covers of a connected, orientable surface of negative Euler
characteristic are isomorphic in terms of which curves have simple elevations.
We also construct hyperbolic surfaces $\displaystyle X$ and $\displaystyle Y$
with the same full unmarked length spectrum but so that for each
$\displaystyle k$, the sets of lengths associated to curves with at most
$\displaystyle k$ self-intersections differ.

##  1\. Introduction

Otal [Ota90] showed that any negatively curved metric on $\displaystyle S$ is
determined up to isotopy by its _marked length spectrum_ , the function
associating each closed curve to its infimal length. More generally, one can
ask this question for different sets of closed curves and/or different
families of metrics (see Fricke, Wolpert [Wol79], and Hamenstädt’s [Ham03]
work on hyperbolic metrics and Duchin–Leininger–Rafi [DLR10] and Loving’s
[Lov20] work on flat metrics).

The roots of this paper lie in a family of interrelated questions
distinguished from the above work by the absence of markings. Concretely,
consider a $\displaystyle\mathrm{Mod}\left\lparen S\right\rparen$-invariant
set $\displaystyle\mathcal{G}$ of closed curves on $\displaystyle S$. For
example, we will consider for each $\displaystyle k\in\mathbb{Z}_{\geq
0}\cup\left\\{\infty\right\\}$ the set $\displaystyle\mathcal{G}_{k}$ of
curves whose self-intersection number is at most $\displaystyle k$. The
_unmarked $\displaystyle\mathcal{G}$-length spectrum_ of a hyperbolic metric
$\displaystyle\mu$ on $\displaystyle S$ is the multi-set of infimal lengths of
closed curves in $\displaystyle\mathcal{G}$ in the metric $\displaystyle\mu$.
Particularly relevant examples are the _unmarked length spectrum_ and the
_unmarked simple length spectrum_ , which arise in the cases when
$\displaystyle\mathcal{G}=\mathcal{G}_{\infty}$ is the set of all closed
curves on $\displaystyle S$ and $\displaystyle\mathcal{G}=\mathcal{G}_{0}$ is
the set of simple curves on $\displaystyle S$, respectively.

######  Question 1.

For which $\displaystyle\mathcal{G}$ is a hyperbolic metric $\displaystyle\mu$
on $\displaystyle S$ determined up to isometry by its unmarked
$\displaystyle\mathcal{G}$-length spectrum?

Unlike in the case of marked spectra, it is well-known that many classes of
metrics on $\displaystyle S$ are not unmarked length spectrally rigid. Indeed,
examples of unmarked length isospectral but non-isometric complete hyperbolic
surface metrics were first constructed by Vignéras [Vig80] in the arithmetic
setting. Since then, a celebrated construction of Sunada [Sun85] yields a
robust method for producing unmarked length isospectral but non-isometric
surface metrics arising as pullbacks of metrics along covers of a fixed
surface.

On the other hand, relatively little is known about the question of unmarked
simple length spectral rigidity for hyperbolic metrics. Baik–Choi–Kim [BCK21]
recently showed that, up to excluding the loci of some length identities, the
family of hyperbolic metrics with the property that lengths in the simple
length spectrum have multiplicity $\displaystyle 1$ (that is, for each
positive real number $\displaystyle L$, there is at most one simple closed
geodesic of length $\displaystyle L$) is simple length spectrally rigid. This
result motivates our examination of pullbacks of hyperbolic metrics along
finite-degree covers, since high multiplicity is common in this regime.

Explicitly, fix for the remainder of this article two finite-degree covers
$\displaystyle p\colon X\xrightarrow[]{}S$ and $\displaystyle q\colon
Y\xrightarrow[]{}S$ of a closed, orientable surface $\displaystyle S$ of
negative Euler characteristic.

question+OurQuestion

If $\displaystyle X$ and $\displaystyle Y$ are generically unmarked simple
length isospectral over $\displaystyle S$, are the covers $\displaystyle p$
and $\displaystyle q$ isomorphic?

Here (and in what follows), we say that a property of hyperbolic metrics on
$\displaystyle X$ and/or $\displaystyle Y$ holds generically over
$\displaystyle S$ if it holds for the pullbacks of hyperbolic metrics in a
dense and full measure subset of the Teichmüller space of $\displaystyle S$.
Our strategy for answering 1 is discussed in prior work of
Aougab–Lahn–Loving–Xiao [Aou+20] and at the beginning of Section 4.

###  1.1. Main results

Our first result implies that there are infinitely many pairs of hyperbolic
surfaces (i.e. topological surfaces equipped with a choice of hyperbolic
metric) which are unmarked length isospectral but not unmarked simple length
isospectral.

theorem+examples

Every closed surface $\displaystyle S$ of negative Euler characteristic admits
a pair of finite-degree covers which are unmarked length isospectral over
$\displaystyle S$ but generically not unmarked simple length isospectral over
$\displaystyle S$.

In fact, the pairs of isospectral hyperbolic surfaces we obtain in the proof
of Section 1.1 have the stronger property that the respective multi-sets of
lengths associated to curves with any bounded number of self intersections do
not coincide.

corollary+k-isospectral

There exist hyperbolic surfaces $\displaystyle X$ and $\displaystyle Y$ which
are unmarked length isospectral but so that for each $\displaystyle k\geq 0$,
$\displaystyle X$ and $\displaystyle Y$ are not unmarked
$\displaystyle\mathcal{G}_{k}$-length isospectral.

The statement of Section 1.1 combines work of Maunchang [Mau13] for a surface
of genus $\displaystyle 2$ with our work for surfaces of genus at least
$\displaystyle 3$ (see Section 4.3). Maungchang uses Sunada’s construction to
build a pair of finite-degree covers $\displaystyle X\xrightarrow[]{}S$ and
$\displaystyle Y\xrightarrow[]{}S$ of a closed surface $\displaystyle S$ of
genus $\displaystyle 2$ so that a generic hyperbolic metric on $\displaystyle
S$ pulls back to metrics on $\displaystyle X$ and $\displaystyle Y$ which have
the same length spectrum but not the same simple length spectrum. Our proof of
Section 1.1 generalizes Maungchang’s technique and relies crucially on the
following strengthening of [Aou+20, Theorem 1.1], which uses entirely new
proof techniques in order to remove the regularity hypothesis, thus resolving
[Aou+20, Conjecture 1.4]. Given a finite cover $\displaystyle p\colon
X\xrightarrow[]{}S$, a curve $\displaystyle\alpha$ on $\displaystyle X$ is an
_elevation_ of a curve $\displaystyle\gamma$ on $\displaystyle S$ along
$\displaystyle p$ if $\displaystyle p\left\lparen\alpha\right\rparen$ is a
nonzero iterate of $\displaystyle\gamma$.

theorem+noneffective

The covers $\displaystyle p\colon X\xrightarrow[]{}S$ and $\displaystyle
q\colon Y\xrightarrow[]{}S$ are isomorphic if and only if for every curve
$\displaystyle\gamma$ on $\displaystyle S$, $\displaystyle\gamma$ has a simple
elevation along $\displaystyle p$ to $\displaystyle X$ if and only if it has a
simple elevation along $\displaystyle q$ to $\displaystyle Y$.

In Section 1.1 and Section 1.1, we use Sunada’s construction to guarantee that
the covers have the same unmarked length spectrum. We choose the covers
somewhat carefully to satisfy the desired properties, but in the spirit of 1,
one might wonder whether _any_ pair of non-isomorphic covers arising from
Sunada’s construction are generically simple length non-isospectral over
$\displaystyle S$.

###  1.2. Applications to simple length isospectrality of covers

Our applications of Section 1.1 aim towards answering 1 for the unmarked
simple length spectra of hyperbolic metrics. More precisely, Section 1.1 also
allows us to specify a finite-degree cover of $\displaystyle S$ by the data of
which curves have simple elevations. When one can find a curve that lifts
simply to one cover but not the other (Section 1.1 guarantees that one can
always do this) and which has no _length twins_ (distinct curves on
$\displaystyle S$ whose lengths are commensurable in any hyperbolic metric,
see Section 2.7), this lifting data is sufficient to distinguish the simple
length spectra of the pullbacks of generic hyperbolic metrics on
$\displaystyle S$ to $\displaystyle X$ and $\displaystyle Y$. In particular,
the following proposition together with Section 1.1 yields a tool for
verifying that such a pair of covers are not simple length isospectral.

proposition+lengthSpectra

Suppose that there is a curve $\displaystyle\gamma$ on $\displaystyle S$ and a
positive integer $\displaystyle m\geq 1$ with the following properties:

  1. (i)

$\displaystyle\gamma$ has no length twins on $\displaystyle S$; and

  2. (ii)

$\displaystyle\gamma$ has different numbers of simple elevations of degree
$\displaystyle m$ along each of the covers $\displaystyle p\colon
X\xrightarrow[]{}S$ and $\displaystyle q\colon Y\xrightarrow[]{}S$.

Then $\displaystyle X$ and $\displaystyle Y$ are generically not unmarked
simple length isospectral over $\displaystyle S$.

We use Section 1.2 to provide an affirmative answer to 1 when $\displaystyle
p$ and $\displaystyle q$ are regular and _simply generated_ , meaning that the
fundamental groups of $\displaystyle X$ and $\displaystyle Y$ are generated by
the set of all elevations of simple closed curves on $\displaystyle S$ (see
Section 4.2 for a formal definition and discussion).

theorem+simplyGeneratedCovers

If $\displaystyle p\colon X\xrightarrow[]{}S$ and $\displaystyle q\colon
Y\xrightarrow[]{}S$ are regular and simply generated, then they are either
isomorphic or generically not unmarked simple length isospectral over
$\displaystyle S$.

It is known by work of Koberda–Santharoubane [KS16] and Malestein–Putman
[MP19] that not all regular covers are simply generated, but their examples
are necessarily of very high degree. We therefore think of Section 1.2 as
addressing 1 in the setting of low-degree regular covers.

We also use Section 1.2 to exhibit a method for identifying when pairs of
covers produced from Sunada’s construction are unmarked simple length non-
isospectral. Sunada’s construction begins with a surjective homomorphism of a
closed surface group onto a finite group, from which one finds finite covers
by pulling back a pair of almost conjugate subgroups. We exhibit a general
strategy for identifying when these covers arising from Sunada’s construction
satisfy the hypotheses of Section 1.2, and are therefore generically not
simple length isospectral over the original surface. We use this method to
produce three distinct infinite families of such pairs. This provides a vast
generalization of the work of Maungchang [Mau13], producing infinitely many
new examples of pairs of isospectral surfaces with distinct simple length
spectra, and proving Section 1.1.

Identifying and controlling the presence of pairs of curves which have the
same geodesic length in any hyperbolic metric presents a major challenge when
attempting to distinguish covers by their simple length spectra. Such curves
were shown to exist and studied extensively by Horowitz [Hor72]. Leininger
[Lei03] proved that any such pair has the property that the modulus of their
traces must coincide under any representation
$\displaystyle\pi_{1}\left\lparen
S\right\rparen\xrightarrow[]{}\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen$.
Indeed, the existence of such curves is a phenomenon that arises from trace
identities specific to
$\displaystyle\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen$. The
existence (or non-existence) of words in the free group $\displaystyle F_{2}$
on two generators for which the traces coincide in any representation to some
$\displaystyle\mathrm{SL}_{n}\left\lparen\mathbb{C}\right\rparen$ with
$\displaystyle n\geq 3$ remains a significant open question [HT10].

###  1.3. A remark on compact surfaces with boundary

Although we phrase our results for closed surfaces of negative Euler
characteristic, they also apply to compact surfaces with boundary. Indeed, a
short argument involving the double of such a surface extends [Lei03, Theorem
1.4], the key ingredient.

###  1.4. Organization of paper

In Section 2, we provide an overview of the relevant preliminary concepts and
results that we will use throughout the paper. In Section 3, we prove Section
1.1 and a corollary relating to covers produced via Sunada’s construction
which are both length isospectral and simple length isospectral. In Section 4,
we discuss and prove Section 1.2 and Section 1.2, and then construct the
aforementioned three infinite families of examples of covers which are length
isospectral but generically simple length non-isospectral, thereby proving
Section 1.1.

###  1.5. Acknowledgements

We gratefully acknowledge support from NSF grants DMS-1939936 (Aougab),
DMS-1906441 (Lahn), DMS-1902729 & DMS-2231286 (Loving), and DMS-2005438 &
2300370 (Miller). We thank Aaron Calderon, Siddhi Krishna, Chris Leininger,
Ian Runnels, Nick Salter, Lukas Scheiwiller, Robert Tang, and Samuel Taylor
for helpful conversations. We would also like to thank Dongryul Kim, Hugo
Parlier, and Andy Putman for their feedback on a draft of this paper.

##  2\. Preliminaries

###  2.1. Curves and their elevations

By a _(closed) curve_ on a surface $\displaystyle S$, we mean an equivalence
class of continuous maps from the circle $\displaystyle\mathbb{S}^{1}$ to
$\displaystyle S$, where two such maps are considered equivalent if one is
freely homotopic to a reparametrization of the other. Note that with this
definition, curves are not oriented. A curve $\displaystyle\gamma$ on a
surface $\displaystyle S$ is:

  1. (i)

_essential_ if it has no freely nullhomotopic representative, and
_inessential_ otherwise;

  2. (ii)

_primitive_ if none of its representatives factors through a non-trivial
finite-degree covering map of the circle $\displaystyle\mathbb{S}^{1}$, and
_non-primitive_ otherwise; and

  3. (iii)

_simple_ if one of its representatives is injective, and _non-simple_
otherwise.

We will implicitly assume that all curves under consideration are essential.
Under this assumption, all simple curves are primitive.

Now consider a finite-degree cover $\displaystyle p\colon X\xrightarrow[]{}S$.
A curve $\displaystyle\alpha$ on $\displaystyle X$ is an _elevation_ of a
curve $\displaystyle\gamma$ on $\displaystyle S$ along $\displaystyle p$ of
degree $\displaystyle d\geq 1$ if there are parametrized representatives such
that $\displaystyle p\circ\alpha$ traces $\displaystyle d$ times along
$\displaystyle\gamma$. More formally, $\displaystyle\alpha$ is an elevation of
$\displaystyle\gamma$ along $\displaystyle p$ of degree $\displaystyle d$ if
there are parametrized representatives so that the following square commutes:

$\displaystyle{\mathbb{S}^{1}}$$\displaystyle{X}$$\displaystyle{\mathbb{S}^{1}}$$\displaystyle{S}$$\displaystyle\scriptstyle{f}$$\displaystyle\scriptstyle{\alpha}$$\displaystyle\scriptstyle{p}$$\displaystyle\scriptstyle{\gamma}$

where $\displaystyle f\colon\mathbb{S}^{1}\xrightarrow[]{}\mathbb{S}^{1}$ is a
covering map of degree $\displaystyle d$. Note that the degree of an elevation
is bounded above by the degree of the cover $\displaystyle p\colon
X\xrightarrow[]{}S$.

The _deck group_ of a finite-degree cover $\displaystyle p\colon
X\xrightarrow[]{}S$ is the group of homeomorphisms of $\displaystyle X$ which
cover the identity homeomorphism of $\displaystyle S$; that is,

$\displaystyle\mathrm{Deck}\left\lparen
p\right\rparen\coloneqq\left\\{\varphi\in\mathrm{Homeo}\left\lparen
X\right\rparen:p\circ\varphi=p\right\\}.$

$\displaystyle\mathrm{Deck}\left\lparen p\right\rparen$ acts freely on each
fiber of $\displaystyle p$ and so is a finite group with
$\displaystyle\left\lvert\mathrm{Deck}\left\lparen
p\right\rparen\right\rvert\leq\deg\left\lparen p\right\rparen$. When
$\displaystyle\left\lvert\mathrm{Deck}\left\lparen
p\right\rparen\right\rvert=\deg\left\lparen p\right\rparen$, then
$\displaystyle\mathrm{Deck}\left\lparen p\right\rparen$ also acts transitively
on each fiber. In this case, the covering map $\displaystyle p\colon
X\xrightarrow[]{}S$ is the quotient associated to this action and is called
_regular_ (these covers are also sometimes called _normal_ or _Galois_). Note
that the action of the deck group on $\displaystyle X$ induces an action on
the set of degree $\displaystyle d$ elevations of any curve
$\displaystyle\gamma$ on $\displaystyle S$.

A choice of basepoint $\displaystyle s\in S$ yields a bijection between the
set of curves on $\displaystyle S$ and the set of equivalence classes of non-
identity elements of the fundamental group $\displaystyle\pi_{1}\left\lparen
S,s\right\rparen$, where two elements
$\displaystyle\alpha,\beta\in\pi_{1}\left\lparen
S,s\right\rparen\setminus\left\\{\text{id}\right\\}$ are considered equivalent
if either $\displaystyle\alpha$ and $\displaystyle\beta$ are conjugate or
$\displaystyle\alpha$ and $\displaystyle\beta^{-1}$ are conjugate. From this
perspective, primitive curves correspond to equivalence classes of elements
which cannot be written non-trivially as a power of another element.

