Dimension of representation and character varieties for two and three-orbifolds
aa r X i v : . [ m a t h . G T ] S e p Dimension of representation and character varietiesfor two and three-orbifolds
Joan Porti ∗ September 8, 2020
Abstract
We consider varieties of representations and characters of 2 and 3-dimensional orbifoldsin 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 holonomycomposed 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 boundfor 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 applythis computation to the dimension growth of varieties of characters of some 3-dimensionalmanifolds in SL( n, C ). In this paper we are interested in varieties of representations and characters of the fundamentalgroup of two and three-dimensional orbifolds O in complex semi-simple algebraic Lie groups G ,denoted respectively by R ( O , G ) = hom( π ( O ) , G ) and X ( O , G ) = R ( O , G ) //G. Since those are not irreducible, we use dim ρ R ( O , G ) and dim [ ρ ] X ( O , G ) to denote the dimensionof a component that contains ρ .We discuss first a particular representation for oriented hyperbolic 3-orbifolds, with holon-omy in Isom + ( H ) ∼ = PSL(2 , C ). Let G be an adjoint complex simple Lie group, eg. PSL( n, C ),PO( n, C ) or PSp(2 m, C ). Let τ : PSL(2 , C ) → G be the principal representation (see § n − : PSL(2 , C ) → PSL( n, C ),whose image is contained in PSp(2 m, C ) for n = 2 m or in PO(2 m + 1 , C ) for n = 2 m + 1. When G R is a split real form of G , it restricts to τ : PGL(2 , R ) → G R , so that the Hitchin componentof a surface or a 2-orbifold is the component of the variety of representations that containsthe composition of τ with a Fuchsian representation. Here we consider orientable hyperbolic3-orbifolds and we compose the holonomy in PSL(2 , C ) with the principal representation τ . Weprove the following generalization of [26, 27]: Theorem 1.1.
Let O be a compact orientable 3-orbifold whose interior is hyperbolic withholonomy hol : π ( O ) → PSL(2 , C ) . Then the character of τ ◦ hol is a smooth point of X ( O , G ) of dimension dim [ τ ◦ hol] X ( O , G ) = 12 dim X ( ∂ O , G ) . ∗ Partially supported by the Micinn/FEDER grant PGC2018-095998-B-I00 urthermore, the map induced by restriction X ( O , G ) → X ( ∂ O , G ) is locally injective at thecharacter of τ ◦ hol . The proof of Theorem 1.1 is based on a vanishing theorem of L -cohomology, as for manifoldsin [26, 27]. Notice that for a Euclidean component O of ∂ O , the restriction of τ ◦ hol can bea singular point of X ( O , G ), and to prove that X ( O , G ) → X ( ∂ O , G ) is locally injective weuse evaluation of characters.A representation ρ : Γ → G is called good if it is irreducible and the centralizer of ρ (Γ) in G isthe center Z ( G ) of G , in particular g ρ (Γ) = 0. We are interested in representations of Euclidean2-orbifolds (eg. χ ( O ) = 0) that may be not good. For instance, nontrivial representationsof a 2-torus are never good by Kolchin’s theorem. A representation of a Euclidean 2-orbifold ρ : π ( O ) → G is called strongly regular if for a maximal torsion free subgroup Γ < π ( O ), itholds: • dim g ρ (Γ ) = rank G , where g ρ (Γ ) = { v ∈ g | Ad ρ ( γ ) ( v ) = v, ∀ γ ∈ Γ } , and • the projection of ρ (Γ ) 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 [9]: Theorem 1.2.
Let O be a compact orientable good 3-orbifold, with boundary ∂ O = ∂ O ⊔· · · ⊔ ∂ k O , and χ ( ∂ i O ) ≤ for i = l, . . . , k . Let ρ : π ( O ) → G be a good representation suchthat: • If χ ( ∂ i O ) < , then ρ | π ( ∂ i O ) is good. • If χ ( ∂ i O ) = 0 , then ρ | π ( ∂ i O ) is strongly regular.Then dim [ ρ ] X ( O , G ) ≥
12 dim X ( ∂ O , G ) . If ∂ O = ∂ O ⊔ · · · ⊔ ∂ k O denotes the decomposition in connected components, then X ( ∂ O , G ) = X ( ∂ O , G ) × · · · × X ( ∂ k O , G ) anddim X ( ∂ O , G ) = dim X ( ∂ O , G ) + · · · + dim X ( ∂ k O , G ) . When the component ∂ i O is a (closed orientable) surface S :dim X ( S, G ) = ( − χ ( S ) dim G if χ ( S ) < , G if χ ( S ) = 0 , which can be rewritten asdim X ( S, G ) = − χ ( S ) dim G + 2 dim g ρ ( π S ) , where g H = { v ∈ g | Ad ρ ( γ ) ( v ) = v, ∀ γ ∈ H } denotes the centralizer in the Lie algebra of asubset H ⊂ G , via the adjoint action. When the ∂ i O is a (closed orientable) 2-orbifold O with branching locus Σ ⊂ O , we prove in Theorems 1.3 and 1.4:dim X ( O , G ) = − χ ( O \ Σ) dim G + X x ∈ Σ dim g ρ (Stab( x )) + 2 dim g ρ ( π ( O )) . (1)Formula (1) suggests that, instead of the usual Euler characteristic of an orbifold, we needanother quantity to compute dimensions of varieties of characters and representations. For2hat purpose, consider K a CW -structure on a compact orbifold O n and ρ : π ( O n ) → G arepresentation. The twisted Euler characteristic is defined (Definition 3.1) as e χ ( O n , Ad ρ ) = X e cell of K ( − dim e dim g Ad( ρ (Stab(˜ e ))) ∈ Z . 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 characteristicof the underlying space of the orbifold times dim G . It is the alternated sum of dimensions ofcohomology groups of O n twisted by Ad ρ (Proposition 3.2): e χ ( O n , Ad ρ ) = X i ( − i dim H i ( O n , Ad ρ ) . In Section 4 we prove the following results, based on Goldman’s work on surfaces [11] (seealso [34]):
Theorem 1.3.
Let O be a compact connected 2-orbifold, with χ ( O ) ≤ . Let ρ ∈ R ( O , G ) be a good representation. Then [ ρ ] is a smooth point of X ( O , G ) of dimension − e χ ( O , Ad ρ ) . Theorem 1.4.
Let O be a closed Euclidean 2-orbifold and ρ : π ( O ) → G a strongly regularrepresentation. Then it belongs to a single component of X ( O , G ) that has dimension(a) − e χ ( O , Ad ρ ) + 2 dim g ρ ( π ( O )) if O is orientable,(b) − ˜ χ ( O , Ad ρ ) + dim g ρ ( π ( f O )) if O is non-orientable, where f O → O denotes the orienta-tion covering of O . Notice that Theorem 1.4 does not conclude smoothness. For instance, a parabolic represen-tation of Z in SL(2 , C ) is strongly regular but its character (the trivial character) is a singularpoint of X ( Z , SL(2 , C )) and X ( Z , PSL(2 , C )).Under the hypothesis of Theorem 1.3 or Theorem 1.4, when O is closed and orientable then e χ ( O , Ad ρ ) is even (Corollary 3.5) and therefore dim [ ρ ] X ( O , G ) is even.Most of the computations apply to real Lie groups, in particular we spend a section discussingapplications to the Hitchin component, that we denote by Hit( O , G R ), where G R is the splitreal form of the adjoint group G . As we consider non-orientable non-connected real forms G R : we require that G R contains the image by the principal representationof PGL(2 , R ), not only of its identity component PSL(2 , R ). For instance G R = PSL( n, R ),PSp ± (2 m ) or PO( n, n + 1).The dimension of Hit( O , G R ) has already been computed in [1] by Alessandrini, Lee andSchaffhauser, but we give a different approach, closer to the one of Long and Thistlethwaite in[24] for turnovers. For instance, we show that (cid:12)(cid:12) dim(Hit( O , PGL( n, R ))) + χ ( O ) dim(PGL( n, R )) (cid:12)(cid:12) ≤ C ( O ) , where C ( O ) is a constant that depends only on O , Proposition 5.9. For PSp ± (2 m, R ) thebound is not uniform but linear on m and we need to introduce another term depending on theorbifold (cid:12)(cid:12) dim(Hit( O , PSp ± (2 m, R ))) + χ ( O ) dim(PSp ± (2 m, R )) + c ( O ) m (cid:12)(cid:12) ≤ C ( O )where C ( O ) and c ( O ) are constants that depend only on O , Proposition 5.11.3e also show that Hit( O , G R ) maximizes the dimension among all components of the varietyof (good) representations of O in G R , for G R = PGL( n, R ), Proposition 5.7, or PSp ± (2 m, R ),Proposition 5.12.For O closed and orientable, the differential form of Atiyah-Bott-Goldman gives naturallya symplectic structure on Hit( O , G R ). Furthermore (see Proposition 5.2): Proposition 1.5.
When O is closed and non-orientable, with orientation covering f O , then Hit( O , G R ) embeds in Hit( f O , G R ) as a Lagrangian submanifold. We apply these computations on 2-orbifolds to estimate the growth of X ( M , SL( n, C )) withrespect 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 − ), for large n X (Γ , SL( n, C )) has atleast components that contain irreducible representations, whose dimension grow respectivelyas n / , n / and n / . The paper is organized as follows. In Section 2 we recall some basic notions and toolson 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 tovarieties of representations and characters. In Section 4 we discuss varieties of representationsof two dimensional orbifolds, and we apply some of the results to discuss dimensions of Hitchincomponents of orbifolds in Section 5. In Section 6 we prove the results on three-dimensionalorbifolds. Finally, some explicit examples are computed in Section 7.
Let Γ be a finitely generated group, we are mainly interested in the fundamental group ofa compact orbifold. Let G be a complex semi-simple algebraic Lie group. The variety ofrepresentations R (Γ , G ) = hom(Γ , G )is an affine algebraic set, perhaps not radical (i.e. possible with non-reduced function ring C [ R (Γ , G )] G ). 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 bedefined as the affine variety this function ring (or affine scheme when the ring is non-reduced): C [ X (Γ , G )] ∼ = C [ R (Γ , G )] G . When Γ = π ( O ), they are denoted respectively by R ( O , G ) and X ( O , G ).Following for instance [20], 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 ), an irreducible representation is also good.Let R good (Γ , G ) denote the subset of good representations in R (Γ , G ). The set of conjugacyclasses R good (Γ , G ) = R good (Γ , G ) /G
4s a Zariski open subset of X (Γ , G ), cf. [20].We discuss also representations in real Lie groups. For G R , the variety of characters is moresubtle, cf. [6], but we just consider R good (Γ , G R ) = R good (Γ , G R ) /G R .For a compact 2 orbifold O , we may also consider the relative character variety , defined as R good ( O , ∂ O , G ) = { [ ρ ] ∈ R good ( O , G ) | [ ρ | ∂ O ] , . . . , [ ρ | ∂ k O ] are constant } , where ∂ O = ∂ O ⊔ · · · ⊔ ∂ k O is the decomposition in connected components and [ ρ | ∂ i O ]denotes the conjugacy class of the restriction to the i -th boundary component. For a representation ρ : Γ → G , its composition with the adjoint representation is denoted byAd ρ : 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 ( γ γ ) = d ( γ ) + Ad ρ ( γ ) ( d ( γ )) , ∀ γ , γ ∈ Γ . The space of crossed morphisms is denoted by Z (Γ , Ad ρ ). A crossed morphism is called inner if there exists a ∈ g such that d ( γ ) = (Ad ρ ( γ ) − a ) , ∀ γ ∈ Γ , and the subspace of inner crossed morphisms is denoted by B (Γ , Ad ρ ). The quotient is thefirst cohomology group: H (Γ , g Ad ρ ) = H (Γ , Ad ρ ) ∼ = Z (Γ , Ad ρ ) /B (Γ , Ad ρ ) . Theorem 2.2 (Weil) . Let T Zarρ R (Γ , G ) be the Zariski tangent space of R (Γ , G ) as a scheme at ρ . There is a natural isomorphism Z (Γ , Ad ρ ) ∼ = T Zarρ R (Γ , G ) that maps B (Γ , Ad ρ ) to the tangent space to the orbit by conjugation. Furthermore, when ρ isgood, H (Γ , Ad ρ ) ∼ = T Zarρ X (Γ , G ) . For a discussion of Theorem 2.2, see for instance [18] and the references therein. We donot need the precise definition of scheme, just mention that the polynomial ideal that defines R (Γ , G ) or X (Γ , G ) may be non-reduced, and this is taken into account in the Zariski tangentspace. In particular, when the dimension of the Zariski tangent space at some representationor character equals the dimension of the component, the variety is smooth (and the scheme isreduced and smooth) at this representation or character.There is a real version of this theorem: when the image of a good representations is containedin a real form G R , then H (Γ , Ad ρ R ) ∼ = T Zarρ R good (Γ , G R ) . Proposition 2.3 ([18]) . For a compact 2-orbifold O , if ρ : π ( O ) → G is good, then T Zarρ R good ( O , ∂ O , G ) ∼ = ker (cid:0) H ( π ( O ) , Ad ρ ) → k M i =1 H ( π ( ∂ i O ) , Ad ρ ) (cid:1) . .2 The principal representation Let G be a simple complex Lie group, with Lie algebra g . An element in g is called regular ifits 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 whoseimage contains the given regular nilpotent element. Such a representation is unique up toconjugacy. Assume that G is an adjoint group, namely that has trivial center. Then theinduced representation of Lie groups τ : PSL(2 , C ) → G is called the principal representation . Lemma 2.4.
The image of the principal representation τ (PSL(2 , C )) is irreducible and hastrivial center in G . This lemma is a particular case of Lemma 2.8 in [1], for the complexification of the Hitchincomponent. Alternatively, Lemma 2.4 can also be proved using only Lie algebras.By construction, the principal representation of the Lie algebra maps nontrivial elements of sl (2 , C ) to regular elements of g , i.e. with centralizer the rank of g . Via the exponential map wehave: Remark 2.5.
