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

Abstract For odd n we construct a path 
$\rho\;:\;\thinspace \Pi_1(S) \to SL(n\mathbb{R})$
 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 
$t\in \mathbb{Q}$
 .


Introduction
Constructing Zariski dense surface subgroups in SL(n, R) has attracted attention as a step to finding thin groups, these are infinite index subgroups of a lattice in SL(n, 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 [BGS10,Sar14].
Though thin subgroups are in a sense generic [Fuc14,FR17], finding particular specimens of thin surface subgroups in a given lattice remains a difficult task.In this direction, in 2011 Long, Reid and Thistlethwaite [LRT11] produced the first infinite family of nonconjugate thin surface groups in SL(3, Z).Their approach relies on parametrising a family of representations ρ t of the triangle group (3, 3, 4) in the Hitchin component, so that for every t ∈ Z the subgroup ρ t ( (3, 3, 4)) is in SL(3, Q) and has integral traces.By results of Bass [Bas80] these two properties together with ρ t ( (3, 3, 4)) being non-solvable and finitely generated guarantee that it is conjugate to a subgroup of SL(3, Z).In 2018 Long and Thistlethwaite [LT18] used a similar approach to obtain an infinite family of non-conjugate Zariski dense surface subgroups in SL(4, Z) and SL(5, Z).
Ballas and Long [BL18] in turn used the idea of "bending" a representation of the fundamental group of a hyperbolic n-manifold π 1 (N) along an embedded totally geodesic and separating hypersurface to obtain thin groups in SL(n + 1, R) which are isomorphic to π 1 (N).The aim of this paper 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 O 3,3,3,3 and every odd n > 1 there exists a path of discrete, faithful and irreducible representations ρ t : π 1 (S) → SL(n, R), so that: (i) ρ 0 (π 1 (S)) < SL(n, Z); C The Author(s), 2024.Published by Cambridge University Press on behalf of Cambridge Philosophical Society.This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/),which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
CARMEN GALAZ GARCÍA (ii) ρ t is Zariski dense for every t > 0; and Every representation ρ t in Theorem 1 is a surface Hitchin representation.Several of its properties are derived from the seminal work of Labourie [Lab06] on Anosov representations, the classification of Zariski closures of surface Hitchin representations by Guichard [Gui], and the recent introduction of orbifold Hitchin representations by Alessandrini, Lee and Schaffhauser [ALS19].We provide an overview of these results in Sections 2 and 3.The discussion in these sections applies to all n, with the assumption of odd n coming into play later in Section 4. At the end of Section 3 we also prove the following criterion for Zariski density, which will be subsequently used to discard Zariski closures.PROPOSITION 2. Let ρ : π 1 (O) → PSL(n, R) be an orbifold Hitchin representation such that: In Section 4 we give a general construction to obtain a path of representations as in Theorem 1.This is based on bending the fundamental group π 1 (O) of a hyperbolic 2dimensional orbifold along a simple closed curve in O with infinite order as an element of π 1 (O).Theorem 1 then follows from applying the results in Section 2 to a suitable representation of the fundamental group of the orbifold 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 5.
Remark.During the finalisation of this project, Long and Thistlethwaite used bending to construct thin surface groups in SL(n, Z) for every odd n [LT20], the even case remains open.

Hitchin representations
In this section we give a short overview of surface and orbifold Hitchin representations.Recall that a subgroup H < GL(n, R) is irreducible if the only invariant subspaces for the action of H on R n are {0} and R n .A representation ρ: → GL(n, R) is said to be irreducible if the image subgroup ρ( ) is irreducible, and it is strongly irreducible if the restriction of ρ to every finite index subgroup is irreducible.These characteristics are defined similarly for projective representations ρ : → PGL(n, R)

2•1. Spaces of representations
Let G be a Lie group and let be a group with a finite presentation α 1 , . . ., α k | r 1 , . . ., r m .Then every relator r i defines a map R i : G k → G.If we let Hom( , G) = ∩ m i=1 R −1 i (Id), then the map φ → (φ(α 1 ), . . ., φ(α k )) is a bijection between the set of all group homomorphisms from to G and Hom( , G).We will regard Hom( , G) as having the subspace topology from G k .
Let Hom + ( , G) be the subset of representations in Hom( , G) which decompose as a direct sum of irreducible representations and let Rep + ( , G) = Hom + ( , G)/G be the quotient space by the conjugation action.With the quotient topology Rep + ( , G) has the structure of an algebraic variety [BGPGW07, section 5•2].
In the following we will frequently use the representation given by the action of SL(2, R) on the vector space P of homogeneous polynomials in 2 variables of degree n − 1.It is known that the representation ωn is absolutely irreducible and is, up to conjugation, the unique irreducible representation from SL(2, R) into SL(n, R).This representation induces a projective representation ω n : PSL(2, R) → PSL(n, R) which is also irreducible and unique up to conjugation.
The space Rep + (π 1 (S), PSL(n, R)) has three topological connected components if n is odd and 6 if n is even [Hit92, 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 R (1−n 2 )(1−g) .When n > 2 is even, both Hitchin components are related by an inner automorphism of PSL(n, R).In the odd case, where there is only one component, we will denote the Hitchin component by Hit(π 1 (S), PSL(n, R)).
Definition 5. Let S be a closed surface of genus greater than one.A representation r : In [Lab06], Labourie introduces Anosov representations and proves that surface Hitchin representations are B-Anosov where B is any Borel subgroup of PSL(n, R).This gives surface Hitchin representations essential algebraic properties, out of which we will use Theorem 7 below.
) is discrete, faithful and strongly irreducible.Moreover, the image of every non-trivial element of π 1 (S) under r is purely loxodromic.

