This is the accepted version of the journal article: Porti, Joan. «Dimension of representation and character varieties for two- and three- orbifolds». Algebraic and Geometric Topology, Vol. 22, Issue 4 (2022), p. 1905-1967. DOI 10.2140/agt.2022.22.1905 This version is available at https://ddd.uab.cat/record/275429 under the terms of the license Dimension of representation and character varieties for two and three-orbifolds Joan Porti∗ February 9, 2023 Abstract We consider varieties of representations and characters of 2 and 3-dimensional orbifolds in semisimple Lie groups, and we focus on computing their dimension. For hyperbolic 3- orbifolds, we consider the component of the variety of characters that contains the holonomy composed with the principal representation, we show that its dimension equals half the di- mension of the variety of characters of the boundary. We also show that this is a lower bound for the dimension of generic components. We furthermore provide tools for computing di- mensions of varieties of characters of 2-orbifolds, including the Hitchin component. We apply this computation to the dimension growth of varieties of characters of some 3-dimensional manifolds in SL(n, C). 1 Introduction In this paper we are interested in varieties of representations and characters of the fundamental group of two and three-dimensional orbifolds O in complex semi-simple algebraic Lie groups G, denoted respectively by R(O, G) = hom(π1 (O), G) and X(O, G) = R(O, G)//G. Since those are not irreducible, we use dimρ R(O, G) and dim[ρ] X(O, G) to denote the dimension of a component that contains ρ. We discuss first a particular representation for oriented hyperbolic 3-orbifolds, with holon- omy in Isom+ (H3 ) ∼= PSL(2, C). Let G be an adjoint complex simple Lie group, eg. PSL(n, C), PO(n, C) or PSp(2m, C). Let τ : PSL(2, C) → G be the principal representation (see §2.2 for the definition), eg. Symn−1 : PSL(2, C) → PSL(n, C), whose image is contained in PSp(2m, C) for n = 2m or in PO(2m + 1, C) for n = 2m + 1. When GR is a split real form of G, it restricts to τ : PGL(2, R) → GR , so that the Hitchin component of a surface or a 2-orbifold is the component of the variety of representations that contains the composition of τ with a Fuchsian representation. Here we consider orientable hyperbolic 3-orbifolds and we compose the holonomy in PSL(2, C) with the principal representation τ . We prove the following generalization of [25, 26]: Theorem 1.1. Let O3 be a compact orientable 3-orbifold whose interior is hyperbolic with holonomy hol : π1 (O3 ) → PSL(2, C). Then the character of τ ◦hol is a smooth point of X(O3 , G) of dimension 1 dim[τ ◦hol] X(O3 , G) = dim X(∂O3 , G). 2 ∗ Partially supported by the Micinn/FEDER grant PGC2018-095998-B-I00 1 Furthermore, the map induced by restriction X(O3 , G) → X(∂O3 , G) is locally injective at the character of τ ◦ hol. The proof of Theorem 1.1 is based on a vanishing theorem of L2 -cohomology, as for manifolds in [25, 26]. Notice that for a Euclidean component O2 of ∂O3 , the restriction of τ ◦ hol can be a singular point of X(O2 , G), and to prove that X(O3 , G) → X(∂O3 , G) is locally injective we use evaluation of characters. A representation ρ : Γ → G is called good if it is irreducible and the centralizer of ρ(Γ) in G is the center Z(G) of G, in particular gρ(Γ) = 0. We are interested in representations of Euclidean 2-orbifolds (eg. χ(O2 ) = 0) that may be not good. For instance, nontrivial representations of a 2-torus are never good by Kolchin’s theorem. A representation of a Euclidean 2-orbifold ρ : π1 (O2 ) → G is called strongly regular if for a maximal torsion free subgroup Γ0 < π1 (O2 ), it holds: • dim gρ(Γ0 ) = rank G, and • the projection of ρ(Γ0 ) is contained in a connected abelian subgroup of G/Z(G). The following is a generalization of a theorem of Falbel and Guilloux for manifolds in [8]: Theorem 1.2. Let O3 be a compact orientable good 3-orbifold, with boundary ∂O3 = ∂1 O3 ⊔ · · · ⊔ ∂k O3 , and χ(∂i O3 ) ≤ 0 for i = l, . . . , k. Let ρ : π1 (O3 ) → G be a good representation such that: • If χ(∂i O3 ) < 0, then ρ|π1 (∂i O2 ) is good. • If χ(∂i O3 ) = 0, then ρ|π1 (∂i O2 ) is strongly regular. Then 1 dim[ρ] X(O3 , G) ≥ dim X(∂O3 , G). 2 If ∂O3 = ∂1 O3 ⊔ · · · ⊔ ∂k O3 denotes the decomposition in connected components, then X(∂O3 , G) = X(∂1 O3 , G) × · · · × X(∂k O3 , G) and dim X(∂O3 , G) = dim X(∂1 O3 , G) + · · · + dim X(∂k O3 , G). When the component ∂i O3 is a (closed orientable) surface S: ( −χ(S) dim G if χ(S) < 0, dim X(S, G) = 2 rank G if χ(S) = 0, which can be rewritten as dim X(S, G) = −χ(S) dim G + 2 dim gρ(π1 S) , where gH denotes the centralizer in the Lie algebra of a subset H ⊂ G, via the adjoint action. When the ∂i O3 is a (closed orientable) 2-orbifold O2 with branching locus Σ ⊂ O2 , we prove in Theorems 1.3 and 1.4: 2 X dim X(O2 , G) = −χ(O2 \ Σ) dim G + dim gρ(Stab(x)) + 2 dim gρ(π1 (O )) . (1) x∈Σ Formula (1) suggests that, instead of the usual Euler characteristic of an orbifold, we need another quantity to compute dimensions of varieties of characters and representations. For 2 that purpose, consider K a CW -structure on a compact orbifold On and ρ : π1 (On ) → G a representation. The twisted Euler characteristic is defined (Definition 3.1) as X e(On , Adρ) = χ (−1)dim e dim gAd(ρ(Stab(ẽ))) ∈ Z. e cell of K This is always an integer and should not be confused with the orbifold Euler characteristic, that is a rational number. For the trivial representation, this is just the Euler characteristic of the underlying space of the orbifold times dim G. It is the alternated sum of dimensions of cohomology groups of On twisted by Adρ (Proposition 3.2): X e(On , Adρ) = χ (−1)i dim H i (On , Adρ). i In Section 4 we prove the following results, based on Goldman’s work on surfaces [10] (see also [32]): Theorem 1.3. Let O2 be a compact connected 2-orbifold, with χ(O2 ) ≤ 0. Let ρ ∈ R(O2 , G) be a good representation. Then [ρ] is a smooth point of X(O2 , G) of dimension −e χ(O2 , Adρ). Theorem 1.4. Let O2 be a closed Euclidean 2-orbifold and ρ : π1 (O2 ) → G a strongly regular representation. Then it belongs to a single component of X(O2 , G) that has dimension 2 )) χ(O2 , Adρ) + 2 dim gρ(π1 (O (a) −e if O2 is orientable, f2 )) (b) −χ̃(O2 , Adρ) + dim gρ(π1 (O if O2 is non-orientable, where O f2 → O2 denotes the orienta- tion covering of O2 . Notice that Theorem 1.4 does not conclude smoothness. For instance, a parabolic represen- tation of Z2 in SL(2, C) is strongly regular but its character (the trivial character) is a singular point of X(Z2 , SL(2, C)) and X(Z2 , PSL(2, C)). Under the hypothesis of Theorem 1.3 or Theorem 1.4, when O2 is closed and orientable then e(O2 , Adρ) is even (Corollary 3.5) and therefore dim[ρ] X(O2 , G) is even. χ Most of the computations apply to real Lie groups, in particular we spend a section discussing applications to the Hitchin component, that we denote by Hit(O2 , GR ), where GR is the split real form of the adjoint group G. As we consider non orientable 2-orbifolds, we deal with non connected real forms GR : we require that GR contains the image by the principal representation of PGL(2, R), not only of its identity component PSL(2, R). For instance GR = PSL(n, R), PSp± (2m) or PO(n, n + 1). The dimension of Hit(O2 , GR ) has already been computed in [1] by Alessandrini, Lee and Schaffhauser, but we give a different approach, closer to the one of Long and Thistlethwaite in [23] for turnovers. For instance, we show that dim(Hit(O2 , PGL(n, R))) + χ(O2 ) dim(PGL(n, R)) ≤ C(O2 ), where C(O2 ) is a constant that depends only on O2 , Proposition 5.10. For PSp± (2m, R) the bound is not uniform but linear on m and we only have a weaker conclusion (Corollary 5.15), cf. [1, Proposition 5.15]: dim(Hit(O2 , PSp± (2m, R))) lim = −χ(O2 ). m→∞ dim(PSp± (2m, R)) We also show that Hit(O2 , GR ) maximizes the dimension among all components of the variety of (good) representations of O2 in GR , for GR = PGL(n, R), Proposition 5.8, or PSp± (2m, R), Proposition 5.16. For O2 closed and orientable, the differential form of Atiyah-Bott-Goldman gives naturally a symplectic structure on Hit(O2 , GR ). Furhermore (see Proposition 5.4): 3 Proposition 1.5. When O2 is closed and non-orientable, with orientation covering O f2 , then 2 2 Hit(O , GR ) embeds in Hit(O , GR ) as a Lagrangian submanifold. f We apply these computations on 2-orbifolds to estimate the growth of X(M 3 , SL(n, C)) with respect to n for some 3-manifolds, in particular for some knot exteriors. For instance we show: Proposition 1.6. Let Γ be the fundamental group of the exterior of the figure eight knot. Besides the canonical component (that has dimension n − 1), for large n X(Γ, SL(n, C)) has at least 3 components that contain irreducible representations, whose dimension grow respectively as n2 /12, n2 /20 and n2 /42. The paper is organized as follows. In Section 2 we recall some basic notions and tools on varieties of representations, and in Section 3 we introduce tools of orbifold cohomology (some well known, some other new) that we use later to compute Zariski tangent spaces to varieties of representations and characters. In Section 4 we discuss varieties of representations of two dimensional orbifolds, and we apply some of the ideas to compute dimensions of Hitchin components of orbifolds in Section 5. In Section 6 we prove the results on three-dimensional orbifolds. Finally, some explicit examples are computed in Section 7. 2 Varieties of representations and characters Let Γ be a finitely generated group, we are mainly interested in the fundamental group of a compact orbifold. Let G be a complex semi-simple algebraic Lie group. The variety of representations R(Γ, G) = hom(Γ, G) is an affine algebraic set and its quotient by conjugation in the algebraic category is the variety of characters X(Γ, G) = R(Γ, G)//G. Namely, the algebra of invariant functions C[R(Γ, G)]G is of finite type and X(Γ, G) can be defined as the affine scheme with this function ring: C[X(Γ, G)] ∼= C[R(Γ, G)]G . When Γ = π1 (O), they are denoted respectively by R(O, G) and X(O, G). Following for instance [19], we define: Definition 2.1. A representation ρ : Γ → G is: (a) irreducible if ρ(Γ) is not contained in a proper parabolic subgroup of G, and (b) good if it is irreducible and the centralizer of the image equals the center of G, Z(G). When G = SL(n, C), irreducible and good are equivalent. Let Rgood (Γ, G) denote the subset of good representations in R(Γ, G). The set of conjugacy classes Rgood (Γ, G) = Rgood (Γ, G)/G is a Zariski open subset of X(Γ, G), cf. [19]. We discuss also representations in real Lie groups. For GR , the variety of characters is more subtle, cf. [5], but we just consider Rgood (Γ, GR ) = Rgood (Γ, GR )/GR . For a compact 2 orbifold O2 , we may also consider the relative character variety, defined as Rgood (O2 , ∂O2 , G) = {[ρ] ∈ Rgood (O2 , G) | [ρ|∂1 O2 ], . . . , [ρ|∂k O2 ] are constant}, where ∂O2 = ∂1 O2 ⊔ · · · ⊔ ∂k O2 is the decomposition in connected components and [ρ|∂i O2 ] denotes the conjugacy class of the restriction to the i-th boundary component. 4 2.1 Zarisi tangent space and cohomology For a representation ρ : Γ → G, its composition with the adjoint representation is denoted by Adρ : Γ → Aut(g). The adjoint representation preserves the Killing form B : g × g → C, that is non-degenerate, as we assume G semi-simple. A derivation or crossed morphism is a C-linear map d : Γ → g that satisfies d(γ1 γ2 ) = d(γ1 ) + Adρ(γ1 ) (d(γ2 )), ∀γ1 , γ2 ∈ Γ. The space of crossed morphisms is denoted by Z 1 (Γ, Adρ). A crossed morphism is called inner if there exists a ∈ g such that d(γ) = (Adρ(γ) − 1)(a), ∀γ ∈ Γ, and the subspace of inner morphism is denoted by B 1 (Γ, Adρ). The quotient is the first coho- mology group: H 1 (Γ, gAdρ ) = H 1 (Γ, Adρ) ∼ = Z 1 (Γ, Adρ)/B 1 (Γ, Adρ). Theorem 2.2 (Weil). Let TρZar R(Γ, G) be the Zariski tangent space of R(Γ, G) as a scheme at ρ. There is a natural isomorphism Z 1 (Γ, Adρ) ∼ = TρZar R(Γ, G) that maps B 1 (Γ, Adρ) to the tangent space to the orbit by conjugation. Furthermore, when ρ is good, H 1 (Γ, Adρ) ∼ = TρZar X(Γ, G). We do not need the precise definition of scheme, just mention that the polynomial ideal that defines R(Γ, G) may be nonreduced, and this is taken into account in the Zariski tangent space. In particular, when the dimension of the Zariski tangent space at some representation or character equals the dimension of the component, the variety is smooth (and the scheme is reduced and smooth) at this representation or character. For a discussion of Theorem 2.2, see for instance [17] and the references therein. There is a real version of this theorem: when the image of a good representations is contained in a real form GR , then H 1 (Γ, AdρR ) ∼ = TρZar Rgood (Γ, GR ). When O2 is orientable, a representation ρ : π1 (O2 ) → G is called ∂-regular if for each γ generator of a peripheral subgroup, ρ(γ) is a regular element (where regular means dim gρ(γ) = rank G). For non-orientable O2 , we call a representation ∂-regular if its restriction to the orientation covering is ∂-regular. Proposition 2.3 ([17]). Assume that O2 is orientable and that ρ : π1 (O2 ) → G is good and ∂-regular. Then k TρZar Rgood (O2 , ∂O2 , G) ∼ M = ker H 1 (π1 (O2 ), Adρ) → H 1 (π1 (∂i O2 ), Adρ) .  