For g ∈ PSL(2 , C ) not elliptic nor trivial, τ ( g ) is regular in G (the centralizer of τ ( g ) in G has dimension rank( G ) ). For G = PSL( n, C ) the principal representation is Sym n − . When defined from SL(2 , C )to SL( n, C ), Sym n − preserves a bilinear form of C n − , that is symmetric for n odd and skew-symmetric for n even, and its restriction yields the principal representations in PSp(2 m, C ) andPO(2 m + 1 , C ).The principal representation restricts to τ : PGL(2 , R ) → G R , for G R the split real formof G , perhaps not connected. As we deal with non-orientable 2-orbifolds, we consider bothcomponents of PGL(2 , R ), hence we need to consider two components of G R . For instancePGL( n, R ) contains two components, according to the sign of the determinant. We also use thenotation Sp ± (2 m ) = { A ∈ SL(2 m, R ) | A t J A = ± J } for J an antisymmetric, non-degenerate, bilinear form; the sign + or − corresponds to oneor the other component of the group. The Hitchin component is the connected component of R good ( π ( O ) , G R ) that contains the composition of τ with the holonomy representation of anyFuchsian structure on O .For the principal representation τ : PSL(2 , C ) → G we have a decompositionAd ◦ τ = r M i =1 Sym d i , (2)where r = rank G , cf. [32]. In particular,dim G = r X i =1 (2 d i + 1) . (3) Definition 2.6.
The d , . . . , d r ∈ N are called the exponents of G . For instance, the exponents of PSL( n, C ) are 1 , . . . , n −
1, becauseAd ◦ Sym n − = n − M i =1 Sym i . The exponents of simple Lie algebras are detailed in Tables 1 and 2, cf. [1].6ie algebra Exponents Rank Dimension sl ( n, C ) 1 , , . . . , n − n − n − sp (2 m, C ) 1 , , , . . . , m − m m + m so (2 m + 1 , C ) 1 , , , . . . , m − m m + m so (2 m, C ) 1 , , , . . . , m − , m − m m − m Table 1: Exponents of simple classical Lie algebrasLie algebra Exponents Rank Dimension g
1, 5 2 14 f
1, 5, 7, 11 4 52 e
1, 4, 5, 7, 8, 11 6 78 e
1, 5, 7, 9, 11, 13, 17 7 133 e
1, 7, 11, 13, 17, 19, 23, 29 8 248Table 2: Exponents of exceptional Lie algebras
Group cohomology is useful to study dimensions of varieties of representations, by Weil’s the-orem (Theorem 2.2). In this section we discuss a combinatorial approach to cohomology. Weare interested in representations in semi-simple complex algebraic groups, but this section ap-plies also to their real forms, by replacing complex dimension by real dimension. We focus inhomology and cohomology with coefficients twisted by the adjoint of a representation, but theconstructions and results can be easily adapted to coefficients twisted by other representations.We recall some basic definitions of orbifolds [2, 33]. 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; • hyperbolic/Euclidean/spherical if it is the quotient of hyperbolic space/Euclidean space/theround sphere by a discrete group of isometries.In a cell decomposition K of an orbifold, we require that isotropy groups or stabilizers areunchanged along open 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 χ ( O ) = X e cell of K | Stab( e ) | ( − dim e ∈ Q . (4)This is well behaved under coverings: if O ′ → O is an orbifold covering of finite index k , then χ ( O ′ ) = kχ ( O ). See [33, 2] for instance. Let O n be a compact n -dimensional good orbifold, possibly not orientable. Fix a CW-complexstructure K on O n . This means that K is a CW-complex with the same underlying space7s O n , so that the branching locus of O n is a subcomplex of K . In particular, K it lifts to aCW-complex structure e K of the universal covering e O n of O n , so that each cell of K lifts to adisjoint union of homeomorphic cells of e K (perhaps with nontrivial stabilizer).The complex of chains on the universal covering is the free Z -module on the cells of e K ,equipped with the usual boundary operator, and it is denoted by C ∗ ( e K, Z ). It has a (non-free) action of Γ = π ( O n ) induced by deck transformations. The twisted chain and cochaincomplexes are defined as: C ∗ ( K, Ad ρ ) = g ⊗ Γ C ( e K, Z ) , (5) C ∗ ( K, Ad ρ ) = hom Γ ( C ( e K, Z ) , g ) . (6)The group Γ acts on g via Ad ρ on the left in (5), and for the tensor product in (6) Γ acts on g on the right using inverses. Those are complexes and cocomplexes of finite-dimensional vectorspaces, and the corresponding homology and cohomology groups are denoted respectively by H ∗ ( K, Ad ρ ) and H ∗ ( K, Ad ρ ) . For a compact manifold M and a representation ρ : π ( M ) → G , as π ( M ) acts freely on theuniversal covering f M , we have χ ( M ) dim( G ) = X i ( − i dim H i ( M, Ad ρ ) = X i ( − i dim H i ( M, Ad ρ ) . To have a similar formula for an orbifold O we need to take into account that Γ = π ( O ) actsnon-freely on e O ; 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 subgroupΓ ⊂ Γ = π ( O ), the space of invariants is g ρ (Γ ) = { v ∈ g | Ad ρ ( g ) ( v ) = v, ∀ g ∈ Γ } , (7)and the quotient of coinvariants , g ρ (Γ ) = g / h Ad ρ ( g ) ( v ) − v | v ∈ g , g ∈ Γ i . (8)As the Killing form on g is nondegenerate ( G is semisimple) and Ad-invariant we have h Ad ρ ( g ) ( v ) − v | v ∈ g , g ∈ Γ i = ( g ρ (Γ ) ) ⊥ . (9)Therefore dim( g ρ (Γ ) ) = dim( g ρ (Γ ) ) . (10) Definition 3.1.
The orbifold Euler characteristic of O twisted by Ad ρ is e χ ( O , Ad ρ ) = X e cell of K ( − dim e dim g ρ (Stab(˜ e )) ∈ Z . Here ˜ e denotes any lift of e to the universal covering of O and Stab(˜ e ) ⊂ 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 e χ ( O , Ad ρ ) is always an integer. Notice also that e χ is not multi-plicative under coverings either; its intended to be a tool to compute dimensions of cohomologygroups: 8 roposition 3.2. For O and ρ as above: e χ ( O , Ad ρ ) = X i ( − i dim H i ( O , Ad ρ ) = X i ( − i dim H i ( O , Ad ρ ) . Proof.
We compute the dimension of C ∗ ( K, Ad ρ ) and C ∗ ( K, Ad ρ ) as C -vector spaces. We aimto show that, for 0 ≤ i ≤ dim O ,dim C i ( K, Ad ρ ) = dim C i ( K, Ad ρ ) = k i X j =1 dim g ρ (Stab(˜ e ij )) (11)where { e i , . . . , e ik i } are the i -cells of K and { ˜ e i , . . . , ˜ e ik i } a choice of lifts in the universal covering.Equation (11) will imply that e χ ( O , Ad ρ ) = X i ( − i dim C i ( K, Ad ρ ) = X i ( − i dim C i ( K, Ad ρ ) (12)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 e K : C i ( e K, Z ) = k i M j =1 Z [Γ]˜ e ij , (13)because the Γ-orbits of the lifts { ˜ e i , . . . , ˜ e ik i } give a partition of the i -cells of e K . Then we applythe natural isomorphisms of C -vector spaces, for each cell ˜ e of e K :hom Γ ( Z [Γ]˜ e, g ) ∼ = g ρ (Stab(˜ e ) θ θ (˜ e ) g ⊗ Γ Z [Γ]˜ e ∼ = g ρ (Stab(˜ e )) v ⊗ ˜ e v (14)whose proof is straightforward and use Equality (10).For 2-dimensional orientable orbifolds, Proposition 3.2 is essentially a formula computed byAndr´e Weil in [34, Sections 6 and 7]. Let O → O be a finite regular covering. Namely, Γ = π ( O ) is a normal subgroup ofΓ = π ( O ) of finite index. Though O is very good, we do not require O to be a manifold inthis subsection (for instance, O can be the orientation covering).The Galois group , or group of deck transformations , of the covering is Γ / Γ . It acts naturallyon the chain and cochain complexes, C ∗ ( K , Ad ρ ) and C ∗ ( K , Ad ρ ). Namely, any γ ∈ Γ mapsa chain m ⊗ c ∈ C ∗ ( K , Ad ρ ) to m · γ − ⊗ γc = Ad ρ ( γ )( m ) ⊗ γc , cf. [28]. The action of Γ is trivial by construction and therefore we have an action of Γ / Γ . Similarly, γ ∈ Γ maps acochain θ ∈ C ∗ ( K , Ad ρ ) to Ad( ρ ( γ )) ◦ θ ◦ γ − , which again induces an action of the Galoisgroup Γ / Γ . The subspace of elements fixed by this action in homology and cohomology aredenoted respectively by H ∗ ( O , Ad ρ ) Γ / Γ and H ∗ ( O , Ad ρ ) Γ / Γ . roposition 3.3. The map π : O → O induces an epimorphism π ∗ : H ∗ ( O , Ad ρ ) ։ H ∗ ( O , Ad ρ ) that restricts to an isomorphism H ∗ ( O , Ad ρ ) Γ / Γ ∼ = H ∗ ( O , Ad ρ ) . Similarly, it induces a monomorphism π ∗ : H ∗ ( O , Ad ρ ) ֒ → H ∗ ( O , Ad ρ ) that yields an isomor-phism H ∗ ( O , Ad ρ ) ∼ = H ∗ ( O , Ad ρ ) Γ / Γ . Proof.
The argument is standard and we just outline it. The proof is based in constructing asection to the maps induced by the projection π . The section at the chain and cochain levelconsists in taking an element of the inverse image of the map induced by π and averaging bythe action of Γ / Γ . 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 cochainsand 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 = π ( O ) in G . The following hold:(a) H i ( O , Ad ρ ) and H i ( O , Ad ρ ) are dual, via the Kronecker pairing.(b) If O is orientable , then the cup product induces a perfect pairing H i ( O , Ad ρ ) × H dim O− i ( O , ∂ O , Ad ρ ) → C . (c) There is a natural isomorphism H (Γ , Ad ρ ) ∼ = H ( O , Ad ρ ) .(d) If O is aspherical (its universal covering is a contractible manifold), then there is a naturalisomorphism H ∗ (Γ , Ad ρ ) ∼ = H ∗ ( O , Ad ρ ) .Proof. For assertion (a), we use the Kronecker pairing between chains and cochains via theKilling form B : C i ( K, Ad ρ ) × C i ( K, Ad ρ ) → 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 / Γ of the corre-sponding properties for a finite regular covering O → O that is a manifold. More precisely,using that O is a manifold, the cup product defines a perfect pairing H i ( O , Ad ρ ) × H dim O − i ( O , ∂ O , Ad ρ ) → H dim O ( O , ∂ O , C ) ∼ = C . This is compatible with the action of Γ / Γ , 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 / Γ on H dim O ( O , ∂ O , C ) ∼ = C is trivial.Next we prove (c). Chose X a K (Γ , π ( X ) ∼ = Γ and trivialhigher homotopy groups. Thus there is a natural isomorphism H ∗ ( X, Ad ρ ) ∼ = H ∗ (Γ , Ad ρ ).Furthermore, the covering corresponding to Γ < Γ, X → X , is a K (Γ , ∗ ( X , Ad ρ ) ∼ = H ∗ (Γ , Ad ρ ). As O is a manifold, a K (Γ ,
1) can be constructed from the2-skeleton of O by adding cells of dimension ≥
3, therefore: H ( O , Ad ρ ) ∼ = H ( X , Ad ρ ) . Thus, by Proposition 3.3 we have natural isomorphisms H ( O , Ad ρ ) ∼ = H ( O , Ad ρ ) Γ / Γ ∼ = H ( X , Ad ρ ) Γ / Γ ∼ = H ( X, Ad ρ ) , and H ( X, Ad ρ ) ∼ = H (Γ , Ad ρ ) because X is a K (Γ , O is already a K (Γ , X , hence H ∗ ( O , Ad ρ ) ∼ = H ∗ ( X , Ad ρ ) . and the conclusion holds for any degree in cohomology.From the duality in Proposition 3.4 (b) we get: Corollary 3.5.
When O n is closed and orientable, of dimension n , for n odd e χ ( O n , Ad ρ ) = 0 ,and for n even e χ ( O n , Ad ρ ) is also even. For a connected non-orientable orbifold O n , there exists a unique orientable covering f O n → O n of index 2, called the orientation covering . Instead of Poincar´e duality we have: Lemma 3.6.
Let O n be a compact, non-orientable, very good orbifold of dimension n withorientation covering f O n → O n . Then dim H i ( O n , Ad ρ ) + dim H n − i ( O n , ∂ O n , Ad ρ ) = dim H i ( f O n , Ad ρ ) . Proof.
The nontrivial deck transformation of the orientation covering is an involution denotedby σ : f O n → f O n . Let σ ∗ be the induced morphism in the cohomology group. By Proposition 3.3: H i ( O n , Ad ρ ) ∼ = H i ( f O n , Ad ρ ) σ ∗ and H i ( O n , ∂ O n , Ad ρ ) ∼ = H i ( f O n , ∂ f O n , Ad ρ ) σ ∗ . (15)The cup product induces a non-degenerate pairing, Proposition 3.4: H i ( f O n , Ad ρ ) × H n − i ( f O n , ∂ f O n , Ad ρ ) → H n ( f O n , ∂ f O n , C ) ∼ = C . (16)Chose C -basis for H i ( f O n , Ad ρ ) and for H n − i ( f O n , ∂ f O n , Ad ρ ), 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 ( f O n , Ad ρ ), and • T j denote the matrix of σ ∗ on H j ( f O n , ∂ f O n , Ad ρ ).Since σ reverses the orientation, σ ∗ acts as minus the identity on H n ( f O n , ∂ f O n , C ):( S i ) t J T n − i = − J. As the pairing is non-degenerate, J is invertible, and since σ is an involution:( S i ) t = − J T n − i J − . T n − i − Id) = dim ker(( S i ) t + Id) = dim ker( S i + Id) . (17)Since ( S i ) = Id: H i ( f O n , Ad ρ ) = 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 ( f O n , Ad ρ ) . With (15) this concludes the proof.
Corollary 3.7. If O n is non-orientable, very good, closed and of even dimension n , then dim H n ( O n , Ad ρ ) = 12 dim H n ( f O n , Ad ρ ) . Remark 3.8.