Moreover, choices of basepoints $\displaystyle x\in X$ and $\displaystyle s\in
S$ so that $\displaystyle p\left\lparen x\right\rparen=s$ give an alternate
characterization of elevations of degree $\displaystyle d$: given non-identity
elements $\displaystyle\alpha\in\pi_{1}\left\lparen X,x\right\rparen$ and
$\displaystyle\gamma\in\pi_{1}\left\lparen S,s\right\rparen$ which cannot be
written non-trivially as a power of another element, the primitive curve on
$\displaystyle X$ represented by $\displaystyle\alpha$ is an elevation of the
primitive curve on $\displaystyle S$ represented by $\displaystyle\gamma$
along $\displaystyle p$ of degree $\displaystyle d$ if and only if
$\displaystyle p_{*}\left\lparen\alpha\right\rparen$ is conjugate in
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ to
$\displaystyle\gamma^{d}$ or $\displaystyle\gamma^{-d}$, where $\displaystyle
p_{*}\colon\pi_{1}\left\lparen
X,x\right\rparen\xhookrightarrow{}\pi_{1}\left\lparen S,s\right\rparen$ is the
injective homomorphism induced by $\displaystyle p$.

###  2.2. Length and the Teichmüller space

The _Teichmüller 111The authors would like to draw the reader’s attention to
increasingly relevant historical context: in addition to his current status as
the namesake of many fundamental objects in the field, Oswald Teichmüller, may
his name be erased, was an ardent Nazi. The authors support ongoing efforts to
rename these objects in ways that do not glorify the perpetrators of genocide.
space_ $\displaystyle\mathscr{T}\left\lparen S\right\rparen$ of a closed
surface $\displaystyle S$ of negative Euler characteristic is the space of
isotopy classes of hyperbolic metrics on $\displaystyle S$. More explicitly,
two hyperbolic metrics $\displaystyle\mu$ and $\displaystyle\nu$ are
_isotopic_ if there is an isometry from $\displaystyle\left\lparen
S,\mu\right\rparen$ to $\displaystyle\left\lparen S,\nu\right\rparen$ which is
isotopic to the identity map on $\displaystyle S$. Teichmüller space is
equipped with _Fenchel-Nielsen coordinates_ [FN02], which parametrize
$\displaystyle\mathscr{T}\left\lparen S\right\rparen$ as a complex manifold
homeomorphic to $\displaystyle\mathbb{R}^{3\left\lvert\chi\left\lparen
S\right\rparen\right\rvert}$. $\displaystyle\mathscr{T}\left\lparen
S\right\rparen$ is also equipped with a proper geodesic metric called the
_Teichmüller metric_ as well as an isometric and properly discontinuous action
of the mapping class group $\displaystyle\mathrm{Mod}\left\lparen
S\right\rparen$, whose associated quotient is the moduli space of hyperbolic
surfaces homeomorphic to $\displaystyle S$.

Pulling back hyperbolic metrics along a finite-degree cover contravariantly
defines an embedding of the corresponding Teichmüller spaces. More explicitly,
given a finite-degree cover $\displaystyle p\colon X\xrightarrow[]{}S$, there
is well-defined map $\displaystyle p^{*}\colon\mathscr{T}\left\lparen
S\right\rparen\xrightarrow[]{}\mathscr{T}\left\lparen X\right\rparen$ defined
by the formula $\displaystyle
p^{*}\left\lparen\left[\mu\right]\right\rparen\coloneqq\left[p^{*}\mu\right]$.
When $\displaystyle\mathscr{T}\left\lparen S\right\rparen$ and
$\displaystyle\mathscr{T}\left\lparen X\right\rparen$ are equipped with their
respective Teichmüller metrics, this map $\displaystyle p^{*}$ is a proper
isometric embedding.

A choice of basepoint $\displaystyle s\in S$ yields another perspective on
$\displaystyle\mathscr{T}\left\lparen S\right\rparen$. Via the association of
a metric with any of its monodromy representations, we may also view
$\displaystyle\mathscr{T}\left\lparen S\right\rparen$ as the space of
$\displaystyle\mathrm{PGL}_{2}\left\lparen\mathbb{R}\right\rparen$-conjugacy
classes of discrete and faithful representations of the fundamental group
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ into
$\displaystyle\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen$. From this
vantage, pulling back a metric along a cover $\displaystyle p\colon
X\xrightarrow[]{}S$ corresponds to the restriction of the monodromy
representation to the appropriate subgroup; that is, given a choice of
basepoints $\displaystyle x\in X$ and $\displaystyle s\in S$ such that
$\displaystyle p\left\lparen x\right\rparen=s$, $\displaystyle
p^{*}\left\lparen\left[\rho\right]\right\rparen=\left[\rho\circ p_{*}\right]$
for all discrete and faithful representations
$\displaystyle\rho\colon\pi_{1}\left\lparen
S,s\right\rparen\xrightarrow[]{}\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen$.

The _(infimal) length_
$\displaystyle\ell_{\mu}\left\lparen\gamma\right\rparen$ of a curve
$\displaystyle\gamma$ with respect to a hyperbolic metric $\displaystyle\mu$
on $\displaystyle S$ is defined to be the minimal length of any of its
parametrized representatives. Equivalently,
$\displaystyle\ell_{\mu}\left\lparen\gamma\right\rparen$ is the length of the
unique (up to reparametrization) geodesic representative of
$\displaystyle\gamma$. Since $\displaystyle\gamma$ has the same length in any
isotopic hyperbolic metric,
$\displaystyle\ell_{\left\lparen\>\cdot\>\right\rparen}\left\lparen\gamma\right\rparen$
descends to a real-valued function on the Teichmüller space
$\displaystyle\mathscr{T}\left\lparen S\right\rparen$ defined by the formula
$\displaystyle\left[\mu\right]\mapsto\ell_{\mu}\left\lparen\gamma\right\rparen$.
In fact, these length functions are real-analytic, with

$\displaystyle\ell_{\left[\rho\right]}\left\lparen\left[\gamma\right]\right\rparen=2\mathrm{cosh}^{-1}\left\lparen\frac{\left\lvert\mathrm{tr}\left\lparen\rho\left\lparen\gamma\right\rparen\right\rparen\right\rvert}{2}\right\rparen,$

for all elements $\displaystyle\gamma\in\pi_{1}\left\lparen S,s\right\rparen$
and discrete and faithful representations
$\displaystyle\rho\colon\pi_{1}\left\lparen
S,s\right\rparen\xhookrightarrow{}\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen$.

###  2.3. The Thurston compactification of Teichmüller space

Both the Teichmüller space $\displaystyle\mathscr{T}\left\lparen
S\right\rparen$ and the curve complex $\displaystyle\mathscr{C}\left\lparen
S\right\rparen$ (which will be introduced in Section 2.4) of a surface
$\displaystyle S$ admit compactifications whose boundary points may be
described in the language of geodesic laminations. A _geodesic lamination_ on
a hyperbolic surface $\displaystyle S$ is a compact subset of $\displaystyle
S$ which is the union of pairwise disjoint simple geodesics (some, all, or
none of which may be closed) called _leaves_. We note that geodesic
laminations may also be defined independently of a reference hyperbolic metric
on $\displaystyle S$.

A _transverse measure_ on a geodesic lamination $\displaystyle\lambda$ is a
positive real-valued function on the set of arcs transverse to
$\displaystyle\lambda$ which is invariant under leaf-preserving isotopy and
absolutely continuous with respect to the Lebesgue measure. A _measured
lamination_ on $\displaystyle S$ is a pair
$\displaystyle\left\lparen\lambda,\mu\right\rparen$, where
$\displaystyle\lambda$ is a geodesic lamination on $\displaystyle S$ and
$\displaystyle\mu$ is a tranverse measure on $\displaystyle\lambda$. We will
denote by $\displaystyle\mathcal{ML}\left\lparen S\right\rparen$ the space of
measured laminations on $\displaystyle S$ topologized by the weak* topology.

Two measured laminations on $\displaystyle S$ are called _projectively
equivalent_ if they have the same underlying geodesic lamination and their
transverse measures differ by multiplication by a positive constant. A
projective equivalence class of measured laminations is called a _projective
measured lamination_. We will denote the space of such projective measured
laminations by $\displaystyle\mathcal{PML}\left\lparen S\right\rparen$, which
is homeomorphic to a sphere of dimension $\displaystyle 6g-7$. Thurston
[Thu82, Theorem 5.3] introduced a compactification
$\displaystyle\overline{\mathscr{T}\left\lparen S\right\rparen}$ of
Teichmüller space, now known as the _Thurston compactification_ , in which
$\displaystyle\overline{\mathscr{T}\left\lparen
S\right\rparen}\setminus\mathscr{T}\left\lparen
S\right\rparen=\mathcal{PML}\left\lparen S\right\rparen$. For this reason,
$\displaystyle\mathcal{PML}\left\lparen S\right\rparen$ is called the
_Thurston boundary_ of Teichmüller space.

Many of the relevant properties of the Teichmüller space extend naturally to
the Thurston boundary and Thurston compactification. For example, the mapping
class group $\displaystyle\mathrm{Mod}\left\lparen S\right\rparen$ acts
naturally on $\displaystyle\mathcal{PML}\left\lparen S\right\rparen$ by
homeomorphisms, and this action along with the action of
$\displaystyle\mathrm{Mod}\left\lparen S\right\rparen$ on
$\displaystyle\mathscr{T}\left\lparen S\right\rparen$ jointly comprise an
action of $\displaystyle\mathrm{Mod}\left\lparen S\right\rparen$ on
$\displaystyle\overline{\mathscr{T}\left\lparen
S\right\rparen}=\mathscr{T}\left\lparen
S\right\rparen\sqcup\mathcal{PML}\left\lparen S\right\rparen$ by
homeomorphisms. Moreover, the isometric embedding of Teichmüller spaces
induced by a finite-degree cover extends to continuous embeddings of their
Thurston boundaries and Thurston compactifications.

######  Theorem 2.1 ([BMN99, Theorem 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 the induced isometric embedding $\displaystyle
p^{*}\colon\mathscr{T}\left\lparen
S\right\rparen\xhookrightarrow{}\mathscr{T}\left\lparen X\right\rparen$
extends to a continuous embedding $\displaystyle
p^{*}\colon\overline{\mathscr{T}\left\lparen
S\right\rparen}\xhookrightarrow{}\overline{\mathscr{T}\left\lparen
X\right\rparen}$.

###  2.4. Intersection numbers and the curve complex

Any pair of curves $\displaystyle\alpha$ and $\displaystyle\beta$ on
$\displaystyle S$ has a _(geometric) intersection number_ $\displaystyle
i\left\lparen\alpha,\beta\right\rparen$, which is the minimal number of
transverse crossings between $\displaystyle\alpha$ and $\displaystyle\beta$
among all parametrized representatives. When $\displaystyle\alpha=\beta$, this
number $\displaystyle i\left\lparen\alpha,\alpha\right\rparen$ is called the
_self-intersection number_. A curve $\displaystyle\alpha$ on $\displaystyle S$
is simple if and only if $\displaystyle
i\left\lparen\alpha,\alpha\right\rparen=0$, and a pair of curves
$\displaystyle\alpha$ and $\displaystyle\beta$ on $\displaystyle S$ are called
_disjoint_ if $\displaystyle i\left\lparen\alpha,\beta\right\rparen=0$.

A choice of parametrized representatives for a set of curves on $\displaystyle
S$ is said to be in _minimal position_ if every representative achieves its
self-intersection number and every pair of representatives achieves its
geometric intersection number. It is well-known that any set of closed
geodesics on a hyperbolic surface are in minimal position. In particular, this
implies that every set of curves on $\displaystyle S$ has a set of
parametrized representatives in minimal position.

The _curve complex_ $\displaystyle\mathscr{C}\left\lparen S\right\rparen$ of a
closed surface $\displaystyle S$ of genus $\displaystyle g\geq 2$ is a flag
simplicial complex introduced by Harvey [Har81]. The simplices in
$\displaystyle\mathscr{C}\left\lparen S\right\rparen$ correspond to _multi-
curves_ on $\displaystyle S$, which are collections of pairwise disjoint
simple curves on $\displaystyle S$; that is, the vertices of
$\displaystyle\mathscr{C}\left\lparen S\right\rparen$ are simple curves on
$\displaystyle S$ and a set of vertices span a simplex if they are pairwise
disjoint. There is a natural simplicial action of the mapping class group
$\displaystyle\mathrm{Mod}\left\lparen S\right\rparen$ on
$\displaystyle\mathscr{C}\left\lparen S\right\rparen$. Moreover,
$\displaystyle\mathscr{C}\left\lparen S\right\rparen$ may be equipped with a
metric $\displaystyle d_{S}$ so that every simplex is a regular Euclidean
simplex of side length $\displaystyle 1$. Harvey [Har81] observed that the
$\displaystyle 1$-skeleton of $\displaystyle\mathscr{C}\left\lparen
S\right\rparen$ is a connected metric graph.

Masur–Minsky [MM99, Theorem 1.1] proved that, considered with the metric
$\displaystyle d_{S}$, the curve complex $\displaystyle\mathscr{C}\left\lparen
S\right\rparen$ is a Gromov hyperbolic geodesic space, and Klarreich [Kla18]
constructed an explicit parametrization of its Gromov boundary by the space
$\displaystyle\mathcal{EL}\left\lparen S\right\rparen$ of ending laminations
of $\displaystyle S$, which we define in Section 2.5.

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.

###  2.5. The Gromov boundary of the curve complex

Recall that a geodesic lamination of a closed hyperbolic surface
$\displaystyle S$ is a closed subset of $\displaystyle S$ foliated by simple
geodesics, each of which is called a _leaf_. We will call a geodesic
lamination on a surface $\displaystyle S$:

  1. (i)

_minimal_ if each leaf is dense in $\displaystyle S$;

  2. (ii)

_filling_ if every component of the complement of $\displaystyle\lambda$ in
$\displaystyle S$ is simply connected; and

  3. (iii)

_ending_ if $\displaystyle\lambda$ is both minimal and filling.

Note that these properties are well-defined irrespective of the choice of
hyperbolic metric and that, necessarily, a minimal geodesic lamination cannot
contain a closed leaf.

The space $\displaystyle\mathcal{EL}\left\lparen S\right\rparen$ of ending
laminations on $\displaystyle S$ is topologized as follows: Denote by
$\displaystyle\mathcal{PMEL}\left\lparen S\right\rparen$ the space of
projective measured laminations whose underlying geodesic lamination is both
minimal and filling. There is a natural surjective map
$\displaystyle\left\lparen\>\cdot\>\right\rparen_{\mathrm{top}}\colon\mathcal{PMEL}\left\lparen
S\right\rparen\xrightarrow[]{}\mathcal{EL}\left\lparen S\right\rparen$ defined
by the formula
$\displaystyle\left[\left\lparen\lambda,\mu\right\rparen\right]_{\mathrm{top}}=\lambda$;
that is, this map sends a projective measured ending lamination to its
underlying geodesic lamination. We equip
$\displaystyle\mathcal{EL}\left\lparen S\right\rparen$ with the quotient
topology associated to this map. An equivalent characterization of this
topology is described by Hamenstädt [Ham06], in which it is called the _coarse
Hausdorff topology_.

As previously mentioned, Klarreich [Kla18] parametrized the Gromov boundary of
the curve complex $\displaystyle\mathscr{C}\left\lparen S\right\rparen$ of a
closed surface $\displaystyle S$ of negative Euler characteristic by the space
$\displaystyle\mathcal{EL}\left\lparen S\right\rparen$ of ending laminations.

######  Theorem 2.3 ([Kla18, Theorem 1.3]).

There is a natural homeomorphism
$\displaystyle\Lambda_{S}\colon\mathscr{C}\left\lparen
S\right\rparen\sqcup\partial\mathscr{C}\left\lparen
S\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen
S\right\rparen\sqcup\mathcal{EL}\left\lparen S\right\rparen$ which is the
identity on $\displaystyle\mathscr{C}\left\lparen S\right\rparen$ and
restricts to a homeomorphism between
$\displaystyle\partial\mathscr{C}\left\lparen S\right\rparen$ and
$\displaystyle\mathcal{EL}\left\lparen S\right\rparen$.

Recall that a finite-degree cover $\displaystyle p\colon X\xrightarrow[]{}S$
induces a coarsely well-defined quasi-isometric embedding
$\displaystyle\widetilde{p}\colon\mathscr{C}\left\lparen
S\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen X\right\rparen$. This in
turn induces a continuous embedding
$\displaystyle\widetilde{p}\colon\partial\mathscr{C}\left\lparen
S\right\rparen\xhookrightarrow{}\partial\mathscr{C}\left\lparen
X\right\rparen$, which is both well-defined and canonical by the discussion in
the previous subsection. By conjugating by the homeomorphisms provided in 2.3,
we obtain a continuous embedding $\displaystyle\mathcal{EL}\left\lparen
S\right\rparen\xhookrightarrow{}\mathcal{EL}\left\lparen X\right\rparen$ so
that the following diagram commutes

$\displaystyle{\partial\mathscr{C}\left\lparen
S\right\rparen}$$\displaystyle{\partial\mathscr{C}\left\lparen
X\right\rparen}$$\displaystyle{\mathcal{EL}\left\lparen
S\right\rparen}$$\displaystyle{\mathcal{EL}\left\lparen
X\right\rparen}$$\displaystyle\scriptstyle{\widetilde{p}}$$\displaystyle\scriptstyle{\Lambda_{S}}$$\displaystyle\scriptstyle{\Lambda_{X}}$

One can verify that in the presence of a fixed hyperbolic metric on
$\displaystyle S$, this map takes an ending lamination
$\displaystyle\lambda\in\mathcal{EL}\left\lparen S\right\rparen$ to its full
pre-image

$\displaystyle\left\lparen\Lambda_{X}\circ\widetilde{p}\circ\Lambda_{S}^{-1}\right\rparen\left\lparen\lambda\right\rparen=\Lambda_{X}\left\lparen\widetilde{p}\left\lparen\Lambda_{S}^{-1}\left\lparen\lambda\right\rparen\right\rparen\right\rparen=p^{-1}\left\lparen\lambda\right\rparen=\left\\{x\in
X:p\left\lparen x\right\rparen\in\lambda\right\\},$

which is an ending lamination on $\displaystyle X$ when $\displaystyle X$ is
equipped with the pullback metric.