i=1 5 2.2 The principal representation Let G be a simple complex Lie group, with Lie algebra g. An element in g is called regular if its centralizer has minimal dimension, that is the rank of g. Given a regular nilpotent element, Jacobson-Morozov theorem provides a representation of Lie algebras τ : sl(2, C) → g whose image contains the given regular nilpotent element. Such a representation is unique up to conjugacy. Assume that G is an adjoint group, namely that has trivial center. Then the induced representation of Lie groups τ : PSL(2, C) → G is called the principal representation. For G = PSL(n, C) the principal representation is Symn−1 . When defined from SL(2, C) to SL(n, C), Symn−1 preserves a bilinear form of Cn−1 , that is symmetric for n odd and skew- symmetric for n even, and its restriction yields the principal representations in PSp(2m, C) and PO(2m + 1, C). The principal representation restricts to τ : PGL(2, R) → GR , for GR the split real form of G, perhaps not connected. The Hitchin component is the connected component of Rgood (π1 (O2 ), GR ) that contains the composition of τ with the holonomy representation of any Fuchsian structure on O2 . We require that the real form contains the image of PGL(2, R) and therefore may be not connected. For instance PGL(n, R) contains two components, according to the sign of the determinant. We also use the notation Sp± (2m) = {A ∈ SL(2m, R) | At JA = ±J} for J an antisymmetric, non-degenerate, bilinear form; the sign + or − corresponds to one or the other component of the group. Remark 2.4. The principal representation is irreducible, and its image consists of regular elements (and the trivial element). For the principal representation τ : PSL(2, C) → G we have a decomposition r M Ad ◦ τ = Sym2di , (2) i=1 where r = rank G. The d1 , . . . , dr ∈ N are denoted the exponents of G. In particular, r X dim G = (2di + 1). (3) i=1 For instance, the exponents of PSL(n, C) are 1, . . . , n − 1, because n−1 M Ad ◦ Symn−1 = Sym2i . i=1 The exponents of simple Lie algebras are detailed in Tables 1 and 2, cf. [1]. The algebra of polynomial functions on the Cartan subalgebra invariant by the Weyl group C[h]W is freely generated by r homogeneous polynomials. By a theorem of Kostant, the degree of these polynomials is d1 + 1, . . . , dr + 1. 6 Lie algebra Exponents Rank Dimension sl(n, C) 1, 2, . . . , n − 1 n−1 n2 − 1 sp(2m, C) 1, 3, 5, . . . , 2m − 1 m 2m2 + m so(2m + 1, C) 1, 3, 5, . . . , 2m − 1 m 2m2 + m so(2m, C) 1, 3, 5, . . . , 2m − 3, m − 1 m 2m2 − m Table 1: Exponents of simple classical Lie algebras Lie algebra Exponents Rank Dimension g2 1, 5 2 14 f4 1, 5, 7, 11 4 52 e6 1, 4, 5, 7, 8, 11 6 78 e7 1, 5, 7, 9, 11, 13, 17 7 133 e8 1, 7, 11, 13, 17, 19, 23, 29 8 248 Table 2: Exponents of exceptional Lie algebras 3 Orbifold cohomology Group cohomology is useful to study dimensions, by Weil’s theorem (Theorem 2.2). In this section we discuss a combinatorial approach to cohomology. We are interested in representations in semi-simple complex algebraic groups, but this section applies also to their real forms, by replacing complex dimension by real dimension. We focus in homology and cohomology with coefficients twisted by the adjoint of a representation, but the constructions and results can be easily adapted to coefficients twisted by other representations. We recall some basic definitions of orbifolds [2, 31]. An orbifold O is called: • good if it has an orbifold cover that is a manifold. • very good it has an orbifold cover of finite index that is a manifold. • aspherical if its universal covering is a contractible manifold. • An orbifold is called hyperbolic/Euclidean/spherical if it is the quotient of hyperbolic space/Euclidean space/the round sphere by a discrete group of isometries. In a cell decomposition K of an orbifold, we require that isotropy groups or stabilizers are unchanged along cells. For a cell e in K this isotropy group is denoted by Stab(e). The orbifold Euler characteristic for a compact orbifold is defined as X 1 χ(O) = (−1)dim e ∈ Q. (4) |Stab(e)| e cell of K This is well behaved under coverings: if O′ → O is an orbifold covering of finite index k, then χ(O′ ) = kχ(O). See [31, 2] for instance. 3.1 Combinatorial homology and cohomology for orbifolds Let On be a compact n-dimensional good orbifold, possibly not orientable. Fix a CW-complex structure K on On . This means that K is a CW-complex with the same underlying space as On , 7 so that the branching locus of On is a subcomplex of K. In particular, it lifts to a CW-complex structure Ke of the universal covering Oen of On , so that each cell of K lifts to a disjoint union of homeomorphic cells of K (perhaps with nontrivial stabilizer). e The complex of chains on the universal covering is the free Z-module on the cells of K, e equipped with the usual boundary operator, and it is denoted by C∗ (K, Z). It has a (non- e free) action of Γ = π1 (On ) induced by deck transformations. The twisted chain and cochain complexes are defined as: C∗ (K, Adρ) = g ⊗Γ C(K, e Z), (5) C ∗ (K, Adρ) = homΓ (C(K, e Z), g). (6) The group Γ acts on g via Adρ on the left, and for the tensor product Γ acts on g on the right using inverses. Those are complexes and cocomplexes of finite-dimensional vector spaces, and the corresponding homology and cohomology groups are denoted respectively by H∗ (K, Adρ) and H ∗ (K, Adρ). 3.2 Twisted orbifold Euler characteristic For a compact manifold M and a representation ρ : π1 (M ) → G, as π1 (M ) acts freely on the universal covering M f, we have X X χ(M ) dim(G) = (−1)i dim Hi (M, Adρ) = (−1)i dim H i (M, Adρ). i i To have a similar formula for an orbifold O we need to take into account that Γ = π1 (O) acts non freely on O;e this motivates the definition of twisted orbifold Euler characteristic below, Definition 3.1. Let O be compact oriented orbifold, very good, with a CW-structure K. Given a subset Γ0 ⊂ Γ = π1 (O), the space of invariants is gρ(Γ0 ) = {v ∈ g | Adρ(g) (v) = v, ∀g ∈ Γ0 }, (7) and the quotient of coinvariants, gρ(Γ0 ) = g/⟨Adρ(g) (v) − v | v ∈ g, g ∈ Γ0 ⟩. (8) As the Killing form on g is nondegenerate (G is semisimple) and Ad-invariant, ⟨Adρ(g) (v) − v | v ∈ g, g ∈ Γ0 ⟩ = (gρ(Γ0 ) )⊥ . (9) Therefore dim(gρ(Γ0 ) ) = dim(gρ(Γ0 ) ). (10) Definition 3.1. The orbifold Euler characteristic of O twisted by Adρ is X χ e(O, Adρ) = (−1)dim e dim gρ(Stab(ẽ)) ∈ Z. e cell of K Here ẽ denotes any lift of e to the universal covering of O and Stab(ẽ) ⊂ Γ its stabilizer, whose conjugacy class in Γ is independent of the choice of the lift. It should not be confused with the usual orbifold Euler characteristic χ(O), recalled in (4), that is a rational number, whilst χe(O, Adρ) is always an integer. Notice also that χ e is not multi- plicative under coverings either; its intended to be a tool to compute dimensions of cohomology groups: 8 Proposition 3.2. For O and ρ as above: X X χ e(O, Adρ) = (−1)i dim Hi (O, Adρ) = (−1)i dim H i (O, Adρ). i i Proof. We compute the dimension of C∗ (K, Adρ) and C ∗ (K, Adρ) as C-vector spaces. We aim to show that, for 0 ≤ i ≤ dim O, ki X i dim Ci (K, Adρ) = dim C i (K, Adρ) = dim gρ(Stab(ẽj )) (11) j=1 where {ei1 , . . . , eiki } are the i-cells of K and {ẽi1 , . . . , ẽiki } a choice of lifts in the universal covering. Equation (11) will imply that X X χe(O, Adρ) = (−1)i dim Ci (K, Adρ) = (−1)i dim C i (K, Adρ) (12) i i and then the proposition will follow from standard arguments in homological algebra. To prove (11), use the decomposition as Z[Γ]-module of the chain complex on K:e ki M Ci (K, e Z) = Z[Γ]ẽij , (13) j=1 because the Γ-orbits of the lifts {ẽi1 , . . . , ẽiki } give a partition of the i-cells of K. e Then we apply the natural isomorphisms of C-vector spaces, for each cell ẽ of K: e homΓ (Z[Γ]ẽ, g) ∼ = gρ(Stab(ẽ) g ⊗Γ Z[Γ]ẽ ∼ = gρ(Stab(ẽ)) (14) θ → 7 θ(ẽ) v ⊗ ẽ → 7 v whose proof is straightforward and use Equality (10). For 2-dimensional orientable orbifolds, Proposition 3.2 is essentially a formula computed by André Weil in [32, Sections 6 and 7]. 3.3 Regular Coverings Let O0 → O be a finite regular covering. Namely, Γ0 = π1 (O0 ) is a normal subgroup of Γ = π1 (O) of finite index. Though O is very good, we do not require O0 to be a manifold in this subsection (for instance, O0 can be the orientation covering). The Galois group, or group of deck transformations, of the covering is Γ/Γ0 . It acts naturally on the chain and cochain complexes, C∗ (K0 , Adρ) and C ∗ (K0 , Adρ). Namely, any γ ∈ Γ maps a chain m ⊗ c ∈ C∗ (K0 , Adρ) to m · γ −1 ⊗ γc = Adρ(γ)(m) ⊗ γc, cf. [27]. The action of Γ0 is trivial by construction and therefore we have an action of Γ/Γ0 . Similarly, γ ∈ Γ maps a cochain θ ∈ C ∗ (K0 , Adρ) to Ad(ρ(γ)) ◦ θ ◦ γ −1 , which again induces an action of the Galois group Γ/Γ0 . The subspace of elements fixed by this action in homology and cohomology are denoted respectively by H∗ (O0 , Adρ)Γ/Γ0 and H ∗ (O0 , Adρ)Γ/Γ0 . 9 Proposition 3.3. The map π : O0 → O induces an epimorphism π∗ : H∗ (O0 , Adρ) ↠ H∗ (O, Adρ) that restricts to an isomorphism H∗ (O0 , Adρ)Γ/Γ0 ∼ = H∗ (O, Adρ). Similarly, it induces a monomorphism π ∗ : H ∗ (O, Adρ) ,→ H ∗ (O0 , Adρ) that yields an isomor- phism H ∗ (O, Adρ) ∼ = H ∗ (O0 , Adρ)Γ/Γ0 . Proof. The argument is standard and we just outline it. The proof is based in constructing a section to the maps induced by the projection π. The section at the chain and cochain level consists in taking an element of the inverse image of the map induced by π and averaging by the action of Γ/Γ0 . One can check that the section for chains is a well defined chain morphism, induces sections in homology and has the required property, as well as for sections of cochains and cohomology. The following proposition summarizes the main properties we need about orbifold cohomol- ogy: Proposition 3.4. Let O be a compact very good orbifold and ρ a representation of Γ = π1 (O) in G. The following hold: (a) H i (O, Adρ) and Hi (O, Adρ) are dual, via the Kronecker pairing. (b) If O is orientable, then the cup product induces a perfect pairing H i (O, Adρ) × H dim O−i (O, ∂O, Adρ) → C. (c) There is a natural isomorphism H 1 (Γ, Adρ) ∼ = H 1 (O, Adρ). (d) If O is aspherical (its universal covering is a contractible manifold), then there is a natural isomorphism H ∗ (Γ, Adρ) ∼ = H ∗ (O, Adρ). Proof. For assertion (a), we use the Kronecker pairing between chains and cochains via the Killing form B: Ci (K, Adρ) × C i (K, Adρ) → C (m ⊗ c, θ) 7→ B(m, θ(c)) One checks that it is well defined and it is not degenerate at the level of chains, using (13), (14), and (9). Hence it induces a perfect pairing between homology and cohomology. For (b), using Proposition 3.3, the strategy is to use the invariance by Γ/Γ0 of the corre- sponding properties for a finite regular covering O0 → O that is a manifold. More precisely, using that O0 is a manifold, the cup product defines a perfect pairing H i (O0 , Adρ) × H dim O0 −i (O0 , ∂O0 , Adρ) → H dim O0 (O0 , ∂O0 , C) ∼ = C. This is compatible with the action of Γ/Γ0 , therefore the assertion follows from Proposition 3.3. Notice that orientability of O is relevant for saying that the action of the Galois group Γ/Γ0 on H dim O0 (O0 , ∂O0 , C) ∼ = C is trivial. Next we prove (c). Chose X a K(Γ, 1), namely a CW-complex with π1 (X) ∼ = Γ and trivial higher homotopy groups. Thus there is a natural isomorphism H (X, Adρ) ∼∗ = H ∗ (Γ, Adρ). Furthermore, the covering corresponding to Γ0 < Γ, X0 → X, is a K(Γ0 , 1); in particular 10 H ∗ (X0 , Adρ) ∼ = H ∗ (Γ0 , Adρ). As O0 is a manifold, a K(Γ0 , 1) can be constructed from the 2-skeleton of O0 by adding cells of dimension ≥ 3, therefore: H 1 (O0 , Adρ) ∼ = H 1 (X0 , Adρ). Thus, by Proposition 3.3 we have natural isomorphisms H 1 (O, Adρ) ∼ = H 1 (O0 , Adρ)Γ/Γ0 ∼ = H 1 (X0 , Adρ)Γ/Γ0 ∼ = H 1 (X, Adρ), and H 1 (X, Adρ) ∼= H 1 (Γ, Adρ) because X is a K(Γ, 1). Finally, we prove (d). In the proof of (c), as O0 is already a K(Γ0 , 1), and therefore homotopy equivalent to X0 , hence H ∗ (O0 , Adρ) ∼ = H ∗ (X0 , Adρ). and the conclusion holds for any degree in cohomology. From the duality in Proposition 3.4 (b) we get: Corollary 3.5. When On is closed and orientable, of dimension n, for n odd χ e(On , Adρ) = 0, e(On , Adρ) is also even. and for n even χ 3.4 Non orientable orbifolds For a connected non-orientable orbifold On , there exists a unique orientable covering O fn → On of index 2, called the orientation covering. Lemma 3.6. Let On be a compact, non-orientable, very good orbifold of dimension n with fn → On . Then orientation covering O dim H i (On , Adρ) + dim H n−i (On , ∂On , Adρ) = dim H i (O fn , Adρ). Proof. The nontrivial deck transformation of the orientation covering is an involution denoted fn . Let σ ∗ be the induced morphism in the cohomology group. By Proposition 3.3: fn → O by σ : O ∗ ∗ H i (On , Adρ) ∼ = H i (O fn , Adρ)σ and H i (On , ∂On , Adρ) ∼ = H i (O fn , Adρ)σ . fn , ∂ O (15) The cup product induces a non-degenerate pairing, Proposition 3.4: H i (O fn , Adρ) × H n−i (O fn , Adρ) → H n (O fn , ∂ O fn , C) ∼ fn , ∂ O = C. (16) Chose C-basis for H i (O fn , Adρ) and for H n−i (O fn , ∂ O fn , Adρ), and use them to get matrices; let • J denote the matrix of the pairing (16), • S i denote the matrix of σ ∗ on H i (O fn , Adρ), and • T j denote the matrix of σ ∗ on H j (O fn , ∂ O fn , Adρ). Since σ reverses the orientation, σ ∗ acts as minus the identity on H n (O fn , ∂ O fn , C): (S i )t JT n−i = −J. As the pairing is non-degenerate, J is invertible, and since σ is an involution: (S i )t = −JT n−i J −1 . 