The pairing in the proof of Lemma 3.6 induces a nondegenerate pairing betweenthe kernels ker (cid:0) H i ( f O n , Ad ρ ) → H i ( ∂ f O n , Ad ρ ) (cid:1) and ker (cid:0) H n − i ( f O n , Ad ρ ) → H n − i ( ∂ f O n , Ad ρ ) (cid:1) [23, 18]. Thus the same argument in the proof of Lemma 3.6 yields dim ker (cid:0) H i ( O n , Ad ρ ) → H i ( ∂ O n , Ad ρ ) (cid:1) + dim ker (cid:0) H n − i ( O n , Ad ρ ) → H n − i ( ∂ O n , Ad ρ ) (cid:1) = dim ker (cid:0) H i ( f O n , Ad ρ ) → H i ( ∂ f O n , Ad ρ ) (cid:1) . We recall the possible stabilizers (or isotropy groups) of a point in a 2-orbifold, so that we fixnotation. Besides the trivial one, the possible stabilizers of a point x ∈ O are: • Stab( x ) ∼ = C k is a cyclic group of rotations of the plane R of order k . Then the singularpoint is isolated and its is called a cone point . Those are the unique possible non-trivialstabilizers for an orientable 2-orbifold (Figure 1 left). • Stab( x ) ∼ = C is a group of order two generated by a reflection of the plane or the half-plane, ie. modeled 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 ) ∼ = D k is a dihedral group of 2 k elements, generated by two reflections at theplane of angle π/k . The point is called a corner reflector (Figure 1 right). xx xx Figure 1: From left to right, a cone point, an interior mirror point, a mirror point in theboundary, and a corner reflector. Mirror points are represented by a double line.The boundary of a compact 2-orbifold is a union of closed 1-orbifolds. There are twopossibilities for a closed 1-orbifold, up to homeomorphism:12
A circle S (Figure 2 left, as it appears in the boundary). Its fundamental group is Z . • An interval with mirror end-points [[0 , S by an orientation reversing involution. Its fundamental group isdihedral infinite D ∞ ∼ = C ∗ C , the free product of the stabilizers of the mirror points. Itis called full S and an interval with mirror end-points [[0 , χ ( O ) ≤ O is verygood and either hyperbolic or Euclidean. In particular a closed 2-orbifold O is aspherical iff χ ( O ) ≤ -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 forsurfaces [11]. In the orientable case, those results go back to Andr´e Weil [34, Sections 6 and 7].We start with the following formulas:
Proposition 4.1.
Let O be a closed aspherical 2-orbifold. Set Γ = π ( O ) and let ρ ∈ R (Γ , G ) .a) If O is orientable, then dim T Zarρ R (Γ , G ) = − e χ ( O , Ad ρ ) + dim G + dim g ρ (Γ) . b) If O is non-orientable, with orientation covering f O , and e Γ = π ( f O ) , then dim T Zarρ R (Γ , G ) = − e χ ( O , Ad ρ ) + dim G + dim g ρ ( e Γ) − dim g ρ (Γ) . Proof.
We first claimdim T Zarρ R (Γ , G ) = − e χ ( O , Ad ρ ) + dim G + dim H ( O , Ad ρ ) . (19)To prove it, we use Weil’s theorem, Theorem 2.2: T Zarρ R (Γ , G ) ∼ = Z (Γ , Ad ρ ) = dim H (Γ , Ad ρ ) + dim B (Γ , Ad ρ ) . (20)From the isomorphism H ∗ ( O , Ad ρ ) ∼ = H ∗ (Γ , Ad ρ ) (as O is aspherical we apply Proposition 3.4(d)), (19) follows from (20) and the following equations:dim H ( O , Ad ρ ) = − e χ ( O , Ad ρ ) + dim H ( O , Ad ρ ) + dim H ( O , Ad ρ ) , dim B (Γ , Ad ρ ) = dim g − dim g ρ (Γ) = dim G − dim H (Γ , Ad ρ ) . H ( O , Ad ρ ). In the orientable case,by duality (Proposition 3.4):dim H ( O , Ad ρ ) = dim H ( O , Ad ρ ) = dim g ρ (Γ) . In the non-orientable case, Lemma 3.6 yieldsdim H ( O , Ad ρ ) = dim H ( f O , Ad ρ ) − dim H ( O , Ad ρ ) = dim g ρ ( e Γ) − dim g ρ (Γ) , and the proposition follows. Proposition 4.2.
Let O be a compact aspherical 2-orbifold and ρ : Γ → G a good representa-tion. When O is non-orientable, assume furthermore that the restriction of ρ to the orientationcovering is also good.Then the conjugacy class [ ρ ] is a smooth point of X ( O , G ) of dimension − e χ ( O , Ad ρ ) , withtangent space naturally isomorphic to H ( O , Ad ρ ) .Proof. As ρ is good, its centralizer is finite, hencedim H ( O , Ad ρ ) = dim H (Γ , Ad ρ ) = dim g ρ (Γ) = 0 . When O is closed then H ( O , Ad ρ ) = 0 (by duality in the orientable case, Proposition 3.4,or by Lemma 3.6 in the non-orientable case). When O is not closed, then it has virtually thehomotopy type of a 1-complex and also H ( O , Ad ρ ) = 0.Moreover, as O is aspherical, by Proposition 3.4 H (Γ , Ad ρ ) = 0. By Goldman’s obstruc-tion theory [11], this implies that [ ρ ] is a smooth point of local dimension dim H (Γ , Ad ρ ) =dim H ( O , Ad ρ ) = − e χ ( O , Ad ρ ).From Proposition 4.2 and Corollary 3.7: Corollary 4.3.
Let O be a closed non-orientable 2-orbifold with χ ( O ) ≤ and ρ : Γ → G .Assume that the restriction of ρ to the orientation covering f O is good. Then dim [ ρ ] X ( O , G ) = 12 dim [ ρ ] X ( f O , G ) . Proposition 4.4.
Let O be a compact aspherical 2-orbifold and ρ : Γ → G a good representa-tion that is ∂ -regular. Assume that O is orientable. Then the cup product defines a C -valuedsymplectic structure on R good ( O , ∂ O , G ) . Furthermore, it is real valued for G R , a real formof G .Proof. From the construction, non-degeneracy and skew symmetry is clear. The non-trivialissue is to check that this differential form is closed in R good ( O , ∂ O , G ). For 2-manifolds withboundary, being closed is due to [14], as explained in [18]. For an orientable orbifold withboundary, we reduce to the manifold case. Let Σ O denote the branching locus of O , it is afinite union of cone points. The restriction map R good ( O , ∂ O , G ) → R good ( O \ N (Σ O ) , ∂ ( O \ N (Σ O )) , G )is a (local) isomorphism, as the conjugacy classes of finite order elements cannot be deformed. Tocheck the isomorphism at the level of tangent spaces, notice that the map induced by inclusion H ( O , Ad ρ ) → H ( O \ N (Σ O ) , Ad ρ )14estricts to an isomorphism of kernels:ker (cid:0) H ( O , Ad ρ ) → H ( ∂ O , Ad ρ ) (cid:1) ∼ = ker (cid:0) H ( O \ N (Σ O ) , Ad ρ ) → H ( ∂ ( O \ N (Σ O ) , Ad ρ ) (cid:1) , that can be proved for instance by using Mayer-Vietoris exact sequence. Furthermore, bynaturality, it can be shown that the isomorphism maps the cup product to the cup product. Proposition 4.5.
Let O be a closed non-orientable 2-orbifold with χ ( O ) ≤ . Let ρ : Γ → G be a representation whose restriction to the orientation covering f O is good and ∂ -regular. Thenthe restriction map induces an immersion around [ ρ ] of R good ( O , ∂ O , G ) in R good ( f O , ∂ f O , G ) as a Lagrangian submanifold.Proof.
Let σ : f O → f O be the orienting reversing involution so that f O /σ = O . By Proposi-tions 3.3 and 4.2 T Zar [ ρ ] R good ( O , G ) ∼ = H ( O , Ad ρ ) ∼ = H ( f O , Ad ρ ) σ ∗ ∼ = (cid:0) T Zar [ ρ ] R good ( f O , G ) (cid:1) σ ∗ . As σ ∗ acts as minus the identity on H ( f O , C ) and by construction of the non-degeneratebilinear form H ( f O , Ad ρ ) × H ( f O , Ad ρ ) → H ( f O , C )(see the proof of Lemma 3.6), the bilinear form must be trivial on the space fixed by σ ∗ , H ( f O , Ad ρ ) σ ∗ . Namely, T Zar [ ρ ] R ( O , G ) is an isotropic subspace. By Corollary 4.3, it has thedimension to be Lagrangian. For an orientable 2-orbifold O , a representation ρ is called ∂ -regular if for each γ generator ofa peripheral subgroup, ρ ( γ ) is a regular element (i.e. dim g ρ ( γ ) = rank G ). For a non-orientable2-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 R good ( O , ∂ O , G ): Proposition 4.6.
Let O be a compact 2-orbifold with boundary and χ ( O ) ≤ . Assume ρ : Γ → G is good and ∂ -regular. When O is non-orientable, assume that the restriction to theorientation covering is good and ∂ -regular. Then [ ρ ] is a smooth point of R good ( O , ∂ O , G ) ofdimension − e χ ( O , Ad ρ ) − ( c + b G ) + 12 e χ ( ∂ O , Ad ρ ) , where c is the number of components of ∂ O that are circles and b the number of componentsof ∂ O that are intervals with mirror points [[0 , . In this proposition, the components of the boundary that are circles do not contribute to e χ ( ∂ O , Ad ρ ). Proof.
The same proof as in [18, Proposition 2.10] applies here, in particular we have smoothnessand the following equalitiesdim( R good ( O , ∂ O , G )) = dim ker (cid:0) H ( O , Ad ρ ) → H ( ∂ O , Ad ρ ) (cid:1) = dim( H ( O , Ad ρ )) − dim( H ( ∂ O , Ad ρ )) . H ( O , Ad ρ ) = H ( O , Ad ρ ) = 0,dim H ( O , Ad ρ ) = − e χ ( O , Ad ρ ) . We count the contribution of each component ∂ i O of ∂ O to dim H ( ∂ O , Ad ρ ). • When ∂ i O ∼ = S , then, by duality and ∂ -regularity:dim H ( ∂ i O , Ad ρ ) = dim H ( ∂ i O , Ad ρ ) = dim g ρ ( ∂ i O ) = rank G. (21) • When ∂ i O ∼ = [[0 , e χ ( ∂ i O , Ad ρ ) = dim H ( ∂ i O , Ad ρ ) − dim H ( ∂ i O , Ad ρ )rank G = dim H ( ∂ i O , Ad ρ ) + dim H ( ∂ i O , Ad ρ ) , where the last line follows from Lemma 3.6 and the assumption that ρ is ∂ -regular on theorientation covering. From this we deducedim H ( ∂ i O , Ad ρ ) = 12 (rank G − e χ ( ∂ i O , Ad ρ )) . (22)From (21) and (22): dim H ( ∂ O , Ad ρ ) = ( c + b G − e χ ( ∂ O , Ad ρ ) , (23)which concludes the proof of the proposition. Proposition 4.7.
Let O be a non-orientable compact 2-orbifold with orientation covering f O .Let ρ : π ( O ) → G be a representation whose restriction to f O is good and ∂ -regular. Then dim [ ρ ] X ( O , G ) = 12 dim ρ X ( f O , G ) + 12 X [[0 , ⊂ ∂ O e χ ([[0 , , Ad ρ ) where the sum runs on the components of ∂ O that are intervals with mirror boundary, [[0 , .Proof. In the proof of Proposition 4.6 we use (based on [18]):dim [ ρ ] X ( O , G ) = dim [ ρ ] R good ( O , ∂ O , G ) + dim H ( ∂ O , Ad ρ ) , dim [ ρ ] X ( f O , G ) = dim [ ρ ] R good ( f O , ∂ O , G ) + dim H ( ∂ f O , Ad ρ ) . Furthermore, by Proposition 4.5:dim [ ρ ] R good ( O , ∂ O , G ) = 12 dim [ ρ ] R good ( f O , ∂ f O , G ) . And from (21) and (23) in the proof of Proposition 4.6:dim H ( ∂ f O , Ad ρ ) = (2 c + b ) rank G, dim H ( ∂ O , Ad ρ ) = ( c + b G − e χ ( ∂ O , Ad ρ ) . Using these formulas, we just need to know that e χ ( ∂ O , Ad ρ ) is the sum of the twisted Eulercharacteristics of each component, and that e χ ( S , Ad ρ ) = 0.16 / Z Z / Z Figure 3: The orbifold [[0 , R /D ∞ .The orbifold fundamental group of [[0 , D ∞ , so that R /D ∞ ∼ =[[0 , D ∞ = h σ , σ | σ = σ = 1 i the stabilizers of the mirror points are the cyclic groups of order 2 generated by σ and σ respectively, see Figure 3. For further applications we need the following lemma: Lemma 4.8.
For any representation ρ : D ∞ = h σ , σ | σ = σ = 1 i → G : e χ ( R /D ∞ , Ad ρ ) = dim g ρ ( σ ) + dim g ρ ( σ ) − dim g , dim H ( R /D ∞ , Ad ρ ) = (cid:0) dim g ρ ( σ σ ) + dim g ρ ( σ ) + dim g ρ ( σ ) − dim g (cid:1) / , dim H ( R /D ∞ , Ad ρ ) = (cid:0) dim g ρ ( σ σ ) − dim g ρ ( σ ) − dim g ρ ( σ ) + dim g (cid:1) / . Proof.
We compute the twisted Euler characteristic by using the simplicial structure of R /D ∞ with one 1-cell (with trivial stabilizers) and two 0-cells (with stabilizers generated by σ and σ respectively): e χ ( R /D ∞ , Ad ρ ) = dim g ρ ( σ ) + dim g ρ ( σ ) − dim g = dim H ( R /D ∞ , Ad ρ ) − dim H ( R /D ∞ , Ad ρ ) . (24)The orientation covering is denoted by S → R /D ∞ , and π ( S ) corresponds to the infinitecyclic subgroup of D ∞ generated by σ σ . As S is a manifold and χ ( S ) = 0:dim H ( S , Ad ρ ) ∼ = dim H ( S , Ad ρ ) ∼ = g ρ ( σ σ ) . (25)By Lemma 3.6: dim H ( R /D ∞ , Ad ρ ) + dim H ( R /D ∞ , Ad ρ ) = dim g ρ ( σ σ ) . (26)Then the lemma follows from (24) and (26).Notice that in the previous lemma we do not require anything for the representation ρ , thatcould be trivial. The first direct application goes to the stabilizer of a corner reflector, thedihedral group D k , by pre-composing any representation D k → G with the natural surjectionfrom the infinite dihedral group D ∞ → D k : Corollary 4.9.