2•3. Hitchin representations of orbifold groups
Let O be a 2-dimensional closed orbifold of negative orbifold Euler characteristic χ(O) and let π 1 (O) be its orbifold fundamental group.In [Thu78] Thurston proves there is a connected component of the representation space Rep(π 1 (O), PGL(2, R)) that parametrizes hyperbolic structures on O.This component is called the Teichmüller space of the orbifold O, we will denote it by T (O).As with surfaces, the orbifold Teichmüller space consists of conjugacy classes of discrete and faithful representations of π 1 (O) into PGL(2, R) ≡ Isom(H 2 ), which we will call Fuchsian representations too.More recently, Alessandrini, Lee, and Schaffhauser used the irreducible representation ω n to define the as the unique connected component in this representation space which contains the connected Fuchsian locus The definition of Anosov representations has been generalized by Guichard  ).An orbifold Hitchin representation r : π 1 (O) → PGL(n, R) is discrete, faithful and strongly irreducible.Moreover, the image of every infinite order element of π 1 (O) under r is purely loxodromic.

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.

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 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 The image of the irreducible representation . This implies that the images of Fuchsian representations are contained in (a conjugate of) PSp(n, R) or in SO(k, k + 1) and, in particular, they are not Zariski dense.More generally, for surface Hitchin representations Guichard [Gui] has announced a classification of Zariski closures of their lifts.An alternative proof of this result has been given recently by Sambarino [Sam20, corallary 1•5] The version of this result we cite here comes from theorem 11•7 in [BCLS15].

3•2. A criterion for Zariski density
Here we prove Proposition 2 which gives us a criterion to find Zariski dense Hitchin representations.
By the previous argument we may lift ρ t ([α]) to a path Ãt ∈ SL(n, R) such that Ã0 has n distinct positive eigenvalues.Since each eigenvalue of Ãt varies continuously and det Ãt = 0, all eigenvalues of Ãt are positive.Moreover, by Theorem 9 the absolute values of the eigenvalues of ρ t ([α]) are distinct.This in turn implies all the eigenvalues of Ãt are distinct.Therefore Ã1 ∈ SL(n, R) is a lift of ρ([α]) 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.
THEOREM 12 ([Cul66, theorem 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 ρ : π 1 (O) → PSL(n, R) with n even be an orbifold Hitchin representation so that ρ(π 1 (O)) is not conjugate to a subgroup of PSp(n, R).If S is a surface finitely covering O then ρ(π 1 (S)) is Zariski dense.
Let [α] ∈ π 1 (O) be an infinite order element.By Lemma 11 we can lift ρ([α]) ∈ PSL(n, R) to a matrix A ∈ SL(n, R) with n positive distinct eigenvalues.Since π 1 (S) has finite index in π 1 (O) there exists a k ∈ N such that ρ([α]) k ∈ ρ(π 1 (S)).Then A k is a lift of ρ([α]) k and A k ∈ Sp( ).Given that A has n positive distinct eigenvalues, by Theorem 12 there is a unique X ∈ M n×n (R) such that exp(X) = A. Then using that exp(kX) = A k preserves we get that exp(kX Applying Theorem 12 now to −1 exp(kX) T we obtain that This implies that kX ∈ sp( ) and thus A = exp(X) ∈ Sp( ).Given that A is a lift of ρ([α]), we have that ρ([α]) ∈ PSp( ).Since π 1 (O) is generated by its infinite order elements we get that ρ(π 1 (O)) ⊂ PSp( ), a contradiction.So it cannot be that ρ(π 1 (S)) is conjugate to a subgroup of PSp(n, R).In particular, if r is a lift of the Hitchin surface representation ρ| π 1 (S) then the Zariski closure of r(π 1 (S)) cannot be conjugate to a subgroup of Sp(n, R).By Theorem 10 it must be that the Zariski closure of r(π 1 (S)) is SL(n, R).Therefore the Zariski closure of ρ(π 1 (S)) is PSL(n, R).
In the case when n = 2k + 1 is odd, by Theorem 10 the Zariski closure of ρ(π 1 (S)) where ρ is a surface Hitchin representation is either conjugate to a subgroup of SO(k, k + 1) or equals SL(n, R).By assuming there exists a symmetric bilinear form J such that ρ(π 1 (S)) ⊂ 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.PROPOSITION 14.Let ρ : π 1 (O) → SL(n, R) with n odd be an orbifold Hitchin representation such that there is no real quadratic form J for which ρ(π 1 (O)) ⊂ SO(J).If S is a surface finitely covering O then ρ(π 1 (S)) is Zariski dense.
Given that any finite index subgroup of π 1 (O) contains a surface subgroup which has finite index in π 1 (O) we obtain the following result.PROPOSITION 15.Let ρ : π 1 (O) → PSL(n, R) be an orbifold Hitchin representation such that:

Bending representations of orbifold groups
Theorem 19 in this section gives a general construction of a path ρ t of Zariski dense Hitchin surface representations into SL(n, R) for odd n.By requiring that the initial representation ρ 0 has image inside SL(n, Q) we obtain Corollary 20, in which every representation ρ t with t ∈ Q also has image in SL(n, Q).
From now onwards we will consider the case where there is a simple closed curve γ ⊂ O, not parallel to a cone point, that divides O into two orbifolds O L and O R which share γ as their common boundary, so that

a representation for which ρ([γ ]) has n distinct positive eigenvalues. Then there exists a path of representations ρ
(iii) ρ t has image in SL(n, Q) for every t ∈ Q.

Proof. The matrix ρ([γ ]
) is conjugate to a diagonal matrix D with entries λ 1 , . . ., λ n > 0 along its diagonal.Now for every t > 0 define ( Notice that det (tρ Then each δ t is in SL(n, R) and we can check that δ t commutes with ρ([γ ]).Since ρ is a rational representation, whenever t ∈ Q the matrix tρ([γ ]) + I has rational entries and non-zero determinant. Let . Notice that ρ 0 = ρ and that for every t ∈ Q the representation ρ t has image in SL(n, 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, R) ≡ PSL(n, R).
LEMMA 17.Let ρ : → SL(n, R) be an irreducible representation and suppose there is a quadratic form J such that ρ( ) ⊂ SO(J).Then J is unique up to scaling.
Proof.By Proposition 16 there are δ t ∈ SL(n, R) that commute with ρ([γ ]), with which we can construct a path of representations t is also a multiple of J.This implies there is a λ ∈ R such that λJ = δ t Jδ T t and then λ n = det (δ t ) 2 = 1.Since n is odd it must be that λ = 1 and we obtain δ t ∈ SO(J).Given that ) is conjugate to a diagonal matrix D whose eigenvalues are all distinct.If μI = ρ([γ ]) −1 + ρ([γ ]) then by conjugating we would obtain that μI = D −1 + D, which is not the case given that n > 2.

4•3. Representations of surface groups
Recall we are assuming that 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 π 1 (S) is a finite index subgroup of π 1 (O).Given a representation ρ : π 1 (O) → G we will denote the restriction of ρ to π 1 (S) by ρ S .
an infinite order element.Let ρ : π 1 (O) → SL(n, R) be an orbifold Fuchsian representation such that the restrictions ρ| π 1 (O L ) and ρ| π 1 (O R ) are irreducible.If S is a surface finitely covering O then there exists a path of representations ρ S t : π 1 (S) → SL(n, R) such that ρ S 0 = ρ S and ρ S t is a Zariski dense surface Hitchin representation for each t > 0.
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 ρ(π 1 (O)) ⊂ SL(n, Q) to obtain that the image of every ρ t is in SL(n, Q) for every t ∈ Q.
COROLLARY 20.Let ρ : π 1 (O) → PSL(n, Q) be a representation satisfying the assumptions of Theorem 19.If S is a surface finitely covering O then there exists a path ρ S t : π 1 (S) → SL(n, R) of Hitchin representations such that ρ S 0 = ρ S , ρ S t is Zariski dense for each t > 0 and ρ S t has image in SL(n, Q) for every t ∈ Q.
Knowing that ρ is an integral orbifold Fuchsian representation, the previous proposition shows ρ satisfies the assumptions of Theorem 19.Thus we obtain the following application of Corollary 20.THEOREM 23.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 ρ t : π 1 (S) → SL(n, R), so that: and Wienhard [GW12, 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, R) [ALS19, proposition 2•26] and thus share some strong algebraic properties.THEOREM 9 ([ALS19, theorem 1•1]

4• 1 .
Bending representationsLet O be a 2-dimensional orientable connected closed orbifold of negative orbifold Euler characteristic and O L , O R be open connected suborbifolds with connected intersection O L ∩ O R .Given a representation ρ : π 1 (O) → G there is a standard way of "bending" ρ by an element δ of the centraliser in