11 Thus dim ker(T n−i − Id) = dim ker((S i )t + Id) = dim ker(S i + Id). (17) Since (S i )2 = Id: H i (O fn , Adρ) = ker(S i + Id) ⊕ ker(S i − Id). (18) Combining (17) and (18): dim ker(T n−i − Id) + dim ker(S i − Id) = dim H i (O fn , Adρ). With (15) this concludes the proof. Corollary 3.7. If On is non-orientable, very good, closed and of even dimension n, then n 1 n dim H 2 (On , Adρ) = fn , Adρ). dim H 2 (O 2 Remark 3.8. The pairing in the proof of Lemma  3.6 induces a nondegenerate pairing between i n i n the kernels ker H (O , Adρ) → H (∂ O , Adρ) and ker H (O f f n−i fn , Adρ) → H n−i (∂ Ofn , Adρ) [22, 17]. Thus the same argument in the proof of Lemma 3.6 yields dim ker H i (On , Adρ) → H i (∂On , Adρ)  + dim ker H n−i (On , Adρ) → H n−i (∂On , Adρ)  = dim ker H i (O fn , Adρ) → H i (∂ O  fn , Adρ) . 4 Varieties of representations and characters of 2-orbifolds We recall the possible stabilizers (or isotropy groups) of a point in a 2-orbifold, so that we fix notation. Besides the trivial one, the possible stabilizers of a point x ∈ O2 are: • Stab(x) ∼= Ck is a cyclic group of rotations of the plane R2 of order k. Then the singular point is isolated and its is called a cone point. Those are the unique possible non-trivial stabilizers for an orientable 2-orbifold (Figure 1 left). • Stab(x) ∼ = C2 is a group of order two generated by a reflection of the plane or the semiplane, ie. modelled in the interior or in the boundary. This is called a mirror point. The singular locus is locally an open segment, or a proper half segment in the boundary (Figure 1, both pictures in the center). • Stab(x) ∼= D2k is a dihedral group of 2k elements, generated by two reflections at the plane of angle π/k. The point is called a corner reflector (Figure 1 right). x x x x Figure 1: From left to right, a cone point, an interior mirror point, a mirror point in the boundary, and a corner reflector. Mirror points are represnetes by a double line. The boundary of a compact 2-orbifold is a union of closed 1-orbifolds. There are two possibilities for a closed 1-orbifold, up to homeomorphism: 12 • A circle S 1 (Figure 2 left). Its fundamental group is Z. • An interval with mirror end-points [[0, 1]] (Figure 2 right). It is the quotient of S 1 by an orientation reversing involution. Its fundamental group is dihedral infinite D∞ ∼ = C2 ∗ C2 , the free product of the stabilizers of the mirror points. It is called full 1-orbifolds in [1]. Figure 2: A circle S 1 and an interval with mirror end-points [[0, 1]] as boundaries. In dimension two, good and very good are equivalent. Furthermore, a closed 2-orbifold is good if and only if it is hyperbolic, Euclidean or spherical. When χ(O2 ) ≤ 0, O2 is very good and either hyperbolic or Euclidean. In particular a closed 2-orbifold O2 is aspherical iff χ(O2 ) ≤ 0. 4.1 Varieties of representations of 2-orbifolds In this subsection we give a couple of results on the tangent space of the variety of represen- tations and characters of a 2-orbifold. Those are orbifold versions of theorems of Goldman for surfaces [10]. In the orientable case, those results go back to André Weil [32, Sections 6 and 7]. We start with the following formulas: Proposition 4.1. Let O2 be a closed aspherical 2-orbifold. Set Γ = π1 (O2 ) and let ρ ∈ R(Γ, G). a) If O2 is orientable, then dim TρZar R(Γ, G) = −e χ(O2 , Adρ) + dim G + dim gρ(Γ) . b) If O2 is non-orientable, with orientation covering O f2 , and Γ f2 ), then e = π1 (O dim TρZar R(Γ, G) = −e χ(O2 , Adρ) + dim G + dim gρ(Γ) − dim gρ(Γ) . e Proof. Using TρZar R(Γ, G) ∼ = Z 1 (Γ, Adρ) = dim H 1 (Γ, Adρ) + dim B 1 (Γ, Adρ), the isomorphism H ∗ (O2 , Adρ) ∼ = H ∗ (Γ, Adρ) (as O2 is aspherical we apply Proposition 3.4 (d)), and: dim H 1 (O2 , Adρ) = −e χ(O2 , Adρ) + dim H 0 (O2 , Adρ) + dim H 2 (O2 , Adρ), dim B 1 (Γ, Adρ) = dim g − dim gρ(Γ) = dim G − dim H 0 (Γ, Adρ). we get dim TρZar R(Γ, G) = −e χ(O2 , Adρ) + dim G + dim H 2 (O2 , Adρ). In the orientable case, by duality (Proposition 3.4): dim H 2 (O2 , Adρ) = dim H 0 (O2 , Adρ) = dim gρ(Γ) . In the non-orientable case, Lemma 3.6 yields dim H 2 (O2 , Adρ) = dim H 2 (O f2 , Adρ) − dim H 0 (O2 , Adρ) = dim gρ(Γ) − dim gρ(Γ) , e and the proposition follows. 13 Proposition 4.2. Let O2 be a compact aspherical 2-orbifold and ρ : Γ → G a good representa- tion. When O2 is non-orientable, assume furthermore that the restriction of ρ to the orientation covering is also good. Then the conjugacy class [ρ] is a smooth point of X(O2 , G) of dimension −e χ(O2 , Adρ). Proof. As ρ is good, its centralizer is finite, hence dim H 0 (O2 , Adρ) = dim H 0 (Γ, Adρ) = dim gρ(Γ) = 0. When O2 is closed then H 2 (O2 , Adρ) = 0 (by duality in the orientable case, Proposition 3.4, or by Lemma 3.6 in the non-orientable case). When O2 is not closed, then it has virtually the homotopy type of a 1-complex and also H 2 (O2 , Adρ) = 0. Moreover, as O2 is aspherical, by Proposition 3.4 H 2 (Γ, Adρ) = 0. By Goldman’s obstruc- tion theory [10], this implies that [ρ] is a smooth point of local dimension dim H 1 (Γ, Adρ) = dim H 1 (O, Adρ) = −e χ(O, Adρ). From Proposition 4.2 and Corollary 3.7: Corollary 4.3. Let O2 be a closed non-orientable 2-orbifold with χ(O2 ) ≤ 0 and ρ : Γ → G. f2 is good. Then Assume that the restriction of ρ to the orientation covering O 1 dim[ρ] X(O2 , G) = f2 , G). dim[ρ] X(O 2 Proposition 4.4. Let O2 be a compact aspherical 2-orbifold and ρ : Γ → G a good representa- tion that is ∂-regular. Assume that O2 is orientable. Then the cup product defines a C-valued symplectic structure on Rgood (O2 , ∂O2 , G). Furthermore, it is real valued for GR , a real form of G. Proof. From the construction, non-degeneracy and skew symmetry is clear. The non-trivial issue is to check that this differental form is closed. For manifolds with boundary, closedness is due to [13], as explained in [17]. For an orientable orbifold with boundary, we reduce to the manifold case. Let ΣO2 denote its branching locus, it is a finite union of cone points. The restriction map Rgood (O2 , ∂O2 , G) → Rgood (O2 \ N (ΣO2 ), ∂(O2 \ N (ΣO2 )), G) is a (local) isomorphism, as the conjugacy classes of finite order elements cannot be deformed. To check the isomorphism at the level of tangent spaces, notice that the map induced by inclusion H 1 (O2 , Adρ) → H 1 (O2 \ N (ΣO2 ), Adρ) yields an isomorphism ker H 1 (O2 , Adρ) → H 1 (∂O2 , Adρ)  ∼ = ker H 1 (O2 \ N (ΣO2 ), Adρ) → H 1 (∂(O2 \ N (ΣO2 ), Adρ) ,  that can be proved for instance by using Mayer-Vietoris exact sequence. Furthermore, by naturality, it can be shown that the isomorphism maps the cup product to the cup product. Proposition 4.5. Let O2 be a closed non-orientable 2-orbifold with χ(O2 ) ≤ 0. Let ρ : Γ → G f2 is good and ∂-regular. Then be a representation whose restriction to the orientation covering O the restriction map induces an immersion around [ρ] of Rgood (O2 , ∂O2 , G) in Rgood (O f2 , ∂ O f2 , G) as a Lagrangian submanifold. 14 f2 → O Proof. Let σ : O f2 /σ = O2 . By Proposi- f2 be the orienting reversing involution so that O tions 3.3 and 4.2 Zar T[ρ] R(O2 , G) ∼ Zar = T[ρ] f2 , G)σ R(O As σ acts as minus the identity on H 2 (O f2 , C) and by construction of the bilinear form (see the Zar R(O 2 , G) proof of Lemma 3.6), the bilinear form must be trivial on the space fixed by σ, ie T[ρ] is an isotropic subspace. By Corollary 4.3, it has the dimension to be Lagrangian. 4.2 Orbifolds with boundary For an orientable 2-orbifold O2 a representation ρ is called ∂-regular if for each γ generator of a peripheral subgroup, ρ(γ) is a regular element (i.e. dim gρ(γ) = rank G). For a non-orientable 2-orbifold, we will consider ∂-regularity on the orientation covering. We start with a result on the dimension of the relative variety of (conjugacy classes of) representations Rgood (O2 , ∂O2 , G): Proposition 4.6. Let O2 be a compact 2-orbifold with boundary and χ(O2 ) ≤ 0. Assume ρ : Γ → G is good and ∂-regular. When O2 is non-orientable, assume that the restriction to the orientation covering is good and ∂-regular. Then [ρ] is a smooth point of Rgood (O2 , ∂O2 , G) of dimension b 1 −eχ(O2 , Adρ) − (c + ) rankG + χ e(∂O2 , Adρ), 2 2 where c is the number of components of ∂O2 that are circles and b the number of components of ∂O2 that are intervals with mirror points [[0, 1]]. In this proposition, the components of the boundary that are circles do not contribute to e(∂O2 , Adρ). χ Proof. The same proof as in [17, Proposition 2.10] applies here, in particular we have smoothness and the following equalities dim(Rgood (O2 , ∂O2 , G)) = dim ker H 1 (O2 , Adρ) → H 1 (∂O2 , Adρ)  = dim H 1 (O2 , Adρ) − dim H 1 (∂O2 , Adρ). As H 0 (O2 , Adρ) = H 2 (O2 , Adρ) = 0, dim H 1 (O2 , Adρ) = −e χ(O2 , Adρ). We count the contribution of each component ∂i O2 of ∂O2 to dim H 1 (∂O2 , Adρ). • When ∂i O2 ∼ = S 1 , then, by duality and ∂-regularity: 2) dim H 1 (∂i O2 , Adρ) = dim H 0 (∂i O2 , Adρ) = dim gρ(∂i O = rank G. (19) • When ∂i O2 ∼ = [[0, 1]], then e(∂i O2 , Adρ) = dim H 0 (∂i O2 , Adρ) − dim H 1 (∂i O2 , Adρ) χ rank G = dim H 0 (∂i O2 , Adρ) + dim H 1 (∂i O2 , Adρ), where the last line follows from Lemma 3.6 and the assumption that ρ is ∂-regular on the orientation covering. From this we deduce 1 dim H 1 (∂i O2 , Adρ) = (rank G − χ e(∂i O2 , Adρ)). (20) 2 15 From (19) and (20): b 1 dim H 1 (∂O2 , Adρ) = (c + ) rankG − χe(∂O2 , Adρ), 2 2 which concludes the proof of the proposition. Proposition 4.7. Let O2 be a non-orientable compact 2-orbifold with orientation covering O f2 . Let ρ : π1 (O2 ) → G be a representation whose restriction to O f2 is good and ∂-regular. Then 1 f2 , G) + 1 X dimρ X(O2 , G) = dimρ X(O χ e([[0, 1]], Adρ) 2 2 [[0,1]]⊂∂O2 where the sum runs on the components of ∂O2 that are intervals with mirror boundary, [[0, 1]]. Proof. In the proof of Proposition 4.6 we use (based on [17]): dim[ρ] X(O2 , G) = dim[ρ] Rgood (O2 , ∂O2 , G) + dim H 1 (∂O2 , Adρ), f2 , G) = dim Rgood (O dim[ρ] X(O f2 , ∂O2 , G) + dim H 1 (∂ O f2 , Adρ). [ρ] Furthermore, by Proposition 4.5: 1 dim[ρ] Rgood (O2 , ∂O2 , G) = dim[ρ] Rgood (O f2 , ∂ O f2 , G). 2 And from (19) and (20) in the proof of Proposition 4.6: b 1 dim H 1 (∂O2 , Adρ) = (c + ) rank G − χ e(∂O2 , Adρ), 2 2 dim H 1 (∂ O f2 , Adρ) = (2c + b) rank G. Using these formulas, we just need to know that χ e(∂O2 , Adρ) is the sum of the twisted Euler e(S 1 , Adρ) = 0. characteristics of each component, and that χ Z/2Z Z/2Z Figure 3: The orbifold [[0, 1]] = R/D∞ . The fundamental group of [[0, 1]] is the infinite dihedral group D∞ , so that R/D∞ ∼ = [[0, 1]]. In a group presentation D∞ = ⟨σ1 , σ2 | σ12 = σ22 = 1⟩ the stabilizers of the mirror points are the cyclic groups of order 2 generated by σ1 and σ2 respectively, see Figure 3. For further applications we need the following lemma: Lemma 4.8. For any representation ρ : D∞ = ⟨σ1 , σ2 | σ12 = σ22 = 1⟩ → G: e(R/D∞ , Adρ) = dim gρ(σ1 ) + dim gρ(σ2 ) − dim g, χ dim H 0 (R/D∞ , Adρ) = dim gρ(σ1 σ2 ) + dim gρ(σ1 ) + dim gρ(σ2 ) − dim g /2,  dim H 1 (R/D∞ , Adρ) = dim gρ(σ1 σ2 ) − dim gρ(σ1 ) − dim gρ(σ2 ) + dim g /2.  16 Proof. We compute the twisted Euler characteristic by using the simplicial structure of R/D∞ with one 1-cell and two 0-cells (with stabilizers generated by σ1 and σ2 respectively): e(R/D∞ , Adρ) = dim gρ(σ1 ) + dim gρ(σ2 ) − dim g χ (21) = dim H 0 (R/D∞ , Adρ) − dim H 1 (R/D∞ , Adρ). The orientation covering is denoted by S 1 → R/D∞ , and π1 (S 1 ) corresponds to the infinite cyclic subgroup of D∞ generated by σ1 σ2 . As S 1 is a manifold and χ(S 1 ) = 0: dim H 1 (S 1 , Adρ) ∼ = dim H 0 (S 1 , Adρ) ∼ = gρ(σ1 σ2 ) . (22) By Lemma 3.6: dim H 0 (R/D∞ , Adρ) + dim H 1 (R/D∞ , Adρ) = dim gρ(σ1 σ2 ) . (23) Then the lemma follows from (21) and (23). Notice that in the previous lemma we do not require anything for the representation ρ, that could be trivial, but G is a semi-simple group. The first direct application goes to the stabilizer of a corner reflector, the dihedral group D2k , by pre-composing any representation D2k → G with the natural surjection from the infinite dihedral group D∞ → D2k : x2 x1 x0 Figure 4: Corner reflector modeled on R2 /D2k , with the notation in Corollary 4.9. Corollary 4.9. Let x0 ∈ O2 be a corner reflector and x1 and x2 mirror points in a neighborhood of x0 separated by x0 , see Figure 4. If Ck ⊂ Stab(x0 ) is the orientation preserving cyclic subgroup of index 2, then: dim gρ(Stab(x0 )) = dim gρ(Ck ) + dim gρ(Stab(x1 )) + dim gρ(Stab(x2 )) − dim g /2.  