In the rest of this article, we will implicitly use the above theorem to
identify the space of ending laminations with the Gromov boundary of the curve
complex, and we will abuse notation by writing
$\displaystyle\widetilde{p}\colon\mathcal{EL}\left\lparen
S\right\rparen\xhookrightarrow{}\mathcal{EL}\left\lparen X\right\rparen$.

###  2.6. The simple length spectrum and Sunada’s construction

The _(unmarked) length spectrum_ (resp. _simple length spectrum_) of a closed
hyperbolic surface $\displaystyle X$ is the multi-set of lengths of closed
geodesics (resp. simple closed geodesics), counted with multiplicity. We will
call two such surfaces _length isospectral_ (resp. _simple length
isospectral_) if they have the same length spectrum (resp. simple length
spectrum).

Most known examples of length isospectral surfaces arise from a construction
due to Sunada [Sun85], which we now describe. Sunada’s construction relies on
a surjective homomorphism $\displaystyle\rho\colon\pi_{1}\left\lparen
S,s\right\rparen\twoheadrightarrow G$, called the _constructing
representation_ , of a closed hyperbolic surface group
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ onto a finite group
$\displaystyle G$ and a pair of subgroups $\displaystyle A,B\leq G$, called
the _constructing subgroups_. The pre-images
$\displaystyle\rho^{-1}\left\lparen A\right\rparen$ and
$\displaystyle\rho^{-1}\left\lparen B\right\rparen$ are finite-index subgroups
of $\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ and so the Galois
correspondence between subgroups of $\displaystyle\pi_{1}\left\lparen
S,s\right\rparen$ and based covers of $\displaystyle\left\lparen
S,s\right\rparen$ yields the commutative diagram of finite-degree based covers
drawn in Figure 1.

$\displaystyle{\pi_{1}\left\lparen
S,s\right\rparen}$$\displaystyle{\left\lparen
Z,z\right\rparen}$$\displaystyle{p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen}$$\displaystyle{q_{*}\left\lparen\pi_{1}\left\lparen
Y,y\right\rparen\right\rparen}$ $\displaystyle{\left\lparen
X,x\right\rparen}$$\displaystyle{\left\lparen
Y,y\right\rparen}$$\displaystyle{\mathrm{ker}\left\lparen\rho\right\rparen}$$\displaystyle{\left\lparen
S,s\right\rparen}$$\displaystyle\scriptstyle{\pi_{X}}$$\displaystyle\scriptstyle{\pi_{Y}}$$\displaystyle\scriptstyle{\pi_{S}}$$\displaystyle\scriptstyle{p}$$\displaystyle\scriptstyle{q}$
Figure 1. A Sunada covering diamond from the Galois correspondence

We will call a commutative diagram of this form a _(based) Sunada covering
diamond_. Note that since
$\displaystyle\mathrm{ker}\left\lparen\rho\right\rparen$ is normal in
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$,
$\displaystyle\pi_{S}\colon Z\xrightarrow[]{}S$ is regular, with deck group

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

where $\displaystyle N_{\pi_{1}\left\lparen
S,s\right\rparen}\left\lparen\mathrm{ker}\left\lparen\rho\right\rparen\right\rparen$
is the normalizer of $\displaystyle\mathrm{ker}\left\lparen\rho\right\rparen$
in $\displaystyle\pi_{1}\left\lparen S,s\right\rparen$. Similarly,
$\displaystyle\pi_{X}$ and $\displaystyle\pi_{Y}$ are regular, with deck
groups $\displaystyle\mathrm{Deck}\left\lparen\pi_{X}\right\rparen\cong A$ and
$\displaystyle\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen\cong B$.

The constructing subgroups $\displaystyle A,B\leq G$ of the finite group
$\displaystyle G$ are called _almost conjugate_ (or _Gassmann equivalent_) if
they contain the same number of elements of every conjugacy class in
$\displaystyle G$; that is, $\displaystyle A$ and $\displaystyle B$ are almost
conjugate in $\displaystyle G$ if $\displaystyle\left\lvert A\cap
C\right\rvert=\left\lvert B\cap C\right\rvert$ for every conjugacy class
$\displaystyle C\subseteq G$. Note that two conjugate subgroups of
$\displaystyle G$ are almost conjugate, but almost conjugate subgroups need
not be conjugate in $\displaystyle G$.

######  Theorem 2.4 ([Sun85]).

Let $\displaystyle p\colon X\xrightarrow[]{}S$ and $\displaystyle q\colon
Y\xrightarrow[]{}S$ be the finite-degree covers of a closed surface
$\displaystyle S$ of negative Euler characteristic associated to a surjective
homomorphism $\displaystyle\rho\colon\pi_{1}\left\lparen
S,s\right\rparen\twoheadrightarrow G$ and a pair of subgroups $\displaystyle
A,B\leq G$ as described above.

  1. 1.

If $\displaystyle A$ and $\displaystyle B$ are almost conjugate in
$\displaystyle G$, then for every $\displaystyle\mu\in\mathscr{T}\left\lparen
S\right\rparen$, $\displaystyle p^{*}\left\lparen\mu\right\rparen$ and
$\displaystyle q^{*}\left\lparen\mu\right\rparen$ are length isospectral; and

  2. 2.

If $\displaystyle A$ and $\displaystyle B$ are almost conjugate but not
conjugate in $\displaystyle G$, then for a generic
$\displaystyle\mu\in\mathscr{T}\left\lparen S\right\rparen$, $\displaystyle
p^{*}\left\lparen\mu\right\rparen$ and $\displaystyle
q^{*}\left\lparen\mu\right\rparen$ are not isometric.

The first part of Sunada’s theorem can be stated in more general contexts
involving actions of finite groups on complete and connected Riemannian
manifolds by isometries with at most finitely many fixed points. A proof of
this more general form (which implies the first part of 2.4) can be found in
[Bus10, 291-294].

Fix two finite-degree covers $\displaystyle p\colon X\xrightarrow[]{}S$ and
$\displaystyle q\colon Y\xrightarrow[]{}S$ of a closed, orientable surface
$\displaystyle S$ of negative Euler characteristic. For a given
$\displaystyle\mu\in\mathscr{T}\left\lparen S\right\rparen$ on $\displaystyle
S$, we will say that $\displaystyle X$ and $\displaystyle Y$ are _simple
length isospectral over $\displaystyle S$_ if $\displaystyle\left\lparen
X,p^{*}\left\lparen\mu\right\rparen\right\rparen$ and
$\displaystyle\left\lparen Y,q^{*}\left\lparen\mu\right\rparen\right\rparen$
are simple length isospectral. The manifold structure on
$\displaystyle\mathscr{T}\left\lparen S\right\rparen$ (or equivalently, the
Weil–Petersson measure on $\displaystyle\mathscr{T}\left\lparen
S\right\rparen$) gives a notion of genericness for these kinds of conditions;
we say that $\displaystyle X$ and $\displaystyle Y$ generically have a certain
property over $\displaystyle S$ if the set of all
$\displaystyle\mu\in\mathscr{T}\left\lparen S\right\rparen$ for which
$\displaystyle\left\lparen X,p^{*}\left\lparen\mu\right\rparen\right\rparen$
and $\displaystyle\left\lparen
Y,q^{*}\left\lparen\mu\right\rparen\right\rparen$ do not have that property
has null measure in $\displaystyle\mathscr{T}\left\lparen S\right\rparen$. In
this language, we can rephrase 2.4 as saying that when the constructing
subgroups are almost conjugate but not conjugate, the resulting covers are
isospectral but generically not isometric.

###  2.7. Trace-commensurability in free and surface groups

The _character_ of an element $\displaystyle\alpha$ in a group
$\displaystyle\Gamma$ is the complex-valued function
$\displaystyle\mathrm{tr}\left\lparen\alpha\right\rparen$ on the space
$\displaystyle\mathrm{hom}\left\lparen\Gamma,\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen\right\rparen$
of linear representations of $\displaystyle\Gamma$ in
$\displaystyle\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen$ defined by
the formula
$\displaystyle\mathrm{tr}\left\lparen\alpha\right\rparen\left\lparen\rho\right\rparen=\mathrm{tr}\left\lparen\rho\left\lparen\alpha\right\rparen\right\rparen$.
We will call two elements $\displaystyle\alpha,\beta\in\Gamma$ _trace-
commensurable_ if there are non-zero integers $\displaystyle m,n$ so that
$\displaystyle\mathrm{tr}\left\lparen\alpha^{m}\right\rparen^{2}=\mathrm{tr}\left\lparen\beta^{n}\right\rparen^{2}$
as functions on
$\displaystyle\mathrm{hom}\left\lparen\Gamma,\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen\right\rparen$.
More explicitly, $\displaystyle\alpha$ and $\displaystyle\beta$ are trace-
commensurable if
$\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\alpha^{m}\right\rparen\right\rparen^{2}=\mathrm{tr}\left\lparen\rho\left\lparen\beta^{n}\right\rparen\right\rparen^{2}$
for all linear representations
$\displaystyle\rho\colon\Gamma\xrightarrow[]{}\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen$.

This equivalence relation on $\displaystyle\Gamma$ is a natural generalization
of the notion of _trace equivalence_ defined in [Lei03], in which
$\displaystyle\Gamma$ is a closed hyperbolic surface group and $\displaystyle
m=n=1$.

######  Observation 2.5.

Consider two elements $\displaystyle\alpha$ and $\displaystyle\beta$ of a
group $\displaystyle\Gamma$.

  1. (i)

If $\displaystyle\alpha$ and $\displaystyle\beta$ are trace-commensurable,
then there are non-zero integers $\displaystyle m$ and $\displaystyle n$ such
that
$\displaystyle\mathrm{tr}\left\lparen\alpha^{m}\right\rparen^{2}=\mathrm{tr}\left\lparen\beta^{n}\right\rparen^{2}$.
However, the trace relation

$\displaystyle\mathrm{tr}\left\lparen\mathbf{A}\right\rparen\cdot\mathrm{tr}\left\lparen\mathbf{B}\right\rparen=\mathrm{tr}\left\lparen\mathbf{A}\mathbf{B}\right\rparen+\mathrm{tr}\left\lparen\mathbf{A}\mathbf{B}^{-1}\right\rparen$

for matrices
$\displaystyle\mathbf{A},\mathbf{B}\in\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen$
implies that

$\displaystyle\mathrm{tr}\left\lparen\alpha^{2m}\right\rparen=\mathrm{tr}\left\lparen\alpha^{m}\right\rparen^{2}-2=\mathrm{tr}\left\lparen\beta^{n}\right\rparen^{2}-2=\mathrm{tr}\left\lparen\beta^{2n}\right\rparen.$

  2. (ii)

Suppose that $\displaystyle\alpha$ and $\displaystyle\beta$ are trace-
commensurable, so that there are non-zero integers $\displaystyle m$ and
$\displaystyle n$ such that
$\displaystyle\mathrm{tr}\left\lparen\alpha^{m}\right\rparen^{2}=\mathrm{tr}\left\lparen\beta^{n}\right\rparen^{2}$.
Let $\displaystyle\varphi\colon\Gamma\xrightarrow[]{}\Delta$ be a group
homomorphism and note that for any linear representation
$\displaystyle\rho\colon\Delta\xrightarrow[]{}\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen$
of $\displaystyle\Delta$, the composition
$\displaystyle\rho\circ\varphi\colon\Gamma\xrightarrow[]{}\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen$
is a linear representation of $\displaystyle\Gamma$, which implies that

$\displaystyle\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\varphi\left\lparen\alpha\right\rparen^{m}\right\rparen\right\rparen^{2}$
$\displaystyle\displaystyle=\mathrm{tr}\left\lparen\rho\left\lparen\varphi\left\lparen\alpha^{m}\right\rparen\right\rparen\right\rparen^{2}=\mathrm{tr}\left\lparen\left\lparen\rho\circ\varphi\right\rparen\left\lparen\alpha^{m}\right\rparen\right\rparen^{2}=\mathrm{tr}\left\lparen\left\lparen\rho\circ\varphi\right\rparen\left\lparen\alpha^{m}\right\rparen^{2}\right\rparen$
$\displaystyle\displaystyle=\mathrm{tr}\left\lparen\left\lparen\rho\circ\varphi\right\rparen\left\lparen\beta^{n}\right\rparen^{2}\right\rparen=\mathrm{tr}\left\lparen\left\lparen\rho\circ\varphi\right\rparen\left\lparen\beta^{n}\right\rparen\right\rparen^{2}=\mathrm{tr}\left\lparen\rho\left\lparen\varphi\left\lparen\beta^{n}\right\rparen\right\rparen\right\rparen^{2}=\mathrm{tr}\left\lparen\rho\left\lparen\varphi\left\lparen\beta\right\rparen^{n}\right\rparen\right\rparen^{2}.$

Thus $\displaystyle\varphi\left\lparen\alpha\right\rparen$ and
$\displaystyle\varphi\left\lparen\beta\right\rparen$ are trace-commensurable
as elements of $\displaystyle\Delta$, with the same commensurability
constants.

We will call two curves $\displaystyle\alpha$ and $\displaystyle\beta$ on
$\displaystyle S$ _(hyperbolic) length-commensurable_ if there are positive
integers $\displaystyle m,n\geq 1$ such that

$\displaystyle
m\cdot\ell_{\mu}\left\lparen\alpha\right\rparen=n\cdot\ell_{\mu}\left\lparen\beta\right\rparen,$

for all points $\displaystyle\mu\in\mathscr{T}\left\lparen S\right\rparen$.
This equivalence relation on the space of currents is a natural generalization
of the notion of _hyperbolic equivalence_ of curves as in [Lei03], in which
$\displaystyle\alpha$ and $\displaystyle\beta$ are primitive curves and
$\displaystyle m=n=1$. Leininger gave an equivalent characterization in terms
of certain characters of representations of the fundamental group, which we
now present.

Fix a basepoint $\displaystyle s\in S$ on a closed, orientable surface
$\displaystyle S$ of negative Euler characteristic, and suppose that two
elements $\displaystyle\alpha,\beta\in\pi_{1}\left\lparen S,s\right\rparen$
are trace-commensurable, so that there are non-zero integers $\displaystyle
m,n$ such that
$\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen\alpha\right\rparen^{m}\right\rparen^{2}=\mathrm{tr}\left\lparen\rho\left\lparen\beta\right\rparen^{n}\right\rparen^{2}$
for all linear representations $\displaystyle\rho\colon\pi_{1}\left\lparen
S,s\right\rparen\xrightarrow[]{}\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen$.
Note that any point $\displaystyle\mu\in\mathscr{T}\left\lparen
S\right\rparen$ is represented by a discrete and faithful representation
$\displaystyle\pi_{1}\left\lparen
S,s\right\rparen\xrightarrow[]{}\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen$,
which lifts to a linear representation $\displaystyle\pi_{1}\left\lparen
S,s\right\rparen\xrightarrow[]{}\mathrm{SL}_{2}\left\lparen\mathbb{R}\right\rparen\subsetneq\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen$.
Thus

$\displaystyle\displaystyle\left\lvert
m\right\rvert\cdot\ell_{\mu}\left\lparen\alpha\right\rparen$
$\displaystyle\displaystyle=\ell_{\mu}\left\lparen\alpha^{m}\right\rparen=2\mathrm{cosh}^{-1}\left\lparen\frac{\left\lvert\mathrm{tr}\left\lparen\rho\left\lparen\alpha^{m}\right\rparen\right\rparen\right\rvert}{2}\right\rparen$
$\displaystyle\displaystyle=2\mathrm{cosh}^{-1}\left\lparen\frac{\left\lvert\mathrm{tr}\left\lparen\rho\left\lparen\beta^{n}\right\rparen\right\rparen\right\rvert}{2}\right\rparen=\ell_{\mu}\left\lparen\beta^{n}\right\rparen=\left\lvert
n\right\rvert\cdot\ell_{\mu}\left\lparen\beta\right\rparen.$

In summary, the above argument shows that trace-commensurable elements of
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ represent length-
commensurable curves on $\displaystyle S$. The following theorem of Leininger
provides a converse. Note that Leininger considers the slightly more
restrictive relationships of trace equivalence and hyperbolic equivalence
mentioned above.

