# Zariski dense surface subgroups in $SL(n,\mathbb{Q})$ with odd $n$

Carmen Galaz-García

###### Abstract.

For odd $n$ we construct a path $\rho_{t}\colon\thinspace\pi_{1}(S)\to
SL(n,\mathbb{R})$ of discrete, faithful and Zariski dense representations of a
surface group such that $\rho_{t}(\pi_{1}(S))\subset SL(n,\mathbb{Q})$ for
every $t\in\mathbb{Q}$.

Constructing Zariski dense surface subgroups in $SL(n,\mathbb{R})$ has
attracted attention as a step to finding thin groups, these are infinite index
subgroups of a lattice in $SL(n,\mathbb{R})$ which are Zariski dense. Finding
thin subgroups inside lattices in a variety of Lie groups has been a topic of
significant interest in recent years, in part from the connections thin groups
have to expanders and the affine sieve of Bourgain, Gamburd, and Sarnak
[1][2].

Though thin subgroups are in a sense generic [3][4], finding particular
specimens of thin surface subgroups in a given lattice remains a difficult
task. In this direction Long, Reid and Thistlethwaite [5] produced in 2011 the
first infinite family of nonconjugate thin surface groups in
$SL(3,\mathbb{Z})$. Their approach relies on parametrizing a family of
representations $\rho_{t}$ of the triangle group $\Delta(3,3,4)$ in the
Hitchin component, so that for every $t\in\mathbb{Z}$ the subgroup
$\rho_{t}(\Delta(3,3,4))$ is in $SL(3,\mathbb{Q})$ and has integral traces. By
results of Bass [6] these two properties together with
$\rho_{t}(\Delta(3,3,4))$ being non-solvable and finitely generated guarantee
that it is conjugate to a subgroup of $SL(3,\mathbb{Z})$. In 2018 Long and
Thistlethwaite [7] used a similar approach to obtain an infinite family of
non-conjugate Zariski dense surface subgroups in $SL(4,\mathbb{Z})$ and
$SL(5,\mathbb{Z})$.

Ballas and Long [8] in turn used the idea of "bending" a representation of the
fundamental group of a hyperbolic $n$-manifold $\pi_{1}(N)$ along an embedded
totally geodesic and separating hypersurface to obtain thin groups in
$SL(n+1,\mathbb{R})$ which are isomorphic to $\pi_{1}(N)$. The goal of this
article is to combine the aforementioned approaches to construct a family of
Zariski dense rational surface group representations by bending orbifold
representations. Our main result is the following:

######  Theorem 1.

For every surface $S$ finitely covering the orbifold $\mathcal{O}_{3,3,3,3}$
and every odd $n>1$ there exists a path of discrete, faithful and irreducible
representations $\rho_{t}\colon\thinspace\pi_{1}(S)\to SL(n,\mathbb{R})$, so
that

  1. (1)

$\rho_{0}(\pi_{1}(S))<SL(n,\mathbb{Z})$,

  2. (2)

$\rho_{t}$ is Zariski dense for every $t>0$, and

  3. (3)

$\rho_{t}(\pi_{1}(S))<SL(n,\mathbb{Q})$ for every $t\in\mathbb{Q}$.

Every representation $\rho_{t}$ in theorem 1 is a surface Hitchin
representation. Several of its properties are derived from the seminal work of
Labourie [9] on Anosov representations, the classification of Zariski closures
of surface Hitchin representations by Guichard [10] and the recent
introduction of orbifold Hitchin representations by Alessandrini, Lee and
Schaffhauser [11]. We provide an overview of these results in sections 1 and
2. At the end of section 2 we also prove the following criterion for Zariski
density, which will be subsequently used to discard Zariski closures.

######  Proposition 2.

Let $\rho\colon\thinspace\pi_{1}(\mathcal{O})\to PSL(n,\mathbb{R})$ be an
orbifold Hitchin representation such that

  * •

if $n=2k$ is even then $\rho(\pi_{1}(\mathcal{O}))$ is not conjugate to a
subgroup of $PSp(2k,\mathbb{R})$ or,

  * •

if $n=2k+1$ is odd then $\rho(\pi_{1}(\mathcal{O}))$ is not conjugate to a
subgroup of $PSO(k,k+1)$.

Then $\rho(H)$ is Zariski dense in $PSL(n,\mathbb{R})$ for every finite index
subgroup $H$ of $\pi_{1}(\mathcal{O})$.

In section 3 we give a general construction to obtain a path of
representations as in theorem 1. This is based on bending the fundamental
group $\pi_{1}(\mathcal{O})$ of a hyperbolic 2-dimensional orbifold along a
simple closed curve in $\mathcal{O}$ with infinite order as an element of
$\pi_{1}(\mathcal{O})$. Theorem 1 then follows from applying the results in
section 2 to a suitable representation of the fundamental group of the
orbifold $\mathcal{O}_{3,3,3,3}$ whose underlying topological space is $S^{2}$
and has four cone points of order 3. This final step is covered in section 4.

Remark. During the finalization of this project, Long and Thistlethwaite used
bending to construct thin surface groups in $SL(n,\mathbb{Z})$ for every odd
$n$ [12], the even case remains open.

##  1\. Hitchin representations

In this section we give a short overview of surface and orbifold Hitchin
representations.

###  1.1. Spaces of representations

Let $G$ be a Lie group and let $\Gamma$ be a group with a finite presentation
$\langle\alpha_{1},\ldots,\alpha_{k}\ |\ r_{1},\ldots,r_{m}\rangle$. Then
every relator $r_{i}$ defines a map $R_{i}\colon\thinspace G^{k}\to G$. If we
let $\text{Hom}(\Gamma,G)=\cap_{i=1}^{m}R_{i}^{-1}(Id)$, then the map
$\phi\mapsto(\phi(\alpha_{1}),\ldots,\phi(\alpha_{k}))$ is a bijection between
the set of all group homomorphisms from $\Gamma$ to $G$ and
$\text{Hom}(\Gamma,G)$. We will regard $\text{Hom}(\Gamma,G)$ as having the
subspace topology from $G^{k}$.

Let $\text{Hom}^{+}(\Gamma,G)$ be the subset of representations in
$\text{Hom}(\Gamma,G)$ which decompose as a direct sum of irreducible
representations and let $\text{Rep}^{+}(\Gamma,G)=\text{Hom}^{+}(\Gamma,G)/G$
be the quotient space by the conjugation action. With the quotient topology
$\text{Rep}^{+}(\Gamma,G)$ has the structure of an algebraic variety ([13]
sec. 5.2).

