Michael Heusener

SL n (C)-REPRESENTATION SPACES OF KNOT GROUPS

Keywords: . 57M25, 57M05, 57M27, 20C99, 14M99 knot group, representation variety, character variety

The first part of this article is a general introduction to the the theory of representation spaces of discrete groups into SL n (C). Special attention is paid to knot groups. In Section 2 we discuss the difference between the tangent space at the representation variety, and the representation scheme. We give an example of Lubotzky and Magid of a non scheme reduced representation (see Example 2.18).

In the second part recent results about the representation and character varieties of knot groups into SL n (C) with n ≥ 3 are presented. This second part concerns mostly joint work with L.

Introduction

Since the foundational work of Thurston [START_REF] Thurston | The Geometry and Topology of Three-Manifolds[END_REF][START_REF] Thurston | Three-dimensional manifolds, Kleinian groups and hyperbolic geometry[END_REF] and Culler and Shalen [START_REF] Culler | Varieties of group representations and splittings of 3 -manifolds[END_REF], the varieties of representations and characters of three-manifold groups in SL 2 (C) have been intensively studied, as they reflect geometric and topological properties of the threemanifold. In particular they have been used to study knots k ⊂ S 3 , by analysing the SL 2 (C)-character variety of the fundamental group of the knot complement S 3k (these are called knot groups).

Much less is known about the character varieties of three-manifold groups in other Lie groups, notably for SL n (C) with n ≥ 3. There has been an increasing interest for those in the last years. For instance, inspired by the A-coordinates in higher Teichmüller theory of Fock and Goncharov [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF], some authors have used the so called Ptolemy coordinates for studying spaces of representations, based on subdivisions of ideal triangulations of the three-manifold. Among others, we mention the work of Dimofty, Gabella, Garoufalidis, Goerner, Goncharov, Thurston, and Zickert [START_REF] Dimofte | K-decompositions and 3d gauge theories[END_REF][START_REF] Dimofte | The quantum content of the gluing equations[END_REF][START_REF] Garoufalidis | The complex volume of SL(n,C)-representations of 3-manifolds[END_REF][START_REF] Garoufalidis | The symplectic properties of the PGL(n,C)-gluing equations[END_REF][START_REF] Garoufalidis | Gluing equations for PGL(n,C)representations of 3-manifolds[END_REF]. Geometric aspects of these representations, including volume and rigidity, have been addressed by Bucher, Burger, and Iozzi in [START_REF] Bucher | Rigidity of representations of hyperbolic lattices Γ < PSL(2, C) into PSL(n, C)[END_REF], and by Bergeron, Falbel, and Guilloux in [START_REF] Bergeron | Tetrahedra of flags, volume and homology of SL(3)[END_REF], who view these representations as holonomies of marked flag structures. We also recall the work Deraux and Deraux-Falbel in [START_REF] Deraux | On spherical CR uniformization of 3-manifolds[END_REF][START_REF] Deraux | A 1-parameter family of spherical CR uniformizations of the figure eight knot complement[END_REF][START_REF] Deraux | Complex hyperbolic geometry of the figure-eight knot[END_REF] to study CR and complex hyperbolic structures.

Background

Definition 2.1. Let k ⊂ S 3 be a knot. The knot group is Γ k ∶= π 1 (S 3 ∖ k). The knot exterior is the compact manifold C k = S 3 ∖ V (k) where V (k) is a tubular neighborhood of k .

In what follows we will make use of the following properties of knot groups:

• We have H 1 (C k ; Z) ≅ Z. A canonical surjection ϕ ∶ Γ k → Z is given by ϕ(γ) = lk(γ, k) where lk denotes the linking number in S 3 (see [12, 3.B]). • The knot exterior is aspherical: we have π n (C k ) = 0 for n > 1 i.e. C k is an Eilenberg-MacLane space K(Γ k , 1) (see [12, 3.F]). As a cosequence, the (co-)homology groups of Γ and C k are naturally identified, and for a given Γ kmodule M we have H * (C k ; M) ≅ H * (Γ k ; M), and H * (C k ; M) ≅ H * (Γ k ; M). It follows that every abelian representation factors through ϕ ∶ Γ k → Z. Here we call ρ abelian if its image is abelian. We obtain for each non-zero complex number η ∈ C * an abelian representation η ϕ ∶ Γ k → GL(1, C) = C * given by γ ↦ η ϕ(γ) .

2.1. Representation varieties. The general reference for representation and character varieties is Lubotzky's and Magid's book [START_REF] Lubotzky | Varieties of representations of finitely generated groups[END_REF]. Let Γ = ⟨γ 1 , . . . , γ m ⟩ be a finitely generated group.

Definition 2.2. A SL n (C)-representation is a homomorphism ρ ∶ Γ → SL n (C). The SL n (C)-representation variety is R n (Γ) = Hom(Γ, SL n (C)) ⊂ SL n (C) m ⊂ M n (C) m ≅ C n 2 m .
The representation variety R n (Γ) is an affine algebraic set. It is contained in SL n (C) m via the inclusion ρ ↦ ρ(γ 1 ), . . . , ρ(γ m ) , and it is the set of solutions of a system of polynomial equations in the matrix coefficients.

2.1.1. Affine algebraic sets. Let k be a field and let F λ = F λ (x 1 , . . . , x n ) ∈ k[x 1 , . . . , x n ], λ ∈ Λ, be a family of polynomials. The set of all common zeros of this family of polynomials is denoted by

V({F λ , λ ∈ Λ}) = {v ∈ k n F λ (v) = 0 for all λ ∈ Λ} .
It is clear that V({F λ , λ ∈ Λ}) = V(I) where I = ({F λ , λ ∈ Λ}) is the ideal generated by the family {F λ } λ∈Λ . Recall that, by Hilbert's basis theorem, each ideal I ⊂ k[x 1 , . . . , x n ] has a finite set of generators. An (affine) algebraic subset in k n is a subset V ⊂ k n consisting of all common zeros of finitely many polynomials with coefficients in k . It is easy to see that arbitrary intersections and finite unions of affine algebraic sets are affine algebraic. Now, given an algebraic subset V ⊂ k n a function f ∶ V → k is called regular if there exists F ∈ k[x 1 , . . . , x n ] such that f (v) = F (v) for all v ∈ V . All regular functions on V form the coordinate ring O(V ) (or k[V ]) of the variety V . Notice that O(V ) is a finitely generated k algebra since there is a surjection k[x 1 , . . . , x n ] → O(V ). The kernel of this surjection is called the ideal of V and is denoted by I(V ), hence

I(V ) = {F ∈ k[x 1 , . . . , x n ] F (v) = 0 for all v ∈ V } and O(V ) ≅ k[x 1 , . . . , x n ] I(V ) .
Notice that in general I V(I) ⊃ I but I V(I) ≠ I is possible. For example, if V ⊂ k is given by the equation x 2 = 0 then I = (x 2 ) ⊊ (x) = I({0}). If k is algebraically closed then Hilbert's Nullstellensatz implies that I V(I) is equal to the radical √

I of I I V(I) = √ I = {F ∈ k[x 1 , . . . , x n ] ∃m ∈ Z, m > 0, such that F m ∈ I } .
Now, two affine algebraic sets V ⊂ k m and W ⊂ k n are isomorphic if and only if there is an algebra isomorphism between O(V ) and O(W ) (see [START_REF] Shafarevich | Basic algebraic geometry[END_REF] for more details).

Example 2.3.

If V = {v = (v 1 , . . . , v n )} ⊂ k n is a point then I({v}) = (x 1 -v 1 , . . . , x n -v n )
and O(V ) ≅ k . In general, O(V ) is finite dimensional as a k -vector space if and only if V is finite, and in this case dim k (O(V )) = #V .

In the next example we investigate some very special representation varieties:

Example 2.4. A homomorphism of Z is determined by the image of the generator 1 ∈ Z and hence R n (Z) ≅ SL n (C). Similar, for a free group

F k of rank k we have R n (F k ) ≅ SL n (C) k .
For the cyclic group Z 2Z of two elements we have R 2 (Z 2Z) = {±I 2 } consists only of two points the identity matrix I 2 and -I 2 . Hence, R 2 (Z 2Z) is not irreducible as an algebraic variety. Even more concretely, a representation ρ ∶ Z 2Z → SL 2 (C) is determined by the image X of a generator. Now, considering X =

C[x 11 , x 12 , x 21 , x 22 ] I ≅ C[x] (x 2 -1) ≅ C[x] (x -1) ⊕ C[x] (x + 1) ≅ C ⊕ C
is the coordinate ring of the union of two points.

General facts. Given two representations ρ

1 ∶ Γ → GL m (C) and ρ 2 ∶ Γ → GL n (C) we define the direct sum ρ 1 ⊕ ρ 2 ∶ Γ → GL m+n (C) and the tensor product ρ 1 ⊗ ρ 2 ∶ Γ → GL mn (C) by ρ 1 ⊕ ρ 2 (γ) = ρ 1 (γ) 0 0 ρ 2 (γ) and ρ 1 ⊗ ρ 2 (γ) = ρ 1 (γ) ⊗ ρ 2 (γ), ∀ γ ∈ Γ,
respectively. Here, A⊗B denotes the Kronecker product of A ∈ GL m (C) and

B ∈ GL n (C). The dual representation ρ * ∶ Γ → GL(n) of ρ ∶ Γ → GL(n) is defined by ρ * (γ) = t ρ(γ) -1
where t A is the transpose of the matrix A. (See also Lemme 4.7.)

Definition 2.5. We call a representation ρ ∶ Γ → GL n (C) reducible if there exists a nontrivial subspace V ⊂ C n , 0 ≠ V ≠ C n , such that V is ρ(Γ)-stable. The representation ρ is called irreducible if it is not reducible. A semisimple representation is a direct sum of irreducible representations.
The group SL n (C) acts by conjugation on R n (Γ). More precisely, for

A ∈ SL n (C) and ρ ∈ R n (Γ) we define (A.ρ)(γ) = Aρ(γ)A -1 for all γ ∈ Γ. Moreover, we let O(ρ) = {A.ρ A ∈ SL n (C)} denote the orbit of ρ.
In what follows we will write ρ ∼ ρ ′ if there exists A ∈ SL n (C) such that ρ ′ = A.ρ, and we will call ρ and ρ ′ equivalent.

For ρ ∈ R n (Γ) we define its character χ ρ ∶ Γ → C by χ ρ (γ) = tr(ρ(γ)). We have ρ ∼ ρ ′ ⇒ χ ρ = χ ρ ′ . Lemma 2.6. Let ρ ∈ R n (Γ) be a representation. The orbit O(ρ) is closed if and only if ρ is semisimple. Moreover, let ρ, ρ ′ be semisimple. Then ρ ∼ ρ ′ if and only if χ ρ = χ ρ ′ .
Proof. See Theorems 1.27 and 1.28 in Lubotzky's and Magid's book [START_REF] Lubotzky | Varieties of representations of finitely generated groups[END_REF].

Example 2.7. We give two examples of a non-semisimple representations:

(1) Let ρ ∶ Z → SL 2 (C) be given by ρ(n) = 1 n 0 1 . The representation ρ is reducible but not semisimple. Notice that the orbit O(ρ) is not closed, O(ρ) contains the trivial representation: lim t→0 t 0 0 t -1

1 n 0 1 t -1 0 0 t = 1 0 0 1 . ( 2
ρ d = ζ ϕ ⊕ ζ -ϕ .
2.2. Character varieties. The algebraic quotient or GIT quotient for the action of SL n (C) on R n (Γ) is called the character variety. This quotient will be denoted by

X n (Γ) = R n (Γ) SL n (C).
The character variety is not necessary an irreducible affine algebraic set.

