# The Goldman Symplectic form on the $\operatorname{\mathrm{PSL}}(V)$-Hitchin
component

Zhe Sun  Yau Mathematical Sciences Center, Tsinghua University, Jingzhai 204,
Haidian district, Beijing, 100084, China<EMAIL_ADDRESS>and  Tengren
Zhang  Mathematics Department, California Institute of Technology, 1200 East
California Boulevard, Mail Code 253-37, Pasadena, CA 91125
<EMAIL_ADDRESS>
###### Abstract.

This article is the second of a pair of articles about the Goldman symplectic
form on the $\operatorname{\mathrm{PSL}}(V)$-Hitchin component. We show that
any ideal triangulation on a closed connected surface of genus at least 2, and
any compatible bridge system determine a symplectic trivialization of the
tangent bundle to the Hitchin component of the surface. Using this, we prove
that a large class of flows defined in the companion paper [SWZ17] are
Hamiltonian. We also construct an explicit collection of Hamiltonian vector
fields on the Hitchin component that give a symplectic basis at every point.

TZ was partially supported by the National Science Foundation under agreements
DMS-1536017.

The authors acknowledge support from U.S. National Science Foundation grants
DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation
varieties” (the GEAR Network). Research was (partially) completed while the
authors were visiting the Institute for Mathematical Sciences, National
University of Singapore in 2016.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Frenet curves, and $\operatorname{\mathrm{PSL}}(V)$-Hitchin representations
    2. 2.2 Ideal triangulations, barriers and bridges
    3. 2.3 Algebraic intersection number
    4. 2.4 The tangent space to $\operatorname{\mathrm{Hit}}_{V}(S)$ and the Goldman symplectic form
    5. 2.5 The tangent cocycle
  3. 3 The vector space of admissible labellings
    1. 3.1 Legal labellings
    2. 3.2 Admissible labellings
  4. 4 The tangent space to $\operatorname{\mathrm{Hit}}_{V}(S)$ using admissible labellings
    1. 4.1 The maps $K_{0}$ and $K_{1}$
    2. 4.2 Obtaining cocycles from admissible labellings
    3. 4.3 Admissible labellings and tangent cocycles
  5. 5 Computing the Goldman symplectic form on $\operatorname{\mathrm{Hit}}_{V}(S)$
    1. 5.1 The $(\mathcal{T},\mathcal{J})$-trivialization of $T\operatorname{\mathrm{Hit}}_{V}(S)$
    2. 5.2 A triangulation of $S$.
    3. 5.3 The cup product of parallel vector fields
  6. 6 A symplectic basis of vector fields on $\operatorname{\mathrm{Hit}}_{V}(S)$.
    1. 6.1 Choice of oriented ideal triangulation and compatible bridge system
    2. 6.2 Special admissible labellings
    3. 6.3 Choice of triangulation on $\Sigma$
    4. 6.4 Computational tools
    5. 6.5 Explicit computations for the proof of Theorem 6.5.
  7. A Proof of Proposition 2.13
  8. B Proof of Proposition 4.6

##  1\. Introduction

Let $S$ be a closed, oriented, connected surface of genus at least $2$, and
let $\Gamma$ denote the fundamental group of $S$. The _Teichmüller space_ of
$S$, denoted $\mathcal{T}(S)$, is the deformation space of hyperbolic
structures on $S$. By considering the holonomy representations of these
hyperbolic structures, one can identify $\mathcal{T}(S)$ as the connected
component

$\left\\{[\rho]\in\mathcal{X}\big{(}\Gamma,\operatorname{\mathrm{PSL}}(2,\mathbb{R})\big{)}:\rho\text{
is discrete and faithful}\right\\}$

of
$\mathcal{X}\big{(}\Gamma,\operatorname{\mathrm{PSL}}(2,\mathbb{R})\big{)}:=\operatorname{\mathrm{Hom}}\big{(}\Gamma,\operatorname{\mathrm{PSL}}(2,\mathbb{R})\big{)}/\operatorname{\mathrm{PGL}}(2,\mathbb{R})$.

This representation theoretic description of $\mathcal{T}(S)$ admits a natural
generalization where we replace $\operatorname{\mathrm{PSL}}(2,\mathbb{R})$
with a higher rank, semisimple Lie group $G$ that is real split. These
generalizations of $\mathcal{T}(S)$ are known as the $G$-_Hitchin components_
, and are central objects of investigation in the study of higher Teichmüller
theory. In this article, we consider the setting when
$G=\operatorname{\mathrm{PSL}}(V)$, where $V$ is an $n$-dimensional real
vector space (see Section 2.1). The $\operatorname{\mathrm{PSL}}(V)$-Hitchin
component, denoted $\operatorname{\mathrm{Hit}}_{V}(S)$, was first studied by
Hitchin [Hit92], who showed that $\operatorname{\mathrm{Hit}}_{V}(S)$ is real-
analytically diffeomorphic to $\mathbb{R}^{(n^{2}-1)(2g-2)}$. Later, Labourie
[Lab06] proved that every representation in
$\operatorname{\mathrm{Hit}}_{V}(S)$ is Anosov, which implies in particular
that these representations are all quasi-isometric embeddings from $\Gamma$
(equipped with any word metric) to $\operatorname{\mathrm{PSL}}(V)$ (equipped
with any left-invariant Riemannian metric).

By specializing a general construction due to Goldman [Gol84], one can equip
$\operatorname{\mathrm{Hit}}_{V}(S)$ with a natural symplectic structure (see
Section 2.4), known as the _Goldman symplectic form_. Goldman also proved
[Gol86] that when $V=\mathbb{R}^{2}$, this symplectic form is a multiple of
the Weil-Petersson symplectic form on $\mathcal{T}(S)$. The Goldman symplectic
form on $\operatorname{\mathrm{Hit}}_{V}(S)$ has been an object of interest in
the last few years, and was previously studied by many other authors,
including [Bri15], [LW16], [Lab12], [Nie13], [Sun17].

Perhaps the most famous result regarding this symplectic form on
$\mathcal{T}(S)$ is a theorem of Wolpert [Wol82],[Wol83], which states that
for any pants decomposition $\mathcal{P}$ on $S$, the Hamiltonian flows of the
Fenchel-Nielsen coordinate functions corresponding to $\mathcal{P}$ are
commuting flows whose tangent vector fields give a symplectic basis of
$T_{[\rho]}\mathcal{T}(S)$ for every $[\rho]\in\mathcal{T}(S)$. This was
partially generalized to $\operatorname{\mathrm{Hit}}_{V}(S)$ when $\dim(V)=3$
by Kim [Kim99], and when $\dim(V)=4$ by H.T. Jung (in preparation). The key
tool they used to prove these results is known as Fox calculus, which is a
technique that allows one to compute the Goldman Poisson pairing between
certain types of functions on $\operatorname{\mathrm{Hit}}_{V}(S)$. However,
the computations required to implement Fox calculus become more and more
complicated as $\dim(V)$ gets large.

This article is the second of a pair of articles about the Goldman symplectic
form on $\operatorname{\mathrm{Hit}}_{V}(S)$, where the objective is to
generalize Wolpert’s theorem to $\operatorname{\mathrm{Hit}}_{V}(S)$ for
general $V$. In the companion article by Sun-Wienhard-Zhang [SWZ17], we show
that given any ideal triangulation $\mathcal{T}$ on $S$ and any compatible
bridge system $\mathcal{J}$ (see Section 2.2 for definitions), one can
construct a family of real-analytic flows on
$\operatorname{\mathrm{Hit}}_{V}(S)$ using the geometry of the representations
in $\operatorname{\mathrm{Hit}}_{V}(S)$. These are called the
$(\mathcal{T},\mathcal{J})$-parallel flows ([SWZ17, Section 5.1]), and satisfy
the following properties:

  * •

Any pair of $(\mathcal{T},\mathcal{J})$-parallel flows commute.

  * •

The set of $(\mathcal{T},\mathcal{J})$-parallel flows form a
$(n^{2}-1)(2g-2)$-dimensional vector subspace of the space of flows on
$\operatorname{\mathrm{Hit}}_{V}(S)$.

  * •

For any $[\rho]\in\operatorname{\mathrm{Hit}}_{V}(S)$ and any $v\in
T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)$, there is a unique
$(\mathcal{T},\mathcal{J})$-parallel flow whose tangent vector at $[\rho]$ is
$v$.

Special cases of these flows were previously constructed by Wienhard-Zhang
[WZ17].

More informally, given a choice of some topological data on $S$ (the choice of
$\mathcal{T}$ and $\mathcal{J}$), the geometry of the representations in
$\operatorname{\mathrm{Hit}}_{V}(S)$ determine a vector space of commuting
flows on $\operatorname{\mathrm{Hit}}_{V}(S)$ that is naturally in bijection
with $T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)$ for every
$[\rho]\in\operatorname{\mathrm{Hit}}_{V}(S)$. In particular, this determines
a trivialization of $T\operatorname{\mathrm{Hit}}_{V}(S)$. The first goal of
this article is to prove that this trivialization is symplectic.

######  Theorem 1.1 (Theorem 5.3).

Let $\mathcal{T}$ be an ideal triangulation on $S$ and let $\mathcal{J}$ be a
compatible bridge system. If $X_{1}$ and $X_{2}$ are the tangent vector fields
for a pair of $(\mathcal{T},\mathcal{J})$-parallel flows on
$\operatorname{\mathrm{Hit}}_{V}(S)$, then the function
$\operatorname{\mathrm{Hit}}_{V}(S)\to\mathbb{R}$ given by

$[\rho]\mapsto\omega(X_{1}[\rho],X_{2}[\rho])$

is constant.

Combining Theorem 1.1 with the properties of
$(\mathcal{T},\mathcal{J})$-parallel flows listed above, one can then prove
the following corollary.

######  Corollary 1.2 (Corollary 5.4).

Let $(\mathcal{T},\mathcal{J})$ be as in Theorem 5.3. Any
$(\mathcal{T},\mathcal{J})$-parallel flow is a Hamiltonian flow on
$\operatorname{\mathrm{Hit}}_{V}(S)$.

By Theorem 1.1, the vector space of $(\mathcal{T},\mathcal{J})$-parallel flows
admits a symplectic pairing induced by the Goldman symplectic form on
$\operatorname{\mathrm{Hit}}_{V}(S)$. Choose any symplectic basis for this
vector space. By Corollary 5.4, the flows in the symplectic basis are
Hamiltonian, so their Hamiltonian functions give global Darboux coordinates
for $\operatorname{\mathrm{Hit}}_{V}(S)$. As a consequence, we have the
following corollary.

######  Corollary 1.3 (Corollary 5.5).

$\operatorname{\mathrm{Hit}}_{V}(S)$ is a complete integrable system.

Behind the proof of Theorem 1.1 is a new method to compute the Goldman
symplectic form, which is similar in flavor to the techniques used by Bonahon-
Sozen [BS01] on $\mathcal{T}(S)$. This method uses the following theorem
obtained by combining the work of Labourie [Lab06] and Guichard [Gui08].

######  Theorem 1.4 (Guichard, Labourie).

There is canonical bijection

$\operatorname{\mathrm{Hit}}_{V}(S)\simeq\left\\{\xi:\partial\Gamma\to\operatorname{\mathrm{PSL}}(V)\bigg{|}\begin{array}[]{l}\xi\text{
is Frenet and }\rho\text{-equivariant }\\\ \text{for some
}\rho\in\operatorname{\mathrm{Hom}}\big{(}\Gamma,\operatorname{\mathrm{PSL}}(V)\big{)}\end{array}\right\\}\bigg{/}\operatorname{\mathrm{PGL}}(V)$

In other words, instead of thinking of $\operatorname{\mathrm{Hit}}_{V}(S)$ as
a space of conjugacy classes of representations, one can think of
$\operatorname{\mathrm{Hit}}_{V}(S)$ as a space of projective classes of
Frenet curves (see Definition 2.3) satisfying an equivariance property.

We will now briefly describe this new computational method. Let
$[\rho]\in\operatorname{\mathrm{Hit}}_{V}(S)$ and let $[\nu]\in
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})\simeq
T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)$. We first show that if one
chooses $(\mathcal{T},\mathcal{J})$, a hyperbolic metric on $S$ to get a
hyperbolic surface $\Sigma$, and a representative $\rho\in[\rho]$, then these
choices naturally determine a cocycle representative
$\mu_{\rho,[\nu]}\in[\nu]$ from the $\rho$-equivariant Frenet curve guaranteed
by Theorem 1.4. We call this $1$-cocycle the
$(\rho,\Sigma,\mathcal{T},\mathcal{J})$-_tangent cocycle_ in $[\nu]$ (see
Section 2.5).

Next, we give a different way to construct the
$(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle by making an
additional choice of an orientation $r$ on $\mathcal{T}$. From $\mathcal{T}$,
we construct an associated _barrier system_ $\widetilde{\mathcal{B}}$ (see
Definition 2.8) which is a set that admits a $\Gamma$-action, and only depends
on the topology of $S$. Using $r$ and $\rho$, we then define the notion of an
_admissible labelling_
$L:\widetilde{\mathcal{B}}\to\mathfrak{s}\mathfrak{l}(V)$ (see Definition
3.7). The space of such admissible labellings, denoted by
$\mathcal{A}(\rho,r,\mathcal{T})$, is a vector space of dimension
$(n^{2}-1)(2g-2)$. We also define an algebraic intersection number (see
Section 2.3) between $1$-simplices in $\widetilde{S}$ and barriers in
$\widetilde{\mathcal{B}}$ by using $r$ and $\Sigma$. Together with
$\mathcal{J}$, this allows us to construct, for every
$L\in\mathcal{A}(\rho,r,\mathcal{T})$, a $1$-cocycle $\mu_{L}\in
C^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
(see Section 4.2). The following theorem relates these $1$-cocycles and the
$(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycles.

######  Theorem 1.5 (Theorem 4.8).

Let $L\in\mathcal{A}(\rho,r,\mathcal{T})$ be an admissible labelling. Then
$\mu_{L}$ is the $(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle in
the cohomology class $[\mu_{L}]$.

From this, we can deduce that the map
$\Phi_{\rho}:\mathcal{A}(\rho,r,\mathcal{T})\to
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
given by $L\mapsto[\mu_{L}]$ is a linear isomorphism that does not depend on
$\Sigma$. Informally, this theorem converts the abstract definition of the
$(\rho,\Sigma,\mathcal{T},\mathcal{J})$-cocycle into a concrete, explicit
definition in terms of admissible labellings, which can be used to do
computations. Hence, we can now compute the Goldman symplectic pairing between
any pair of first cohomology classes in
$H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
by using the cup product formula from simplicial cohomology.

The computational procedure above is the main tool used to prove Theorem 1.1.
More precisely, for every $L\in\mathcal{A}(\rho,r,\mathcal{T})$, we define the
_coeffecients_ of $L$ (see Definition 6.4). These are a collection of real
numbers that determine $L$ once we are given the $\rho$-equivariant Frenet
curve. We then used Theorem 1.5 to compute that for any
$L_{1},L_{2}\in\mathcal{A}(\rho,r,\mathcal{T})$,
$\omega([\mu_{L_{1}}],[\mu_{L_{2}}])$ depends only on the coeffecients of
$L_{1}$ and $L_{2}$. Theorem 1.1 then follows from the observation that the
coeffecients of the admissible labellings corresponding to tangent vectors to
any $(\mathcal{T},\mathcal{J})$-parallel flow are constant on
$\operatorname{\mathrm{Hit}}_{V}(S)$. For more details, see the proof of
Theorem 5.3.

The second goal of this article is to use the computational method described
above to explicitly find a collection of $(n^{2}-1)(2g-2)$ Hamiltonian flows
on $\operatorname{\mathrm{Hit}}_{V}(S)$ that commute, so that their tangent
vector fields give a symplectic basis at every
$[\rho]\in\operatorname{\mathrm{Hit}}_{V}(S)$. To that end, we specialize to
certain types of ideal triangulations $\mathcal{T}$ on $S$ that contain a
pants decomposition $\mathcal{P}$ of $S$ (see Section 6.1). For these ideal
triangulations, we found an explicit family of $(n^{2}-1)(2g-2)$ admissible
labellings in $\mathcal{A}(\rho,r,\mathcal{T})$, called _special admissible
labellings_ , that are defined by specifying their coeffecients. These consist
of

  * •

$(g-1)(n-1)(n-2)$ _eruption labellings_ , $\frac{(n-1)(n-2)}{2}$ for each
$P\in\mathbb{P}$,

  * •

$(g-1)(n-1)(n-2)$ _hexagon labellings_ , $\frac{(n-1)(n-2)}{2}$ for each
$P\in\mathbb{P}$,

  * •

$(3g-3)(n-1)$ _twist labellings_ , $(n-1)$ for each $c\in\mathcal{P}$,

  * •

$(3g-3)(n-1)$ _length labellings_ , $(n-1)$ for each $c\in\mathcal{P}$.

Here, $\mathbb{P}$ denotes the set of pairs of pants given by $\mathcal{P}$.
See Definition 6.2 and Definition 6.3 for the explicit description of these
admissible labellings. We then prove the following theorem.

######  Theorem 1.6 (Theorem 6.5).

For any pants decomposition $\mathcal{P}$ of $S$, there is an ideal
triangulation $\mathcal{T}$ of $S$ containing $\mathcal{P}$ so that for any
compatible bridge system $\mathcal{J}$ and any
$[\rho]\in\operatorname{\mathrm{Hit}}_{V}(S)$, the set

$\\{[\mu_{L}]\in
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho}):L\in\mathcal{A}(\rho,r,\mathcal{T})\text{
is a special admissible labelling}\\}$

is a symplectic basis of
$H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$.

Each tangent vector in this symplectic basis corresponds to a unique
$(\mathcal{T},\mathcal{J})$-parallel flow, which is Hamiltonian by Corollary
1.2. In the companion paper [SWZ17], we used this to give explicit global
Darboux coordinates for $\operatorname{\mathrm{Hit}}_{V}(S)$ by computing the
Hamiltonian functions of the special $(\mathcal{T},\mathcal{J})$-parallel
flows. If we specialize to the case when $\dim(V)=2$, then the Hamiltonian
flows we found are essentially the Hamiltonian flows of the $6g-6$ Fenchel-
Nielsen coordinate functions. This thus gives a generalization of Wolpert’s
theorem.

The rest of this paper is organized in the following way. In Section 2, we
formally define the basic objects we need, including Frenet curves, Hitchin
representations, ideal triangulations, barriers systems, and bridge systems.
In this section, we also define a notion of algebraic intersection number
between $1$-simplices in $S$ and oriented barriers, and construct the
$(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle mentioned above. The
notion of an admissible labelling is defined in Section 3. In Section 4, we
construct the linear map $\Phi_{\rho}:\mathcal{A}(\rho,r,\mathcal{T})\to
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
described above, prove that it does not depend on the choice of $\Sigma$ on
$S$, and that it is in fact an isomorphism. In Section 5, we use $\Phi_{\rho}$
to construct a symplectic trivialization of
$T\operatorname{\mathrm{Hit}}_{V}(S)$, and prove Theorem 1.1. Then in Section
6, we define the special admissible labellings and prove Theorem 1.6.

Some of the proofs in this article require long but elementary computations,
which are completely written up in the appendices attached. Also, one should
note that the results in this article depend only on the first five sections
(but not Section 6) of the companion article [SWZ17].

Acknowledgements: The authors thank Sara Maloni, Sam Ballas, Frederic Palesi,
Francois Labourie, and Francis Bonahon for many useful conversations that
helped them develop their understanding of this subject. They especially thank
Bill Goldman for asking the question that motivated this project. They also
thank Anna Wienhard, their collaborator in the companion paper, for suggesting
that they implement their computational methods explicitly.

This project started when the first author visited the second author at the
California Institute of Technology in the spring of 2016. The bulk of the work
in this project was done when both authors visited the Institute for
Mathematical Sciences in the National University of Singapore during the
summer of 2016, and also during the second author’s visit to the Yau
Mathematical Sciences Center in Tsinghua University during the spring and
summer of 2017. The authors are grateful towards these institutions for their
hospitality. The first author would also like to thank Yi Huang for some
financial support for this project.

##  2\. Background

###  2.1. Frenet curves, and $\operatorname{\mathrm{PSL}}(V)$-Hitchin
representations

In this section, we will recall some well-known facts about
$\operatorname{\mathrm{PSL}}(V)$-Hitchin representations. We begin with the
definition of these representations.

######  Definition 2.1.

A representation $\rho:\Gamma\to\operatorname{\mathrm{PSL}}(V)$ is a
$\operatorname{\mathrm{PSL}}(V)$-_Hitchin representation_ if it can be
continuously deformed in
$\operatorname{\mathrm{Hom}}(\Gamma,\operatorname{\mathrm{PSL}}(V))$ to a
faithful representation whose image is a discrete subgroup that lies in the
image of an irreducible representation from
$\operatorname{\mathrm{PSL}}_{2}(\mathbb{R})$ to
$\operatorname{\mathrm{PSL}}(V)$.

