Deformations of reducible SL(n,C) representations of fibered 3-manifold groups
aa r X i v : . [ m a t h . G T ] S e p DEFORMATIONS OF REDUCIBLE
SL( n, C ) REPRESENTATIONS OF FIBERED 3-MANIFOLD GROUPS
KENJI KOZAI
Abstract.
Let M φ be a surface bundle over a circle with monodromy φ : S → S . We study deformations of certain reducible representa-tions of π ( M φ ) into SL( n, C ), obtained by composing a reducible rep-resentation into SL(2 , C ) with the irreducible representation SL(2 , C ) → SL( n, C ). In particular, we show that under conditions on the eigenval-ues of φ ∗ , the reducible representation is contained in a ( n +1+ k )( n − k is thenumber of components of ∂M φ . Moreover, the reducible representationis the limit of a path of irreducible representations. Introduction
Suppose that S = S g,p is a surface of genus g with p ≥ g + p >
2, i.e. S admits a hyperbolic structure. If φ : S → S is a homeomor-phism, we can form the mapping torus M φ = S × [0 , / ( x, ∼ ( φ ( x ) , λ is an eigenvalue of φ ∗ : H ( S ) → H ( S ) with eigenvector( a , . . . , a g + p − ) T with respect to a generating set { [ γ ] , . . . , [ γ g + p − ] } of H ( S ), we obtain a reducible representation ρ : π ( M φ ) → SL(2 , C ) bydefining, ρ λ ( γ i ) = (cid:18) a i (cid:19) ρ λ ( τ ) = (cid:18) λ λ − (cid:19) , where τ is the generator of the fundamental group of the S base of the fiberbundle S → M φ → S . (Recall that a representation ρ : G → GL( n, C ) is reducible if the image ρ ( G ) preserves a proper subspace of C n , and otherwiseis called irreducible .)When M φ is the complement of a knot K in S , this observation wasoriginally made by Burde [3] and de Rham [4]. Furthermore, the Alexanderpolynomial is the characteristic polynomial of φ ∗ , so the condition on λ is equivalent to the condition that λ is a simple root of the Alexanderpolynomial ∆ K ( t ). It was shown in [7] that the non-abelian, metabelian,reducible representation ρ λ is the limit of irreducible representations if λ isa simple root of ∆ K ( t ). Recently, Heusener and Medjerab [6] have shownthat the conclusion still holds in SL( n, C ), n ≥
3, if ρ λ is composed withthe irreducible representation r n : SL(2 , C ) → SL( n, C ). These results apply even if the knot complement is not fibered, as long as λ is a simple root of∆ K ( t ).In this paper, we apply some of the techniques in [6] to show that re-ducible SL( n, C ) representations of fibered 3-manifolds groups obtained asthe composition ρ λ,n = r n ◦ ρ λ can be deformed to irreducible representa-tions. If the punctures form a single orbit under φ and the complement isthe complement of a fibered knot, then the results of [7] and [6] apply. Themain result in Theorem 1.1 also covers the cases where M φ is the comple-ment of a fibered link L with k ≥ L , . . . , L k , or a k -cuspedfibered manifold which is not a link complement. In the statement of The-orem 1.1, ¯ φ is the homemorphism on ¯ S = S g, obtained from φ by filling inthe p punctures of S g,p . This gives a homeomorphism ¯ φ : ¯ S → ¯ S . Theorem 1.1.
Suppose that λ is a simple eigenvalue of φ ∗ . If | λ | 6 = 1 , ¯ φ ∗ : H ( ¯ S ) → H ( ¯ S ) does not have 1 as an eigenvalue, and if for each ≤ j ≤ n ,we have that λ j is not an eigenvalue of φ ∗ , then ρ λ,n is a limit of irreducible SL( n, C ) representations and is a smooth point of the representation variety R ( π ( M φ ) , SL( n, C )) , contained in a unique component of dimension ( n +1 + k )( n − . When φ is a pseudo-Anosov element of the mapping class group, λ is thedilatation factor of φ , and the p punctures are exactly the singular points ofthe invariant foliations of φ , ρ λ is shown to have deformations to irreduciblerepresentations under some additional conditions on the eigenvalues of ¯ φ ∗ ,the map on the closed surface S g , in [9]. We show that under the samehypotheses, the same holds for ρ λ,n . Theorem 1.2.