For an introduction to algebraic invariant theory see Dolgachev's book [START_REF] Dolgachev | Lectures on invariant theory[END_REF]. Roughly speaking, geometric invariant theory is concerned with an algebraic action of a group G on an algebraic variety V . Classical invariant theory addresses the situation when V is a vector space and G is either a finite group, or one of the classical Lie groups that acts linearly on V . The action of G on V induces an action of G on the coordinate algebra O(V ) of the variety V given by g ⋅ f (v) = f (g -1 v), for all g ∈ G, and v ∈ V .

The invariant functions of the G-action on V are

O(V ) G = {f ∈ O(V ) g ⋅ f = f for all g ∈ G} .
The invariant functions O(V ) G form a commutative algebra, and this algebra is interpreted as the algebra of functions on the GIT quotient V G. The main problem is to prove that the algebra O(V ) G is finitely generated. This is necessary if one wantes the quotient to be an affine algebraic variety. We are only interested in affine varieties V and in reductive groups G, and in this situation Nagata's theorem applies (see [START_REF] Dolgachev | Lectures on invariant theory[END_REF]Sec. 3.4]).

Reductive groups include all finite groups and all classical groups (see [START_REF] Dolgachev | Lectures on invariant theory[END_REF]Chap. 3]). Geometrically, the GIT quotient V G parametrizes the set of closed orbits (see [START_REF] Dolgachev | Lectures on invariant theory[END_REF]Corollary 6.1]). For a point v ∈ V the orbit G v will be denoted by O(v). If f 1 , . . . , f N generate the algebra O(V ) G then a model for the quotient is given by the image of the

map t ∶ V → V G ⊂ C N given by t(v) = (f 1 (v), . . . , f N (v)).
[x 1 , x 2 ] C * consist only of the constant functions i.e. C[x 1 , x 2 ] C * ≅ C. The GIT quotient C 2 C * ≅ { * } is just one point, and dim(C 2 C * ) < dim(C 2 ) -dim(C * ). (2) Let C * act on C 2 by λ.(z 1 , z 2 ) = (λ z 1 , 1 λ z 2 ). The topological quotient C 2 C *
is again non-Hausdorff topological space. More precisely, O(1, 0) and O(0, 1) are not closed and disjoint, but the closed orbit {(0, 0)} is contained in the closure of both orbits. Now, in order to determine

C[x 1 , x 2 ] C * we consider the space R n ⊂ C[x 1 , x 2 ]
of homogeneous polynomials of degree n. The set R n is a vector space of dimension n+1 with basis x i 1 x j 2 , i+j = n, and it is stable by the action of C * . Now,

λ ⋅ (x i 1 x j 2 ) = λ i-j x i 1 x j 2
, and the algebra of invariant functions is generated by Work of C. Procesi [START_REF] Procesi | The invariant theory of n × n matrices[END_REF] implies that there exists a finite number of group elements

x 1 x 2 . Hence C[x 1 , x 2 ] C * = C[x 1 x 2 ] ≅ C[x]. It follows that GIT quotient C 2 C * ≅ C. The quotient map t ∶ C 2 → C is given by the invariant functions t(z 1 , z 2 ) = z 1 z 2 . The whole "non-hausdorff" part O(1, 0) ∪ {(0, 0)} ∪ O(0, 1) ≅ C × {0} ∪ {0} × C is identified,
{γ i 1 ≤ i ≤ M} ⊂ Γ such that the image of t ∶ R n (Γ) → C M given by t(ρ) = χ ρ (γ 1 ), . . . , χ ρ (γ M )
can be identified with the affine algebraic set X n (Γ) ≅ t(R n (Γ)), see also [47, p. 27]. This justifies the name character variety.

Example 2.9.

(1) Let F 2 be the free group on the two generators x and y . Then it is possible to show that X 2 (F 2 ) ≅ C 3 and t ∶ R 2 (F 2 ) ≅ → C 3 given by t(ρ) = χ ρ (x), χ ρ (y), χ ρ (xy) . See Goldman's article [START_REF]Handbook of Teichmüller theory[END_REF]Chap. 15] and the article of Gonzàlez-Acuña and Montesinos-Amilibia [START_REF] González-Acuña | On the character variety of group representations in SL(2, C) and PSL(2, C)[END_REF] for more details.

(2) We obtain X 3 (Z) ≅ C 2 More precisely, R 3 (Z) ≅ SL 3 (C) and t ∶ R 3 (Z) ≅ → C 2 is given by t(A) = (tr(A), tr(A -1 )).

(3) Explicit coordinates for X 3 (F 2 ) are also known: X 3 (F 2 ) is isomorphic to a degree 6 affine hyper-surface in C 9 (see Lawton [START_REF] Lawton | Generators, relations and symmetries in pairs of 3×3 unimodular matrices[END_REF]). (4) If Γ is a finite group then X n (Γ) is finite for all n. This follows since Γ has up to equivalence only finitely many irreducible representations, and every representation of a finite group is semisimple (see [START_REF] Serre | Linear representations of finite groups[END_REF]).

2.3. Tangent spaces and group cohomology. The general reference for group cohomology is Brown's book [START_REF] Brown | Cohomology of groups[END_REF]. In order to shorten notation we will sometimes write SL(n) and sl(n) instead of SL n (C), and sl n (C).

The following construction was presented by A. Weil [START_REF] Weil | Remarks on the cohomology of groups[END_REF]. For ρ ∈ R n (Γ) the Lie algebra sl(n) turns into a Γ-module via Ad ○ρ, i.e. for X ∈ sl(n) and γ ∈ Γ we have γ ⋅ X = Ad ρ(γ) (X) = ρ(γ)Xρ(γ) -1 . In what follows this Γ-module will be denoted by sl(n) Ad ρ . We obtain an inclusion T Zar R n (Γ) ↪ Z 1 (Γ, sl(n) Ad ρ ): for a smooth family of representations ρ t with ρ 0 = ρ we obtain a map u ∶ Γ → sl(n) given by ( 1)

u(γ) = dρ t (γ) dt t=0 ρ(γ) -1 .
The map u verifies: u(γ

1 γ 2 ) = u(γ 1 ) + γ 1 ⋅ u(γ 2 ) i.e. u ∈ Z 1 (Γ, sl(n) Ad ρ ) is a cocycle or derivation in group cohomology. If ρ t = Ad At ○ρ is contained in O(ρ) where A t , A 0 = I n ,
is a path of matrices, then the corresponding cocycle is a coboundary i.e. there exists

X ∈ sl(n) such that u(γ) = (1 -γ) ⋅ X = X -Ad ρ(γ) (X). Attention! The inclusion T Zar R n (Γ) ↪ Z 1 (Γ, sl(n) Ad ρ ) might
dim ρ R n (Γ) ≤ dim T Zar ρ R n (Γ) ≤ dim Z 1 (Γ, sl(n) Ad ρ ) where dim ρ R n (Γ) denotes the local dimension of R n (Γ) at ρ i.e. the maximum of the dimensions of the irreducible components of R n (Γ) containing ρ.
In what follows, will call ρ a regular or scheme smooth point of R n (Γ) if the equality dim ρ R n (Γ) = dim Z 1 (Γ, sl(n) Ad ρ ) holds. In this case every derivation is integrable, and we obtain the following:

Lemma 2.11 (see [38, Lemma 2.6]). Let ρ ∈ R n (Γ) be a representation. If ρ is regular, then ρ is a smooth point of the representation variety R n (Γ), and ρ is contained in a unique component of R n (Γ) of dimension dim Z 1 (Γ; sl(n) Ad ρ ).
Example 2.12. Central representations are smooth points of R n (Γ k ). Let Γ k be a knot group and

ρ 0 ∈ R n (Γ k ) be a central representation i.e. ρ 0 (γ) = ζ ϕ(γ) Id n where ζ ∈ C * , ζ n = 1. Then sl(n) is a trivial Γ k -module and Z 1 (Γ, sl(n)) = H 1 (Γ, sl(n)) = H 1 (Γ, Z) ⊗ sl(n) has dimension n 2 -1.
On the other hand the surjection ϕ

∶ Γ k → Z induces an injection ϕ * ∶ R n (Z) ↪ R n (Γ k ) where R n (Z) ≅ SL n (C). Therefore, n 2 -1 ≤ dim ρ 0 R n (Γ k ) ≤ dim Z 1 (Γ, sl(n)) = n 2 -1, and ρ 0 ∈ R n (Γ k ) is a regular point which is contained in an unique (n 2 -1)-dimensional component of R n (Γ k ) (the component consist of abelian representations).
We give an example where the first inequality of (2) is strict, and the second is an equality. In this case the representation ρ is a singular point of the representation variety, but we will see that in our example

χ ρ ∈ X 2 (Γ) is a smooth point. Example 2.13. Let Γ = D(3, 3, 3) = ⟨a, b, c a 3 , b 3 , c 3 , abc⟩ ≅ ⟨a, b a 3 , b 3 , (ab) 3 ⟩ be the van Dyck group. We consider the representation ρ 0 ∶ Γ → SL(2) given by ρ 0 (a) = ρ 0 (b) = A = ω 0 0 ω
where ω is a primitive third root of unity.

Let F (a, b) denote the free group of rank two and consider the canonical surjection

κ ∶ F (a, b) → Γ. We consider sl(2) as a F (a, b)-module via Ad ○ρ 0 ○ κ. Now, for every X, Y ∈ sl(2) we obtain a cocycle z ∶ F (a, b) → sl(2) such that z(a) = X and z(b) = Y . By using Fox-calculus [12, Chapter 9], we obtain for w ∈ F (a, b) z(w) = ∂w ∂a ⋅ X + ∂w ∂b ⋅ Y .
This cocycle factors through κ if and only if z(a

3 ) = z(b 3 ) = z((ab) 3 ) = 0. Writing X = x 11 x 12 x 21 -x 11 and Y = y 11 y 12 y 21 -y 11 the equation z(a 3 ) = 0 gives 0 = (1 + a + a 2 ) ⋅ X = X + Ad A (X) + Ad 2 A (X)
and hence x 11 = 0. Similar z(b 3 ) = 0 gives y 11 = 0. The equation z((ab) 3 ) = 0 gives no further restrictions. Hence the space of cocycles Z 1 (Γ, sl(2) Ad ρ 0 ) is 4-dimensional. The space B 1 (Γ, sl(2) Ad ρ 0 ) is 2-dimensional and generated by b 1 and b 2 which are given by b 1 ∶ a, b ↦ 0 1 0 0 and b 2 ∶ a, b ↦ 0 0 1 0 . Two non-principal derivations are given by z 1 (a) = 0 0 0 0 , z 1 (b) = 0 1 0 0 , and z 2 (a) = 0 0 0 0 , z 2 (b) = 0 0 1 0 . These two derivations are integrable, more precisely the two families

ρ i (t) ∶ Γ → SL(2) are given by ρ i (t)(γ) = (I 2 + tz i (γ))ρ(γ), or explicitly by ρ 1 (t) ∶ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a ↦ ω 0 0 ω , b ↦ 1 t 0 1 ω 0 0 ω = ω tω 0 ω , ρ 2 (t) ∶ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a ↦ ω 0 0 ω , b ↦ 1 0 t 1 ω 0 0 ω = ω 0 tω ω . It follows that dim T Zar ρ 0 R n (Γ) = dim Z 1 (Γ, sl(n) Ad ρ 0 ). Now notice that if A ∈ SL(2) verifies A 3 = I 2 and A ≠ I 2 , then A is conjugate to ω 0 0 ω
where ω is a third root of unity, ω 2 + ω + 1 = 0. Hence we have for all A ∈ SL(2):

A 3 = I 2 ⇐⇒ A = I 2 or tr A = -1 . Lemma 2.14. All representation ρ ∶ D(3, 3, 3) → SL(2) are reducible. More precisely, if ρ(a) = I 2 or ρ(b) = I 2 or ρ(ab)) = I 2 is trivial then ρ is conjugate to a diagonal representation. If tr ρ(a) = tr ρ(b) = tr ρ(ab) = -1 then χ ρ = χ ρ 0 ,

Notice that a cocycle

z = c 1 z 1 + c 2 z 2 , c 1 , c 2 ∈ C, is integrable if and only if c 1 c 2 = 0 i.e.
only multiples of z 1 and multiples of z 2 are integrable.