######  Theorem 2.6 ([Lei03, Theorem 1.4]).

Fix a basepoint $\displaystyle s\in S$ on a closed, orientable surface
$\displaystyle S$ of negative Euler characteristic. Two elements
$\displaystyle\alpha,\beta\in\pi_{1}\left\lparen S,s\right\rparen$ are trace-
commensurable in $\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ if and
only if the curves $\displaystyle\left[\alpha\right]$ and
$\displaystyle\left[\beta\right]$ they represent are length-commensurable.

Moreover, for any two trace-commensurable elements
$\displaystyle\alpha,\beta\in\pi_{1}\left\lparen S,s\right\rparen$
representing primitive curves and any simple curve $\displaystyle\gamma$ on
$\displaystyle S$, $\displaystyle
i\left\lparen\left[\alpha\right],\gamma\right\rparen=0$ if and only if
$\displaystyle i\left\lparen\left[\beta\right],\gamma\right\rparen=0$.

Consider elements $\displaystyle\alpha$ and $\displaystyle\beta$ of a group
$\displaystyle\Gamma$. If some non-zero powers $\displaystyle\alpha^{m}$ and
$\displaystyle\beta^{n}$ are conjugate in $\displaystyle\Gamma$, then the
conjugation-invariance of the trace implies that $\displaystyle\alpha$ and
$\displaystyle\beta$ are trace-commensurable. However, not all pairs of trace-
commensurable elements need be of this form. We will call such a pair of
elements, which are trace-commensurable but not for the aforementioned reason,
_trace twins_.

It may be surprising to learn that examples of trace twins exist in many
groups, including free groups and surface groups. For example, consider the
free group $\displaystyle F_{2}=\left\langle a,b\mid\right\rangle$ on two
generators $\displaystyle a$ and $\displaystyle b$. Horowitz computes [Hor72,
Example 5.2] that

$\displaystyle\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen
a^{2}b^{-1}ab\right\rparen\right\rparen^{2}$
$\displaystyle\displaystyle=\mathrm{tr}\left\lparen\rho\left\lparen
a^{2}bab^{-1}\right\rparen\right\rparen^{2}$
$\displaystyle\displaystyle\mathrm{tr}\left\lparen\rho\left\lparen
ab^{2}a^{2}b\right\rparen\right\rparen^{2}$
$\displaystyle\displaystyle=\mathrm{tr}\left\lparen\rho\left\lparen
aba^{2}b^{2}\right\rparen\right\rparen^{2},$

under any representation $\displaystyle\rho\colon
F_{2}\xrightarrow[]{}\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen$, but
no powers of these elements are conjugate to each other. Thus $\displaystyle
a^{2}b^{-1}ab$ and $\displaystyle a^{2}bab^{-1}$ are trace twins in
$\displaystyle F_{2}$ and so are $\displaystyle ab^{2}a^{2}b$ and
$\displaystyle aba^{2}b^{2}$.

By 2.6 above, trace twins in a surface group yield homotopically distinct
curves on the surface whose lengths are commensurable in any hyperbolic
metric. As described in greater detail in Section 4, we will call such a pair
of curves _length twins_. We will find it very useful to find elements in a
group which have no trace twins. Horowitz gives useful characterizations of
such elements in free groups and uses them to build a wealth of examples.

######  Theorem 2.7 ([Hor72, Corollary 7.2]).

Let $\displaystyle a,b\in F$ be primitive elements of a free group
$\displaystyle F$. For any integers $\displaystyle j,k\in\mathbb{Z}$ and
element $\displaystyle u\in F$, if $\displaystyle a^{j}b^{k}$ and
$\displaystyle u$ are trace equivalent in $\displaystyle F$, then
$\displaystyle u$ is conjugate in $\displaystyle F$ to either $\displaystyle
a^{j}b^{k}$ or its inverse.

In Section 4.1, we extend Horowitz’s original argument to show that elements
of the form $\displaystyle a^{j}b^{k}$ have no trace twins in $\displaystyle
F$ (this involves dealing with commensurability constants other than
$\displaystyle 1$), and then derive a consequence for the lengths of certain
curves on surfaces represented by such elements.

##  3\. Covers and simple elevations

In this section, we prove Section 1.1 and then deduce 3.6, a corollary related
to Sunada’s theorem. In Section 3.1 and Section 3.2, we prove some preliminary
technical lemmas concerning the space of projective measured laminations and
the deck group of a finite-degree cover, respectively. We use these results in
the proof of Section 1.1 in Section 3.3.

###  3.1. Uniquely ergodic ending laminations

A geodesic lamination is _uniquely ergodic_ if it admits a single projective
equivalence class of transverse measures. Masur’s criterion provides a
sufficient condition for the endpoint of a Teichmüller geodesic to be uniquely
ergodic based on recurrence to the thick part of Teichmüller space, which we
now define: given a positive real number $\displaystyle\epsilon>0$, the
_$\displaystyle\epsilon$ -thick part_ of $\displaystyle\mathscr{T}\left\lparen
X\right\rparen$ is the subset corresponding to metrics with injectivity radius
strictly greater than $\displaystyle\epsilon$. In other words, a point
$\displaystyle\left[\mu\right]\in\mathscr{T}\left\lparen X\right\rparen$ is in
the $\displaystyle\epsilon$-thick part if $\displaystyle\left\lparen
X,\mu\right\rparen$ has no closed geodesic with length at most
$\displaystyle\epsilon$.

######  Theorem 3.1 ([Mas92]).

Let $\displaystyle R$ be a geodesic ray in
$\displaystyle\mathscr{T}\left\lparen X\right\rparen$ with endpoint
$\displaystyle\lambda\in\mathcal{PML}\left\lparen X\right\rparen$. If there is
a positive real number $\displaystyle\epsilon>0$ such that $\displaystyle R$
recurs infinitely often to the $\displaystyle\epsilon$-thick part of
$\displaystyle\mathscr{T}\left\lparen X\right\rparen$, then the underlying
lamination $\displaystyle\lambda_{\mathrm{top}}$ is uniquely ergodic.

The following result of Lindenstrauss–Mirzakhani implies that the mapping
class group orbit of any uniquely ergodic measured lamination is dense in the
space of measured laminations.

######  Theorem 3.2 ([LM08, Theorem 8.9]).

A measured lamination $\displaystyle\lambda\in\mathcal{ML}\left\lparen
X\right\rparen$ on a closed, orientable surface $\displaystyle X$ of negative
Euler characteristic has dense $\displaystyle\mathrm{Mod}\left\lparen
X\right\rparen$-orbit if and only if its support contains no simple closed
curves.

We now use the above two results to show that the space
$\displaystyle\mathcal{PMEL}_{\epsilon}\left\lparen X\right\rparen$ of
projective measured ending laminations whose underlying laminations are the
endpoints of Teichmüller geodesic rays in
$\displaystyle\mathscr{T}\left\lparen X\right\rparen$ which recur infinitely
often to the $\displaystyle\epsilon$-thick part is dense in
$\displaystyle\mathcal{PML}\left\lparen X\right\rparen$. We will use this in
the proof of Section 1.1.

######  Lemma 3.3.

For any closed, orientable surface $\displaystyle X$ of negative Euler
characteristic, there is a positive real number $\displaystyle\epsilon_{X}>0$
such that $\displaystyle\mathcal{PMEL}_{\epsilon}\left\lparen X\right\rparen$
is dense in $\displaystyle\mathcal{PML}\left\lparen X\right\rparen$ for any
$\displaystyle 0<\epsilon\leq\epsilon_{X}$.

###### Proof.

For any positive real number $\displaystyle\epsilon>0$, let
$\displaystyle\mathcal{MEL}_{\epsilon}\left\lparen X\right\rparen\coloneqq
F^{-1}\left\lparen\mathcal{PMEL}_{\epsilon}\left\lparen
X\right\rparen\right\rparen$, where $\displaystyle
F\colon\mathcal{ML}\left\lparen
X\right\rparen\twoheadrightarrow\mathcal{PML}\left\lparen X\right\rparen$ is
the quotient map. We will first show that there exists
$\displaystyle\epsilon_{X}>0$ so that
$\displaystyle\mathcal{MEL}_{\epsilon_{X}}\left\lparen X\right\rparen$ is
dense in $\displaystyle\mathcal{ML}\left\lparen X\right\rparen$.

Let $\displaystyle\phi\in\mathrm{Mod}\left\lparen X\right\rparen$ be a pseudo-
Anosov mapping class with stable lamination
$\displaystyle\lambda_{\mathrm{top}}\in\mathcal{EL}\left\lparen
X\right\rparen$. There is a positive real number
$\displaystyle\epsilon_{X}>0$, which we fix for the remainder of this proof,
so that the axis $\displaystyle R$ of $\displaystyle\phi$ in
$\displaystyle\mathscr{T}\left\lparen X\right\rparen$ is contained in the
$\displaystyle\epsilon_{X}$-thick part of
$\displaystyle\mathscr{T}\left\lparen X\right\rparen$. This implies that any
measured lamination $\displaystyle\lambda$ with underlying lamination
$\displaystyle\lambda_{\mathrm{top}}$ has the property that $\displaystyle
F(\lambda)$ lies in $\displaystyle\mathcal{PMEL}_{\epsilon_{X}}\left\lparen
X\right\rparen$. Fix such a measured lamination $\displaystyle\lambda$.

Since $\displaystyle\mathrm{Mod}\left\lparen X\right\rparen$ acts by
isometries of the Teichmüller metric which preserve the
$\displaystyle\epsilon_{X}$-thick part of
$\displaystyle\mathscr{T}\left\lparen X\right\rparen$,
$\displaystyle\mathcal{MEL}_{\epsilon_{X}}\left\lparen X\right\rparen$ is
invariant under this action. Indeed, for any mapping class $\displaystyle
g\in\mathrm{Mod}\left\lparen X\right\rparen$, $\displaystyle F\left\lparen
g\cdot\lambda\right\rparen=g\cdot F\left\lparen\lambda\right\rparen$ is an
endpoint of the Teichmüller geodesic $\displaystyle g\cdot R$, which is
contained in the $\displaystyle\epsilon_{X}$-thick part of
$\displaystyle\mathscr{T}\left\lparen X\right\rparen$. Thus,
$\displaystyle\mathcal{MEL}_{\epsilon_{X}}\left\lparen X\right\rparen$
contains the entire orbit $\displaystyle\mathrm{Mod}\left\lparen
X\right\rparen\cdot\lambda$.

3.1 implies that the underlying lamination
$\displaystyle\lambda_{\mathrm{top}}$ of $\displaystyle\lambda$ is uniquely
ergodic, and so 3.2 implies that its orbit under the mapping class group
$\displaystyle\mathrm{Mod}\left\lparen X\right\rparen$ is dense in
$\displaystyle\mathcal{ML}\left\lparen X\right\rparen$. Since
$\displaystyle\mathcal{MEL}_{\epsilon_{X}}\left\lparen X\right\rparen$
contains this orbit by the above argument, it is also dense in
$\displaystyle\mathcal{ML}\left\lparen X\right\rparen$. Therefore, since
$\displaystyle\mathcal{PMEL}_{\epsilon_{X}}\left\lparen
X\right\rparen=F\left\lparen\mathcal{MEL}_{\epsilon_{X}}\left\lparen
X\right\rparen\right\rparen$ is the image of a dense subset under a continuous
surjection, it is dense in $\displaystyle\mathcal{PML}\left\lparen
X\right\rparen$. Finally, note that for any $\displaystyle
0<\epsilon\leq\epsilon_{X}$,
$\displaystyle\mathcal{PMEL}_{\epsilon_{X}}\left\lparen
X\right\rparen\subseteq\mathcal{PMEL}_{\epsilon}\left\lparen X\right\rparen$
and so $\displaystyle\mathcal{PMEL}_{\epsilon}\left\lparen X\right\rparen$ is
also dense in $\displaystyle\mathcal{PML}\left\lparen X\right\rparen$, as
desired. ∎

###  3.2. The deck group of a finite-degree cover

The pullback along a finite-degree regular cover of a hyperbolic metric on the
base is a hyperbolic metric on the total space for which all deck
transformations are isometries. If the base is a closed surface of genus at
least three, these deck transformations generically comprise the entire
isometry group.

######  Lemma 3.4.

For any finite-degree regular cover $\displaystyle\pi\colon
Z\xrightarrow[]{}X$ of a closed, orientable surface $\displaystyle X$ of genus
at least $\displaystyle 3$, there is a hyperbolic metric $\displaystyle\mu$ on
$\displaystyle X$ such that $\displaystyle\mathrm{Isom}\left\lparen
Z,\pi^{*}\mu\right\rparen=\mathrm{Deck}\left\lparen\pi\right\rparen$.

###### Proof.

For any hyperbolic metric $\displaystyle\mu$ on $\displaystyle X$, it follows
directly from the definitions of the deck group and the pullback metric that
$\displaystyle\mathrm{Deck}\left\lparen\pi\right\rparen\leq\mathrm{Isom}\left\lparen
Z,\pi^{*}\mu\right\rparen$. It therefore suffices to show there is a
hyperbolic metric $\displaystyle\mu$ on $\displaystyle X$ so that the reverse
containment holds.

Fix basepoints $\displaystyle x\in X$ and $\displaystyle z\in Z$ such that
$\displaystyle\pi\left\lparen z\right\rparen=x$, and let
$\displaystyle\rho_{\mu}\colon\pi_{1}\left\lparen
X,x\right\rparen\xhookrightarrow{}\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen$
be an associated holonomy representation. Such a choice is well-defined up to
conjugation by an element of
$\displaystyle\mathrm{PGL}_{2}\left\lparen\mathbb{R}\right\rparen$. We will
denote

$\displaystyle\displaystyle\Gamma$
$\displaystyle\displaystyle\coloneqq\rho_{\mu}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen,$ $\displaystyle\displaystyle\Delta$
$\displaystyle\displaystyle\coloneqq\rho_{\pi^{*}\mu}\left\lparen\pi_{1}\left\lparen
Z,z\right\rparen\right\rparen,$

where
$\displaystyle\rho_{\pi^{*}\mu}\coloneqq\rho_{\mu}\circ\pi_{*}\colon\pi_{1}\left\lparen
Z,z\right\rparen\xhookrightarrow{}\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen$
is a holonomy representation of the cover $\displaystyle\left\lparen
Z,\pi^{*}\mu\right\rparen$. Recall that the _commensurator_ of
$\displaystyle\Gamma$ in
$\displaystyle\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen$ is defined
by

$\displaystyle\mathrm{Comm}\left\lparen\Gamma\right\rparen\coloneqq\left\\{A\in\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen:\Gamma\cap
A\Gamma A^{-1}\textrm{ has finite index in both }\Gamma\textrm{ and }A\Gamma
A^{-1}\right\\},$

and that when $\displaystyle\Gamma$ is non-arithmetic,
$\displaystyle\mathrm{Comm}\left\lparen\Gamma\right\rparen$ is a lattice, and
the quotient
$\displaystyle\mathbb{H}^{2}/\mathrm{Comm}\left\lparen\Gamma\right\rparen$ is
the unique minimal area representative in the commensurability class of
$\displaystyle\Gamma$. Moreover, since
$\displaystyle\Gamma\leq\mathrm{Comm}\left\lparen\Gamma\right\rparen$ and
$\displaystyle\mathrm{Comm}\left\lparen\Delta\right\rparen=\mathrm{Comm}\left\lparen\Gamma\right\rparen$,
there are covering maps
$\displaystyle\mathbb{H}^{2}/\Gamma\xrightarrow[]{}\mathbb{H}^{2}/\mathrm{Comm}\left\lparen\Gamma\right\rparen$
and
$\displaystyle\mathbb{H}^{2}/\Delta\xrightarrow[]{}\mathbb{H}^{2}/\mathrm{Comm}\left\lparen\Gamma\right\rparen$.

For the rest of the proof, we fix a metric $\displaystyle\mu$ on
$\displaystyle X$ for which
$\displaystyle\mathrm{Comm}\left\lparen\Gamma\right\rparen=\Gamma$. By [Gre63,
Theorems 2, 3A], the set of points in Teichmüller space represented by such
metrics is a dense open subset of $\displaystyle\mathscr{T}\left\lparen
X\right\rparen$ when $\displaystyle X$ has genus at least $\displaystyle 3$.
Note that with this choice of $\displaystyle\mu$, $\displaystyle\Gamma$ is
necessarily non-arithmetic by the Margulis dichotomy [Mar75], as otherwise
$\displaystyle\mathrm{Comm}\left\lparen\Gamma\right\rparen$ would be dense in
$\displaystyle\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen$.

