2·1. 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 $R_i \;:\;\thinspace G^k \to G$ . 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 [ Reference Bradlow, Garca-Prad, Goldman and WienhardBGPGW07 , section 5·2].

In the following we will frequently use the representation

(1) \begin{equation} \tilde\omega_n \;:\;\thinspace SL(2,\mathbb{R}) \longrightarrow SL(n,\mathbb{R})\end{equation}

given by the action of $SL(2,\mathbb{R})$ on the vector space $\mathcal{P}$ of homogeneous polynomials in 2 variables of degree $n-1$ . It is 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 $\omega_n \;:\;\thinspace PSL(2,\mathbb{R}) \to PSL(n,\mathbb{R})$ which is also irreducible and unique up to conjugation.

2·2. 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 [ Reference GoldmanGol88 , theorem A] [ Reference HitchinHit92 , theorem 10·2]. 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 $\rho\;:\;\thinspace \pi_1(S) \to PSL(2,\mathbb{R})\equiv\text{Isom}^+(\mathbb{H}^{2} )$ .

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 [ Reference HitchinHit92 , theorem 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 $\text{Hit}(\pi_1(S),PSL(n,\mathbb{R}))$ .

In [ Reference LabourieLab06 ], 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.

2·3. 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 [ Reference ThurstonThu78 ] 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 $\omega_n(\mathcal{T}({\mathcal{O}}))$ [ Reference Alessandrini, Lee and SchaffhauserALS19 , definition 2·3] and prove $\text{Hit}(\pi_1({\mathcal{O}}),PGL(n,\mathbb{R}))$ is homeomorphic to an open ball [ Reference Alessandrini, Lee and SchaffhauserALS19 , theorem 1·2].

The definition of Anosov representations has been generalized by Guichard and Wienhard [ Reference Guichard and WienhardGW12 , definition 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 $PGL(n,\mathbb{R})$ [ Reference Alessandrini, Lee and SchaffhauserALS19 , proposition 2·26] and thus share some strong algebraic properties.

3·1. 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 $r\;:\;\thinspace \Gamma \to G$ is Zariski dense if $r(\Gamma)$ is Zariski dense in G.

The image of the irreducible representation $\omega_n\;:\;\thinspace PSL(2,\mathbb{R}) \to PSL(n,\mathbb{R})$ is contained, if n is even, in a conjugate of $PSp(n, \mathbb{R})$ , which is the projectivisation 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 [ Reference GuichardGui ] has announced a classification of Zariski closures of their lifts. An alternative proof of this result has been given recently by Sambarino [ Reference SambarinoSam20 , corallary 1·5] The version of this result we cite here comes from theorem 11·7 in [ Reference Bridgeman, Canary, Labourie and SambarinoBCLS15 ].

3·2. A criterion for Zariski density

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

Now consider a Hitchin representation $\rho\;:\;\thinspace \pi_1({\mathcal{O}}) \to PSL(n,\mathbb{R})$ . 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.

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

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

\begin{eqnarray*} \exp\!(kX)^T\Omega \exp\!(kX) = \Omega &\Longrightarrow& \Omega^{-1}\exp\!(kX)^T\Omega = \exp\!(kX)^{-1}\\[5pt] &\Longrightarrow& \exp\!(\Omega^{-1}(kX)^T\Omega) = \Omega^{-1}\exp\!(kX)^T\Omega =\exp\!(\!-kX).\end{eqnarray*}

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

\begin{eqnarray*} \Omega^{-1}(kX)^T\Omega = -kX &\Rightarrow& -\Omega(kX)^T\Omega = -kX\\[5pt] &\Rightarrow& \Omega(kX)^T\Omega = kX.\end{eqnarray*}

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 $PSL(n,\mathbb{R})$ .

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 Proposition 13 to get a criterion for Zariski density of surface Hitchin representations in the odd case.

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.