In the following we will frequently use the representation

$\displaystyle\tilde{\omega}_{n}\colon\thinspace SL(2,\mathbb{R})\to
SL(n,\mathbb{R})$ (1)

given by the action of $SL(2,\mathbb{R})$ on the vector space $\mathcal{P}$ of
homogeneous polynomials in 2 variables of degree $n-1$. If $n=2k$ is even, the
image of $\tilde{\omega}_{n}$ is contained in the symplectic group
$Sp(2k,\mathbb{R})$, and if $n=2k+1$ is odd, it is contained in a group
isomorphic to $SO(k,k+1)$.

It is well known that the representation $\tilde{\omega}_{n}$ is absolutely
irreducible and is, up to conjugation, the unique irreducible representation
from $SL(2,\mathbb{R})$ into $SL(n,\mathbb{R})$. This representation induces a
projective representation $\omega_{n}\colon\thinspace PSL(2,\mathbb{R})\to
PSL(n,\mathbb{R})$ which is also irreducible and unique up to conjugation.

###  1.2. Hitchin representations of surface groups

Let $S$ be a closed surface of genus $g>1$. In 1988 Goldman proved that
$\text{Rep}^{+}(\pi_{1}(S),PSL(2,\mathbb{R}))$ has $4g-3$ connected
components, two of which are diffeomorphic to $\mathbb{R}^{6g-6}$ and called
these Teichmüller spaces ([14] thm. A, see also note at end of thm. 10.2 in
[15]). The two Teichmüller spaces $\mathcal{T}^{\pm}(S)$ are precisely the
sets of conjugacy classes by $PSL(2,\mathbb{R})$ of Fuchsian representations,
which are discrete and faithful representations
$\rho\colon\thinspace\pi_{1}(S)\to
PSL(2,\mathbb{R})\equiv\text{Isom}^{+}(\mathbb{H}^{2})$.

######  Definition 3.

For $n>2$ a representation $r\colon\thinspace\pi_{1}(S)\to PSL(n,\mathbb{R})$
is called Fuchsian if it can be decomposed as $r=\omega_{n}\circ r_{0}$ where
$r_{0}\colon\thinspace\pi_{1}(S)\to PSL(2,\mathbb{R})$ is discrete and
faithful, and $\omega_{n}\colon\thinspace PSL(2,\mathbb{R})\to
PSL(n,\mathbb{R})$ is the unique irreducible representation introduced in
section 1.1.

######  Definition 4.

The Fuchsian locus is the set of all $PSL(n,\mathbb{R})$ conjugacy classes of
Fuchsian representations, namely the set $\omega_{n}(\mathcal{T}^{\pm}(S))$.

The space $\text{Rep}^{+}(\pi_{1}(S),PSL(n,\mathbb{R}))$ has three topological
connected components if $n$ is odd and 6 if $n$ is even ([15], thm. 10.2). The
Fuchsian locus is contained in one component in the odd case and in two
components in the even case. Each of these distinguished components, called
Hitchin components, is diffeomorphic to $\mathbb{R}^{(1-n^{2})(1-g)}$. When
$n>2$ is even, both Hitchin components are related by an inner automorphism of
$PSL(n,\mathbb{R})$. In the odd case, where there is only one component, we
will denote the Hitchin component by
$\text{Hit}(\pi_{1}(S),PSL(n,\mathbb{R}))$.

######  Definition 5.

Let $S$ be a closed surface of genus greater than one. A representation
$r\colon\thinspace\pi_{1}(S)\to PSL(n,\mathbb{R})$ is a surface Hitchin
representation if its $PSL(n,\mathbb{R})$-conjugacy class belongs to a Hitchin
component of $\text{Rep}^{+}(\pi_{1}(S),PSL(n,\mathbb{R}))$.

In [9], Labourie introduces Anosov representations and proves that surface
Hitchin representations are $B$-Anosov where $B$ is any Borel subgroup of
$PSL(n,\mathbb{R})$. This gives surface Hitchin representations essential
algebraic properties, out of which we will use theorem 7 below.

######  Definition 6 ([16] sec. 2.2).

A matrix $A\in SL(n,\mathbb{R})$ is purely loxodromic if it is diagonalizable
over $\mathbb{R}$ with eigenvalues of distinct modulus. If $A\in
PSL(n,\mathbb{R})$ then we say $A$ is purely loxodromic if any lift of $A$ to
an element of $SL(n,\mathbb{R})$ is purely loxodromic.

######  Theorem 7 ([9] thm. 1.5, lemma 10.1).

A surface Hitchin representation $r\colon\thinspace\pi_{1}(S)\to
PSL(n,\mathbb{R})$ is discrete, faithful and strongly irreducible. Moreover,
the image of every non-trivial element of $\pi_{1}(S)$ under $r$ is purely
loxodromic.

###  1.3. Hitchin representations of orbifold groups

Let $\mathcal{O}$ be a 2-dimensional closed orbifold of negative orbifold
Euler characteristic $\chi(\mathcal{O})$ and let $\pi_{1}(\mathcal{O})$ be its
orbifold fundamental group. In [17] Thurston proves there is a connected
component of the representation space
$\text{Rep}(\pi_{1}(\mathcal{O}),PGL(2,\mathbb{R}))$ that parametrizes
hyperbolic structures on $\mathcal{O}$. This component is called the
Teichmüller space of the orbifold $\mathcal{O}$, we will denote it by
$\mathcal{T}(\mathcal{O})$. As with surfaces, the orbifold Teichmüller space
consists of conjugacy classes of discrete and faithful representations of
$\pi_{1}(\mathcal{O})$ into
$PGL(2,\mathbb{R})\equiv\text{Isom}(\mathbb{H}^{2})$, which we will call
Fuchsian representations too. More recently, Alessandrini, Lee, and
Schaffhauser used the irreducible representation $\omega_{n}$ to define the
Hitchin component $\text{Hit}(\pi_{1}(\mathcal{O}),PGL(n,\mathbb{R}))$ of
$\text{Rep}(\pi_{1}(\mathcal{O}),PGL(n,\mathbb{R}))$ as the unique connected
component in this representation space which contains the connected Fuchsian
locus $\omega_{n}(\mathcal{T}(\mathcal{O}))$ ([11] def. 2.3) and prove
$\text{Hit}(\pi_{1}(\mathcal{O}),PGL(n,\mathbb{R}))$ is homeomorphic to an
open ball ([11] thm. 1.2).