Suppose that λ is the dilatation of a pseudo-Anosov map φ such that the stable and unstable foliations are orientable, and the singularpoints coincide with the punctures of S . Suppose also that is not an eigen-value of ¯ φ ∗ . Then ρ λ,n is a limit of irreducible SL( n, C ) representations andis a smooth point of R ( π ( M φ ) , SL( n, C )) , contained in a unique componentof dimension ( n + 1 + k )( n − . In Section 2, we give the basic definitions and background about repre-sentations of SL(2 , C ) into SL( n, C ). Section 3 discusses the general theoryof deformations, and Section 4 contains the main results, including relevantcohomological calculations and the irreducibility of nearby representations.2. Representations of
SL(2 , C )For notational convenience, we denote SL( n ) = SL( n, C ), sl ( n ) = sl ( n, C ),GL( n ) = GL( n, C ), and Γ φ = π ( M φ ). A more general version of the dis-cussion in this section can be found in [6, Section 4]. EFORMATIONS OF REDUCIBLE SL( n, C ) REPRESENTATIONS 3 Let R = C [ X, Y ] be the polynomial algebra on two variables. We havean action of SL(2) on R by, (cid:18) a bc d (cid:19) · X = dX − bY (cid:18) a bc d (cid:19) · Y = − cX + aY, for (cid:0) a bc d (cid:1) ∈ SL(2). Let R n − ⊂ R denote the n -dimensional subspace ofhomogenous polynomials of degree n −
1, generated by X l − Y n − l , ≤ l ≤ n . The action of SL(2) leaves R n − invariant, turning R n − into a SL(2)module, and we obtain a representation r n : SL(2) → GL( R n − ). We canidentify R n − with C n by identifying the basis elements { X l − Y n − l } withthe standard basis elements { e l } of C n . The induced isomorphism turns r n into a representation r n : SL(2) → GL( n ), which we will also call r n . Therepresentation r n is rational , that is the coefficients of the matrix coordinatesof r n (cid:0) a bc d (cid:1) are polynomials in a, b, c, d .We have the following two well-known results about r n . Lemma 2.1. [16, Lemma 3.1.3(ii)]
The representation r n is irreducible. Lemma 2.2. [16, Lemma 3.2.1]
Any irreducible rational representation of
SL(2 , C ) is conjugate to some r n . It is easy to check that r n maps the unipotent matrices (cid:0) b (cid:1) and ( c )to unipotent elements of SL( R n − ), and the diagonal element (cid:0) a a − (cid:1) ismapped to the diagonal element diag( a n − , a n − , . . . , a − n +1 ). Hence, theimage of r n lies in SL( R n − ) ∼ = SL( n ).We now define ρ λ,n = r n ◦ ρ λ . As we will only be considering the case when λ is a simple eigenvalue of φ ∗ , and the above lemmas imply the uniquenessof r n , this gives a well-defined and unique (up to conjugation) representation ρ λ,n : Γ φ → SL( n ).One can also show via explicit calculation that r n (cid:18) a a − b a − (cid:19) · X l − Y n − l = ( a − X − a − bY ) l − ( aY ) n − l = a n − l +1 ( X − bY ) l − Y n − l = a n − l +1 l − X j =0 ( − b ) j (cid:18) l − j (cid:19) X l − j − Y n − ( l − j ) . In particular, this implies that the space spanned by X Y n − is invariantunder the subgroup of upper triangular matrices in SL(2). Specifically, r n (cid:18) a a − b a − (cid:19) · X Y n − = a n − X Y n − . KENJI KOZAI As ρ λ is an upper triangular representation, this action turns R n − into a Γ φ module, with γ ∈ Γ φ acting by r n ◦ ρ λ ( γ ). Under this action, < X Y n − > is an invariant submodule. Definition 2.3.
Let ψ : Γ φ → Z denote the canonical surjection which isdual to the fiber. For a non-zero complex number α ∈ C ∗ , we define C α tobe the Γ φ module C , where the action of γ ∈ Γ φ is defined by x α ψ ( γ ) x .By the previously defined action of Γ φ , we have that < X Y n − > isisomorphic to C λ n − . Let ¯ R n − be the quotient R n − / < X Y n − > . Wewill need the following facts about the relationship between R n − , ¯ R n − , and C λ n − . Lemma 2.4. [6, Equations (4.3) and (4.4)]
There are short exact sequencesof Γ φ -modules (2.1) 0 → C λ n − → R n − → ¯ R n − → , and, (2.2) 0 → R n − → ¯ R n − → C λ − n +1 → . By composing ρ λ,n with the adjoint representation, we also obtain anaction of Γ φ on sl ( n ), turning it into a Γ φ module. The following decom-position is a consequence of the Clebsch-Gordan formula (see, for example,[12, Lemma 1.4]). Lemma 2.5.