Let $\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ denote the set of Hitchin
representations in
$\operatorname{\mathrm{Hom}}(\Gamma,\operatorname{\mathrm{PSL}}(V))$. Since
$\operatorname{\mathrm{PSL}}(V)$ is a real algebraic group,
$\operatorname{\mathrm{Hom}}(\Gamma,\operatorname{\mathrm{PSL}}(V))$ is
naturally a real algebraic variety. One can also topologize
$\operatorname{\mathrm{Hom}}(\Gamma,\operatorname{\mathrm{PSL}}(V))$ by the
compact-open topology. It is clear that
$\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ is a union of connected
components of
$\operatorname{\mathrm{Hom}}(\Gamma,\operatorname{\mathrm{PSL}}(V))$.
Furthermore, since the condition for being a
$\operatorname{\mathrm{PSL}}(V)$-Hitchin representation is invariant under
conjugation by elements in $\operatorname{\mathrm{PGL}}(V)$, we can consider
$\operatorname{\mathrm{Hit}}_{V}(S):=\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)/\operatorname{\mathrm{PGL}}(V)$.
This was first studied by Hitchin [Hit92], who used Higgs bundle techniques to
prove that $\operatorname{\mathrm{Hit}}_{V}(S)$ is homeomorphic to
$\mathbb{R}^{(2g-2)(n^{2}-1)}$. In particular,
$\operatorname{\mathrm{Hit}}_{V}(S)$ is connected, and is commonly known as
the $\operatorname{\mathrm{PSL}}(V)$-_Hitchin component_.

Note that in the case when $\dim(V)=2$, $\operatorname{\mathrm{Hit}}_{V}(S)$
is exactly the space of conjugacy classes of faithful representations from
$\Gamma$ to $\operatorname{\mathrm{PSL}}(V)$ with discrete image. It is also
classically known that every such conjugacy class is the holonomy of a unique
hyperbolic structure on $S$, so $\operatorname{\mathrm{Hit}}_{V}(S)$ can
naturally be identified with the deformation space of hyperbolic structures on
$S$. This phenomena was generalized by Choi-Goldman [CG93], who proved that
when $\dim(V)=3$, the $\operatorname{\mathrm{PSL}}(V)$-Hitchin representations
are exactly holonomies of convex real projective structures on $S$.

However, the geometric properties of $\operatorname{\mathrm{PSL}}(V)$-Hitchin
representations in general were poorly understood until a breakthrough by
Labourie [Lab06], who used Frenet curves to study them. We will now give a
brief account of Labourie’s result.

Let $V$ be an $n$-dimensional real vector space. A _(complete) flag_ $F$ in
$V$ is a nested sequence of subspaces

$F^{(1)}\subset F^{(2)}\subset\dots\subset F^{(n-1)}\subset V$

so that $\dim F^{(i)}=i$. Let $\mathcal{F}(V)$ denote the space of flags in
$V$. It is well-known that $\mathcal{F}(V)$ naturally has the structure of a
real-analytic manifold, and $\operatorname{\mathrm{PGL}}(V)$ acts transitively
and real-analytically on $\mathcal{F}(V)$. We say that a $k$-tuple of flags
$(F_{1},\dots,F_{k})\in\mathcal{F}(V)^{k}$ is _transverse_ if
$F_{1}^{(i_{1})}+\dots+F_{k}^{(i_{k})}=V$ for all $i_{1},\dots,i_{k}\geq 0$ so
that $i_{1}+\dots+i_{k}=n$. Let $\mathcal{F}(V)^{(k)}$ denote the space of
transverse $k$-tuples of flags. Note that this is an open set in
$\mathcal{F}(V)^{k}$, and so naturally inherits a real-analytic structure
which descends to a real-analytic structure on
$\mathcal{F}(V)^{(k)}/\operatorname{\mathrm{PGL}}(V)$.

######  Remark 2.2.

In the rest of this paper, we will repeatedly use the following two well-known
observations.

  1. (1)

Let $(F,G),(F^{\prime},G^{\prime})\in\mathcal{F}(V)^{(2)}$ and let
$p,p^{\prime}\in\mathbb{P}(V)$ so that

$p+F^{(i)}+G^{(n-i-1)}=V=p^{\prime}+F^{\prime(i)}+G^{\prime(n-i-1)}$

for all $i=0,\dots,n-1$. Then there is a unique projective transformation
$g\in\operatorname{\mathrm{PGL}}(V)$ so that $g\cdot F=F^{\prime}$, $g\cdot
G=G^{\prime}$ and $g\cdot p=p^{\prime}$.

  2. (2)

Let $(F,G),(F,G^{\prime})\in\mathcal{F}(V)^{(2)}$. Then there is a unique
unipotent projective transformation $u\in\operatorname{\mathrm{PGL}}(V)$ so
that $u\cdot F=F$ and $u\cdot G=G^{\prime}$.

######  Definition 2.3.

A continuous map $\xi:S^{1}\to\mathcal{F}(V)$ is _Frenet_ if for all
$k=1,\dots,n$, the following hold:

  * •

If $x_{1},\dots,x_{k}\in S^{1}$ is a pairwise distinct $k$-tuple, then
$\xi(x_{1}),\dots,\xi(x_{k})$ is a transverse $k$-tuple of flags in
$\mathcal{F}(V)$.

  * •

Let $\\{(x_{1,i},\dots,x_{k,i})\\}_{i=1}^{\infty}$ be a sequence of pairwise
distinct $k$-tuples in $S^{1}$, so that $\lim_{i\to\infty}x_{j,i}=x\in S^{1}$
for all $j=1,\dots,k$. For any $i_{1},\dots,i_{k}\in\mathbb{Z}^{+}$ so that
$i_{1}+\dots+i_{k}=m\leq n$, we have

$\lim_{i\to\infty}\xi(x_{1,i})^{(i_{1})}+\dots+\xi(x_{k,i})^{(i_{k})}=\xi^{(m)}(x).$

It is well-known that $\Gamma$ is a hyperbolic group, and that its Gromov
boundary $\partial\Gamma$ is topologically a circle. Hence, it makes sense to
say when a continuous map $\xi:\partial\Gamma\to\mathcal{F}(V)$ is Frenet.
Labourie proved that every $\operatorname{\mathrm{PSL}}(V)$-Hitchin
representation preserves a unique $\rho$-equivariant Frenet curve
$\xi_{\rho}:\partial\Gamma\to\mathcal{F}(V)$. Using this, he deduces that
every $\operatorname{\mathrm{PSL}}(V)$-Hitchin representation is an
irreducible quasi-isometric embedding (with respect to any word metric on
$\Gamma$ and any left-invariant Riemannian metric on
$\operatorname{\mathrm{PSL}}(V)$). He also conjectured that for any
representation $\rho:\Gamma\to\operatorname{\mathrm{PSL}}(V)$, the existence
of such a $\rho$-equivariant Frenet curve implies that $\rho$ is a
$\operatorname{\mathrm{PSL}}(V)$-Hitchin representation. This was later proven
by Guichard, thus establishing the following theorem.

######  Theorem 2.4.

[Gui08, Theorem 1], [Lab06, Theorem 1.4] A representation
$\rho:\Gamma\to\operatorname{\mathrm{PSL}}(V)$ is a
$\operatorname{\mathrm{PSL}}(V)$-_Hitchin representation_ if and only if there
exists a $\rho$-equivariant Frenet curve
$\xi_{\rho}:\partial\Gamma\to\mathcal{F}(V)$. If such a $\rho$-equivariant
Frenet curve exists, then it is unique. Furthermore, for any pair
$\rho,\rho^{\prime}$ of $\operatorname{\mathrm{PSL}}(V)$-Hitchin
representations, $\xi_{\rho}=\xi_{\rho^{\prime}}$ if and only if
$\rho=\rho^{\prime}$.

In particular, this establishes a natural bijection between
$\operatorname{\mathrm{Hit}}_{V}(S)$ and the set of projective classes of
Frenet curves $\xi:\partial\Gamma\to\mathcal{F}(V)$ that are
$\rho$-equivariant for some representation
$\rho:\Gamma\to\operatorname{\mathrm{PSL}}(V)$. At about the same time, Fock-
Goncharov [FG06] also gave a more algebraic version of the criterion in
Theorem 2.4, where they replace the Frenet property with the notion of a
“positive map”.

Since every $\operatorname{\mathrm{PSL}}(V)$-Hitchin representation is
irreducible, we see that $\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ lies
in the smooth locus of the algebraic variety
$\operatorname{\mathrm{Hom}}(\Gamma,\operatorname{\mathrm{PSL}}(V))$. As such,
$\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ is naturally equipped with a
real-analytic structure, which descends to a real-analytic structure on
$\operatorname{\mathrm{Hit}}_{V}(S)$. With this real-analytic structure,
Hitchin’s proof in fact shows that $\operatorname{\mathrm{Hit}}_{V}(S)$ is
real-analytically diffeomorphic to $\mathbb{R}^{(2g-2)(n^{2}-1)}$.

The description of Hitchin representations via Frenet curves also allows one
to prove a striking positivity feature for Hitchin representations, which was
first observed by Fock-Goncharov [FG06]. To describe this, we will define two
well-known projective invariants associated to transverse pairs and triples of
flags in $\mathcal{F}(V)$.

######  Definition 2.5.

  * •

Let $E,F,G,H\in\mathcal{F}(V)$ be a quadruple of flags so that
$(E,F,H),(E,G,H)\in\mathcal{F}(V)^{(3)}$. For any $i=1,\dots,n-1$, define the
_cross ratio_ by

$C_{i}(E,F,G,H):=\frac{E^{(i)}\wedge H^{(n-i-1)}\wedge G^{(1)}}{E^{(i)}\wedge
H^{(n-i-1)}\wedge F^{(1)}}\cdot\frac{E^{(i-1)}\wedge H^{(n-i)}\wedge
F^{(1)}}{E^{(i-1)}\wedge H^{(n-i)}\wedge G^{(1)}}.$

  * •

Let $(F,G,H)\in\mathcal{F}(V)^{(3)}$ and let $i,j,k\in\mathbb{Z}^{+}$ so that
$i+j+k=n$. Then define the _triple ratio_

$T_{i,j,k}(F,G,H):=\frac{F^{(i+1)}\wedge G^{(j)}\wedge
H^{(k-1)}}{F^{(i+1)}\wedge G^{(j-1)}\wedge H^{(k)}}\cdot\frac{F^{(i-1)}\wedge
G^{(j+1)}\wedge H^{(k)}}{F^{(i)}\wedge G^{(j+1)}\wedge
H^{(k-1)}}\cdot\frac{F^{(i)}\wedge G^{(j-1)}\wedge H^{(k+1)}}{F^{(i-1)}\wedge
G^{(j)}\wedge H^{(k+1)}}.$

We now explain the formulas used in Definition 2.5. Given any triple of flags
$F,G,H$, choose three bases $\\{f_{1},\dots,f_{n}\\}$,
$\\{g_{1},\dots,g_{n}\\}$, $\\{h_{1},\dots,h_{n}\\}$ of $V$ so that
$F^{(l)}=\operatorname{\mathrm{Span}}_{\mathbb{R}}(f_{1},\dots,f_{l})$,
$G^{(l)}=\operatorname{\mathrm{Span}}_{\mathbb{R}}(g_{1},\dots,g_{l})$ and
$H^{(l)}=\operatorname{\mathrm{Span}}_{\mathbb{R}}(h_{1},\dots,h_{l})$ for all
$l=1,\dots,n-1$. For each $i,j,k\in\mathbb{Z}^{+}$ so that $i+j+k=n$, define

$F^{(i)}\wedge G^{(j)}\wedge
H^{(k)}:=\det(f_{1},\dots,f_{i},g_{1},\dots,g_{j},h_{1},\dots,h_{k}),$

and note that if $(F,G,H)\in\mathcal{F}(V)^{(3)}$, then $F^{(i)}\wedge
G^{(j)}\wedge H^{(k)}\neq 0$. Of course, the definition of $F^{(i)}\wedge
G^{(j)}\wedge H^{(k)}$ depends on the choice of the three bases, but one can
easily verify that the cross ratio and triple ratio do not depend on these
choices.

An important (but easily verified) property of the cross ratio and triple
ratio is their projective invariance, i.e. $C_{i}(E,F,G,H)=C_{i}(g\cdot
E,g\cdot F,g\cdot G,g\cdot H)$ and $T_{i,j,k}(F,G,H)=T_{i,j,k}(g\cdot F,g\cdot
G,g\cdot H)$ for any $g\in\operatorname{\mathrm{PGL}}(V)$. It is also easy to
check the following symmetries from the definition:

  * •

$\displaystyle C_{i}(E,F,G,H)=C_{n-i}(H,G,F,E)=\frac{1}{C_{i}(E,G,F,H)}$,

  * •

$\displaystyle T_{i,j,k}(F,G,H)=T_{j,k,i}(G,H,F)=\frac{1}{T_{i,k,j}(F,H,G)}$.

By definition, $\big{(}\xi(x),\xi(y),\xi(z)\big{)}\in\mathcal{F}(V)^{(3)}$ for
any triple $x,y,z\in S^{1}$ and any Frenet curve $\xi:S^{1}\to\mathcal{F}(V)$.
This allows us to evaluate the cross ratios and triple ratios along the image
of $\xi$. The following theorem is essentially due to Fock-Goncharov [FG06]
(also see [LM09, Appendix B] or [Zha15, Proposition 2.5.7]).

######  Theorem 2.6.

Let $\xi:S^{1}\to\mathcal{F}(V)$ be Frenet.

  1. (1)

For any pairwise distinct triple of points $x,y,z\in S^{1}$, and for any
$i,j,k\in\mathbb{Z}^{+}$ so that $i+j+k=n$, we have

$T_{i,j,k}\big{(}\xi(x),\xi(y),\xi(z)\big{)}>0.$

  2. (2)

For any pairwise distinct quadruple of points $x,y,z,w\in S^{1}$ so that
$x<y<z<w<x$ in the cyclic ordering on $S^{1}$, and for any $i=1,\dots,n-1$, we
have

$C_{i}\big{(}\xi(x),\xi(y),\xi(w),\xi(z)\big{)}<0.$

Another important feature of the triple ratio is that one can use them to
parameterize the space of projective classes of triples of transverse flags.
The following proposition was observed by Fock-Goncharov [FG06], and its proof
is a straightforward computation which we omit.

######  Proposition 2.7.

The map
$\mathcal{F}(V)^{(3)}/\operatorname{\mathrm{PGL}}(V)\to(\mathbb{R}\setminus\\{0\\})^{\frac{(n-1)(n-2)}{2}}$
given by

$[F,G,H]\mapsto\big{(}T_{i,j,k}(F,G,H)\big{)}_{i,j,k}$

is a real-analytic diffeomorphism.

###  2.2. Ideal triangulations, barriers and bridges

In this section, we introduce some terminology and notation regarding
topological choices on $S$ that we will make in this paper. While it is often
convenient to choose a hyperbolic metric on $S$ to realize these topological
choices as geometric objects on $S$ equipped with the chosen hyperbolic
metric, it is important for our purposes that these are defined purely
topologically.

For the rest of the article, if we choose a hyperbolic metric $m$ on $S$, then
we will use the notation $\Sigma:=(S,m)$, while reserving the notation $S$ to
be used only to denote a topological surface. We will also abuse terminology
by referring to $\Sigma$ as a hyperbolic metric on $S$.

Let $\widetilde{S}$ denote the universal cover of $S$ and
$\pi:\widetilde{S}\to S$ the covering map. An _oriented geodesic in_
$\widetilde{S}$ is an ordered pair in
$\mathcal{G}^{o}:=\\{(x,y)\in\partial\Gamma^{2}:x\neq y\\}$, and a _geodesic
in $\widetilde{S}$_ is an unordered pair in
$\mathcal{G}:=\mathcal{G}^{o}/\big{(}(x,y)\sim(y,x)\big{)}$. We will use the
set notation $\\{x,y\\}$ to denote the equivalence class in $\mathcal{G}$
containing $(x,y)$. Two geodesics
$\\{x,y\\},\\{x^{\prime},y^{\prime}\\}\in\mathcal{G}$ are _transverse_ if the
two connected components of $\partial\Gamma\setminus\\{x,y\\}$ each contain a
point in the set $\\{x^{\prime},y^{\prime}\\}$.

If we choose a hyperbolic metric $\Sigma$ on $S$, then this induces a
hyperbolic metric $\widetilde{\Sigma}$ on $\widetilde{S}$, and the visual
boundary $\partial\widetilde{\Sigma}$ of $\widetilde{\Sigma}$ is canonically
identified with $\partial\Gamma$. This in turn induces an identification of
$\mathcal{G}^{o}$ and $\mathcal{G}$ respectively with the set of oriented
hyperbolic geodesics in $\widetilde{\Sigma}$ and the set of hyperbolic
geodesics in $\widetilde{\Sigma}$. More explicitly, $\\{x,y\\}\in\mathcal{G}$
is identified with the hyperbolic geodesic in $\widetilde{\Sigma}$ whose
endpoints in $\partial\widetilde{\Sigma}=\partial\Gamma$ are $x$ and $y$, and
$(x,y)\in\mathcal{G}^{o}$ is identified with the oriented hyperbolic geodesic
in $\widetilde{\Sigma}$ whose backward and forward endpoints are $x$ and $y$
respectively.

An _ideal triangulation of $\widetilde{S}$_ is a maximal $\Gamma$-invariant
subset $\widetilde{\mathcal{T}}\subset\mathcal{G}$ so that any two geodesics
$\\{x,y\\},\\{x^{\prime},y^{\prime}\\}\in\widetilde{\mathcal{T}}$ are not
transverse, and for any geodesic $\\{x,y\\}\in\widetilde{\mathcal{T}}$, one of
the following must hold:

  1. (1)

There is some $z\in\partial\Gamma$ so that $\\{x,z\\}$ and $\\{y,z\\}$ are
geodesics in $\widetilde{\mathcal{T}}$,

  2. (2)

There is some group element $\gamma\in\Gamma$ so that $\gamma\cdot x=x$ and
$\gamma\cdot y=y$.

The elements in $\widetilde{\mathcal{T}}$ are _edges_. If (1) holds, then
$\\{x,y\\}$ is an _isolated edge_ of $\widetilde{\mathcal{T}}$, and if (2)
holds, then $\\{x,y\\}$ is a _closed edge_ of $\widetilde{\mathcal{T}}$. Let
$\widetilde{\mathcal{Q}}=\widetilde{\mathcal{Q}}_{\widetilde{\mathcal{T}}}$
denote the set of isolated edges in $\widetilde{\mathcal{T}}$ and let
$\widetilde{\mathcal{P}}=\widetilde{\mathcal{P}}_{\widetilde{\mathcal{T}}}$
denote the set of closed edges in $\widetilde{\mathcal{T}}$. For any
$\\{x,y\\}\in\widetilde{\mathcal{T}}$, $x$ and $y$ are the _vertices_ of the
edge $\\{x,y\\}$. Also, any triple of $\\{x,y,z\\}$ of points in
$\partial\Gamma$ so that
$\\{x,y\\},\\{y,z\\},\\{z,x\\}\in\widetilde{\mathcal{T}}$ is an _ideal
triangle_ of $\widetilde{\mathcal{T}}$. Denote the set of ideal triangles of
$\widetilde{\mathcal{T}}$ by
$\widetilde{\Theta}_{\widetilde{\mathcal{T}}}=\widetilde{\Theta}$. If
$e_{1},e_{2},e_{3}\in\widetilde{\mathcal{T}}$ are the edges of an ideal
triangle $T\in\widetilde{\Theta}$, we will also use the notation
$T=\\{e_{1},e_{2},e_{3}\\}$.

If $\widetilde{\mathcal{T}}$ is an ideal triangulation of $\widetilde{S}$,
then the quotient $\mathcal{T}:=\widetilde{\mathcal{T}}/\Gamma$ is an _ideal
triangulation of $S$_. Similarly, the elements in $\mathcal{T}$,
$\mathcal{Q}=\mathcal{Q}_{\mathcal{T}}:=\widetilde{\mathcal{Q}}/\Gamma$ and
$\mathcal{P}=\mathcal{P}_{\mathcal{T}}:=\widetilde{\mathcal{P}}/\Gamma$ are
called _edges_ , _isolated edges_ , and _closed edges_ respectively. The
quotient $\Theta=\Theta_{\mathcal{T}}:=\widetilde{\Theta}/\Gamma$ is the set
of _ideal triangles_ of $\mathcal{T}$.

By choosing a hyperbolic metric $\Sigma$ on $S$, $\mathcal{T}$ is realized as
an ideal triangulation of the hyperbolic surface $\Sigma$ in the classical
sense. Hence, $\widetilde{\mathcal{T}}$ is realized as a $\Gamma$-invariant
lift of $\mathcal{T}$ to $\widetilde{\Sigma}$. This implies that
$|\Theta|=4g-4$, $|\mathcal{Q}|=9g-9$, and $1\leq|\mathcal{P}|\leq 3g-3$,
where $g$ is the genus of $S$.

