A note on Reidemeister Torsion of G-Anosov Representations
aa r X i v : . [ m a t h . G T ] J a n A NOTE ON REIDEMEISTER TORSION OF G-ANOSOVREPRESENTATIONS
HAT˙ICE ZEYBEK AND YAS¸AR S ¨OZEN
Abstract.
This article considers G -Anosov representations of a fixed closedoriented Riemann surface Σ of genus at least 2. Here, G is the Lie groupPSp(2 n, R ), PSO( n, n ) or PSO( n, n + 1). It proves that Reidemeister torsion(R-torsion) associated to Σ with coefficients in the adjoint bundle represen-tations of such representations is well-defined. Moreover, by using symplecticchain complex method, it establishes a novel formula for R-torsion of suchrepresentations in terms of the Atiyah-Bott-Goldman symplectic form corre-sponding to the Lie group G . Furtermore, it applies the results to Hitchincomponents, in particular, Teichm¨uller space. Introduction
Throughout the paper, Σ is a closed Riemann surface of genus g ≥ . Teichm¨ullerspace
Teich(Σ) of Σ is the space of isotopy classes of complex structures on Σ . Itis a differentiable manifold and diffeomorphic to a ball of dimension 6 g − . By theUniformization Theorem, it can be interpreted as the isotopy classes of hyperbolicmetrics on Σ , i.e. Riemannian metrics of constant Gaussian curvature ( − . It canalso be interpreted as Hom df ( π (Σ) , PSL(2 , R )) discrete, faithful representations ofthe fundamental group π (Σ) of Σ to PSL(2 , R ) . This is a connected componentof the space Rep( π (Σ) , PSL(2 , R )) = Hom + ( π (Σ) , PSL(2 , R )) / PSL(2 , R ) of allreductive representations of π (Σ) to PSL(2 , R ) . In the paper [15], N. Hitchin proved the existence of an analogous component ofRep( π (Σ) , G ) , where G is a split real semi-simple Lie group, such as PSL( n, R ) , PSp(2 n, R ) , PO( n, n +1) , and PO( n, n ) . He called this component
Teichm¨uller com-ponent but now it is called
Hitchin component . He proved that Hitchin componentof Hom( π (Σ) , G ) /G is diffeomorphic to R (6 g −
6) dim G . He also paused the problemabout the geometric significance of this component.We already mentioned the geometric significance of the Hitchin component for G = PSL(2 , R ) . Namely, the hyperbolic structures on Σ . For G = PSL(3 , R ) , S.Choi and W.M. Goldman proved that the Hitchin component is diffeomorphic toconvex real projective structures on Σ [5]. F. Labourie introduced the notion ofAnosov representations in his investigation of Hitchin component by dynamicalsystem method [19], where he also proved that such representations are purelyloxodromic, discrete, faithful, and irreducible.The problem of giving a geometric interpretation of Hitchin components wascompletely solved by O. Guichard and A. Wienhard [14]. To be more precise,they proved that the Hitchin component of Hom( π (Σ) , G ) /G parametrizes the Mathematics Subject Classification.
Primary 32G15; Secondary 57R30.
Key words and phrases.
Reidemeister torsion, symplectic chain complex, G-Anosov, Hitchinrepresentations, Riemann surfaces. deformation space of (
G, X ) − structures on a compact manifold M. Here, if X = RP n − then G denotes PSL(2 n, R ) , PSp(2 n, R ) when n ≥
2, PSO( n, n ) when n ≥ X = F , n ( R n +1 ) = { ( D, H ) ∈ RP n × (cid:0) RP n (cid:1) ∗ ; D ⊂ H } then G denotesPSL(2 n + 1 , R ) when n ≥ , PSO( n, n + 1) when n ≥ G -Anosov representations where the Lie group G belongs to { PSp(2 n, R ), PSO( n, n ) , PSO( n, n + 1) } . We showed that the Rei-demeister torsion of such representations is well-defined (Proposition 4.1). Fur-thermore, with the help of symplectic chain complex method, we establish a novelformula for R-torsion of such representation in terms of the Atiyah-Bott-Goldmansymplectic form corresponding to the Lie group G (Theorem 4.3).2. The Reidemeister Torsion
For more information and the detailed proofs, we refer the reader to [23, 27, 33,34, 36], and the references therein.Suppose C ∗ = ( C n ∂ n → C n − → · · · → C ∂ → C →