With the Γ φ module structure, sl ( n ) ∼ = ⊕ n − j =1 R j . Infinitesimal deformations
In this section, let M be a 3-manifold, Γ = π ( M ), and ∂ Γ = π ( ∂M )).Let R (Γ , SL( n )) = Hom(Γ , SL( n )) be the variety of representations of Γinto SL( n ) and X (Γ , SL( n )) = R (Γ , SL( n )) // SL( n ) be the SL( n ) charactervariety, where the quotient is the GIT quotient as SL( n ) acts by conjugation.Suppose ρ : Γ → SL( n ) is a representation. The group of twisted cocycles Z (Γ; sl ( n ) ρ ) is defined as the set of maps z : Γ → sl ( n ) that satisfy thetwisted cocycle condition(3.1) z ( ab ) = z ( a ) + Ad ρ ( a ) z ( b ) , which can be interpreted as the derivative of the homomorphism conditionfor a smooth family of representation ρ t at ρ . The derivative of the triv-iality condition that ρ t is a smooth family of representations obtained byconjugating ρ gives the coboundary condition,(3.2) z ( γ ) = u − Ad ρ ( γ ) u, and B (Γ; sl ( n ) ρ ) is defined as the set of coboundaries, or the cocycles sat-isfying Equation (3.2). The quotient is defined to be H (Γ; sl ( n ) ρ ) = Z (Γ; sl ( n ) ρ ) /B (Γ; sl ( n ) ρ ) . EFORMATIONS OF REDUCIBLE SL( n, C ) REPRESENTATIONS 5 Weil [17, 10] has noted that Z (Γ; sl ( n ) ρ ) contains the tangent space to R (Γ , SL( n )) at ρ as a subspace. The following tools can be used to determineif the representation variety is smooth at ρ , so that we can study thespace of cocycles to determine the first order behavior of deformations of arepresentation ρ . In the following proposition, C (Γ; sl ( n )) denotes the setof cochains { c : Γ → sl ( n ) } . Proposition 3.1 ([6], Lemma 3.2; [7], Proposition 3.1) . Let ρ ∈ R (Γ , SL( n )) and u i ∈ C (Γ; sl ( n ) ρ ) , ≤ i ≤ j be given. If ρ j ( γ ) = exp( j X i =1 t i u i ( γ )) ρ ( γ ) is a homomorphism into SL( n, C [[ t ]]) modulo t j +1 , then there exists an ob-struction class ζ ( u ,...,u k ) j +1 ∈ H (Γ; sl ( n ) ρ ) such that: (1) There is a cochain u j +1 : Γ → sl ( n ) such that ρ j +1 ( γ ) = exp( j +1 X i =1 t i u i ( γ )) ρ ( γ ) is a homomorphism modulo t j +2 if and only if ζ j +1 = 0 . (2) The obstruction ζ j +1 is natural, i.e. if f is a homomorphism then f ∗ ρ j := ρ j ◦ f is also a homomorphism modulo t j +1 and f ∗ ( ζ ( u ,...,u j ) j +1 ) = ζ ( f ∗ u ,...,f ∗ u j ) j +1 . We will apply the previous proposition to the restriction map i ∗ on coho-mology, which is induced by the inclusion map i : ∂ Γ → Γ. As ∂M φ consistsof a disjoint union of tori, we will need to understand H ( π ( T ); sl ( n ) r n ◦ ρ ). Lemma 3.2.
Suppose ρ : π ( T ) → SL(2) contains a hyperbolic element inits image. Then dim H ( π ( T ); sl ( n ) r n ◦ ρ ) = 2( n − .Proof. Suppose γ ∈ π ( T ) such that ρ ( γ ) is a hyperbolic element in SL(2).Then, up to conjugation, ρ ( γ ) = (cid:18) a a − (cid:19) . The image of such an element under the irreducible representation r n :SL(2) → SL( n ) is conjugate to a diagonal matrix with n distinct eigenvalues.Hence, for any nearby representation ρ ′ : π ( T ) → SL( n ), ρ ′ ( γ ) is conjugateto a diagonal matrix with distinct entries. In other words, up to coboundary,we can assume that any class [ z ] ∈ H ( π ( T ); sl ( n ) r n ◦ ρ ) has the form of adiagonal matrix z ( γ ) = diag( y , y , . . . , y n ) where tr z ( γ ) = 0. Since for anyother γ ′ ∈ π ( T ), we have that γ ′ commutes with γ , z ( γ ′ ) must also bediagonal, so the dimension of H ( π ( T ); sl ( n ) r n ◦ ρ ) is 2( n − (cid:3) KENJI KOZAI
Lemma 3.3.