Let
$\widetilde{\mathcal{T}}^{o}:=\\{(x,y)\in\mathcal{G}^{o}:\\{x,y\\}\in\widetilde{\mathcal{T}}\\}$,
and note that $\widetilde{\mathcal{T}}^{o}$ admits a natural $\Gamma$-action
given by $\gamma\cdot(x,y)=(\gamma\cdot x,\gamma\cdot y)$ for any
$\gamma\in\Gamma$ and any $(x,y)\in\widetilde{\mathcal{T}}^{o}$. Also, observe
that there is an obvious two-to-one, $\Gamma$-equivariant map
$\widetilde{\Pi}:\widetilde{\mathcal{T}}^{o}\to\widetilde{\mathcal{T}}$ given
by $\widetilde{\Pi}(x,y)=\\{x,y\\}$. Let
$\mathcal{T}^{o}:=\widetilde{\mathcal{T}}^{o}/\Gamma$, and let
$\Pi:\mathcal{T}^{o}\to\mathcal{T}$ denote the quotient map induced by
$\widetilde{\Pi}$. We will also denote
$\widetilde{\mathcal{P}}^{o}:=\\{(x,y)\in\mathcal{G}^{o}:\\{x,y\\}\in\widetilde{\mathcal{P}}\\}$,
$\widetilde{\mathcal{Q}}^{o}:=\\{(x,y)\in\mathcal{G}^{o}:\\{x,y\\}\in\widetilde{\mathcal{Q}}\\}$,
$\mathcal{P}^{o}:=\widetilde{\mathcal{P}}^{o}/\Gamma$ and
$\mathcal{Q}^{o}:=\widetilde{\mathcal{Q}}^{o}/\Gamma$. An _orientation_ on
$\widetilde{\mathcal{T}}$ is a $\Gamma$-equivariant map
$\widetilde{r}:\widetilde{\mathcal{T}}\to\widetilde{\mathcal{T}}^{o}$ so that
$\widetilde{\Pi}\circ\widetilde{r}:\widetilde{\mathcal{T}}\to\widetilde{\mathcal{T}}$
is the identity. This descends to a map $r:\mathcal{T}\to\mathcal{T}^{o}$,
which is called an _orientation on $\mathcal{T}$_.

Given an ideal triangulation $\mathcal{T}$ of $S$, let

$\widetilde{\mathcal{M}}=\widetilde{\mathcal{M}}_{\widetilde{\mathcal{T}}}:=\\{(v,T):T\in\widetilde{\Theta},v\text{
a vertex of }T\\}.$

This set also admits an obvious $\Gamma$ action, so we can consider
$\mathcal{M}=\mathcal{M}_{\mathcal{T}}:=\Gamma\backslash\widetilde{\mathcal{M}}$.

Figure 1. This is a drawing of $\widetilde{\mathcal{B}}$ and a compatible
$\widetilde{\mathcal{J}}$ in the Klein model of $\widetilde{\Sigma}$. The
closed edges in $\widetilde{\mathcal{P}}$ are drawn in red, the isolated edges
in $\widetilde{\mathcal{Q}}$ are drawn in blue, and the non-edge barriers in
$\widetilde{\mathcal{M}}$ are drawn in yellow. The bridges in
$\widetilde{\mathcal{J}}$ are drawn in green. One ideal triangle in
$\widetilde{\Theta}$ is drawn in violet, and one fragmented ideal triangle is
drawn in turquoise.

######  Definition 2.8.

  1. (1)

Let $\widetilde{\mathcal{T}}$ be an ideal triangulation of $\widetilde{S}$.
The _barrier system_ associated to $\widetilde{\mathcal{T}}$ is
$\widetilde{\mathcal{B}}=\widetilde{\mathcal{B}}_{\widetilde{\mathcal{T}}}:=\widetilde{\mathcal{T}}\cup\widetilde{\mathcal{M}}$.
If $\widetilde{r}$ is an orientation on $\widetilde{\mathcal{T}}$, the pair
$(\widetilde{r},\widetilde{\mathcal{B}})$ is the _oriented barrier system_
associated to $(\widetilde{r},\widetilde{\mathcal{T}})$. Similarly, denote
$\mathcal{B}=\mathcal{B}_{\mathcal{T}}:=\widetilde{\mathcal{B}}/\Gamma$, and
call $\mathcal{B}$ and $(r,\mathcal{B})$ the _barrier system_ and _oriented
barrier system_ associated to $\mathcal{T}$ and $(r,\mathcal{T})$
respectively.

  2. (2)

The elements in $\mathcal{B}$ and $\widetilde{\mathcal{B}}$ are called
_barriers_ , and the elements in $\mathcal{M}$ and $\widetilde{\mathcal{M}}$
are called _non-edge barriers_.

  3. (3)

For any closed edge $\\{x,y\\}\in\widetilde{\mathcal{P}}$, let
$\partial\Gamma_{1}$ and $\partial\Gamma_{2}$ be the connected components of
$\partial\Gamma\setminus\\{x,y\\}$. A _bridge_ across $\\{x,y\\}$ is an
unordered pair $\\{T_{1},T_{2}\\}$ so that for all $m=1,2$,

     * •

$T_{m}\in\widetilde{\Theta}$

     * •

one of the vertices of $T_{m}$ is either $x$ or $y$,

     * •

the other two vertices of $T_{m}$ lie in $\partial\Gamma_{m}$.

  4. (4)

A _bridge system compatible with $\widetilde{\mathcal{T}}$_ is a minimal
$\Gamma$-invariant collection of bridges
$\widetilde{\mathcal{J}}=\widetilde{\mathcal{J}}_{\widetilde{\mathcal{T}}}$ so
that for every closed edge $e\in\widetilde{\mathcal{P}}$, there is a bridge
$\\{T_{1},T_{2}\\}\in\widetilde{\mathcal{J}}$ across $e$. If
$\widetilde{\mathcal{J}}$ is a bridge system compatible with
$\widetilde{\mathcal{T}}$, then
$\mathcal{J}=\mathcal{J}_{\mathcal{T}}:=\widetilde{\mathcal{J}}/\Gamma$ is a
_bridge system compatible with $\mathcal{T}$_.

Observe that while the barrier system associated to $\widetilde{\mathcal{T}}$
(resp. $\mathcal{T}$) is determined by $\widetilde{\mathcal{T}}$ (resp.
$\mathcal{T}$), this is not true for bridge systems compatible with
$\widetilde{\mathcal{T}}$ (resp. $\mathcal{T}$).

If we choose a hyperbolic metric $\Sigma$ on $S$, then each ideal triangle
$\widehat{T}\in\Theta$ is realized as a hyperbolic ideal triangle in $\Sigma$
and each $T\in\widetilde{\Theta}$ is realized as a hyperbolic ideal triangle
in $\widetilde{\Sigma}$. For each hyperbolic ideal triangle
$T\in\widetilde{\Theta}$, let $p_{T}$ be the baricenter of $T$. Observe that
this allows us to realize each non-edge barrier
$(v,T)\in\widetilde{\mathcal{M}}$ as a geodesic ray in $\widetilde{\Sigma}$
from $p_{T}$ to $v$ (see Figure 1). Any non-edge barrier $[v,T]\in\mathcal{M}$
is then realized as a geodesic ray with source $p_{\widehat{T}}$, the
baricenter of $\widehat{T}$, that “spirals” towards the closed geodesic on
$\Sigma$ that corresponds to some $\gamma\in\Gamma$ that has $v$ as a fixed
point in $\partial\Gamma$.

Note that $\widetilde{\Sigma}\setminus\widetilde{\mathcal{B}}$ is a union of
triangles, each of which has two vertices in $\partial\widetilde{\Sigma}$ and
a vertex in $\widetilde{\Sigma}$. We call any such triangle a _fragmented
ideal triangle_ (see Figure 1). Any fragmented ideal triangle $T^{\prime}$
lies in a unique ideal triangle $T$ in $\widetilde{\Theta}$, and $T^{\prime}$,
$T$ have two vertices in common. Similarly, a connected component of
$\Sigma\setminus\mathcal{B}$ is also called a _fragmented ideal triangle_.

$p_{1}=q_{2}$$z_{1}$$w_{1}$$w_{2}$$z_{2}$$q_{1}=p_{2}$$p_{1}=p_{2}$$z_{1}$$w_{1}$$z_{2}$$w_{2}$$q_{1}=q_{2}$$T_{1}$$T_{1}$$T_{2}$$T_{2}$

Figure 2. Notation 2.9

######  Notation 2.9.

For each bridge $J=\\{T_{1},T_{2}\\}$ across
$\\{x,y\\}\in\widetilde{\mathcal{P}}$, let $p_{m}$ denote the point in
$\\{x,y\\}$ that is a vertex of $T_{m}$ for $m=1,2$, and let $q_{m}$ denote
the point in $\\{x,y\\}$ that is not a vertex of $T_{m}$. Also, let $z_{m}$
and $w_{m}$ be the vertices of $T_{m}$ that are not $p_{m}$, so that
$\\{w_{m},p_{m}\\}$ intersects $\\{z_{m},q_{m}\\}$ transversely (see Figure
2).

With our choice of a hyperbolic metric $\Sigma$ on $S$, the bridge
$\\{T_{1},T_{2}\\}$ can be realized as an unoriented geodesic segment in
$\widetilde{\Sigma}$ with one endpoint in the fragmented ideal triangle
$T^{\prime}_{1}\subset\widetilde{\Sigma}$ with vertices
$p_{T_{1}},p_{1},w_{1}$ and the other endpoint in the fragmented ideal
triangle $T^{\prime}_{2}\subset\widetilde{\Sigma}$ with vertices
$p_{T_{2}},p_{2},w_{2}$ (see Figure 1).

We can define orientations on a bridge system in the same way we defined
orientations on ideal triangulations. Namely, let
$\widetilde{\mathcal{J}}^{o}:=\\{(T_{1},T_{2}):\\{T_{1},T_{2}\\}\in\widetilde{\mathcal{J}}\\}$,
then $\Gamma$ acts diagonally on $\widetilde{\mathcal{J}}^{o}$, and the map
$\widetilde{\Pi}:\widetilde{\mathcal{J}}^{o}\to\widetilde{\mathcal{J}}$ given
by $\widetilde{\Pi}(T_{1},T_{2})=\\{T_{1},T_{2}\\}$ is a $\Gamma$-equivariant
two-to-one projection. Hence, if we define
$\mathcal{J}^{o}:=\Gamma\backslash\widetilde{\mathcal{J}}^{o}$, then
$\widetilde{\Pi}$ descends to $\Pi:\mathcal{J}^{o}\to\mathcal{J}$.

######  Definition 2.10.

An _orientation_ on a bridge system $\widetilde{\mathcal{J}}$ is a
$\Gamma$-equivariant map
$\widetilde{s}:\widetilde{\mathcal{J}}\to\widetilde{\mathcal{J}}^{o}$ so that
$\widetilde{\Pi}\circ\widetilde{s}:\widetilde{\mathcal{J}}\to\widetilde{\mathcal{J}}$
is the identity. The orientation
$\widetilde{s}:\widetilde{\mathcal{J}}\to\widetilde{\mathcal{J}}^{o}$ descends
to a map $s:\mathcal{J}\to\mathcal{J}^{o}$, which is called an _orientation on
$\mathcal{J}$_.

###  2.3. Algebraic intersection number

Choose an oriented barrier system $(r,\mathcal{B})$ on $S$, a compatible
bridge system $\mathcal{J}$, and a hyperbolic metric $\Sigma$ on $S$. The
choice of $\Sigma$ realizes the oriented edges in
$\widetilde{\mathcal{T}}^{o}$, non-edge barriers in $\widetilde{\mathcal{M}}$
and oriented bridges in $\widetilde{\mathcal{J}}^{o}$ as oriented geodesics,
geodesic rays, and oriented geodesic segments in $\widetilde{\Sigma}$. Using
this, we define the notion of an algebraic intersection between $1$-simplices
$g:[0,1]\to\widetilde{\Sigma}$ and elements
$X\in\widetilde{\mathcal{J}}^{o}\cup\widetilde{\mathcal{T}}^{o}\cup\widetilde{\mathcal{M}}$
that satisfy $\big{|}g[0,1]\cap X\big{|}<\infty$.

Let $I\subset g[0,1]$ be a closed subinterval whose interior is a component of

$g[0,1]\setminus\bigcup_{X\in\widetilde{\mathcal{J}}^{o}\cup\widetilde{\mathcal{T}}^{o}\cup\widetilde{\mathcal{M}}}X.$

Denote the backward and forward endpoints of $I$ by $I^{-}$ and $I^{+}$
respectively. For any $p\in\\{I^{-},I^{+}\\}$, we say that $I$ lies _to the
right (resp. left) of $X$ at $p$_ if $p\in X$, and there is some closed
subinterval $I^{\prime}\subset I$ containing $p$ so that $I^{\prime}$ lies to
the right (resp. left) of the oriented (bi-infinite) geodesic containing $X$.
If $I^{-}$ is not the source of a non-edge barrier, define

$\epsilon(I^{-},X):=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{2}&\text{if
}I\text{ lies to the right of }X\text{ at }I^{-};\\\
\displaystyle-\frac{1}{2}&\text{if }I\text{ lies to the left of }X\text{ at
}I^{-};\\\ 0&\text{if }I^{-}\notin X.\end{array}\right.$

Similarly, if $I^{+}$ is not the source of a non-edge barrier, define

$\epsilon(I^{+},X):=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{2}&\text{if
}I\text{ lies to the left of }X\text{ at }I^{+};\\\
\displaystyle-\frac{1}{2}&\text{if }I\text{ lies to the right of }X\text{ at
}I^{+};\\\ 0&\text{if }I^{+}\notin X.\end{array}\right.$

On the other hand, suppose that $p\in\\{I^{-},I^{+}\\}$ is the source of a
non-edge barrier. Let $X_{1},X_{2},X_{3}\in\widetilde{\mathcal{M}}$ be the
three non-edge barriers whose source is $p$, so that the endpoint $v_{i}$ of
$X_{i}$ in $\partial\Gamma$ satisfy $v_{1}<v_{2}<v_{3}<v_{1}$. Let
$U_{1,p},U_{2,p},U_{3,p}$ be the three connected components of
$\widetilde{\Sigma}\setminus(X_{1}\cup X_{2}\cup X_{3})$ so that
$X_{i-1},X_{i+1}\subset\overline{U_{i,p}}$. Here, the arithmetic in the
subscripts is done modulo $3$. Then define

$\epsilon(I^{-},X):=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{3}&\text{if
}X=X_{i}\text{ and }I\subset U_{i-1,I^{-}};\\\
\displaystyle-\frac{1}{3}&\text{if }X=X_{i}\text{ and }I\subset
U_{i+1,I^{-}};\\\ 0&\text{otherwise.}\end{array}\right.$

and

$\epsilon(I^{+},X):=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{3}&\text{if
}X=X_{i}\text{ and }I\subset U_{i+1,I^{+}};\\\
\displaystyle-\frac{1}{3}&\text{if }X=X_{i}\text{ and }I\subset
U_{i-1,I^{+}};\\\ 0&\text{otherwise.}\end{array}\right.$

######  Definition 2.11.

Let $g:[0,1]\to\widetilde{\Sigma}$ be a $1$-simplex and
$X\in\widetilde{\mathcal{J}}^{o}\cup\widetilde{\mathcal{T}}^{o}\cup\widetilde{\mathcal{M}}$
satisfying $\big{|}g[0,1]\cap X\big{|}<\infty$. Let $I_{1},\dots,I_{k}$ be the
closed subintervals of $g[0,1]$ whose interiors are the connected components
of $g[0,1]\setminus X$. The _algebraic intersection number_ (see Figure 3)
between any $g$ and $X$ is defined to be

$\hat{i}(g,X)=\hat{i}_{\Sigma}(g,X):=\sum_{l=1}^{k}\epsilon(I_{k}^{-},X)+\epsilon(I_{k}^{+},X).$

$g$$b_{1}$$b_{2}$$b_{3}$$b_{4}$$e_{1}$$e_{2}$

Figure 3. $\hat{i}(g,b_{1})=1$, $\hat{i}(g,b_{2})=-\frac{2}{3}$,
$\hat{i}(g,b_{3})=\frac{1}{3}$, $\hat{i}(g,b_{4})=\frac{1}{3}$,
$\hat{i}(g,e_{1})=-1$, $\hat{i}(g,e_{2})=\frac{1}{2}$.

Let $\pi:\widetilde{S}\to S$ be the covering map. Observe that when
$g:[0,1]\to\widetilde{S}$ is a $1$-simplex so that $\pi\circ g:[0,1]\to S$ is
a closed curve and $e\in\widetilde{\mathcal{P}}^{o}$, $\hat{i}(g,e)$ is the
usual algebraic intersection number on $H_{1}(S,\mathbb{Z})$ between the
homology classes $[\pi\circ g]$ and $[\pi\circ e]$, which is in particular
independent of $\Sigma$. However, $\hat{i}$ in general depends on the choice
of $\Sigma$.

###  2.4. The tangent space to $\operatorname{\mathrm{Hit}}_{V}(S)$ and the
Goldman symplectic form

Recall that the Lie algebra $\mathfrak{s}\mathfrak{l}(V)$ of
$\operatorname{\mathrm{PSL}}(V)$ is naturally identified with the traceless
endomorphisms of $V$. Hence, one can define the _trace form_

$B:\mathfrak{s}\mathfrak{l}(V)\times\mathfrak{s}\mathfrak{l}(V)\to\mathbb{R}$

by $B(X,Y):=\operatorname{\mathrm{tr}}(XY)$. This pairing is well-known to be
bilinear, symmetric and non-degenerate (it is a multiple of the Killing form
on $\mathfrak{s}\mathfrak{l}(V)$).

Using the trace form, we can apply a general construction by Goldman [Gol84]
to define a symplectic form on $\operatorname{\mathrm{Hit}}_{V}(S)$, which we
will refer to as the _Goldman symplectic form_. From the work of Atiyah-Bott
[AB83], one also has a description of this symplectic form using the language
of flat bundles over $S$. In the case when $\dim(V)=2$, Goldman [Gol84,
Proposition 2.5] showed that this symplectic form agrees with the Weil-
Petersson symplectic form on the Teichmüller space of $S$.

We will now recall the construction of the Goldman symplectic form. First,
Goldman observed that for any $[\rho]\in\operatorname{\mathrm{Hit}}_{V}(S)$
and any representative $\rho$ of $[\rho]$, one has the following canonical
isomorphisms:

$T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)\simeq
H^{1}(\Gamma,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})\simeq
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$

The first linear isomorphism
$T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)\simeq
H^{1}(\Gamma,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
can be described explicitly in the following way. Let $t\mapsto[\rho_{t}]$ be
a smooth curve in $\operatorname{\mathrm{Hit}}_{V}(S)$ so that
$[\rho_{0}]=[\rho]$, and denote its tangent vector at $[\rho_{0}]$ by $X\in
T_{[\rho_{0}]}\operatorname{\mathrm{Hit}}_{V}(S)$. Choose any smooth curve
$t\mapsto\rho_{t}$ in $\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ so that
$\Pi(\rho_{t})=[\rho_{t}]$, where
$\Pi:\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)\to\operatorname{\mathrm{Hit}}_{V}(S)$
is the quotient map. For any $\gamma\in\Gamma$, let $t\mapsto g_{t,\gamma}$ be
the smooth curve in $\operatorname{\mathrm{PSL}}(V)$ defined by
$\rho_{t}(\gamma)=g_{t,\gamma}\cdot\rho_{0}(\gamma)$. The fact that $\rho_{t}$
is a homomorphism for all $t$ then implies that the map
$\mu_{X}:\Gamma\to\mathfrak{s}\mathfrak{l}(V)$ defined by
$\mu_{X}:\gamma\mapsto\frac{d}{dt}|_{t=0}g_{t,\gamma}$ is a group cocycle, and
thus defines a cohomology class $[\mu_{X}]\in
H^{1}(\Gamma,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$.
Goldman [Gol84, Section 1.4] then showed that the map
$T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)\to
H^{1}(\Gamma,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
given by $X\mapsto[\mu_{X}]$ is a well-defined linear isomorphism.

The second linear isomorphism
$H^{1}(\Gamma,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})\simeq
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
can also be described explicitly. Let $\mu\in
C^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$ be
a cocycle in the cohomology class $[\mu]\in
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$,
and let
$\widetilde{\mu}:C_{1}(\widetilde{S},\mathbb{Z})\to\mathfrak{s}\mathfrak{l}(V)$
be its $\operatorname{\mathrm{Ad}}\circ\rho$-equivariant lift. Pick any point
$p\in\widetilde{S}$. For any $\gamma\in\Gamma$, let
$h_{\gamma}:[0,1]\to\widetilde{S}$ be a $1$-simplex so that $h_{\gamma}(0)=p$
and $h_{\gamma}(1)=\gamma\cdot p$. Then define
$\bar{\mu}:\Gamma\to\mathfrak{s}\mathfrak{l}(V)$ by
$\bar{\mu}:\gamma\mapsto\mu(h_{\gamma})$. One can verify that $\bar{\mu}$ is a
group cocycle, and that the map
$H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})\to
H^{1}(\Gamma,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
given by $[\mu]\mapsto[\bar{\mu}]$ is a well-defined linear isomorphism.

With these identifications, we can define the Goldman symplectic form.

######  Definition 2.12.

The _Goldman symplectic form_ is the pairing

$\omega:H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})\times
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})\to
H^{2}(S,\mathbb{R})\to\mathbb{R}$

where the first map is the cup product on cohomology while pairing the
$\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho}$
coeffecients using the trace form, and the second map is taking the cap
product with the fundamental class $[S]\in H_{2}(S,\mathbb{R})$.

Goldman [Gol84, Theorem 1.7] proved that this bilinear pairing defines a
symplectic form on $\operatorname{\mathrm{Hit}}_{V}(S)$.