Let x ∈ O be a corner reflector and x and x mirror points in a neighborhoodof x separated by x , see Figure 4. If C k ⊂ Stab( x ) is the orientation preserving cyclicsubgroup of index 2, then: dim g ρ (Stab( x )) = (cid:0) dim g ρ ( C k ) + dim g ρ (Stab( x )) + dim g ρ (Stab( x )) − dim g (cid:1) / . x x Figure 4: Corner reflector modeled on R /D k , with the notation in Corollary 4.9. Along this subsection, let O be a Euclidean two orbifold without boundary. Set Γ = π ( O ).As Isom( R ) is the semi-direct product R ⋊ O(2), we have an exact sequence (Bieberbachtheorem): 1 → Γ → Γ → Λ → , (27)with Γ ∼ = Z and Λ ⊂ O(2) the linear part. When O is orientable, Λ is cyclic and when O is non-orientable, Λ is dihedral (or cyclic of order 2). Furthermore, when O is orientable ora Coxeter group, Γ is the maximal torsion free subgroup and the sequence (27) splits, as Λcorresponds to the stabilizer of a point in R . A Coxeter group in Isom( R ) is a group generatedby reflections along a square or a Euclidean triangle with angles an integer divisor of π . T S (2 , , ,
2) 3 33 S (3 , ,
3) 2 44 S (2 , ,
4) 2 36 S (2 , , • the 2-torus T , • a 2-sphere with 4 cone points of order 2, S (2 , , , • three 2-spheres with cone points of order p , q and r satisfying p + q + r = 1, S ( p, q, r )for ( p, q, r ) = (3 , , , ,
4) and (2 , , S (2 , , ,
2) is sometimes called a pillowcase , geometrically is the double of arectangle, and S ( p, q, r ) is called a turnover , as it is the double of a triangle. For differentvalues of the ramifications, turnovers can also be spherical or Euclidean. Table 3 gives thecardinality of Λ ∼ = Γ / Γ for these five Euclidean orbifolds.As Γ = π ( O ) 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 : π ( O ) → G is called stronglyregular if for the maximal normal subgroup Γ < π ( O ) , with Γ ∼ = Z ,(a) dim g ρ (Γ ) = rank G , and | Γ / Γ | T S (2 , , ,
2) 2 S (3 , ,
3) 3 S (2 , ,
4) 4 S (2 , ,
6) 6Table 3: Values of k ( O ) = | Γ / Γ | when O is orientable, for Γ maximal torsion free. (b) the projection of ρ (Γ ) is contained in a connected abelian subgroup of G/ Z ( G ) . Theorem 4.11.
Assume that O is compact Euclidean and that ρ : π ( O ) → G is stronglyregular . Then:(i) ρ is a smooth point of R (Γ , G ) of dimension − e χ ( O , Ad ρ ) + dim G + dim g ρ (Γ) . (ii) The component X (Γ , G ) of X (Γ , G ) that contains the character of ρ has dimension: dim X (Γ , G ) = − e χ ( O , Ad ρ ) + 2 dim g ρ (Γ) = dim H ( O , Ad ρ ) . Proof. (i) First consider the case O = T , here we adapt an argument of [17]. In this case e χ ( O , Ad ρ ) = 0, and by a theorem of Richardson [30, Theorem B], ρ is in the closure of T × T 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 g ρ (Γ) = rank G , byProposition 4.1, dim T Zarρ R (Γ , G ) = dim G + rank G . Thereforedim ρ R (Γ , G ) ≥ dim G + rank G = dim T Zarρ R (Γ , G ) . Using that the dimension of the Zariski tangent space is always larger than or equal to thedimension of a variety, with equality only at smooth points, it follows that dim ρ R (Γ , G ) =dim T Zarρ R (Γ , G ) and ρ is a smooth point.For general O , notice that − e χ ( O , Ad ρ ) + dim G + dim g ρ (Γ) = dim T Zarρ 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 thatthe variety of representations of Γ at ρ | Γ is smooth. As Goldman obstructions are natural[19, § , then the sequence of obstructions to integrability is also equivariant. Thus, byProposition 3.3 and Proposition 3.4, the sequence of obstructions to integrability of Γ vanishes.This proves (i).For (ii), notice that dim g ρ (Γ) = dim H (Γ , Ad ρ ) is upper semi-continuous on R (Γ , G ) [15,Ch. III, Theorem 12.8]. By smoothness, the dimension of the Zariski tangent space reaches itsminimum (along the irreducible component) at ρ . Hence, asdim G + dim g ρ (Γ) − e χ ( O , Ad ρ ) = dim T Zarρ R (Γ , G )by Proposition 4.1, dim g ρ (Γ) reaches its minimum along the irreducible component at ρ . (Herewe use that e χ ( O , Ad ρ ) is constant along components, because the elements of finite ordercannot be deformed.) Equivalently the dimension of the orbit by conjugationdim G − dim g ρ (Γ) ρ . It follows (for instance from [7, Section 6.3]) thatdim X (Γ , G ) = dim R (Γ , G ) − (cid:0) dim G − dim g ρ (Γ) (cid:1) , which proves (ii). Remark 4.12.
It does not follow from Theorem 4.11 that the variety of characters X (Γ , G ) issmooth at the character of ρ . To explain Remark 4.12, we point out that even if ρ is strongly regular, its orbit may be notclosed, so ρ could be smooth but its character not. For instance, the parabolic representationof Z in SL(2 , C ) is a smooth point of R ( Z , SL(2 , C )), but its character is a singular point of X ( Z , SL(2 , C )). The orbit of the parabolic representation is not closed because it accumulatesat the trivial representation (that it is not regular).Orientable Euclidean 2-orbifolds appear as peripheral subgroups of hyperbolic three-orbifoldsof finite volume. Thus a natural representation of Γ = π ( O ) in PSL(2 , C ) occurs as theholonomy of horospherical cusps, homeomorphic to O × [0 , + ∞ ). We are interested in thecomposition of this holonomy with the principal representation τ : PSL(2 , C ) → G , in view offurther computations for hyperbolic three-orbifolds. Proposition 4.13.
Let O be a closed orientable Euclidean 2-orbifold. Consider the holonomy hol : π ( O ) → PSL(2 , C ) as the horospherical section of a cusp, composed with the principalrepresentation τ : PSL(2 , C ) → G . Then dim [ τ ◦ hol] X ( O , G ) is given by Table 4. Furthermore,the dimension of g τ ◦ hol(Γ) is given by Table 5. ❍❍❍❍❍❍ G O T S (2 , , , S (3 , , S (2 , , S (2 , , n, C ) 2( n −
1) 2 ⌊ n ⌋ ⌊ n ⌋ ⌊ n ⌋ ⌊ n ⌋ PSp(2 m, C ) 2 m m ⌊ m ⌋ ⌊ m ⌋ ⌊ m ⌋ PO(2 m + 1 , C ) 2 m m ⌊ m ⌋ ⌊ m ⌋ ⌊ m ⌋ PO(2 m, C ) 2 m + 2 4 ⌊ m ⌋ ⌊ m ⌋ ⌊ m ⌋ + ⌊ m +14 ⌋ ) 2( ⌊ m ⌋ + ⌊ m +26 ⌋ )G
12 8 4 4 6E
14 14 6 4 6E
16 16 8 8 8Table 4: Dimension of X ( O , G ) at [ τ ◦ hol], for hol : Γ → PGL(2 , C ) the holonomy of ahorospherical cusp and τ : PGL(2 , C ) → G the principal representation. Proof.
The principal representation τ maps regular elements to regular elements. Furthermorethe parabolic holonomy of the maximal torsion-free subgroup of π ( O ) is contained in a con-nected abelian subgroup of PGL(2 , C ), hence τ ◦ hol | π ( O ) is strongly regular and Theorem 4.11applies. The computation of dimension is based on the decomposition of Ad ◦ τ as a sum ofSym d α (2), according to the exponents d , . . . , d r of G , where r is the rank of G . Let V d α isthe space of the representation Sym d α . We use the following three facts:1. dim V d α = 2 d α + 1, by (3). 20 ❍❍❍❍❍ g O T S (2 , , , S (3 , , S (2 , , S (2 , , sl ( n, C ) n − ⌊ n − ⌋ ⌊ n − ⌋ ⌊ n − ⌋ ⌊ n − ⌋ sp (2 m, C ) m ⌊ m +13 ⌋ so (2 m + 1 , C ) m ⌊ m +13 ⌋ so (2 m, C ) m δ m − Z ⌊ m ⌋ + δ m − Z δ m − Z δ m − Z g f e e e g τ ◦ hol(Γ) for τ ◦ hol as in Table 4, where δ ij Z = 1 if i ∈ j Z and δ ij Z = 0otherwise.2. For a cyclic group of rotations C k < PSL(2 , C ) of order k , with hyperbolic rotation anglesin πk Z , dim V C k d α = 2 ⌊ d α k ⌋ + 1.3. dim V Γ2 d α = δ ik Z , Γ = π ( O ), Γ ∼ = Z a maximal torsion-free subgroup and k = | Γ / Γ | ,where δ ik Z = 1 if i ∈ k Z and 0 if i k Z .From this, the computation is elementary. Lemma 4.14.
Let O = S (2 , , , , O = S (3 , , , O = S (2 , , , or O = S (2 , , .There is an irreducible representation of Γ = π ( O ) in PSL( k, C ) iff and only if k = | Γ / Γ | .Proof. We first construct the irreducible representation of Γ in PSL( k, C ), for k = | Γ / Γ | . Fromthe exact sequence (27), start with a diagonal representation of Γ , then represent the cyclicgroup Γ / Γ as a cyclic permutation of the k coordinates of C k . (In each case, Proposition 4.2applies and the dimension of the variety of characters is 2.)Let ρ : Γ → PSL( k, C ) be an irreducible representation. Let Fix( ρ (Γ )) ⊂ CP k − denote thefixed point set of ρ (Γ ), that is nonempty by Kolchin’s theorem. We consider the action of thecyclic group Γ / Γ on Fix( ρ (Γ )). It can be checked that any possibility other than k = | Γ / Γ | and that Fix( ρ (Γ )) has k points in generic position that are permuted cyclically by Γ / Γ yieldsa contradiction with irreducibility. Example 4.15.
We consider planar Euclidean Coxeter groups, namely generated by reflectionsalong a square or a Euclidean triangle with angles an integer divisor of π , denoted by: Q (2 , , , , T (3 , , , T (2 , , , T (2 , , . The respective orientation coverings are S (2 , , , , S (3 , , , S (2 , , and S (2 , , . Thegroup Γ = π ( O ) acts naturally in the Euclidean plane, and one can construct representationsin G by realizing this action on a flat of the symmetric space associated to G . On this flat, oneallows translations, but the rotational part is restricted to the action of the Weyl group, whichin 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 m generatedby reflections along two lines at angle π/m , the walls of the Weyl chamber. For type A theangle is π/ , for B , π/ , and for G , π/ . This yields discrete faithful representations of the riangle group T (3 , , in SL(3 , C ) and G ( C ) , of T (2 , , in Sp(4 , C ) , of T (2 , , in G ( C ) and of the quadrilateral group Q (2 , , , in Sp(4 , C ) or G ( C ) (or the corresponding split realforms).The triangle groups constructed here have a one-parameter space of deformations (homotetiesin the plane) and the quadrilateral group a 2-parameter space. Those are half the dimension ofthe deformation space of their orientable cover. 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 O be a spherical 2-orbifold and ρ : π ( O ) → G a representation. Then dim H i ( O , Ad ρ ) = ( dim g ρ ( π ( O )) , for i = 0 , , for i = 1 , and thus e χ ( O , Ad ρ ) = 2 dim g ρ ( π ( O )) . (28) Proof.
Let O be a spherical 2-orbifold. As it is finitely covered by S , by Proposition 3.3 wehave H i ( O , Ad ρ ) ∼ = H i ( S , g ) π ( O ) ∼ = H i ( S , Z ) ⊗ g ρ ( π ( O )) . Hence the lemma follows.
Remark 4.17.
For a spherical orbifold, the component of R ( O , G ) that contains ρ is an orbitby conjugation of dimension dim G − dim g ρ ( π ( O )) . In particular, by (28) it is a smooth variety of dimension − e χ ( O , Ad ρ ) + dim G + dim g ρ ( π ( O )) . Assume that O is hyperbolic, namely that the standard orbifold-Euler characteristic is negative, χ ( O ) <
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 ) → G R , for G R a (non-connected) split real form of G . The Hitchin componentHit( O , G R )is the connected component of R good ( O , G R ) that contains the composition of τ with the holon-omy representation of any Fuchsian structure on O . Hitchin components for surface groupshave been intensively studied, here we just mention that Alessandrini, Lee, and Schaffhauser [1]have introduced them for (possibly non-orientable) 2-orbifolds; furthermore they have shownthat they are homeomorphic to R N , as for surfaces.The purpose of this section is to provide formulas for the dimension of the Hitchin componentof orbifolds, using the tools of Section 4. Some of the formulas are equivalent to the ones alreadycomputed in [1], but the approach and presentation of results is different. In particular we givesome applications in Propositions 5.9 and 5.11 to the dimension growth of some families ofHitchin representations. Long and Thistlethwaite in [24] have also computed the dimension ofHitchin components in PGL( n, R ) for turnovers (2-spheres with three cone points).22 emark 5.1. For O closed and orientable, Atiyah-Bott-Goldman differential form defines asymplectic structure on Hit( O , G R ) , Proposition 4.4. The following is Proposition 1.5 from the introduction:
Proposition 5.2.
For O closed and non-orientable, with orientation covering f O , via therestriction map Hit( O , G R ) embeds in Hit( f O , G R ) as a Lagrangian submanifold.Proof. By Proposition 4.5 the restriction map yields a Lagrangian immersion from Hit( O , G R )to Hit( f O , G R ). Furthermore, using the irreducibility of Hitchin representations, one can provethat this restriction map is injective (see for instance Lemma 2.9 and Proposition 2.10 in [1]).The restriction map to a finite index subgroup is proper, therefore it is an embedding (oralternatively one can quote [1, Corollary 2.13]). Remark 5.3.