######  Definition 8 ([11] def. 2.4).

Let $\mathcal{O}$ be a 2-dimensional connected closed orbifold with negative
orbifold Euler characteristic. A representation
$r\colon\thinspace\pi_{1}(\mathcal{O})\to PGL(n,\mathbb{R})$ is an orbifold
Hitchin representation if its $PGL(n,\mathbb{R})$-conjugacy class belongs to
the Hitchin component $\text{Hit}(\pi_{1}(\mathcal{O}),PGL(n,\mathbb{R}))$ of
$\text{Rep}(\pi_{1}(\mathcal{O}),PGL(n,\mathbb{R}))$.

The definition of Anosov representations has been generalized by Guichard and
Wienhard ([18] def. 2.10) to include representations of word hyperbolic groups
into semisimple Lie groups. With this more general definition, and just as
their surface counterparts, orbifold Hitchin representations are also
$B$-Anosov where $B$ is a Borel subgroup of $PGL(n,\mathbb{R})$ ([11] prop.
2.16) and thus share some strong algebraic properties.

######  Theorem 9 ([11] thm. 1.1).

An orbifold Hitchin representation $r\colon\thinspace\pi_{1}(\mathcal{O})\to
PGL(n,\mathbb{R})$ is discrete, faithful and strongly irreducible. Moreover,
the image of every infinite order element of $\pi_{1}(\mathcal{O})$ under $r$
is purely loxodromic.

##  2\. Zariski dense Hitchin representations

In this section we focus on Zariski density of Hitchin representations and
prove corollary 15 which gives a criterion to determine when the image of a
finite index subgroup of an orbifold group under a Hitchin representation is
Zariski dense.

###  2.1. Zariski closures of Hitchin representations

Let $G$ be an algebraic matrix Lie group, then $G$ has both its standard
topology as a subset of some $\mathbb{R}^{N}$ and the Zariski topology. If $X$
is a subset of $G$ then its Zariski closure is the closure of $X$ in $G$ with
respect to the Zariski topology. We say a subgroup $H<G$ is Zariski dense in
$G$ if its Zariski closure equals $G$. A representation
$r\colon\thinspace\Gamma\to G$ is Zariski dense if $r(\Gamma)$ is Zariski
dense in $G$.

The image of the irreducible representation $\omega_{n}\colon\thinspace
PSL(2,\mathbb{R})\to PSL(n,\mathbb{R})$ is contained, if $n$ is even, in a
conjugate of $PSp(n,\mathbb{R})$, which is the projectivization of the
symplectic group $Sp(n,\mathbb{R})$. If $n=2k+1$ is odd, the image of
$\omega_{n}$ is contained in a conjugate of the orthogonal group
$SO(k,k+1)=PSO(k,k+1)$. This implies that the images of Fuchsian
representations are contained in (a conjugate of) $PSp(n,\mathbb{R})$ or in
$SO(k,k+1)$ and, in particular, they are not Zariski dense. More generally,
for surface Hitchin representations Guichard [10] has announced a
classification of Zariski closures of their lifts. An alternative proof of
this result has been given recently by Sambarino ([19] cor. 1.5). The version
of this result we cite here comes from theorem 11.7 in [20].

######  Theorem 10 ([10], [19]).

If $r\colon\thinspace\pi_{1}(S)\to SL(n,\mathbb{R})$ is the lift of a surface
Hitchin representation and $H$ is the Zariski closure of $r(\pi_{1}(S))$, then

  * •

If $n=2k$ is even, $H$ is conjugate to either $\omega_{n}(SL(2,\mathbb{R}))$,
$Sp(2k,\mathbb{R})$ or $SL(2k,\mathbb{R})$.

  * •

If $n=2k+1$ is odd and $n\neq 7$, then $H$ is conjugate to either
$\omega_{n}(SL(2,\mathbb{R}))$, $SO(k,k+1)$ or $SL(2k+1,\mathbb{R})$.

  * •

If $n=7$, then $H$ is conjugate to either $\omega_{7}(SL(2,\mathbb{R}))$,
$G_{2}$, $SO(3,4)$ or $SL(7,\mathbb{R})$.

###  2.2. A criterion for Zariski density

Here we prove proposition 2 which gives us a criterion to find Zariski dense
Hitchin representations.

######  Lemma 11.

Let $\rho\colon\thinspace\pi_{1}(\mathcal{O})\to PSL(n,\mathbb{R})$ with $n$
even be an orbifold Hitchin representation. Then for every
$[\alpha]\in\pi_{1}(\mathcal{O})$ of infinite order there is a lift $A\in
SL(n,\mathbb{R})$ of $\rho([\alpha])$ which has $n$ positive distinct
eigenvalues.

Proof. First consider a Fuchsian representation
$\sigma\colon\thinspace\pi_{1}(\mathcal{O})\to PSL(2,\mathbb{R})$ and
$[\alpha]$ an infinite order element of $\pi_{1}(\mathcal{O})$. Since
$\mathcal{O}$ is a hyperbolic orbifold, $\sigma([\alpha])$ is conjugate to a
hyperbolic element $\begin{bmatrix}\lambda&0\\\
0&\frac{1}{\lambda}\end{bmatrix}\in PSL(2,\mathbb{R})$. We can lift this
element to a matrix $\begin{pmatrix}\lambda&0\\\
0&\frac{1}{\lambda}\end{pmatrix}\in SL(2,\mathbb{R})$ with $\lambda>0$. Let
$\tilde{\omega}_{n}\colon\thinspace SL(2,\mathbb{R})\to SL(n,\mathbb{R})$ be
the unique irreducible representation in (1), then
$\tilde{\omega}_{n}\begin{pmatrix}\lambda&0\\\
0&\frac{1}{\lambda}\end{pmatrix}\in SL(n,\mathbb{R})$ has $n$ distinct
positive eigenvalues
$\lambda^{n-1},\lambda^{n-3},\ldots,\lambda^{-(n-3)},\lambda^{-(n-1)}$ and is
a lift of $\omega_{n}\circ\sigma([\alpha])\in PSL(n,\mathbb{R})$.

Now consider a Hitchin representation
$\rho\colon\thinspace\pi_{1}(\mathcal{O})\to PSL(n,\mathbb{R})$. Let
$\rho_{t}$ be a path of Hitchin representations such that $\rho_{0}$ is
Fuchsian and $\rho_{1}=\rho$. This induces a path $\rho_{t}([\alpha])\subset
PSL(n,\mathbb{R})$. By the previous argument we may lift $\rho_{t}([\alpha])$
to a path $\tilde{A}_{t}\in SL(n,\mathbb{R})$ such that $\tilde{A}_{0}$ has
$n$ distinct positive eigenvalues. Since each eigenvalue of $\tilde{A}_{t}$
varies continuously and $\det\tilde{A}_{t}\neq 0$, all eigenvalues of
$\tilde{A}_{t}$ are positive. Moreover, by theorem 9 the absolute values of
the eigenvalues of $\rho_{t}([\alpha])$ are distinct. This in turn implies all
the eigenvalues of $\tilde{A}_{t}$ are distinct. Therefore $\tilde{A}_{1}\in
SL(n,\mathbb{R})$ is a lift of $\rho([\alpha])$ with $n$ positive distinct
eigenvalues.