4.3 Euclidean 2-orbifolds Along this subsection, let O2 be a Euclidean two orbifold without boundary. Set Γ = π1 (O2 ). As Isom+ (R2 ) is the semidirect product R2 ⋊ O(2), we have an exact sequence (Bieberbach theorem): 1 → Γ0 → Γ → Λ → 1, (24) with Γ0 ∼= Z2 and Λ ⊂ O(2) the linear part. When O2 is orientable, Λ is cyclic and when O2 is non-orientable, Λ is dihedral (or cyclic of order 2). Furthermore, when O2 is orientable or a Coxeter group, Γ0 is the maximal torsion free subgroup and the sequence (24) splits, as Λ corresponds to the stabilizer of a point in R2 . A Euclidean Coxeter group is a group generated by reflections along a square or a Euclidean triangle with angles an integer divisor of π. There are precisely five closed orientable Euclidean 2-orbifolds (Figure 5): • the 2-torus T 2 , 17 2 2 3 4 6 2 2 3 3 2 4 2 3 T2 S 2 (2, 2, 2, 2) S 2 (3, 3, 3) S 2 (2, 4, 4) S 2 (2, 3, 6) Figure 5: The five closed, orientable, and Euclidean 2-orbifolds. • a 2-sphere with 4 cone points of order 2, S 2 (2, 2, 2, 2), and • three 2-spheres with cone points of order p, q and r satisfying 1 p + 1 q + 1 r = 1, S 2 (p, q, r) for (p, q, r) = (3, 3, 3), (2, 4, 4) and (2, 3, 6). The orbifold S 2 (2, 2, 2, 2) is sometimes called a pillowcase, geometrically is the double of a rectangle, and S 2 (p, q, r) is called a turnover, as it is the double of a triangle. For different values of the ramifications, turnovers can also be spherical or Euclidean. Table 3 gives the cardinality of Λ ∼ = Γ/Γ0 for these five Euclidean orbifolds. O2 |Γ/Γ0 | T2 1 S 2 (2, 2, 2, 2) 2 S 2 (3, 3, 3) 3 S 2 (2, 4, 4) 4 S 2 (2, 3, 6) 6 Table 3: Values of k(O2 ) = |Γ/Γ0 | when O2 is orientable, for Γ0 maximal torsion free. As Γ = π1 (O2 ) is virtually abelian, most of the representations we consider are not irre- ducible. Instead, we deal with strong regularity: Definition 4.10. A representation of a Euclidean 2-orbifold ρ : π1 (O2 ) → G is called strongly regular if for a normal subgroup Γ0 < π1 (O2 ), Γ0 ∼ = Z2 , (a) dim gρ(Γ0 ) = rank G, and (b) the projection of ρ(Γ0 ) is contained in a connected abelian subgroup of G/Z(G). Theorem 4.11. Assume that O2 is compact Euclidean and that ρ : π1 (O2 ) → G is strongly regular. Then: (i) ρ is a smooth point of R(Γ, G) of dimension χ(O2 , Adρ) + dim G + dim gρ(Γ) . −e (ii) The component X0 (Γ, G) of X(Γ, G) that contains the character of ρ has dimension: χ(O2 , Adρ) + 2 dim gρ(Γ) = dim H 1 (O2 , Adρ). dim X0 (Γ, G) = −e Proof. (i) First consider the case O2 = T 2 , here we adapt an argument of [16]. In this case e(O2 , Adρ) = 0, and by a theorem of Richardson [29, Theorem B], ρ is in the closure of T × T χ 18 for T ⊂ G a maximal torus, T ∼ = (C∗ )rank(G) (meaning torus of an algebraic group). Therefore ρ is contained in a component of dimension at least dim G + rank G. As dim gρ(Γ) = rank G, by Proposition 4.1, dim TρZar R(Γ, G) = dim G + rank G. Therefore dimρ R(Γ, G) ≥ dim G + rank G = dim TρZar R(Γ, G). Using that the dimension of the Zariski tangent space is always larger than or equal to the dimension of a variety, with equality only at smooth points, it follows that dimρ R(Γ, G) = dim TρZar R(Γ, G) and ρ is a smooth point. For general O2 , notice that −e χ(O2 , Adρ) + dim G + dim gρ(Γ) = dim TρZar R(Γ, G), by Propo- sition 4.1 and all we need to show is that ρ is a smooth point. For that purpose, we use that the variety of representations of Γ0 at ρ|Γ0 is smooth. As Goldman obstructions are natural [18, §3], they are also equivariant. In particular if we start with a equivariant infinitesimal de- formation of Γ0 , then the sequence of obstructions to integrability is also equivariant. Thus, by Proposition 3.3 and Proposition 3.4, the sequence of obstructions to integrability of Γ vanishes. This proves (i). For (ii), notice that dim gρ(Γ) = dim H 0 (Γ, Adρ) is upper semicontinuous on R(Γ, G) [14, Ch. III, Theorem 12.8]. By smoothness, the dimension of the Zariski tangent space reaches its minimum (along the irreducible component) at ρ. Hence, as dim G + dim gρ(Γ) − χ e(O2 , Adρ) = dim TρZar R(Γ, G) by Proposition 4.1, dim gρ(Γ) reaches its minimum along the irreducible component at ρ. (Here we use that χe(O2 , Adρ) is constant along components, because the elements of finite order cannot be deformed.) Equivalently the dimension of the orbit by conjugation dim G − dim gρ(Γ) reaches its maximum at ρ. It follows (for instance from [6, Section 6.3]) that dim X0 (Γ, G) = dim R0 (Γ, G) − dim G − dim gρ(Γ) ,  which proves (ii). Remark 4.12. It does not follow from Theorem 4.11 that the variety of characters X(Γ, G) is smooth at the character of ρ. To explain Remark 4.12, we point out that even if ρ is strongly regular, its orbit may be not closed, so ρ could be smooth but its character not. For instance, the parabolic representation of Z2 in SL(2, C) is a smooth point of R(Z2 , SL(2, C)), but its character is a singular point of X(Z2 , SL(2, C)). The orbit of the parabolic representation is not closed because it accumulates to the trivial representation (that it is not regular). Orientable Euclidean 2-orbifolds appear as peripheral subgroups of hyperbolic three-orbifolds of finite volume. Thus a natural representation of Γ = π1 (O2 ) in PSL(2, C) occurs as the holonomy of horospherical cusps, homeomorphic to O2 × [0, +∞). We are interested in the composition of this holonomy with the principal representation τ : PSL(2, C) → G, in view of further computations for hyperbolic three-orbifolds. Proposition 4.13. Let O2 be a closed orientable Euclidean 2-orbifold. Consider the holonomy hol : π1 (O2 ) → PSL(2, C) as the horospherical section of a cusp, composed with the principal representation τ : PSL(2, C) → G. Then dim[τ ◦hol] X(O2 , G) is given by Table 4. Furthermore, the dimension of gτ ◦hol(Γ) is given by Table 5. 19 HH O 2 H T2 S 2 (2, 2, 2, 2) S 2 (3, 3, 3) S 2 (2, 4, 4) S 2 (2, 3, 6) G HHH PSL(n, C) 2(n − 1) 2⌊ n2 ⌋ 2⌊ n3 ⌋ 2⌊ n4 ⌋ 2⌊ n6 ⌋ PSp(2m, C) 2m 2m 2⌊ m3⌋ 2⌊ m 2⌋ 2⌊ m 3⌋ PO(2m + 1, C) 2m 2m 2⌊ m3⌋ 2⌊ m 2⌋ 2⌊ m 3⌋ PO(2m, C) 2m + 2 4⌊ m2⌋ m 2⌊ 3 ⌋ 2(⌊ 4 ⌋ + ⌊ m+1 m 4 ⌋) 2(⌊ 6 ⌋ + ⌊ m+2 m 6 ⌋) G2 4 4 2 0 2 F4 8 8 4 4 4 E6 12 8 4 4 6 E7 14 14 6 4 6 E8 16 16 8 8 8 Table 4: Dimension of X(O2 , G) at [τ ◦ hol], for hol : Γ → PGL(2, C) the holonomy of a horospherical cusp and τ : PGL(2, C) → G the principal representation. HH O 2 HH T2 S 2 (2, 2, 2, 2) S 2 (3, 3, 3) S 2 (2, 4, 4) S 2 (2, 3, 6) g H H sl(n, C) n−1 ⌊ n−1 2 ⌋ ⌊ n−1 3 ⌋ ⌊ n−1 4 ⌋ ⌊ n−1 6 ⌋ m+1 sp(2m, C) m 0 ⌊ 3 ⌋ 0 0 so(2m + 1, C) m 0 ⌊ m+1 3 ⌋ 0 0 m−1 m m−1 m−1 m−1 so(2m, C) m δ2Z ⌊ 3 ⌋ + δ3Z δ4Z δ6Z g2 2 0 0 0 0 f4 4 0 0 0 0 e6 6 2 0 2 0 e7 7 0 1 0 0 e8 8 0 0 0 0 Table 5: Dimension of gτ ◦hol(Γ) for τ ◦ hol as in Table 4, where δjZ i = 1 if i ∈ jZ and δ i = 0 jZ otherwise. Proof. The principal representation τ maps regular elements to regular elements. Furthermore the parabolic holonomy of the maximal torsion-free subgroup of π1 (O2 ) is contained in a con- nected abelian subgroup of PGL(2, C), hence τ ◦ hol|π1 (O2 ) is strongly regular and Theorem 4.11 applies. The computation of dimension is based on the decomposition of Ad ◦ τ as a sum of Sym2dα , according to the exponents d1 , . . . , dr of G, where r is the rank of G. Let V2dα is the space of the representation Sym2dα . We use the following three facts: 1. dim V2dα = 2dα + 1. 2. For a cyclic group of rotations Ck < PSL(2, C) of order k, with hyperbolic rotation angles Ck dα in 2π k Z, dim V2dα = 2⌊ k ⌋ + 1. Γ = δ i , Γ = π (O 2 ), Γ ∼ Z2 a maximal torsion-free subgroup and k = |Γ/Γ |, 3. dim V2dα kZ 1 0 = 0 i = 1 if i ∈ kZ and 0 if i ̸∈ kZ. where δkZ From this, the computation is elementary. Lemma 4.14. Let O2 = S 2 (2, 2, 2, 2), O2 = S 2 (3, 3, 3), O2 = S 2 (2, 4, 4), or O2 = S 2 (2, 3, 6). There is an irreducible representation of Γ = π1 (O2 ) in PSL(k, C) iff and only if k = |Γ/Γ0 |. 20 Proof. We first construct the irreducible representation of Γ in PSL(k, C), for k = |Γ/Γ0 |. From the exact sequence (24), start with a diagonal representation of Γ0 , then represent the cyclic group Γ/Γ0 as a cyclic permutation the k coordinates of Ck . (In each case, Proposition 4.2 applies and the dimension of the variety of characters is 2.) Let ρ : Γ → PSL(k, C) be an irreducible representation. Let Fix(ρ(Γ0 )) ⊂ CPk−1 denote the fixed point set of ρ(Γ0 ), that is nonempty by Kolchin’s theorem. We consider the action of the cyclic group Γ/Γ0 on Fix(ρ(Γ0 )). It can be checked that any possibility other than k = |Γ/Γ0 | and that Fix(ρ(Γ0 )) has k points in generic position that are permuted cyclically by Γ/Γ0 yields a contradiction with irreducibility. Example 4.15. We consider planar Euclidean Coxeter groups, namely generated by reflections along a square or a Euclidean triangle with angles an integer divisor of π, denoted by: Q(2, 2, 2, 2), T (3, 3, 3), T (2, 4, 4), T (2, 3, 6). The respective orientation coverings are S 2 (2, 2, 2, 2), S 2 (3, 3, 3), S 2 (2, 4, 4) and S 2 (2, 3, 6). The group Γ = π1 (O) acts naturally in the Euclidean plane, and one can construct representations in G by realizing this action on a flat of the symmetric space associated to G. On this flat, one allows translations, but the rotational part is restricted to the action of the Weyl group, which in its turn is a Coxeter group that acts faithfully on a maximal flat (eg the Cartan subalgebra). Hence one can construct representations by analyzing the Weyl group. We discuss the case of rank 2. Here the Weyl group is a dihedral group of order 2m generated by reflections along two lines at angle π/m, the walls of the Weyl chamber. For type A2 the angle is π/3, for B2 , π/4, and for G2 , π/6. This yields discrete faithful representations of the triangle group T (3, 3, 3) in SL(3, C) and G2 (C), of T (2, 4, 4) in Sp(4, C), of T (2, 3, 6) in G2 (C) and of the quadrilateral group Q(2, 2, 2, 2) in Sp(4, C) or G2 (C) (or the corresponding split real forms). The triangle groups constructed here have a one-parameter space of deformations (homoteties in the plane) and the quadrilateral group a 2-parameter space. Those are half the dimension of the deformation space of their orientable cover. 4.4 Spherical 2-orbifolds Finally, we also need to consider spherical two-orbifolds, namely finitely covered by the 2-sphere. They are rigid, but we require a cohomological computation later in the paper. Lemma 4.16. Let O2 be a spherical 2-orbifold and ρ : π1 (O2 ) → G a representation. Then ( 2 i 2 dim gρ(π1 (O )) , for i = 0, 2 dim H (O , Adρ) = , 0, for i = 1 and thus 2 )) e(O2 , Adρ) = 2 dim gρ(π1 (O χ . (25) Proof. Let O2 be a spherical 2-orbifold. As it is finitely covered by S 2 , by Proposition 3.3 we have 2 2 H i (O2 , Adρ) ∼ = H i (S 2 , g)π1 (O ) ∼ = H i (S 2 , Z) ⊗ gρ(π1 (O )) . Hence the lemma follows. 21 Remark 4.17. For a spherical orbifold, the component of R(O2 , G) that contains ρ is an orbit by conjugation of dimension 2 dim G − dim gρ(π1 (O )) . In particular, by (25) it is a smooth variety of dimension 2 )) χ(O2 , Adρ) + dim G + dim gρ(π1 (O −e . 5 Computing dimensions of the Hitchin component Assume that O2 is hyperbolic, namely that the standard orbifold-Euler characteristic is negative, χ(O2 ) < 0. For G a simple complex adjoint group, the principal representation τ : PSL(2, C) → G is constructed from Jacobson-Mozorov theorem, and it restricts to τ : PGL(2, R) → GR , for GR a (non connected) split real form of G. The Hitchin component Hit(O2 , GR ) is the connected component of Rgood (O2 , GR ) that contains the composition of τ with the holonomy representation of any Fuchsian structure on O2 . Hitchin componnents for surface have been intensively studied. here just mention that Alessandrini, Lee, and Schaffhauser [1] have introduced them for (perhaps non-orientable) 2-orbifolds; furthermore they have shown that they are homeomorphic to balls, as for surfaces. The purpose of this section is to provide formulas for the dimension of the Hitchin component of orbifolds, using the tools of Section 4. Many of the formulas are equivalent to the ones already computed in [1], but the approach and presentation of results is different. In particular we give some applications in Proposition 5.10 and Corollary 5.15 to the dimension growth of some families of Hitchin representations. Long and Thistlethwaite in [23] have also computed the dimension of Hitchin components in PGL(n, R) for turnovers (2-spheres with three cone points). Remark 5.1. The representation τ ◦ hol is good, Remark 2.4, so the real dimension of the Hitchin component of O2 in GR is precisely the complex dimension of X(O2 , G) at the character of τ ◦ hol, namely −χ̃(O2 , Adρ); we often use: dimR Hit(O2 , GR ) = dimC,[τ ◦hol] X(O2 , G) = −χ̃(O2 , Adρ). Remark 5.2. By Corollary 3.5, for O2 orientable dimR Hit(O2 , GR ) is even. Remark 5.3. For O2 closed and orientable, Atiyah-Bott-Goldmann defines a symplectic struc- ture on Hit(O2 , GR ), Proposition 4.4. The following is Proposition 1.5 from the introduction: Proposition 5.4. For O2 closed and non-orientable, with orientation covering O f2 , via the restriction map Hit(O2 , GR ) embeds in Hit(O f2 , GR ) as a Lagrangian submanifold. Proof. By Proposition 4.5 the restriction map yields a Lagrangian immersion from Hit(O2 , GR ) f2 , GR ). Furthermore, using the irreducibility of Hitchin representations, one can prove to Hit(O that this restriction map is injective (see for instance Lemma 2.9 and Proposition 2.10 in ??). The restriction map to a finite index subgroup is proper, therefore it is an embedding (or one can quote [1, Corollary 2.13]) 22 5.1 Hitchin component for PGL(n, R) . Following Long and Thistlethwaite [23], for n, k positive integers we set σ(n, k) = qn + (q + 1)r, where q and r are nonnegative integers such that n = q k + r (the quotient and reminder of integer division). Proposition 5.5. Let x ∈ O2 and τ = Symn−1 . For ρ = τ ◦ hol : π1 (O2 ) → PGL(n, R), cone point with Stab(x) ∼   σ(n, k) − 1 if x is a = Ck , ∼  (σ(n, k) − 1)/2 if x is a corner with Stab(x) = D2k and n is odd,    dim g ρ(Stab(x)) = (σ(n, k) − 2)/2 if x is a corner with Stab(x) ∼ = D2k and n is even,  (n2 − 1)/2 if x is a mirror point and n is odd,       2 (n − 2)/2 if x is a mirror point and n is even. Proof. For an elliptic element γ ∈ Γ of order k, we may compute dim gρ(γ) using that, up to conjugacy: πi ! n−1 e k 0 πi πi πi ρ(γ) = ± Sym − πi = ±diag(e k (n−1) , e k (n−3) , . . . , e k (n−1) ). 