The Goldman symplectic form can be described more explicitly if one makes some
topological choices. Choose a finite triangulation $\mathbb{T}$ (NOT an ideal
triangulation described in Section 2.2) of $S$ so that the three vertices of
every triangle given by $\mathbb{T}$ are pairwise distinct. Enumerate the
vertices of $\mathbb{T}$ by $v_{1},\dots,v_{k}$. Then for any triangle
$\delta\in\mathbb{T}$ with vertices $v_{a},v_{b},v_{c}$ so that $a<b<c$, let
$e_{\delta,1}$ denote the edge of $\delta$ with endpoints $v_{a},v_{b}$,
oriented from $v_{a}$ to $v_{b}$. Similarly, let $e_{\delta,2}$ denote the
edge of $\delta$ with endpoints $v_{b},v_{c}$, oriented from $v_{b}$ to
$v_{c}$. Let $\widetilde{\delta}$ denote a triangle in $\widetilde{S}$ so that
$\pi(\widetilde{\delta})=\delta$ (recall that $\pi:\widetilde{S}\to S$ is the
covering map), and for $m=1,2$, let $e_{\widetilde{\delta},m}$ denote the
oriented boundary edge of $\widetilde{\delta}$ so that
$\pi(e_{\widetilde{\delta},m})=e_{\delta,m}$.

If $\mu_{1},\mu_{2}\in
C^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
are cocycles with $\operatorname{\mathrm{Ad}}\circ\rho$-equivariant lifts
$\widetilde{\mu}_{1},\widetilde{\mu}_{2}:C_{1}(\widetilde{S},\mathbb{Z})\to\mathfrak{s}\mathfrak{l}(V)$,
then the cup product formula in simplicial cohomology allows us to write the
Goldman symplectic form as

(2.1)
$\omega\big{(}[\mu_{1}],[\mu_{2}]\big{)}=\sum_{\delta\in\mathbb{T}}\operatorname{sgn}(\delta)\cdot\operatorname{\mathrm{tr}}\big{(}\widetilde{\mu}_{1}(e_{\widetilde{\delta},1})\cdot\widetilde{\mu}_{2}(e_{\widetilde{\delta},2})\big{)}.$

Here, $\operatorname{sgn}(\delta)$ is $1$ if the enumeration of the vertices
of $\delta$ is increasing in the clockwise order, and is $-1$ otherwise. It is
a standard exercise in algebraic topology that this formula does not depend on
any of the choices made.

###  2.5. The tangent cocycle

The formula for the Goldman symplectic form given by (2.1) shows that if we
can find explicit $\operatorname{\mathrm{Ad}}\circ\rho$-equivariant cocycles
$\widetilde{\mu}_{1},\widetilde{\mu}_{2}:C_{1}(\widetilde{S},\mathbb{Z})\to\mathfrak{s}\mathfrak{l}(V)$
representing the cohomology classes $[\mu_{1}],[\mu_{2}]\in
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$,
then we can explicitly compute the Goldman symplectic form between the tangent
vectors in $T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)$ corresponding to
$[\mu_{1}]$ and $[\mu_{2}]$. It turns out that one way to systematically find
these cocycle representatives is to use Theorem 2.4 to view the cohomology
classes in
$H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
as describing infinitesimal deformations of Frenet curves instead of
representations. This gives rise to the notion of the tangent cocycle which we
will now define.

Choose an oriented ideal triangulation $\widetilde{\mathcal{T}}$ on
$\widetilde{S}$, let $\widetilde{\mathcal{B}}$ be the associated barrier
system, and choose a compatible bridge system $\widetilde{\mathcal{J}}$. Also,
choose a hyperbolic metric $\Sigma$ on $S$. This realizes the elements in
$\widetilde{\mathcal{B}}$ as geodesics (edges in $\widetilde{\mathcal{T}}$) or
geodesic rays (non-edge barriers in $\widetilde{\mathcal{M}}$) in
$\widetilde{\Sigma}$, and also realizes the elements in
$\widetilde{\mathcal{J}}$ as geodesic segments that are transverse to the
closed edges in $\widetilde{\mathcal{P}}$ (see Section 2.2). Given
$\rho\in\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ and a cohomology class
$[\nu]\in
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$,
we will use these choices to define a particular cocycle representative
$\mu_{\rho,[\nu]}\in[\nu]$.

It is sufficient to define $\widetilde{\mu}_{\rho,[\nu]}$ on $1$-simplices in
$\widetilde{S}$. We will do so in four steps. In the first step, we define
$\widetilde{\mu}_{\rho,[\nu]}(h)$ when $h$ is a $1$-simplex with endpoints
that do not lie in any of the barriers in $\widetilde{\mathcal{B}}$. In the
second step, we extend the definition of $\widetilde{\mu}_{\rho,[\nu]}(h)$ to
the case when $h$ has some endpoint in an isolated edge in
$\widetilde{\mathcal{Q}}$ or in the interior of a non-edge barrier in
$\widetilde{\mathcal{M}}$. Then in the third step, we further extend
$\widetilde{\mu}_{\rho,[\nu]}(h)$ to the case when $h$ has some endpoint as
the source of a non-edge barrier. Finally, in the fourth step we extend the
definition of $\widetilde{\mu}_{\rho,[\nu]}(h)$ to the case when $h$ has an
endpoint in a closed edge.

Choose a smooth path $[\rho_{t}]\in\operatorname{\mathrm{Hit}}_{V}(S)$ so that
$[\rho_{0}]=[\rho]$ and

$\frac{d}{dt}\bigg{|}_{t=0}[\rho_{t}]=[\nu]\in
T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)=H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho}).$

It will be clear from our definition that $\widetilde{\mu}_{\rho,[\nu]}$ will
not depend on the choice of $[\rho_{t}]$, but only on the tangent vector
$[\nu]$ to $[\rho_{t}]$ at $t=0$.

Step 1: Suppose that $h:[0,1]\to\widetilde{\Sigma}$ is a $1$-simplex with the
property that the endpoints of $h$ do not lie in any of the geodesics or
geodesic rays in $\widetilde{\mathcal{B}}$. Observe that for such $h$, there
are unique fragmented ideal triangles $T_{h,0}^{\prime}$ and
$T_{h,1}^{\prime}$ that contain $h(0)$ and $h(1)$ respectively. For $m=0,1$,
let $T_{h,m}\in\widetilde{\Theta}$ be the ideal triangle that contains
$T_{h,m}^{\prime}$, and let $x_{h,m},y_{h,m},z_{h,m}$ be the vertices of
$T_{h,m}$ so that $x_{h,m}<y_{h,m}<z_{h,m}<x_{h,m}$ along $\partial\Gamma$,
and $z_{h,m}$ is not a vertex of $T_{h,m}^{\prime}$ (see Figure 4).

$x_{h,0}$$y_{h,0}$$z_{h,0}$$x_{h,1}$$y_{h,1}$$z_{h,1}$$h$$T^{\prime}_{h,0}$$T^{\prime}_{h,1}$

Figure 4. $1$-simplices considered in Step 1.

Let $\rho_{t}\in\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ be the smooth
lift of $[\rho_{t}]$ so that $\rho_{0}=\rho$, and the $\rho_{t}$-equivariant
Frenet curve $\xi_{t}:\partial\Gamma\to\mathcal{F}(V)$ guaranteed by Theorem
2.4 satisfy $\xi_{t}(x_{h,0})=\xi_{0}(x_{h,0})$,
$\xi_{t}(y_{h,0})=\xi_{0}(y_{h,0})$, and
$\xi_{t}^{(1)}(z_{h,0})=\xi_{0}^{(1)}(z_{h,0})$. Such a lift exists uniquely
by Remark 2.2(1). Again by Remark 2.2(1), there is a unique projective
transformation $g_{t,h}\in\operatorname{\mathrm{PSL}}(V)$ so that

$g_{t,h}\cdot\xi_{0}(x_{h,1})=\xi_{t}(x_{h,1}),\,\,\,\,g_{t,h}\cdot\xi_{0}(y_{h,1})=\xi_{t}(y_{h,1}),\,\,\,\,g_{t,h}\cdot\xi_{0}^{(1)}(z_{h,1})=\xi_{t}^{(1)}(z_{h,1}).$

Observe that $t\mapsto g_{t,h}$ is a smooth path in
$\operatorname{\mathrm{PSL}}(V)$ with $g_{0,h}=\operatorname{\mathrm{id}}$. We
define

$\widetilde{\mu}_{\rho,[\nu]}(h):=\frac{d}{dt}\bigg{|}_{t=0}g_{t,h}.$

Figure 5. In Step 2, $\widetilde{\mu}_{\rho,[\nu]}$ evaluated on the black
$1$-simplex is the average of $\widetilde{\mu}_{\rho,[\nu]}$ evaluated on the
violet $1$-simplices, which was defined in Step 1.

Step 2: Suppose now that $h:[0,1]\to\widetilde{\Sigma}$ is a $1$-simplex so
that $h(m)$ lies in the interior of an oriented geodesic (isolated edge) or
geodesic ray (bridge) $b\in\widetilde{\mathcal{M}}\cup\widetilde{\mathcal{Q}}$
for some $m=0,1$. Let $h_{1},h_{2}:[0,1]\to\widetilde{\Sigma}$ be
$1$-simplices so that (see Figure 5)

  * •

$h_{1}(1-m)=h_{2}(1-m)=h(1-m)$,

  * •

$h_{1}(m)$ and $h_{2}(m)$ lie in the two fragmented ideal triangles that
contain $h(m)$ in their boundaries

and define
$\widetilde{\mu}_{\rho,[\nu]}(h):=\frac{1}{2}\left(\widetilde{\mu}_{\rho,[\nu]}(h_{1})+\widetilde{\mu}_{\rho,[\nu]}(h_{2})\right)$.
Note that $\widetilde{\mu}_{\rho,[\nu]}(h)$ does not depend on the choice of
$h_{1}$ or $h_{2}$. This, together with Step 1, iteratively defines
$\widetilde{\mu}_{\rho,[\nu]}(h)$ for all $1$-simplices
$h:[0,1]\to\widetilde{\Sigma}$ where neither $h(0)$ nor $h(1)$ is the source
of a non-edge barrier in $\widetilde{\mathcal{M}}$, or lies in a closed edge
in $\widetilde{\mathcal{P}}$.

Step 3: Next, consider the case where $h:[0,1]\to\widetilde{\Sigma}$ is a
$1$-simplex so that $h(m)$ is the source of a non-edge barrier in
$\widetilde{\mathcal{M}}$ for some $m=0,1$. Let $b_{1}=(v_{1},T)$,
$b_{2}=(v_{2},T)$, $b_{3}=(v_{3},T)$ be the three non-edge barriers whose
source is $h(m)$ so that $v_{1}<v_{2}<v_{3}<v_{1}$ in $\partial\Gamma$. Also,
let $U_{1,h(m)},U_{2,h(m)},U_{3,h(m)}$ be the three connected components of
$\widetilde{\Sigma}\setminus\left(b_{1}\cup b_{2}\cup b_{3}\right)$ so that
the closure $\overline{U_{i,h(m)}}$ of $U_{i,h(m)}$ in $\widetilde{\Sigma}$
contains both $b_{i-1}$ and $b_{i+1}$.

If $h(0)=h(1)$, then define $\widetilde{\mu}_{\rho,[\nu]}(h):=0$. If $h(1-m)$
lies in $U_{i,h(m)}$, let $\bar{h}_{m}$ be a $1$-simplex so that
$\bar{h}_{m}(1-m)=h(1-m)$ and $\bar{h}_{m}(m)$ lies in the fragmented ideal
triangle $U_{i,h(m)}\cap T$. Also, for $i,j=1,2,3$, let
$l_{i,j,h(m)}:[0,1]\to\widetilde{S}$ be the $1$-simplex with
$l_{i,j,h(m)}(0)\in U_{i,h(m)}\cap T$ and $l_{i,j,h(m)}(1)\in U_{j,h(m)}\cap
T$ (see Figure 6). If $m=0$ and $h(1)$ lies in $U_{i,h(0)}$, define

$\widetilde{\mu}_{\rho,[\nu]}(h):=\widetilde{\mu}_{\rho,[\nu]}(\bar{h}_{0})+\frac{1}{3}\left(\widetilde{\mu}_{\rho,[\nu]}(l_{i-1,i,h(0)})+\widetilde{\mu}_{\rho,[\nu]}(l_{i+1,i,h(0)})\right),$

and if $m=1$ and $h(0)$ lies in $U_{i,h(1)}$, define

$\widetilde{\mu}_{\rho,[\nu]}(h):=\widetilde{\mu}_{\rho,[\nu]}(\bar{h}_{1})+\frac{1}{3}\left(\widetilde{\mu}_{\rho,[\nu]}(l_{i,i-1,h(1)})+\widetilde{\mu}_{\rho,[\nu]}(l_{i,i+1,h(1)})\right),$

Here, arithmetic in the subscripts is done modulo $3$. This, together with
Step 1 and Step 2, iteratively defines $\widetilde{\mu}_{\rho,[\nu]}(h)$ for
all $1$-simplices $h:[0,1]\to\widetilde{\Sigma}$ whose endpoints do not lie in
a closed edge in $\widetilde{\mathcal{P}}$.

Figure 6. In Step 3, $\widetilde{\mu}_{\rho,[\nu]}$ evaluated on the black
$1$-simplex is the sum of $\widetilde{\mu}_{\rho,[\nu]}$ evaluated on the
violet $1$-simplex, and a third of the sum of $\widetilde{\mu}_{\rho,[\nu]}$
evaluated on the turquoise $1$-simplices.

Step 4: We will now consider the case where $h:[0,1]\to\widetilde{\Sigma}$ is
a $1$-simplex so that $h(m)$ lies in a closed edge
$e=\\{x,y\\}\in\widetilde{\mathcal{P}}$ for some $m=0,1$. The strategy to
define $\widetilde{\mu}_{\rho,[\nu]}(h)$ in this case is to first define
$\widetilde{\mu}_{\rho,[\nu]}(h)$ when $h[0,1]$ lies entirely in $e$ or
entirely in a bridge across $e$. These are defined in Step 4.1 and Step 4.2
below respectively. Then using this and Step 3, we can extend the definition
of $\widetilde{\mu}_{\rho,[\nu]}(h)$ to all $1$-simplices with an endpoint in
$e$.

Step 4.1: Suppose that $h(0)$ and $h(1)$ both lie in $e$. Let
$\gamma\in\Gamma$ be the primitive group element whose axis is $e$, so that
$\gamma$ translates along $e$ in the the direction of $h$. Also, choose a
smooth lift $\rho_{t}\in\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ of
$[\rho_{t}]$ so that $\rho_{0}=\rho$ and the $\rho_{t}$-equivariant Frenet
curve $\xi_{t}$ satisfies $\xi_{t}(x)=\xi_{0}(x)$ and $\xi_{t}(y)=\xi_{0}(y)$.
Note that the (possibly empty) set of bridges that intersect $h(0)$ and $h(1)$
are naturally ordered according to the orientation of $h$. Let $k_{1}$ be the
number of bridges that intersect the interior of $h$, let $k_{2}$ be the
number of bridges that contain the endpoints of $h$, and let
$k:=k_{1}+\frac{k_{2}}{2}$. Then define

$\widetilde{\mu}_{\rho,[\nu]}(h):=k\cdot\frac{d}{dt}\bigg{|}_{t=0}\rho_{t}(\gamma)\rho(\gamma)^{-1}\in\mathfrak{s}\mathfrak{l}(V).$

It is easy to see that this does not depend on the choice $\rho_{t}$.

Step 4.2: Suppose that $h(0)$ is the endpoint of a bridge
$J\in\widetilde{\mathcal{J}}$ and $h(1)$ is the intersection of $J$ with a
closed edge $\\{x,y\\}\in\widetilde{\mathcal{P}}$. Let
$h^{\prime}:[0,1]\to\widetilde{\Sigma}$ be the $1$-simplex so that
$h^{\prime}(0)$ is the endpoint of $J$ that is not $h(0)$, and
$h^{\prime}(1)=h(1)$. Observe that the concatenation of
$h\cdot{h^{\prime}}^{-1}$ is $J$ equipped with the orientation from $h(0)$ to
$h^{\prime}(0)$ (see Figure 7). Let
$\rho_{t},\rho_{t}^{\prime}\in\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$
be the smooth lifts of $[\rho_{t}]$ so that $\rho_{0}=\rho_{0}^{\prime}=\rho$,
and the corresponding equivariant Frenet curves $\xi_{t}$, $\xi_{t}^{\prime}$
respectively satisfy

$\begin{array}[]{lll}\xi_{t}(x_{h,0})=\xi_{0}(x_{h,0}),&\xi_{t}(y_{h,0})=\xi_{0}(y_{h,0}),&\xi_{t}^{(1)}(z_{h,0})=\xi_{0}^{(1)}(z_{h,0}),\\\
\xi_{t}^{\prime}(x_{h^{\prime},0})=\xi_{0}^{\prime}(x_{h^{\prime},0}),&\xi_{t}^{\prime}(y_{h^{\prime},0})=\xi_{0}^{\prime}(y_{h^{\prime},0}),&\xi_{t}^{\prime(1)}(z_{h^{\prime},0})=\xi_{0}^{\prime(1)}(z_{h^{\prime},0}).\end{array}$

$h(0)$$h^{\prime}(0)$$p=q^{\prime}=x_{h,0}$$p^{\prime}=q=x_{h^{\prime},0}$$y_{h,0}$$z_{h,0}$$y_{h^{\prime},0}$$z_{h^{\prime},0}$

Figure 7. The $1$-simplex $h$ in Step 4.2.

Let $p$ be the common vertex shared by $e$ and $T_{h,0}$, and let $q$ be the
other endpoint of $e$. Similarly, let $p^{\prime}$ be the common vertex shared
by $e$ and $T_{h^{\prime},0}$ and let $q^{\prime}$ be the other endpoint of
$e$. Then let $u_{t,h},u_{t,h^{\prime}}\in\operatorname{\mathrm{PSL}}(V)$ be
the unique unipotent projective transformations so that

$u_{t,h}\cdot\xi_{0}(p)=\xi_{t}(p)=\xi_{0}(p)\,\,\,\text{ and
}\,\,\,u_{t,h}\cdot\xi_{0}(q)=\xi_{t}(q),$
$u_{t,h^{\prime}}\cdot\xi_{0}^{\prime}(p^{\prime})=\xi_{t}^{\prime}(p^{\prime})=\xi_{0}^{\prime}(p^{\prime})\,\,\,\text{
and
}\,\,\,u_{t,h^{\prime}}\cdot\xi_{0}^{\prime}(q^{\prime})=\xi_{t}^{\prime}(q^{\prime}).$

Let $d_{t,h}\in\operatorname{\mathrm{PGL}}(V)$ be the unique projective
transformation that sends $u_{t,h^{\prime}}^{-1}\cdot\xi_{t}^{\prime}$ to
$u_{t,h}^{-1}\cdot\xi_{t}$, and observe that $d_{t,h}$ fixes both $\xi(x)$ and
$\xi(y)$. Define

$\widetilde{\mu}_{\rho,[\nu]}(h):=\frac{d}{dt}\bigg{|}_{t=0}u_{t,h}+\frac{1}{2}\cdot\frac{d}{dt}\bigg{|}_{t=0}d_{t,h}\in\mathfrak{s}\mathfrak{l}(V).$

Finally, for any arbitrary $h:[0,1]\to\widetilde{\Sigma}$ so that $h(1)$ lies
in a closed edge $e\in\widetilde{\mathcal{P}}$, we can homotope $h$ to the
concatenation $h_{1}\cdot h_{2}\cdot h_{3}$, where

  * •

$h_{1}(0)=h(0)$,

  * •

$h_{1}(1)=h_{2}(0)$ is the endpoint of a bridge $J\in\widetilde{\mathcal{J}}$
that intersects $e$,

  * •

$h_{2}(1)=h_{3}(0)$ is the intersection of $J$ with $e$

  * •

$h_{3}(1)=h(1)$.

Then define

$\widetilde{\mu}_{\rho,[\nu]}(h):=\widetilde{\mu}_{\rho,[\nu]}(h_{1})+\widetilde{\mu}_{\rho,[\nu]}(h_{2})+\widetilde{\mu}_{\rho,[\nu]}(h_{3})$

and define
$\widetilde{\mu}_{\rho,[\nu]}(h^{-1}):=-\widetilde{\mu}_{\rho,[\nu]}(h)$. Of
course, one needs to verify that $\widetilde{\mu}_{\rho,[\nu]}(h)$ does not
depend on the choice of $h_{1}$, $h_{2}$ and $h_{3}$. In fact, we prove the
following.

######  Proposition 2.13.

The map
$\widetilde{\mu}_{\rho,[\nu]}:C_{1}(\widetilde{S},\mathbb{Z})\to\mathfrak{s}\mathfrak{l}(V)$
as described above is a well-defined,
$\operatorname{\mathrm{Ad}}\circ\rho$-equivariant $1$-cocycle.

The proof of Proposition 2.13 is a straightforward argument which we include
as Appendix A. By the description of the linear isomorphisms

$T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)\simeq
H^{1}(\Gamma,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})\simeq
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$

given above, it is clear that $\left[\mu_{\rho,[\nu]}\right]=[\nu]$. Hence, we
can make the following definition.