$\Box$

To prove our criterion for Zariski density (propositions 13 and 14) we will
make use of the following theorem by Culver.

######  Theorem 12 ([21] thm. 2).

Let $C$ be a real square matrix. Then the equation $C=\exp(X)$ has a unique
real solution $X$ if and only if all the eigenvalues of $C$ are positive real
and no elementary divisor (Jordan block) of C belonging to any eigenvalue
appears more than once.

######  Proposition 13.

Let $\rho\colon\thinspace\pi_{1}(\mathcal{O})\to PSL(n,\mathbb{R})$ with $n$
even be an orbifold Hitchin representation so that
$\rho(\pi_{1}(\mathcal{O}))$ is not conjugate to a subgroup of
$PSp(n,\mathbb{R})$. If $S$ is a surface finitely covering $\mathcal{O}$ then
$\rho(\pi_{1}(S))$ is Zariski dense.

Proof. Let $S$ be a surface finitely covering $\mathcal{O}$ and suppose that
$\rho(\pi_{1}(S))$ is conjugate to a subgroup of $PSp(n,\mathbb{R})$. Then
there exists an alternating form $\Omega\in SL(n,\mathbb{R})$ such that
$Sp(\Omega)=\\{g\in SL(n,\mathbb{R})\ |\ g^{T}\Omega g=\Omega\\}$ and
$\rho(\pi_{1}(S))\subset PSp(\Omega)=Sp(\Omega)/\pm I$.

Let $[\alpha]\in\pi_{1}(\mathcal{O})$ be an infinite order element. By lemma
11 we can lift $\rho([\alpha])\in PSL(n,\mathbb{R})$ to a matrix $A\in
SL(n,\mathbb{R})$ with $n$ positive distinct eigenvalues. Since $\pi_{1}(S)$
has finite index in $\pi_{1}(\mathcal{O})$ there exists a $k\in\mathbb{N}$
such that $\rho([\alpha])^{k}\in\rho(\pi_{1}(S))$. Then $A^{k}$ is a lift of
$\rho([\alpha])^{k}$ and $A^{k}\in Sp(\Omega)$. Given that $A$ has $n$
positive distinct eigenvalues, by theorem 12 there is a unique $X\in
M_{n\times n}(\mathbb{R})$ such that $\exp(X)=A$. Then using that
$\exp(kX)=A^{k}$ preserves $\Omega$ we get that

$\displaystyle\exp(kX)^{T}\Omega\exp(kX)=\Omega$ $\displaystyle\Rightarrow$
$\displaystyle\Omega^{-1}\exp(kX)^{T}\Omega=\exp(kX)^{-1}$
$\displaystyle\Rightarrow$
$\displaystyle\exp(\Omega^{-1}(kX)^{T}\Omega)=\Omega^{-1}\exp(kX)^{T}\Omega=\exp(-kX).$

Applying theorem 12 now to $\Omega^{-1}\exp(kX)^{T}\Omega$ we obtain that

$\displaystyle\Omega^{-1}(kX)^{T}\Omega=-kX$ $\displaystyle\Rightarrow$
$\displaystyle-\Omega(kX)^{T}\Omega=-kX$ $\displaystyle\Rightarrow$
$\displaystyle\Omega(kX)^{T}\Omega=kX.$

This implies that $kX\in\mathfrak{sp}(\Omega)$ and thus $A=\exp(X)\in
Sp(\Omega)$. Given that $A$ is a lift of $\rho([\alpha])$, we have that
$\rho([\alpha])\in PSp(\Omega)$. Since $\pi_{1}(\mathcal{O})$ is generated by
its infinite order elements we get that $\rho(\pi_{1}(\mathcal{O}))\subset
PSp(\Omega)$, a contradiction. So it cannot be that $\rho(\pi_{1}(S))$ is
conjugate to a subgroup of $PSp(n,\mathbb{R})$. In particular, if $r$ is a
lift of the Hitchin surface representation $\rho|_{\pi_{1}(S)}$ then the
Zariski closure of $r(\pi_{1}(S))$ cannot be conjugate to a subgroup of
$Sp(n,\mathbb{R})$. By theorem 10 it must be that the Zariski closure of
$r(\pi_{1}(S))$ is $SL(n,\mathbb{R})$. Therefore the Zariski closure of
$\rho(\pi_{1}(S))$ is $PSL(n,\mathbb{R})$.

$\Box$

In the case when $n=2k+1$ is odd, by theorem 10 the Zariski closure of
$\rho(\pi_{1}(S))$ where $\rho$ is a surface Hitchin representation is either
conjugate to a subgroup of $SO(k,k+1)$ or equals $SL(n,\mathbb{R})$. By
assuming there exists a symmetric bilinear form $J$ such that
$\rho(\pi_{1}(S))\subset SO(J)$ we have an analogous proof to that of 13 to
get a criterion for Zariski density of surface Hitchin representations in the
odd case.

######  Proposition 14.

Let $\rho\colon\thinspace\pi_{1}(\mathcal{O})\to SL(n,\mathbb{R})$ with $n$
odd be an orbifold Hitchin representation such that there is no real quadratic
form $J$ for which $\rho(\pi_{1}(\mathcal{O}))\subset SO(J)$. If $S$ is a
surface finitely covering $\mathcal{O}$ then $\rho(\pi_{1}(S))$ is Zariski
dense.

Given that any finite index subgroup of $\pi_{1}(\mathcal{O})$ contains a
surface subgroup which has finite index in $\pi_{1}(\mathcal{O})$ we obtain
the following result.

######  Proposition 15.

Let $\rho\colon\thinspace\pi_{1}(\mathcal{O})\to PSL(n,\mathbb{R})$ be an
orbifold Hitchin representation such that

  * •

if $n=2k$ is even then $\rho(\pi_{1}(\mathcal{O}))$ is not conjugate to a
subgroup of $PSp(2k,\mathbb{R})$ or,

  * •

if $n=2k+1$ is odd then $\rho(\pi_{1}(\mathcal{O}))$ is not conjugate to a
subgroup of $PSO(k,k+1)$.

Then for every finite index subgroup $H$ of $\pi_{1}(\mathcal{O})$ the image
$\rho(H)$ is Zariski dense in $PSL(n,\mathbb{R})$.