4·1. 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 $\rho \;:\;\thinspace \pi_1({\mathcal{O}}) \to G$ there is a standard way of “bending” $\rho$ by an element $\delta$ of the centraliser in G of $\rho(\pi_1({\mathcal{O}}_L\cap {\mathcal{O}}_R))$ to obtain a representation $\rho_\delta \;:\;\thinspace \pi_1({\mathcal{O}}) \simeq \pi_1({\mathcal{O}}_L)\ast_{\pi_1({\mathcal{O}}_L\cap{\mathcal{O}}_R)}\pi_1({\mathcal{O}}_R)\to G$ 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 [ Reference GoldmanGol87 , section 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 $\pi_1({\mathcal{O}}) \simeq \pi_1({\mathcal{O}}_L)\ast_{\langle [\gamma] \rangle}\pi_1({\mathcal{O}}_R).$

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

(2) \begin{equation} \delta_t = (t\rho([\gamma]) + I)\det(t\rho([\gamma]) + I)^{-\frac{1}{n}}.\end{equation}

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 $\rho_t \;:\;\thinspace \pi_1({\mathcal{O}}) \to SL(n,\mathbb{R})$ 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 $SL(n,\mathbb{Q})$ .

4·2. Discarding Zariski closures

For the rest of Section 4 we focus on the case where $n=2k+1$ is odd. Recall that in this case $SL(n,\mathbb{R}) \equiv PSL(n,\mathbb{R})$ .

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 $\rho(\gamma)J_1J_2^{-1} = J_1J_2^{-1}\rho(\gamma).$ Since n is odd, $J_1J_2^{-1}$ has a real eigenvalue $\lambda$ . Then $\text{Ker} (J_1J_2^{-1} - \lambda I)$ is a non-zero invariant subspace for the irreducible representation $\rho$ , which implies $J_1 = \lambda J_2$ .

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

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

having $J = \delta_tJ\delta_t^T$ 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 $n>2$ .

4·3. 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 $\rho \;:\;\thinspace \pi_1({\mathcal{O}}) \to G$ we will denote the restriction of $\rho$ to $\pi_1(S)$ by $\rho^S$ .

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 $t\in \mathbb{Q}$ .

5·1. 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 $\{{\pi}/{3}$ , ${\pi}/{4}$ , ${\pi}/{4}\}$ , then the generators of $\Delta(3,4,4)$ are the rotations x and y by ${2\pi}/{3}$ and ${\pi}/{2}$ around the corresponding vertices of T. This group has presentation

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

The fundamental domain for the action of $\Delta(3,4,4)$ on $\mathbb{H}^{2} $ is a quadrilateral with angles $\{{\pi}/{2},{\pi}/{3},{\pi}/{2},{\pi}/{3}\}$ . 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 (Figure 1). This defines, up to conjugation, an isomorphism $\pi_1(S^2(3,4,4))\to \Delta(3,4,4)\subset PSL(2,\mathbb{R})$ .

Fig. 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

(4) \begin{equation} \sigma \;:\;\thinspace \pi_1(\mathcal{O}_{3,3,3,3}) \longrightarrow \Delta(3,4,4) < PSL(2,\mathbb{R}).\end{equation}

It follows from the Borel density theorem that $\sigma$ is a Zariski dense representation.

5·2. 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 \;:\;\thinspace \pi_1(\mathcal{O}_{3,3,3,3}) \to SL(n,\mathbb{R}),$ where $\sigma$ is the representation defined in (4) and $\omega_n\;:\;\thinspace PSL(2,\mathbb{R}) \to PSL(n,\mathbb{R}) =SL(n,\mathbb{R})$ the irreducible representation introduced in Section 3·1. 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

(5) \begin{equation} \rho \;:\;\thinspace \pi_1(\mathcal{O}_{3,3,3,3}) \longrightarrow SL(n,\mathbb{Z}) < SL(n,\mathbb{R}).\end{equation}

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 $\pi_1({\mathcal{O}}_{3,3,3,3}) \simeq \pi_1({\mathcal{O}}_L)\ast_{\langle [\gamma] \rangle}\pi_1({\mathcal{O}}_R).$

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.