Recall that $\displaystyle\mathrm{Isom}\left\lparen Z,\pi^{*}\mu\right\rparen$
is canonically isomorphic to $\displaystyle
N_{\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen}\left\lparen\Delta\right\rparen/\Delta$,
where

$\displaystyle
N_{\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen}\left\lparen\Delta\right\rparen\coloneqq\left\\{A\in\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen:A\Delta
A^{-1}=\Delta\right\\},$

is the normalizer of $\displaystyle\Delta$ in
$\displaystyle\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen$. Note that

$\displaystyle\Delta\leq
N_{\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen}\left\lparen\Delta\right\rparen\leq\mathrm{Comm}\left\lparen\Delta\right\rparen=\mathrm{Comm}\left\lparen\Gamma\right\rparen=\Gamma.$

Since $\displaystyle\pi$ is a regular cover, $\displaystyle\Delta$ is normal
in each of these groups, and so

$\displaystyle\mathrm{Isom}\left\lparen
Z,\pi^{*}\mu\right\rparen\cong\frac{N_{\mathrm{PSL}_{2}\left\lparen\mathbb{R}\right\rparen}\left\lparen\Delta\right\rparen}{\Delta}\leq\frac{\mathrm{Comm}\left\lparen\Gamma\right\rparen}{\Delta}=\frac{\Gamma}{\Delta}\cong\mathrm{Deck}\left\lparen\pi\right\rparen.$

By the remark at the beginning of the proof, we see that for any such choice
of $\displaystyle\mu$ as above, $\displaystyle\mathrm{Isom}\left\lparen
Z,\pi^{*}\mu\right\rparen=\mathrm{Deck}\left\lparen\pi\right\rparen$ as
required. ∎

We remark that the key reason for the hypothesis in 3.4 that $\displaystyle X$
has genus at least $\displaystyle 3$ is due to the hyper-elliptic involution
of surfaces of genus $\displaystyle 2$. Specifically, due to the presence of
this isometry, we cannot conclude that
$\displaystyle\mathrm{Comm}\left\lparen\Gamma\right\rparen=\Gamma$ for a
generic metric; in this case, we can only show that
$\displaystyle\left[{\mathrm{Comm}\left\lparen\Gamma\right\rparen}:{\Gamma}\right]\leq
2$ and therefore that $\displaystyle\mathrm{Isom}\left\lparen
Z,\pi^{*}\left\lparen\mu\right\rparen\right\rparen$ is at worst a
$\displaystyle\mathbb{Z}/2\mathbb{Z}$ extension of
$\displaystyle\mathrm{Deck}\left\lparen\pi\right\rparen$ for generic
$\displaystyle\mu\in\mathscr{T}\left\lparen X\right\rparen$.

3.4 has the following consequences for the collection of mapping classes
corresponding to deck transformations of a finite-degree cover.

######  Proposition 3.5.

Let $\displaystyle\pi\colon Z\xrightarrow[]{}X$ be a finite-degree cover of a
closed, orientable surface $\displaystyle X$ of negative Euler characteristic
and consider the quotient map $\displaystyle
Q\colon\mathrm{Homeo}^{+}\left\lparen
Z\right\rparen\twoheadrightarrow\mathrm{Mod}\left\lparen Z\right\rparen$. The
restriction of $\displaystyle Q$ to the deck group
$\displaystyle\mathrm{Deck}\left\lparen\pi\right\rparen$ is injective.
Moreover, if $\displaystyle\pi$ is regular and $\displaystyle X$ has genus at
least $\displaystyle 3$, then $\displaystyle Q$ has image

$\displaystyle
Q\left\lparen\mathrm{Deck}\left\lparen\pi\right\rparen\right\rparen=\left\\{\varphi\in\mathrm{Mod}\left\lparen
Z\right\rparen:\varphi\cdot\pi^{*}\left\lparen\mu\right\rparen=\pi^{*}\left\lparen\mu\right\rparen\textrm{
for all }\mu\in\mathscr{T}\left\lparen X\right\rparen\right\\}.$

###### Proof.

Let $\displaystyle\varphi\in\mathrm{Deck}\left\lparen\pi\right\rparen$, fix an
orientation and a hyperbolic metric on $\displaystyle X$, which pullback along
$\displaystyle\pi$ to an orientation and a hyperbolic metric on $\displaystyle
Z$. Then any deck transformation of $\displaystyle\pi$ is an orientation-
preserving isometry of $\displaystyle X$.

Fix a non-separating simple closed geodesic $\displaystyle\gamma$ on
$\displaystyle X$ and a component $\displaystyle\alpha$ of its full pre-image
on $\displaystyle Z$, which is also a non-separating simple closed geodesic.
Since $\displaystyle\alpha$ is simple and non-separating, there is a simple
closed geodesic $\displaystyle\beta$ on $\displaystyle Z$ which intersects it
exactly once. Since $\displaystyle\varphi$ is an isometry,
$\displaystyle\varphi\left\lparen\alpha\right\rparen$ and
$\displaystyle\varphi\left\lparen\beta\right\rparen$ are simple closed
geodesics on $\displaystyle Z$ which intersect exactly once.

Suppose that $\displaystyle Q\left\lparen\varphi\right\rparen$ is trivial in
$\displaystyle\mathrm{Mod}\left\lparen Z\right\rparen$. Then $\displaystyle
Q\left\lparen\varphi\right\rparen$ acts trivially on the homotopy classes of
$\displaystyle\alpha$ and $\displaystyle\beta$, and so
$\displaystyle\varphi\left\lparen\alpha\right\rparen$ and
$\displaystyle\varphi\left\lparen\beta\right\rparen$ are homotopic to
$\displaystyle\alpha$ and $\displaystyle\beta$, respectively. Since the metric
on $\displaystyle Z$ is negatively curved, closed geodesics are unique within
their homotopy class, and hence $\displaystyle\varphi$ preserves
$\displaystyle\alpha$ and $\displaystyle\beta$. This implies that
$\displaystyle\varphi$ fixes the unique intersection point between
$\displaystyle\alpha$ and $\displaystyle\beta$. Since any non-trivial element
of the deck group $\displaystyle\mathrm{Deck}\left\lparen\pi\right\rparen$
acts freely on the total space $\displaystyle Z$, this implies that
$\displaystyle\varphi=\mathrm{id}_{Z}$ is trivial in
$\displaystyle\mathrm{Homeo}^{+}\left\lparen Z\right\rparen$.

Now suppose that $\displaystyle\pi$ is regular and that $\displaystyle X$ has
genus at least $\displaystyle 3$. The containment

$\displaystyle
Q\left\lparen\mathrm{Deck}\left\lparen\pi\right\rparen\right\rparen\subseteq\left\\{\varphi\in\mathrm{Mod}\left\lparen
Z\right\rparen:\varphi\cdot\pi^{*}\left\lparen\mu\right\rparen=\pi^{*}\left\lparen\mu\right\rparen\textrm{
for all }\mu\in\mathscr{T}\left\lparen X\right\rparen\right\\},$

is clear, since pullbacks of metrics along $\displaystyle\pi$ are invariant
under the deck group $\displaystyle\mathrm{Deck}\left\lparen\pi\right\rparen$.
To show the other containment, fix a hyperbolic metric $\displaystyle\mu$ on
$\displaystyle X$ as in 3.4; that is, we require
$\displaystyle\mathrm{Isom}\left\lparen
Z,\pi^{*}\mu\right\rparen=\mathrm{Deck}\left\lparen\pi\right\rparen$. Let
$\displaystyle\varphi\in\mathrm{Homeo}^{+}\left\lparen Z\right\rparen$, and
suppose that $\displaystyle
Q\left\lparen\varphi\right\rparen\cdot\pi^{*}\left\lparen\left[\mu\right]\right\rparen=\pi^{*}\left\lparen\left[\mu\right]\right\rparen$.
Then

$\displaystyle\displaystyle\left[\left\lparen\pi\circ\varphi\right\rparen^{*}\mu\right]$
$\displaystyle\displaystyle=\left[\varphi^{*}\left\lparen\pi^{*}\mu\right\rparen\right]=Q\left\lparen\varphi^{-1}\right\rparen\cdot\left[\pi^{*}\mu\right]=Q\left\lparen\varphi^{-1}\right\rparen\cdot\left\lparen
Q\left\lparen\varphi\right\rparen\cdot\left[\pi^{*}\mu\right]\right\rparen$
$\displaystyle\displaystyle=Q\left\lparen\varphi^{-1}\circ\varphi\right\rparen\cdot\left[\pi^{*}\mu\right]=Q\left\lparen\mathrm{id}_{Z}\right\rparen\cdot\left[\pi^{*}\mu\right]=\left[\pi^{*}\mu\right],$

and so there is a homeomorphism
$\displaystyle\psi\in\mathrm{Homeo}^{+}_{0}\left\lparen Z\right\rparen$ such
that

$\displaystyle\left\lparen\varphi\circ\psi\right\rparen^{*}\left\lparen\pi^{*}\mu\right\rparen=\left\lparen\pi\circ\varphi\circ\psi\right\rparen^{*}\mu=\pi^{*}\mu.$

Thus $\displaystyle\varphi\circ\psi\in\mathrm{Isom}\left\lparen
Z,\pi^{*}\mu\right\rparen=\mathrm{Deck}\left\lparen\pi\right\rparen$ is an
isometry of $\displaystyle\pi^{*}\mu$ and therefore a deck transformation of
$\displaystyle\pi$. In particular, this implies that $\displaystyle
Q\left\lparen\varphi\right\rparen=Q\left\lparen\varphi\circ\psi\right\rparen\in
Q\left\lparen\mathrm{Deck}\left\lparen\pi\right\rparen\right\rparen$. ∎

###  3.3. The proof of Section 1.1 and a corollary

We are now ready to prove Section 1.1, which we restate below for the reader’s
convenience. Recall that $\displaystyle p\colon X\xrightarrow[]{}S$ and
$\displaystyle q\colon Y\xrightarrow[]{}S$ are finite-degree covers of a
closed surface $\displaystyle S$ of negative Euler characteristic.

*

Since the proof is fairly involved, we begin with a brief outline of our
strategy. Given covers $\displaystyle p\colon X\xrightarrow[]{}S$ and
$\displaystyle q\colon Y\xrightarrow[]{}S$ which satisfy the simple elevation
criteria, we first fix a regular cover $\displaystyle\pi_{S}\colon
Z\xrightarrow[]{}S$ which factors through both $\displaystyle X$ and
$\displaystyle Y$ (see Figure 1). We then show in 1 that the simple elevation
criteria implies that the orbits of the curve complexes of $\displaystyle X$
and $\displaystyle Y$, considered as subsets of the curve complex of
$\displaystyle Z$, under the action of
$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ coincide. 2, 3,
and 4 show that similar statements hold for
$\displaystyle\mathcal{EL}\left\lparen X\right\rparen$ and
$\displaystyle\mathcal{EL}\left\lparen Y\right\rparen$; for
$\displaystyle\mathcal{PML}\left\lparen X\right\rparen$ and
$\displaystyle\mathcal{PML}\left\lparen Y\right\rparen$; and then finally for
$\displaystyle\mathscr{T}\left\lparen X\right\rparen$ and
$\displaystyle\mathscr{T}\left\lparen Y\right\rparen$, respectively.

As $\displaystyle\mathscr{T}\left\lparen X\right\rparen$ and
$\displaystyle\mathscr{T}\left\lparen Y\right\rparen$ are properly embedded
submanifolds of $\displaystyle\mathscr{T}\left\lparen Z\right\rparen$, the
dimensionality argument in 5 shows that if the orbits of the isometrically
embedded Teichmüller spaces coincide, these Teichmüller spaces must be related
by a single deck transformation. From this, we conclude in 6 that the deck
groups of $\displaystyle p\colon X\xrightarrow[]{}S$ and $\displaystyle
q\colon Y\xrightarrow[]{}S$ are conjugate in
$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ (by this very
deck transformation) and therefore that the corresponding covers are
equivalent.

###### Proof of Section 1.1.

First, suppose that for all curves $\displaystyle\gamma$ on $\displaystyle S$,
$\displaystyle\gamma$ has a simple elevation along $\displaystyle p$ to
$\displaystyle X$ if and only if it has a simple elevation along
$\displaystyle q$ to $\displaystyle Y$. Fix basepoints $\displaystyle x\in X$,
$\displaystyle y\in Y$, and $\displaystyle s\in S$ such that $\displaystyle
p\left\lparen x\right\rparen=q\left\lparen y\right\rparen=s$, and let
$\displaystyle K$ be a finite-index normal subgroup of
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ contained in both
$\displaystyle p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen$ and $\displaystyle
q_{*}\left\lparen\pi_{1}\left\lparen Y,y\right\rparen\right\rparen$. For
example, we may take $\displaystyle K$ to 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
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$; that is,

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

The Galois correspondence between finite-index subgroups of
$\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ and finite-degree based
covers then yields the based Sunada covering diamond drawn in Figure 1.

As discussed in Section 2.6, $\displaystyle\pi_{S}$, $\displaystyle\pi_{X}$,
and $\displaystyle\pi_{Y}$ are regular covers with deck groups
$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cong G$,
$\displaystyle\mathrm{Deck}\left\lparen\pi_{X}\right\rparen\cong A$, and
$\displaystyle\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen\cong B$,
respectively. By 3.5 above, we may identify the deck groups of these covers by
their isomorphic images in the mapping class group
$\displaystyle\mathrm{Mod}\left\lparen Z\right\rparen$.

For the remainder of the proof, it may be useful for the reader to refer back
to Section 2 for detailed definitions and relevant notation.

######  Claim 1.

$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\widetilde{\pi_{X}}\left\lparen\mathscr{C}\left\lparen
X\right\rparen\right\rparen=\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\widetilde{\pi_{Y}}\left\lparen\mathscr{C}\left\lparen
Y\right\rparen\right\rparen$ as subsets of
$\displaystyle\mathscr{C}\left\lparen Z\right\rparen$.

###### Proof of 1.

Let $\displaystyle\alpha\in\mathscr{C}\left\lparen X\right\rparen$ be a simple
curve on $\displaystyle X$. Then $\displaystyle\alpha$ is a simple elevation
of $\displaystyle p\left\lparen\alpha\right\rparen$ along $\displaystyle p$ to
$\displaystyle X$ and so by assumption, $\displaystyle
p\left\lparen\alpha\right\rparen$ has a simple elevation
$\displaystyle\beta\in\mathscr{C}\left\lparen Y\right\rparen$ along
$\displaystyle q$ to $\displaystyle Y$. Note that

$\displaystyle\pi_{S}\left\lparen
g\cdot\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen\right\rparen=\pi_{S}\left\lparen\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen\right\rparen=p\left\lparen\pi_{X}\left\lparen\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen\right\rparen\right\rparen=p\left\lparen\alpha\right\rparen=q\left\lparen\beta\right\rparen=q\left\lparen\pi_{Y}\left\lparen\widetilde{\pi_{Y}}\left\lparen\beta\right\rparen\right\rparen\right\rparen=\pi_{S}\left\lparen\widetilde{\pi_{Y}}\left\lparen\beta\right\rparen\right\rparen,$

for any $\displaystyle g\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$.
Since $\displaystyle\pi_{S}\colon Z\xrightarrow[]{}S$ is a regular cover,
$\displaystyle
g\cdot\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen=h\cdot\widetilde{\pi_{Y}}\left\lparen\beta\right\rparen$
for some deck transformation $\displaystyle
h\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$, and thus
$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\widetilde{\pi_{X}}\left\lparen\mathscr{C}\left\lparen
X\right\rparen\right\rparen\subseteq\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\widetilde{\pi_{Y}}\left\lparen\mathscr{C}\left\lparen
Y\right\rparen\right\rparen$. By symmetry, the other containment also holds. ∎

######  Claim 2.

$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\widetilde{\pi_{X}}\left\lparen\mathcal{EL}\left\lparen
X\right\rparen\right\rparen=\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\widetilde{\pi_{Y}}\left\lparen\mathcal{EL}\left\lparen
Y\right\rparen\right\rparen$ as subsets of
$\displaystyle\mathcal{EL}\left\lparen Z\right\rparen$.

###### Proof of 2.

Let $\displaystyle g\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ and
$\displaystyle\alpha\in\mathcal{EL}\left\lparen X\right\rparen$. Since
$\displaystyle\mathcal{EL}\left\lparen X\right\rparen$ is the Gromov boundary
of $\displaystyle\mathscr{C}\left\lparen X\right\rparen$ (as discussed in
Section 2.5), some sequence
$\displaystyle\left\lparen\alpha_{n}\right\rparen_{n=0}^{\infty}$ in
$\displaystyle\mathscr{C}\left\lparen X\right\rparen$ converges to
$\displaystyle\alpha$. Thus $\displaystyle\left\lparen
g\cdot\widetilde{\pi_{X}}\left\lparen\alpha_{n}\right\rparen\right\rparen_{n=0}^{\infty}$
converges to $\displaystyle
g\cdot\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen$ in
$\displaystyle\mathscr{C}\left\lparen Z\right\rparen$.

By 1, for each $\displaystyle n$ there is a deck transformation $\displaystyle
h_{n}\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ and a simple curve
$\displaystyle\beta_{n}\in\mathscr{C}\left\lparen Y\right\rparen$ so that
$\displaystyle
g\cdot\widetilde{\pi_{X}}\left\lparen\alpha_{n}\right\rparen=h_{n}\cdot\widetilde{\pi_{Y}}\left\lparen\beta_{n}\right\rparen$.
Since $\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ is finite,
some element $\displaystyle h$ appears in the sequence
$\displaystyle\left\lparen h_{n}\right\rparen_{n=0}^{\infty}$ infinitely often
and so, by passing to a subsequence, we may assume without loss of generality
that $\displaystyle h_{n}=h$ is a fixed deck transformation for all indices
$\displaystyle n\in\mathbb{N}$.