Let M be a 3-manifold with torus boundary components ∂M = ⊔ ki =1 T i . Let ρ : π ( M ) → SL(2) be a non-abelian representation such that ρ ( π ( T i )) contains a hyperbolic element for each component T i of ∂M . If dim H (Γ; sl (2) r n ◦ ρ ) = k ( n − where k is the number of components of ∂M ,then i ∗ : H ( M ; sl ( n ) r n ◦ ρ ) → H ( ∂M ; sl ( n ) r n ◦ ρ ) is injective.Proof. We have the cohomology exact sequence for the pair (
M, ∂M ) H ( M, ∂M ) −−−−→ H ( M ) α −−−−→ H ( ∂M ) β −−−−→ H ( M, ∂M ) −−−−→ H ( M ) i ∗ −−−−→ H ( ∂M ) −−−−→ H ( M, ∂M ) −−−−→ where all cohomology groups are taken to be with the twisted coefficients sl ( n ) r n ◦ ρ . A standard Poincar´e duality argument [7, 8, 14] gives that α hashalf-dimensional image. By Lemma 3.2,dim H ( π ( T i ); sl ( n ) r n ◦ ρ ) = 2( n − , as long as ρ ( π ( T i )) contains a hyperbolic element. Hence, α is injective.Since β is dual to α under Poincar´e duality, then β is surjective. This impliesthat i ∗ is injective. (cid:3) We now utilize the previous facts to determine sufficient conditions fordeforming representations.
Proposition 3.4.
Let M be a 3-manifold with torus boundary components ∂M = ⊔ ki =1 T i . Let ρ : Γ → SL(2) be a non-abelian representation such that ρ ( π ( T i )) contains a hyperbolic element for each component T i of ∂M . If H (Γ; sl (2) r n ◦ ρ ) = k ( n − where k is the number of components of ∂M ,then r n ◦ ρ is a smooth point of the representation variety R (Γ , SL( n )) , andit is contained in a unique component of dimension ( n + 1 + k )( n − − dim H (Γ; sl ( n ) r n ◦ ρ ) .Proof. We begin by showing that every cocyle in Z (Γ; sl ( n ) r n ◦ ρ ) is inte-grable.Suppose we have u , . . . , u j : Γ → sl ( n ) such that ρ jn ( γ ) = exp( j X i =1 t i u i ( γ )) ρ ( γ )is a homomorphism modulo t j +1 . By Lemma 3.2 and [15], the restriction of ρ n to π ( T i ) is a smooth point of the representation variety R ( π ( T i ) , SL( n )).Hence ρ jn | π ( T i ) extends to a formal deformation of order j + 1 by the for-mal implicit function theorem (see [7], Lemma 3.7). This implies that therestriction of ζ ( u ,...,u j ) j +1 to each component H ( T i ) < H ( ∂N φ ) vanishes.As H ( ∂N φ ) = ⊕ ki =1 H ( T i ), hence, i ∗ ζ ( u ,...,u j ) j +1 = ζ ( i ∗ u ,...,i ∗ u j ) j +1 = 0. Theinjectivity of i ∗ follows from Lemma 3.3 and implies that ζ ( u ,...,u j ) j +1 = 0. EFORMATIONS OF REDUCIBLE SL( n, C ) REPRESENTATIONS 7 Hence, the homomorphism can be extended to a deformation ( r n ◦ ρ ) j +1 oforder j + 1, and inductively to a formal deformation ( r n ◦ ρ ) ∞ .Applying [7, Proposition 3.6] to the formal deformation ( r n ◦ ρ ) ∞ resultsin a convergent deformation. Hence, r n ◦ ρ is a smooth point of the repre-sentation variety.As in [6], we note that the exactness of1 → H (Γ; sl ( n ) r n ◦ ρ ) → sl ( n ) r n ◦ ρ → B (Γ; sl ( n ) r n ◦ ρ )implies thatdim B (Γ; sl ( n ) r n ◦ ρ ) = n − − dim H (Γ; sl ( n ) r n ◦ ρ ) . Thus, we conclude that the local dimension of R (Γ , SL( n )) isdim Z (Γ; sl ( n ) r n ◦ ρ ) = ( n + 1 + k )( n − − dim H (Γ; sl ( n ) r n ◦ ρ ) . That it is in a unique component follows from [7, Lemma 2.6]. (cid:3) Deforming ρ λ,n We will now show that ρ λ,n satisfies the conditions in Proposition 3.4, sothat ρ λ,n can be deformed. This will entail a computation of the dimensionof the cohomology group H (Γ φ ; sl ( n ) ρ λ,n ).To simplify the computations which follow, we give a presentation ofΓ φ with an additional generator γ g + p . We will choose γ , . . . , γ g to bestandard generators of the fundamental group for the closed surface S g , and γ g +1 , . . . , γ g + p to be curves around the p punctures of S . Then π (Γ φ ) hasa presentation of the form: < γ , . . . , γ g + p , τ | τ γ i τ − = φ ( γ i ) , Π gi =1 [ γ i − , γ i ] = Π pj =1 γ g + j > . Up to a choice of generators for π ( S ), φ ∗ : H ( S ) → H ( S ) can be writtenas a block matrix (cid:18) [ ¯ φ ∗ ] [ ∗ ]0 [ P ] (cid:19) where ¯ φ ∗ : H ( ¯ S ) → H ( ¯ S ) is the induced map on the first cohomology ofthe closed surface ¯ S obtained by filling in the p punctures of S , and P = ( p ij )is a permutation matrix denoting the permutation of the punctures on S . Inparticular, p jk j = 1 if and only if τ δ j τ − is conjugate to δ k j , with p jk j = 0otherwise. We have that ¯ φ ∗ is a symplectic matrix preserving the intersectionform ω on ¯ S . The eigenvalues of P are roots of unity, with 1 occurring asan eigenvalue for each cycle in the permutation.The following inductive step is based on in [6, Lemma 4.4]. Along withLemma 2.5, it will allow us to compute the cohomological dimension forarbitrary n from the case when n = 2. Lemma 4.1.
Let λ ∈ C ∗ and n > . Suppose λ n − is not an eigenvalue of φ ∗ and λ n − = 1 . Then, H ∗ (Γ φ ; R n − ) ∼ = H ∗ (Γ φ ; R n − ) . KENJI KOZAI
Proof.
The short exact sequence in Equation (2.1) induces a long exactsequence [2, III.6], H k ( π (Γ φ ; C λ n − ) → H k (Γ φ ; R n − ) → H k (Γ φ ; ¯ R n − ) → H k +1 (Γ φ ; C λ n − ) , which is exact for k = 0 , , . Since λ n − = 1, 0 is the only point of C λ n − fixed by Γ φ . Consequently, H (Γ φ ; C λ n − ) = 0. By the universal coefficienttheorem, H (Γ φ ; C λ n − ) ∼ = Hom(Γ φ ; C ); C λ n − ) . As λ n − is not an eigenvalue of φ ∗ , it is also not an eigenvalue of φ ∗ , implyingthat H (Γ φ ; C λ n − ) = 0. Since M φ has non-empty boundary and Eulercharacteristic 0, then it must also follow that H (Γ φ ; C λ n − ) = 0. Thus, weconclude that H k (Γ φ ; R n − ) ∼ = H k (Γ φ ; ¯ R n − ) , for k = 0 , , H k (Γ φ ; R n − ) ∼ = H k (Γ φ ; ¯ R n − ) , for k = 0 , , λ − ( n − is an eigenvalue of φ ∗ if and only if λ n − is aneigenvalue. (cid:3) We now compute the cohomological dimension when n = 2. The argu-ment generalizes [9, Theorem 4.1]. Proposition 4.2.