##  3\. Bending representations of orbifold groups

Theorem 19 in this section gives a general construction of a path $\rho_{t}$
of Zariski dense Hitchin surface representations into $SL(n,\mathbb{R})$ for
odd $n$. By requiring that the initial representation $\rho_{0}$ has image
inside $SL(n,\mathbb{Q})$ we obtain corollary 20, in which every
representation $\rho_{t}$ with $t\in\mathbb{Q}$ also has image in
$SL(n,\mathbb{Q})$.

###  3.1. Bending representations

Let $\mathcal{O}$ be a 2-dimensional orientable connected closed orbifold of
negative orbifold Euler characteristic and $\mathcal{O}_{L}$,
$\mathcal{O}_{R}$ be open connected suborbifolds with connected intersection
$\mathcal{O}_{L}\cap\mathcal{O}_{R}$. Given a representation
$\rho\colon\thinspace\pi_{1}(\mathcal{O})\to G$ there is a standard way of
"bending" $\rho$ by an element $\delta$ of the centralizer in $G$ of
$\rho(\pi_{1}(\mathcal{O}_{L}\cap\mathcal{O}_{R}))$ to obtain a representation
$\rho_{\delta}\colon\thinspace\pi_{1}(\mathcal{O})\simeq\pi_{1}(\mathcal{O}_{L})\ast_{\pi_{1}(\mathcal{O}_{L}\cap\mathcal{O}_{R})}\pi_{1}(\mathcal{O}_{R})\to
G$ so that
$\rho_{\delta}(\pi_{1}(\mathcal{O}))=\langle\rho(\pi_{1}(\mathcal{O}_{L})),\delta\rho(\pi_{1}(\mathcal{O}_{R}))\delta^{-1}\rangle$
(see for example [22] sec. 5).

From now onwards we will consider the case where there is a simple closed
curve $\gamma\subset\mathcal{O}$, not parallel to a cone point, that divides
$\mathcal{O}$ into two orbifolds $\mathcal{O}_{L}$ and $\mathcal{O}_{R}$ which
share $\gamma$ as their common boundary, so that
$\pi_{1}(\mathcal{O})\simeq\pi_{1}(\mathcal{O}_{L})\ast_{\langle[\gamma]\rangle}\pi_{1}(\mathcal{O}_{R}).$

######  Proposition 16.

Let
$\rho\colon\thinspace\pi_{1}(\mathcal{O})\simeq\pi_{1}(\mathcal{O}_{L})\ast_{\langle[\gamma]\rangle}\pi_{1}(\mathcal{O}_{R})\to
SL(n,\mathbb{Q})$ be a representation for which $\rho([\gamma])$ has $n$
distinct positive eigenvalues. Then there exists a path of representations
$\rho_{t}\colon\thinspace\pi_{1}(\mathcal{O})\to SL(n,\mathbb{R})$ with $t\geq
0$ such that

  1. (1)

$\rho_{0}=\rho$,

  2. (2)

$\rho_{t}(\pi_{1}(\mathcal{O}))=\langle\rho(\pi_{1}(\mathcal{O}_{L})),\delta_{t}\rho(\pi_{1}(\mathcal{O}_{R}))\delta_{t}^{-1}\rangle$
for some $\delta_{t}\in SL(n,\mathbb{R})$ which commutes with
$\rho([\gamma])$, and

  3. (3)

$\rho_{t}$ has image in $SL(n,\mathbb{Q})$ for every $t\in\mathbb{Q}$.

Proof. The matrix $\rho([\gamma])$ is conjugate to a diagonal matrix $D$ with
entries $\lambda_{1},\ldots,\lambda_{n}>0$ along its diagonal. Now for every
$t>0$ define

$\delta_{t}=(t\rho([\gamma])+I)\det(t\rho([\gamma])+I)^{-\frac{1}{n}}$ (2)

Notice that
$\det(t\rho([\gamma])+I)=\det(tD+I)=\Pi_{k=1}^{n}(t\lambda_{i}+1)>0$, so
$t\rho([\gamma])+I$ is invertible for all $t$. Then each $\delta_{t}$ is in
$SL(n,\mathbb{R})$ and we can check that $\delta_{t}$ commutes with
$\rho([\gamma])$. Since $\rho$ is a rational representation, whenever
$t\in\mathbb{Q}$ the matrix $t\rho([\gamma])+I$ has rational entries and non-
zero determinant.

Let $\rho_{t}\colon\thinspace\pi_{1}(\mathcal{O})\to SL(n,\mathbb{R})$ be the
representation such that
$\rho_{t}(\pi_{1}(\mathcal{O}))=\langle\rho(\pi_{1}(\mathcal{O}_{L})),\delta_{t}\rho(\pi_{1}(\mathcal{O}_{R}))\delta_{t}^{-1}\rangle$.
Notice that $\rho_{0}=\rho$ and that for every $t\in\mathbb{Q}$ the
representation $\rho_{t}$ has image in $SL(n,\mathbb{Q})$.

$\Box$

###  3.2. Discarding Zariski closures

For the rest of section 3 we focus on the case where $n=2k+1$ is odd. Recall
that in this case $SL(n,\mathbb{R})\equiv PSL(n,\mathbb{R})$.

######  Lemma 17.

Let $\rho\colon\thinspace\Gamma\to SL(n,\mathbb{R})$ be an irreducible
representation and suppose there is a quadratic form $J$ such that
$\rho(\Gamma)\subset SO(J)$. Then $J$ is unique up to scaling.

Proof. Suppose $\rho(\Gamma)<SO(J_{1})\cap SO(J_{2})$. Then for any
$\rho(\gamma)\in\rho(\Gamma)$ we have that

$J_{1}^{-1}\rho(\gamma)J_{1}=\rho(\gamma)^{-T}=J_{2}^{-1}\rho(\gamma)J_{2},$

which implies that $\rho(\gamma)J_{1}J_{2}^{-1}=J_{1}J_{2}^{-1}\rho(\gamma).$
Since $n$ is odd, $J_{1}J_{2}^{-1}$ has a real eigenvalue $\lambda$. Then
$\text{Ker}(J_{1}J_{2}^{-1}-\lambda I)$ is a non-zero invariant subspace for
the irreducible representation $\rho$, which implies $J_{1}=\lambda J_{2}$.