######  Definition 2.14.

Let $\Sigma$ be a hyperbolic metric on $S$, let $\mathcal{T}$ be an ideal
triangulation of $S$ and let $\mathcal{J}$ be a compatible bridge system.
Also, let $\rho\in\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ and let
$[\nu]\in
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$.
The cocycle $\mu_{\rho,[\nu]}$ defined above is the
$(\rho,\Sigma,\mathcal{T},\mathcal{J})$-_tangent cocycle_ in $[\nu]$.

For any choice of a hyperbolic metric $\Sigma$ on $S$, an oriented ideal
triangulation $\mathcal{T}$ for $S$ and a compatible bridge system
$\mathcal{J}$, we see that every cohomology class in
$H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
contains a unique $(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle.
Later in Section 4, we will prove that these tangent cocycles arise from
admissible labellings, and thus can be used to explicitly compute the Goldman
symplectic form on $\operatorname{\mathrm{Hit}}_{V}(S)$.

##  3\. The vector space of admissible labellings

The goal of this section is to define the vector space of admissible
labellings. An admissible labelling is a particular way to label any barrier
system on $\widetilde{S}$ with elements in
$\mathfrak{s}\mathfrak{l}(V)=\\{X\in\operatorname{\mathrm{End}}(V):\operatorname{\mathrm{tr}}(X)=0\\}$.
These admissible labellings will eventually give us an explicit way to
describe all tangent cocycles. As such, it gives us an explicit description of
the tangent space to $\operatorname{\mathrm{Hit}}_{V}(S)$, which allows one to
explicitly compute the Goldman symplectic pairing between any two vectors that
are given to us as admissible labellings. We will later (Section 5) also show
that the admissible labellings give us a symplectic identification between the
tangent spaces to any pair of points in $\operatorname{\mathrm{Hit}}_{V}(S)$,
which allows us to define large families of Hamiltonian vector fields for the
Goldman symplectic form.

###  3.1. Legal labellings

To define admissible labellings, we first give a preliminary definition of a
legal labelling. An important piece of this definition is a particular family
of elements in $\mathfrak{s}\mathfrak{l}(V)$, called eruption and shearing
endomorphisms.

First, we define the eruption endomorphisms.

######  Definition 3.1.

The _$(i,j,k)$ -eruption endomorphism_ with respect to the triple
$(F,G,H)\in\mathcal{F}(V)^{(3)}$ is the Lie algebra element
$A^{i,j,k}_{F,G,H}\in\mathfrak{s}\mathfrak{l}(V)$ with eigenspaces $F^{(i)}$
and $G^{(j)}+H^{(k)}$ corresponding to eigenvalues $\frac{n-i}{n}$ and
$-\frac{i}{n}$ respectively.

For the triple $(F,G,H)\in\mathcal{F}(V)^{(3)}$, and let
$\\{f_{1},\dots,f_{n}\\}$, $\\{g_{1},\dots,g_{n}\\}$,
$\\{h_{1},\dots,h_{n}\\}$ be three bases for $V$ so that for all
$l=1,\dots,n$,

$F^{(l)}=\operatorname{\mathrm{Span}}_{\mathbb{R}}\\{f_{1},\dots,f_{l}\\},\,\,\,\,G^{(l)}=\operatorname{\mathrm{Span}}_{\mathbb{R}}\\{g_{1},\dots,g_{l}\\},\,\,\,\,H^{(l)}=\operatorname{\mathrm{Span}}_{\mathbb{R}}\\{h_{1},\dots,h_{l}\\}.$

Since $(F,G,H)\in\mathcal{F}(V)^{(3)}$,
$B^{i,j,k}_{F,G,H}:=\\{f_{1},\dots,f_{i},g_{1},\dots,g_{j},h_{1},\dots,h_{k}\\}$
is an ordered basis of $V$ for any $i,j,k\in\mathbb{Z}^{+}$ so that $i+j+k=n$.
In this ordered basis, we can write $A^{i,j,k}_{F,G,H}$ as the matrix

$A^{i,j,k}_{F,G,H}=\left[\begin{array}[]{cc}\frac{n-i}{n}\cdot\operatorname{\mathrm{id}}_{i}&0\\\
0&-\frac{i}{n}\cdot\operatorname{\mathrm{id}}_{n-i}\end{array}\right],$

where $\operatorname{\mathrm{id}}_{l}$ is the $l\times l$ identity matrix.

The next proposition states some important properties of $A^{i,j,k}_{F,G,H}$.

######  Proposition 3.2.

Let $(F,G,H)\in\mathcal{F}(V)^{(3)}$ and let $i,j,k\in\mathbb{Z}^{+}$ so that
$i+j+k=n$. The following statements hold.

  1. (1)

$A^{i,j,k}_{F,G,H}$ fixes $F$.

  2. (2)

There is a projective transformation $g\in\operatorname{\mathrm{PGL}}(V)$ so
that

$\bigg{(}F,\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot
G,H\bigg{)}=g\cdot\bigg{(}F,G,\exp\left(tA^{j,k,i}_{G,H,F}\right)\cdot
H\bigg{)}.$

  3. (3)

For all $i^{\prime},j^{\prime},k^{\prime}\in\mathbb{Z}^{+}$ so that
$i^{\prime}+j^{\prime}+k^{\prime}=n$,

$T_{i^{\prime},j^{\prime},k^{\prime}}\bigg{(}F,\exp\left(tA^{i,j,k}_{F,G,H}\right)G,H\bigg{)}=\left\\{\begin{array}[]{ll}T_{i^{\prime},j^{\prime},k^{\prime}}(F,G,H)&\text{if
}(i,j,k)\neq(i^{\prime},j^{\prime},k^{\prime})\\\ e^{t}\cdot
T_{i,j,k}(F,G,H)&\text{if
}(i,j,k)=(i^{\prime},j^{\prime},k^{\prime})\end{array}\right..$

  4. (4)

$\left\\{A^{i,j,k}_{F,G,H}:i,j,k\in\mathbb{Z}^{+},i+j+k=n\right\\}$ is
linearly independent in $\mathfrak{s}\mathfrak{l}(V)$.

###### Proof.

(1) It is clear that $A^{i,j,k}_{F,G,H}$ fixes $F^{(m)}$ for all
$m=1,\dots,i$. On the other hand, for any $m>i$, write the vector $f_{m}$ in
the basis $B^{i,j,k}_{F,G,H}$, i.e.

$f_{m}=\sum_{l=1}^{i}a_{l}f_{l}+\sum_{l=1}^{j}b_{l}g_{l}+\sum_{l=1}^{k}c_{l}h_{l}$

for some constants $a_{l},b_{l},c_{l}\in\mathbb{R}$. Then

$\displaystyle A_{F,G,H}^{i,j,k}\cdot f_{m}$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{i}\frac{n-i}{n}a_{l}f_{l}-\sum_{l=1}^{j}\frac{i}{n}b_{l}g_{l}-\sum_{l=1}^{k}\frac{i}{n}c_{l}h_{l}$
$\displaystyle=$ $\displaystyle\sum_{l=1}^{i}a_{l}f_{l}-\frac{i}{n}f_{m}.$

This implies that for all $m=1,\dots,n-1$, $A^{i,j,k}_{F,G,H}\cdot
F^{(m)}=F^{(m)}$.

(2) For any $i,j,k\in\mathbb{Z}^{+}$ so that $i+j+k=n$ and any
$(F,G,H)\in\mathcal{F}(V)^{(3)}$, let
$X^{i,j,k}_{F,G,H}\in\mathfrak{s}\mathfrak{l}(V)$ be the endomorphism that is
the zero endomorphism on $F^{(i)}$, scales $G^{(j)}$ by $\frac{k}{n}$ and
scales $H^{(k)}$ by $-\frac{j}{n}$. In other words, in the basis
$B^{i,j,k}_{F,G,H}$, $X^{i,j,k}_{F,G,H}$ is represented by the matrix

$X^{i,j,k}_{F,G,H}=\left[\begin{array}[]{ccc}0&0&0\\\
0&\frac{k}{n}\cdot\operatorname{\mathrm{id}}_{j}&0\\\
0&0&-\frac{j}{n}\cdot\operatorname{\mathrm{id}}_{k}\end{array}\right].$

One can then compute in the basis $B_{F,G,H}^{i,j,k}$ that

$X^{k,i,j}_{H,F,G}-X^{j,k,i}_{G,H,F}=\left[\begin{array}[]{ccc}\frac{j}{n}\operatorname{\mathrm{id}}_{i}&0&0\\\
0&-\frac{i}{n}\cdot\operatorname{\mathrm{id}}_{j}&0\\\
0&0&0\end{array}\right]-\left[\begin{array}[]{ccc}-\frac{k}{n}\operatorname{\mathrm{id}}_{i}&0&0\\\
0&0&0\\\
0&0&\frac{i}{n}\cdot\operatorname{\mathrm{id}}_{k}\end{array}\right]=A^{i,j,k}_{F,G,H}.$

It is clear that $X^{k,i,j}_{H,F,G}$ and $X^{j,k,i}_{G,H,F}$ commute, so (1)
implies that the product
$\exp\left(tX^{k,i,j}_{H,F,G}\right)\exp\left(-tX^{j,k,i}_{G,H,F}\right)=\exp\left(tA^{i,j,k}_{F,G,H}\right)\in\operatorname{\mathrm{PSL}}(V)$
fixes the flag $F$, i.e. $\exp\left(tX^{k,i,j}_{H,F,G}\right)\cdot
F=\exp\left(tX^{j,k,i}_{G,H,F}\right)\cdot F$. Similarly,
$\exp\left(tX^{i,j,k}_{F,G,H}\right)\cdot
G=\exp\left(tX^{k,i,j}_{H,F,G}\right)\cdot G$ and
$\exp\left(tX^{j,k,i}_{G,H,F}\right)\cdot
H=\exp\left(tX^{i,j,k}_{F,G,H}\right)\cdot H$. Hence,

$\displaystyle\bigg{(}F,\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot G,H\bigg{)}$
$\displaystyle=$
$\displaystyle\bigg{(}\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot
F,\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot G,H\bigg{)}$ $\displaystyle=$
$\displaystyle\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot\bigg{(}F,G,\exp\left(-tA^{i,j,k}_{F,G,H}\right)\exp\left(tA^{k,i,j}_{H,F,G}\right)\cdot
H\bigg{)}$ $\displaystyle=$
$\displaystyle\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot\bigg{(}F,G,\exp\left(tA^{j,k,i}_{G,H,F}\right)\cdot
H\bigg{)}.$

(3) First, consider the case where
$(i,j,k)\neq(i^{\prime},j^{\prime},k^{\prime})$, then either $i>i^{\prime}$,
$j>j^{\prime}$ or $k>k^{\prime}$. Suppose that $j>j^{\prime}$, and observe
that

$\exp(tA^{i,j,k}_{F,G,H})=\left[\begin{array}[]{cc}e^{\frac{(n-i)t}{n}}\cdot\operatorname{\mathrm{id}}_{i}&0\\\
0&e^{-\frac{it}{n}}\cdot\operatorname{\mathrm{id}}_{n-i}\end{array}\right]$

when written as a matrix in the basis $B^{i,j,k}_{F,G,H}$. In particular,
$\exp\left(tA^{i,j,k}_{F,G,H}\right)$ fixes the subspaces
$G^{(j^{\prime}+1)}$, $G^{(j^{\prime})}$ and $G^{(j^{\prime}-1)}$. This means
that

$\displaystyle
T_{i^{\prime},j^{\prime},k^{\prime}}\bigg{(}F,\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot
G,H\bigg{)}$ $\displaystyle=$ $\displaystyle\frac{F^{(i^{\prime}+1)}\wedge
G^{(j^{\prime})}\wedge H^{(k^{\prime}-1)}}{F^{(i^{\prime}+1)}\wedge
G^{(j^{\prime}-1)}\wedge H^{(k^{\prime})}}\cdot\frac{F^{(i^{\prime}-1)}\wedge
G^{(j^{\prime}+1)}\wedge H^{(k^{\prime})}}{F^{(i^{\prime})}\wedge
G^{(j^{\prime}+1)}\wedge H^{(k^{\prime}-1)}}\cdot\frac{F^{(i^{\prime})}\wedge
G^{(j^{\prime}-1)}\wedge H^{(k^{\prime}+1)}}{F^{(i^{\prime}-1)}\wedge
G^{(j^{\prime})}\wedge H^{(k^{\prime}+1)}}$ $\displaystyle=$ $\displaystyle
T_{i^{\prime},j^{\prime},k^{\prime}}(F,G,H).$

Since the triple ratio is a projective invariant, (2) implies that

$\displaystyle
T_{i^{\prime},j^{\prime},k^{\prime}}\bigg{(}F,\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot
G,H\bigg{)}$ $\displaystyle=$ $\displaystyle
T_{i^{\prime},j^{\prime},k^{\prime}}\bigg{(}F,G,\exp\left(tA^{j,k,i}_{G,H,F}\right)\cdot
H\bigg{)}$ $\displaystyle=$ $\displaystyle
T_{i^{\prime},j^{\prime},k^{\prime}}\bigg{(}\exp\left(tA^{k,i,j}_{H,F,G}\right)\cdot
F,G,H\bigg{)}.$

Thus, if $k>k^{\prime}$ or $i>i^{\prime}$, we can run the same argument as we
did above, using
$T_{i^{\prime},j^{\prime},k^{\prime}}\bigg{(}F,G,\exp\left(tA^{j,k,i}_{G,H,F}\right)\cdot
H\bigg{)}$ or
$T_{i^{\prime},j^{\prime},k^{\prime}}\bigg{(}\exp\left(tA^{k,i,j}_{H,F,G}\right)\cdot
F,G,H\bigg{)}$ respectively in place of
$T_{i^{\prime},j^{\prime},k^{\prime}}\bigg{(}F,\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot
G,H\bigg{)}$. This will prove that if
$(i,j,k)\neq(i^{\prime},j^{\prime},k^{\prime})$, then
$T_{i^{\prime},j^{\prime},k^{\prime}}\bigg{(}F,\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot
G,H\bigg{)}=T_{i^{\prime},j^{\prime},k^{\prime}}(F,G,H)$.

Next, consider the case where $(i^{\prime},j^{\prime},k^{\prime})=(i,j,k)$. In
this case, observe that

$\displaystyle T_{i,j,k}\bigg{(}F,\exp\left(tA^{i,j,k}_{F,G,H}\right)\cdot
G,H\bigg{)}$ $\displaystyle=$ $\displaystyle\frac{F^{(i+1)}\wedge
G^{(j)}\wedge H^{(k-1)}}{F^{(i+1)}\wedge G^{(j-1)}\wedge
H^{(k)}}\cdot\frac{e^{\frac{(n-i)t}{n}}\cdot F^{(i-1)}\wedge G^{(j+1)}\wedge
H^{(k)}}{e^{-\frac{it}{n}}\cdot F^{(i)}\wedge G^{(j+1)}\wedge
H^{(k-1)}}\cdot\frac{F^{(i)}\wedge G^{(j-1)}\wedge H^{(k+1)}}{F^{(i-1)}\wedge
G^{(j)}\wedge H^{(k+1)}}$ $\displaystyle=$ $\displaystyle e^{t}\cdot
T_{i,j,k}(F,G,H).$

(4) Suppose for contradiction that there is some
$i_{0},j_{0},k_{0}\in\mathbb{Z}^{+}$ so that $i_{0}+j_{0}+k_{0}=n$ and

(3.2)
$A^{i_{0},j_{0},k_{0}}_{F,G,H}=\sum_{(i,j,k)\neq(i_{0},j_{0},k_{0})}\alpha^{i,j,k}A^{i,j,k}_{F,G,H}$

for some $\alpha^{i,j,k}\in\mathbb{R}$. By (3), we have that

$T_{i,j,k}\left(F,\exp\left(tA^{i_{0},j_{0},k_{0}}_{F,G,H}\right)\cdot
G,H\right)=\left\\{\begin{array}[]{ll}T_{i,j,k}(F,G,H)&\text{if
}(i,j,k)\neq(i_{0},j_{0},k_{0});\\\ e^{t}\cdot
T_{i_{0},j_{0},k_{0}}(F,G,H)&\text{if
}(i,j,k)=(i_{0},j_{0},k_{0}).\end{array}\right.$

On the other hand, (3.2) and (3) together imply that

$T_{i,j,k}\left(F,\exp\left(tA^{i_{0},j_{0},k_{0}}_{F,G,H}\right)\cdot
G,H\right)=\left\\{\begin{array}[]{ll}e^{\alpha^{i,j,k}t}\cdot
T_{i,j,k}(F,G,H)&\text{if }(i,j,k)\neq(i_{0},j_{0},k_{0});\\\
T_{i_{0},j_{0},k_{0}}(F,G,H)&\text{if
}(i,j,k)=(i_{0},j_{0},k_{0}).\end{array}\right.$

This is clearly a contradiction. ∎

Similarly, since $(F,G)\in\mathcal{F}(V)^{(2)}$, we see that
$B^{i,n-i}_{F,G}:=\\{f_{1},\dots,f_{i},g_{1},\dots,g_{n-i}\\}$ is a basis of
$V$ as well. This allows us to define the shearing endomorphisms.

######  Definition 3.3.

The $(i,n-i)$-_shearing endomorphism_ with respect to the pair
$(F,G)\in\mathcal{F}(V)^{(2)}$ is the endomorphism
$A^{i,n-i}_{F,G}\in\mathfrak{s}\mathfrak{l}(V)$ with eigenspaces $F^{(i)}$ and
$G^{(n-i)}$ corresponding to eigenvalues $\frac{n-i}{n}$ and $-\frac{i}{n}$
respectively. Also, if $F,G,H\in\mathcal{F}(V)^{(3)}$, we will use the
notation $A^{i,n-i,0}_{F,G,H}:=A^{i,n-i}_{F,G}$.

If we write $A^{i,n-i}_{F,G}$ as a matrix in the basis $B^{i,n-i}_{F,G}$, then

$A^{i,n-i}_{F,G}=\left[\begin{array}[]{cc}\frac{n-i}{n}\cdot\operatorname{\mathrm{id}}_{i}&0\\\
0&-\frac{i}{n}\cdot\operatorname{\mathrm{id}}_{n-i}\end{array}\right].$

Furthermore, it is clear that the following proposition holds.

######  Proposition 3.4.

Let $F$ and $G$ be a transverse pair of flags in $\mathcal{F}(V)$, then the
following hold.

  1. (1)

For all $i=1,\dots,n-1$, $A^{i,n-i}_{F,G}$ preserve the flags $F$ and $G$.

  2. (2)

$\left\\{A^{i,n-i}_{F,G}:i=1,\dots,n-1\right\\}$ is a linearly independent
collection in $\mathfrak{s}\mathfrak{l}(V)$.

We are now ready to use the eruption and shearing endomophisms to define legal
labellings. To do so, choose an ideal triangulation $\mathcal{T}$ on $S$. This
determines a barrier system $\mathcal{B}$ on $S$, which lifts to a barrier
system $\widetilde{\mathcal{B}}$. Note that the orientation on $S$ induces a
cyclic ordering on $\partial\Gamma$.

######  Definition 3.5.

Let $\rho:\Gamma\to\operatorname{\mathrm{PSL}}(V)$ be a Hitchin representation
with Frenet curve $\xi:\partial\Gamma\to\mathcal{F}(V)$. A _legal labeling at
$\rho$_ is a $\operatorname{\mathrm{Ad}}\circ\rho$-equivariant map
$L:\widetilde{\mathcal{B}}\to\mathfrak{s}\mathfrak{l}(V)$ so that the
following conditions hold:

  1. (1)

For every non-edge barrier $(x,T)\in\widetilde{\mathcal{M}}$ where
$T=\\{x,y,z\\}$ so that $x<y<z<x$ in the cyclic ordering on $\partial\Gamma$,
we have

$L(x,T)=\sum_{i+j+k=n;i,j,k\in\mathbb{Z}^{+}}a_{i,j,k}(x,T)\cdot
A^{i,j,k}_{\xi(x),\xi(y),\xi(z)},$

for some $a_{i,j,k}(x,T)=a_{i,j,k}(x,T,L)\in\mathbb{R}$.

  2. (2)

For all $T=\\{x,y,z\\}\in\widetilde{\Theta}$ so that $x<y<z<x$ in the cyclic
ordering on $\partial\Gamma$, $a_{i,j,k}(x,T)=a_{j,k,i}(y,T)=a_{k,i,j}(z,T)$.

  3. (3)

For every edge $e=\\{x,y\\}\in\widetilde{\mathcal{T}}$, we have

$L(e)=\sum_{i=1}^{n-1}a_{i}(y,x)\cdot
A^{i,n-i}_{\xi(x),\xi(y)}=\sum_{i=1}^{n-1}a_{i}(x,y)\cdot
A^{i,n-i}_{\xi(y),\xi(z)},$

for some $a_{i}(y,x)=a_{i}(y,x,L)=-a_{n-i}(y,x)=-a_{n-i}(y,x,L)\in\mathbb{R}$.

Denote the set of legal labellings at $\rho$ by
$\mathcal{L}(\rho)=\mathcal{L}(\rho,\mathcal{T})$. We refer to the numbers

$\\{a_{i,j,k}(x,T,L):(x,T)\in\widetilde{\mathcal{M}}\\}_{i+j+k=n;i,j,k>0}\cup\\{a_{i}(y,x,L):(y,x)\in\widetilde{\mathcal{T}}^{o}\\}_{i=1,\dots,n-1}$

as the _coeffecients_ of $L$.