Let φ : S → S be a homeomorphism, with λ a simpleeigenvalue of φ ∗ . Suppose also that | λ | 6 = 1 and ¯ φ ∗ : H ( ¯ S ) → H ( ¯ S ) doesnot have 1 as an eigenvalue. Then dim H (Γ φ , sl (2) ρ λ ) = k where k is thenumber of components of ∂M φ .Proof. Let z ∈ Z (Γ φ , sl (2) ρ λ ). Then z is determined by its values on γ , . . . , γ g + p , and τ , subject to the cocycle condition (3.1) imposed by therelations in Γ φ . These can be computed via the Fox calculus [10, Chapter3]. Differentiating the relations τ γ i τ − = φ ( γ i ) , yields ∂ [ φ ( γ i ) τ γ − i τ − ] ∂γ i = ∂φ ( γ i ) ∂γ i − φ ( γ i ) τ γ − i = ∂φ ( γ i ) ∂γ i − τ∂ [ φ ( γ i ) τ γ − i τ − ] ∂γ j = ∂φ ( γ i ) ∂γ j , i = j∂ [ φ ( γ i ) τ γ − i τ − ] ∂τ = φ ( γ i ) − φ ( γ i ) τ γ − i τ − = φ ( γ i ) − . (4.1) EFORMATIONS OF REDUCIBLE SL( n, C ) REPRESENTATIONS 9 Choosing the basis, e = (cid:18) (cid:19) , e = (cid:18) − (cid:19) , e = (cid:18) (cid:19) for sl (2), the values z ( γ i ) can be expressed in coordinates ( x i , y i , z i ), where z ( γ i ) is the matrix z ( γ i ) = (cid:18) y i x i z i − y i (cid:19) , and we similarly let z ( τ ) be given in the coordinates ( x , y , z ). The set ofcoboundaries can be computed from Equation (3.2), as the set of cocycle z ′ satisfying, z ′ ( γ i ) = (cid:18) − a i z a i y + a i z a i z (cid:19) z ′ ( τ ) = (cid:18) x − λxz − λ − z (cid:19) , where x, y, z ∈ C parametrize B (Γ φ , sl (2) ρ λ ). In particular, adding theappropriate coboundary z ′ to z , we can assume x = z = 0, so that z ( τ )has the form z ( τ ) = (cid:18) y − y (cid:19) . We first note that if W is a word in the γ i , then ρ ( W ) = (cid:0) A (cid:1) for somereal number A . Then, under the chosen basis for sl (2), ρ λ ( W ) acts by − A − A A . We obtain one term from ∂φ ( γ i ) ∂γ j for each instance of γ j in φ ( γ i ), and itsnegation for each instance of γ − j in φ ( γ i ).Similarly, we can compute that ρ λ ( τ ) acts on sl (2) via λ λ − . Then z is determined, as in [7], by a vector( x , . . . , x g + p , y , y , . . . , y g + p , z , . . . , z g + p ) T in the kernel of the matrix S = φ ∗ − λ I − λa ... − λa g + p K C φ ∗ − I D φ ∗ − λ − I . As λ is a simple eigenvalue, ¯ φ ∗ is symplectic, and the eigenvalues of P areroots of unity, φ ∗ − λ I and φ ∗ − λ − I have 1 dimensional kernel. Further-more, since 1 is not an eigenvalue of ¯ φ ∗ , φ ∗ − I has kernel whose dimension isequal to the number of disjoint cycles of the permutation of the punctures.This is equal to the number of components of ∂M φ . Hence, the kernel of S has dimension at most 2 + k + 1, where the additional dimension comesfrom the column vector − λ ( a , . . . , a g + p , , . . . , T , in S , and k = ∂M φ . Consider the upper left portion of the matrix S . U = φ ∗ − λ I − λa ... − λa g + n K φ ∗ − I . If null( S ) > k , then we must have that null( U ) > k + 1.Since λ is a simple eigenvalue of φ ∗ and ( a , . . . , a g ) T is an eigenvector ofthe λ eigenspace, ( a , . . . , a g ) T is not in the image of φ ∗ − λ I . Hence, forany y = ( y , . . . , y g + p ) T in the kernel of φ ∗ − I , there is a unique y such that Ky − y ( a , . . . , a g ) T is in the image of φ ∗ − λI . Therefore, null( U ) = k + 1Hence null( R ) = 2 + k . However, the solution arising from the kernel of φ ∗ − λ I is the eigenvector( a , . . . , a g + n , , . . . , , , . . . , T which is a coboundary. So we have that dim H (Γ φ ; sl (2) ρ λ ) ≤ k +1. Finally,there is one further redundancy sinceΠ gi =1 [ γ i − , γ i ] = Π pj =1 γ g + j . EFORMATIONS OF REDUCIBLE SL( n, C ) REPRESENTATIONS 11 From the φ ∗ − I block, we can see that y g +1 , . . . , y g + p can be freely chosenas long as y g + j = y g + k j whenever γ g + j and γ g + k j are in the same cycleof P . Hence, the upper-left entry of z (Π nj =1 γ g + j ) can be chosen to be anyquantity(4.2) y g +1 + y g +2 + . . . y g + p . The relation Π gi =1 [ γ i , γ i +1 ] = Π pj =1 γ g + j relates the sum