It follows from Lemma 2.14 that the families ρ 1 (t) and ρ 2 (t) together form a slice étale S 0 through the representation ρ 0 (see [START_REF] Ben | Variété des caractères et slice étale de l'espace des représentations d'un groupe[END_REF] for more details). The slice S 0 is isomorphic to the union of the two coordinate axes in C 2 ,

S 0 ≅ C × {0} ∪ {0} × C ⊂ C 2 , ρ 1 (s) ↦ (s, 0) and ρ 2 (t) ↦ (0, t) . It follows form [3, Prop. 2.8] that H 1 (Γ, sl(2) Ad ρ 0 ) is isomorphic to the tangent space T Zar ρ 0 S 0 , and that T Zar χρ 0 X 2 (Γ) is isomorphic to T Zar χρ 0 S 0 Stab(ρ 0 ) . Now, Stab(ρ 0 ) ≅ C * consits of diagonal matrices, and λ ∈ C * acts as follows λ ⋅ ρ 1 (s) = ρ 1 (λ 2 s) and λ ⋅ ρ 2 (t) = ρ 1 (λ -2 t) (see Example 2.8.2). It follows that S 0 Stab(ρ 0 ) ≅ {0} is just a point, and that T Zar χρ 0 X 2 (Γ) vanishes. Notice that all representations ρ i (t) are equivalent to ρ i (1). On the other hand ρ 1 (1) and ρ 2 (1) are not equivalent. Again, O(ρ i (t)) is not closed, but O(ρ 0 ) = O(ρ 1 (t)) ∩ O(ρ 2 (t)).
All representations ρ 0 , ρ 1 (t), and ρ 2 (t) have the same character.

Notice also that H 1 (Γ, sl(2) Ad ρ 0 ) is isomorphic to the tangent space of the slice étale, and that H 1 (Γ, sl(2) Ad ρ 0 ) Stab(ρ) ≅ C. This shows that in [START_REF] Sikora | Character varieties[END_REF]Theorem 53] the hypothesis scheme smooth can not be omitted.

Remark 2.15. Example 2.13 can be generalized to a representation of the fundamental group of the closed 3-dimensional Seifert fibred manifold M which is an oriented Seifertbundle over the orbifold S 2 (3, 3, 3). The fundamental group π 1 (M) is a central extension of D(3, 3, 3) with presentation

π 1 M = ⟨a, b, c, z a 3 = b 3 = c 3 = abc = z, [a, z], [b, z], [c, z]⟩ ≅ ⟨a, b, c a 3 = b 3 = c 3 = abc⟩ . It is easy to see that a diagonal representation ρ t ∶ π 1 M → SL(2) given by ρ t ∶ a, b, c ↦ t 0 0 t -1
is a singular point of the representation variety if and only if 1 + t 2 + t 4 = 0.

2.4. The scheme R n (Γ). Let R be a commutative and unitary ring. A radical ideal is an ideal I ⊂ R such that I = √ I = {r ∈ R r k ∈ I for some positive integer k}. Notice that I ⊂ R is radical if and only if the quotient ring R I is reduced i.e. R I has no non-zero nilpotent elements. By virtue of Hilbert's Nullstellensatz there is a bijection between algebraic subsets in C N and radical ideals of C[x 1 , . . . , x N ] (see [START_REF] Shafarevich | Basic algebraic geometry[END_REF][START_REF] Shafarevich | Basic algebraic geometry[END_REF]). Recall that over C a vanishing ideal I(V ) is always radical (see Section 2.1.1). Now, the ideal generated by the algebraic equations defining the representation variety may be non-radical (see Example 2.18). Therefore, one considers the underlying affine scheme R n (Γ) ∶= R(Γ, SL n (C)) with a possible non-reduced coordinate ring. Weil's construction gives an isomorphism

T Zar ρ R(Γ, SL n (C)) ≅ Z 1 (Γ; sl(n) Ad ρ ) . Each d ∈ Z 1 (Γ; sl n (C)) gives the infinitesimal deformation γ ↦ (1 + ε d(γ))ρ(γ), ∀γ ∈ Γ, which satisfies the defining equations for R n (Γ) up to terms in the ideal (ε 2 ) of C[ε],
i.e. a Zariski tangent vector to R n (Γ) (see [START_REF] Lubotzky | Varieties of representations of finitely generated groups[END_REF]Prop. 2.2] and [START_REF] Ben | Variété des caractères et slice étale de l'espace des représentations d'un groupe[END_REF]).

2.4.1.

The difference between a scheme and a variety -heuristics and examples. We start with some heuristics. For more details see Shafarevich's book [56, 5.1]. Here we are only interested in affine schemes which correspond to rings of the form

R = C[x 1 , . . . , x N ] I , for an ideal I ⊂ C[x 1 , . . . , x N ]. It may happen that R = C[x 1 , . . . , x N ] I is not reduced. In this case can consider the reduced ring R red = C[x 1 , . . . , x N ] √
I which is the coordinate ring of the variety V = V(I) ⊂ C N . The underlying space of the scheme corresponding to R is Spec R the set of prime ideals of R. Since the kernel of π ∶ R ↠ R red is the nilradical, it follows that π * ∶ Spec R red → Spec R is a homeomorphism of topological spaces (the two spectra are equipped with the Zariski-topology, see [56, 5.1.2]). Now, the points of V correspond to the maximal ideals of R red which in turn correspond to the maximal ideals in R.

On the other hand, the regular functions on Spec R red and Spec R are different: a nonzero nilpotent element f ∈ R gives a non-zero function on Spec R, but π(f ) is zero in R red . This means that there are non-zero functions on Spec R which take the value zero on every point of V . These functions may affect the calculation of the tangent space. One can visualize the scheme corresponding to R as containing some extra normal material which is actually not tangent to a dimension present in the variety.

f ∈ (T ) ⇔ f (0) = 0.
On the other hand, the restriction of f onto Spec R vanishes if and only if f ∈ (T 2 ) ⇔ f (0) = 0 and f ′ (0) = 0. Hence there are non-zero regular functions on Spec R which are zero on every point of Spec R. This affect the calculation of the Zariski tangent space:

T Zar 0 Spec R red = 0, but T Zar 0 Spec R ≅ C . Notice that Spec C[x] (x 2 ) appears naturally if we intersect the parabola V(y -x 2 )
with the coordinate axis V(y) in C 2 . See [START_REF] Eisenbud | The geometry of schemes[END_REF]II.3] for a detailed discussion.

Remark 2.17. There is also an associated character scheme

X (Γ, SL n (C)) = R(Γ, SL n (C)) SL n (C) .
In general the relation between the cohomology group H 1 (Γ, sl(n) Ad ρ ) and the tangent space T Zar χρ X (Γ, SL n (C)) is more complicate. However, if ρ is an irreducible regular representation then we have for the character variety [START_REF] Lubotzky | Varieties of representations of finitely generated groups[END_REF]Lemma 2.18], and [58, Section 13] for a generalisation to completely reducible regular representations.)

T Zar χρ X n (Γ) ≅ H 1 (Γ, sl(n) Ad ρ ) ≅ T Zar χρ X (Γ, SL n (C)) . (See
The next example is a representation

ρ ∶ Γ → SL(2) such that dim ρ R 2 (Γ) = dim T Zar ρ R 2 (Γ) and dim T Zar ρ R 2 (Γ) < dim Z 1 (Γ, sl(2) Ad ρ ).
Hence the coordinate ring of the associated scheme has nilpotent elements.

Example 2.18. Following Lubotzky and Magid [47, pp. 40-43] we give an example of a finitely presented group Γ and a representation ρ ∶ Γ → SL(2) with non reduced coordinate ring.

For motivation we start with the dihedral group D 3 = ⟨a, s a 3 , s 2 , sas -1 = a -1 ⟩, and a representation r ∶ D 3 → Iso(C). Recall that a transformation σ ∈ Iso(C) is of the form

σ ∶ z ↦ ζ z + α or σ ∶ z ↦ ζ z + α where α ∈ C, and ζ ∈ C * , ζ = 1, is a complex number of norm 1. A homomorphism r ∶ D 3 → Iso(C) is given by r(a) ∶ z ↦ ω z and r(s) ∶ z ↦ z
where ω is a third root of unity ω 2 +ω +1 = 0. The image r(D 3 ) is contained in Iso(C) 0 ∶= {σ ∈ Iso(C) σ(0) = 0}. Notice also that C ⋊ Iso(C) 0 = Iso(C) where C is identified with the subgroup of translations. Let us consider the two translations τ 1 , τ 2 ∶ C → C given by

τ 1 ∶ z ↦ z + (1 + η) and τ 2 ∶ z ↦ z + (1 + η) = z + (2 -η)
where η is a primitive 6-th root of unity, η 2 = ω . An elementary calculation shows that

r(s) τ 1 r(s) -1 = τ 2 , r(a) τ 1 r(a) -1 = τ -1 2 r(s) τ 2 r(s) -1 = τ 1 , r(a) τ 2 r(a) -1 = τ 1 τ -1 2 .
Finally we define the group Γ = (Z × Z) ⋊ Dic 3 where Dic 3 = ⟨a, s a 6 , s 2 = a 3 , sas -1 = a -1 ⟩ is the binary dihedral group of order 12. The group Γ has the following presentation: ⟨a, s, t 1 , t 2 a 6 , s 2 = a 3 , sas 2) is given by ρ(a) = η 0 0 η , ρ(s) = 0 1 -1 0 , and ρ(t 1 ) = ρ(t 2 ) = I 2 . An elementary but tedious calculation shows that z ∶ Γ → sl(2) given by z(a) = z(s) = 0, and z(t 1 ) = 0 1+η

-1 = a -1 , [t 1 , t 2 ], st 1 s -1 = t 2 , st 2 s -1 = t 1 , at 1 a -1 = t -1 2 , at 2 a -1 = t 1 t -1 2 ⟩ . A homomorphism ρ ∶ Γ → SL(
-(1+η) 0 , z(t 2 ) = 0 1+η -(1+η) 0 is a derivation. The derivation z is non-principal since for each principal derivation b ∶ Γ → sl(2) we have b(t 1 ) = b(t 2 ) = 0 since ρ(t 1 ) = ρ(t 2 ) = I 2 are trivial. Hence, H 1 (Γ, sl(2) Ad ρ ) is non-trivial. More precisely, H 1 (Γ, sl(2) Ad ρ ) ≅ C is generated by the cohomology class of z .
On the other hand, it can be shown directly that each representation of Γ into SL(2) factors through the finite group Γ ′ = Γ ⟪t 1 t 2 ⟫ ≅ (Z 3Z)⋊Dic 3 . More generally, this follows also from [START_REF] Lubotzky | Varieties of representations of finitely generated groups[END_REF]Example 2.10]. Therefore, X 2 (Γ) is finite and

χ ρ ∈ X 2 (Γ) is an isolated point. It follows that the coordinate ring O(R 2 (Γ)) is non-reduced.
More concretely, we can use SageMath [START_REF] Stein | Sage Mathematics Software (Version 6.9)[END_REF] to compute the ideal I generated by algebraic equations of the SL(2)-representation variety R 2 (Γ) ⊂ C 16 [START_REF] Heusener | SL r (C)-representation spaces of knot groups: Sage worksheet[END_REF]. It turns out that √ I is generated by I and the equations given by the relation

t 1 t 2 = 1. Therefore, we obtain O(R 2 (Γ)) red ≅ O(R 2 (Γ ′ )) ≅ O(R 2 (Γ ′ )).
If we impose the corresponding relations i.e. if we consider the representation ρ ′ ∶ Γ → SL(2) given by ρ ′ (a) = η 0 0 η , ρ ′ (s) = 0 1 -1 0 , ρ(t 1 ) = ω 0 0 ω and ρ(t 2 ) = ω 0 0 ω then we obtain H 1 (Γ, sl(2) Ad ρ ′ ) = 0.