0 e k As n = qk + r, among the k eigenvalues of this matrix, (k − r) eigenvalues have multiplicity q and r eigenvalues have multiplicity q + 1. Thus dim gρ(γ) = (k − r)q 2 + r(q + 1)2 − 1 = qn + (q + 1)r − 1 = σ(n, k) − 1. This computation yields the formula for cone points and mirror points. For corner points, apply Corollary 4.9 and the previous computations. We then have: Proposition 5.6. Let O2 be a connected hyperbolic 2-orbifold, with underlying surface |O2 |. Then cp cr 2 2 2 X X n2 − σ(n, lj ) n2 dim Hit(O , PGL(n, R)) = −(n − 1)χ(|O |) + (n2 − σ(n, ki )) + + b⌊ ⌋, 2 2 i=1 j=1 where the cone points have order k1 , k2 , . . . , kcp , the corner reflectors have order 2l1 , 2l2 , . . . , 2lcr and b is the number of components of ∂O2 homeomorphic to R/D∞ , an interval with mirror boundary. Proof. We apply Proposition 4.2 to get that the dimension of the Hitchin component is −χ(O2 , Adρ), which, by Proposition 5.5, equals: cp ( P σ(n,lj )−2 2 2 2 X − crj=1 2 + (me − mp) n 2−2 if n even −(n − 1)χ(O \ Σ) − (σ(n, ki ) − 1) + σ(n,lj )−1 2 (26) − cr + (me − mp) n 2−1 if n odd P i=1 j=1 2 23 where me is the number of mirror edges (joining corner reflectors or mirror points in the bound- ary), and mp the number of boundary points that are also mirrors. Next we use the formulas (from additivity of the Euler characteristic): χ(|O2 |) = χ(O2 \ Σ) + cp + cr + mp − me, 0 = χ(∂|O2 |) = − me − b + cr + mp, that are equivalent to −χ(O2 \ Σ) = − χ(|O2 |) + cp + b, (27) me − mp = cr − b. (28) Using (27) and (28), (26) becomes: cp cr ( 2 X X n2 − σ(n, lj ) b n2 if n even −(n2 − 1)χ(|O2 |) + (n2 − σ(n, ki )) + + 2 i=1 j=1 2 b n 2−1 if n odd which proves the proposition. Example 5.7 (Proved in [23] when cp = 3). Let S 2 (k1 , . . . , kcp ) be a sphere with cp ≥ 3 cone points of order k1 , . . . , kcp respectively (and satisfying k11 + k12 + k13 < 1 for cp = 3). Then, the dimension of the Hitchin component of S 2 (k1 , . . . , kcp ) in PGL(n, R) is cp X 2 n (cp − 2) + 2 − σ(n, ki ). i=1 Consider now P (k1 , . . . , kcr ) the orbifold generated by reflections on a hyperbolic polygon with angles kπ1 , . . . , kπcr . In particular S 2 (k1 , . . . , kcr ) is its orientation covering and therefore the dimension of the Hitchin component of P (k1 , . . . , kcr ) in PGL(n, R) is n2 (cr − 2) + 2 − cr P i=1 σ(n, ki ) . 2 Proposition 5.8. Every component of Rgood (O2 , PGL(n, R)) that contains representations that are boundary regular has dimension at most the dimension of the Hitchin component. Proof. Assume first that ∂O2 = ∅. By Corollary 4.3, we may assume that O2 is orientable, hence the singular locus is a finite union of cone points, that have cyclic stabilizers. We claim that for an elliptic element γ, dim(gρ(γ) ) is minimized for ρ in the Hitchin component. Assuming the claim, χ e(O2 , ρ) is minimized for ρ in the Hitchin component, and apply Proposition 4.2. To prove the claim, since ρ(γ) has order k, its eigenvalues have multiplicities {n1 , . . . , nk }, with n1 + · · · + nk = n. Therefore dim(gρ(γ) ) = n21 + · · · + n2k − 1. This function is minimized when |ni − nj | ≤ 1, because if we replace n1 by n1 + 1 and n2 by n2 − 1, then n21 + n22 increases by 2(n1 − n2 + 1). As, for a rotation of angle 2π/k, Symn−1 (ρ(γ)) satisfies |ni − nj | ≤ 1, it minimizes the required dimension. When ∂O2 ̸= ∅, in the orientable case we apply Proposition 4.6, knowing that by ori- entability ∂O2 is a union of circles and χ e(∂O2 , Adρ) = 0. In the non-orientable case we apply Proposition 4.7 and use the claim we have proved, that for an elliptic element γ, dim(gρ(γ) ) is minimized for ρ in the Hitchin component. Example 5.9. In Table 6 we compute the dimension of Hit(S 2 (3, 3, 4)PSL(n, R)). Notice that the difference with n2 /12 is bounded, this is a particular case of the next proposition. 24 n mod 12 dim Hit(S 2 (3, 3, 4), PGL(n, R)) 0 n2 /12 + 2 ±1, ±5 (n2 − 1)/12 ±2 (n2 − 4)/12 ±3 (n2 + 15)/12 ±4 (n2 + 8)/12 6 n2 /12 + 1 Table 6: Dimension of the Hitchin component of S 2 (3, 3, 4). The dimension for the group generated by reflections on a triangle of angles ( π3 , π3 , π4 ) is half of it. Proposition 5.10. There exists a uniform constant C(O2 ) depending only on O2 such that dim Hit(O2 , PGL(n, R)) + χ(O2 )(n2 − 1) ≤ C(O2 ). Here 0 > χ(O2 ) ∈ Q denotes the (untwisted) orbifold-Euler characteristic. Proof. Let ρ = τ ◦ hol. Assume first that O2 is orientable. If γ is an elliptic element of order k, 2 we shall prove that n k−1 − dim gρ(γ) is bounded uniformly on n. Namely, if n = kq + r, then dim gρ(γ) = kq 2 + 2rq + r − 1. On the other hand n2 = k 2 q 2 + 2rkq + r2 , thus: n2 − 1 r2 − 1   ρ(γ) r+1 − dim g = − r + 1 = (r − 1) 1 − ≤ k. k k k In the orientable case, as all stabilizers are cyclic this estimate tells that the difference χ̃(O2 , Adρ)− χ(O2 )(n2 − 1) is uniformly bounded, for ρ = Symn−1 ◦ hol. With Proposition 4.2, this yields the proposition for O2 orientable. In the non-orientable case, consider O f2 → O2 the orientation cover. By Proposition 4.7 1 f2 , PGL(n, R)) ≤ 1 dim G − 2 dim gρ(C2 ) b(O2 ) dim Hit(O2 , PGL(n, R)) − dim Hit(O 2 2 where ρ(C2 ) denotes the image of any cyclic stabilizer of order 2 and b(O2 ) is the number of components of ∂O2 homeomorphic to R/D∞ = [[0, 1]] (an interval with mirror end-points). As ( 1 n 2−1 1/2 for even n dim G − 2 dim gρ(C2 ) =  − (σ(n, 2) − 1) = 2 2 0 for odd n the proposition follows. 5.2 Exponents of a simple Lie algebra Recall from Proposition 5.5 that the exponents d1 , . . . , dr ∈ N of the Lie algebra g are defined by the equation Mr Ad ◦ τ = Sym2dα , α=1 Pr Equation (2), where τ is the principal representation. Here r = rank g and α=1 (2dα + 1) = dim g. 25 Lemma 5.11. For ρ ∈ Hit(O2 , G) and x ∈ ΣO2 a point in the branching locus with nontrivial stabilizer Γx : (P ρ(Γx ) r dα α=1 (2⌊ k ⌋ + 1) if Γx ∼ = Ck dim g = r dα if Γx ∼ P  α=1 ⌊ k ⌋ + #{dα | dα is even} = D2k where Ck is cyclic of order k, and D2k dihedral of order 2k, r = rank(g), and {d1 , . . . , dr } are the exponents of the Lie algebra. Proof. For a Fuchsian representation, the image of the cyclic group Ck is generated by a matrix conjugate to ! πi ek 0 ± πi . (29) 0 e− k Using (2), the contribution of the α-th exponent to dim gρ(Γx ) is the multiplicity of eigen- value 1 in the 2dα -th symmetric power of (29). Namely, the number of appearances of 1 in 2πi 2πi 2πi {e k dα , e k (dα −1) , . . . , e k (−dα ) } which is precisely 2⌊ dkα ⌋ + 1. For the dihedral group, Corollary 4.9, the cyclic case, and (3) yield that the dimension is r ρ(Dk ) ρ(C2 ) ρ(Ck ) X dα dα dim g = (2 dim g + dim g − dim g)/2 = (2⌊ ⌋ + 1 + ⌊ ⌋ − dα ) 2 k α=1 ( 1 for dα even, and 2⌊ d2α ⌋ + 1 − dα = 0 for dα odd. Lemma 5.11 allows to give a new proof of Proposition 5.5. As the exponents for sp(2m) and s0(m, m + 1) are the same, (1, 3, 5 · · · , 2m − 1), by Lemma 5.11 we have: Remark 5.12. Let ρ1 ∈ Hit(O2 , PSp± (2m)) and ρ2 ∈ Hit(O2 , PO(m + 1, m)). For any x ∈ O2 , dim gρ1 (Γx ) = dim gρ2 (Γx ) . Another consequence of Lemma 5.11 is the following. Corollary 5.13 ([1]). Let O2 be a connected hyperbolic 2-orbifold. If {d1 , . . . , drankG } are the exponents of G. Then the dimension of Hit(O2 , G) is   rankG cp cr X X dα X dα dα + 1  −χ(|O2 |) dim G +  2(dα − ⌊ ⌋) + (dα − ⌊ ⌋) + 2b ⌊ ⌋ ki lj 2 α=1 i=1 j=1 where the cone points have order k1 , k2 , . . . , kcp , the corner reflectors have order 2l1 , 2l2 , . . . , 2lcr , and b is the number of components of ∂O2 homeomorphic to R/D∞ , an interval with mirror boundary. i = 1 if i ∈ jZ and Proof. We follow the same scheme as the proof of Proposition 5.6. Set δjZ i = 0 otherwise. Firstly we apply Proposition 4.2 and Lemma 5.11 to get that the dimension δjZ of the Hitchin component is −χ(O2 , Adρ), which equals:   rankG cp cr X X dα X dα dα dα −χ(O \ Σ) dim G + − (2⌊ ⌋ + 1) − (⌊ ⌋ + δ2Z ) + (me − mp)(2⌊ ⌋ + 1) ki lj 2 α=1 i=1 j=1 where me is the number of mirror edges (joining corner reflectors or mirror points in the bound- dα ary), and mp the number of boundary points that are also mirrors, and δ2Z = 1 if dα is even, and 0 if dα is odd. Then the corollary follows from (27) and (28). 26 This computation of dimensions is contained in [1, Theorem 1.2], where it is proved that the Hitchin component is a cell. For exceptional groups the dimension can be directly computed from this corollary and the exponents in Table 2. For instance, for (the real split form of the adjoint group) G2 , as the indices are 1 and 5 the dimension of the Hitchin component is −14χ(|O2 |) + 8cp2 + 10cp3,4,5 + 12cp≥6 + 4cr4 + 5cr6,8,20 + 6cr≥12 + 8b, where cpi is the number of cone points with stabilizer Ci and cr2i of corner reflectors with stabilizer D2i . It can also be checked that for G2 and F4 that dimension of the variety of characters is maximized by the Hitchin component, but this is a rather tedious computation. 5.3 Hitchin component for PSp± (2m) and PO(m, m + 1) For both G = PSp± (2m) and G = PO(m, m + 1), the principal representation τ : PGL(2, R) → G is the restriction of Symn−1 : PGL(2, R) → PGL(n, R), with n = 2m and n = 2m + 1, respectively. Since SL(2, R) preserves the skew bilinear form   0 1 J= , −1 0 Symn−1 (SL(2, R)) preserves the symmetric power of J, which is symmetric or skew, according to the parity. Similarly one easily argues that Symn−1 (SL(2, R)) is contained in either PSp± (2m) or PO(m, m + 1), according to wheter n = 2m or n = 2m + 1. Write 2m = kq + r with q, m ∈ Z, q, r ≥ 0 and r < k. Recall that: σ(2m, k) = 2mq + r(q + 1). Proposition 5.14. Let ρ ∈ Hit(O2 , G), for G = PSp± (2m) or PO(m, m + 1) and let x ∈ O2 . 1. If x is a cone point with Stab(x) ∼ = Ck then ( σ(2m,k) for k even, dim gρ(Ck ) = 2 σ(2m,k) 2 + ⌊ q+1 2 ⌋ for k odd. 2. If x is a mirror point with Stab(x) ∼ = C2 then dim gρ(C2 ) = m2 . 3. If x is a corner reflector with Stab(x) ∼= D2k then ( σ(2m,k) −m for k even, dim gρ(Dk ) = σ(2m,k) 4 2 4 − 2 + 12 ⌊ q+1 m 2 ⌋ for k odd. Proof. By Remark 5.12, it is sufficient to make the computations for G = PSp(2m, C). The bilinear form Sym2m−1 (J) has matrix   0 ··· 0 0 1  0 · · · 0 −1 0    0 · · · 1 0 0  .  .. . . .. .. ..   . . . . . −1 · · · 0 0 0 27 thus sp(2m) = {(ai,j ∈ M2m,2m (C) | ai,j = (−1)i+j+1 a2m+1−j,2m+1−i }. From this expression, as a diagonal matrix D ∈ PSp(2m, C) is −1 D = diag(λ1 , . . . , λm , λ−1 m , . . . , λ1 ), we have that 1 dim(sp(2m)D ) = dim(gl(2m)D ) + dim(adD ) 2 where ad = {(ai,j ∈ M2m,2m (C) | aij = 0 if i + j ̸= 2m + 1} ⊂ sp(2m) is the anti-diagonal, namely the subspace of matrices   0 ··· 0 0 ∗ 0 · · · 0 ∗ 0   0 · · · ∗ 0 0  .  .. . . .. .. ..  . . . . . ∗ ··· 0 0 0 πi ! 2π ek 0 A rotation of angle k in PGL(2, C) is conjugate to ± πi and 0 e− k πi !! ek 0 D = Sym2m−1 ± πi 0 e− k πi πi πi πi πi = ± diag(e k (2m−1) , e k (2m−3) , . . . , e k , e− k , . . . , e− k (2m−1) ). Therefore, the (adjoint) action of D, on the antidiagonal ad has eigenvalues 2πi 2πi 2πi 2πi 2πi {e k (2m−1) ,e k (2m−3) ,...,e k , e− k , . . . , e− k (2m−1) }. Thus, dim adD is the number appearances of 1 in this list of eigenvalues: ( D 0 for k even, dim ad = 2m−1+k q+1 2⌊ 2k ⌋ = 2⌊ 2 ⌋ for k odd. On the other hand, gl(2m)D = 2mq + r(q + 1) (because D has r eigenvalues with multiplicity q + 1 and k − r eigenvalues with multiplicity q), what concludes item 1. Item 2 is a particular case of item 1, and item 3 follows from Corollary 4.9 and the previous cases. If we look at the growth of dim Hit(O2 , PSp± (2m)), the difference | dim Hit(O2 , PSp± (2m)) + χ(O2 ) dim(PSp± (2m))| may be unbounded because of cyclic stabilizers of odd order (and dihedral stabilizers of order a multiple of 4). But instead we have (already proved in [1]): Corollary 5.15 ([1]). For a compact hyperbolic two orbifold O2 : dim Hit(O2 , PSp± (2m)) dim Hit(O2 , PO(m + 1, m)) lim = lim = −χ(O2 ). m→∞ dim(PSp± (2m)) m→∞ dim(PO(m + 1, m)) 28 Proof. For every stabilizer Stab(x) we check from Proposition 5.14 that dim gρ(Stab(x)) 1 lim 2 = , m→∞ 2m | Stab(x)| because 2m/q → k. The corollary follows using Proposition 4.2. Proposition 5.16. Every component of Rgood (O2 , PSp± (2m)) that contains representations that are boundary regular has dimension at most the dimension of the Hitchin component. Proof. We use the same argument as in the proof of Proposition 5.8. The same discussion applies here, the difference being in the proof that the Hitchin component minimizes the dimension of the centralizer of elliptic elements of given order k. We follow the description in the proof of Proposition 5.14, in particular a diagonal element in PSp(2m, C) is −1 −1 D = diag(λ1 , λ2 , . . . , λm , λ−1 m , . . . , λ2 , λ1 ). Hence {± Id} is the center of Sp(2m, C). To compute dim Sp(2m, C)D , we distinguish the case k even from the case k odd. For k even we assume first that λki = −1, then there are no elements in the antidiagonal ad fixed by D. Therefore dim Sp(2m, C)D = 21 dim gl(2m, C)D . Hence if ni is the multiplicity of λi plus the multiplicity of λ−1i (in this notation we chose half of the eigenvalues {λ1 , . . . , λ k } 2 disjoint form their inverses): dim Sp(2m, C)D = n21 + n22 + · · · + n2k . As seen in the proof of Proposition 5.8 this is minimized when |ni − nj | ≤ 1. Next assume that k is even and that λki = 1. Then ±1 are possible eigenvalues of D, and they yield fixed elements in the antidiagonal. Let n0 be the multiplicity of 1, n1 the multiplicity of −1, and let for i = 2, . . . , k2 the addition of the multiplicities of each eigenvalue and its inverse. Then dim Sp(2m, C)D = 2n20 + n0 + 2n21 + n1 + n22 + · · · + n2k An elementary computation shows that 2n20 + n0 + 2n21 + n1 ≥ (n0 + n1 )2 (with equality iff n0 = n1 = 0), therefore the dimension of the invariant subspace is higher than when λki = −1. For k odd, up to changing D by −D we may assume that λki = 1 (and −1 is not an eigenvalue). We count again multiplicities: we take n0 the multiplicity of 1. We group the remaining eigenvalues in pairs, each eigenvalue with its inverse, and we add the multiplicities of each pair to get n1 , . . . , n(k−1)/2 . Then a computation yields dim Sp(2m, C)D = 2n20 + n0 + n21 + n22 + · · · + n2(k−1)/2 . To minimize it, we notice that when we replace n0 by n0 + 1 and n1 by n1 − 1 then this quantity increases by 2(2n0 − n1 + 2). Furthermore, if we replace n0 by n0 − 1 and n1 by n1 + 1 then it increases by 2(−2n0 + n1 + 1). Combined with the previous computations, this quantity is minimized when 2n0 − 1 ≤ ni ≤ 2n0 + 2 and |ni − nj | ≤ 1, for 1 ≤ i, j ≤ (k − 1)/2. This is realized by the symmetric power of a fuchsian element of order k. Remark 5.17. Proposition 5.16 also holds true for PO(m+1, m) with a similar proof. However the argument does not work for PO(m + 1, m + 1). 