$\Box$

######  Proposition 18.

Let
$\rho\colon\thinspace\pi_{1}(\mathcal{O})\simeq\pi_{1}(\mathcal{O}_{L})\ast_{\langle[\gamma]\rangle}\pi_{1}(\mathcal{O}_{R})\to
SL(n,\mathbb{R})$ be a representation in which the restrictions
$\rho|_{\pi_{1}(\mathcal{O}_{L})}$ and $\rho|_{\pi_{1}(\mathcal{O}_{R})}$ are
irreducible and $\rho([\gamma])$ has $n$ positive distinct eigenvalues.
Suppose there is a quadratic form $J$ such that
$\rho(\pi_{1}(\mathcal{O}))\subset SO(J)$. Then there exists a path of
representations $\rho_{t}\colon\thinspace\pi_{1}(\mathcal{O})\to
SL(n,\mathbb{R})$ such that

  1. (1)

$\rho_{0}=\rho$ and

  2. (2)

for each $t>0$ there is no quadratic form $\tilde{J}$ such that
$\rho_{t}(\pi_{1}(\mathcal{O}))\subset SO(\tilde{J})$.

Proof. By proposition 16 there are $\delta_{t}\in SL(n,\mathbb{R})$ that
commute with $\rho([\gamma])$, with which we can construct a path of
representations $\rho_{t}\colon\thinspace\pi_{1}(\mathcal{O})\to
SL(n,\mathbb{R})$ such that $\rho_{0}=\rho$ and
$\rho_{t}(\pi_{1}(\mathcal{O}))=\langle\rho(\pi_{1}(\mathcal{O}_{L})),\delta_{t}\rho(\pi_{1}(\mathcal{O}_{R}))\delta_{t}^{-1}\rangle$.

Now fix $t>0$. Suppose there exists a quadratic form $\tilde{J}$ such that
$\rho_{t}(\pi_{1}(\mathcal{O}))\subset SO(\tilde{J})$. Since
$\rho(\pi_{1}(\mathcal{O}))\subset SO(J)$, in particular
$\rho_{t}(\pi_{1}(\mathcal{O}_{L}))=\rho_{0}(\pi_{1}(\mathcal{O}_{L}))\subset
SO(J)\cap SO(\tilde{J})$. The restriction
$\rho_{t}|_{\pi_{1}(\mathcal{O}_{L})}$ is irreducible, so by lemma 17 $J$ is a
real multiple of $\tilde{J}$. Similarly, by construction
$\rho_{t}(\pi_{1}(\mathcal{O}_{R}))\subset SO(\delta_{t}J\delta_{t}^{T})\cap
SO(\tilde{J})$ and $\rho_{t}|_{\pi_{1}(\mathcal{O}_{R})}$ is irreducible too.
Thus $\delta_{t}J\delta_{t}^{T}$ is also a multiple of $\tilde{J}$. This
implies there is a $\lambda\in\mathbb{R}$ such that $\lambda
J=\delta_{t}J\delta_{t}^{T}$ and then $\lambda^{n}=\det(\delta_{t})^{2}=1.$
Since $n$ is odd it must be that $\lambda=1$ and we obtain $\delta_{t}\in
SO(J)$. Given that

$(t\rho([\gamma])+I)J(t\rho([\gamma])^{T}+I)\ =\ t^{2}J\ +\
tJ(\rho([\gamma])^{T})^{-1}\ +\ tJ\rho([\gamma])^{T}\ +\ J,$

having $J=\delta_{t}J\delta_{t}^{T}$ would imply that $\mu
I=\rho([\gamma])^{-1}+\rho([\gamma])$ for some $\mu\in\mathbb{R}$. Recall that
$\rho([\gamma])$ is conjugate to a diagonal matrix $D$ whose eigenvalues are
all distinct. If $\mu I=\rho([\gamma])^{-1}+\rho([\gamma])$ then by
conjugating we would obtain that $\mu I=D^{-1}+D$, which is not the case given
that $n>2$.

$\Box$

###  3.3. Representations of surface groups

Recall we are assuming that $\mathcal{O}$ is a 2-dimensional orientable
connected closed orbifold of negative orbifold Euler characteristic. Such
orbifolds are always finitely covered by a surface $S$ of genus greater than
one, so $\pi_{1}(S)$ is a finite index subgroup of $\pi_{1}(\mathcal{O})$.
Given a representation $\rho\colon\thinspace\pi_{1}(\mathcal{O})\to G$ we will
denote the restriction of $\rho$ to $\pi_{1}(S)$ by $\rho^{S}$.

######  Theorem 19.

Suppose
$\pi_{1}(\mathcal{O})\simeq\pi_{1}(\mathcal{O}_{L})\ast_{\langle[\gamma]\rangle}\pi_{1}(\mathcal{O}_{R})$
with $[\gamma]$ an infinite order element. Let
$\rho\colon\thinspace\pi_{1}(\mathcal{O})\to SL(n,\mathbb{R})$ be an orbifold
Fuchsian representation such that the restrictions
$\rho|_{\pi_{1}(\mathcal{O}_{L})}$ and $\rho|_{\pi_{1}(\mathcal{O}_{R})}$ are
irreducible. If $S$ is a surface finitely covering $\mathcal{O}$ then there
exists a path of representations $\rho^{S}_{t}\colon\thinspace\pi_{1}(S)\to
SL(n,\mathbb{R})$ such that $\rho^{S}_{0}=\rho^{S}$ and $\rho_{t}^{S}$ is a
Zariski dense surface Hitchin representation for each $t>0$.

Proof. Since $\rho\colon\thinspace\pi_{1}(\mathcal{O})\to SL(n,\mathbb{R})$ is
an orbifold Hitchin representation with odd $n=2k+1$ and $[\gamma]$ has
infinite order, then $\rho([\gamma])$ has $n$ positive distinct real
eigenvalues. Moreover, since $\rho$ is Fuchsian its image is contained in a
conjugate of $SO(k,k+1)$. Using proposition 18 we obtain a path of
representations $\rho_{t}\colon\thinspace\pi_{1}(\mathcal{O})\to
SL(n,\mathbb{R})$ such that $\rho_{0}=\rho$ and for each $t>0$ there is no
real quadratic form $J$ such that $\rho_{t}(\pi_{1}(\mathcal{O}))\subset
SO(J)$. By proposition 14 each $\rho_{t}(\pi_{1}(S))$ is Zariski dense in
$SL(n,\mathbb{R})$.