Remark 2.19. M. Kapovich and J. Millson proved in [START_REF] Kapovich | On representation varieties of 3 -manifold groups[END_REF] that there are essentially no restrictions on the local geometry of representation schemes of 3-manifold groups to SL 2 (C).

Deformations of representations

One way to prove that a certain representation ρ ∈ R n (Γ) is a smooth point of the representation variety is to show that every cocycle u ∈ Z 1 (Γ; sl(n) Ad ρ ) is integrable (see Lemma 2.11). In order to do this, we use the classical approach, i.e. we first solve the corresponding formal problem, and then apply a theorem of Artin [START_REF] Artin | On the solutions of analytic equations[END_REF].

The formal deformations of a representation ρ ∶ Γ → SL n (C) are in general determined by an infinite sequence of obstructions (see [START_REF] William | The symplectic nature of fundamental groups of surfaces[END_REF][START_REF] Ben | Espace des représentations du groupe d'un noeud classique dans un groupe de Lie[END_REF][START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into SL 2 (C)[END_REF]). In what follows we let

C 1 (Γ; sl(n) Ad ρ ) ∶= {c ∶ Γ → sl(n) Ad ρ } denote the 1-cochains of Γ with coefficients in sl(n) (see [10, p.59]). Let ρ ∶ Γ → SL(n) be a representation. A formal deformation of ρ is a homomorphism ρ ∞ ∶ Γ → SL n (C t ) ρ ∞ (γ) = exp ∞ i=1 t i u i (γ) ρ(γ) , u i ∈ C 1 (Γ; sl(n)) such that ev 0 ○ ρ ∞ = ρ. Here ev 0 ∶ SL n (C t ) → SL n (C)
is the evaluation homomorphism at t = 0, and C t denotes the ring of formal power series.

We will say that ρ ∞ is a formal deformation up to order k of ρ if ρ ∞ is a homomorphism modulo t k+1 .

An easy calculation gives that ρ ∞ is a homomorphism up to first order if and only if u 1 ∈ Z 1 (Γ; sl(n) Ad ρ ) is a cocycle. We call a cocycle u 1 ∈ Z 1 (Γ; sl(n) Ad ρ ) formally integrable if there is a formal deformation of ρ with leading term u 1 .

Lemma 3.1. Let u 1 , . . . , u k ∈ C 1 (Γ; sl(n)) such that ρ k (γ) = exp k i=1 t i u i (γ) ρ(γ)
is a homomorphism into SL n C t (t k+1 ) . Then there exists an obstruction class

ζ k+1 ∶= ζ (u 1 ,...,u k ) k+1 ∈ H 2 (Γ, sl(n) Ad ρ ) with the following properties: (i) There is a cochain u k+1 ∶ Γ → sl(n) such that ρ k+1 (γ) = exp k+1 i=1 t i u i (γ) ρ(γ) is a homomorphism modulo t k+2 if and only if ζ k+1 = 0. (ii) The obstruction ζ k+1 is natural, i.e. if f ∶ Γ 1 → Γ is a homomorphism then f * ρ k ∶= ρ k ○ f is also a homomorphism modulo t k+1 and f * (ζ (u 1 ,...,u k ) k+1 ) = ζ (f * u 1 ,...,f * u k ) k+1 ∈ H 2 (Γ 1 ; sl(n) Ad f * ρ ).
Proof. The proof is completely analogous to the proof of Proposition 3.1 in [START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into SL 2 (C)[END_REF]. We replace SL(2) and sl(2) by SL(n) and sl(n) respectively.

The following result streamlines the arguments given in [START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into PSL 2 (C)[END_REF] and [START_REF] Ben Abdelghani | Deformations of metabelian representations of knot groups into SL(3, C)[END_REF]. It is a slight generalization of Proposition 3.3 in [START_REF] Heusener | Deformations of reducible representations of knot groups into SL(n, C)[END_REF]. Proposition 3.2. Let M be a connected, compact, orientable 3-manifold with toroidal boundary ∂M = T 1 ∪ ⋯ ∪ T k , and let ρ ∶ π 1 M → SL(n) be a representation.

If

dim H 1 (π 1 M; sl(n) Ad ρ ) = k(n -1) then ρ is a smooth point of the SL(n)- representation variety R n (π 1 M). Moreover, ρ is contained in a unique component of dimension n 2 -1 + k(n -1) -dim H 0 (π 1 M; sl(n) Ad ρ ).
Proof. First we will show that the map

ι * ∶ H 2 (π 1 M; sl(n) Ad ρ ) → H 2 (π 1 ∂M; sl(n) Ad ρ ) induced by the inclusion ι ∶ ∂M ↪ M is injective.
Recall that for any CW-complex X with π 1 (X) ≅ π 1 (M) and for any π 1 M -module A there are natural morphisms H i (π 1 M; A) → H i (X; A) which are isomorphisms for i = 0, 1 and an injection for i = 2 (see [START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into PSL 2 (C)[END_REF]Lemma 3.3]). Note also that T j ≅ S 1 × S 1 is aspherical and hence H * (π 1 T j ; A) → H * (T j ; A) is an isomorphism.

For every representation ̺ ∈ R n (Z ⊕ Z) we have

(3) dim H 0 (Z ⊕ Z; sl(n) Ad ̺ ) = 1 2 dim H 1 (Z ⊕ Z; sl(n) Ad ̺ ) ≥ n -1 ,
and ̺ ∈ R n (Z ⊕ Z) is regular if and only if equality holds in [START_REF] Ben | Variété des caractères et slice étale de l'espace des représentations d'un groupe[END_REF]. A prove of this statement can be found in the of Proposition 3.3 in [START_REF] Heusener | Deformations of reducible representations of knot groups into SL(n, C)[END_REF].

Now, the exact cohomology sequence of the pair (M, ∂M) gives

→ H 1 (M, ∂M; sl(n) Ad ρ ) → H 1 (M; sl(n) Ad ρ ) α → H 1 (∂M; sl(n) Ad ρ ) β → H 2 (M, ∂M; sl(n) Ad ρ ) → H 2 (M; sl(n) Ad ρ ) ι * → H 2 (∂M; sl(n) Ad ρ ) → H 3 (M, ∂M; sl(n) Ad ρ ) → 0 .
Poincaré-Lefschetz duality implies that α and β are dual to each other. Therefore, we have rk α = rk β , and from the exactness it follows that 2 rk α = dim H 1 (∂M; sl(n) Ad ρ ). Moreover, we have

H 1 (∂M; sl(n) Ad ρ ) ≅ ⊕ k j=1 H 1 (T j ; sl(n) Ad ι * j ρ ) where ι j ∶ T j → M de- notes the inclusion. Equation (3) implies that dim H 0 (T j ; sl(n) Ad ̺ ) ≥ n -1 for all ̺ ∈ R n (π 1 T j ). Hence k(n -1) = dim H 1 (M; sl(n) Ad ρ ) ≥ rk(α) = 1 2 dim H 1 (∂M; sl(n) Ad ρ ) (4) = k j=1 1 2 dim H 1 (T j ; sl(n) Ad ι * j ρ ) = k j=1 dim H 0 (T j ; sl(n) Ad ι * j ρ ) ≥ k(n -1) .
Therefore, equality holds everywhere in (4). This implies that α is injective, hence β is surjective, and

ι * ∶ H 2 (M; sl(n) Ad ρ ) → H 2 (∂M; sl(n) Ad ρ ) ≅ k ⊕ j=1 H 2 (T j ; sl(n) Ad ι * j ρ )
is injective. Moreover, Equation ( 4) implies that dim H 0 (T j ; sl(n) Ad ι * j ρ ) = n -1 holds for all j = 1, . . . , k , and consequently ι * j ρ = ρ ○ ι j# ∈ R n (π 1 T j ) is regular. We obtain the following commutative diagram:

H 2 (M; sl(n) Ad ρ ) ι * → H 2 (∂M; sl(n) Ad ρ ) ≅ H 2 (π 1 M; sl(n) Ad ρ ) ⊕ k j=1 ι * j → ⊕ k j=1 H 2 (π 1 T j ; sl(n) Ad ι * j ρ
) . In order to prove that ρ ∈ R n (π 1 M) is regular, we first show that all cocycles in Z 1 (π 1 M, sl(n) Ad ρ ) are formally integrable. We will prove that all obstructions vanish, by using the fact that the obstructions vanish on the boundary. Let u 1 , . . . , u k ∶ π 1 M → sl(n) be given such that

ρ k (γ) = exp k i=1 t i u i (γ) ρ(γ)
is a homomorphism modulo t k+1 . Then the restriction ι * j ρ k ∶ π 1 T j → SL n (C t ) is also a formal deformation of order k . Since ι * j ρ is a regular point of the representation variety R n (π 1 T j ), the formal implicit function theorem gives that ι * j ρ k extends to a formal deformation of order k + 1 (see [START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into SL 2 (C)[END_REF]Lemma 3.7]). Therefore, we have that

0 = ζ (ι * j u 1 ,...,ι * j u k ) k+1 = ι * j ζ (u 1 ,...,u k ) k+1
Now, ⊕ k j=1 ι * j is injective and the obstruction ζ

(u 1 ,...,u k ) k+1
vanishes. Hence all cocycles in Z 1 (Γ, sl(n) Ad ρ ) are formally integrable. By applying Artin's theorem [START_REF] Artin | On the solutions of analytic equations[END_REF] we obtain from a formal deformation of ρ a convergent deformation (see [START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into SL 2 (C)[END_REF]

, Lemma 3.3] or [2, § 4.2]). Thus ρ ∈ R n (π 1 M) is a regular point, and dim H 1 (π 1 M; sl(n) Ad ̺ ) = k(n -1). The exactness of 0 → H 0 (π 1 M; sl(n) Ad ̺ ) → sl(n) → B 1 (π 1 M; sl(n) Ad ̺ ) → 0,
and the regularity of ρ ∈ R n (π 1 M) imply:

dim ρ R n (π 1 M) = dim Z 1 (π 1 M; sl(n) Ad ̺ ) = dim H 1 (π 1 M; sl(n) Ad ̺ ) + dim B 1 (π 1 M; sl(n) Ad ̺ ) = k(n -1) + n 2 -1 -dim H 0 (π 1 M; sl(n) Ad ̺ ) .
Finally, the proposition follows from Lemma 2.11. Definition 3.3. Let M be a connected, compact, orientable 3-manifold with toroidal boundary

∂M = T 1 ∪ ⋯ ∪ T k . We call a representation ρ ∶ π 1 M → SL n (C) infinitesimally regular if dim H 1 (π 1 M; sl(n) Ad ρ ) = k(n -1).
Remark 3.4. It follows from Proposition 3.2 that infinitesimally regular representations are regular points on the representation variety.

Example 3.5. Let Γ k be a knot group and let D = diag(λ 1 , . . . , λ n ) ∈ SL(n) be a diagonal matrix. We define the diagonal representation ρ D by ρ D (γ) = D ϕ(γ) . Now, ρ D is the direct sum of the one-dimensional representations λ ϕ i , and the Γ k -module sl(n) Ad ρ D decomposes as:

sl(n) Ad ρ A = ⊕ i≠j C λ i λ j ⊕ C n-1 .
Now, for all α ∈ C * we have H

Existence of irreducible representations of knot groups

Let k ⊂ S 3 be a knot, and let Γ k be the knot group. Given representations of Γ k into SL(2) there are several constructions which give higher dimensional representations. The most obvious is probably the direct sum of two representatios.