29 6 Representations of 3-orbifolds We restrict to orientable 3-orbifolds. In particular their singular locus is a union of circles, proper segments, and trivalent graphs, where the isotropy groups or stabilizers of edges are cyclic, and the isotropy groups or stabilizers of trivalent vertices are non-cyclic finite subgroups of SO(3) (hence dihedral, tetrahedral, octahedral or icosahedral). See Figure 6. 2 2 2 2 n n n 3 3 3 4 3 5 Cn D2n T12 O24 I60 Figure 6: Local models for the branching locus, with the corresponding isotropy groups: the cyclic group Cn , the dihedral group D2n , the tetrahedral group T12 , the octahedral group O24 , and the icosahedral group I60 . The subindex denotes the order of the group. The label on an edge denotes the order of the cyclic isotropy group of points in the interior of the edge. 6.1 Geometric representations Let O3 denote a hyperbolic 3-orbifold of finite type, possibly of infinite volume. In particular it has a compactification O3 that is an orbifold with boundary. Let ∂O3 = ∂1 O3 ⊔ · · · ⊔ ∂k O3 denote its decomposition in connected components. Consider the composition of the holonomy hol : π1 (O3 ) → PSL(2, C) with the principal representation τ : PSL(2, C) → G. The following is part of Theorem 1.1. Theorem 6.1. The character of ρ = τ ◦ hol is a smooth point of X(O3 , G). Furthermore k k 1X X 3 3 dim[ρ] X(O , G) = − e(∂i O3 , Adρ) + χ dim gρ(π1 (∂i O )) 2 i=1 i=1 k 1 X 1 = dim[ρ|∂i ] X(∂i O3 , G) = dim[ρ|∂] X(∂O3 , G). 2 2 i=1 Proof. We follow the proof for the manifold case in [26] for representations in PSL(n, C), or [20] in PSL(2, C). In particular we use Selberg’s lemma: there exists O03 → O3 a finite regular covering which is a manifold. The first step is to prove that the inclusion of ∂O3 in O3 induces an injection in cohomology H 1 (O3 , Adρ) ,→ H 1 (∂O3 , Adρ). When O3 is a manifold, then this is proved in [26] using a theorem of vanishing in L2 -cohomology (as, using de Rham cohomology and a metric on the bundle and differential forms, every element in the kernel is represented by an L2 -form), by using the decomposition (2) of Ad ◦ τ . For orbifolds in general, using Selberg’s lemma, the inclusion follows from the manifold case and Proposition 3.3. 30 The next step is to prove that 1 dim H 1 (O3 , Adρ) = dim H 1 (∂O3 , Adρ). 2 As in the manifold case, this follows from Poincaré duality, Proposition 3.4, and the long exact sequence of the pair (O3 , ∂O3 ). Namely, if the morphisms in the long exact sequence in cohomology are i∗ β H 1 (O3 , Adρ) → H 1 (∂O3 , Adρ) → H 2 (O3 , ∂O3 ; Adρ) then ⟨i∗ (x), y⟩∂O3 = ⟨x, β(y)⟩O3 ∀x ∈ H 1 (O3 , Adρ), ∀y ∈ H 1 (∂O3 , Adρ), where ⟨, ⟩ denotes the perfect pairing in Proposition 3.4, and the pairing on ∂O3 is the sum of pairings on each connected component. As the pairings are nondegenerate and ker β = im i∗ , the claim on the dimension follows from a linear algebra argument. Finally, as H 1 (O3 , Adρ) ∼ = H 1 (Γ, Adρ) by Proposition 3.4, with Γ = π1 (O3 ), we use Gold- man’s obstruction theory to integrability to prove that [ρ] is a smooth point of the character variety of local dimension H 1 (O3 , Adρ). For Γ0 < Γ a torsion-free subgroup, any infinitesimal deformation in H 1 (Γ, Adρ) yields a Γ/Γ0 -equivariant infinitesimal deformation of Γ0 , and we may apply the same argument as in Theorem 4.11. Alternatively as X(Γ0 , G) is analytically smooth at the character of the restriction of ρ, by Cartan’s linearization [4, Lemma 1] there exist local analytic coordinates that linearize the action of Γ/Γ0 in a neighborhood of the character of ρ|Γ0 in X(Γ0 , G). Hence the fixed point set of Γ/Γ0 is a smooth subvariety and, as ρ|Γ0 is good this fixed point set can be locally identified with X(Γ, G). Definition 6.2. The component of Theorem 6.1 is called the canonical or distinguished com- ponent. To complete the proof of Theorem 1.1 we need to show that X(O3 , G) → X(∂O3 , G) is locally an injection. This does not follow directly from the injection H 1 (O3 , Adρ) → H 1 (∂O3 , Adρ), as X(∂O3 , G) can be non smooth because of rank 2 cusps. (Rank one cusps are part of components with negative Euler characteristic and we discuss only rank 2 cusps.) Consider a nonsingular (manifold) cusp, T 2 × [0, +∞) equipped with the warped product metric e−2t gT 2 + dt2 , where gT 2 is a flat metric on the torus and t ∈ [0, +∞) is the Busemann function coordinate. Set ρ = τ ◦ hol : π1 (T 2 ) → G. Using de Rham cohomology, we may talk about L2 -forms on T 2 × [0, +∞). Lemma 6.3. For any non-contractible loop l ⊂ T 2 , the restriction induces a surjection H 1 (T 2 × [0, +∞), Adρ) → H 1 (l, Adρ) whose kernel consists of the cohomology classes represented by L2 -forms. Proof. Since Ad ◦ τ = ⊕α Sym2dα by (2) and Ad ◦ Symn−1 = ⊕n−1 2i i=1 Sym , it suffices to prove the lemma for τ = Symn−1 and g = sl(n, C). In [25] precisely the situation when τ = Symn−1 is discussed. In particular by [25, Lemma 3.3] the image of the map induced by the inclusion 2 sl(n, C)π1 (T ) ⊂ sl(n, C), 2 H 1 (T 2 , sl(n, C)π1 (T ) ) → H 1 (T 2 , sl(n, C)Adρ ) (30) is represented by L2 -forms. (Here sl(n, C)Adρ denotes the coefficients on the Lie algebra, usually 2 just denoted by Adρ, whilst on the invariant subspace sl(n, C)π1 (T ) the action is trivial by construction.) Furthermore, either by [25, Lemma 3.4] or by a straightforward computation, H 1 (T 2 , sl(n, C)Adρ ) → H 1 (l, sl(n, C)Adρ ) 31 is a surjection with kernel precisely the image of (30). Remark 6.4. Let γ ∈ π1 (T 2 ) be an element represented by the loop l as in Lemma 6.3. We have a natural isomorphism: H 1 (l, Adρ) ∼ = H0 (l, Adρ) ∼ = H0 (⟨γ⟩, Adρ) ∼ = gρ(γ) = gρ(π1 (T 2 )) Let G e be the universal covering of G and let χ1 , . . . , χr denote its fundamental characters (the characters of the fundamental representations). E.g. for G e = SL(n, C) the fundamental characters are the symmetric functions on the eigenvalues. By a theorem of Steinberg, they define an isomorphism: e reg /G ∼ (χ1 , . . . , χr ) : G = Cr , (31) where G e reg denotes the set of regular elements in G. e When the group G is not simply connected, the characters (χ1 , . . . , χr ) still define local functions in a neighborhood of ρ(γ) in G, where γ ∈ π1 (T 2 ) is represented by the loop l. We identify the variety of representations of the cyclic group ⟨γ⟩ ∼ = Z with G via the image of γ. In particular the differential form dχi : g → C may be viewed as a linear map dχi : H 1 (⟨γ⟩, Adρ) ∼ = gρ(γ) → C, because the characters χi are constant on orbits by conjugation (hence the characters are trivial on coboundaries). This uses the natural identification between H 1 (⟨γ⟩, Adρ) and the space of coinvariants gρ(γ) in Remark 6.4. Lemma 6.5. (dχ1 , . . . , dχr ) : H 1 (⟨γ⟩, Adρ) → Cr is an isomorphism. Proof. We follow the proof of [17, Corollary 19]. Steinberg has shown that the map (31) has a smooth section, Cr → G e reg [30]. In particular (dχ1 , . . . , dχr ) : g → Cr is surjective and so is the induced map (dχ1 , . . . , dχr ) : H 1 (⟨γ⟩, Adρ) ∼ = gρ(γ) → Cr . Hence the lemma follows from equality of dimensions. Consider ∂hyp O3 the union of components of ∂O3 that have negative Euler characteristic. In particular, for a manifold O3 = M 3 , ∂M 3 = ∂hyp M 3 ⊔ T12 ⊔ · · · ⊔ Tk2 where k is the number of rank-2 cusps of M 3 . Proposition 6.6. When M 3 = O3 is a manifold, then the restriction to the boundary and the characters of peripheral curves of cusps yield a local embedding X(M 3 , G) → X(∂hyp M 3 , G) × Crk , where k is the number of rank-2 cusps and the coordinates in Crk are (locally defined) funda- mental characters of chosen peripheral elements, one for each cusp. Proof. The proof of Theorem 6.1 is based in the vanishing of L2 -cohomology, here we use the fact that a form is L2 on M 3 if it is so in the restriction to each end. Hence using Lemmas 6.3 and 6.5 the kernel of the differential of the map of the proposition is represented by L2 -forms, hence trivial. The following concludes the proof of Theorem 1.1. Corollary 6.7. The restriction to the boundary yields a local embedding X(O3 , G) → X(∂O3 , G). 32 Proof. In the manifold case, we use that the injection in Proposition 6.6 factors through the restriction to X(∂M 3 , G): X(M 3 , G) → X(∂M 3 , G) → X(∂hyp M 3 , G) × Crk , Thus the map induced by restriction X(M 3 , G) → X(∂M 3 , G) must be locally an injection. For an orbifold O3 , consider a finite manifold covering O03 → O3 , we have the commutative diagram: X(O03 , G) −−−−→ X(∂O03 , G) x x     X(O3 , G) −−−−→ X(∂O3 , G) As O03 is a manifold, we have alreay shown that X(O03 , G) → X(∂O03 , G) is locally an inclusion. Furthermore, as the restriction of ρ to O03 is good, the restriction map X(O3 , G) → X(O03 , G) is also locally injective, and we are done from the commutativity of the diagram. A particular case of Proposition 6.6 is the following corollary (proved for representations of manifold groups in SL(2, C) in [3, 20], and for representations of manifold groups in SL(n, C) in [25]). Corollary 6.8. When M 3 is a manifold whose interior has finite volume, chose γ1 , . . . , γk a nontrivial peripheral element for each cusp. Let χ1 , . . . , χr denote the fundamental characters of G. e Then (χ1,γ1 , . . . , χr,γk ) : U ⊂ X(M 3 , G) → Ckr . define a local homeomorphism for a neighborhood U ⊂ X(M 3 , G) of [τ ◦ hol]. Recall that the adjoint or Sym2 gives an isomorphism PSL(2, C) ∼ = SO(3, C). By an argu- ment on dimensions (see Table 4): Corollary 6.9. Let O3 be a compact orientable three-orbifold, with hyperbolic interior of finite volume. If all components of ∂O3 are homeomorphic to S 2 (2, 2, 2, 2), S 2 (2, 4, 4) or S 2 (2, 3, 6), then X0 (O3 , SL(3, C)) ∼ = X0 (O3 , SO(3, C)) ∼ = X0 (O3 , PSL(2, C)), where X0 denotes the distinguished component. 6.2 Lower bound of the dimension In this subsection we provide a lower bound à la Thurston (for PSL(2, C)) or à la Falbel-Guilloux (for general G), in both cases for manifolds. For a compact three orbifold O3 , let ∂O3 = ∂1 O3 ⊔ · · · ⊔ ∂k O3 denote the decomposition in connected components of its boundary. Theorem 6.10. Let G be semi-simple C-algebraic Lie group, O3 a compact, orientable very good orbifold. Let ρ : π1 (O3 ) → G be a good representation. Assume that: (a) if ∂i O3 is a hyperbolic boundary component, then the centralizer of its image is zero- dimensional; (b) if ∂i O3 is a Euclidean boundary component, then the restriction ρ|π1 (O3 ) is strongly regular. 33 Then k 1X dim[ρ] X0 (Γ, G) ≥ dim[ρ|∂i ] X0 (∂i O3 , G) 2 i=1 k k 1 X X 3 ))) =− 3 e(∂i O , Adρ) + χ dim gρ(π1 (∂i O . 2 i=1 i=1 Proof. The branching locus Σ of the compact orbifold O3 is a trivalent graph. Chose a finite subset Σ0 ⊂ Σ as follows: for each component of Σ that is a circle chose precisely one point, and chose also the trivalent vertices of Σ. So Σ0 has the smallest cardinality so that Σ \ Σ0 is a disjoint union of edges (open or not, as they can meet ∂O3 ). Consider the orbifold O0 = O3 \ N (Σ0 ), ie. remove a tubular neighborhood for each point in Σ0 . The branching locus of O0 is a union of (proper) edges (it contains no vertices nor circles). The boundary components of O0 are either the boundary components of O or the boundary of a neighborhood of a point in Σ0 (spherical). For each Euclidean or spherical boundary component ∂i O0 , chose ki disjoint embedded loops γi,1 , . . . , γi,ki in O0 based at ∂i O0 so that gρ(π1 (∂O0 )) ∩ gρ(γi,1 ) ∩ · · · ∩ gρ(γi,ki ) = 0. Namely, the centralizer in g of the image of the group generated by π1 (∂i O0 ) and γi,1 , . . . , γi,ki is trivial. The next step in the construction is to remove a tubular neighborhood of the γi,j : [ O1 = O0 \ N (γi,j ). i,j Finally remove a tubular neighborhood of the branching locus of O1 (which is a union of edges): O2 = O1 \ N (ΣO1 ). In particular O2 is a manifold. The connected 3-manifold with non-empty boundary O2 retracts to a 2-dimensional CW- complex with a single 0-cell. This gives a presentation of π1 (O2 ) from this CW-complex (1-cells are generators and 2-cells relations). Each generator contributes with a copy of G in the variety of representations, and each relation decreases at most dim G the dimension; this gives the standard bound:   χ(∂O2 ) dim R(O2 , G) ≥ (1 − χ(O2 )) dim G = 1 − dim G 2 1 =− χ e(∂O2 , Adρ)) + dim G. (32) 2 Next we compute lower bounds for the orbifolds O1 , O0 , and finally for O. We shall use the following key lemma of Falbel and Guilloux, in fact it is a local version of Proposition 1 in [8]: Lemma 6.11 ([8]). Let W be a smooth complex (analytic) variety and W ′ ⊂ W a smooth subvariety. Let X be a complex variety with an analytic map f : X → W . For p ∈ X there exists a neighborhood U ⊂ X of p such that codimp (f −1 (W ′ ) ∩ U, X) ≤ codimf (p) (W ′ , W ). 