Note that the sequence
$\displaystyle\left\lparen\beta_{n}\right\rparen_{n=0}^{\infty}$ is unbounded
in $\displaystyle\mathscr{C}\left\lparen Y\right\rparen$. Indeed, since
$\displaystyle\widetilde{\pi_{X}}\colon\mathscr{C}\left\lparen
X\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen Z\right\rparen$ is a
quasi-isometric embedding, and $\displaystyle g\colon\mathscr{C}\left\lparen
Z\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen Z\right\rparen$ is an
isometry, it follows that the sequence $\displaystyle\left\lparen
g\cdot\widetilde{\pi_{X}}\left\lparen\alpha_{n}\right\rparen\right\rparen_{n=0}^{\infty}=\left\lparen
h\cdot\widetilde{\pi_{Y}}\left\lparen\beta_{n}\right\rparen\right\rparen_{n=0}^{\infty}$
is unbounded in $\displaystyle\mathscr{C}\left\lparen Z\right\rparen$.
Similarly, since
$\displaystyle\widetilde{\pi_{Y}}\colon\mathscr{C}\left\lparen
Y\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen Z\right\rparen$ is a
quasi-isometric embedding and $\displaystyle h\colon\mathscr{C}\left\lparen
Z\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen Z\right\rparen$ is an
isometry, it follows that
$\displaystyle\left\lparen\beta_{n}\right\rparen_{n=0}^{\infty}$ is unbounded
in $\displaystyle\mathscr{C}\left\lparen Y\right\rparen$.

Thus by passing to yet another subsequence, we may assume that
$\displaystyle\left\lparen\beta_{n}\right\rparen_{n=0}^{\infty}$ converges to
an ending lamination $\displaystyle\beta\in\mathcal{EL}\left\lparen
Y\right\rparen$ on $\displaystyle Y$. In particular, by continuity of
$\displaystyle\widetilde{\pi_{X}}\colon\mathcal{EL}\left\lparen
X\right\rparen\xrightarrow[]{}\mathcal{EL}\left\lparen Z\right\rparen$ and
$\displaystyle\widetilde{\pi_{Y}}\colon\mathcal{EL}\left\lparen
Y\right\rparen\xrightarrow[]{}\mathcal{EL}\left\lparen Z\right\rparen$,

$\displaystyle\displaystyle
g\cdot\widetilde{\pi_{X}}\left\lparen\alpha\right\rparen$
$\displaystyle\displaystyle=g\cdot\widetilde{\pi_{X}}\left\lparen\lim_{n\xrightarrow[]{}\infty}{\alpha_{n}}\right\rparen=\lim_{n\xrightarrow[]{}\infty}{g\cdot\widetilde{\pi_{X}}\left\lparen\alpha_{n}\right\rparen}$
$\displaystyle\displaystyle=\lim_{n\xrightarrow[]{}\infty}{h\cdot\widetilde{\pi_{Y}}\left\lparen\beta_{n}\right\rparen}=h\cdot\widetilde{\pi_{Y}}\left\lparen\lim_{n\xrightarrow[]{}\infty}{\beta_{n}}\right\rparen=h\cdot\widetilde{\pi_{Y}}\left\lparen\beta\right\rparen,$

and so
$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\widetilde{\pi_{X}}\left\lparen\mathcal{EL}\left\lparen
X\right\rparen\right\rparen\subseteq\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\widetilde{\pi_{Y}}\left\lparen\mathcal{EL}\left\lparen
Y\right\rparen\right\rparen$. By symmetry, the other containment also holds. ∎

######  Claim 3.

$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{X}^{*}\left\lparen\mathcal{PML}\left\lparen
X\right\rparen\right\rparen=\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{Y}^{*}\left\lparen\mathcal{PML}\left\lparen
Y\right\rparen\right\rparen$ as subsets of
$\displaystyle\mathcal{PML}\left\lparen Z\right\rparen$.

###### Proof of 3.

Fix
$\displaystyle\epsilon\coloneqq\min\left\lparen\epsilon_{X},\epsilon_{Y}\right\rparen>0$
as in 3.3, let $\displaystyle
g\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ and consider a projective
measured ending lamination
$\displaystyle\alpha\in\mathcal{PMEL}_{\epsilon}\left\lparen X\right\rparen$
on $\displaystyle X$ whose underlying geodesic lamination
$\displaystyle\alpha_{\mathrm{top}}\in\mathcal{EL}\left\lparen X\right\rparen$
is the endpoint of a geodesic ray $\displaystyle R$ in
$\displaystyle\mathscr{T}\left\lparen X\right\rparen$ which recurs infinitely
often to the $\displaystyle\epsilon$-thick part. Since lower bounds on
injectivity radius are preserved by local isometries,
$\displaystyle\pi_{X}^{*}\left\lparen R\right\rparen$ is a geodesic ray in
$\displaystyle\mathscr{T}\left\lparen Z\right\rparen$ which recurs infinitely
often to the $\displaystyle\epsilon$-thick part, with endpoint

$\displaystyle\left\lparen\pi_{X}^{*}\left\lparen\alpha\right\rparen\right\rparen_{\mathrm{top}}=\pi_{X}^{-1}\left\lparen\alpha_{\mathrm{top}}\right\rparen=\widetilde{\pi_{X}}\left\lparen\alpha_{\mathrm{top}}\right\rparen\in\mathcal{EL}\left\lparen
Z\right\rparen.$

In particular, Masur’s criterion 3.1 implies that this ending lamination on
$\displaystyle Z$ is uniquely ergodic. By 2, $\displaystyle
g\cdot\widetilde{\pi_{X}}\left\lparen\alpha_{\mathrm{top}}\right\rparen=h\cdot\widetilde{\pi_{Y}}\left\lparen\beta_{\mathrm{top}}\right\rparen$
for some deck transformation $\displaystyle
h\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ and some ending lamination
$\displaystyle\beta_{\mathrm{top}}\in\mathcal{EL}\left\lparen Y\right\rparen$
on $\displaystyle Y$. Since
$\displaystyle\widetilde{\pi_{X}}\left\lparen\alpha_{\mathrm{top}}\right\rparen$
is uniquely ergodic, so too is its image
$\displaystyle\widetilde{\pi_{Y}}\left\lparen\beta_{\mathrm{top}}\right\rparen=h^{-1}g\cdot\widetilde{\pi_{X}}\left\lparen\alpha_{\mathrm{top}}\right\rparen$
under the deck transformation $\displaystyle h^{-1}g$.

As
$\displaystyle\widetilde{\pi_{Y}}\left\lparen\beta_{\mathrm{top}}\right\rparen$
comes from pullback of an ending lamination on $\displaystyle Y$, it is
invariant under the action of
$\displaystyle\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen$, and so any
representative of the unique projective class of transverse measures on
$\displaystyle\widetilde{\pi_{Y}}\left\lparen\beta_{\mathrm{top}}\right\rparen$
descends to a transverse measure on $\displaystyle\beta_{\mathrm{top}}$. Let
$\displaystyle\beta\in\mathcal{PML}\left\lparen Y\right\rparen$ be the
corresponding projective measured lamination, and note that $\displaystyle
h\cdot\pi_{Y}^{*}\left\lparen\beta\right\rparen$ is a projective measured
lamination on $\displaystyle Z$ with underlying geodesic lamination

$\displaystyle\left\lparen
h\cdot\pi_{Y}^{*}\beta\right\rparen_{\mathrm{top}}=h\cdot\left\lparen\pi_{Y}^{*}\left\lparen\beta\right\rparen\right\rparen_{\mathrm{top}}=h\cdot\widetilde{\pi_{Y}}\left\lparen\beta_{\mathrm{top}}\right\rparen=g\cdot\widetilde{\pi_{X}}\left\lparen\alpha_{\mathrm{top}}\right\rparen=g\cdot\left\lparen\pi_{X}^{*}\left\lparen\alpha\right\rparen\right\rparen_{\mathrm{top}}=\left\lparen
g\cdot\pi_{X}^{*}\left\lparen\alpha\right\rparen\right\rparen_{\mathrm{top}}.$

Since $\displaystyle g\cdot\pi_{X}^{*}\left\lparen\alpha\right\rparen$ is
uniquely ergodic, this implies that $\displaystyle
g\cdot\pi_{X}^{*}\left\lparen\alpha\right\rparen=h\cdot\pi_{Y}^{*}\left\lparen\beta\right\rparen$,
and so

$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{X}^{*}\left\lparen\mathcal{PMEL}_{\epsilon}\left\lparen
X\right\rparen\right\rparen\subseteq\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{Y}^{*}\left\lparen\mathcal{PML}\left\lparen
Y\right\rparen\right\rparen.$

Note that for each deck transformation $\displaystyle
h\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$, $\displaystyle
h\cdot\pi_{Y}^{*}\colon\mathcal{PML}\left\lparen
Y\right\rparen\xrightarrow[]{}\mathcal{PML}\left\lparen Z\right\rparen$ is a
topological embedding, and so $\displaystyle
h\cdot\pi_{Y}^{*}\left\lparen\mathcal{PML}\left\lparen
Y\right\rparen\right\rparen$ is compact and therefore closed in the Hausdorff
space $\displaystyle\mathcal{PML}\left\lparen Z\right\rparen$. Thus

$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{Y}^{*}\left\lparen\mathcal{PML}\left\lparen
Y\right\rparen\right\rparen=\bigcup_{\mathclap{h\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen}}{h\cdot\pi_{Y}^{*}\left\lparen\mathcal{PML}\left\lparen
Y\right\rparen\right\rparen},$

is closed in $\displaystyle\mathcal{PML}\left\lparen Z\right\rparen$, since
$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ is finite.
Finally, since $\displaystyle\mathcal{PMEL}_{\epsilon}\left\lparen
X\right\rparen$ is dense in $\displaystyle\mathcal{PML}\left\lparen
X\right\rparen$ by 3.3, $\displaystyle\pi_{X}^{*}$ is a topological embedding,
and $\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ is finite

$\displaystyle\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{X}^{*}\left\lparen\mathcal{PML}\left\lparen
X\right\rparen\right\rparen$
$\displaystyle\displaystyle=\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{X}^{*}\left\lparen\overline{\mathcal{PMEL}_{\epsilon}\left\lparen
X\right\rparen}\right\rparen$
$\displaystyle\displaystyle\subseteq\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\overline{\pi_{X}^{*}\left\lparen\mathcal{PMEL}_{\epsilon}\left\lparen
X\right\rparen\right\rparen}=\overline{\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{X}^{*}\left\lparen\mathcal{PMEL}_{\epsilon}\left\lparen
X\right\rparen\right\rparen}$
$\displaystyle\displaystyle\subseteq\overline{\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{Y}^{*}\left\lparen\mathcal{PML}\left\lparen
Y\right\rparen\right\rparen}=\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{Y}^{*}\left\lparen\mathcal{PML}\left\lparen
Y\right\rparen\right\rparen.$

By symmetry, the other containment also holds. ∎

######  Claim 4.

$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen=\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$ as a subset of $\displaystyle\mathscr{T}(Z)$.

###### Proof of 4.

Let $\displaystyle g\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ and
$\displaystyle\mu\in\mathscr{T}\left\lparen X\right\rparen$. Fix a projective
measured lamination $\displaystyle\kappa\in\mathcal{PML}\left\lparen
S\right\rparen$, and let $\displaystyle\alpha^{+}\coloneqq
p^{*}\left\lparen\kappa\right\rparen\in\mathcal{PML}\left\lparen
X\right\rparen$ and $\displaystyle\beta^{+}\coloneqq
q^{*}\left\lparen\kappa\right\rparen\in\mathcal{PML}\left\lparen
Y\right\rparen$. By [HM79], $\displaystyle\alpha^{+}$ is one endpoint of a
geodesic $\displaystyle L_{X}\subseteq\mathscr{T}\left\lparen X\right\rparen$
through $\displaystyle\mu$. Let $\displaystyle\alpha^{-}$ be the other
endpoint of $\displaystyle L_{X}$. Since
$\displaystyle\pi_{X}^{*}\colon\mathscr{T}\left\lparen
X\right\rparen\xhookrightarrow{}\mathscr{T}\left\lparen Z\right\rparen$ is an
isometric embedding and $\displaystyle g\colon\mathscr{T}\left\lparen
Z\right\rparen\xrightarrow[]{}\mathscr{T}\left\lparen Z\right\rparen$ is an
isometry, $\displaystyle g\cdot\pi_{X}^{*}\left\lparen L_{X}\right\rparen$ is
a geodesic in $\displaystyle\mathscr{T}\left\lparen Z\right\rparen$ containing
$\displaystyle g\cdot\pi_{X}^{*}\left\lparen\mu\right\rparen$ with endpoints
$\displaystyle g\cdot\pi_{X}^{*}\left\lparen\alpha^{+}\right\rparen$ and
$\displaystyle g\cdot\pi_{X}^{*}\left\lparen\alpha^{-}\right\rparen$.

By 3, $\displaystyle
g\cdot\pi_{X}^{*}\left\lparen\alpha^{-}\right\rparen=h\cdot\pi_{Y}^{*}\left\lparen\beta^{-}\right\rparen$
for some deck transformation $\displaystyle
h\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ and projective measured
lamination $\displaystyle\beta^{-}\in\mathcal{PML}\left\lparen
Y\right\rparen$. On the other hand,
$\displaystyle\pi_{X}^{*}\left\lparen\alpha^{+}\right\rparen=\pi_{S}^{*}\left\lparen\kappa\right\rparen$
is $\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$-invariant, and
so

$\displaystyle
g\cdot\pi_{X}^{*}\left\lparen\alpha^{+}\right\rparen=g\cdot\pi_{S}^{*}\left\lparen\kappa\right\rparen=\pi_{S}^{*}\left\lparen\kappa\right\rparen=h\cdot\pi_{S}^{*}\left\lparen\kappa\right\rparen=h\cdot\pi_{Y}^{*}\left\lparen\beta^{+}\right\rparen.$

In summary, $\displaystyle g\cdot\pi_{X}^{*}\left\lparen L_{X}\right\rparen$
is a geodesic in $\displaystyle\mathscr{T}\left\lparen X\right\rparen$
containing $\displaystyle g\cdot\pi_{X}^{*}\left\lparen\mu\right\rparen$ with
endpoints $\displaystyle
g\cdot\pi_{X}^{*}\left\lparen\alpha^{+}\right\rparen=h\cdot\pi_{Y}^{*}\left\lparen\beta^{+}\right\rparen$
and $\displaystyle
g\cdot\pi_{Y}^{*}\left\lparen\alpha^{-}\right\rparen=h\cdot\pi_{Y}^{*}\left\lparen\beta^{-}\right\rparen$.

We now show that $\displaystyle\beta^{+}$ and $\displaystyle\beta^{-}$ jointly
fill $\displaystyle Y$ and therefore are endpoints of a geodesic
$\displaystyle L_{Y}\subseteq\mathscr{T}\left\lparen Y\right\rparen$. Indeed,
if $\displaystyle\lambda\in\mathcal{PML}\left\lparen Y\right\rparen$ is
disjoint from $\displaystyle\beta^{+}$ and $\displaystyle\beta^{-}$, then
$\displaystyle\pi_{Y}^{*}\left\lparen\lambda\right\rparen$ is a projective
measured lamination on $\displaystyle Z$ which is disjoint from
$\displaystyle\pi_{Y}^{*}\left\lparen\beta^{+}\right\rparen$ and
$\displaystyle\pi_{Y}^{*}\left\lparen\beta^{-}\right\rparen$. However, the
latter projective measured laminations are the endpoints of the geodesic
$\displaystyle h^{-1}g\cdot\pi_{X}^{*}\left\lparen L_{X}\right\rparen$ in
$\displaystyle\mathscr{T}\left\lparen Z\right\rparen$, hence they jointly fill
$\displaystyle Z$. Therefore, no such $\displaystyle\lambda$ exists, and so
$\displaystyle\beta^{+}$ and $\displaystyle\beta^{-}$ jointly fill
$\displaystyle Y$ as required.

In particular, $\displaystyle g\cdot\pi_{X}^{*}\left\lparen
L_{X}\right\rparen$ and $\displaystyle h\cdot\pi_{Y}^{*}\left\lparen
L_{Y}\right\rparen$ are geodesics in $\displaystyle\mathscr{T}\left\lparen
Z\right\rparen$ with the same endpoints, and hence are equal. Therefore

$\displaystyle g\cdot\pi_{X}^{*}\left\lparen\mu\right\rparen\in
g\cdot\pi_{X}^{*}\left\lparen L_{X}\right\rparen=h\cdot\pi_{Y}^{*}\left\lparen
L_{Y}\right\rparen\subseteq\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen,$

and so
$\displaystyle\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen\subseteq\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$. By symmetry, the other containment also holds. ∎

######  Claim 5.

$\displaystyle g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen=\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$ for some deck transformation $\displaystyle
g\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$.