in Equation (4.2)to the upper-left entry of Π gi =1 [ γ i , γ i +1 ], which has no dependence on y g + j ,for 1 ≤ j ≤ p . This imposes a 1-dimensional relation on the space of cocycles,and we conclude that dim H (Γ φ , sl (2) ρ λ ) = k. (cid:3) We will need one final technical lemma in order to show that the re-ducible representation is a limit of irreducible representations. Let ρ t bea smooth family of representations such that ρ = ρ λ,n . Since ρ λ,n ( τ ) isdiagonal with distinct eigenvalues, it follows that up to conjugation, ρ t isdiagonal for t sufficiently small. Thus, we can assume that A ( t ) = ρ t ( τ ) =diag( a ( t ) , a ( t ) , . . . , a nn ( t )), with A (0) = diag( λ n − , λ n − , . . . , λ − n +1 ). Let B ( t ) = ρ t ( γ i ) for some i such that a i = 0. Denote B ( t ) = ( b jl ( t )). We havethat B (0) = − a i (cid:0) (cid:1) ( − a i ) (cid:0) (cid:1) ( − a i ) (cid:0) (cid:1) · · · ( − a i ) n − (cid:0) n − n − (cid:1) − a i ) (cid:0) (cid:1) ( − a i ) (cid:0) (cid:1) · · · ...0 0 1 ( − a i ) (cid:0) (cid:1) · · · ...0 0 0 . . .0 0 0 0 · · · . The following gives a condition for irreducibility of the representation, andis similar to the argument in [1, Proposition 5.4].
Lemma 4.3.
Suppose A ( t ) and B ( t ) are matrices as defined above. Supposealso that b ( n − n = 0 and b ( k ) n = 0 for all < k < n − . Then for sufficientlysmall t = 0 , A ( t ) and B ( t ) generate the full matrix algebra M ( n, C ) .Proof. Consider the row vectors,(1 , , . . . , A (0) , (1 , , . . . , B (0) , (1 , , . . . , B (0) , . . . , (1 , , . . . , B n − (0) . We have thatdet λ n − . . . − a i ( − a i ) . . . ( − a i ) n − − a i ( − a i ) . . . ( − a i ) n − ... ...1 − ( n − a i ( − ( n − a i ) . . . ( − ( n − a i ) n − = λ n − det − a i ( − a i ) . . . ( − a i ) n − − a i ( − a i ) . . . ( − a i ) n − ... ... − ( n − a i ( − ( n − a i ) . . . ( − ( n − a i ) n − . Note that the second determinant is 0 if and only if there exist constants c , . . . , c n − , not all equal to 0, such that f ( x ) = c ( − a i ) x + c ( − a i x ) + · · · + c n − ( − a i x ) n − = 0 for x = 1 , . . . , n −
1. But we can also see that f (0) = 0, so that f ( x ) has n roots, so must be identically 0. Hence,it must be that (1 , , . . . , A (0), (1 , , . . . , B (0), (1 , , . . . , B (0), . . . ,(1 , , . . . , B n − (0) are linearly independent, so(1 , , . . . , A ( t ) , (1 , , . . . , B ( t ) , (1 , , . . . , B ( t ) , . . . , (1 , , . . . , B n − ( t )generate C n for sufficiently small t .Now let g ( t ) be the determinant of the matrix consisting of the columnvectors, a ( t ) = A ( t ) = a ( t )0...0 ,b j ( t ) = B ( t ) j = B ( t ) j − b ( t ) b ( t )... b n ( t ) , j = 1 , . . . , n − . Then, g ( k ) ( t ) = X k + k + ··· + k n = k k ! k ! k ! · · · k n ! det( a ( k ) ( t ) , b ( k )1 ( t ) , . . . , b ( k n − ) n − ( t )) . Since b ( k ) n (0) = 0 for k ≤ n −
1, we see that g ( k ) (0) = 0 for k < ( n − , and(( n − n − ( n − n + 1)! g ( n − n +1) = det( a (0) , b ( n − (0) , B (0) b ( n − (0) , . . . , B n − (0) b ( n − (0)) . EFORMATIONS OF REDUCIBLE SL( n, C ) REPRESENTATIONS 13 Noting that B (0) = I + N where N is a nilpotent matrix, we can see thatif b ( n − n = 0, then(( n − n − ( n − n + 1)! g ( n − n +1) (0)= det( a (0) , b ( n − (0) , N b ( n − (0) , . . . , N n − b ( n − (0)) = 0 . Hence, for sufficiently small t = 0, we have that A ( t ) and B ( t ) generate C n .Let P t ( x ) = ( x − a ( t ))( x − a ( t )) · · · ( x − a nn ( t )). Then, P t ( A ( t )) P t ( a ( t )) = · · ·
00 0 · · · · · · = ⊗ (1 , , . . . , . Since every rank one matrix can be written as v ⊗ w , and since for anymatrix M , we have that M ( v ⊗ w ) = M v ⊗ w = v ⊗ wM , it follows that A ( t )and B ( t ) generate all rank one matrices for sufficiently small t = 0. Everymatrix is a sum of rank one matrices, then we conclude that A ( t ) and B ( t )generate the full matrix algebra. (cid:3) We now prove Theorem 1.1.