Deformations of the direct sum of two representations. Starting from two representations

α ∶ Γ k → SL a (C) and β ∶ Γ k → SL b (C) such that a + b = n, we obtain a family of representations ρ λ ∈ R n (Γ k ), λ ∈ C * , by ρ λ = (λ bϕ ⊗ α) ⊕ (λ -aϕ ⊗ β) ∈ R n (Γ k ) i.e. for all γ ∈ Γ k (5) ρ λ (γ) = λ bϕ(γ) α(γ) 0 0 λ -aϕ(γ) β(γ) .
Recall that λ ϕ ∶ Γ k → C * is given by γ ↦ λ ϕ(γ) . Throughout this section we will assume that α and β are both irreducible and infinitesimal regular.

The natural question which arises is if ρ λ can be deformed to irreducible representations, and if this would be possible what could we say about the local structure of X n (Γ k ) at χ ρ λ ?

The easiest case. A very special case is

α = β ∶ Γ k → SL 1 (C) = {1} are trivial. Then ρ λ = λ ϕ ⊕ λ -ϕ ∈ R 2 (Γ k ) i.e. for all γ ∈ Γ k (6) ρ λ (γ) = λ ϕ(γ) 0 0 λ -ϕ(γ) .
Example 4.1. Let us consider the trefoil knot k = 3 1 . The knot group of the trefoil knot is given by Γ 3 1 = ⟨S, T ST S = T ST ⟩ = ⟨x, y x 2 = y 3 ⟩ where x = ST S and y = T S . A meridian is m = S = xy -1 . For every irreducible representation ρ ∈ R 2 (Γ 3 1 ) there exists a unique s ∈ C such that ρ ∼ α s , where

α s (x) = i 0 s -i and α s (y) = η η -η 0 η ,
and η 2η + 1 = 0 is a primitive 6-th root of unity. Moreover, α s is irreducible if and only if s ≠ 0, 2i (see [START_REF] Heusener | Representations of knot groups into SL n (C) and twisted Alexander polynomials[END_REF]Lemma 9.1] for a proof). Now, if s = 0 then the one parameter group P (t) = diag(t, t -1 ), t ∈ C * , verifies that lim t→0 P (t).α 0 exists, and is the diagonal representation ρ ζ where ζ = iη is a primitive 12-th root of unity. If s = 2i we can take P (t) = t -1 -t -1 t 0

, t ∈ C * , and we obtain lim t→0 P (t).α 2i = ρ -ζ . Therefore, the two diagonal representations ρ ±ζ are limit of irreducible representations. Notice also that (±ζ) 2 = η is a primitive 6-th roof of unity and that ∆ 3

1 (η) = 0.
This examples shows a general phenomena which goes back to work of E. Klassen [START_REF] Paul Klassen | Representations of knot groups in SU(2)[END_REF]. Proof. In general the function R n (Γ) → Z given by ρ ↦ dim Z 1 (Γ, sl(n) Ad ρ ) is upper-semi continuous which means that for every k ∈ Z the set {ρ ∈ R n (Γ) dim Z 1 (Γ, sl(n) Ad ρ ) ≥ k} is closed. Notice that Z 1 (Γ, sl(n) Ad ρ ) is the kernel of a linear map which depends algebraically on ρ.

Moreover, if the representation ρ λ ∈ R 2 (Γ k ) can be deformed into irreducible representations then dim Z 1 (Γ k , sl(2) Ad ρ λ ) ≥ 4 (see [START_REF] Heusener | Representations of knot groups into SL n (C) and twisted Alexander polynomials[END_REF]Lemma 5.1]). The Γ k -module sl(2

) Ad ρ λ ≅ C ⊕ C λ 2 ⊕ C λ -2 decomposes into one-dimensional modules (see Example 3.5). Now, H 1 (Γ k , C) ≅ C and for λ 2 ≠ 1 we have B 1 (Γ k , C λ 2 ) ≅ C. Hence, dim Z 1 (Γ k , sl(2) Ad ρ λ ) ≥ 4 implies that H 1 (Γ k , C λ ±2 ) ≠ 0 or H 1 (Γ k , C λ -2 ) ≠ 0. Finally, H 1 (Γ k , C λ ±2 ) ≠ 0 and λ ±2 ≠ 1 implies that ∆ k (λ ±2 ) = 0 (see Example3.5). Remark 4.3. Notice that ∆ k (t) ≐ ∆ k (t -1 ) is symmetric and hence H 1 (Γ k , C λ -2 ) ≠ 0 if and only if H 1 (Γ k , C λ 2 ) ≠ 0.
Here p ≐ q means that p, q ∈ C[t ±1 ] are associated elements, i.e. there exists some unit c t k ∈ C[t ±1 ], with c ∈ C * and k ∈ Z, such that p = c t k q .

In general, it is still a conjecture that the necessary condition in Theorem 4.2 is also sufficient i.e. infinitesimal deformation implies deformation. Nevertheless, we have the following result [START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into SL 2 (C)[END_REF]: Theorem 4.4. Let k ⊂ S 3 be a knot and let λ ∈ C * . If λ 2 is a simple root of ∆ k (t) then ρ λ is the limit of irreducible representation.

More precisely, the character χ λ of ρ λ is contained in exactly two components. One component Y 2 ≅ C only contains characters of abelian (diagonal representations), and the second component X λ contains characters of irreducible representations. Moreover, we have Y 2 and X λ intersect transversally at χ ρ , and χ λ is a smooth point on Y 2 and X λ . Remark 4.5. Related results, also for other Lie groups are: Shors [START_REF] Shors | Deforming Reducible Representations of Knot Groups in SL(C)[END_REF], Frohman-Klassen [START_REF] Frohman | Deforming representations of knot groups in SU(2)[END_REF], Herald [START_REF] Christopher | Existence of irreducible representations for knot complements with nonconstant equivariant signature[END_REF], Heusener-Kroll [START_REF] Heusener | Deforming abelian SU(2)-representations of knot groups[END_REF], Ben Abdelghani [START_REF] Ben | Espace des représentations du groupe d'un noeud classique dans un groupe de Lie[END_REF][START_REF] Ben | Tangent cones and local geometry of the representation and character varieties of knot groups[END_REF], Heusener-Porti [START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into PSL 2 (C)[END_REF]. 4.1.2. The general case. Let us go back to the representation ρ λ = (λ bϕ ⊗ α) ⊕ (λ -aϕ ⊗ β) ∈ R n (Γ k ) given by Equation ( 5):

ρ λ (γ) = λ bϕ(γ) α(γ) 0 0 λ -aϕ(γ) β(γ) .
The following generalization of Theorem 4.2 was proved in [START_REF] Heusener | Representations of knot groups into SL n (C) and twisted Alexander polynomials[END_REF]:

Theorem 4.6. Let α ∶ Γ k → SL a (C) and β ∶ Γ k → SL b (C) be irreducible, a + b = n,
and assume that α and β are infinitesimal regular. If

ρ λ ∈ R n (Γ k ) is a limit of irreducible representations then ∆ α⊗β * 1 (λ n ) = ∆ β⊗α * 1 (λ -n ) = 0.
Let us recall some facts about the twisted Alexander polynomial. For more details see [START_REF] Wada | Twisted Alexander polynomial for finitely presentable groups[END_REF][START_REF] Kitano | Twisted Alexander polynomial and Reidemeister torsion[END_REF][START_REF] Kirk | Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants[END_REF][START_REF] Wada | Twisted alexander polynomial revisited[END_REF][START_REF] Heusener | Representations of knot groups into SL n (C) and twisted Alexander polynomials[END_REF]. Let V be a complex vector space, and ρ ∶ Γ k → GL(V ) a representation. We let C ∞ → C k denote the infinite cyclic covering of the knot exterior. The twisted Alexander module is the The following theorem is proved in [START_REF] Heusener | Representations of knot groups into SL n (C) and twisted Alexander polynomials[END_REF]:

C[Z] ≅ C[t ±1 ]-module H i (C ∞ , V ). A generator ∆ ρ i (t) of its order ideal is called the twisted Alexander polynomial ∆ ρ i (t) ∈ C[t ± 1]. Notice that H i (C ∞ , V ) ≅ H i (C k , V [Z]) ≅ H i (Γ k , V [Z]) where V [Z] = V ⊗ C[Γ] C[Z] is a Γ k module via ρ ⊗ t ϕ . The dual representation ρ * ∶ Γ → GL(V * ) is given by ρ * (γ)(f ) = f ○ ρ(γ) -1 for f ∈ V * = Hom(V, C) and γ ∈ Γ. In particular, if ρ ∶ Γ → GL(n) then ρ * (γ) = t ρ(γ) -1 for all γ ∈ Γ k .
Theorem 4.9. If ρ ∶ Γ k → GL(V ) is a semisimple representation then ∆ ρ * i (t) ≐ ∆ ρ i (t -1
). Now, the proof of Theorem 4.6 follows the proof of Theorem 4.2. First, we have to understand the Γ k -module sl(n) Ad ρ λ . Let M a,b (C) the vector space of a × b matrices over the complex numbers. The group

Γ k acts on M a,b (C) via α ⊗ β * i.e. for all γ ∈ Γ k and X ∈ M a,b (C) we have α ⊗ β * (γ)(X) = α(γ)Xβ(γ -1 ) .
Similarly, we obtain a representation β ⊗ α * ∶ Γ k → M b,a (C). The proof of the following lemma is given in [START_REF] Heusener | Representations of knot groups into SL n (C) and twisted Alexander polynomials[END_REF]:

Lemma 4.10. If α ∶ Γ k → SL a (C) and β ∶ Γ k → SL b (C) are irreducible then the represen- tation α * ∶ Γ k → SL a (C) is also irreducible. Moreover, α ⊗ β and β ⊗ α * are semisimple.
In what follows we let M + t and M - t denote the Γ k -modules

M + t = M a,b (C) ⊗ C[t, t -1 ] and M - t = M b,a (C) ⊗ C[t, t -1 ] where Γ k acts via α ⊗ β * ⊕ t ϕ and β ⊗ α * ⊗ t ϕ repectively. Corollary 4.11. If α ∶ Γ k → SL a (C) and β ∶ Γ k → SL b (C) are irreducible then ∆ α⊗β * i (t) ≐ ∆ β⊗α * i (t -1 ) .
Now the Γ k -module sl(n) Ad ρ λ decomposes into a direct sum: [START_REF] Bergeron | Tetrahedra of flags, volume and homology of SL(3)[END_REF] sl(n

) Ad ρ λ = sl a (C) Ad α ⊕ sl b (C) Ad β ⊕ C ⊕ M + λ n ⊕ M - λ -n .
This can be visualized as

sl(n) Ad ρ λ = sl(a) Ad α M + λ n M - λ -n sl(b) Ad β ⊕ C b Id a 0 0 -a Id b .
For every λ ∈ C * we have a non-degenerate Γ k -invariant bilinear form:

Ψ ∶ M - λ -n × M + λ n → C given by Ψ(Y, X) ↦ tr(Y X).
As an immediate consequence, we have Poincaré and Kronecker dualities:

H i (C; M ± λ ±n ) ≅ H 3-i (C, ∂C; M ∓ λ ∓n ) * ; H i (C; M ± λ ±n ) ≅ H 3-i (C, ∂C; M ∓ λ ∓n ) * ; (8) H i (C; M ± λ ±n ) ≅ H i (C; M ∓ λ ∓n ) * . Lemma 4.12. If α ∶ Γ k → SL a (C) and β ∶ Γ k → SL b (C) are irreducible then α * ∶ Γ k → SL a (C) is irreducible, and α ⊗ β is semisimple.
Proof of Theorem 4.6. As in the proof of Theorem 4.2 it follows from Lemma 5.1 in [START_REF] Heusener | Representations of knot groups into SL n (C) and twisted Alexander polynomials[END_REF] that if ρ λ is limit of irreducible representations then

dim Z 1 (Γ k , sl(2) Ad ρ λ ) ≥ n 2 + n -2 .
Now, consider the decomposition (7) of sl(n) Ad ρ λ . Claim: If α and β are infinitesimal regular and irreducible then dim