###### Proof of 5.

For each deck transformation $\displaystyle
g\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ of
$\displaystyle\pi_{S}\colon Z\xrightarrow[]{}S$, let

$\displaystyle I_{g}\coloneqq\left\lparen
g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen\right\rparen\cap\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen\subset\mathscr{T}(Z).$

Since $\displaystyle\pi_{X}^{*}$ and $\displaystyle\pi_{Y}^{*}$ are proper
isometric embeddings and $\displaystyle g\colon\mathscr{T}\left\lparen
Z\right\rparen\xrightarrow[]{}\mathscr{T}\left\lparen Z\right\rparen$ is an
isometry, it follows that $\displaystyle
g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen$ and
$\displaystyle\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$ are properly embedded submanifolds of
$\displaystyle\mathscr{T}\left\lparen Z\right\rparen$. Therefore,
$\displaystyle I_{g}$ is a properly embedded submanifold of both
$\displaystyle g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen$ and of
$\displaystyle\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$. Note that

$\displaystyle\displaystyle\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$
$\displaystyle\displaystyle=\left\lparen\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen\right\rparen\cap\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen=\left\lparen\mathrm{Deck}\left\lparen\pi_{S}\right\rparen\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen\right\rparen\cap\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$
$\displaystyle\displaystyle=\left\lparen\bigcup_{g}{\left\lparen
g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen\right\rparen}\right\rparen\cap\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen=\bigcup_{g}{I_{g}},$

where in the above unions $\displaystyle g$ ranges over elements of
$\displaystyle\mathrm{Deck}(\pi_{S})$. So
$\displaystyle\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$ is covered by finitely many properly embedded
submanifolds.

For each deck transformation $\displaystyle
g\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$, if the containment
$\displaystyle I_{g}\subsetneq\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$ is strict, then $\displaystyle I_{g}$ has
positive codimension in
$\displaystyle\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$, since it is a properly embedded submanifold of a
connected manifold. Since
$\displaystyle\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$ is covered by finitely many of these
submanifolds, this cannot be the case for all such deck transformations and so
$\displaystyle\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen=I_{g}$ for some deck transformation $\displaystyle
g\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$. In particular,

$\displaystyle\displaystyle\mathrm{dim}\left\lparen\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen\right\rparen$
$\displaystyle\displaystyle=\mathrm{dim}\left\lparen
I_{g}\right\rparen=\mathrm{dim}\left\lparen\left\lparen
g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen\right\rparen\cap\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen\right\rparen$
$\displaystyle\displaystyle\leq\mathrm{dim}\left\lparen
g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen\right\rparen=\mathrm{dim}\left\lparen\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen\right\rparen.$

By symmetry, the reverse inequality also holds, so that the above inequality
is an equality. Thus
$\displaystyle\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen=I_{g}$ is a properly embedded open submanifold of
$\displaystyle g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen$. Since $\displaystyle
g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen$ is connected, this implies that $\displaystyle
g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen=\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$. ∎

For the remainder of the proof of Section 1.1, fix a deck transformation
$\displaystyle g\in\mathrm{Deck}\left\lparen\pi_{S}\right\rparen$ so that
$\displaystyle g\cdot\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen=\pi_{Y}^{*}\left\lparen\mathscr{T}\left\lparen
Y\right\rparen\right\rparen$, the existence of which is guaranteed by 5.

######  Claim 6.

$\displaystyle
g\mathrm{Deck}\left\lparen\pi_{X}\right\rparen=\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen
g$.

###### Proof of 6.

Consider a deck transformation $\displaystyle
h\in\mathrm{Deck}\left\lparen\pi_{X}\right\rparen$, and note that for each
point $\displaystyle\mu\in\mathscr{T}\left\lparen Y\right\rparen$,
$\displaystyle
g^{-1}\cdot\pi_{Y}^{*}\left\lparen\mu\right\rparen\in\pi_{X}^{*}\left\lparen\mathscr{T}\left\lparen
X\right\rparen\right\rparen$, so that

$\displaystyle ghg^{-1}\cdot\pi_{Y}^{*}(\mu)=g\cdot\left\lparen
h\cdot\left\lparen
g^{-1}\cdot\pi_{Y}^{*}(\mu)\right\rparen\right\rparen=g\cdot\left\lparen
g^{-1}\cdot\pi_{Y}^{*}(\mu)\right\rparen=\pi_{Y}^{*}(\mu).$

By 3.5 above, this implies that $\displaystyle
ghg^{-1}\in\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen$, so that
$\displaystyle
g\mathrm{Deck}\left\lparen\pi_{X}\right\rparen\subseteq\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen
g$. Conversely, consider a deck transformation $\displaystyle
k\in\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen$, and note that for each
point $\displaystyle\mu\in\mathscr{T}\left\lparen X\right\rparen$,
$\displaystyle g\cdot\pi_{X}^{*}(\mu)\in\pi_{Y}^{*}\mathscr{T}\left\lparen
Y\right\rparen$, so that

$\displaystyle g^{-1}kg\cdot\pi_{X}^{*}(\mu)=g^{-1}\cdot\left\lparen
k\cdot\left\lparen
g\cdot\pi_{X}^{*}(\mu)\right\rparen\right\rparen=g^{-1}\cdot\left\lparen
g\cdot\pi_{X}^{*}(\mu)\right\rparen=\pi_{X}^{*}(\mu).$

Again by 3.5, this implies that $\displaystyle
g^{-1}kg\in\mathrm{Deck}\left\lparen\pi_{X}\right\rparen$, so that the reverse
containment $\displaystyle\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen
g\subseteq g\mathrm{Deck}\left\lparen\pi_{X}\right\rparen$ also holds. ∎

We are now ready to prove that $\displaystyle p\colon X\xrightarrow[]{}S$ and
$\displaystyle q\colon Y\xrightarrow[]{}S$ are isomorphic as covers of
$\displaystyle S$. To that end, consider the map
$\displaystyle\widetilde{\varphi}\coloneqq\pi_{Y}\circ g\colon
Z\xrightarrow[]{}Y$. 6 implies that for all deck transformations
$\displaystyle h\in\mathrm{Deck}\left\lparen\pi_{X}\right\rparen$, there is a
deck transformation $\displaystyle
k\in\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen$ with $\displaystyle gh=kg$,
so that

$\displaystyle\displaystyle\widetilde{\varphi}\left\lparen h\left\lparen
z\right\rparen\right\rparen=\pi_{Y}\left\lparen g\left\lparen h\left\lparen
z\right\rparen\right\rparen\right\rparen=\pi_{Y}\left\lparen k\left\lparen
g\left\lparen z\right\rparen\right\rparen\right\rparen=\pi_{Y}\left\lparen
g\left\lparen z\right\rparen\right\rparen=\widetilde{\varphi}\left\lparen
z\right\rparen$

for all points $\displaystyle z\in Z$. In other words,
$\displaystyle\widetilde{\varphi}$ is invariant under the action of the deck
group $\displaystyle\mathrm{Deck}\left\lparen\pi_{X}\right\rparen$, and so
descends to a continuous map $\displaystyle\varphi\colon X\xrightarrow[]{}Y$
such that $\displaystyle\varphi\circ\pi_{X}=\widetilde{\varphi}=\pi_{Y}\circ
g$. Similarly, the map $\displaystyle\widetilde{\psi}\coloneqq\pi_{X}\circ
g^{-1}\colon Z\xrightarrow[]{}X$ is invariant under the action of the deck
group $\displaystyle\mathrm{Deck}\left\lparen\pi_{Y}\right\rparen$, and so
descends to a continuous map $\displaystyle\psi\colon Y\xrightarrow[]{}X$ such
that $\displaystyle\psi\circ\pi_{Y}=\widetilde{\psi}=\pi_{X}\circ g^{-1}$.
Note that

$\displaystyle\displaystyle\pi_{X}$ $\displaystyle\displaystyle=\pi_{X}\circ
g^{-1}\circ g=\psi\circ\pi_{Y}\circ g=\psi\circ\varphi\circ\pi_{X}$

and

$\displaystyle\displaystyle\pi_{Y}$ $\displaystyle\displaystyle=\pi_{Y}\circ
g\circ g^{-1}=\varphi\circ\pi_{X}\circ g^{-1}=\varphi\circ\psi\circ\pi_{Y}.$

Since $\displaystyle\pi_{X}$ and $\displaystyle\pi_{Y}$ are surjective, this
implies that $\displaystyle\psi\circ\varphi=\mathrm{id}_{X}$ and
$\displaystyle\varphi\circ\psi=\mathrm{id}_{Y}$. Thus $\displaystyle\varphi$
and $\displaystyle\psi$ are mutually inverse homeomorphisms. Note that

$\displaystyle\displaystyle p\circ\pi_{X}=\pi_{S}=\pi_{S}\circ
g=q\circ\pi_{Y}\circ g=q\circ\varphi\circ\pi_{X}.$

Since $\displaystyle\pi_{X}$ is surjective, this implies that $\displaystyle
p=q\circ\varphi$ and so $\displaystyle\varphi$ is an isomorphism of covers.

Conversely, now suppose that $\displaystyle p$ and $\displaystyle q$ are
isomorphic as covers of $\displaystyle S$, so that $\displaystyle
p=q\circ\varphi$ for some homeomorphism $\displaystyle\varphi\colon
X\xrightarrow[]{}Y$ and let $\displaystyle\gamma$ be a primitive curve on
$\displaystyle S$. If $\displaystyle\alpha$ is a simple elevation of
$\displaystyle\gamma$ along $\displaystyle p$ to $\displaystyle X$, then
$\displaystyle\varphi\left\lparen\alpha\right\rparen$ is a simple elevation of
$\displaystyle\gamma$ along $\displaystyle q$ to $\displaystyle Y$. On the
other hand, if $\displaystyle\beta$ is a simple elevation of
$\displaystyle\gamma$ along $\displaystyle q$ to $\displaystyle Y$, then
$\displaystyle\varphi^{-1}\left\lparen\beta\right\rparen$ is a simple
elevation of $\displaystyle\gamma$ along $\displaystyle p$ to $\displaystyle
X$. ∎

The following corollary can be derived fairly directly from Section 1.1 and
the partial proof of 2.4 found in [Bus10, 291-294].

######  Corollary 3.6.

Suppose that two closed, orientable surfaces $\displaystyle X$ and
$\displaystyle Y$ arise from Sunada’s construction. If the resulting length-
preserving bijection between curves on $\displaystyle X$ and curves on
$\displaystyle Y$ restricts to a bijection
$\displaystyle\mathscr{C}\left\lparen
X\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen Y\right\rparen$, then
the constructing subgroups are conjugate and hence $\displaystyle X$ and
$\displaystyle Y$ are isometric over $\displaystyle S$.

###### Proof.

Let $\displaystyle p\colon\left\lparen
X,x\right\rparen\xrightarrow[]{}\left\lparen S,s\right\rparen$ and
$\displaystyle q\colon\left\lparen
Y,y\right\rparen\xrightarrow[]{}\left\lparen S,s\right\rparen$ be finite-
degree based covers of a closed surface $\displaystyle S$ of negative Euler
characteristic arising from Sunada’s construction on a surjective homomorphism
$\displaystyle\rho\colon\pi_{1}\left\lparen S,s\right\rparen\twoheadrightarrow
G$ and almost conjugate subgroups $\displaystyle A,B\leq G$. 2.4 provides a
bijection between the set of curves on $\displaystyle X$ and the set of curves
on $\displaystyle Y$. Moreover, as can be seen from the proof in [Bus10,
291-294], for each curve $\displaystyle\gamma$ on $\displaystyle X$ and each
positive integer $\displaystyle d\geq 1$, this bijection restricts to a
bijection between the set of degree $\displaystyle d$ elevations of
$\displaystyle\gamma$ along $\displaystyle p$ to $\displaystyle X$ and the set
of degree $\displaystyle d$ elevations of $\displaystyle\gamma$ along
$\displaystyle q$ to $\displaystyle Y$.

Suppose that the above bijection restricts to a bijection
$\displaystyle\psi\colon\mathscr{C}\left\lparen
X\right\rparen\xrightarrow[]{}\mathscr{C}\left\lparen Y\right\rparen$, and
consider a primitive curve $\displaystyle\gamma$ on $\displaystyle S$. If
$\displaystyle\alpha\in\mathscr{C}\left\lparen X\right\rparen$ is a simple
elevation of $\displaystyle\gamma$ along $\displaystyle p$ to $\displaystyle
X$, then
$\displaystyle\psi\left\lparen\alpha\right\rparen\in\mathscr{C}\left\lparen
Y\right\rparen$ is a simple elevation of $\displaystyle\gamma$ along
$\displaystyle q$ to $\displaystyle Y$. Conversely, if
$\displaystyle\beta\in\mathscr{C}\left\lparen Y\right\rparen$ is a simple
elevation of $\displaystyle\gamma$ along $\displaystyle q$ to $\displaystyle
Y$, then
$\displaystyle\psi^{-1}\left\lparen\beta\right\rparen\in\mathscr{C}\left\lparen
X\right\rparen$ is a simple elevation of $\displaystyle\gamma$ along
$\displaystyle p$ to $\displaystyle X$.

Section 1.1 now implies that $\displaystyle p\colon X\xrightarrow[]{}S$ and
$\displaystyle q\colon Y\xrightarrow[]{}S$ are isomorphic as covers of
$\displaystyle S$. The Galois correspondence yields an element
$\displaystyle\gamma\in\pi_{1}\left\lparen S,s\right\rparen$ such that
$\displaystyle\gamma\left\lparen p_{*}\left\lparen\pi_{1}\left\lparen
X,x\right\rparen\right\rparen\right\rparen\gamma^{-1}=q_{*}\left\lparen\pi_{1}\left\lparen
Y,y\right\rparen\right\rparen$. Thus
$\displaystyle\rho\left\lparen\gamma\right\rparen
A\rho\left\lparen\gamma\right\rparen^{-1}=B$. Moreover, any isomorphism of
covers is an isometry with respect to the pullbacks of any Riemannian metric
on $\displaystyle S$. ∎

##  4\. Applications to simple length isospectrality

Recall that we call two elements of a group _trace twins_ if they are trace-
commensurable but no non-zero powers of these elements are conjugate. When the
group in question is a closed surface group, we will call the curves on the
underlying surface represented by such a pair of elements _(hyperbolic) length
twins_. 2.6 gives an equivalent characterization of these curves; a pair of
curves are length twins if they are length-commensurable but not powers of the
same primitive curve.

Our current methods of distinguishing covers by their simple length spectra
rely on finding curves with no length twins and comparing their elevations to
each cover. Section 1.2, which we restate below for the reader’s convenience,
makes this notion precise.

*

###### Proof of Section 1.2.

Since properties (i) and (ii) also hold (possibly with a different positive
integer $\displaystyle m$) if we replace $\displaystyle\gamma$ with a
primitive element which has $\displaystyle\gamma$ as a non-zero power, we may
assume without loss of generality that $\displaystyle\gamma$ is primitive.
Suppose that $\displaystyle\gamma$ has $\displaystyle r\geq 1$ simple
elevations of degree $\displaystyle m$ along $\displaystyle p$ to
$\displaystyle X$, which is strictly more than the number of simple elevations
of degree $\displaystyle m$ along $\displaystyle q$ to $\displaystyle Y$.

Let $\displaystyle\mathscr{U}$ be the subset of
$\displaystyle\mathscr{T}\left\lparen S\right\rparen$ consisting of all
hyperbolic metrics on $\displaystyle S$ for which $\displaystyle X$ and
$\displaystyle Y$ are simple length isospectral. We will show that
$\displaystyle\mathscr{U}$ has null Lebesgue measure in
$\displaystyle\mathscr{T}\left\lparen S\right\rparen$, and that its complement
is dense. To that end, for each primitive curve $\displaystyle\delta$ on
$\displaystyle S$ and each positive integer $\displaystyle n\geq 1$, let
$\displaystyle Z_{n}\left\lparen\delta\right\rparen$ be the zero set of the
function $\displaystyle
m\cdot\ell_{\left\lparen\>\cdot\>\right\rparen}\left\lparen\gamma\right\rparen-n\cdot\ell_{\left\lparen\>\cdot\>\right\rparen}\left\lparen\delta\right\rparen$;
that is,

$\displaystyle
Z_{n}\left\lparen\delta\right\rparen=\left\\{\mu\in\mathscr{T}\left\lparen
S\right\rparen:m\cdot\ell_{\mu}\left\lparen\gamma\right\rparen=n\cdot\ell_{\mu}\left\lparen\delta\right\rparen\right\\}.$

This function $\displaystyle
m\cdot\ell_{\left\lparen\>\cdot\>\right\rparen}\left\lparen\gamma\right\rparen-n\cdot\ell_{\left\lparen\>\cdot\>\right\rparen}\left\lparen\delta\right\rparen$
is real-analytic on $\displaystyle\mathscr{T}\left\lparen S\right\rparen$, and
since $\displaystyle\gamma$ is primitive and has no length twins, it can only
be identically zero if $\displaystyle n$ divides $\displaystyle m$ and
$\displaystyle\delta$ is the
$\displaystyle\left\lparen\frac{m}{n}\right\rparen$th power of
$\displaystyle\gamma$. For all other curves $\displaystyle\delta$,
$\displaystyle Z_{n}\left\lparen\delta\right\rparen$ is a set of null Lebesgue
measure whose complement is open and dense. In particular, as a countable
union of null measure sets,

$\displaystyle
Z\coloneqq\bigcup_{\delta\neq\gamma}\bigcup_{n=1}^{\infty}{Z_{n}\left\lparen\delta\right\rparen},$

has null Lebesgue measure, where in the above union, $\displaystyle\delta$
varies over all primitive curves except $\displaystyle\gamma$. Moreover, the
complement of $\displaystyle Z$ is dense, since
$\displaystyle\mathscr{T}\left\lparen S\right\rparen$ is a Baire space. It
therefore suffices to show that $\displaystyle\mathscr{U}\subseteq Z$.