Let $L\in\mathcal{L}(\rho)$ be any legal labelling. Definition 3.5(2) and
Proposition 3.2(4) ensure that for any ideal triangle
$T=\\{x,y,z\\}\in\widetilde{\Theta}$, any one of $L(x,T)$, $L(y,T)$, $L(z,T)$
determines the other two.

The set $\mathcal{L}(\rho)$ is naturally a real vector space, whose dimension
we can compute. Suppose that $\mathcal{T}$ has $k$ closed edges, then it is
easy to see that $|\mathcal{T}|=6g-6+k$ and $|\mathcal{M}|=12g-12$. By
Proposition 3.2(4) and Proposition 3.4(2),

$\displaystyle\dim\mathcal{L}(\rho)$ $\displaystyle=$
$\displaystyle(6g-6+k)(n-1)+\frac{1}{3}(12g-12)\frac{(n-1)(n-2)}{2}$
$\displaystyle=$ $\displaystyle(n^{2}-1)(2g-2)+k(n-1).$

###  3.2. Admissible labellings

Next, we will define the notion of an admissible labelling. For that, we
further choose an orientation $r$ on $\mathcal{T}$. This lifts to an
orientation $\widetilde{r}$ on $\widetilde{\mathcal{T}}$. We will also need
the indicator function $\alpha$ defined as follows. For any edge
$e=\\{x,y\\}\in\widetilde{\mathcal{T}}$, define

$\alpha(y,x):=\left\\{\begin{array}[]{ll}1&\text{if
}\widetilde{r}(e)=(y,x);\\\ -1&\text{if
}\widetilde{r}(e)=(x,y).\end{array}\right.$

######  Notation 3.6.

For every closed edge $e=\\{x,y\\}\in\widetilde{\mathcal{P}}$, let
$J=\\{T_{1},T_{2}\\}$ be any bridge across $e$ so that $T_{1}$ and $T_{2}$ lie
to the right and left of $\widetilde{r}(e)=(x,y)$ respectively. For $m=1,2$,
let $\gamma_{m}\in\Gamma$ be the primitive group element that has $q_{m}$ as
its attracting fixed point and $p_{m}$ as its repelling fixed point (see
Notation 2.9). Let $T_{m}=:T_{m,1},\dots,T_{m,C_{m}+1}:=\gamma_{m}\cdot T_{m}$
be the finite sequence of triangles in $\widetilde{\Theta}$ so that $T_{m,c}$
shares a common edge with $T_{m,c+1}$ for all $c=1,\dots,C_{m}$ (see Figure
8). Also, let $e_{m,c}$ be the common edge of $T_{m,c}$ and $T_{m,c+1}$. Then
define

$\widetilde{\Theta}(J,T_{m}):=\\{T_{m,1},\dots,T_{m,C_{m}}\\}\,\,\,\,\text{
and
}\,\,\,\,\widetilde{\mathcal{T}}(J,T_{m}):=\\{e_{m,1},\dots,e_{m,C_{m}}\\}.$

$T_{2,1}$$T_{2,2}$$\gamma_{2}\cdot T_{2,1}$$\gamma_{1}\cdot
T_{1,1}$$T_{1,3}$$T_{1,2}$$T_{1,1}$$T_{2,1}$$T_{2,2}$$T_{2,3}$$\gamma_{2}\cdot
T_{2,1}$$T_{1,1}$$T_{1,2}$$T_{1,3}$$T_{1,4}$$\gamma_{1}\cdot
T_{1,1}$$x$$y$$x$$y$

Figure 8. In both diagrams above, the closed edge $e=\\{x,y\\}$ is drawn in
red, and the $\langle\gamma_{1}\rangle=\langle\gamma_{2}\rangle$ orbit of the
bridge $J=\\{T_{1},T_{2}\\}$ is drawn in green. On the left, $C_{1}=3$ and
$C_{2}=2$. On the right, $C_{1}=4$ and $C_{2}=3$.

Observe that in Notation 3.6, the triangles in $\widetilde{\Theta}(J,T_{m})$
and the edges in $\widetilde{\mathcal{T}}(J,T_{m})$ all share $p_{m}$ as a
common vertex.

######  Definition 3.7.

A legal labelling $L:\widetilde{\mathcal{B}}\to\mathfrak{s}\mathfrak{l}(V)$ at
$\rho$ is _admissible_ if for any closed edge
$e=\\{x,y\\}\in\widetilde{\mathcal{P}}$ and any bridge $J=\\{T_{1},T_{2}\\}$
across $e$ so that $T_{1}$ and $T_{2}$ lie to the right and left of
$\widetilde{r}(e)=(x,y)$ respectively, the following hold for all
$i=1,\dots,n-1$:

  * •

If $p_{1}\neq p_{2}$, then

$\displaystyle\sum_{\\{p_{1},z\\}\in\widetilde{\mathcal{T}}(J,T_{1})}\alpha(z,p_{1})a_{i}(z,p_{1},L)+\sum_{T\in\widetilde{\Theta}(J,T_{1})}\sum_{j+k=n-i}a_{i,j,k}(p_{1},T,L)$
$\displaystyle=$
$\displaystyle\sum_{\\{p_{2},z\\}\in\widetilde{\mathcal{T}}(J,T_{2})}\alpha(z,p_{2})a_{n-i}(z,p_{2},L)+\sum_{T\in\widetilde{\Theta}(J,T_{2})}\sum_{j+k=i}a_{n-i,j,k}(p_{2},T,L).$

  * •

If $p_{1}=p_{2}$, then

$\displaystyle\sum_{\\{p_{1},z\\}\in\widetilde{\mathcal{T}}(J,T_{1})}\alpha(z,p_{1})a_{i}(z,p_{1},L)+\sum_{T\in\widetilde{\Theta}(J,T_{1})}\sum_{j+k=n-i}a_{i,j,k}(p_{1},T,L)$
$\displaystyle=$
$\displaystyle-\left(\sum_{\\{p_{2},z\\}\in\widetilde{\mathcal{T}}(J,T_{2})}\alpha(z,p_{2})a_{i}(z,p_{2},L)+\sum_{T\in\widetilde{\Theta}(J,T_{2})}\sum_{j+k=n-i}a_{i,j,k}(p_{2},T,L)\right).$

Let $\mathcal{A}(\rho)=\mathcal{A}(\rho,r,\mathcal{T})$ denote the set of
admissible labellings at $\rho$.

Observe that the $\operatorname{\mathrm{Ad}}\circ\rho$-equivariance of $L$ and
the $\Gamma$-invariance of the orientation $\widetilde{r}$ ensure that if the
admissibility condition holds for some bridge across the closed edge $e$, then
it automatically holds for all bridges across $e$. We will now explain the
admissibility condition geometrically.

$h_{m}$$h_{3-m}$$h_{3-m}$$h_{m}$$p_{m}$$q_{m}$$z_{m}$$w_{m}$$p_{m}$$q_{m}$$z_{m}$$w_{m}$

Figure 9. The case where $p_{1}=p_{2}$ is drawn on the left, while the case
where $p_{1}=q_{2}$ is drawn on the right.

Let $e=\\{x,y\\}\in\widetilde{\mathcal{P}}$ be any closed edge and let
$J=\\{T_{1},T_{2}\\}$ be any bridge across $e$ so that $T_{1}$ and $T_{2}$ lie
to the right and left of $\widetilde{r}(e)$ respectively. If
$q_{m}<w_{m}<z_{m}<p_{m}<q_{m}$ (see Notation 2.9), define

$B(J,T_{m}):=\sum_{\\{p_{m},z\\}\in\widetilde{\mathcal{T}}(J,T_{m})}\alpha(z,p_{m})L\\{p_{m},z\\}+\sum_{T\in\widetilde{\Theta}(J,T_{m})}L(p_{m},T),$

and if $p_{m}<z_{m}<w_{m}<q_{m}<p_{m}$, define

$B(J,T_{m}):=-\left(\sum_{\\{p_{m},z\\}\in\widetilde{\mathcal{T}}(J,T_{m})}\alpha(z,p_{m})L\\{p_{m},z\\}+\sum_{T\in\widetilde{\Theta}(J,T_{m})}L(p_{m},T)\right).$

More geometrically, $B(J,T_{m})$ can be described as follows. Consider a
geodesic $1$-simplex $h_{m}:[0,1]\to\widetilde{\Sigma}$ so that $h_{m}(0)$
lies in the fragmented ideal triangle in $T_{m,1}$ that has $e_{m,1}$ as an
edge, and $h_{m}(1)=\gamma_{m}\cdot h_{m}(0)$ (see Figure 9). Then

$B(J,T_{m})=\sum_{e\in\widetilde{\mathcal{T}}(J,T_{m})}\hat{i}\big{(}h_{m},\widetilde{r}(e)\big{)}L(e)+\sum_{T\in\widetilde{\Theta}(J,T_{m})}\hat{i}\big{(}h_{m},(p_{m},T)\big{)}L(p_{m},T).$

By Proposition 3.2(1) and Proposition 3.4(1), we see that $B(J,T_{m})$ is an
endomorphism that preserves the flag $\xi(p_{m})$, i.e.
$B(J,T_{m})\cdot\xi^{(l)}(p_{m})\subset\xi^{(l)}(p_{m})$ for all
$l=1,\dots,n-1$. In particular, we can write

$B(J,T_{m})=N(J,T_{m})+D(J,T_{m}),$

where $N(J,T_{m})$ is a nilpotent endomorphism that preserves $\xi(p_{m})$,
and $D(J,T_{m})$ is an endomorphism that preserves both $\xi(x)$ and $\xi(y)$.
The next proposition explains the reason we impose the admissibility condition
on the space of legal labellings.

######  Proposition 3.8.

Let $L\in\mathcal{L}(\rho)$ be any legal labelling, let
$e=\\{x,y\\}\in\widetilde{\mathcal{P}}$, and let $J=\\{T_{1},T_{2}\\}$ be any
bridge across $e$ so that $T_{1}$ and $T_{2}$ lie to the right and left of
$\widetilde{r}(e)$ respectively. For any $m=1,2$,

$\displaystyle D(J,T_{m})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\left(\sum_{\\{p_{m},z\\}\in\widetilde{\mathcal{T}}(J,T_{m})}\alpha(z,p_{m})a_{i}(z,p_{m})\right.$
$\displaystyle\hskip
71.13188pt\left.+\sum_{T\in\widetilde{\Theta}(J,T_{m})}\sum_{j+k=n-i}a_{i,j,k}(p_{m},T)\right)\cdot
A^{i,n-i}_{\xi(p_{m}),\xi(q_{m})}.$

if $q_{m}<w_{m}<z_{m}<p_{m}<q_{m}$, and

$\displaystyle D(J,T_{m})$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n-1}\left(\sum_{\\{p_{m},z\\}\in\widetilde{\mathcal{T}}(J,T_{m})}\alpha(z,p_{m})a_{i}(z,p_{m})\right.$
$\displaystyle\hskip
71.13188pt\left.+\sum_{T\in\widetilde{\Theta}(J,T_{m})}\sum_{j+k=n-i}a_{i,j,k}(p_{m},T)\right)\cdot
A^{i,n-i}_{\xi(p_{m}),\xi(q_{m})}.$

if $p_{m}<z_{m}<w_{m}<q_{m}<p_{m}$.

######  Remark 3.9.

In particular, unlike $B(J,T_{m})$ and $N(J,T_{m})$, $D(J,T_{m})$ does not
depend on the choice of $J$ nor $T_{m}$, but only on the closed edge $e$. To
emphasize this, we will henceforth denote $D(J,T_{m})$ by
$D_{m}(e)=D_{m}(e,L)$. Furthermore, Proposition 3.8 implies that the
admissibility condition on a legal labelling $L\in\mathcal{A}(\rho)$ is simply
requiring

$D_{1}(e)=\left\\{\begin{array}[]{ll}-D_{2}(e)&\text{if }p_{1}\neq p_{2};\\\
D_{2}(e)&\text{if }p_{1}=p_{2}.\end{array}\right.$

###### Proof.

For all $l=1,\dots,n$ and $m=1,2$, let $f_{l,m}\in V$ so that
$\operatorname{\mathrm{Span}}_{\mathbb{R}}(f_{l,m})=\xi^{(l)}(p_{m})\cap\xi^{(n-l+1)}(q_{m})$.
For each $T\in\widetilde{\Theta}(J,T_{m})$, let $a=a(T)$ and $b=b(T)$ be the
two vertices of $T$ that are not $p_{m}$, so that $p_{m}<a<b<p_{m}$ along
$\partial\Gamma$. Let $A_{i}(p_{m})\in\mathfrak{s}\mathfrak{l}(V)$ be any
endomorphism of the form $A^{i,j,k}_{\xi(p_{m}),\xi(a),\xi(b)}$ or
$A^{i,n-i}_{\xi(p_{m}),\xi(a)}$.

By Proposition 3.2(1) and Proposition 3.4(1), we see that the endomorphism
$A_{i}(p_{m})$ preserves the flag $\xi(p_{m})$. Thus, as a matrix in the basis
$\\{f_{1,m},\dots,f_{m,n}\\}$, $A_{i}(p_{m})$ is upper triangular.
Furthermore, $A_{i}(p_{m})$ has an eigenbasis in $V$ whose corresponding
eigenvalues are either $\frac{n-i}{n}$ or $-\frac{i}{n}$, and it acts as
scaling by $\frac{n-i}{n}$ on $\xi^{(i)}(p_{m})$. In other words, the diagonal
entries of $A_{i}(p_{m})$ give the matrix representing the endomorphism
$A^{i,n-i}_{\xi(p_{m}),\xi(q_{m})}$ in the basis
$\\{f_{1,m},\dots,f_{m,n}\\}$.

This implies that for all $i=1,\dots,n-1$, the endomorphism

$\displaystyle S_{i}$ $\displaystyle:=$
$\displaystyle\left(\sum_{\\{p_{m},z\\}\in\widetilde{\mathcal{T}}(J,T_{m})}\alpha(z,p_{m})a_{i}(z,p_{m})A^{i,n-i}_{\xi(p_{m}),\xi(z)}\right.$
$\displaystyle\hskip
85.35826pt\left.+\sum_{T\in\widetilde{\Theta}(J,T_{m})}\sum_{j+k=n-i}a_{i,j,k}(p_{m},T)A^{i,j,k}_{\xi(p_{m}),\xi(a(T)),\xi(b(T))}\right)$

when written as a matrix in the basis $\\{f_{1,m},\dots,f_{n,m}\\}$, is upper
triangular, and its diagonal is of the form

$\left(\sum_{\\{p_{m},z\\}\in\widetilde{\mathcal{T}}(J,T_{m})}\alpha(z,p_{m})a_{i}(z,p_{m})+\sum_{T\in\widetilde{\Theta}(J,T_{m})}\sum_{j+k=n-i}a_{i,j,k}(p_{m},T)\right)\cdot
A^{i,n-i}_{\xi(p_{m}),\xi(q_{m})}.$

Since

$B(J,T_{m})=\left\\{\begin{array}[]{ll}\displaystyle\sum_{i=1}^{n-1}S_{i}&\text{if
}q_{m}<w_{m}<z_{m}<p_{m}<q_{m};\\\
\displaystyle-\sum_{i=1}^{n-1}S_{i}&\text{if
}p_{m}<z_{m}<w_{m}<q_{m}<p_{m}.\end{array}\right.$

and $D(J,T_{m})$ is the diagonal part of $B(J,T_{m})$, the proposition
follows. ∎

The $\rho$-equivariance of $L$ ensures that if $e=\rho(\gamma)\cdot
e^{\prime}\in\widetilde{\mathcal{P}}$ for some $\gamma\in\Gamma$, then the
$n-1$ equations associated to $e$ and $e^{\prime}$ in Definition 3.7 agree.
Hence, if $\mathcal{T}$ has $k$ closed edges, then
$\mathcal{A}(\rho)\subset\mathcal{L}(\rho)$ is a vector subspace determined by
$k(n-1)$ linear equations. In particular,
$\dim\mathcal{A}(\rho)\geq\dim\mathcal{L}(\rho)-k(n-1)\geq(n^{2}-1)(2g-2)$. In
fact, we will later show that $\dim\mathcal{A}(\rho)=(n^{2}-1)(2g-2)$, so
these $k(n-1)$ linear equations are linearly independent. Also, we will verify
in Section 4.3 that the linear equations determined by the admissibility
condition, when interpreted appropriately, are exactly the closed leaf
equalities found by Bonahon-Dreyer [BD14].

##  4\. The tangent space to $\operatorname{\mathrm{Hit}}_{V}(S)$ using
admissible labellings

Let $\rho:\Gamma\to\operatorname{\mathrm{PSL}}(V)$ be a Hitchin
representation. We saw previously that by choosing an oriented ideal
triangulation $(r,\mathcal{T})$ of $S$, we can define the vector space of
admissible labellings $\mathcal{A}(\rho)$. The goal of this section is to
establish a linear isomorphism between $\mathcal{A}(\rho)$ and
$T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)$ that depends on an additional
choice of a bridge system $\mathcal{J}$ that is compatible with $\mathcal{T}$.
To do so, choose a hyperbolic metric $\Sigma$ on $S$. We then use the data of
$\Sigma$, $(r,\mathcal{T})$ and $\mathcal{J}$ to construct a cocycle
$\mu_{L}\in
C^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
from any admissible labelling $L\in\mathcal{A}(\rho)$. We will later see
(Section 4.3) that even though the cocycle $\mu_{L}$ depends on $\Sigma$, it’s
cohomology class $[\mu_{L}]\in
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
does not. This thus defines a linear map $\Phi_{\rho}:\mathcal{A}(\rho)\to
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})=T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)$
by $\Phi_{\rho}:L\mapsto[\mu_{L}]$ that depends only on $(r,\mathcal{T})$ and
$\mathcal{J}$.

The rough idea behind constructing the cocycle $\mu_{L}\in
C^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$,
or equivalently an $\operatorname{\mathrm{Ad}}\circ\rho$-equivariant linear
map
$\widetilde{\mu}_{L}:C_{1}(\widetilde{S},\mathbb{Z})\to\mathfrak{s}\mathfrak{l}(V)$,
is the following. Let $\widetilde{\mathcal{B}}$ be the barrier system
associated to $(r,\widetilde{\mathcal{T}})$, and let
$h:[0,1]\to\widetilde{\Sigma}$ be a $1$-simplex. We want to define
$\widetilde{\mu}_{L}(h)$ to be the sum

$\widetilde{\mu}_{L}(h)=\sum_{b\in\widetilde{\mathcal{B}}}\hat{i}(h,\widetilde{r}(b))L(b).$

In general however, this is an infinite sum that does not converge.
Fortunately, we can use our choice of the bridge system
$\widetilde{\mathcal{J}}$ to tweak the definition of $\widetilde{\mu}_{L}$ in
order to ensure it is well-defined (see Section 4.2).

To prove that $\Phi_{\rho}$ is indeed an isomorphism of vector spaces, we will
show that each $\mu_{L}$ is the
$(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle in the cohomology
class $[\mu_{L}]$. This will imply that the map $\Phi_{\rho}$ is an injective
linear map, which in turn implies that $\Phi_{\rho}$ is an isomorphism of
vector spaces because we have already established that
$\dim\mathcal{A}(\rho)\leq(n^{2}-1)(2g-2)=\dim
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$.
The proof that each $\mu_{L}$ is a
$(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle heavily depends on the
notion of a $(\mathcal{T},\mathcal{J})$-parallel flow on
$\operatorname{\mathrm{Hit}}_{V}(S)$, which was defined in Sun-Wienhard-Zhang
[SWZ17].

For the rest of this section, we will fix an oriented ideal triangulation
$(r,\mathcal{T})$ on $S$, a compatible bridge system
$\widetilde{\mathcal{J}}$, and a hyperbolic metric $\Sigma$.

###  4.1. The maps $K_{0}$ and $K_{1}$

In order to formally define the cocycle $\widetilde{\mu}_{L}$, it is
convenient to consider the pair of maps

$K_{0}:\widetilde{\mathcal{P}}^{o}\to\mathfrak{s}\mathfrak{l}(V)\,\,\,\,\text{
and }\,\,\,\,K_{1}:\widetilde{\mathcal{J}}^{o}\to\mathfrak{s}\mathfrak{l}(V)$

which we will now define. For any $e=\\{x,y\\}\in\widetilde{\mathcal{P}}$, let
$J=\\{T_{1},T_{2}\\}\in\widetilde{\mathcal{J}}$ be a bridge across $e$ so that
$T_{1}$ and $T_{2}$ are on the right and left of $\widetilde{r}(e)$
respectively. Also, for $m=1,2$, let $p_{m},q_{m}\in\partial\Gamma$ be as
defined in Notation 2.9 and let $\gamma_{m}\in\Gamma$ be as defined in
Notation 3.6. Define $B(J,T_{m})$, $N(J,T_{m})$ and $D_{m}(e)$ as we did in
Section 3.2.