Proof of Theorem 1.1.
By Lemma 2.5, sl ( n ) is the direct sum of R j , j =1 , . . . , n −
1. The conditions on the eigenvalues of φ ∗ and Lemma 4.1 implythat for each j , dim H (Γ φ ; R ) = dim H (Γ φ ; R ). By Proposition 4.2, weknow that dim H (Γ φ ; R ) = k , hence H (Γ φ , sl ( n ) ρ λ,n ) = k ( n − R (Γ φ , SL( n ) at ρ λ,n . Since ρ λ,n is non-abelian, it has trivial infinitesimal centralizer, so H (Γ φ ; R ) = 0, sothat the local dimension is ( n + 1 + k )( n − n = 2 case gives a path of representations ρ t where ρ = ρ λ and ρ t ( γ i ) = (cid:16) a ( t ) b ( t ) c ( t ) d ( t ) (cid:17) satisfies that c ′ (0) = 0, since i is chosen so that the i th coordinate of the eigenvector of φ ∗ corresponding to the eigenvalue λ − isnon-zero. A straightforward computation shows that B ( t ) = r n ◦ ρ t ( γ i ) hasits b n ( t ) coordinate equal to ( − c ( t )) n − , so that the hypotheses of Lemma4.3 are satisfied. By Burnside’s theorem on matrix algebras, it follows that r n ◦ ρ t is irreducible for sufficiently small t = 0. (cid:3) We note this result strengthens the conclusions of [6], where it was shownthat the image of an irreducible SL(2) representation under r n is genericallyirreducible as an SL( n ) representation. Theorem 1.1 shows that for suffi-ciently small t >
0, a path ρ t of irreducible representations limits to ρ λ,n .We obtain the special case in Theorem 1.2 when λ is the dilatation of apseudo-Anosov map φ . When the stable and unstable foliations of φ are Figure 1.
The curves α , α , β , β which form the basis for H ( S ), and γ .orientable, it is a well-known fact that the dilatation is a simple eigenvalueand the largest eigenvalue of φ ∗ (see [5], [11], [13]).The genus 2 example φ : S , → S , from [9], obtained from taking the leftDehn twists T β , T β , T γ , followed by the right Dehn twists T − α , T − α , satisfiesthe hypotheses of Theorem 1.1. Each component of S \ { α , β , α , β , γ } contains one of the two punctures. The map on cohomology ¯ φ ∗ has twosimple eigenvalues λ = √ and λ = √ , along with their recipro-cals λ − and λ − . The reducible representations ρ λ i ,n are smooth pointsof R (Γ φ , SL( n )), each on a component of dimension ( n + 3)( n − X (Γ φ , SL( n )),which is the image of a two-dimensional family of irreducible representationsin X (Γ φ , SL (2)) under r n , limiting to ρ λ i ,n .The proof of Theorem 1.1 guarantees that these are irreducible, how-ever, it is an interesting question whether there are families of irreduciblerepresentations that limit to ρ λ,n which are not the image of SL(2) repre-sentations. One can show that when n > b ′ n (0) = 0 for any family ofrepresentations ρ t near ρ λ,n . An explicit calculation of higher order deriva-tives of b n ( t ) is difficult, but it seems possible that there is a larger familyof irreducible representations limiting to ρ λ,n . References
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