To that end, let $\displaystyle\mu\in\mathscr{U}$ and $\displaystyle
L=m\cdot\ell_{\mu}\left\lparen\gamma\right\rparen$. Note that each elevation
$\displaystyle\alpha$ of $\displaystyle\gamma$ of degree $\displaystyle m$
along $\displaystyle p$ to $\displaystyle X$ has length

$\displaystyle\ell_{p^{*}\left\lparen\mu\right\rparen}\left\lparen\alpha\right\rparen=m\cdot\ell_{\mu}\left\lparen\gamma\right\rparen=L.$

In particular, $\displaystyle\left\lparen
X,p^{*}\left\lparen\mu\right\rparen\right\rparen$ has at least $\displaystyle
r$ simple curves of length $\displaystyle L$. Since $\displaystyle\left\lparen
X,p^{*}\left\lparen\mu\right\rparen\right\rparen$ and
$\displaystyle\left\lparen Y,q^{*}\left\lparen\mu\right\rparen\right\rparen$
are simple length isospectral, this implies that $\displaystyle\left\lparen
Y,q^{*}\left\lparen\mu\right\rparen\right\rparen$ also has at least
$\displaystyle r$ simple curves of length $\displaystyle L$. However, since
$\displaystyle\gamma$ does not have $\displaystyle r$ simple elevations of
degree $\displaystyle m$ along $\displaystyle q$ to $\displaystyle Y$, some of
these curves are not elevations of $\displaystyle\gamma$ along $\displaystyle
q$.

Fix such a simple curve $\displaystyle\beta$ on $\displaystyle Y$ which has
length
$\displaystyle\ell_{q^{*}\left\lparen\mu\right\rparen}\left\lparen\beta\right\rparen=L$
but is not an elevation of $\displaystyle\gamma$ along $\displaystyle q$. Let
$\displaystyle\delta$ be the primitive curve corresponding to $\displaystyle
q\left\lparen\beta\right\rparen$, so that $\displaystyle\beta$ is a simple
elevation of $\displaystyle\delta$ along $\displaystyle q$ to $\displaystyle
Y$ of degree some positive integer $\displaystyle 1\leq n\leq\deg\left\lparen
q\right\rparen$. Then

$\displaystyle\displaystyle
m\cdot\ell_{\mu}\left\lparen\gamma\right\rparen=\ell_{p^{*}\left\lparen\mu\right\rparen}\left\lparen\alpha\right\rparen=L=\ell_{q^{*}\left\lparen\mu\right\rparen}\left\lparen\beta\right\rparen=n\cdot\ell_{\mu}\left\lparen\delta\right\rparen.$

$\displaystyle\gamma\neq\delta$ by assumption, and so $\displaystyle\mu\in
Z_{n}\left\lparen\delta\right\rparen\subseteq Z$. Thus
$\displaystyle\mathscr{U}\subseteq Z$, and so $\displaystyle\mathscr{U}$ has
null Lebesgue measure in $\displaystyle\mathscr{T}\left\lparen S\right\rparen$
and its complement is dense. ∎

In light of Section 1.2, we would like to find examples of curves with no
length twins. Section 4.1 is dedicated to constructing many examples of such
curves.

Given a finite-degree cover $\displaystyle p\colon X\xrightarrow[]{}S$ of
$\displaystyle S$, we will denote by $\displaystyle\mathscr{L}\left\lparen
p\right\rparen$ the set of simple elevations of curves on $\displaystyle S$
with no length twins. More knowledge of the distribution of
$\displaystyle\mathscr{L}\left\lparen p\right\rparen$ in
$\displaystyle\mathscr{C}\left\lparen X\right\rparen$ has the potential to
strengthen Section 1.1. Towards this end, we pose the following question.

######  Question 4.1.

Given a finite-degree cover $\displaystyle p\colon X\xrightarrow[]{}S$, which
points in $\displaystyle\mathcal{EL}\left\lparen
X\right\rparen=\partial\mathscr{C}\left\lparen X\right\rparen$ are
accumulation points of $\displaystyle\mathscr{L}\left\lparen p\right\rparen$?

If the answer to 4.1 is that $\displaystyle\mathscr{L}\left\lparen
p\right\rparen$ accumulates at every point of
$\displaystyle\mathcal{EL}\left\lparen X\right\rparen$, then we may replace
$\displaystyle\mathscr{C}\left\lparen X\right\rparen$ and
$\displaystyle\mathscr{C}\left\lparen Y\right\rparen$ with
$\displaystyle\mathscr{L}\left\lparen p\right\rparen$ and
$\displaystyle\mathscr{L}\left\lparen q\right\rparen$, respectively, in 1 of
the proof of Section 1.1 in Section 3.3. Combined with Section 1.2 above, we
would thus obtain an affirmative answer to 1, which we restate below for the
reader’s convenience:

*

Even without an answer to 4.1, we can still provide partial affirmative
answers to 1 for simply generated covers in Section 4.2. Further evidence for
an affirmative answer is given in Section 4.3, where we show that every closed
surface of negative Euler characteristic has covers that are length
isospectral but generically not simple length isospectral.

###  4.1. Curves with no length twins

As outlined above, a main component of our program of distinguishing covers
involves finding curves which have no length twins. The goal of this section
is to prove the following proposition, which provides many examples of such
curves.

######  Proposition 4.2.

Let $\displaystyle S$ be a closed surface of genus $\displaystyle g\geq 2$,
and suppose that elements $\displaystyle\alpha,\beta\in\pi_{1}\left\lparen
S,s\right\rparen$ representing distinct, disjoint, simple, non-separating
curves have injective parametrized representatives
$\displaystyle\overline{\alpha}$ and $\displaystyle\overline{\beta}$ meeting
exactly once (at the basepoint $\displaystyle s$).

  1. 1.

If $\displaystyle g=2$, then for any integers $\displaystyle
j,k\in\mathbb{Z}$, the only length twin of the curve represented by
$\displaystyle\alpha^{j}\beta^{k}$ is its image under the hyperelliptic
involution; and

  2. 2.

If $\displaystyle g\geq 3$, then for any integers $\displaystyle
j,k\in\mathbb{Z}$, the curve represented by $\displaystyle\alpha^{j}\beta^{k}$
has no length twins.

For example, each of the curves drawn in Figure 2 can be realized by elements
of the corresponding surface group which satisfy the hypotheses of 4.2, and
therefore have no length twins.

Figure 2. Curves with no length twins

Before beginning the proof of 4.2, we will establish two preliminary lemmas,
the first of which states that powers of simple curves have no length twins.

######  Lemma 4.3.

Powers of simple curves have no length twins.

###### Proof.

Let $\displaystyle\alpha$ and $\displaystyle\beta$ be a pair of length-
commensurable curves on a closed surface $\displaystyle S$ of negative Euler
characteristic. Suppose that $\displaystyle\alpha$ is a power of a simple
curve $\displaystyle\gamma$, $\displaystyle\beta$ is a power of a primitive
curve $\displaystyle\delta$, and that $\displaystyle\gamma$ and
$\displaystyle\delta$ are length-commensurable.

Since $\displaystyle\gamma$ and $\displaystyle\delta$ are primitive and
$\displaystyle i\left\lparen\gamma,\gamma\right\rparen=0$, 2.6 implies that
$\displaystyle i\left\lparen\gamma,\delta\right\rparen=0$. If
$\displaystyle\gamma\neq\delta$, take a simple curve
$\displaystyle\eta\neq\gamma$ obtained from $\displaystyle\delta$ via surgery.
Note that $\displaystyle i\left\lparen\gamma,\eta\right\rparen=0$, and so
Fenchel-Nielsen coordinates yield, for any two positive real numbers
$\displaystyle c$ and $\displaystyle e$, a hyperbolic metric
$\displaystyle\mu$ on $\displaystyle S$ for which
$\displaystyle\ell_{\mu}\left\lparen\gamma\right\rparen=c$ and
$\displaystyle\ell_{\mu}\left\lparen\eta\right\rparen=e$. Note that
$\displaystyle\ell_{\mu}\left\lparen\delta\right\rparen\geq\ell_{\mu}\left\lparen\eta\right\rparen=e$.
This contradicts our assumption that $\displaystyle\gamma$ and
$\displaystyle\delta$ are length-commensurable, and thus
$\displaystyle\delta=\gamma$. We have shown that any curve which is length-
commensurable to $\displaystyle\alpha$ is also a power of
$\displaystyle\gamma$, and so $\displaystyle\alpha$ has no length twins. ∎

Recall 2.6, which states that length-commensurable curves on a surface are
represented by trace-commensurable elements in the corresponding surface
group. Finding curves without length twins is thus equivalent to finding
elements with no trace twins. The following argument augments Horowitz’s proof
of 2.7 in [Hor72, Corollary 7.2] to find examples of such elements in free
groups.

######  Lemma 4.4.

For any integers $\displaystyle j,k\in\mathbb{Z}$ and primitive elements
$\displaystyle a,b\in F$ of a free group $\displaystyle F$, the element
$\displaystyle w=a^{j}b^{k}$ has no trace twins in $\displaystyle F$.

###### Proof.

Note that [Hor72, Theorem 1.1] implies that we may assume that $\displaystyle
F$ has finite rank. If $\displaystyle j=0$, $\displaystyle k=0$, or
$\displaystyle a=b$, then the desired result follows immediately from 2.7.
Thus we may assume that $\displaystyle j$ and $\displaystyle k$ are both non-
zero and that $\displaystyle a\neq b$. Suppose that $\displaystyle
w=a^{j}b^{k}$ and some element $\displaystyle x\in F$ are trace-commensurable,
so that

$\displaystyle\mathrm{tr}\big{\lparen}\underbrace{a^{j}b^{k}\dotsm
a^{j}b^{k}}_{m}\big{\rparen}^{2}=\mathrm{tr}\left\lparen\big{\lparen}a^{j}b^{k}\big{\rparen}^{m}\right\rparen^{2}=\mathrm{tr}\left\lparen
w^{m}\right\rparen^{2}=\mathrm{tr}\left\lparen x^{n}\right\rparen^{2}$

for some non-zero integers $\displaystyle
m,n\in\mathbb{Z}\setminus\left\\{0\right\\}$. Extend
$\displaystyle\left\\{a,b\right\\}$ to a basis $\displaystyle B$ for
$\displaystyle F$, and let $\displaystyle y=x^{n}$.

[Hor72, Theorem 3.1] gives a way to express
$\displaystyle\mathrm{tr}\left\lparen w^{m}\right\rparen$ as a polynomial in
$\displaystyle\mathrm{tr}\left\lparen a\right\rparen$,
$\displaystyle\mathrm{tr}\left\lparen b\right\rparen$, and
$\displaystyle\mathrm{tr}\left\lparen ab\right\rparen$. Similarly, this
theorem gives a way to express $\displaystyle\mathrm{tr}\left\lparen
y\right\rparen$ as a polynomial in products of the elements of $\displaystyle
B$ which appear in a freely reduced word representing $\displaystyle y$. Since
these traces must be equal for all linear representations of $\displaystyle
F$, this implies that $\displaystyle y\in\left\langle a,b\right\rangle$, so
that $\displaystyle y$ is conjugate to a word of the form

$\displaystyle z=a^{e_{1}}b^{f_{1}}a^{e_{2}}b^{f_{2}}\dotsm
a^{e_{p}}b^{f_{p}}$

for some integers $\displaystyle
e_{1},\dotsc,e_{p},f_{1},\dotsc,f_{p}\in\mathbb{Z}$, at most one of which is
zero. We will show that $\displaystyle w$ is either $\displaystyle z$ or
$\displaystyle z^{-1}$. Since the trace is invariant under conjugation,

$\displaystyle\mathrm{tr}\big{\lparen}\underbrace{a^{j}b^{k}\dotsm
a^{j}b^{k}}_{m}\big{\rparen}^{2}=\mathrm{tr}\left\lparen
x^{n}\right\rparen^{2}=\mathrm{tr}\left\lparen
y\right\rparen^{2}=\mathrm{tr}\left\lparen a^{e_{1}}b^{f_{1}}\dotsm
a^{e_{p}}b^{f_{p}}\right\rparen^{2}.$

[Hor72, Lemma 6.1] implies that $\displaystyle m=p$, $\displaystyle\left\lvert
e_{1}\right\rvert=\dotsb=\left\lvert e_{p}\right\rvert=\left\lvert
j\right\rvert$, and $\displaystyle\left\lvert
f_{1}\right\rvert=\dotsb=\left\lvert f_{p}\right\rvert=\left\lvert
k\right\rvert$. Let $\displaystyle\varphi\colon
F_{2}\xrightarrow[]{}\mathbb{Z}$ be the homomorphism defined by the conditions
$\displaystyle\varphi\left\lparen a\right\rparen=\mathrm{sgn}\left\lparen
j\right\rparen$ and $\displaystyle\varphi\left\lparen
b\right\rparen=\mathrm{sgn}\left\lparen k\right\rparen$, where

$\displaystyle\mathrm{sgn}\left\lparen t\right\rparen=\begin{dcases*}-1,&if
$\displaystyle x<0$;\\\ 0,&if $\displaystyle x=0$; and\\\ 1,&otherwise if
$\displaystyle x>0$.\end{dcases*}$

Since $\displaystyle\varphi$ is a homomorphism,
$\displaystyle\mathrm{tr}\left\lparen\varphi\left\lparen
w^{m}\right\rparen\right\rparen^{2}=\mathrm{tr}\left\lparen\varphi\left\lparen
z\right\rparen\right\rparen^{2}$ as functions on
$\displaystyle\mathrm{hom}\left\lparen\mathbb{Z},\mathrm{SL}_{2}\left\lparen\mathbb{C}\right\rparen\right\rparen$,
and so $\displaystyle\left\lvert\varphi\left\lparen
w^{m}\right\rparen\right\rvert=\left\lvert\varphi\left\lparen
z\right\rparen\right\rvert$ by 2.7. Thus

$\displaystyle\displaystyle\left\lvert
m\right\rvert\cdot\big{\lvert}\left\lvert j\right\rvert+\left\lvert
k\right\rvert\big{\rvert}$ $\displaystyle\displaystyle=\left\lvert
m\right\rvert\cdot\left\lvert j\cdot\mathrm{sgn}\left\lparen
j\right\rparen+k\cdot\mathrm{sgn}\left\lparen
k\right\rparen\right\rvert=\big{\lvert}m\cdot\varphi\big{\lparen}a^{j}b^{k}\big{\rparen}\big{\rvert}=\left\lvert
m\cdot\varphi\left\lparen
w\right\rparen\right\rvert=\left\lvert\varphi\left\lparen
w^{m}\right\rparen\right\rvert$
$\displaystyle\displaystyle=\left\lvert\varphi\left\lparen
z\right\rparen\right\rvert=\left\lvert\varphi\left\lparen
a^{e_{1}}b^{f_{1}}\dotsm
a^{e_{m}}b^{f_{m}}\right\rparen\right\rvert=\left\lvert\mathrm{sgn}\left\lparen
j\right\rparen\sum_{i=1}^{m}{e_{i}}+\mathrm{sgn}\left\lparen
k\right\rparen\sum_{i=1}^{m}{f_{i}}\right\rvert.$

Since $\displaystyle\left\lvert e_{1}\right\rvert=\dotsb=\left\lvert
e_{m}\right\rvert=\left\lvert j\right\rvert$ and $\displaystyle\left\lvert
f_{1}\right\rvert=\dotsb=\left\lvert f_{m}\right\rvert=\left\lvert
k\right\rvert$, the only way for this to happen is for there to be no
cancellation in the sum appearing in the bottom right of the above equations.
Thus there are two cases; either $\displaystyle e_{1}=\dotsb=e_{m}=j$ and
$\displaystyle f_{1}=\dotsb=f_{m}=k$, in which case $\displaystyle w=z$, or
alternatively $\displaystyle e_{1}=\dotsb=e_{m}=-j$ and $\displaystyle
f_{1}=\dotsb=f_{m}=-k$, in which case $\displaystyle w$ is conjugate to
$\displaystyle z^{-1}$.

In summary, we have shown that for any element $\displaystyle x\in F$, if
$\displaystyle w$ and $\displaystyle x$ are trace-commensurable, then some
non-zero powers of $\displaystyle w$ and $\displaystyle x$ are conjugate in
$\displaystyle F$ (and in fact conjugate in $\displaystyle\left\langle
a,b\right\rangle$). Therefore, $\displaystyle w=a^{j}b^{k}$ has no trace twins
in $\displaystyle F$. ∎

We are now ready to prove 4.2. Note that we will make use of the second part
of 2.5, that the images of trace-commensurable elements of a group under a
homomorphism are themselves trace-commensurable.

###### Proof of 4.2.

For ease of readability, we will suppress the distinction between an element
of $\displaystyle\pi_{1}\left\lparen S,s\right\rparen$ and the curve on