Recall that Proposition 3.8 ensures

$D_{1}(e)=\left\\{\begin{array}[]{ll}-D_{2}(e)&\text{if }p_{1}\neq p_{2};\\\
D_{2}(e)&\text{if }p_{1}=p_{2}.\end{array}\right.$

so we can define $K_{0}$ by

$K_{0}(x,y):=\left\\{\begin{array}[]{ll}D_{m}(e)&\text{if
}(x,y)=(p_{m},q_{m});\\\ -D_{m}(e)&\text{if
}(x,y)=(q_{m},p_{m}).\end{array}\right.$

Also, for $m=1,2$, define $K_{1}$ by

$K_{1}(T_{m},T_{3-m}):=\sum_{l=0}^{\infty}\rho(\gamma_{m}^{l})N(J,T_{m})\rho(\gamma_{m}^{-l}).$

######  Proposition 4.1.

The map $K_{1}$ is well-defined, i.e. for any bridge
$\\{T_{1},T_{2}\\}\in\widetilde{\mathcal{J}}$, the sum $K_{1}(T_{m},T_{3-m})$
converge for $m=1,2$.

###### Proof.

Recall that $N(J,T_{m})$ is a nilpotent endomorphism that preserves
$\xi(p_{m})$, and $\rho(\gamma_{m})$ has $\xi(p_{m})$ and $\xi(q_{m})$ as its
repelling and attracting flags respectively. Let $\\{f_{1,m},\dots,f_{n,m}\\}$
be a basis of $V$ so that
$\operatorname{\mathrm{Span}}_{\mathbb{R}}(f_{l,m})=\xi^{(l)}(p_{m})\cap\xi^{(n-l+1)}(q_{m})$
for all $l=1,\dots,n$. In this basis, $N(J,T_{m})$ and $\rho(\gamma_{m})$ can
be represented by the matrices

$N(J,T_{m})=\left(\begin{array}[]{ccccc}0&a_{1,2}&a_{1,3}&\dots&a_{1,n}\\\
0&0&a_{2,3}&\dots&a_{2,n}\\\ 0&0&0&\dots&a_{3,n}\\\
\vdots&\vdots&\vdots&&\vdots\\\ 0&0&0&\dots&0\\\
\end{array}\right),\,\,\,\,\rho(\gamma_{m})=\left(\begin{array}[]{ccccc}\lambda_{1}&0&0&\dots&0\\\
0&\lambda_{2}&0&\dots&0\\\ 0&0&\lambda_{3}&\dots&0\\\
\vdots&\vdots&\vdots&&\vdots\\\ 0&0&0&\dots&\lambda_{n}\\\
\end{array}\right),$

where $a_{i,j}\in\mathbb{R}$ for all $i<j$ and
$\lambda_{1}<\dots<\lambda_{n}$. One can then compute that

$\rho(\gamma^{l})\cdot
N(J,T_{m})\cdot\rho(\gamma^{-l})=\left(\begin{array}[]{ccccc}0&\left(\frac{\lambda_{1}}{\lambda_{2}}\right)^{l}\cdot
a_{1,2}&\left(\frac{\lambda_{1}}{\lambda_{3}}\right)^{l}\cdot
a_{1,3}&\dots&\left(\frac{\lambda_{1}}{\lambda_{n}}\right)^{l}\cdot a_{1,n}\\\
0&0&\left(\frac{\lambda_{2}}{\lambda_{3}}\right)^{l}\cdot
a_{2,3}&\dots&\left(\frac{\lambda_{2}}{\lambda_{n}}\right)^{l}\cdot a_{2,n}\\\
0&0&0&\dots&\left(\frac{\lambda_{3}}{\lambda_{n}}\right)^{l}\cdot a_{3,n}\\\
\vdots&\vdots&\vdots&&\vdots\\\ 0&0&0&\dots&0\\\ \end{array}\right)$

for all $l=0,\dots,\infty$, which implies that

$K_{1}(T_{m},T_{3-m})=\left(\begin{array}[]{ccccc}0&\frac{\lambda_{2}}{\lambda_{2}-\lambda_{1}}\cdot
a_{1,2}&\frac{\lambda_{3}}{\lambda_{3}-\lambda_{1}}\cdot
a_{1,3}&\dots&\frac{\lambda_{n}}{\lambda_{n}-\lambda_{1}}\cdot a_{1,n}\\\
0&0&\frac{\lambda_{3}}{\lambda_{3}-\lambda_{2}}\cdot
a_{2,3}&\dots&\frac{\lambda_{n}}{\lambda_{n}-\lambda_{2}}\cdot a_{2,n}\\\
0&0&0&\dots&\frac{\lambda_{n}}{\lambda_{n}-\lambda_{3}}\cdot a_{3,n}\\\
\vdots&\vdots&\vdots&&\vdots\\\ 0&0&0&\dots&0\\\ \end{array}\right).$

∎

The $\rho$-equivariance of $L$ ensures that $K_{0}$ and $K_{1}$ are also
$\rho$-equivariant.

###  4.2. Obtaining cocycles from admissible labellings

Let $L\in\mathcal{A}(\rho)$. We will now construct the cocycle $\mu_{L}\in
C^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$
that depends on $\Sigma$, $(r,\mathcal{T})$ and $\mathcal{J}$. Observe that
constructing a cochain $\mu_{L}\in
C^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$ is
equivalent to constructing an
$\operatorname{\mathrm{Ad}}\circ\rho$-equivariant linear map
$\widetilde{\mu}_{L}:C_{1}(\widetilde{S},\mathbb{Z})\to\mathfrak{s}\mathfrak{l}(V)$.
It is sufficient to define $\widetilde{\mu}_{L}$ on the set of $1$-simplices,
and extend this to $C_{1}(\widetilde{S},\mathbb{Z})$ by linearity.

Let $h:[0,1]\to\widetilde{\Sigma}$ be a $1$-simplex. Before defining
$\widetilde{\mu}_{L}(h)$ formally, we will give more colloquial description of
$\widetilde{\mu}_{L}(h)$ to motivate the formalism we use. Homotope $h$
(relative end points) to a smooth curve $f:[0,1]\to\widetilde{\Sigma}$ that
intersects every barrier in $\widetilde{\mathcal{B}}$ only finitely many
times. In the case when $h[0,1]$ does not intersect any
$e\in\widetilde{\mathcal{P}}$, then $f[0,1]$ only intersects finitely barriers
in $\widetilde{\mathcal{B}}$, so we can define

(4.1)
$\widetilde{\mu}_{L}(h):=\sum_{b\in\widetilde{\mathcal{B}}}\hat{i}(f,\widetilde{r}(b))L(b).$

Of course, one needs to check that $\widetilde{\mu}_{L}(h)$ does not depend on
the choice of $f$. This is a consequence of Proposition 4.6.

On the other hand, if $h[0,1]$ intersects $e\in\widetilde{\mathcal{P}}$, then
any $1$-simplex that is homotopic to $h$ (relative endpoints) will intersect
infinitely many barriers in $\widetilde{\mathcal{B}}$, and the sum given in
$(\ref{eqn:fake})$ does not converge for a general admissible labelling $L$.
In this case, we further homotope $h$ to a piecewise geodesic path
$f:[0,1]\to\widetilde{\Sigma}$ with the property that if $t\in[0,1]$ and
$f(t)\in e$, then there is some subinterval of $I\subset[0,1]$ so that $t\in
I$ and $f(I)=J$ for some bridge $J\in\widetilde{\mathcal{J}}$. More
informally, we require that $f$ crosses closed edges only along bridges.

This allows us to decompose $f[0,1]$ into subsegments $f_{1},\dots,f_{k}$ that
are bridges, and subsegments $f^{\prime}_{1},\dots,f^{\prime}_{l}$ that do not
intersect any closed edges. We then use (4.1) to assign to each
$f_{i}^{\prime}$ the vector
$\widetilde{\mu}_{L}(f_{i}^{\prime})\in\mathfrak{s}\mathfrak{l}(V)$, and we
use the functions $K_{0}$ and $K_{1}$ to assign to each $f_{i}$ a vector
$\widetilde{\mu}_{L}(f_{i})\in\mathfrak{s}\mathfrak{l}(V)$. We can then define

$\widetilde{\mu}_{L}(h):=\sum_{i=1}^{k}\widetilde{\mu}_{L}(f_{i})+\sum_{i=1}^{l}\widetilde{\mu}_{L}(f_{i}^{\prime}).$

Again, one needs to prove that $\widetilde{\mu}_{L}(h)$ does not depend on the
choice of $f$, and that $\widetilde{\mu}_{L}$ defined this way is indeed
$\operatorname{\mathrm{Ad}}\circ\rho$-equivariant, and satisfies the cocycle
condition.

We will now formally define $\widetilde{\mu}_{L}(h)$. Note that the
orientation $r:\mathcal{T}\to\mathcal{T}^{o}$ induces an orientation
$s:\mathcal{J}\to\mathcal{J}^{o}$ with the property that if
$\widetilde{s}:\widetilde{\mathcal{J}}\to\widetilde{\mathcal{J}}^{o}$ is the
lift of $s$, then for all $J\in\widetilde{\mathcal{J}}$ that intersects the
closed edge $e\in\widetilde{\mathcal{P}}$, $\widetilde{s}(J)$ crosses from the
right to the left of $\widetilde{r}(e)$. Recall that we have chosen a
hyperbolic metric $\Sigma$ on $S$ to define the algebraic intersection number
(see Section 2.3). We first give the definition of $\widetilde{\mu}_{L}(h)$ in
the three special cases described below.

Figure 10. In the above picture, the red edge is a closed edge, the blue edges
are isolated edges, and the bridges are drawn in green. A Type A, Type B, Type
C $1$-simplex is drawn in turqoise, violet and yellow respectively.

######  Definition 4.2.

A $1$-simplex $h:[0,1]\to\widetilde{\Sigma}$ is a _Type A $1$-simplex_ if it
is a geodesic segment that lies in some closed edge
$e=\\{x,y\\}\in\widetilde{P}$ (see Figure 10).

If $h$ is a Type A $1$-simplex that lies in the closed edge
$e=\\{x,y\\}\in\widetilde{P}$, define

$\widetilde{\mu}_{L}(h):=\sum_{J\in\widetilde{\mathcal{J}}}\hat{i}\big{(}h,\widetilde{s}(J)\big{)}K_{0}(\widetilde{r}(e)).$

######  Definition 4.3.

A $1$-simplex $h:[0,1]\to\widetilde{\Sigma}$ is a _Type B $1$-simplex_ if it
is a geodesic segment that lies in a bridge $J\in\widetilde{\mathcal{J}}$ that
intersects the closed edge $e\in\widetilde{P}$, so that one of the endpoints
of $h$ is an endpoint of $J$, and the other endpoint of $h$ is the point of
intersection of $e$ and $J$ (see Figure 10).

If $h$ is a Type B $1$-simplex that lies in the bridge $J$ which intersects
the closed edge $e\in\widetilde{\mathcal{P}}$, let $v$ be the common endpoint
of $J$ and $g[0,1]$, and equip $J$ with the orientation $\overrightarrow{J}$
so that $v$ is the backward vertex of $\overrightarrow{J}$. Then define

$\widetilde{\mu}_{L}(h):=\left\\{\begin{array}[]{ll}K_{1}(\overrightarrow{J})+\hat{i}\big{(}h,\widetilde{r}(e)\big{)}\cdot
L(e)&\text{if }h(0)=v\text{ and }h(1)=e\cap J;\\\
-K_{1}(\overrightarrow{J})+\hat{i}\big{(}h,\widetilde{r}(e)\big{)}\cdot
L(e)&\text{if }h(0)=e\cap J\text{ and }h(1)=v.\end{array}\right.$

######  Definition 4.4.

A $1$-simplex $h:[0,1]\to\widetilde{\Sigma}$ is a _Type C $1$-simplex_ if it
satisfies the following properties (see Figure 10):

  * •

$h[0,1]$ intersects only finitely many barriers
$b_{1},\dots,b_{k}\in\widetilde{\mathcal{B}}$,

  * •

$\big{|}h[0,1]\cap b_{l}\big{|}<\infty$ for all $l=1,\dots,k$.

In particular, Type C $1$-simplices do not intersect closed edges in
$\widetilde{\mathcal{P}}$. If $h$ is a Type C $1$-simplex, define

$\widetilde{\mu}_{L}(h):=\sum_{b\in\widetilde{\mathcal{B}}}\hat{i}\big{(}h,\widetilde{r}(b)\big{)}\cdot
L(b).$

######  Definition 4.5.

The Type A, Type B and Type C $1$-simplices are collectively called
_elementary $1$-simplices_.

Using the elementary $1$-simplices, we will now define
$\widetilde{\mu}_{L}(h)$ for a general $1$-simplex
$h:[0,1]\to\widetilde{\Sigma}$. Note that $h$ can be homotoped relative
endpoints to a concatenation $h_{1}\cdot\ldots\cdot h_{k}$, where each $h_{i}$
is an elementary $1$-simplex. Define

$\widetilde{\mu}_{L}(h):=\sum_{i=1}^{k}\widetilde{\mu}_{L}(h_{i}).$

######  Proposition 4.6.

For any simplex $h:[0,1]\to\widetilde{\Sigma}$, $\widetilde{\mu}_{L}(h)$ is
well-defined. Furthermore, if $h_{1},h_{2},h_{3}$ are $1$-simplices so that
$h_{i+1}(0)=h_{i}(1)$ for $i=1,2,3$, then $\widetilde{\mu}_{L}(h_{1}\cdot
h_{2}\cdot h_{3})=0$. In particular, $\widetilde{\mu}_{L}\in
C^{1}(\widetilde{S},\mathfrak{s}\mathfrak{l}(V))$ is a cocycle.

In the Proposition 4.6, arithmetic in the subscripts are done modulo $3$. The
proof of Proposition 4.6 is a long but elementary argument, which we postpone
to Appendix B. The $\operatorname{\mathrm{Ad}}\circ\rho$-equivariance of $L$
implies that $\widetilde{\mu}_{L}$ is an
$\operatorname{\mathrm{Ad}}\circ\rho$-equivariant cocycle, so
$\widetilde{\mu}_{L}$ descends to a cocycle $\mu_{L}\in
C^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$.
Hence, we can define

$\Phi_{\rho}=\Phi_{\rho,r,\mathcal{T},\mathcal{J}}:\mathcal{A}(\rho)\to
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})=T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)$

by $L\mapsto[\mu_{L}]$. It is easy to see that $\Phi_{\rho}$ is a linear map.

Recall that in the construction of the cocycle $\mu_{L}$, we chose a
hyperbolic metric $\Sigma$ on $S$. We will now prove that the cohomology class
$[\mu_{L}]$ does not depend on this choice. In particular, the linear map
$\Phi_{\rho}$ does not depend on the choice of $\Sigma$.

######  Proposition 4.7.

Let $L\in\mathcal{A}(\rho)$ and let $\Sigma,\Sigma^{\prime}$ be two hyperbolic
metrics on $S$. Also, let $\mu_{L},\mu^{\prime}_{L}\in
C^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$ be
the two cocycles constructed as above using $\Sigma,\Sigma^{\prime}$
respectively. Then $[\mu_{L}]=[\mu^{\prime}_{L}]$.

###### Proof.

Let $\gamma\in\Gamma\setminus\\{\operatorname{\mathrm{id}}\\}$ and let
$\gamma^{-},\gamma^{+}\in\partial\Gamma$ denote the repelling and attracting
fixed points of $\gamma$ respectively. Let $e\in\widetilde{\mathcal{T}}$ be an
edge that intersects $\\{\gamma^{-},\gamma^{+}\\}$ transversely, and let
$e_{\Sigma}$, $e_{\Sigma^{\prime}}$ be the hyperbolic geodesics in $\Sigma$,
$\Sigma^{\prime}$ respectively that realize the geodesic $e$. Then let
$p_{\Sigma}$ be the point of intersection between $e_{\Sigma}$ and the axis of
$\gamma$ in $\Sigma$. Similarly, let $p_{\Sigma^{\prime}}$ be the point of
intersection between $e_{\Sigma^{\prime}}$ and the axis of $\gamma$ in
$\Sigma^{\prime}$. Define $g_{\Sigma}:[0,1]\to\widetilde{\Sigma}$ to be the
geodesic segment with endpoints $p_{\Sigma}$ and $\gamma\cdot p_{\Sigma}$ and
$g_{\Sigma^{\prime}}:[0,1]\to\widetilde{\Sigma^{\prime}}$ to be the geodesic
segment with endpoints $p_{\Sigma^{\prime}}$ and $\gamma\cdot
p_{\Sigma^{\prime}}$.

From the coboundary condition, it is sufficient to show that
$\widetilde{\mu}_{L}(g_{\Sigma})=\widetilde{\mu}_{L}^{\prime}(g_{\Sigma^{\prime}})$.
Let $z,w\in\partial\Gamma$ so that $z$ and $w$ lie to the left and right of
$(\gamma^{-},\gamma^{+})$ respectively, and $\\{z,w\\}=e$. By definition,
$\gamma\cdot e_{\Sigma}$ contains $g_{\Sigma}(1)$ and $\gamma\cdot
e_{\Sigma^{\prime}}$ contains $g_{\Sigma^{\prime}}(1)$. Hence, if
$e^{\prime}\in\widetilde{\mathcal{T}}$, and
$\widetilde{r}(e^{\prime})=(q^{-},q^{+})$, then

$\hat{i}_{\Sigma}(g_{\Sigma},\widetilde{r}(e^{\prime}))=\hat{i}_{\Sigma^{\prime}}(g_{\Sigma^{\prime}},\widetilde{r}(e^{\prime}))=\left\\{\begin{array}[]{ll}1&\text{if
}z<q^{+}<\gamma\cdot z<\gamma\cdot w<q^{-}<w;\\\ -1&\text{if
}z<q^{-}<\gamma\cdot z<\gamma\cdot w<q^{+}<w;\\\
\displaystyle\frac{1}{2}&\text{if }\widetilde{r}(e)=(w,z)\text{ or
}\gamma\cdot(w,z);\\\ \displaystyle-\frac{1}{2}&\text{if
}\widetilde{r}(e)=(z,w)\text{ or }\gamma\cdot(z,w);\\\
0&\text{otherwise.}\end{array}\right.$

Similarly, if $T\in\widetilde{\Theta}$ is an ideal triangle with vertices
$v_{1}<v_{2}<v_{3}<v_{1}$, let $b_{i}:=(T,v_{i})\in\widetilde{\mathcal{M}}$.
Then

$\displaystyle\sum_{i=1}^{3}\hat{i}_{\Sigma}(g_{\Sigma},b_{i})L(b_{i})$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}L(b_{i})&\text{if
}z<v_{i}<\gamma\cdot z<\gamma\cdot w<v_{i+1}<v_{i-1}<w<z\text{ for some
}i=1,2,3;\\\ -L(b_{i})&\text{if }z<v_{i+1}<v_{i-1}<\gamma\cdot z<\gamma\cdot
w<v_{i}<w<z\text{ for some }i=1,2,3;\\\ 0&\text{otherwise.}\end{array}\right.$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\hat{i}_{\Sigma^{\prime}}(g_{\Sigma^{\prime}},b_{i})L(b_{i}).$

It immediately follows from the definition of $\widetilde{\mu}_{L}$ and
$\widetilde{\mu}^{\prime}_{L}$ that
$\widetilde{\mu}_{L}(g_{\Sigma})=\widetilde{\mu}_{L}^{\prime}(g_{\Sigma^{\prime}})$.
∎

###  4.3. Admissible labellings and tangent cocycles

Our next goal is to prove the following theorem.

######  Theorem 4.8.

For any admissible labelling $L\in\mathcal{A}(\rho,r,\mathcal{T})$, $\mu_{L}$
is the $(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle in the
cohomology class $[\mu_{L}]$.

Theorem 4.8 has the following corollary, which is the key that allows us to
compute the Goldman symplectic pairing between tangent vectors to
$\operatorname{\mathrm{Hit}}_{V}(S)$.

######  Corollary 4.9.

The linear map $\Phi_{\rho}:\mathcal{A}(\rho)\to
T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)$ is a linear isomorphism.

###### Proof.

Recall we previously computed that $\dim\mathcal{A}(\rho)\geq\dim
T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)$, so it is sufficient to show
that $\ker\Phi_{\rho}=\\{0\\}$. Suppose that $L\in\ker\Phi_{\rho}$. By Theorem
4.8, the cocycle $\mu_{L}$ is a
$(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle whose cohomology class
is the tangent vector to the constant path $t\mapsto[\rho]$ in
$\operatorname{\mathrm{Hit}}_{V}(S)$ at $t=0$. It then follows from the
definition of the $(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle that
$\mu_{L}=0$, which in turn implies that $L=0$. ∎

To prove that $\mu_{L}$ is the $(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent
cocycle in $[\mu_{L}]$, it is convenient to specify a path
$t\mapsto[\rho_{t}]$ in the Hitchin component that is tangential to
$[\mu_{L}]\in
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})\simeq
T_{[\rho]}\operatorname{\mathrm{Hit}}_{V}(S)$ so that $\rho$ is a
representative of $[\rho_{0}]$. It turns out that good candidates for such
paths are the flow lines of the $(\mathcal{T},\mathcal{J})$-parallel flows
that are the main objects studied in the companion paper [SWZ17]. We will now
give a brief description of these flows. For more details, see Section 5 of
[SWZ17].