Now consider the continuous path
$[\rho_{t}]\in\text{Rep}(\pi_{1}(\mathcal{O}),PGL(n,\mathbb{R}))$ for $t\geq
0$. Its image is connected so all $PGL(n,\mathbb{R})$-conjugacy classes
$[\rho_{t}]$ are contained in the same connected component of
$\text{Rep}(\pi_{1}(\mathcal{O}),PGL(n,\mathbb{R}))$. Because the
representation $\rho_{0}=\rho$ is Fuchsian, $[\rho_{0}]$ is in the Hitchin
component $\text{Hit}(\pi_{1}(\mathcal{O}),PGL(n,\mathbb{R}))$ and so is every
$[\rho_{t}]$. Thus, by theorem 9, each $\rho_{t}$ is discrete, faithful and
strongly irreducible. Since $\pi_{1}(S)$ has finite index in
$\pi_{1}(\mathcal{O})$, each restriction
$\rho^{S}_{t}\colon\thinspace\pi_{1}(S)\to SL(n,\mathbb{R})$ is irreducible.
In particular $\rho_{0}^{S}$ is a surface Fuchsian representation. Then
$[\rho^{S}_{t}]$ is a continuous path in
$\text{Rep}^{+}(\pi_{1}(S),SL(n,\mathbb{R}))$ with
$[\rho_{0}^{S}]\in\text{Hit}(\pi_{1}(S),SL(n,\mathbb{R}))$. Since the Hitchin
component is path connected
$[\rho^{S}_{t}]\in\text{Hit}(\pi_{1}(S),SL(n,\mathbb{R}))$ for all $t\geq 0$.

$\Box$

To finish this section notice that the construction of the path of Zariski
dense representations in the previous theorem is based on proposition 16, so
we may add the assumption of $\rho(\pi_{1}(\mathcal{O}))\subset
SL(n,\mathbb{Q})$ to obtain that the image of every $\rho_{t}$ is in
$SL(n,\mathbb{Q})$ for every $t\in\mathbb{Q}$.

######  Corollary 20.

Let $\rho\colon\thinspace\pi_{1}(\mathcal{O})\to PSL(n,\mathbb{Q})$ be a
representation satisfying the assumptions of theorem 19. If $S$ is a surface
finitely covering $\mathcal{O}$ then there exists a path
$\rho^{S}_{t}\colon\thinspace\pi_{1}(S)\to SL(n,\mathbb{R})$ of Hitchin
representations such that $\rho^{S}_{0}=\rho^{S}$, $\rho_{t}^{S}$ is Zariski
dense for each $t>0$ and $\rho_{t}^{S}$ has image in $SL(n,\mathbb{Q})$ for
every $t\in\mathbb{Q}$.

##  4\. Representations of $\pi_{1}(\mathcal{O}_{3,3,3,3})$

In this section we look at the orbifold $\mathcal{O}_{3,3,3,3}$ and find a
Fuchsian representation $\rho\colon\thinspace\pi_{1}(\mathcal{O}_{3,3,3,3})\to
SL(n,\mathbb{Z})$ satisfying the assumptions of corollary 20.

###  4.1. The orbifold $\mathcal{O}_{3,3,3,3}$

In what follows we focus on the triangle group $\Delta(3,4,4)\subset
PSL(2,\mathbb{R})$. If we let $T$ be the hyperbolic triangle with angles
$\\{\frac{\pi}{3}$, $\frac{\pi}{4}$, $\frac{\pi}{4}\\}$, then the generators
of $\Delta(3,4,4)$ are the rotations $x$ and $y$ by $\frac{2\pi}{3}$ and
$\frac{\pi}{2}$ around the corresponding vertices of $T$. This group has
presentation

$\Delta(3,4,4)=\langle x,y\ |\ x^{3}=y^{4}=(xy)^{4}=1\rangle.$ (3)

The fundamental domain for the action of $\Delta(3,4,4)$ on $\mathbb{H}^{2}$
is a quadrilateral with angles
$\\{\frac{\pi}{2},\frac{\pi}{3},\frac{\pi}{2},\frac{\pi}{3}\\}$. The quotient
$\mathbb{H}^{2}/\Delta(3,4,4)$ is homeomorphic to the orbifold $S^{2}(3,4,4)$
whose underlying topological space is $S^{2}$ and has three cone points of
orders $3,\ 4$ and $4$. This defines, up to conjugation, an isomorphism
$\pi_{1}(S^{2}(3,4,4))\to\Delta(3,4,4)\subset PSL(2,\mathbb{R})$.

Figure 1. Orbifold $S^{2}(3,4,4)$

Let $\theta_{1}=x$ and $\theta_{i}=y\theta_{i-1}y^{-1}$ for $i=2,3,4$, then
$\langle\theta_{1},\ldots,\theta_{4}\rangle$ the quotient of $\mathbb{H}^{2}$
by the action of $\langle\theta_{1},\ldots,\theta_{4}\rangle$ is homeomorphic
to the orbifold $\mathcal{O}_{3,3,3,3}$ with underlying topological space
$S^{2}$ and 4 cone points of order 3. By construction, we obtain that
$\mathcal{O}_{3,3,3,3}$ is an index four orbifold covering of $S^{2}(3,4,4)$.
If $\gamma_{1},\ldots,\gamma_{4}$ are loops around the cone points of
$\mathcal{O}_{3,3,3,3}$, then the orbifold fundamental group has the
presentation

$\pi_{1}(\mathcal{O}_{3,3,3,3})=\langle\gamma_{1},\ldots,\gamma_{4}\ |\
\gamma_{1}^{3}=\ldots=\gamma_{4}^{3}=\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}=1\rangle.$

Identifying each $\gamma_{i}$ with the rotation $\theta_{i}$ gives an
isomorphism
$\pi_{1}(\mathcal{O}_{3,3,3,3})\cong\langle\theta_{1},\ldots,\theta_{4}\rangle$
which defines (up to conjugation) a discrete and faithful representation

$\sigma\colon\thinspace\pi_{1}(\mathcal{O}_{3,3,3,3})\to\Delta(3,4,4)<PSL(2,\mathbb{R}).$
(4)

######  Lemma 21.

The representation $\sigma\colon\thinspace\pi_{1}(\mathcal{O}_{3,3,3,3})\to
PSL(2,\mathbb{R})$ defined in (4) is Zariski dense.