The representation τ ◦ hol is good, Lemma 2.4, so the real dimension of theHitchin component of O in G R is precisely the complex dimension of X ( O , G ) at the characterof τ ◦ hol , namely − ˜ χ ( O , Ad ρ ) ; we often use: dim R Hit( O , G R ) = dim C , [ τ ◦ hol] X ( O , G ) = − ˜ χ ( O , Ad ρ ) . PGL( n, R ) . Following Long and Thistlethwaite [24], 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 ofinteger division). Proposition 5.4.
Let x ∈ O and τ = Sym n − . For ρ = τ ◦ hol : π ( O ) → PGL( n, R ) , dim g ρ (Stab( x )) = σ ( n, k ) − if x is a cone point with Stab( x ) ∼ = C k , ( σ ( n, k ) − / if x is a corner with Stab( x ) ∼ = D k and n is odd , ( σ ( n, k ) − / if x is a corner with Stab( x ) ∼ = D k and n is even , ( n − / if x is a mirror point and n is odd , ( n − / if x is a mirror point and n is even . Proof.
For an elliptic element γ ∈ Γ of order k , we may compute dim g ρ ( γ ) using that, up toconjugacy: ρ ( γ ) = ± Sym n − e πik e − πik ! = ± diag( e πik ( n − , e πik ( n − , . . . , e πik ( n − ) . As n = qk + r , among the k eigenvalues of this matrix, ( k − r ) eigenvalues have multiplicity q and r eigenvalues have multiplicity q + 1. Thusdim g ρ ( γ ) = ( k − r ) q + r ( q + 1) − qn + ( q + 1) r − σ ( n, k ) − . This computation yields the formula for cone points and mirror points. For corner points, applyCorollary 4.9 and the previous computations. 23e then have:
Proposition 5.5.
Let O be a connected hyperbolic 2-orbifold, with underlying surface |O | .Then dim Hit( O , PGL( n, R )) = − ( n − χ ( |O | ) + cp X i =1 ( n − σ ( n, k i )) + cr X j =1 n − σ ( n, l j )2 + b ⌊ n ⌋ , where the cone points have order k , k , . . . , k cp , the corner reflectors have order l , l , . . . , l cr and b is the number of components of ∂ O homeomorphic to R /D ∞ , an interval with mirrorboundary.Proof. We apply Proposition 4.2 to get that the dimension of the Hitchin component is − χ ( O , Ad ρ ),which, by Proposition 5.4, equals: − ( n − χ ( O \ Σ) − cp X i =1 ( σ ( n, k i ) −
1) + ( − P cr j =1 σ ( n,l j ) − + ( me − mp ) n − if n even − P cr j =1 σ ( n,l j ) − + ( me − mp ) n − if n odd (29)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 ( |O | ) = χ ( O \ Σ) + cp + cr + mp − me , χ ( ∂ |O | ) = − me − b + cr + mp , that are equivalent to − χ ( O \ Σ) = − χ ( |O | ) + cp + b , (30) me − mp = cr − b . (31)Using (30) and (31), (29) becomes: − ( n − χ ( |O | ) + cp X i =1 ( n − σ ( n, k i )) + cr X j =1 n − σ ( n, l j )2 + ( b n if n even b n − if n oddwhich proves the proposition. Example 5.6 (Proved in [24] when cp = 3) . Let S ( k , . . . , k cp ) be a sphere with cp ≥ conepoints of order k , . . . , k cp respectively (and satisfying k + k + k < for cp = 3 ). Then, thedimension of the Hitchin component of S ( k , . . . , k cp ) in PGL( n, R ) is n ( cp −
2) + 2 − cp X i =1 σ ( n, k i ) . Consider next P ( k , . . . , k cr ) the orbifold generated by reflections on a hyperbolic polygon withangles πk , . . . , πk cr . In particular S ( k , . . . , k cr ) is its orientation covering and therefore thedimension of the Hitchin component of P ( k , . . . , k cr ) in PGL( n, R ) is n ( cr −
2) + 2 − P cr i =1 σ ( n, k i )2 . Proposition 5.7.
Every component of R good ( O , PGL( n, R )) that contains representations thatare boundary regular has dimension at most the dimension of the Hitchin component. roof. Assume first that O 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( g ρ ( γ ) ) is minimized for ρ in the Hitchin component. Assuming the claim, e χ ( O , ρ ) is minimized for ρ in the Hitchincomponent, and apply Proposition 4.2. To prove the claim, since ρ ( γ ) has order k , its eigenvalueshave multiplicities { n , . . . , n k } , with n + · · · + n k = n . Therefore dim( g ρ ( γ ) ) = n + · · · + n k − | n i − n j | ≤
1, because if we replace n by n + 1 and n by n −
1, then n + n increases by 2( n − n + 1). As, for a rotation of angle 2 π/k , Sym n − ( ρ ( γ ))satisfies | n i − n j | ≤
1, it minimizes the required dimension.In the non-orientable case we apply Proposition 4.7 and Lemma 4.8 and use the claimwe have proved, that for an elliptic element γ , dim( g ρ ( γ ) ) is minimized for ρ in the Hitchincomponent. Example 5.8.
In Table 6 we compute the dimension of
Hit( S (3 , , , PSL( n, R )) . Notice thatthe difference with n / is bounded, this is a particular case of the next proposition. n mod 12 dim Hit( S (3 , , , PGL( n, R ))0 n /
12 + 2 ± , ± n − / ± n − / ± n + 15) / ± n + 8) / n /
12 + 1Table 6: Dimension of the Hitchin component of S (3 , , π , π , π ) is half of it. Proposition 5.9.
There exists a uniform constant C ( O ) depending only on O such that (cid:12)(cid:12) dim Hit( O , PGL( n, R )) + χ ( O )( n − (cid:12)(cid:12) ≤ C ( O ) . Here > χ ( O ) ∈ Q denotes the (untwisted) orbifold-Euler characteristic.Proof. Let ρ = τ ◦ hol. Assume first that O is orientable. If γ is an elliptic element of order k ,we shall prove that (cid:12)(cid:12)(cid:12) n − k − dim g ρ ( γ ) (cid:12)(cid:12)(cid:12) is bounded uniformly on n . Namely, if n = kq + r , thendim g ρ ( γ ) = kq + 2 rq + r −
1. On the other hand n = k q + 2 rkq + r , thus: (cid:12)(cid:12)(cid:12)(cid:12) n − k − dim g ρ ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) r − k − r + 1 (cid:12)(cid:12)(cid:12)(cid:12) = ( r − (cid:18) − r + 1 k (cid:19) ≤ k. In the orientable case, as all stabilizers are cyclic this estimate tells that the difference ˜ χ ( O , Ad ρ ) − χ ( O )( n −
1) is uniformly bounded, for ρ = Sym n − ◦ hol. With Proposition 4.2, this yieldsthe proposition for O orientable.In the non-orientable case, consider f O → O the orientation covering. By Proposition 4.7and Lemma 4.8 (cid:12)(cid:12)(cid:12)(cid:12) dim Hit( O , PGL( n, R )) −
12 dim Hit( f O , PGL( n, R )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:0) dim G − g ρ ( C ) (cid:1) b ( O )25here ρ ( C ) denotes the image of any cyclic stabilizer of order 2 and b ( O ) is the number ofcomponents of ∂ O homeomorphic to R /D ∞ = [[0 , (cid:0) dim G − g ρ ( C ) (cid:1) = n − − ( σ ( n, −
1) = ( / n n the proposition follows. PSp ± (2 m ) For G = PSp ± (2 m ) the principal representation τ : PGL(2 , R ) → G is the restriction ofSym m − : PGL(2 , R ) → PGL(2 m − , R ). Let J = (cid:18) − (cid:19) . As A t J A = det( A ) J, for every A ∈ GL(2 , R ), by restricting to matrices with determinant ±
1, we get the inclu-sion of Sym m − (PGL(2 , R )) in G = PSp ± (2 m ). Furthermore, the antisymmetric matrix isSym m − ( J ).Write 2 m = kq + r with q, m ∈ Z , q, r ≥ r < k . Recall that: σ (2 m, k ) = 2 mq + r ( q + 1) . Proposition 5.10.
Let ρ ∈ Hit( O , G ) , for G = PSp ± (2 m ) or PO( m, m + 1) and let x ∈ O .1. If x is a cone point with Stab( x ) ∼ = C k then dim g ρ ( C k ) = ( σ (2 m,k )2 for k even, σ (2 m,k )2 + ⌊ q +12 ⌋ for k odd.2. If x is a mirror point with Stab( x ) ∼ = C then dim g ρ ( C ) = m .3. If x is a corner reflector with Stab( x ) ∼ = D k then dim g ρ ( D k ) = ( σ (2 m,k )4 − m for k even, σ (2 m,k )4 − m + ⌊ q +12 ⌋ for k odd.Proof. By Remark 5.14, it is sufficient to make the computations for G = PSp(2 m, C ). Thebilinear form Sym m − ( J ) has matrix · · · · · · − · · · − · · · . sp (2 m ) = { ( a i,j ∈ M m, m ( C ) | a i,j = ( − i + j +1 a m +1 − j, m +1 − i } . From this expression, as a diagonal matrix D ∈ PSp(2 m, C ) may be written as D = diag( λ , . . . , λ m , λ − m , . . . , λ − ) , we have that dim( sp (2 m ) D ) = 12 (cid:0) dim( gl (2 m ) D ) + dim( ad D ) (cid:1) where ad = { ( a i,j ∈ M m, m ( C ) | a ij = 0 if i + j = 2 m + 1 } ⊂ sp (2 m ) is the anti-diagonal,namely the subspace of matrices · · · ∗ · · · ∗ · · · ∗ ∗ · · · . A rotation of angle πk in PGL(2 , C ) is conjugate to ± e πik e − πik ! and D = Sym m − ± e πik e − πik !! = ± diag( e πik (2 m − , e πik (2 m − , . . . , e πik , e − πik , . . . , e − πik (2 m − ) . Therefore, the (adjoint) action of D , on the antidiagonal ad has eigenvalues { e πik (2 m − , e πik (2 m − , . . . , e πik , e − πik , . . . , e − πik (2 m − } . Thus, dim ad D is the number appearances of 1 in this list of eigenvalues:dim ad D = ( k even , ⌊ m − k k ⌋ = 2 ⌊ q +12 ⌋ for k odd . On the other hand, gl (2 m ) D = 2 mq + 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 previouscases.With a similar proof as for Proposition 5.9, we can show: Proposition 5.11.
There exists a uniform constant C ( O ) depending only on O such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dim Hit( O , PSp ± (2 m )) + χ ( O ) dim(PSp ± (2 m )) + X k even cp k k + X l even cr l l + b ! m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( O ) , where χ ( O ) denotes the untwisted Euler characteristic, cp k the number of cone points withstabilizer of order k , cr k the number of corner reflectors with stabilizer of order l , and b thenumber of components of ∂ O homeomorphic to R /D ∞ , an interval with mirror boundary. Using the same argument as in the proof of Proposition 5.7, and longer computations, onecan also prove:
Proposition 5.12.
Every component of R good ( O , PSp ± (2 m )) that contains representationsthat are boundary regular has dimension at most the dimension of the Hitchin component. .3 Exponents of a simple Lie algebra Recall from Subsection 2.2 that the exponents d , . . . , d r ∈ N of the Lie algebra g are definedby the equation Ad ◦ τ = r M α =1 Sym d α , Equation (2), where τ is the principal representation. Here r = rank g and P rα =1 (2 d α + 1) =dim g . Lemma 5.13.
For ρ ∈ Hit( O , G ) and x ∈ Σ O a point in the branching locus with nontrivialstabilizer Γ x : dim g ρ (Γ x ) = (P rα =1 (2 ⌊ d α k ⌋ + 1) if Γ x ∼ = C k (cid:0)P rα =1 ⌊ d α k ⌋ (cid:1) + { d α | d α is even } if Γ x ∼ = D k where C k is cyclic of order k , and D k dihedral of order k , r = rank( g ) , and { d , . . . , d r } arethe exponents of the Lie algebra.Proof. For a Fuchsian representation, the image of the cyclic group C k is generated by a matrixconjugate to ± e πik e − πik ! . (32)Using (2), the contribution of the α -th exponent to dim g ρ (Γ x ) is the multiplicity of eigen-value 1 in the 2 d α -th symmetric power of (32). Namely, the number of appearances of 1 in { e πik d α , e πik ( d α − , . . . , e πik ( − d α ) } which is precisely 2 ⌊ d α k ⌋ + 1.For the dihedral group, Corollary 4.9, the cyclic case, and (3) yield that the dimension isdim g ρ ( D k ) = (2 dim g ρ ( C ) + dim g ρ ( C k ) − dim g ) / r X α =1 (2 ⌊ d α ⌋ + 1 + ⌊ d α k ⌋ − d α )and 2 ⌊ d α ⌋ + 1 − d α = ( d α even,0 for d α odd.Lemma 5.13 allows to give a new proof of Proposition 5.4.As the exponents for sp (2 m ) and s0 ( m, m + 1) are the same, (1 , , · · · , m − Remark 5.14.
Let ρ ∈ Hit( O , PSp ± (2 m )) and ρ ∈ Hit( O , PO( m + 1 , m )) . For any x ∈ O , dim g ρ (Γ x ) = dim g ρ (Γ x ) . Another consequence of Lemma 5.13 is the following.