H 1 (Γ k , M + λ n ) > dim H 0 (Γ k , M + λ n ) or dim H 1 (Γ k , M - λ -n ) > dim H 0 (Γ k , M - λ -n ) . Proof of the Claim. For each Γ-module m, we use the formula dim Z 1 (Γ; m) = dim H 1 (Γ; m) + dim B 1 (Γ; m) = dim H 1 (Γ; m) + dim m -dim H 0 (Γ; m) . (9)
Ordering the terms as they appear in (9):

dim Z 1 (Γ; sl a (C) Ad α ) = (a -1) + (a 2 -1) -0, dim Z 1 (Γ; sl b (C) Ad α ) = (b -1) + (b 2 -1) -0, dim Z 1 (Γ; C) = 1 + 1 -1, dim Z 1 (Γ; M ± λ ±n ) = dim H 1 (Γ; M ± λ ±n ) + a b -dim H 0 (Γ; M ± λ ±n ). Hence the decomposition (7) together with n 2 + n -2 ≤ dim Z 1 (Γ k , sl(2) Ad ρ λ ) gives: n 2 + n -2 ≤ n 2 + n -3 + dim H 1 (Γ k , M - λ -n ) -dim H 0 (Γ k , M - λ -n ) + dim H 1 (Γ k , M + λ n ) -dim H 0 (Γ k , M + λ n ) . Now, it follows from Kronecker duality (8) that dim H 1 (Γ k , M - λ -n ) > dim H 0 (Γ k , M - λ -n ) ⇔ dim H 1 (Γ k , M + λ n ) > dim H 0 (Γ k , M + λ n ) . The short exact sequence of Γ k -modules 0 → M + t (t-λ n )⋅ → M + t → M + λ n → 0 gives a long exact homology sequence [10, III. §6]: . . . → H 1 (Γ; M + t ) (t-λ -n )⋅ → H 1 (Γ; M + t ) → H 1 (Γ; M + λ n ) ∂ → H 0 (Γ; M + t ) (t-λ -n )⋅ → H 0 (Γ; M + t ) → H 0 (Γ; M + λ n ) → 0 . This implies that dim H 1 (Γ; M + λ n ) ≥ rk(∂) = dim H 0 (Γ; M + λ n ) with equality if and only if H 1 (Γ; M + λ n ) has no (t -λ n )-torsion. This in turn is equivalent to ∆ α⊗β * 1 (λ n ) ≠ 0.
Hence we have:

∆ α⊗β * i (λ n ) = 0 ⇐⇒ dim H 1 (Γ k , M - λ -n ) > dim H 0 (Γ k , M - λ -n ) . A similar argument applies if dim H 1 (Γ k , M + λ n ) > dim H 0 (Γ k , M + λ n ). Remark 4.13. Notice that ∆ α⊗β * 1 (t) ≐ ∆ β⊗α * 1 (t -1
), and hence

dim H 1 (Γ k , M - λ -n ) > dim H 0 (Γ k , M - λ -n ) ⇔ ∆ α⊗β * 1 (λ n ) = 0 ⇔ ∆ β⊗α * 1 (λ -n ) = 0 ⇔ dim H 1 (Γ k , M + λ n ) > dim H 0 (Γ k , M + λ n ) .
There is a partial converse of Theorem 4.6 which was proved in [START_REF] Heusener | Representations of knot groups into SL n (C) and twisted Alexander polynomials[END_REF]:

Theorem 4.14. Let α ∶ Γ k → SL a (C) and β ∶ Γ k → SL b (C) be irreducible, a + b = n,
and assume that α and β are infinitesimal regular.

Assume that

∆ α⊗β * 0 (λ n ) ≠ 0 and that λ n is a simple root of ∆ α⊗β * 1 (t). Then ρ λ ∈ R n (Γ k )
can be deformed to irreducible representations. Moreover, the character χ λ ∈ X n (Γ k ) belongs to precisely two irreducible components Y and Z of X n (Γ). Both components Y and Z have dimension n-1 and meet transversally at χ λ along a subvariety of dimension n-2. The component Y contains characters of irreducible representations and Z consists only of characters of reducible ones.

Sketch of proof. Use Luna's Slice Theorem, and study the quadratic cone of the representation ρ λ by identifying the second obstruction to integrability. This relies heavily on the hypothesis about the simple root of the Alexander polynomial.

Deformation of reducible metabelian representations.

In this subsection we will consider certain reducible metabelian representations and their deformations. The general assumption will be that α ∈ C * is a zero of the Alexander polynomial of k , and hence

H 1 (C ∞ ; C) has a direct summand of the form C[t ±1 ] (t -α) n-1 , n ∈ Z, n > 1.
Recall that a knot group Γ is isomorphic to the semi-direct product Γ ≅ Γ ′ ⋊ Z. Every metabelian representation of Γ factors through the metabelian group

(Γ ′ Γ ′′ ) ⋊ Z. Notice that H 1 (C ∞ ; C) ≅ C ⊗ Γ ′ Γ ′′ . Hence we obtain a homomorphism Γ → (Γ ′ Γ ′′ ) ⋊ Z → (C ⊗ Γ ′ Γ ′′ ) ⋊ Z → C[t ±1 ] (t -α) n-1 ⋊ Z . The multiplication on C[t ±1 ] (t-α) n-1 ⋊Z is given by (p 1 , n 1 )(p 2 , n 2 ) = (p 1 +t n 1 p 2 , n 1 +n 2 ).
Let I n ∈ SL(n) and N n ∈ GL(n) denote the identity matrix and the upper triangular Jordan normal form of a nilpotent matrix of degree n respectively. For later use we note the following lemma which follows easily from the Jordan normal form theorem: Lemma 4.15. Let α ∈ C * be a nonzero complex number and let C n be the C[t ±1 ]-module with the action of t k given by (10)

t k a = α k a J k n where a ∈ C n and J n = I n + N n . Then C n ≅ C[t ±1 ] (t -α) n as C[t ±1 ]-modules.
There is a direct method to construct a reducible metabelian representation of the group

C[t ±1 ] (t -α) n-1 ⋊ Z into GL(n) (see [8, Proposition 3.13]). A direct calculation gives that (a, 0) ↦ 1 a 0 I n-1 , (0, 1) ↦ α 0 0 J -1 n-1 defines a faithful representation C[t ±1 ] (t -α) n-1 ⋊ Z → GL(n).
Therefore, we obtain a reducible, metabelian, non-abelian representation

̺ ∶ Γ → GL(n) if the Alexander module H 1 (C ∞ , C) has a direct summand of the form C[t ±1 ] (t -α) s with s ≥ n -1 ≥ 1: ̺ ∶ Γ → C[t ±1 ] (t -α) s ⋊ Z → C[t ±1 ] (t -α) n-1 ⋊ Z → GL(n)
given by ( 11)

̺(γ) = 1 z(γ) 0 I n-1 α ϕ(γ) 0 0 J -ϕ(γ) n-1
.

It is easy to see that a map ̺ ∶ Γ → GL(n) given by ( 11) is a homomorphism if and only if z ∶ Γ → C n-1 is a cocycle i.e. for all γ 1 , γ 2 ∈ Γ we have

(12) z(γ 1 γ 2 ) = z(γ 1 ) + α ϕ(γ 1 ) z(γ 2 )J ϕ(γ 1 )
n-1 . The unipotent matrices J n and J -1 n are similar: a direct calculation shows that P n J n P -1 n = J -1 n where P n = (p ij ), p ij = (-1) j j i . The matrix P n is upper triangular with ±1 in the diagonal and P 2 n is the identity matrix, and therefore P n = P -1 n . Hence ̺ is conjugate to a representation ̺ ∶ Γ → GL(n) given by ( 13)

̺(γ) = α h(γ) z(γ) 0 J h(γ) n-1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ α h(γ) z 1 (γ) z 2 (γ) . . . z n-1 (γ) 0 1 h 1 (γ) . . . h n-2 (γ) ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ 1 h 1 (γ) 0 . . . . . . 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ where z = (z 1 , . . . , z n-1 ) ∶ Γ → C n-1 satisfies z(γ 1 γ 2 ) = α h(γ 1 ) z(γ 2 ) + z(γ 1 )J h(γ 2 ) n-1 .
It follows directly that z(γ) = z(γ)P n-1 J h(γ)

n-1 and in particular z 1 = -z 1 . We choose an n-th root λ of α = λ n and we define a reducible metabelian representation ̺ λ ∶ Γ → SL(n) by [START_REF] Deraux | A 1-parameter family of spherical CR uniformizations of the figure eight knot complement[END_REF] ̺ λ (γ) = λ -ϕ(γ) ̺(γ).

The following theorem generalizes the results of [START_REF] Ben Abdelghani | Deformations of metabelian representations of knot groups into SL(3, C)[END_REF] where the case n = 3 was investigated. It also applies in the case n = 2 which was studied in [START_REF] Ben | Espace des représentations du groupe d'un noeud classique dans un groupe de Lie[END_REF] and [38, Theorem 1.1]. Theorem 4.16. Let k be a knot in the 3-sphere S 3 . If the (tα)-torsion τ α of the Alexander module H 1 (C; C[t ±1 ]) is cyclic of the form C[t ±1 ] (tα) n-1 , n ≥ 2, then for each λ ∈ C * such that λ n = α there exists a reducible metabelian representation ̺ λ ∶ Γ k → SL(n). Moreover, the representation ̺ λ is a smooth point of the representation variety

R n (Γ). It is contained in a unique (n 2 + n -2)-dimensional component R ̺ λ of R n (Γ)
which contains irreducible non-metabelian representations which deform ̺ λ .

The main part of the proof of this theorem is a cohomological calculation [START_REF] Ben | Irreducible representations of knot groups into SL(n,C)[END_REF]: for the representation

̺ λ ∶ Γ → SL(n) we have H 0 (Γ; sl(n) Ad ○̺ λ ) = 0 and dim H 1 (Γ; sl(n) Ad ○̺ λ ) = dim H 2 (Γ; sl(n) Ad ○̺ λ ) = n -1 .
Then we apply Proposition 3.2.

Remark 4.17. Let ρ λ ∶ Γ → SL(n) be the diagonal representation given by ρ λ (m) = diag(λ n-1 , λ -1 , . . . , λ -1 ) ∈ SL(n) where m is a meridian of k . The orbit O(ρ λ ) of ρ λ under the action of conjugation of SL(n) is contained in the closure O(̺ λ ). Hence ̺ λ and ρ λ project to the same point χ λ of the variety of characters

X n (Γ k ) = R n (Γ k ) SL(n).
It would be natural to study the local picture of the variety of characters [36, § 8]. Unfortunately, there are much more technical difficulties since in this case the quadratic cone Q(ρ λ ) coincides with the Zariski tangent space Z 1 (Γ; sl(n) Ad ρ λ ). Therefore the third obstruction has to be considered.

X n (Γ k ) = R n (Γ k ) SL(n) at χ λ as done in

4.3.

The irreducible representation r n ∶ SL(2) → SL(n). It is interesting to study the behavior of representations ρ ∈ R 2 (Γ) under the composition with the n-dimensional, irreducible, rational representation r n ∶ SL(2) → SL(n). The representation r n is equivalent to (n -1)-fold symmetric power Sym n-1 of the standard representation (see [START_REF] Springer | Invariant theory[END_REF][START_REF] Fulton | Representation theory[END_REF] and [START_REF] Heusener | Deformations of reducible representations of knot groups into SL(n, C)[END_REF] for more details). In particular, r 1 is trivial, r 2 is equivalent to the standard representation, and r 3 is equivalent to Ad ∶ SL(2) → O(sl(2)) ⊂ SL(3). If k is odd then r k is not injective since it factors trough the projection SL(2) → PSL(2). W. Müller [START_REF] Müller | The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds[END_REF] studied the case where ρ ∶ π 1 (M) → SL(2) is the lift of the holonomy representation of a compact hyperbolic manifold. This study was extended by P. Menal-Ferrer and J. Porti [START_REF] Menal | Twisted cohomology for hyperbolic three manifolds[END_REF][START_REF] Menal | Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds[END_REF] to the case of non-compact finite volume hyperbolic manifolds. (For more details see Section 5.1.)