Proof. We will check that the group
$\sigma(\pi_{1}(\mathcal{O}_{3,3,3,}))=\langle\theta_{1},\ldots,\theta_{4}\rangle<\Delta(3,4,4)$
is Zariski dense. Hyperbolic triangles with the same angles are isometric, so
we can fix the hyperbolic triangle with angles
$\\{\frac{\pi}{3},\frac{\pi}{4},\frac{\pi}{4}\\}$ by placing it symmetrically
along the $y$-axis in the upper-half plane. By having the generators $x,y$ of
$\Delta(3,4,4)$ defined in (3) in rational canonical form we obtain that:

$\displaystyle x=\begin{bmatrix}0&-1\\\ 1&1\end{bmatrix}\ \text{ and }\
y=\begin{bmatrix}0&-1-\sqrt{2}\\\ -1+\sqrt{2}&\sqrt{2}\end{bmatrix}.$ (5)

This choice of generators fixes a representative in the conjugacy class of the
representation $\sigma$. Notice that $\theta_{2}\theta_{1}=yxy^{-1}x$ is an
infinite order element in $\Delta(3,4,4)$ and is therefore hyperbolic. By
using the matrices in (5) we can explicitly find $P,D\in PGL(n,\mathbb{R})$
with $D$ diagonal so that $P^{-1}(\theta_{2}\theta_{1})P=D$. It suffices then
to see that the conjugated representation $P^{-1}\sigma P$ is Zariski dense.
Let $H$ be the Zariski closure of
$P^{-1}\sigma(\pi_{1}(\mathcal{O}_{3,3,3,3}))P$ in $PSL(2,\mathbb{R})$ and
$\mathfrak{h}$ its Lie algebra. First notice that the Zariski closure of
$\langle D\rangle$ is the algebraic torus whose Lie algebra is the span of
$X_{1}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$. Taking
$X_{2}=\text{Ad}_{P^{-1}\theta_{1}\theta_{2}P}(X_{1})$ and
$X_{3}=\text{Ad}_{P^{-1}\theta_{1}^{2}\theta_{2}P}(X_{1})$ we obtain three
linearly independent vectors in $\mathfrak{h}$. Then
$\dim(\mathfrak{h})=3=\dim(\mathfrak{sl}(2,\mathbb{R}))$ so the two algebras
must coincide and so $H=PSL(2,\mathbb{R})$.

$\Box$

###  4.2. Rational representations of $\pi_{1}(\mathcal{O}_{3,3,3,3})$

We will now focus on the case $n=2k+1$ and the representation
$\omega_{n}\circ\sigma\colon\thinspace\pi_{1}(\mathcal{O}_{3,3,3,3})\to
SL(n,\mathbb{R}),$ where $\sigma$ is the representation defined in (4) and
$\omega_{n}\colon\thinspace PSL(2,\mathbb{R})\to
PSL(n,\mathbb{R})=SL(n,\mathbb{R})$ the irreducible representation introduced
in 2.1. Since $\omega_{n}\circ\sigma$ is an orbifold Fuchsian representation,
it is irreducible. The following result implies that we can conjugate
$\omega_{n}\circ\sigma$ to obtain an integral representation

$\displaystyle\rho\colon\thinspace\pi_{1}(\mathcal{O}_{3,3,3,3})\to
SL(n,\mathbb{Z})<SL(n,\mathbb{R}).$ (6)

######  Proposition 22 ([12] thm. 2.1 ).

Let $\omega_{n}\colon\thinspace PSL(2,\mathbb{R})\to PSL(n,\mathbb{R})$ be the
unique irreducible representation between these groups. Then for every odd $n$
the restriction $\phi_{n}=\omega_{n}|_{\Delta(3,4,4)}$ is conjugate to a
representation $\rho_{n}\colon\thinspace\Delta(3,4,4)\to PSL(n,\mathbb{Z})$.

Now let $\gamma\subset\mathcal{O}_{3,3,3,3}$ be a simple closed loop dividing
$\mathcal{O}_{3,3,3,3}$ into two orbifolds $\mathcal{O}_{L}$ and
$\mathcal{O}_{R}$ which share $\gamma$ as their common boundary and have two
cone points of order 3 each. Then $[\gamma]\in\pi_{1}(\mathcal{O}_{3,3,3,3})$
is an infinite order element and
$\pi_{1}(\mathcal{O}_{3,3,3,3})\simeq\pi_{1}(\mathcal{O}_{L})\ast_{\langle[\gamma]\rangle}\pi_{1}(\mathcal{O}_{R}).$

######  Proposition 23.

Let
$\rho\colon\thinspace\pi_{1}(\mathcal{O}_{3,3,3,3})\simeq\pi_{1}(\mathcal{O}_{L})\ast_{\langle[\gamma]\rangle}\pi_{1}(\mathcal{O}_{R})\to
PSL(n,\mathbb{Z})$ be the representation defined in (6). Then the restrictions
of $\rho$ to $\pi_{1}(\mathcal{O}_{L})$ and $\pi_{1}(\mathcal{O}_{R})$ are
irreducible.

Proof. To see that $\rho|_{\pi_{1}(\mathcal{O}_{L})}$ is irreducible it
suffices to see that the restriction of $\omega_{n}\circ\sigma$ to
$\pi_{1}(\mathcal{O}_{L})$ is irreducible. By the proof of lemma 21 we have
that $\sigma(\pi_{1}(\mathcal{O}_{L}))$ is Zariski dense in
$PSL(2,\mathbb{R})$. To see that the representation
$\omega_{n}\colon\thinspace\sigma(\pi_{1}(\mathcal{O}_{L}))\to
PSL(n,\mathbb{R})$ is irreducible, it is enough to check that the Zariski
closure of its image is irreducible. This holds since
$\omega_{n}\colon\thinspace PSL(2,\mathbb{R})\to PSL(n,\mathbb{R})$ is an
irreducible representation and a morphism of algebraic groups, so
$\omega_{n}(PSL(2,\mathbb{R}))=\omega_{n}(\overline{\sigma(\pi_{1}(\mathcal{O}_{L})})\subseteq\overline{\omega_{n}\circ\sigma(\pi_{1}(\mathcal{O}_{L}))}$.

To see $\rho|_{\pi_{1}(\mathcal{O}_{R})}$ is irreducible it is enough to
notice that the proof of 21 also holds for $\pi_{1}(\mathcal{O}_{R})$ by using
the generators $\theta_{3}$ and $\theta_{3}$ instead of $\theta_{1}$ and
$\theta_{2}$.

$\Box$

Knowing that $\rho$ is an integral orbifold Fuchsian representation, the
previous proposition shows $\rho$ satisfies the assumptions of theorem 19.
Thus we obtain the following application of corollary 20.

######  Theorem 24.

For every surface $S$ finitely covering the orbifold $O_{3,3,3,3}$ and every
odd $n>1$ there exists a path of Hitchin representations
$\rho_{t}\colon\thinspace\pi_{1}(S)\to SL(n,\mathbb{R})$, so that

  1. (1)

$\rho_{0}(\pi_{1}(S))\subset SL(n,\mathbb{Z})$,

  2. (2)

$\rho_{t}$ is Zariski dense for every $t>0$ and

  3. (3)

$\rho_{t}(\pi_{1}(S))\subset SL(n,\mathbb{Q})$ for every $t\in\mathbb{Q}$.

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