Corollary 5.15 ([1]) . Let O be a connected hyperbolic 2-orbifold. If { d , . . . , d rank G } are theexponents of G . Then the dimension of Hit( O , G ) is − χ ( |O | ) dim G + rank G X α =1 cp X i =1 d α − ⌊ d α k i ⌋ ) + cr X j =1 ( d α − ⌊ d α l j ⌋ ) + 2 b ⌊ d α + 12 ⌋ where the cone points have order k , k , . . . , k cp , the corner reflectors have order l , l , . . . , l cr ,and b is the number of components of ∂ O homeomorphic to R /D ∞ , an interval with mirrorboundary. roof. We follow the same scheme as the proof of Proposition 5.5. Set δ ij Z = 1 if i ∈ j Z and δ ij Z = 0 otherwise. Firstly we apply Proposition 4.2 and Lemma 5.13 to get that the dimensionof the Hitchin component is − χ ( O , Ad ρ ), which equals: − χ ( O \
Σ) dim G + rank G X α =1 − cp X i =1 (2 ⌊ d α k i ⌋ + 1) − cr X j =1 ( ⌊ d α l j ⌋ + δ d α Z ) + ( me − mp )(2 ⌊ d α ⌋ + 1) 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, and δ d α Z = 1 if d α is even,and 0 if d α is odd. Then the corollary follows from (30) and (31).This computation of dimensions is contained in [1, Theorem 1.2], where it is proved thatthe Hitchin component is a cell.For exceptional groups the dimension can be directly computed from this corollary and theexponents in Table 2. For instance, for (the real split form of the adjoint group) G , as theindices are 1 and 5 the dimension of the Hitchin component is − χ ( |O | ) + 8 cp + 10 cp , , + 12 cp ≥ + 4 cr + 5 cr , , + 6 cr ≥ + 8 b, where cp i is the number of cone points with stabilizer C i and cr i of corner reflectors withstabilizer D i . We restrict to orientable 3-orbifolds. In particular their singular locus is a union of circles,proper segments, and trivalent graphs. The isotropy groups or stabilizers of edges are cyclic,and the isotropy groups or stabilizers of trivalent vertices are non-cyclic finite subgroups ofSO(3) (hence dihedral, tetrahedral, octahedral or icosahedral). See Figure 6. nC n n nD n
23 3 T
23 4 O
23 5 I Figure 6: Local models for the branching locus, with the corresponding isotropy groups: thecyclic group C n , the dihedral group D n , the tetrahedral group T , the octahedral group O ,and the icosahedral group I . The subindex denotes the order of the group. The label on anedge denotes the order of the cyclic isotropy group of points in the interior of the edge. Let O denote an orientable hyperbolic 3-orbifold of finite type, possibly of infinite volume, andnon-elementary as Klenian group (hence the Zariski closure of its holonomy is PSL(2 , C )). Inparticular it has a compactification O that is an orbifold with boundary. Let ∂ O = ∂ O ⊔ · · · ⊔ ∂ k O denote its decomposition in connected components.29onsider the composition of the holonomy hol : π ( O ) → PSL(2 , C ) with the principalrepresentation τ : PSL(2 , C ) → G . As we assume that O is non-elementary, by Lemma 2.4 anda standard argument on the Zariski closure, we have: Remark 6.1.
The representation ρ = τ ◦ hol ∈ R ( O , G ) is good. The following is part of Theorem 1.1.
Theorem 6.2.
The character of ρ = τ ◦ hol is a smooth point of X ( O , G ) . Furthermore dim [ ρ ] X ( O , G ) = − k X i =1 e χ ( ∂ i O , Ad ρ ) + k X i =1 dim g ρ ( π ( ∂ i O )) = 12 k X i =1 dim [ ρ | ∂ i ] X ( ∂ i O , G ) = 12 dim [ ρ | ∂ ] X ( ∂ O , G ) . Proof.
We follow the proof for the manifold case in [27] for representations in PSL( n, C ), or[21] in PSL(2 , C ). In particular we use Selberg’s lemma: there exists O → O a finite regularcovering which is a manifold.The first step is to prove that the inclusion of ∂ O in O induces an injection in cohomology H ( O , Ad ρ ) ֒ → H ( ∂ O , Ad ρ ) . When O is a manifold, then this is proved in [27] using a theorem of vanishing in L -cohomology(as, using de Rham cohomology and a metric on the bundle and differential forms, every elementin the kernel is represented by an L -form), by using the decomposition (2) of Ad ◦ τ . Fororbifolds in general, using Selberg’s lemma, the inclusion follows from the manifold case andProposition 3.3.The next step is to prove thatdim H ( O , Ad ρ ) = 12 dim H ( ∂ O , Ad ρ ) . As in the manifold case, this follows from Poincar´e duality, Proposition 3.4, and the longexact sequence of the pair ( O , ∂ O ). Namely, if the morphisms in the long exact sequence incohomology are H ( O , Ad ρ ) i ∗ → H ( ∂ O , Ad ρ ) β → H ( O , ∂ O ; Ad ρ )then h i ∗ ( x ) , y i ∂ O = h x, β ( y ) i O ∀ x ∈ H ( O , Ad ρ ) , ∀ y ∈ H ( ∂ O , Ad ρ ) , where h , i denotes the perfect pairing in Proposition 3.4, and the pairing on ∂ O is the sum ofpairings 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 ( O , Ad ρ ) ∼ = H (Γ , Ad ρ ) by Proposition 3.4, with Γ = π ( O ), we use Gold-man’s obstruction theory to integrability to prove that [ ρ ] is a smooth point of the charactervariety of local dimension H ( O , Ad ρ ). For Γ < Γ a torsion-free subgroup, any infinitesimaldeformation in H (Γ , Ad ρ ) yields a Γ / Γ -equivariant infinitesimal deformation of Γ , and wemay apply the same argument as in Theorem 4.11. Alternatively as X (Γ , G ) is analyticallysmooth at the character of the restriction of ρ , by Cartan’s linearization [5, Lemma 1] there existlocal analytic coordinates that linearize the action of Γ / Γ in a neighborhood of the characterof ρ | Γ in X (Γ , G ). Hence the fixed point set of Γ / Γ is a smooth subvariety and, as ρ | Γ isgood, this fixed point set can be locally identified with X (Γ , G ).30 efinition 6.3. The component of Theorem 6.2 is called the canonical or distinguished com-ponent. To complete the proof of Theorem 1.1 we need to show that X ( O , G ) → X ( ∂ O , G ) is locallyan injection. This does not follow directly from the injection H ( O , Ad ρ ) → H ( ∂ O , Ad ρ ), as X ( ∂ O , G ) can be non-smooth because of rank 2 cusps. (Rank one cusps are part of componentswith negative Euler characteristic and we discuss only rank 2 cusps.) Consider a nonsingular(manifold) cusp, T × [0 , + ∞ ) equipped with the warped product metric e − t g T + dt , where g T is a flat metric on the torus and t ∈ [0 , + ∞ ) is the Busemann function coordinate. Set ρ = τ ◦ hol : π ( T ) → G . Using de Rham cohomology, we may talk about L -forms on T × [0 , + ∞ ). Lemma 6.4.
For any non-contractible loop l ⊂ T , the restriction induces a surjection H ( T × [0 , + ∞ ) , Ad ρ ) → H ( l, Ad ρ ) whose kernel consists of the cohomology classes represented by L -forms.Proof. Since Ad ◦ τ = ⊕ α Sym d α by (2) and Ad ◦ Sym n − = ⊕ n − i =1 Sym i , it suffices to provethe lemma for τ = Sym n − and g = sl ( n, C ). In [26] precisely the situation when τ = Sym n − is discussed. In particular by [26, Lemma 3.3] the image of the map induced by the inclusion sl ( n, C ) π ( T ) ⊂ sl ( n, C ), H ( T , sl ( n, C ) π ( T ) ) → H ( T , sl ( n, C ) Ad ρ ) (33)is represented by L -forms. (Here sl ( n, C ) Ad ρ denotes the coefficients on the Lie algebra, usuallyjust denoted by Ad ρ , whilst on the invariant subspace sl ( n, C ) π ( T ) the action is trivial byconstruction.) Furthermore, either by [26, Lemma 3.4] or by a straightforward computation,the map H ( T , sl ( n, C ) Ad ρ ) → H ( l, sl ( n, C ) Ad ρ )is a surjection with kernel precisely the image of (33). Remark 6.5.
Let γ ∈ π ( T ) be an element represented by the loop l as in Lemma 6.4. Wehave a natural isomorphism: H ( l, Ad ρ ) ∼ = H ( l, Ad ρ ) ∼ = H ( h γ i , Ad ρ ) ∼ = g ρ ( γ ) = g ρ ( π ( T )) Let e G be the universal covering of G and let χ , . . . , χ r denote its fundamental characters(the characters of the fundamental representations). E.g. for e G = SL( n, C ) the fundamentalcharacters are the symmetric functions on the eigenvalues. By a theorem of Steinberg [31], theydefine an isomorphism: ( χ , . . . , χ r ) : e G reg /G ∼ = C r , (34)where e G reg denotes the set of regular elements in e G . When the group G is not simply connected,the characters ( χ , . . . , χ r ) still define local functions in a neighborhood of ρ ( γ ) in G , where γ ∈ π ( T ) is represented by the loop l . We identify the variety of representations of the cyclicgroup h γ i ∼ = Z with G via the image of γ . In particular the differential form dχ i : g → C maybe viewed as a linear map dχ i : H ( h γ i , Ad ρ ) ∼ = g ρ ( γ ) → C , because the characters χ i are constant on orbits by conjugation (hence the characters are trivialon coboundaries). This uses the natural identification between H ( h γ i , Ad ρ ) and the space ofcoinvariants g ρ ( γ ) in Remark 6.5. 31 emma 6.6. ( dχ , . . . , dχ r ) : H ( h γ i , Ad ρ ) → C r is an isomorphism.Proof. We follow the proof of [18, Corollary 19]. Steinberg has shown that the map (34) hasa smooth section, C r → e G reg [31]. In particular ( dχ , . . . , dχ r ) : g → C r is surjective and so isthe induced map ( dχ , . . . , dχ r ) : H ( h γ i , Ad ρ ) ∼ = g ρ ( γ ) → C r . Hence the lemma follows fromequality of dimensions.Consider ∂ hyp O the union of components of ∂ O that have negative Euler characteristic.In particular, for a manifold O = M , ∂M = ∂ hyp M ⊔ T ⊔ · · · ⊔ T k where k is the number of rank-2 cusps of M . Proposition 6.7.
When M = O is a manifold, then the restriction to the boundary and thecharacters of peripheral curves of cusps yield a local embedding X ( M , G ) → X ( ∂ hyp M , G ) × C rk , where k is the number of rank-2 cusps and the coordinates in C rk are (locally defined) funda-mental characters of chosen peripheral elements, one for each cusp.Proof. The proof of Theorem 6.2 is based in the vanishing of L -cohomology, here we use thefact that a form is L on M if it is so in the restriction to each end. Hence using Lemmas 6.4and 6.6 the kernel of the differential of the map of the proposition is represented by L -forms,hence trivial.The following concludes the proof of Theorem 1.1. Corollary 6.8.
The restriction to the boundary yields a local embedding X ( O , G ) → X ( ∂ O , G ) .Proof. In the manifold case, we use that the local injection in Proposition 6.7 factors throughthe restriction to X ( ∂M , G ): X ( M , G ) → X ( ∂M , G ) → X ( ∂ hyp M , G ) × C rk , Thus the map induced by restriction X ( M , G ) → X ( ∂M , G ) must be locally an injection.For an orbifold O , consider a finite manifold covering O → O , we have the commutativediagram: X ( O , G ) −−−−→ X ( ∂ O , G ) x x X ( O , G ) −−−−→ X ( ∂ O , G )As O is a manifold, we have already shown that X ( O , G ) → X ( ∂ O , G ) is locally injective.Furthermore, every character [ ρ ′ ] in a neighborhood of [ ρ ] = [ τ ◦ hol] is good, because beinggood is an open property in the variety of characters and τ ◦ hol is good by Remark 6.1. As O is also non-elementary, the restriction of [ ρ ′ ] to π ( O ) is also good, and its centralizer in G istrivial and therefore the possible extension of the restricted representation ρ ′ | π ( O ) to π ( O )is unique. Namely, the restriction map X ( O , G ) → X ( O , G ) is also locally injective, and weare done from the commutativity of the diagram.A particular case of Proposition 6.7 is the following corollary (proved for representations ofmanifold groups in SL(2 , C ) in [4, 21], and for representations of manifold groups in SL( n, C )in [26]). 32 orollary 6.9. When M is a manifold whose interior has finite volume, chose γ , . . . , γ k anontrivial peripheral element for each cusp. Let χ , . . . , χ r denote the fundamental charactersof e G . Then ( χ ,γ , . . . , χ r,γ k ) : U ⊂ X ( M , G ) → C kr . define a local homeomorphism for a neighborhood U ⊂ X ( M , G ) of [ τ ◦ hol] . Recall that the adjoint or Sym gives an isomorphism PSL(2 , C ) ∼ = SO(3 , C ). By an argu-ment on dimensions (see Table 4): Corollary 6.10.
Let O be a compact orientable three-orbifold, with hyperbolic interior of finitevolume. If all components of ∂ O are homeomorphic to S (2 , , , , S (2 , , or S (2 , , ,then X ( O , SL(3 , C )) ∼ = X ( O , SO(3 , C )) ∼ = X ( O , PSL(2 , C )) , where X denotes the distinguished component. In this subsection we provide a lower bound `a la Thurston (for PSL(2 , C )) or `a la Falbel-Guilloux(for general G ), in both cases for manifolds. For a compact three orbifold O , let ∂ O = ∂ O ⊔ · · · ⊔ ∂ k O denote the decomposition in connected components of its boundary. Theorem 6.11.
Let G be semi-simple C -algebraic Lie group, O a compact, orientable verygood orbifold. Let ρ : π ( O ) → G be a good representation. Assume that:(a) if ∂ i O is a hyperbolic boundary component, then the centralizer of its image is zero-dimensional;(b) if ∂ i O is a Euclidean boundary component, then the restriction ρ | π ( O ) is strongly regular.Then dim [ ρ ] X (Γ , G ) ≥ k X i =1 dim [ ρ | ∂ i ] X ( ∂ i O , G )= − k X i =1 e χ ( ∂ i O , Ad ρ ) + k X i =1 dim g ρ ( π ( ∂ i O ))) . Proof.