Following the work of Fock-Goncharov [FG06] and Bonahon-Dreyer [BD14], one can
define some real-analytic functions on $\operatorname{\mathrm{Hit}}_{V}(S)$
associated to the ideal triangles in $\Theta$ and the edges in $\mathcal{T}$.
First, for any ideal triangle $\widehat{T}=[x,y,z]\in\Theta$, choose a
representative $\\{x,y,z\\}\in\widetilde{\mathcal{T}}$ of $[x,y,z]$, and
assume without loss of generality that $x<y<z<x$ in $\partial\Gamma$. Then for
any $i,j,k\in\mathbb{Z}^{+}$ so that $i+j+k=n$, define the function
$\tau^{i,j,k}_{x,y,z}:\operatorname{\mathrm{Hit}}_{V}(S)\to\mathbb{R}$ by

$\tau^{i,j,k}_{x,y,z}[\rho]:=\log\left(T_{i,j,k}\big{(}\xi_{\rho}(x),\xi_{\rho}(y),\xi_{\rho}(z)\big{)}\right),$

where $\xi_{\rho}$ is the $\rho$-equivariant Frenet curve for some
representative $\rho$ in $[\rho]$ and $T_{i,j,k}$ is the triple ratio defined
in Definition 2.5. This is well-defined because
$T_{i,j,k}\big{(}\xi_{\rho}(x),\xi_{\rho}(y),\xi_{\rho}(z)\big{)}>0$ by
Theorem 2.6. Furthermore, the projective invariance of the triple ratio
implies that $\tau^{i,j,k}_{x,y,z}[\rho]$ does not depend on the choice of
representative $\\{x,y,z\\}$ of $[x,y,z]$, nor on the choice of representative
$\rho$ of $[\rho]$, and the symmetries of the triple ratio imply that the
$\tau^{i,j,k}_{x,y,z}[\rho]=\tau^{j,k,i}_{y,z,x}[\rho]=\tau^{i,k,j}_{x,z,y}[\rho]$.
We refer to any such function as a _triangle invariant_.

Similarly, for any isolated edge $e=[x,y]\in\mathcal{Q}$, choose a
representative $\\{x,y\\}\in\widetilde{\mathcal{Q}}$ of $[x,y]$, and let
$z,w\in\partial\Gamma$ so that $\\{x,y,z\\},\\{x,y,w\\}\in\widetilde{\Theta}$
and $x<z<y<w<x$ in $\partial\Gamma$. Then for any $i=1,\dots,n-1$, define the
function
$\sigma^{i,n-i}_{x,y}:\operatorname{\mathrm{Hit}}_{V}(S)\to\mathbb{R}$ by

$\sigma^{i,n-i}_{x,y}[\rho]:=\log\left(-C_{i}\big{(}\xi_{\rho}(x),\xi_{\rho}(z),\xi_{\rho}(w),\xi_{\rho}(y)\big{)}\right),$

where $\xi_{\rho}$ is as above. Again, Theorem 2.6 implies that
$\sigma^{i,n-i}_{x,y}[\rho]$ is well-defined. Also, the projective invariance
of the cross ratio implies that $\sigma^{i,n-i}_{x,y}[\rho]$ does not depend
on the choice of representatives $\\{x,y\\}$ of $[x,y]$ and $\rho$ of
$[\rho]$, and the symmetries of the cross ratio imply that
$\sigma^{i,n-i}_{x,y}[\rho]=\sigma^{n-i,i}_{y,x}[\rho]$. We refer to any such
function as an _edge invariant_ along the isolated edge $e$.

Finally, for any closed edge $e=[x,y]\in\mathcal{P}$ and any $i=1,\dots,n-1$,
choose a representative $\\{x,y\\}\in\widetilde{\mathcal{Q}}$ of $[x,y]$ and a
bridge $J=\\{T_{1},T_{2}\\}\in\widetilde{\mathcal{J}}$ so that $T_{1}$ and
$T_{2}$ lie to the right and left of $(x,y)$ respectively. For $m=1,2$, let
$p_{m}$, $q_{m}$, $z_{m}$, $w_{m}$ be as defined in Notation 2.9, and let
$u_{m}$ be the unique unipotent projective transformation in
$\operatorname{\mathrm{PSL}}(V)$ that fixes $p_{m}$ and sends $z_{m}$ to
$q_{m}$. Then define the function
$\alpha^{i,n-i}_{x,y}:\operatorname{\mathrm{Hit}}_{V}(S)\to\mathbb{R}$ by

$\alpha^{i,n-i}_{x,y}[\rho]:=\log\left(-C_{i}\big{(}\xi_{\rho}(x),u_{1}\cdot\xi_{\rho}(w_{1}),u_{2}\cdot\xi_{\rho}(w_{2}),\xi_{\rho}(y)\big{)}\right).$

By [SWZ17, Proposition 4.11] that $\alpha^{i,n-i}_{x,y}[\rho]$ is well-
defined, and the projective invariance of the cross ratio implies that
$\alpha^{i,n-i}_{x,y}[\rho]$ does not depend on the choices of $\\{x,y\\}$,
$\xi_{\rho}$ and $J$. Furthermore, the symmetries of the cross ratio also
implies that $\alpha^{i,n-i}_{x,y}[\rho]=\alpha^{n-i,i}_{y,x}[\rho]$. This
function is called the _symplectic closed edge invariant_ along the closed
edge $e$.

By slightly modifying the work of Bonahon-Dreyer [BD14] it was shown in
[SWZ17] that the triangle invariant, edge invariant along isolated edges, and
symplectic closed edge invariants along closed edges give a real-analytic
parameterization of $\operatorname{\mathrm{Hit}}_{V}(S)$. More precisely, the
map

$\begin{array}[]{crcl}\Omega=\Omega_{\mathcal{T},\mathcal{J}}&:\operatorname{\mathrm{Hit}}_{V}(S)&\to&\mathbb{R}^{(n^{2}-1)(2g-2)+|\mathcal{P}|(n-1)}\\\
&[\rho]&\mapsto&(\Sigma_{1},\Sigma_{2},\Sigma_{3})\\\ \end{array}$

where

$\displaystyle\Sigma_{1}$ $\displaystyle:=$
$\displaystyle\left(\sigma_{x,y}^{i,n-i}[\xi]\right)_{i=1,\dots,n-1;[x,y]\in\mathcal{Q}}$
$\displaystyle\Sigma_{2}$ $\displaystyle:=$
$\displaystyle\left(\alpha_{x,y}^{i,n-i}[\xi]\right)_{i=1,\dots,n-1;[x,y]\in\mathcal{P}}$
$\displaystyle\Sigma_{3}$ $\displaystyle:=$
$\displaystyle\left(\tau^{i,j,k}_{x,y,z}[\xi]\right)_{i,j,k\in\mathbb{Z}^{+};i+j+k=n;[x,y,z]\in\Theta}.$

is a real-analytic diffeomorphism onto a convex polytope
$H_{\mathcal{T},\mathcal{J}}=H$ of dimension $(n^{2}-1)(2g-2)$.

Let $W_{\mathcal{T},\mathcal{J}}=W$ be the $(n^{2}-1)(2g-2)$-dimensional
subspace of $\mathbb{R}^{(n^{2}-1)(2g-2)+|\mathcal{P}|(n-1)}$ containing $H$,
and note that $W$ can be thought of as the tangent space to any point in
$\Omega\big{(}\operatorname{\mathrm{Hit}}_{V}(S)\big{)}$. The
$|\mathcal{P}|(n-1)$ linear equations on
$\mathbb{R}^{(n^{2}-1)(2g-2)+|\mathcal{P}|(n-1)}$ that cut out $W$ are known
as the _closed leaf equalities_ , and were computed explicitly by Bonahon-
Dreyer [BD14]. We will now describe these equalities.

For any closed edge $\\{x,y\\}\in\widetilde{\mathcal{P}}$, let
$\\{T_{1},T_{2}\\}$ be any bridge across $\\{x,y\\}$ so that $T_{1}$ and
$T_{2}$ lie to the right and left of $(x,y)$ respectively. For $m=1,2$, let
$p_{m}$, $q_{m}$, $z_{m}$, $w_{m}$ be as defined in Notation 2.9, and let
$\widetilde{\mathcal{T}}(J,T_{m})$, $\widetilde{\Theta}(J,T_{m})$ and $C_{m}$
be as defined in Notation 3.6.

######  Notation 4.10.

For $m=1,2$, let $z_{m,0},z_{m,1},\dots,z_{m,C_{m}}\in\partial\Gamma$ be the
points so that $T_{m,c}=\\{p_{m},z_{m,c},z_{m,c-1}\\}$ (see Notation 3.6) for
all $c=1,\dots,C_{m}$.

Denote an arbitrary vector
$\mu\in\mathbb{R}^{(n^{2}-1)(2g-2)+|\mathcal{P}|(n-1)}$ in coordinates by

$\begin{array}[]{l}\mu=\left(\left(\mu^{i,n-i}_{x,y}=\mu^{n-i,i}_{y,x}\right)_{i=1,\dots,n-1;[x,y]\in\mathcal{Q}},\left(\mu^{i,n-i}_{x,y}=\mu^{n-i,i}_{y,x}\right)_{i=1,\dots,n-1;[x,y]\in\mathcal{P}},\right.\\\
\hskip
113.81102pt\left.\left(\mu^{i,j,k}_{x,y,z}=\mu^{j,k,i}_{y,z,x}=\mu^{k,i,j}_{z,x,y}\right)_{i,j,k\in\mathbb{Z}^{+};i+j+k=n;[x,y,z]\in\Theta}\right).\end{array}$

For all $i=1,\dots,n-1$, the following linear equation on
$\mathbb{R}^{(n^{2}-1)(2g-2)+|\mathcal{P}|(n-1)}$ is called a _closed leaf
equality_.

  1. (1)

If $p_{1}=x_{1}$ and $p_{2}=x_{2}$, then for all $i=1,\dots,n-1$,

$\sum_{c=1}^{C_{1}}\left(\mu_{p_{1},z_{1,c}}^{i,n-i}+\sum_{j+k=n-i}\mu_{p_{1},z_{1,c},z_{1,c-1}}^{i,j,k}\right)=\sum_{c=1}^{C_{2}}\left(\mu_{p_{2},z_{2,c}}^{n-i,i}+\sum_{j+k=i}\mu_{p_{2},z_{2,c},z_{2,c-1}}^{n-i,j,k}\right).$

  2. (2)

If $p_{1}=x_{2}$ and $p_{2}=x_{1}$, then for all $i=1,\dots,n-1$,

$\sum_{c=1}^{C_{1}}\left(\mu_{p_{1},z_{1,c}}^{i,n-i}+\sum_{j+k=n-i}\mu_{p_{1},z_{1,c-1},z_{1,c}}^{i,j,k}\right)=\sum_{c=1}^{C_{2}}\left(\mu_{p_{2},z_{2,c}}^{n-i,i}+\sum_{j+k=i}\mu_{p_{2},z_{2,c-1},z_{2,c}}^{n-i,j,k}\right).$

  3. (3)

If $p_{1}=p_{2}=x_{1}$, then for all $i=1,\dots,n-1$,

$\sum_{c=1}^{C_{1}}\left(\mu_{p_{1},z_{1,c}}^{i,n-i}+\sum_{j+k=n-i}\mu_{p_{1},z_{1,c},z_{1,c-1}}^{i,j,k}\right)=-\sum_{c=1}^{C_{2}}\left(\mu_{p_{2},z_{2,c}}^{i,n-i}+\sum_{j+k=n-i}\mu_{p_{2},z_{2,c-1},z_{2,c}}^{i,j,k}\right).$

  4. (4)

If $p_{1}=p_{2}=x_{2}$, then for all $i=1,\dots,n-1$,

$\sum_{c=1}^{C_{1}}\left(\mu_{p_{1},z_{1,c}}^{i,n-i}+\sum_{j+k=n-i}\mu_{p_{1},z_{1,c-1},z_{1,c}}^{i,j,k}\right)=-\sum_{c=1}^{C_{2}}\left(\mu_{p_{2},z_{2,c}}^{i,n-i}+\sum_{j+k=n-i}\mu_{p_{2},z_{2,c},z_{2,c-1}}^{i,j,k}\right).$

There are $|\mathcal{P}|(n-1)$ such equations.

######  Theorem 4.11.

[BD14] The $|\mathcal{P}|(n-1)$ closed leaf equalities cut out the vector
space $W\subset\mathbb{R}^{(n^{2}-1)(2g-2)+|\mathcal{P}|(n-1)}$. In
particular, they are linearly independent.

Given any ideal triangulation $\mathcal{T}$ on $S$ and any compatible bridge
system $\mathcal{J}$, Sun-Wienhard-Zhang [SWZ17, Theorem 5.8] constructed, for
every $\mu\in W_{\mathcal{T},\mathcal{J}}$, a particular real-analytic flow

$\left(\phi^{\mu}_{\mathcal{T},\mathcal{J}}\right)_{t}=\phi^{\mu}_{t}:\operatorname{\mathrm{Hit}}_{V}(S)\to\operatorname{\mathrm{Hit}}_{V}(S)$

satisfying $\phi^{\mu}_{t}[\rho]=\Omega^{-1}(\Omega[\rho]+t\mu)$ using the
Frenet curve $\xi$ corresponding to $\rho$. This flow was called the
$(\mathcal{T},\mathcal{J})$-_parallel flow_ corresponding to $\mu\in W$. In
particular, $\frac{d}{dt}|_{t=0}\Omega\circ\phi^{\mu}_{t}[\rho]=\mu$ for every
$[\rho]\in\operatorname{\mathrm{Hit}}_{V}(S)$. This flow is known as the
_$(\mathcal{T},\mathcal{J})$ -parallel flow_ corresponding to $\mu$. For the
purposes of this paper, the actual construction of this flow is not so
important, so we will not describe it here. The main properties we need are
summarized in the following pair of propositions. The first is an immediate
consequence of [SWZ17, Proposition 5.11] while the second follows easily from
[SWZ17, Proposition 5.12].

######  Proposition 4.12.

Let
$\\{x,y,z\\}=T,\\{x^{\prime},y^{\prime},z^{\prime}\\}=T^{\prime}\in\widetilde{\Theta}$,
let $\rho\in\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$, and let $\mu\in
W$. Also, let $t\mapsto\rho_{t}$ be a lift in
$\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ of
$t\mapsto\phi^{\mu}_{t}[\rho]$ in $\operatorname{\mathrm{Hit}}_{V}(S)$ so that
$\rho_{0}=\rho$, and the $\rho_{t}$-equivariant Frenet curve $\xi_{t}$
satisfies $\xi_{t}(x)=\xi(x)$, $\xi_{t}(y)=\xi(y)$, and
$\xi_{t}^{(1)}(z)=\xi^{(1)}(z)$. Let $g_{t}\in\operatorname{\mathrm{PSL}}(V)$
be the group element so that $\xi_{t}(x^{\prime})=g_{t}\cdot\xi(x^{\prime})$,
$\xi_{t}(y^{\prime})=g_{t}\cdot\xi(y^{\prime})$,
$\xi_{t}^{(1)}(z^{\prime})=g_{t}\cdot\xi^{(1)}(z^{\prime})$.

  1. (1)

If $e=\\{x,y\\}=\\{x^{\prime},y^{\prime}\\}$, so that $z$ and $z^{\prime}$
lies to the right and left of $(x,y)$ then

$\frac{d}{dt}\bigg{|}_{t=0}g_{t}=\sum_{i=1}^{n-1}\mu^{i,n-i}_{x,y}A^{i,n-i}_{\xi(x),\xi(y)}.$

  2. (2)

Suppose that $x<y<z<x$ and $x^{\prime}=y$, $y^{\prime}=z$ and $z^{\prime}=x$.
Then

$\frac{d}{dt}\bigg{|}_{t=0}g_{t}=\sum_{i+j+k=n}\mu^{j,k,i}_{y,z,x}A^{j,k,i}_{\xi(y),\xi(z),\xi(x)}.$

######  Proposition 4.13.

Let $\\{T_{1},T_{2}\\}\in\widetilde{\mathcal{J}}$ be a bridge across the
closed edge $e=\\{x,y\\}$ so that $T_{1}$ and $T_{2}$ lie to the right and
left of $(x,y)$ respectively. Also, for $m=1,2$, let $p_{m}$, $q_{m}$,
$z_{m}$, $w_{m}$ be as defined in Notation 2.9, and let
$\widetilde{\Theta}(J,T_{m})$ and $\widetilde{\mathcal{T}}(J,T_{m})$ be as
defined in Notation 3.6. Let $t\mapsto\rho_{t}$ be a lift in
$\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ of
$t\mapsto\phi^{\mu}_{t}[\rho]$ in $\operatorname{\mathrm{Hit}}_{V}(S)$ so that
$\rho_{0}=\rho$, and the $\rho_{t}$-equivariant Frenet curve $\xi_{t}$
satisfies $\xi_{t}(p_{1})=\xi(p_{1})$, $\xi_{t}(w_{1})=\xi(w_{1})$, and
$\xi_{t}^{(1)}(z_{1})=\xi^{(1)}(z_{1})$. Let
$g_{t},u_{t}\in\operatorname{\mathrm{PSL}}(V)$ be the group elements so that
$\xi_{t}(p_{2})=g_{t}\cdot\xi(p_{2})$, $\xi_{t}(w_{2})=g_{t}\cdot\xi(w_{2})$,
$\xi_{t}^{(1)}(z_{2})=g_{t}\cdot\xi^{(1)}(z_{2})$,
$\xi_{t}(p_{1})=u_{t}\cdot\xi(p_{1})=\xi(p_{1})$ and
$\xi_{t}(q_{1})=u_{t}\cdot\xi(q_{1})$. Then

$\frac{d}{dt}\bigg{|}_{t=0}u_{t}=K_{1}(T_{1},T_{2})$

and

$\frac{d}{dt}\bigg{|}_{t=0}g_{t}=K_{1}(T_{1},T_{2})-K_{1}(T_{2},T_{1})+\sum_{i=1}^{n-1}\mu^{i,n-i}_{x,y}A^{i,n-i}_{\xi(x),\xi(y)}.$

######  Notation 4.14.

For any admissible labelling $L\in\mathcal{A}(\rho)$, let $\mu=\mu(L)\in W$ be
the vector defined in the following way:

  * •

For every $T=\\{x,y,z\\}\in\widetilde{\Theta}$ so that $x<y<z<x$, define

$\mu^{i,j,k}_{x,y,z}:=a_{i,j,k}(x,T,L),$

  * •

For every $\\{x,y\\}\in\widetilde{\mathcal{T}}$, define

$\mu^{i,n-i}_{x,y}=\alpha(x,y)a_{i,n-i}\big{(}x,y,L\big{)}.$

Recall from Definition 3.5 that
$a_{i,j,k}(x,T,L)=a_{j,k,i}(y,T,L)=a_{k,i,j}(z,T,L)$ and
$\alpha(x,y)a_{i,n-i}\big{(}x,y,L\big{)}=\alpha(y,x)a_{n-i,i}\big{(}y,x,L\big{)}$.
Also, it is easy to verify that the condition of $L\in\mathcal{A}(\rho)$ is
equivalent to the condition that $\mu(L)\in W$, so Notation 4.14 defines a
linear bijection between $\mathcal{A}(\rho)$ and $W$. As a corollary of
Proposition 4.12, Proposition 4.13 and the definition of the tangent cocycle,
we have the following.

######  Corollary 4.15.

Let $\rho\in\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$, $\Sigma$ be a
hyperbolic metric on $S$, $(r,\mathcal{T})$ be an oriented ideal triangulation
on $S$, and $\mathcal{J}$ a compatible bridge system. For any admissible
labelling $L\in\mathcal{A}(\rho,r,\mathcal{T})$, let $\mu(L)\in W$ be the
vector defined as above. Then the cocycle $\mu_{L}\in
C^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho})$ is
the $(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle in the cohomology
class

$[\nu]:=\frac{d}{dt}\bigg{|}_{t=0}\phi^{\mu(L)}_{t}[\rho]\in
H^{1}(S,\mathfrak{s}\mathfrak{l}(V)_{\operatorname{\mathrm{Ad}}\circ\rho}).$

###### Proof.

One needs only to verify that $\mu_{L}$ and the
$(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent cocycle in $[\nu]$ agree on
Type B and Type C $1$-simplices (see Section 4.2). The fact that they agree on
Type B $1$-simplices follows immediately from Proposition 4.13 and the
definition (Step 4) of the $(\rho,\Sigma,\mathcal{T},\mathcal{J})$-tangent
cocycle, while the fact that they agree on Type C $1$-simplices is an
immediate consequence of Proposition 4.12. ∎

Theorem 4.8 follows immediately from this corollary.

##  5\. Computing the Goldman symplectic form on
$\operatorname{\mathrm{Hit}}_{V}(S)$

Let $[\rho],[\rho^{\prime}]$ be a pair of representations in
$\operatorname{\mathrm{Hit}}_{V}(S)$. In this section, we will describe a
trivialization of $T\operatorname{\mathrm{Hit}}_{V}(S)$ that depends only on
the choice of an ideal triangulation $\mathcal{T}$ on $S$ and a compatible the
bridge system $\mathcal{J}$. We will also prove that this trivialization is
symplectic with respect to the Goldman symplectic form.

###  5.1. The $(\mathcal{T},\mathcal{J})$-trivialization of
$T\operatorname{\mathrm{Hit}}_{V}(S)$

Let $\rho,\rho^{\prime}\in\widetilde{\operatorname{\mathrm{Hit}}}_{V}(S)$ and
let $\xi,\xi^{\prime}:\partial\Gamma\to\mathcal{F}(V)$ be the respective
equivariant Frenet curves. Fix an ideal triangulation $\mathcal{T}$ on $S$ and
a compatible bridge system $\mathcal{J}$. Choose an orientation $r$ on