34 Proof. By the implicit function theorem, there exists a neighborhood V ⊂ W of f (p) and an analytic map F : V → Ck such that V ∩ W ′ = F −1 (0), where k = codimf (p) (W ′ , W ). Then, for a neighborhood of p, U ⊂ f −1 (V ) ⊂ X, f −1 (W ′ ) ∩ U = (F ◦ f )−1 (0) ∩ U, and the estimate follows, because f −1 (W ′ ) ∩ U is a fiber of a map to Ck . To get O1 from O2 , we add the edges of ΣO1 . Topologically, we add 2-handles whose co-core is an edge of ΣO1 . For each singular edge e of ΣO1 consider a meridian µ ∈ π1 (O2 ), which is represented by the attaching circle of the 2-handle, and has a finite order in π1 (O1 ). By homogeneity, the conjugation orbit G · ρ(µ) = O(ρ(µ)) ⊂ G is a smooth analytic submanifold of G, of codimension equal the dimension of the centralizer dim(gρ(µ) ). Thus, by Lemma 6.11, when we add the edge e to O2 , the dimension of the variety of representations decreases at most by dim(gρ(µ) ): X dim R(O2 , G) − dim R(O1 , G) = codim(R(O1 , G), R(O2 , G)) ≤ dim(gρ(µ) ). (33) µ Furthermore, adding the 2-handle to O2 corresponding to µ means replacing an annulus by two cone points with cyclic stabilizer generated by µ. Hence, the twisted Euler characteristic increases by 2 dim(gρ(µ) ). Counting the contribution of all edges of ΣO1 : X χ e(∂O1 , Adρ) = χ e(∂O2 , Adρ) + 2 dim(gρ(µ) ). (34) µ Hence from (32), (33), and (34): 1 dim R(O1 , G) ≥ − χe(∂O1 , Adρ) + dim G. (35) 2 To get O0 from O1 , we add the neighborhoods of the loops γi,j , i.e. we add 2-handles (without singular co-core). Consider the i-th boundary component ∂i O1 . The corresponding boundary component ∂i O0 is obtained from ∂i O1 by the surgery corresponding to the addition of ki 2-handles. Namely ki annuli in ∂i O1 corresponds to 2ki smooth disks in ∂i O0 . Thus e(∂i O0 , Adρ) = χ χ e(∂i O1 , Adρ) + 2ki dim G. (36) On the other hand, following the idea of [8], we apply Lemma 6.11 to ∂i O1 and the free product π1 (∂i O0 ) ∗ ⟨γi,1 ⟩ ∗ · · · ∗ ⟨γi,ki ⟩, that is, the group obtained by filling the ki meridians of the surface ∂ i O1 : W ′ = R(π1 (∂i O0 ) ∗ ⟨γi,1 ⟩ ∗ · · · ∗ ⟨γi,ki ⟩, G) ⊂ R(∂i O1 , G) = W. (37) By Proposition 4.1, and since we assume that the centralizer of ρ(π1 (∂i O1 )) in g is trivial, dim W = dim R(∂i O1 , G) = −e χ(∂i O1 , Adρ) + dim G. (38) By Proposition 4.1 when ∂i O0 is hyperbolic, Theorem 4.11 when Euclidean, and Remark 4.17 when spherical, R(π1 (∂i O0 ), G) is smooth and χ(∂i O0 , Adρ) + dim G + dim gρ(π1 (∂i O0 )) . dim R(∂i O0 , G) = −e (39) As the variety of representations of a free product is the Cartesian product of varieties or representations of its factors, W ′ is smooth and: dim W ′ = dim R(π1 (∂i O0 ) ∗ ⟨γi,1 ⟩ ∗ · · · ∗ ⟨γi,ki ⟩, G) = dim R(∂i O0 , G) + ki dim G. (40) 35 From (39) and (40): dim W ′ = −e χ(∂i O0 , Adρ) + (ki + 1) dim G + dim gρ(π1 (∂i O0 )) . (41) It follows from (38) and (41) that the codimension of the inclusion (37) is codim(W ′ , W ) = dim W − dim W ′ = χ e(∂i O1 , Adρ) − ki dim G − dim gρ(π1 (∂i O0 )) , e(∂i O0 , Adρ) − χ that combined with (36) yields codim(W ′ , W ) = ki dim G − dim gρ(π1 (∂i O0 )) . (42) So when we apply Lemma 6.11 we get from the contribution (42) of each boundary component: X  codim(R(O0 , G), R(O1 , G)) ≤ ki dim G − dim gρ(π1 (∂i O0 )) . (43) i Thus, with (35) and (36), (43) becomes: X  dim R(O0 , G) ≥ dim R(O1 , G) − ki dim G − dim gρ(π1 (∂i O0 )) i X1  ≥ − χe(∂i O1 , Adρ)) − ki dim G + dim gρ(π1 (∂i O0 ) + dim G 2 i X 1  = − χe(∂i O0 , Adρ)) + dim gρ(π1 (∂i O0 ) + dim G. 2 i Finally, for each spherical boundary component ∂i O0 , by (25) 1 − χe(∂i O0 , Adρ)) + dim gρ(π1 (∂i O0 ) = 0. 2 Hence we can get rid of the contribution of the neighborhoods of vertices and get the initial orbifold O3 : X 1 3  dim R(O3 , G) ≥ − χ e(∂i O3 , Adρ)) + dim gρ(π1 (∂i O ) + dim G 2 i = dim X(∂O3 , G) + dim G. As dim X(O3 , G) = dim R(O3 , G) − dim G, because the representation is good, this concludes the proof of the theorem. 6.3 Dimension growth Let M 3 be an orientable hyperbolic 3-manifold of finite volume with k ≥ 1 cusps. As M 3 is a manifold, its holonomy lifts to SL(2, C). Let X0 (M, SL(n, C)) be the canonical or distinguished component of X(M, SL(n, C)) (Def- inition 6.2), i.e. the component that contains the composition of a lift of the holonomy with Symn−1 . By Theorem 6.1, dim X0 (M, SL(n, C)) = (n − 1)k, where k is the number of cusps. This linear growth differs from the quadratic growth of 2- orbifolds in Section 5. Those 2-orbifolds may appear as basis of Seifert fibered Dehn filling and yield components in the variety of characters of higher dimension. Next we discuss two examples, the figure eight knot and the Whitehead link exteriors. The following is Proposition 1.6 from the introduction. 36 Proposition 6.12. Let Γ be the fundamental group of the exterior of the figure eight knot. Besides the canonical component (that has dimension n − 1), for large n X(Γ, SL(n, C)) has at least 3 components that contain irreducible representations, whose dimension grow respectively as n2 /12, n2 /20 and n2 /42. Proof. Let Kp/q denote the Dehn surgery on the figure eight knot with coefficients p/q. There are 6 Dehn fillings on the figure eight knot that yield small Seifert manifolds [11, 24]: K±3 are small Seifert manifolds fibered over the 2-orbifold O32 = S 2 (3, 3, 4), K±2 over O22 = S 2 (2, 4, 5), and K±1 over O12 = S 2 (2, 3, 7). They induce representations in SL(2, R) of the filled 3-manifolds Kp , that map the fiber to − Id and induce the holonomy representation of the (unique) hyper- bolic structure on Op2 . The composition with Symn−1 is a representation that maps the fiber to (−1)n−1 Id and it induces a representation from π1 (Op2 ) to PSL(n, R) in the Hitchin com- ponent. Therefore the complexification of the Hitchin component of Op2 yields a component of X(Kp , SL(n, C)) of the same dimension, and hence a subvariety of X(Γ, SL(n, C)). Let Xp ⊂ X(M, SL(n, C)) be the component of X(Γ, SL(n, C)) that contains the subvariety in- duced by X(Kp , SL(n, C)). Next we estimate the dimension of X1 , X2 and X3 , which allow to distinguish them, but does not allow to distinguish the component induced by Xp from X−p , as the dimension estimates are the same. The lower bound on the dimension of Xp is given by Proposition 5.10: dim Xp ≥ −χ(Op2 )n2 − c(Op2 ). Aiming to find an upper bound of dim Xp , we bound above the dimension of its Zariski tangent space, which is isomorphic to H 1 (Γ, Adρ) ∼ = H 1 (M, Adρ) for M the (compact) exterior of the figure eight knot, M = S \ N (K). In particular ∂M ∼ 3 = T 2 . Here ρ = Symn−1 ◦ρ0 , for ρ0 : Γ → SL(2, R) the representation that factors through Kp . First we bound dim H 1 (∂M, Adρ). Notice that since χ(T 2 ) = 0, dim H 1 (∂M, Adρ) = 2 dim H 0 (∂M, Adρ) = 2 dim gAdρ(π1 (∂M )) . To compute dim gAdρ(π1 (∂M )) , one may check explicitly that the image of ρ0 (π1 (∂M )) contains an element of SL(2, R) of infinite order (one may use for instance the A-polynomial). As the symmetric power of a hyperbolic matrix in SL(2, C) is regular, ρ(π1 (∂M )) contains regular elements and dim gAdρ(π1 (∂M )) = rank G = n − 1, Thus dim H 1 (∂M, Adρ) = 2(n − 1). Let i∗ : H 1 (M, Adρ) → H 1 (∂M, Adρ) be the morphism induced by inclusion. Using the long exact sequence of the pair (M, ∂M ) and Poincaré duality as in the proof of Theorem 6.1, its rank is rank(i∗ ) = n − 1. (44) Furthermore, by Mayer-Vietoris exact sequence applied to the decomposition Kp = M ∪∂M (D2 × S 1 ), we get that dim ker i∗ ≤ dim H 1 (Kp , Adρ). (45) Next we need a lemma: Lemma 6.13. The projection π1 (Kp ) → π1 (Op2 ) induces an isomorphism H 1 (π1 (Kp ), Adρ) ∼ = H 1 (π1 (Op2 ), Adρ). (46) 37 Assuming the lemma, we conclude the proof of Proposition 6.12. Putting together (44), (45) and (46): dim H 1 (M, Adρ) ≤ dim H 1 (Op2 , Adρ) + (n − 1). It follows that −n2 χ(Op2 ) − c(Op2 ) ≤ dim Xp ≤ −n2 χ(Op2 ) + c(Op2 ) + n − 1. For large n this allows to distinguish three components, according to the different values for χ(Op2 ). Proof of Lemma 6.13. We use the central exact sequence 1 → Z(π1 (Kp )) → π1 (Kp ) → π1 (Op2 ) → 1, where the center Z(π1 (Kp )) ∼= Z is the group generated by the fiber. To prove the lemma, we use crossed morphisms or derivations, as in Subsection 2.1. Since π1 (Kp ) → π1 (Op2 ) is a surjection, we have an injection of spaces of cocycles Z 1 (π1 (Op2 ), g) ,→ Z 1 (π1 (Kp ), g). (47) We prove surjectivity of the map in (47): let d : π1 (Kp ) → g be a crossed morphism (namely, satisfying d(γ1 γ2 ) = d(γ1 ) + Adρ(γ1 ) d(γ2 ), ∀γ1 , γ2 ∈ π1 (Kp )). Let t ∈ π1 (Kp ) be a generator of the center Z(π1 (Kp )) ∼= Z; from the relation tγt−1 = γ ∀γ ∈ π1 (Kp ), we deduce d(t) + Adρ(t) d(γ) − Adρ(tγt−1 ) d(t) = d(γ), ∀γ ∈ π1 (Kp ). Since ρ(t) = ± Id, Adρ(t) is the identity on g, hence (1 − Adρ(γ) )d(t) = 0, ∀γ ∈ π1 (Kp ). This equality implies that d(t) = 0, because gρ(π1 (Kp )) = 0 by irreducibility of ρ. This proves that every crossed morphism of π1 (Kp ) factors through a crossed morphism of π1 (O2 ), hence (47) is surjective. This isomorphism between cocycle spaces induces an isomorphism of coboundary space and this proves the lemma. Remark 6.14. For small values of n, the arguments do not apply, but for n = 3 the canonical component has dimension 2, and according to Table 6, (±3)-Dehn fillings also provide two sub- varieties of dimension 2. It is proved in [9] and [15] that the canonical component and the sub- varieties induced by the (±3)-Dehn filling are precisely the three components of X(Γ, SL(3, C)) that contain irreducible representations. Remark 6.15. At the moment, we do not know whether for large n the component that contains representations that factor through Kp is the same as the component corresponding to K−p , for p = 1, 2, 3. To distinguish them would require a better upper bound of the dimension of their Zariski tangent space. This would allow also to distinguish components corresponding to Galois conjugates (5-th or 7-th roots of unity). Remark 6.16. An analog of Proposition 6.12 can be obtained for the variety of characters of the figure eight knot in Sp(2m, C), for large m. 38 Proposition 6.17. Let Γ be the fundamental group of the exterior of the Whitehead link. Besides the canonical component (that has dimension 2n − 2), for large n, X(Γ, SL(n, C)) has at least 3 components that contain irreducible characters, whose dimension grow respectively as n2 /3, n2 /4 and n2 /6. The proof of Proposition 6.17 is the same as Proposition 6.12. Here there are partial Dehn fillings that yield Seifert fibered manifolds, with basis a disc with two cone points: the (−3)- filling fibers over D2 (3, 3), the (−2)-filling over D2 (2, 4), and the (−1)-filling is fibered over D2 (2, 3) [24, Table A.1]. This yields the three rational orbifold Euler characteristics, 1/3, 1/4 and 1/6. As the basis orbifolds are rather simple, it is also easy to state the dimension of their Hitchin component. The dimension of the Hitchin component of D2 (3, 3) is ( 2 n 3 + 1 for n ≡ 0 mod 3 n2 −1 3 for n ̸≡ 0 mod 3 The dimension of the Hitchin component of D2 (2, 4) is  2 n  4 + 1 for n ≡ 0 mod 4  n2 −1 4 for n ≡ 1 mod 2  n2  4 for n ≡ 2 mod 4 The dimension of the Hitchin component of D2 (2, 3) is  2 n  6 + 1 for n ≡ 0 mod 6    n2 −1  for n ≡ ±1 mod 6 6 n2 +2    6 for n ≡ 2 mod 6  n2 +3  for n ≡ 3 mod 6 6 Remark 6.18. When we apply these computation to SL(3, C), for D2 (2, 4) and D2 (2, 3) it yields dimension 2, but for D2 (3, 3) it has a component of dimension 4, the same as the dimension of the canonical component. It has been proved by Guilloux and Will [12] that this is in fact a whole component of X(Γ, SL(3, C)). For Montesinos links, the same arguments yield the following: Proposition 6.19. Let L ⊂ S 3 be a Montesinos link. Assume that either it has at least 4 tangles, or that it has three tangles (eg. a pretzel link) with indices α1 , α2 , and α3 satisfying 1 1 1 3 α1 + α2 + α3 ≤ 1. Then dim X(S \ L, SL(n, C)) grows quadratically with n. 7 Representations in SL(3, C) In this section we give explicit computations of some varieties of characters in SL(3, C). We compute varieties of characters of groups generated by two elements, using the description of X(F2 , SL(3, C)) due to Lawton [21], where F2 = ⟨a, b |⟩ is the free group of rank two. Setting coordinates x = ta , y = tb , z = tab , r = tab−1 , τ = t[a,b] , (48) u = ta−1 , v = tb−1 , w = t(ab)−1 , s = ta−1 b , Lawton has proved: 39 Theorem 7.1 ([21]). X(F2 , SL(3, C)) is the hypersurface of C9 defined by τ2 − Pτ + Q = 0 for some polynomials P, Q ∈ Z[x, y, z, u, v, w, r, s]. The solutions of τ 2 − P τ + Q = 0 are t[a,b] and t[b,a] . Namely t[a,b] + t[b,a] = P and t[a,b] t[b,a] = Q. The polynomials P and Q in Theorem 7.1 can be found in [21]: P = xuyv − uyr − xvs − uvz − xyw + rs + xu + yv + zw − 3 and Q = vu2 x2 y + uv 2 y 2 x − ryxu2 − uxvrw − ry 2 vu + rx2 v 2 − rzyxv + su2 y 2 − swvuy − svux2 − uxysz − sv 2 yx − u3 vy + u2 v 2 w − zu2 xv − uv 3 x − zv 2 yu − wx2 uy − uy 3 x − wy 2 vx − x3 vy + x2 y 2 z + uwr2 − 2 r2 xv + r2 zy + surx + svry + wszr + rzu2 + ruv 2 + rvz 2 + xrw2 + rwy 2 + x2 yr − 2 s2 uy + s2 wv + xzs2 + u2 vs + usz 2 + szv 2 + syw2 + swx2 + sxy 2 + u2 wy − 2 w2 vu + xuyv + wuzx + uy 2 z + xv 2 w + wvzy + x2 zv − 2 z 2 yx + r3 + 3 uyr − 3 rvw − 3 rzx + s3 − 3 swu + 3 xvs − 3 syz + u3 + 3 uvz + v 3 + w3 + 3 xyw + x3 + y 3 + z 3 − 6 rs − 6 xu − 6 yv − 6 zw + 9 This theorem tells that an SL(3, C)-character of F2 is a polynomial on the coordinates x, y, z, u, v, w, r, s, τ . Its expression can be computed by using basic identities on traces, including Cayley-Hamilton’s formula A3 − tr(A)A2 + tr(A−1 )A − Id = 0, ∀A ∈ SL(3, C), (49) and elementary identities on traces. 