In [START_REF] Heusener | Deformations of reducible representations of knot groups into SL(n, C)[END_REF] the authors studied the case related to Theorem 4.4. Let Γ k be a knot group. We define ρ n,λ ∶ Γ k → SL(n) by ρ n,λ ∶= r n ○ ρ λ where ρ λ is given by Equation ( 6). Proposition 4.18. Let k ⊂ S 3 be a knot, and assume that ρ 0 ∶ Γ k → SL(2) is irreducible. Then R n (Γ k ) contains irreducible representations.

Proof. It was proved by Thurston that there is at least a 4-dimensional irreducible component R 0 ⊂ R 2 (Γ k ) which contains the irreducible representation ρ 0 (see [13, 3.2.1]).

Let Γ be a discrete group and let ρ ∶ Γ → SL(2) be an irreducible representation. By virtue of Burnside's Theorem on matrix algebras, being irreducible is an open property for representations in R n (Γ). If the image ρ(Γ) ⊂ SL(2) is Zariski-dense then the representation ρ n ∶= r n ○ ρ ∈ R n (Γ) is irreducible. In order to prove the proposition we will show that there is a neighborhood

U = U(ρ 0 ) ⊂ R 0 ⊂ R 2 (Γ k ) such that ρ(Γ) ⊂ SL(2) is Zariski-dense for each irreducible ρ ∈ U .
Let now ρ ∶ Γ k → SL(2) be any irreducible representation and let G ⊂ SL(2) denote the Zariski-closure of ρ(Γ k ). Suppose that G ≠ SL(2). Since ρ is irreducible it follows that G is, up to conjugation, not a subgroup of upper-triangular matrices of SL [START_REF] Ben | Espace des représentations du groupe d'un noeud classique dans un groupe de Lie[END_REF]. Then by [START_REF] Kovacic | An algorithm for solving second order linear homogeneous differential equations[END_REF]Sec. 1.4] and [START_REF] Kaplansky | An introduction to differential algebra[END_REF]Theorem 4.12] there are, up to conjugation, only two cases left:

• G is a subgroup of the infinite dihedral group . As in the dihedral case these finitely many orbits are closed and 3-dimensional. Hence all the irreducible ρ ∈ R 0 such that r n ○ ρ is reducible are contained in a Zariski-closed subset of R 0 . Hence generically

D ∞ = α 0 0 α -1 α ∈ C * ∪ 0 α -α -1 0 α ∈ C * . • G is
ρ n = r n ○ ρ is irreducible for ρ ∈ R 0 .
Remark 4.19. Recall that a finite group has only finitely many irreducible representations (see [START_REF] Serre | Linear representations of finite groups[END_REF][START_REF] Fulton | Representation theory[END_REF]). Hence, the restriction of r n to the groups A is reducible, for all but finitely many n ∈ N.

Let k ⊂ S 3 be a knot, and let λ 2 ∈ C a simple root of ∆ k (t). We let R λ ⊂ R n (Γ k ) denote the 4-dimensional component which maps onto the component Theorem 4.4). We obtain: Corollary 4.20. Let k ⊂ S 3 be a knot, and λ 2 ∈ C a simple root of ∆ k (t). Then the diagonal representation ρ λ,n = r n ○ρ λ ∶ Γ k → SL(n) is the limit of irreducible representations in R n (Γ k ). More precisely, generically a representation ρ n = r n ○ ρ, ρ ∈ R λ , is irreducible. Corollary 4.20 can be made more precise (see [START_REF] Heusener | Deformations of reducible representations of knot groups into SL(n, C)[END_REF]):

X λ ⊂ X 2 (Γ k ) under t ∶ R n (Γ) → X n (Γ) (see
Theorem 4.21. If λ 2 is a simple root of ∆ k (t) and if ∆ k (λ 2i ) ≠ 0 for all 2 ≤ i ≤ n-1 then the reducible diagonal representation ρ λ,n = r n ○ ρ λ is a limit of irreducible representations. More precisely, there is a unique (n + 2)(n -1)-dimensional component R λ,n ⊂ R n (Γ k )
which contains ρ λ,n and irreducible representations. Remark 4.22. Under the assumptions of Corollary 4.20 it is possible to study the tangent cone of R n (Γ k ) at ρ λ,n , and thereby to determine the local structure of R n (Γ). There are 2 n-1 branches of various dimensions of R n (Γ k ) passing through ρ λ . Nevertheless, only the component R λ,n contains irreducible representations. This will be studied in a forthcoming paper.

The global structure of character varieties of knot groups

Not much is known about the global structure of the character varieties of knot groups. In this section we will present some facts and some examples.

(Γ k ). A representation ρ ∶ Γ k → SL(n)
which is the direct sum of one-dimensional representations is equivalent to a diagonal representation. The image of a diagonal representation is abelian. Hence it factors through ϕ∶ Γ k → Z. Therefore, the characters of diagonal representations coincide with the characters

X n (Z) ↪ X n (Γ k ). Recall that X n (Z) ≅ C n-1 .
5.1. The distinguished components for hyperbolic knots. Let k ⊂ S 3 be a hyperbolic knot i.e. S 3 ∖ k has a hyperbolic metric of finite volume. Then there exists, up to complex conjugation, a unique one-dimensional component X 0 ⊂ X(Γ k , PSL 2 (C)) which contains the character of the holonomy representation (see [START_REF] Kapovich | Hyperbolic manifolds and discrete groups[END_REF]Theorem 8.44]). The holonomy representation lifts to a representation ρ∶ Γ k → SL(2) (not unique) since H 2 (Γ k ; Z 2Z) = 0. By composing any lift with the rational, irreducible, r -dimensional representation Sym r-1 ∶ SL(2) → SL(r) we obtain an irreducible representation ρ r ∶ Γ k → SL(r). It follows from work of Menal-Ferrer and Porti [START_REF] Menal | Twisted cohomology for hyperbolic three manifolds[END_REF] that χ ρr ∈ X r (Γ k ) is a scheme smooth point contained in a unique (r -1)-dimensional component of X r (Γ k ).

We will call such a component a distinguished component of X r (Γ k ). For odd r , as Sym r-1 ∶ SL(2) → SL(r) factors through PSL(2), there is a unique distinguished component in X r (Γ) up to complex conjugation.

5.2.

Examples. The aim of this subsection is to describe the components of the SL(3)character varieties of the trefoil knot and the figure eight knot, see [START_REF] Heusener | Representations of knot groups into SL n (C) and twisted Alexander polynomials[END_REF][START_REF] Heusener | The SL(3, C)-character variety of the figure eight knot[END_REF]. If ρ is irreducible then it follows from Schur's Lemma that the matrix

A 2 = B 3 ∈ {id 3 , ω id 3 , ω 2 id 3 } is a central element of SL(3). Lemma 5.2. If ρ ∶ Γ → SL(3) is irreducible then A 2 = B 3 = id 3 .
Proof. The matrix A has an eigenvalue of multiplicity two, and hence A has a twodimensional eigenspace. Therefore, B has only one-dimensional eigenspaces, otherwise ρ would not be irreducible. This implies that B has three different eigenvalues: λ, λω , λω 2 where λ 3 ∈ {1, ω, ω 2 }. We obtain det(B) = 1 = λ 3 . Therefore B 3 = Id 3 .

Lemma 5.2 implies that the matrices A and B are conjugate to

A ∼ 1 -1 -1 and B ∼ 1 ω ω 2 .
The corresponding eigenspaces are the plane E A (-1), and the lines E A (1), E B (1), E B (ω), and E B (ω 2 ). Now, these eigenspaces determine the representation completely, as they determine the matrices A and B , that have fixed eigenvalues. Of course we have E A (1) ∩ E A (-1) = 0 and E B (1), E B (ω), and E B (ω 2 ) are also in general position. Since ρ is irreducible, the five eigenspaces are in general position. For instance

E A (1) ∩ (E B (1) ⊕ E B (ω)) = 0, because otherwise E B (1) ⊕ E B (ω) = E A (1) ⊕ (E A (-1) ∩ (E B (1) ⊕ E B (ω))
) would be a proper invariant subspace.

We now give a parametrization of the conjugacy classes of the irreducible representations. The invariant lines correspond to fixed points in the projective plane P 2 , and E A (-1) determines a projective line.

• The first normalization: the line E A (-1) corresponds to the line at infinity:

P 1 = E A (-1) = ⟨[0 ∶ 1 ∶ 0], [0 ∶ 0 ∶ 1]⟩
The four invariant lines E A (1), E B (1), E B (ω), and E B (ω 2 ) are points in the affine plane C 2 = P 2 ∖ P 1 . They are in general position.

• We fix the three fixed points of B , corresponding to the following affine frame. We obtain the following lemma:

Lemma 5.3. Every irreducible representation ρ ∶ Γ 3 1 → SL(3) is equivalent to exactly one representation ρ s,t . Moreover, ρ s,t is reducible if and only if (s, t) is contained in one of the three affine lines given by s = 0, t = 0, and s + t = 2. If (s, t) ∈ {(0, 0), (0, 2), (2, 0)} is the intersection point of two of those lines then ρ s,t fixes a complete flag, and has the character of a diagonal representation.

The following theorem follows from the above considerations (see [START_REF] Heusener | Representations of knot groups into SL n (C) and twisted Alexander polynomials[END_REF]Theorem 9.10] for more details). We let R irr n (Γ) ⊂ R n (Γ) denote the Zariski-open subset of irreducible representation Theorem 5.4. The GIT quotient X = R irr 3 (Γ) SL(3) of the trefoil knot group Γ is isomorphic to C 2 . Moreover, the Zariski open subset R irr 3 (Γ) is SL(3)-invariant and its GIT quotient is isomorphic to the complement of three affine lines in general position in C 2 .

Remark 5.5. The same arguments as above apply to torus knots T (p, 2), p odd, to prove that the variety of irreducible SL 3 (C)-characters consist of (p -1)(p -2) 2 disjoint components isomorphic to C 2 , (p -1) 2 components of characters of partial reducible representations, and the component of characters of diagonal representations.

In general, the SL(3)-character variety for torus knots was studied by Muñoz and Porti [START_REF] Muñoz | Geometry of the SL(3, C)-character variety of torus knots[END_REF]. In the general case T (p, q), p, q > 2 there are 4-dimensional components in X 3 (Γ T (p,q) ) corresponding to the configuration of 6 points in the projective plane. 5.2.2. The SL(3)-character variety of the figure eight knot. Let Γ = Γ 4 1 be the group of the figure eight knot. The figure eight knot has genus one, and its complement fibres over the circle [START_REF] Burde | Knots[END_REF]. Hence the commutator group of Γ is a free group of rank two, and a presentation is given by Γ ≅ ⟨t, a, b tat -1 = ab, tbt -1 = bab⟩ .

A peripheral system is given by (m, ℓ) = (t, [a, b]). The amphicheirality of the figure eight knot implies that its group has an automorphism h ∶ Γ → Γ which maps the peripheral system (m, ℓ) to (m -1 , ℓ) up to conjugation. Such an automorphism is explicitly given by h(t) = ta -1 t -1 at -1 ∼ t -1 , h(a) = a -1 tab -1 a -1 t -1 a ∼ b -1 , h(b) = a -1 tat -1 a ∼ a Notice that we obtain h(m) = ta -1 m -1 t -1 a and h(ℓ) = h([a, b]) = a -1 ta[b -1 , a]a -1 t -1 a. The relation t -1 a -1 t = ba -2 gives that the peripheral system (h(m), h(ℓ)) is conjugated to (m -1 , ℓ) as desired.