The branching locus Σ of the compact orbifold O is a trivalent graph. Chose a finitesubset Σ ⊂ 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 Σ has the smallest cardinality so that Σ \ Σ isa disjoint union of edges (open or not, as they can meet ∂ O ). Consider the orbifold O = O \ N (Σ ) , ie. remove an open tubular neighborhood for each point in Σ . The branching locus of O isa union of (proper) edges (it contains no vertices nor circles). The boundary components of O are either the boundary components of O or the boundary of a neighborhood of a pointin Σ (spherical). For each Euclidean or spherical boundary component ∂ i O , chose k i disjointembedded loops γ i, , . . . , γ i,k i in O based at ∂ i O so that g ρ ( π ( ∂ O )) ∩ g ρ ( γ i, ) ∩ · · · ∩ g ρ ( γ i,ki ) = 0 . g of the image of the group generated by π ( ∂ i O ) and γ i, , . . . , γ i,k i is trivial. The next step in the construction is to remove an open tubular neighborhood of the γ i,j : O = O \ [ i,j N ( γ i,j ) . Finally remove an open tubular neighborhood of the branching locus of O (which is a union ofedges): O = O \ N (Σ O ) . In particular O is a manifold.The connected 3-manifold with non-empty boundary O retracts to a 2-dimensional CW-complex with a single 0-cell. This gives a presentation of π ( O ) from this CW-complex (1-cellsare generators and 2-cells relations). Each generator contributes with a copy of G in the varietyof representations, and each relation decreases at most dim G the dimension; this gives thestandard bound:dim R ( O , G ) ≥ (1 − χ ( O )) dim G = (cid:18) − χ ( ∂ O )2 (cid:19) dim G = − e χ ( ∂ O , Ad ρ )) + dim G, (35)using that ∂ O is a surface.Next we compute lower bounds for the dimension of the variety of representations of theorbifolds O , O , 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 [9]: Lemma 6.12 ([9]) . Let W be a smooth complex (analytic) variety and W ′ ⊂ W a smoothsubvariety. Let X be a complex variety with an analytic map f : X → W . For p ∈ X with f ( p ) ∈ X , there exists a neighborhood U ⊂ X of p such that codim p ( f − ( W ′ ) ∩ U, X ) ≤ codim f ( p ) ( W ′ , W ) . Proof.
By the implicit function theorem, there exists a neighborhood V ⊂ W of f ( p ) and ananalytic map F : V → C k such that V ∩ W ′ = F − (0), where k = codim f ( p ) ( W ′ , W ). Then, fora neighborhood of p , U ⊂ f − ( V ) ⊂ X , f − ( W ′ ) ∩ U = ( F ◦ f ) − (0) ∩ U, and the estimate follows, because f − ( W ′ ) ∩ U is a fiber of a map F ◦ f : U → C k .To get O from O , we add the edges of Σ O . Topologically, we add 2-handles whose co-coreis an edge of Σ O , hence “2-handles with singular co-core”. For each singular edge e of Σ O consider a meridian µ ∈ π ( O ), which is represented by the attaching circle of the 2-handle,and has finite order in π ( O ). 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 centralizerdim( g ρ ( µ ) ). Thus, by Lemma 6.12, when we add the edge e to O , the dimension of the varietyof representations decreases at most by dim( g ρ ( µ ) ):dim R ( O , G ) − dim R ( O , G ) = codim( R ( O , G ) , R ( O , G )) ≤ X µ codim( O ( ρ ( µ )) , G )= X µ dim( g ρ ( µ ) ) . (36)34urthermore, adding the 2-handle to O corresponding to µ means replacing an annulus in ∂ O by two cone points with cyclic stabilizer generated by µ . Hence, the twisted Euler characteristicof the boundary increases by 2 dim( g ρ ( µ ) ). Counting the contribution of all edges of Σ O : e χ ( ∂ O , Ad ρ ) = e χ ( ∂ O , Ad ρ ) + 2 X µ dim( g ρ ( µ ) ) . (37)Hence from (35), (36), and (37):dim R ( O , G ) ≥ − e χ ( ∂ O , Ad ρ ) + dim G. (38)To get O from O , we add the neighborhoods of the loops γ i,j , i.e. we add 2-handles(without singular co-core). Consider the i -th boundary component ∂ i O . The correspondingboundary component ∂ i O is obtained from ∂ i O by the surgery corresponding to the additionof k i k i annuli in ∂ i O corresponds to 2 k i smooth disks in ∂ i O . Thus e χ ( ∂ i O , Ad ρ ) = e χ ( ∂ i O , Ad ρ ) + 2 k i dim G. (39)On the other hand, following the idea of [9], we apply Lemma 6.12 to ∂ i O and the free product π ( ∂ i O ) ∗ h γ i, i ∗ · · · ∗ h γ i,k i i , that is, the group obtained by filling the k i meridians of the surface ∂ i O : W ′ = R ( π ( ∂ i O ) ∗ h γ i, i ∗ · · · ∗ h γ i,k i i , G ) ⊂ R ( ∂ i O , G ) = W. (40)By Proposition 4.1, and since we assume that the centralizer of ρ ( π ( ∂ i O )) in g is trivial, W is smooth and dim W = dim R ( ∂ i O , G ) = − e χ ( ∂ i O , Ad ρ ) + dim G. (41)By Proposition 4.1 when ∂ i O is hyperbolic, Theorem 4.11 when Euclidean, and Remark 4.17when spherical, R ( π ( ∂ i O ) , G ) is smooth anddim R ( ∂ i O , G ) = − e χ ( ∂ i O , Ad ρ ) + dim G + dim g ρ ( π ( ∂ i O )) . (42)As the variety of representations of a free product is the Cartesian product of varieties orrepresentations of its factors, W ′ is smooth and:dim W ′ = dim R ( π ( ∂ i O ) ∗ h γ i, i ∗ · · · ∗ h γ i,k i i , G ) = dim R ( ∂ i O , G ) + k i dim G. (43)From (42) and (43):dim W ′ = − e χ ( ∂ i O , Ad ρ ) + ( k i + 1) dim G + dim g ρ ( π ( ∂ i O )) . (44)It follows from (41) and (44) that the codimension of the inclusion (40) iscodim( W ′ , W ) = dim W − dim W ′ = e χ ( ∂ i O , Ad ρ ) − e χ ( ∂ i O , Ad ρ ) − k i dim G − dim g ρ ( π ( ∂ i O )) , that combined with (39) yieldscodim( W ′ , W ) = k i dim G − dim g ρ ( π ( ∂ i O )) . (45)So when we apply Lemma 6.12 we get from the contribution (45) of each boundary component:codim( R ( O , G ) , R ( O , G )) ≤ X i (cid:16) k i dim G − dim g ρ ( π ( ∂ i O )) (cid:17) . (46)35hus, with (38) and (39), (46) becomes:dim R ( O , G ) ≥ dim R ( O , G ) − X i (cid:16) k i dim G − dim g ρ ( π ( ∂ i O )) (cid:17) ≥ X i (cid:16) − e χ ( ∂ i O , Ad ρ )) − k i dim G + dim g ρ ( π ( ∂ i O ) (cid:17) + dim G = X i (cid:16) − e χ ( ∂ i O , Ad ρ )) + dim g ρ ( π ( ∂ i O ) (cid:17) + dim G. Finally, for each spherical boundary component ∂ i O , by (28) − e χ ( ∂ i O , Ad ρ )) + dim g ρ ( π ( ∂ i O ) = 0 . Hence we can get rid of the contribution of the neighborhoods of vertices and get the initialorbifold O : dim R ( O , G ) ≥ X i (cid:16) − e χ ( ∂ i O , Ad ρ )) + dim g ρ ( π ( ∂ i O ) (cid:17) + dim G = dim X ( ∂ O , G ) + dim G. As dim X ( O , G ) = dim R ( O , G ) − dim G , because the representation is good, this concludesthe proof of the theorem. Let M be an orientable hyperbolic 3-manifold of finite volume with k ≥ M is amanifold, its holonomy lifts to SL(2 , C ).Let X ( M, SL( n, C )) be the canonical or distinguished component of X ( M, SL( n, C )) (Def-inition 6.3), i.e. the component that contains the composition of a lift of the holonomy withSym n − . By Theorem 6.2, dim X ( M, SL( n, C )) = ( n − 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 andyield 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 fromthe introduction. Proposition 6.13.
Let Γ be the fundamental group of the exterior of the figure eight knot.Besides the canonical component (that has dimension n − ), for large n X (Γ , SL( n, C )) has atleast components that contain irreducible representations, whose dimension grow respectivelyas n / , n / and n / .Proof. Let K p/q denote the Dehn surgery on the figure eight knot with coefficients p/q . Thereare 6 Dehn fillings on the figure eight knot that yield small Seifert manifolds [12, 25]: K ± aresmall Seifert manifolds fibered over the 2-orbifold O = S (3 , , K ± over O = S (2 , , K ± over O = S (2 , , , R ) of the filled 3-manifolds K p , that map the fiber to − Id and induce the holonomy representation of the (unique) hyper-bolic structure on O p . The composition with Sym n − is a representation that maps the fiberto ( − n − Id and it induces a representation from π ( O p ) to PSL( n, R ) in the Hitchin com-ponent. Therefore the complexification of the Hitchin component of O p yields a component36f X ( K p , SL( n, C )) of the same dimension, and hence a subvariety of X (Γ , SL( n, C )). Let X p ⊂ X ( M, SL( n, C )) be the component of X (Γ , SL( n, C )) that contains the subvariety in-duced by X ( K p , SL( n, C )). Next we estimate the dimension of X , X and X , which allow todistinguish them, but does not allow to distinguish the component induced by X p from X − p ,as the dimension estimates are the same.The lower bound on the dimension of X p is given by Proposition 5.9:dim X p ≥ − χ ( O p ) n − c ( O p ) . Aiming to find an upper bound of dim X p , we bound above the dimension of its Zariski tangentspace, which is isomorphic to H (Γ , Ad ρ ) ∼ = H ( M, Ad ρ ) for M the (compact) exterior ofthe figure eight knot, M = S \ N ( K ). In particular ∂M ∼ = T . Here ρ = Sym n − ◦ ρ , for ρ : Γ → SL(2 , R ) the representation that factors through K p .First we bound dim H ( ∂M, Ad ρ ). Notice that since χ ( T ) = 0,dim H ( ∂M, Ad ρ ) = 2 dim H ( ∂M, Ad ρ ) = 2 dim g Ad ρ ( π ( ∂M )) . To compute dim g Ad ρ ( π ( ∂M )) , one may check explicitly that the image of ρ ( π ( ∂M )) containsan element of SL(2 , R ) of infinite order (one may use for instance the A-polynomial). As thesymmetric power of a hyperbolic matrix in SL(2 , C ) is regular, ρ ( π ( ∂M )) contains regularelements and dim g Ad ρ ( π ( ∂M )) = rank G = n −
1, Thus dim H ( ∂M, Ad ρ ) = 2( n − i ∗ : H ( M, Ad ρ ) → H ( ∂M, Ad ρ )be the morphism induced by inclusion. Using the long exact sequence of the pair ( M, ∂M ) andPoincar´e duality as in the proof of Theorem 6.2, its rank isrank( i ∗ ) = n − . (47)Furthermore, by Mayer-Vietoris exact sequence applied to the decomposition K p = M ∪ ∂M ( D × S ), we get that dim ker i ∗ ≤ dim H ( K p , Ad ρ ) . (48)Next we need a lemma: Lemma 6.14.
The projection π ( K p ) → π ( O p ) induces an isomorphism H ( π ( K p ) , Ad ρ ) ∼ = H ( π ( O p ) , Ad ρ ) . (49)Assuming the lemma, we conclude the proof of Proposition 6.13. Putting together (47), (48)and (49): dim H ( M, Ad ρ ) ≤ dim H ( O p , Ad ρ ) + ( n − . It follows that − n χ ( O p ) − c ( O p ) ≤ dim X p ≤ − n χ ( O p ) + c ( O p ) + n − . For large n this allows to distinguish three components, according to the different values for χ ( O p ). Proof of Lemma 6.14.
We use the central exact sequence1 → Z ( π ( K p )) → π ( K p ) → π ( O p ) → , Z ( π ( K p )) ∼ = Z is the group generated by the fiber. We prove the lemma, withcrossed morphisms or derivations, as in Subsection 2.1. Since π ( K p ) → π ( O p ) is a surjection,we have an injection of spaces of derivations Z ( π ( O p ) , g ) ֒ → Z ( π ( K p ) , g ) . (50)We prove surjectivity of the map in (50): let d ∈ Z ( π ( K p ) , g ) be a derivation (namely a map d : π ( K p ) → g satisfying d ( γ γ ) = d ( γ ) + Ad ρ ( γ ) d ( γ ), ∀ γ , γ ∈ π ( K p )). Let t ∈ π ( K p ) bea generator of the center Z ( π ( K p )) ∼ = Z ; from the relation tγt − = γ, ∀ γ ∈ π ( K p ) , we deduce d ( t ) + Ad ρ ( t ) d ( γ ) − Ad ρ ( tγt − ) d ( t ) = d ( γ ) , ∀ γ ∈ π ( K p ) . Since ρ ( t ) = ± Id, Ad ρ ( t ) is the identity on g , hence(1 − Ad ρ ( γ ) ) d ( t ) = 0 , ∀ γ ∈ π ( K p ) . This equality implies that d ( t ) = 0, because g ρ ( π ( K p )) = 0 by irreducibility of ρ . This proves thatevery crossed morphism of π ( K p ) factors through a crossed morphism of π ( O ), hence (50)is surjective. This isomorphism between cocycle spaces induces an isomorphism of coboundaryspace and this proves the lemma. Remark 6.15.
For small values of n , the arguments are not useful, but for n = 3 the canon-ical component has dimension 2, and, according to Table 6, ( ± -Dehn fillings also providetwo subvarieties of dimension 2. It is proved in [10] and [16] that the canonical componentand the subvarieties induced by the ( ± -Dehn filling are precisely the three components of X (Γ , SL(3 , C )) that contain irreducible representations. Remark 6.16.
At the moment, we do not know whether for large n the component that containsrepresentations that factor through K p is the same as the component corresponding to K − p , for p = 1 , , . To distinguish them would require a better upper bound of the dimension of theirZariski tangent space. This would allow also to distinguish components corresponding to Galoisconjugates (5-th or 7-th roots of unity). Remark 6.17.
An analog of Proposition 6.13 can be obtained for the variety of characters ofthe figure eight knot in
Sp(2 m, C ) , for large m . Twist knots have Dehn fillings that are small Seifert manifolds [3], and therefore there is asimilar behavior. Twist knots are obtained by Dehn filling on one component of the Whiteheadlink, that we discuss next.
Proposition 6.18.