7.1 Two dimensional examples Let us compute the variety of SL(3, C)-characters of some 2-orbifolds. We start with the turnover S 2 (3, 3, 4), Figure 7. Its fundamental group has presentation π1 (S 2 (3, 3, 4)) ∼ = ⟨a, b | a3 = b3 = (ab)4 = 1⟩ We look at the possible eigenvalues of the elements of finite order: (a) A matrix in SL(3, C) of order 3 is either central or has eigenvalues {1, ω, ω 2 }, for ω ∈ C a primitive third root of unity: ω 3 = 1 and ω ̸= 1. (b) A matrix in SL(3, C) of order 4 is either trivial or has eigenvalues {1, i, −i}, {−1, i, i}, {−1, −i, −i}, or {1, −1, −1}. Remark 7.2. Let ρ : π1 (S 2 (3, 3, 4)) → SL(3, C) be an irreducible representation. The eigen- values of ρ(a) and ρ(b) are {1, ω, ω 2 } and the eigenvalues of ρ(ab) are {1, i, −i}, {−1, i, i}, {−1, −i, −i}, or {1, −1, −1}. Furthermore, using that for an irreducible representation dim[ρ] X(S 2 (3, 3, 4), SL(3, C)) = −e χ(S 2 (3, 3, 4), Adρ), and dim gρ(ab) depends on whether ρ(ab) has repeated eigenvalues or not, we have: ( 2 2 if the eigenvalues of ρ(ab) are {1, i, −i}, dim[ρ] X(S (3, 3, 4), SL(3, C)) = 0 otherwise. 40 4 4 3 4 3 3 3 3 S 2 (3, 3, 4) T (3, 3, 4) D(3; 4) Figure 7: The turnover S 2 (3, 3, 4) and two nonorientable quotients: a triangle with mirror boundary T (3, 3, 4), which is a Coxeter group, and a disc with a cone point, mirror boundary, and a corner reflector, D(3; 4). Using the coordinates (48), the two dimensional components of the variety of characters are obtained by setting x = y = u = v = 0 and z = w = 1, because this fixes the eigenvalues of ρ(a), ρ(b) and ρ(ab). By replacing those values in Theorem 7.1, we obtain: Example 7.3. The component of X(S 2 (3, 3, 4), SL(3, C)) that has positive dimension is iso- morphic to {(r, s, τ ) ∈ C3 | τ 2 − (rs − 2)τ + (r3 + s3 − 5 rs + 5) = 0}. For the components that are isolated points, we set, again x = y = u = v = 0 and, according to the eigenvalues of ρ(ab): (z, w) = (−1 + 2 i, −1 − 2 i), (z, w) = (−1 − 2 i, −1 + 2 i), or (z, w) = (−1, −1). Notice that this does not fix the conjugacy class of ρ(ab), because it has eigenvalues of multi- plicity 2. We need to find further restrictions on the traces: by taking traces on the relation (ab)−2 a = (ab)2 a, using (49) and replacing x = y = u = v = 0 we get s = zs and r = wr. (50) For the 2-dimensional component it holds z = w = 1, hence (50) do not give any further information. For the isolated components, this yields r = s = 0. For these values P 2 − 4Q = 0 and therefore there is a unique value for τ . Hence we get: Example 7.4. There are three components of X(S 2 (3, 3, 4), SL(3, C)) that contain irreducible representations and are zero-dimensional. They have coordinates x = y = u = v = r = s = 0 and (z, w, τ ) = (−1 + 2 i, −1 − 2 i, 1), (−1 − 2 i, −1 + 2 i, 1), or (−1, −1, −1). Next we describe the Hitchin component Hit(S 2 (3, 3, 4), PSL(3, R)). Notice that the repre- sentation Sym2 : SL(2, R) → SL(3, R) factors through PSL(2, R), hence Hit(S 2 (3, 3, 4), PSL(3, R)) lifts to a component of X(S 2 (3, 3, 4), SL(3, R)). We consider the 2-dimensional component of Example 7.3 and we require that its coordinates are real: {(r, s, τ ) ∈ R3 | τ 2 − (rs − 2)τ + (r3 + s3 − 5 rs + 5) = 0}. By looking at the discriminant of this equation, this real set has has three components. One of them is the isolated point (r, s, τ ) = (2, 2, 1), which is in fact a singular points of the defining surface. The other two components are homeomorphic to R2 , one of them is the lift of the Hitchin component. By looking at the symmetric power of the holonomy, we deduce: 41 Example 7.5. The Hitchin component Hit(S 2 (3, 3, 4), PSL(3, R)) is isomorphic to the compo- nent of {(r, s, τ ) ∈ R3 | τ 2 − (rs − 2)τ + (r3 + s3 − 5 rs + 5) = 0} √ √ √ that contains the point of coordinates (r, s, τ ) = (2 + 2 2, 2 + 2 2, 5 + 4 2). Next we consider the turnover T (3, 3, 4), Figure 7. Its fundamental group is an extension of π1 (S 2 (3, 3, 4)) by an involution σ, that satisfies σ 2 = 1, σaσ = a−1 , and σbσ = b−1 . (51) Denote by σ∗ the involution induced on the fundamental group: σ∗ (γ) = σγσ −1 , ∀γ ∈ π1 (S 2 (3, 3, 4)). This involution σ∗ permutes the coordinates r and s and preserves τ , the trace of the commu- tator. Therefore, the component of X(S 2 (3, 3, 4), SL(3, C))σ∗ of positive dimension is {(r, τ ) ∈ C2 | τ 2 − (r2 − 2)τ + (2r3 − 5 r2 + 5) = 0}. This curve is singular at precisely at (r, τ ) = (2, 1), and the singularity is an ordinary double point (a self intersection with transverse tangents). Lemma 7.6. The restriction map X(T (3, 3, 4), SL(3, C)) → X(S 2 (3, 3, 4), SL(3, C))σ∗ desingularizes the curve τ 2 − (r2 − 2)τ + (2r3 − 5 r2 + 5) = 0. The component of X(T (3, 3, 4), SL(3, C)) of positive dimension that contains irreducible rep- resentations is this desingularization. Proof. We look at the fibre of the restriction map. We show that: (a) the fibre of an irreducible character consists precisely of one point, (b) (r, τ ) = (2, 1) is the only reducible character in X(S 2 (3, 3, 4), SL(3, C))σ∗ , and (c) the fibre of (r, τ ) = (2, 1) consists precisely of 2 points. Those three items prove the lemma, because the singularity is an ordinary double point. We prove (a): For an irreducible representation ρ of π1 (S 2 (3, 3, 4)) in SL(3, C) we show that there is a unique choice of A ∈ SL(3, C) such that mapping σ to A defines an extension of ρ to π1 (T (3, 3, 4)). By irreducibility of ρ, as ρ and ρ ◦ σ∗ have the same character, there exists a matrix A ∈ SL(3, C) such that AρA−1 = ρ ◦ σ∗ . Furthermore, A is unique up multiplication by a matrix in the center (the center of SL(3, C) is {Id, ωId, ω 2 Id}). As σ is an involution, A2 commutes with ρ, that is irreducible, and therefore A2 ∈ {Id, ωId, ω 2 Id}. Hence among A, ωA or ω 2 A there is a unique choice whose square is the identity. Thus there is a unique choice that satisfies (51). To prove (b) we use that for a reducible representation the trace of a commutator and its inverse are the same, thus it must be a zero of the discriminant of the defining equation. This discriminant is (r − 2)2 (r2 − 4r − 4) (52) √ √ and its zeros are r = 2 and r = 2 ± 2 2. The value r = 2 ± 2 2 corresponds to the symmetriza- tion of the fuchsian holonomy (and its Galois conjugate), hence it is irreducible. Thus r = 2 is the only reducible character. 42 To prove (c), we check that r = 2 corresponds to a representation ρ = ρ2 ⊕ Id, for ρ2 : π1 (S 2 (3, 3, 4)) → SL(2, C) an irreducible representation (for a reducible representation in SL(2, C) the trace of any commutator is 2 and, as τ = 1, trace(ρ2 ([a, b])) = τ − 1 ̸= 2). Then, reproducing the argument of the irreducible case (a), there exist a matrix A ∈ SL(2, C) such that Aρ2 A−1 = ρ2 ◦ σ∗ and, by irreducibility of ρ2 , A is unique up to sign. Furthermore, again by irreducibility of ρ2 , A2 is central in SL(2, C), namely A2 = ± Id. The case A2 = + Id does not occur, because this would imply that A = ± Id and with (51) this would yield that ρ2 itself would be central, but it is irreducible. Therefore A2 = − Id and the choices for ρ(σ) are   0 ±i A ρ(σ) =  0 . 0 0 −1 This concludes the proof of (c) and of the lemma. Next we look for the Hitchin component of the turnover. We look therefore at the real points {(r, τ ) ∈ R2 | τ 2 − (r2 − 2)τ + (2r3 − 5 r2 + 5) = 0}. By looking at the discriminant (52), the set of real points √ has three components, √ the isolated √ point (r, τ ) = (2, 1) an two lines, defined by r ≤ 2 − 2 2 and r ≥ 2 + 2 2. As r = 2 + 2 2 is the symmetric power of the holonomy: Example 7.7. The Hitchin component Hit(T (3, 3, 4), PSL(3, R)) is isomorphic (via the restric- tion to S 2 (3, 4, 4)) to the line √ {(r, τ ) ∈ R2 | τ 2 − (r2 − 2)τ + (2r3 − 5 r2 + 5) = 0, r ≥ 2 + 2 2}. Finally, we consider D(3; 4), the disc with a cone point of order 3, mirror boundary, and a corner reflector of order 4 (with isotropy group the dihedral group of 8 elements). It is again the quotient of S 2 (3, 3, 4) by an involution. This involution maps a to b−1 and b to a−1 . Therefore it fixes the coordinates r and s but permutes the trace of [a, b] with the trace of its inverse [b, a]. The fixed point set of this involution in X(S 2 (3, 3, 4), SL(3, C)) is obtained by looking at the zero locus of the discriminant of the variable τ : {(r, s) ∈ C2 | r2 s2 − 4 r3 − 4 s3 + 16 rs − 16 = 0}. With the very same discussion as in Lemma 7.6: Lemma 7.8. The restriction map ′ X(D(3; 4), SL(3, C)) → X(S 2 (3, 3, 4), SL(3, C))σ desingularizes the curve r2 s2 − 4 r3 − 4 s3 + 16 rs − 16 = 0. The component of X(D(3; 4), SL(3, C)) of positive dimension that contains irreducible char- acters is this desingularization. The set of real points of r2 s2 − 4 r3 − 4 s3 + 16 rs − 16 = 0 is the discriminant locus of the set of real points for S 2 (3, 3, 4): it has three components, the isolated point r = s = 2 and two unbounded curves. Thus, similarly to S 2 (3, 3, 4) we have: Example 7.9. The Hitchin component Hit(D(3; 4), PSL(3, R)) is isomorphic (via the restriction to S 2 (3, 4, 4)) to the component of {(r, s) ∈ R2 | r2 s2 − 4 r3 − 4 s3 + 16 rs − 16 = 0} √ that contains r = s = 2 + 2 2. 43 7.2 A family of 3-dimensional orbifolds Consider the family of examples with underlying space S 3 (perhaps punctured twice) and three branching arcs as in Figure 8, with branching labels n0 , n1 and n2 ≥ 2. Whether a singular vertex is included or not depends on the orders of the adjacent branching locus: if 1/n0 +2/n1 > 1, then the three edges meet at a vertex, otherwise the vertex is removed, and similarly for the other vertex. In other words, we consider the twice punctured 3-sphere, and we add the vertices needed for the orbifold to be irreducible. n2 n1 n0 Figure 8: The underlying space of the orbifold is a 3-sphere, possibly punctured once or twice, according to the branching indices, the branching locus consists of three arcs as in the picture. Let O3 denote this orbifold. The geometry of O3 depends on the indices: (a) When n2 = 2 and 1/n0 +2/n1 ≤ 1, it is an interval bundle, over a disc with mirror boundary, with an interior cone point of order n1 and a corner reflector of order 2n0 . It is doubly covered by the product of a turnover S 2 (n0 , n1 , n1 ) by an interval. (b) When n2 = 2 and 1/n0 + 2/n1 > 1, it is obtained by coning the boundary of the previ- ous example, and it is spherical. It is doubly covered by the suspension over a turnover S 2 (n0 , n1 , n1 ). (c) When (n0 , n1 , n2 ) = (2, 3, 3) the orbifold is spherical [7]. (d) If n1 , n2 ̸= 2 and (n0 , n1 , n2 ) ̸= (2, 3, 3), then it is hyperbolic. The hyperbolicity is realized by a tetrahedron as in Figure 9. The vertices are possibly finite, ideal or exterior (hence truncated by geodesic triangles perpendicular to the other sides). Poincaré fundamental domain theorem yields the following presentation for the fundamental group π1 (O3 ) ∼ = ⟨a, b | an1 = bn2 = [a, b]n0 = 1⟩. In case (a) the variety of characters of O3 in SL(3, C) is the variety of characters of a non- orientable 2-orbifold. It has dimension 1 or 0 according to the values of n0 and n1 . Cases (b) and (c) are spherical, hence the variety of characters is finite. In the hyperbolic case, (d), when n0 = 2 then the canonical component is an isolated point, by Theorem 6.1. For ni ≥ 3, it has dimension 2. Example 7.10. Consider the case n0 = n1 = n2 = 3. The fundamental group is π1 (O3 ) = ⟨a, b | a3 = b3 = [a, b]3 = 1⟩. On the canonical component, the image of any of a, b and [a, b] is not central, therefore by Theorem 7.1, using the notation in (48) we impose the following equalities: x = y = u = v = P = Q = 0. Thus the canonical component X0 (O3 , SL(3, C)) is the following complex surface:  rs + zw − 3 = 0 r3 + s3 + z 3 + w3 + rszw − 6rs − 6zw + 9 = 0 44 π 2n0 2π n2 π 2n0 π 2π A 2n0 n1 B π 2n0 Figure 9: The fundamental domain of the orbifold in this subsection. It is an ideal tetrahedron in hyperbolic 3-space, with vertices perhaps ideal or hyperideal. The side parings, a and b, are rotations of order n1 and n2 around two opposite edges. Those edges have dihedral angle 2π/n1 and 2π/n2 respectively, the remaining four edges have dihedral angle π/(2n0 ). The symmetric power of the complete structure has coordinates √ √ −3 + −3 −3 − −3 r=s= , z=w= 2 2 (The complex conjugate corresponds to a change of orientation). There are other components of X(O3 , SL(3, C)) but it may be checked that they are isolated points. For instance, if ρ(a) is central, then ρ(b) has order three at this gives finitely many conjugacy classes for ρ (whether ρ(b) is central or not). If ρ(a) and ρ(b) are noncentral but ρ([a, b]) is central, then it can be computed that x = y = z = r = u = v = w = s = 0. 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