The structure of the SL(3)-character variety of the figure eight knot had been studied in detail in [START_REF] Heusener | The SL(3, C)-character variety of the figure eight knot[END_REF], see also [START_REF] Falbel | Character varieties for SL(3, C): the figure eight knot[END_REF]. The character variety X 3 (Γ 4 1 ) has 5 components:

• the component containing the characters of abelian representations;

• one component containing the characters of the representations

ρ λ = α ⊗ λ ϕ ⊕ λ -2ϕ
where α ∈ R 2 (Γ 4 1 ) is irreducible (compare Equation 5 with β trivial); • three components V 0 , V 1 and V 2 containing characters of irreducible representations. The component V 0 is the distinguished component (see Section 5.1). The two other components which come from a surjection Γ Remark 5.6. The surjection φ ∶ Γ ↠ D [START_REF] Ben | Variété des caractères et slice étale de l'espace des représentations d'un groupe[END_REF][START_REF] Ben | Variété des caractères et slice étale de l'espace des représentations d'un groupe[END_REF][START_REF] Ben | Tangent cones and local geometry of the representation and character varieties of knot groups[END_REF] is related to an exceptional Dehn filling on the figure-eight knot K (see [START_REF] Gordon | Dehn surgery and 3-manifolds[END_REF]). In particular, the Dehn filling manifold K(±3) is a small Seifert fibered manifold, and K(±3) fibers over S 2 [START_REF] Ben | Variété des caractères et slice étale de l'espace des représentations d'un groupe[END_REF][START_REF] Ben | Variété des caractères et slice étale de l'espace des représentations d'un groupe[END_REF][START_REF] Ben | Tangent cones and local geometry of the representation and character varieties of knot groups[END_REF]. The orbifold fundamental group π O 1 (S 2 (3, 3, 4)) is isomorphic to the von Dyck group π O 1 (S 2 (3, 3, 4)) ≅ D(3, 3, 4). Hence the surjection Γ → π 1 (K(±3)) ↠ π 1 (K(±3)) center ≅ D(3, 3, 4) is natural.

The center of π 1 (K(±3)) is generated by a regular fibre. Any irreducible representation of π 1 (K(±3)) → SL(3) maps the fibre to the center of SL [START_REF] Ben | Variété des caractères et slice étale de l'espace des représentations d'un groupe[END_REF]. By using the description of X 3 (F 2 ) given by Lawton [START_REF] Lawton | Generators, relations and symmetries in pairs of 3×3 unimodular matrices[END_REF]) it quite elementary to determine

0 0 1 ζ

 01 ) Let Γ = ⟨S, T ST S = T ST ⟩ be the group of the trefoil knot, and let ζ ∈ C be a primitive 12-th root of unity, ζ 4ζ 2 + 1 = 0. We define ρ ∶ Γ → SL 2 (C) by ρ(S) = ζ , and ρ(T ) = ζ 1 0 1 ζ . The representation is reducible but not semisimple. Again the orbit O(ρ) is not closed, O(ρ) contains the diagonal representation

Example 2 . 8 .

 28 We will give three basic examples of GIT quotients: (1) Let C * act on C 2 by λ.(z 1 , z 2 ) = (λz 1 , λz 2 ). The topological quotient C 2 C * is a non-Hausdorff topological space. More precisely, only the orbit O(0, 0) = {(0, 0)} is closed, and (0, 0) is contained in the closure of every orbit. The algebra O(C 2 ) is isomorphic to the polynomial ring in two variables C[x 1 , x 2 ], and C

1 ω ω b 11 b 22 = - 1 - 1 .

 12211 and ρ is conjugate to an upper/lower triangular representation. Proof. Let ρ ∶ D(3, 3, 3) → SL(2) be a representation. Then ρ is trivial if and only if ρ(a) = ρ(b) = I 2 . If ρ(a) = I 2 and ρ(b) ≠ I 2 then up to conjugation we may assume that ρ(b) = ω 0 0 ω . We obtain that ρ is a diagonal representation. A similar argument applies if ρ(b) = I 2 or ρ(ab) = I 2 . Now suppose that tr ρ(a) = tr ρ(b) = tr ρ(ab) = -1. Up to conjugation we obtain that ρ(a) = ω 0 0 ω and ρ(b) = b 11 b 12 b 21 b 22 where b 11 + b 22 = -1 and b 11 b 22b 12 b 21 = 1. The equation tr ρ(ab) = -1 then implies 1 This system has the unique solution (b 11 , b 22 ) = (ω, ω). Finally, b 11 b 22b 12 b 21 = 1 implies that b 12 b 21 = 0 and ρ is a triangular representation.

Example 2 . 16 .

 216 The ring R = C[T ] (T 2 ) is not reduced, R red = C[T ] (T ). Both rings have only one maximal ideal (T ) ⊂ R, and (0) ⊂ R red ≅ C. The zero locus of (T 2 ) and (T ) is the same, it is just the point {0} ∈ C. The projections C[T ] → C[T ] (T 2 ) and C[T ] → C[T ] (T ) give inclusions Spec R ↪ C, and Spec R red ↪ C. Now, the restriction of a function f ∈ C[T ] vanishes on R red if and only if

Theorem 4 . 2 .

 42 If the diagonal representation ρ λ ∈ R 2 (Γ k ) can be deformed to irreducible representations then ∆ k (λ 2 ) = 0.

Lemma 4 . 7 .

 47 The representations ρ and ρ * are equivalent if and only if there exists a Γ-invariant, non-degenerated bilinear form V ⊗ V → C. Example 4.8. If ρ ∶ Γ → O(n) or ρ ∶ Γ → SL 2 (C) then ρ and ρ * are equivalent.

5 (

 5 one of the groups A the icosahedral group). These groups are the preimages in SL(2) of the subgroups A 4 , S 4 , A 5 ⊂ PSL(2, C). By a result of E. Klassen[START_REF] Paul Klassen | Representations of knot groups in SU(2)[END_REF] Theorem 10] there are up to conjugation only finitely many irreducible representations of a knot group into D ∞ . Moreover, the orbit of each of those irreducible representation is 3-dimensional. Therefore, there exists a Zariski-open subset U ⊂ R 0 which does not contain representations of Γ k into D ∞ .For the second case there are up to conjugation only finitely many irreducible representations of Γ k onto the subgroups A

Example 5 . 1 (

 51 Diagonal representations). The characters of diagonal representations of a knot group Γ k form an algebraic component of X n

5. 2 . 1 .

 21 Irreducible SL(3)-representations of the trefoil knot group. Let k ⊂ S 3 be the trefoil knot and Γ = Γ 3 1 . We use the presentationΓ ≅ ⟨x, y x 2 = y 3 ⟩ .The center of Γ is the cyclic group generated by z = x 2 = y 3 . The abelianization map ϕ ∶ Γ → Z satisfies ϕ(x) = 3, ϕ(y) = 2, and a meridian of the trefoil is given by m = xy -1 . Let ω denote a primitive third root of unity, ω 2 + ω + 1 = 0.For a given representation ρ ∈ R 3 (Γ) we put ρ(x) = A and ρ(y) = B .

E B ( 1 )

 1 = [1 ∶ 0 ∶ 0], E B (ω) = [1 ∶ 1 ∶ 0], and E B (ω 2 ) = [1 ∶ 0 ∶ 1]. • The fourth point (the line E A (1)) is a point in C 2 which does not lie in the affine lines spanned by any two of the fixed points of B :E A (1) = [2 ∶ s ∶ t] where s ≠ 0, t ≠ 0, or s + t ≠ 2This gives rise to the subvariety {ρ s,t ∈ R(Γ, SL(3)) (s, t) ∈ C 2 }, where ρ s,t (x)

4 1 ↠

 1 D(3, 3, 4) onto a triangle group. Let us describe the components V 1 and V 2 without going too much into the technical details. An epimorphism φ ∶ Γ ↠ D(3, 3, 4) = ⟨k, l l 3 , k 3 , (kl) 4 ⟩ is given by φ(a) = k -1 l -1 kl, φ(b) = kl and φ(t) = klk . It satisfies φ(b) 4 = 1 and φ(m 3 ℓ) = 1. Notice that the surjection φ induces an injection φ * ∶ X 3 (D(3, 3, 4)) ↪ X 3 (Γ) .

  X 3 (D(3, 3, 4)) explicitly. The proof of the next lemma can be found in [35, Lemma 10.1]: Lemma 5.7. The variety X irr (D(3, 3, 4), SL(3, C)) has a component W of dimension 2 and three isolated points. The variety W is isomorphic to the hypersurface in C 3 given by the equation ζ 2 -(ν ν -2)ζ + ν 3 + ν3 -5ν ν + 5 = 0 . Here, the parameters are ν = χ(k -1 l), ν = χ(kl -1 ) and ζ = χ([k, l]). For every χ ∈ W , χ(k ±1 ) = χ(l ±1 ) = 0 and χ((kl) ±1 ) = 1.

  x 11 x 12 x 21 x 22 ∈ M 2 (C[x 11 , x 12 , x 21 , x 22 ]) the relation X 2 = 1 0 0 1 and det X = 1 give the equations x 11 x 22x 12 x 21 = 1, and x 2 11 + x 12 x 21 = 1, x 11 x 12 + x 12 x 22 = 0, x 11 x 21 + x 21 x 22 = 0, x 12 x 21 + x 2

22 = 1 . The ideal I has a much simple r set of generators: I = (x 2 22 -1, x 11x 22 , x 12 , x 21 ), and hence

  be strict (see Example 2.18). More precisely, the space Z 1 (Γ; sl(n) Ad ρ ) is the Zariski tangent space to the scheme R(Γ, SL n (C)) at ρ (see Section 2.4).

	Definition 2.10. Let ρ ∶ Γ → SL(n) be a representation. A derivation u ∈ Z 1 (Γ; sl(n) Ad ρ ) is called integrable if there exists a family of representations ρ t ∶ Γ → SL(n) such that ρ 0 = ρ and (1) holds.
	2.3.1. Detecting smooth points. The following is a quite useful observation [47, p. iv] for
	detecting smooth points of the representation variety. In general not every cocycle is
	integrable and there are different reasons for this. We have the following inequalities
	(2)

Acknowledgements. I like to thank the organizers of the workshop Topology, Geometry and Algebra of low-dimensional manifolds for inviting me. In particular, I am very grateful to Teruaki Kitano and Takayuki Morifuji for their hospitality (before and after the workshop). Also, thanks to Leila Ben Abdelghani, Paul Kirk, and Simon Riche for very helpful discussions during the writing of this article. Also, I am pleased to acknowledge support from the ANR projects SGT and ModGroup.

Moreover, all characters in W are characters of irreducible representations except for the three points (ν, ν, ζ) = (2, 2, 1), (2̟, 2̟ 2 , 1), (2̟ 2 , 2̟, 1), ̟ = e 2πi 3 . Now, the components V 1 and V 2 are given by

The components V 1 and V 2 are swapped by h * ∶ X 3 (Γ 4 1 ) → X 3 (Γ 4 1 ), and V 0 is preserved.

Further details in the proof of Lemma 5.7 allow to describe those three isolated points. Composing with φ * , they correspond to the three characters of irreducible metabelian representations in X 3 (Γ 4 1 ) that do not lie in V 2 . Altogether, there are five characters of irreducible metabelian representations (see [START_REF] Hans | Metabelian SL(n, C) representations of knot groups[END_REF]). All these metabelian characters are scheme smooth (see [START_REF] Hans | Metabelian SL(n, C) representations of knot groups, III: deformations[END_REF]). The character corresponding to a point of V 0 comes from a surjection Γ 4 1 ↠ A 4 composed with the irreducible representation A 4 → SL(3).

Proposition 5.8. The components V 1 and V 2 are characters of representations which factor through the surjections Γ ↠ π 1 (K(±3)) respectively. These components are isomorphic to the hypersurface

Here, the parameters are

All characters are irreducible except for the three points (ν, ν, ζ) = (2, 2, 1), (2̟, 2̟ 2 , 1), (2̟ 2 , 2̟, 1), with ̟ = e 2πi 3 , that correspond to the inter-