Let Γ be the fundamental group of the exterior of the Whitehead link.Besides the canonical component (that has dimension n − ), for large n , X (Γ , SL( n, C )) hasat least components that contain irreducible characters, whose dimension grow respectively as n / , n / and n / . The proof of Proposition 6.18 is the same as Proposition 6.13. Here there are partial Dehnfillings that yield Seifert fibered manifolds, with basis a disc with two cone points: the ( − D (3 , − D (2 , − D (2 ,
3) [25, Table A.1]. This yields the three rational orbifold Euler characteristics, 1 /
3, 1 / /
6. 38s the basis orbifolds are rather simple, it is also easy to state the dimension of their Hitchincomponent. The dimension of the Hitchin component of D (3 ,
3) is ( n + 1 for n ≡ n − for n D (2 ,
4) is n + 1 for n ≡ n − for n ≡ n for n ≡ D (2 ,
3) is n + 1 for n ≡ n − for n ≡ ± n +26 for n ≡ n +36 for n ≡ Remark 6.19.
When we apply these computation to
SL(3 , C ) , for D (2 , and D (2 , it yieldsdimension , but for D (3 , it has a component of dimension , the same as the dimensionof the canonical component. It has been proved by Guilloux and Will [13] that this is in fact awhole component of X (Γ , SL(3 , C )) . For Montesinos links, the same arguments yield the following:
Proposition 6.20.
Let L ⊂ S be a Montesinos link. Assume that either it has at least 4tangles, or that it has three tangles (eg. a pretzel link) with indices α , α , and α satisfying α + α + α ≤ . Then dim X ( S \ L, SL( n, C )) grows quadratically with n . (3 , C ) In this section we give explicit computations of some varieties of characters in SL(3 , C ). Wecompute varieties of characters of groups generated by two elements, using the description of X ( F , SL(3 , C )) due to Lawton [22], where F = h a, b |i is the free group of rank two.Setting coordinates x = t a , y = t b , z = t ab , r = t ab − , τ = t [ a,b ] ,u = t a − , v = t b − , w = t ( ab ) − , s = t a − b , (51)Lawton has proved: Theorem 7.1 ([22]) . X ( F , SL(3 , C )) is isomorphic to the hypersurface of C defined by τ − P τ + Q = 0 for some polynomials P, Q ∈ Z [ x, y, z, u, v, w, r, s ] . The solutions of τ − 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 . P and Q in Theorem 7.1 are: P = xuyv − uyr − xvs − uvz − xyw + rs + xu + yv + zw − Q = vu x y + uv y x − ryxu − uxvrw − ry vu + rx v − rzyxv + su y − swvuy − svux − uxysz − sv yx − u vy + u v w − zu xv − uv x − zv yu − wx uy − uy x − wy vx − x vy + x y z + uwr − r xv + r zy + surx + svry + wszr + rzu + ruv + rvz + xrw + rwy + x yr − s uy + s wv + xzs + u vs + usz + szv + syw + swx + sxy + u wy − w vu + xuyv + wuzx + uy z + xv w + wvzy + x zv − z yx + r + 3 uyr − rvw − rzx + s − swu + 3 xvs − syz + u + 3 uvz + v + w + 3 xyw + x + y + z − rs − xu − yv − zw + 9 This theorem tells that an SL(3 , C )-character of F 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, includingCayley-Hamilton’s formula A − tr( A ) A + tr( A − ) A − Id = 0 , ∀ A ∈ SL(3 , C ) , (52)and elementary identities on traces. Let us compute the variety of SL(3 , C )-characters of some 2-orbifolds.We start with the turnover S (3 , , π ( S (3 , , ∼ = h a, b | a = b = ( ab ) = 1 i− 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 { , ω, ω } , for ω ∈ C aprimitive third root of unity: ω = 1 and ω = 1.(b) A matrix in SL(3 , C ) of order 4 is either trivial or has eigenvalues { , i, − i } , {− , i, i } , {− , − i, − i } , or { , − , − } . S (3 , ,
4) 3 34 T (3 , ,
4) 3 4 D (3; 4)Figure 7: The turnover S (3 , ,
4) and two non-orientable quotients: a triangle with mirrorboundary T (3 , , D (3; 4). 40 emark 7.2. Let ρ : π ( S (3 , , → SL(3 , C ) be an irreducible representation. The eigen-values of ρ ( a ) and ρ ( b ) are { , ω, ω } and the eigenvalues of ρ ( ab ) are { , i, − i } , {− , i, i } , {− , − i, − i } , or { , − , − } . Furthermore, using that for an irreducible representation dim [ ρ ] X ( S (3 , , , SL(3 , C )) = − e χ ( S (3 , , , Ad ρ ) , and dim g ρ ( ab ) depends on whether ρ ( ab ) has repeated eigenvalues or not, we have: dim [ ρ ] X ( S (3 , , , SL(3 , C )) = ( if the eigenvalues of ρ ( ab ) are { , i, − i } , otherwise. Using the coordinates (51), the two dimensional components of the variety of characters areobtained 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 (3 , , , SL(3 , C )) that has positive dimension is iso-morphic to { ( r, s, τ ) ∈ C | τ − ( rs − τ + ( r + s − rs + 5) = 0 } . For the components that are isolated points, we set, again x = y = u = v = 0 and, accordingto the eigenvalues of ρ ( ab ):( z, w ) = ( − i, − − i ) , ( z, w ) = ( − − i, − i ) , or ( z, w ) = ( − , − . 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 ) − a = ( ab ) a , using (52) and replacing x = y = u = v = 0 we get s = zs and r = wr. (53)For the 2-dimensional component it holds z = w = 1, hence (53) do not give any furtherinformation. For the isolated components, (53) yield r = s = 0. For these values P − Q = 0and therefore there is a unique value for τ . Hence we get: Example 7.4.
There are three components of X ( S (3 , , , SL(3 , C )) that contain irreduciblerepresentations and are zero-dimensional. They have coordinates x = y = u = v = r = s = 0 and ( z, w, τ ) = ( − i, − − i, , ( − − i, − i, , or ( − , − , − . Next we describe the Hitchin component Hit( S (3 , , , PSL(3 , R )). Notice that the repre-sentation Sym : SL(2 , R ) → SL(3 , R ) factors through PSL(2 , R ), hence Hit( S (3 , , , PSL(3 , R ))lifts to a component of X ( S (3 , , , SL(3 , R )). We consider the 2-dimensional component ofExample 7.3 and we require that its coordinates are real: { ( r, s, τ ) ∈ R | τ − ( rs − τ + ( r + s − rs + 5) = 0 } . By looking at the discriminant of this equation, this real set has has three components. Oneof them is the isolated point ( r, s, τ ) = (2 , , R , one of them is the lift ofthe Hitchin component. By looking at the symmetric power of the holonomy, we deduce:41 xample 7.5. The Hitchin component
Hit( S (3 , , , PSL(3 , R )) is isomorphic to the compo-nent of { ( r, s, τ ) ∈ R | τ − ( rs − τ + ( r + s − rs + 5) = 0 } that contains the point of coordinates ( r, s, τ ) = (2 + 2 √ , √ , √ . Next we consider the turnover T (3 , , π ( S (3 , , σ , that satisfies σ = 1 , σaσ = a − , and σbσ = b − . (54)Denote by σ ∗ the involution induced on the fundamental group: σ ∗ ( γ ) = σγσ − , ∀ γ ∈ π ( S (3 , , . This involution σ ∗ permutes the coordinates r and s and preserves τ , the trace of the commu-tator. Therefore, the component of X ( S (3 , , , SL(3 , C )) σ ∗ of positive dimension is { ( r, τ ) ∈ C | τ − ( r − τ + (2 r − r + 5) = 0 } . This curve is singular precisely at ( r, τ ) = (2 , Lemma 7.6.
The restriction map X ( T (3 , , , SL(3 , C )) → X ( S (3 , , , SL(3 , C )) σ ∗ desingularizes the curve τ − ( r − τ + (2 r − r + 5) = 0 .The component of X ( T (3 , , , 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 (3 , , , 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 π ( S (3 , , , C ) we show thatthere is a unique choice of A ∈ SL(3 , C ) such that mapping σ to A defines an extension of ρ to π ( T (3 , , ρ , as ρ and ρ ◦ σ ∗ have the same character, there existsa matrix A ∈ SL(3 , C ) such that AρA − = ρ ◦ σ ∗ . Furthermore, A is unique up multiplicationby a matrix in the center (the center of SL(3 , C ) is { Id , ω Id , ω Id } ). As σ is an involution, A commutes with ρ , that is irreducible, and therefore A ∈ { Id , ω Id , ω Id } . Hence among A , ωA or ω A there is a unique choice whose square is the identity. Thus there is a unique choice thatsatisfies (54).To prove (b) we use that for a reducible representation the trace of a commutator and itsinverse are the same, thus it must be a zero of the discriminant of the defining equation. Thisdiscriminant is ( r − ( r − r −
4) (55)and its zeros are r = 2 and r = 2 ± √
2. The value r = 2 ± √ r = 2 isthe only reducible character. 42o prove (c), we check first that r = 2 corresponds to a representation ρ = ρ ⊕ Id, for ρ : π ( S (3 , , → SL(2 , C ) an irreducible representation (for a reducible representation inSL(2 , C ) the trace of any commutator is 2 and, as τ = 1, trace( ρ ([ a, b ])) = τ − = 2). Then,reproducing the argument of the irreducible case (a), there exist a matrix A ∈ SL(2 , C ) suchthat Aρ A − = ρ ◦ σ ∗ and, by irreducibility of ρ , A is unique up to sign. Furthermore, againby irreducibility of ρ , A is central in SL(2 , C ), namely A = ± Id. The case A = + Id doesnot occur, because this would imply that A = ± Id and with (54) this would yield that ρ itselfwould be central, but it is irreducible. Therefore A = − Id and the choices for ρ ( σ ) are ρ ( σ ) = ± i A
000 0 − . This concludes the proof of (c) and of the lemma.Next we describe the Hitchin component of the turnover. We look therefore at the realpoints { ( r, τ ) ∈ R | τ − ( r − τ + (2 r − r + 5) = 0 } . By looking at the discriminant (55), the set of real points has three components, the isolatedpoint ( r, τ ) = (2 ,
1) an two lines, defined by r ≤ − √ r ≥ √
2. As r = 2 + 2 √ Example 7.7.
The Hitchin component
Hit( T (3 , , , PSL(3 , R )) is isomorphic (via the restric-tion to S (3 , , ) to the line { ( r, τ ) ∈ R | τ − ( r − τ + (2 r − r + 5) = 0 , r ≥ √ } . Finally, we consider D (3; 4), the disc with a cone point of order 3, mirror boundary, and acorner reflector of order 4 (with isotropy group the dihedral group of 8 elements), Figure 7. Itis again the quotient of S (3 , ,
4) by an involution. This involution maps a to b − and b to a − . Therefore it fixes the coordinates r and s but permutes the trace of [ a, b ] with the trace ofits inverse [ b, a ]. The fixed point set of this involution in X ( S (3 , , , SL(3 , C )) is obtained bylooking at the zero locus of the discriminant of the variable τ : { ( r, s ) ∈ C | r s − r − s + 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 (3 , , , SL(3 , C )) σ ′ desingularizes the curve r s − r − s + 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 r s − r − s + 16 rs −
16 = 0 is the discriminant locus of theset of real points for S (3 , , r = s = 2 and twounbounded curves. Thus, similarly to S (3 , ,
4) we have:
Example 7.9.
The Hitchin component
Hit( D (3; 4) , PSL(3 , R )) is isomorphic (via the restrictionto S (3 , , ) to the component of { ( r, s ) ∈ R | r s − r − s + 16 rs −
16 = 0 } that contains r = s = 2 + 2 √ . .2 A family of 3-dimensional orbifolds Consider the family of examples with underlying space S (perhaps punctured twice) and threebranching arcs as in Figure 8, with branching labels n , n and n ≥
2. Whether a singularvertex is included or not depends on the orders of the adjacent branching locus: if 1 /n +2 /n >
1, then the three edges meet at a vertex, otherwise the vertex is removed, and similarly for theother vertex. In other words, we consider the twice punctured 3-sphere, and we add the verticesneeded for the orbifold to be irreducible. n n n 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 O denote this orbifold. The geometry of O depends on the indices:(a) When n = 2 and 1 /n +2 /n ≤
1, it is an interval bundle, over a disc with mirror boundary,with an interior cone point of order n and a corner reflector of order 2 n . It is doublycovered by the product of a turnover S ( n , n , n ) by an interval.(b) When n = 2 and 1 /n + 2 /n >
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 ( n , n , n ).(c) When ( n , n , n ) = (2 , ,
3) the orbifold is spherical [8].(d) If n , n = 2 and ( n , n , n ) = (2 , , π ( O ) ∼ = h a, b | a n = b n = [ a, b ] n = 1 i . In case (a) the variety of characters of O 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 n and n . Cases (b)and (c) are spherical, hence the variety of characters is finite. In the hyperbolic case, (d), when n = 2 then the canonical component is an isolated point, by Theorem 6.2. For n i ≥
3, it hasdimension 2.
Example 7.10.
Consider the case n = n = n = 3 . The fundamental group is π ( O ) = h a, b | a = b = [ a, b ] = 1 i . On the canonical component, the image of any of a , b and [ a, b ] is not central, therefore byTheorem 7.1, using the notation in (51) we impose the following equalities: x = y = u = v = P = Q = 0 . Thus the canonical component X ( O , SL(3 , C )) is the following complex surface: (cid:26) rs + zw − r + s + z + w + rszw − rs − zw + 9 = 044 n πn πn π n π n π n A B
Figure 9: The fundamental domain of the orbifold in this subsection. It is an ideal tetrahedronin hyperbolic 3-space, with vertices perhaps ideal or hyperideal. The side parings, a and b , arerotations of order n and n around two opposite edges. Those edges have dihedral angle 2 π/n and 2 π/n respectively, the remaining four edges have dihedral angle π/ (2 n ). The symmetric power of the complete structure has coordinates r = s = − √− , z = w = − − √− (The complex conjugate corresponds to a change of orientation).There are other components of X ( O , SL(3 , C )) but it may be checked that they are isolatedpoints. For instance, if ρ ( a ) is central, then ρ ( b ) has order three at this gives finitely manyconjugacy 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 . This isrealized by the representation ρ ( a ) = ω
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Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, and Centrede Recerca Matem`atica (UAB-CRM), 08193 Cerdanyola del Vall`es, Spain [email protected]@mat.uab.cat