0) is a chain complex ofa finite dimensional vector spaces over the field R of real numbers. Let H p ( C ∗ )= Z p ( C ∗ ) /B p ( C ∗ ) denote the p − th homology group of C ∗ , p = 0 , . . . , n, where B p ( C ∗ ) = Im ∂ p +1 and Z p ( C ∗ ) = Ker ∂ p . Assume that c p , b p , and h p are bases of C p , B p ( C ∗ ) , and H p ( C ∗ ) , respec-tively, and that ℓ p : H p ( C ∗ ) → Z p ( C ∗ ) , s p : B p − ( C ∗ ) → C p are sections of Z p ( C ∗ ) → H p ( C ∗ ) , C p → B p − ( C ∗ ) , respectively, p = 0 , . . . , n. The definitionof Z p ( C ∗ ) , B p ( C ∗ ) , and H p ( C ∗ ) result the following short-exact sequences:0 → Z p ( C ∗ ) ֒ → C p ։ B p − ( C ∗ ) → , (2.1)0 → B p ( C ∗ ) ֒ → Z p ( C ∗ ) ։ H p ( C ∗ ) → . (2.2)These short-exact sequences yield a new basis b p ⊔ ℓ p ( h p ) ⊔ s p ( b p − ) of C p . The
Reidemeister torsion of C ∗ with respect to bases { c p } np =0 , { h p } np =0 is definedby T ( C ∗ , { c p } n , { h p } n ) = n Y p =0 [ b p ⊔ ℓ p ( h p ) ⊔ s p ( b p − ) , c p ] ( − ( p +1) , where [ e p , f p ] is the determinant of the change-base-matrix from f p to e p . The Reidemeister torsion T ( C ∗ , { c p } n , { h p } n ) is independent of the bases b p , sections s p , ℓ p [21]. If c ′ p , h ′ p are also bases respectively for C p , H p ( C ∗ ) , then aneasy computation results the following change-base-formula: T ( C ∗ , { c ′ p } n , { h ′ p } n ) = n Y p =0 (cid:18) [ c ′ p , c p ][ h ′ p , h p ] (cid:19) ( − p T ( C ∗ , { c p } n , { h p } n ) . (2.3)If 0 → A ∗ ı → B ∗ π → D ∗ → n + 2 H ∗ : · · · → H p ( A ∗ ) ı p → H p ( B ∗ ) π p → H p ( D ∗ ) δ p → H p − ( A ∗ ) → · · · , (2.5)where H p = H p ( D ∗ ) , H p +1 = H p ( A ∗ ) , and H p +2 = H p ( B ∗ ) . NOTE ON REIDEMEISTER TORSION OF G-ANOSOV REPRESENTATIONS 3
The bases h Dp , h Ap , and h Bp are clearly bases for H p , H p +1 , and H p +2 , respec-tively. Considering sequences (2.4) and (2.5), we have the following result of J.Milnor: Theorem 2.1. ( [21] ) Let c Ap , c Bp , c Dp , h Ap , h Bp , and h Dp be bases of A p , B p , D p ,H p ( A ∗ ) , H p ( B ∗ ) , and H p ( D ∗ ) , respectively. Let c Ap , c Bp , and c Dp be compatible inthe sense that [ c Bp , c Ap ⊕ f c Dp ] = ± , where π (cid:16) f c Dp (cid:17) = c Dp . Then, T ( B ∗ , { c Bp } n , { h Bp } n ) = T ( A ∗ , { c Ap } n , { h Ap } n ) T ( D ∗ , { c Dp } np =0 , { h Dp } n ) × T ( H ∗ , { c p } n +20 , { } n +20 ) . ✷ Considering the short exact sequence0 → A ∗ ı → A ∗ ⊕ D ∗ π → D ∗ → , where for p = 0 , . . . , n, ı p : A p → A p ⊕ D p denotes the inclusion, π p : A p ⊕ D p → D p denotes the projection, and the compatible bases c Ap , c Ap ⊕ c Dp , and c Dp , where weconsider the inclusion as a section of π p : A p ⊕ D p → D p , then by Theorem 2.1 weget: Lemma 2.2. ( [30] ) If A ∗ , D ∗ are two chain complexes, c Ap , c Dp , h Ap , and h Dp arebases of A p , D p , H p ( A ∗ ) , and H p ( D ∗ ) , respectively, then T ( A ∗ ⊕ D ∗ , { c Ap ⊕ c Dp } n , { h Ap ⊕ h Dp } n ) = T ( A ∗ , { c Ap } n , { h Ap } n ) T ( D ∗ , { c Dp } n , { h Dp } n ) . ✷ Note that one can split a general chain complex as a direct sum of an exactand a ∂ − zero chain complexes. Moreover, Reidemeister torsion T ( C ∗ ) of a generalcomplex C ∗ is as an element of ⊗ np =0 (det( H p ( C ∗ ))) ( − p +1 , where det( H p ( C ∗ )) = V dim R H p ( C ∗ ) H p ( C ∗ ) denotes the top exterior power of H p ( C ∗ ) , and det( H p ( C ∗ )) − is the dual of det( H p ( C ∗ )) . We refer the reader [27, 36] for more information andthe detailed proofs.A symplectic chain complex is a chain complex of finite dimensional real vectorspaces C ∗ : 0 → C n ∂ n → C n − → · · · → C n → · · · → C ∂ → C → n ( n odd) together with for each p = 0 , . . . , n, a ∂ − compatible anti-symmetricnon-degenerate bilinear form ω p, n − p : C p × C n − p → R . Namely, ω p, n − p ( ∂a, b ) = ( − p +1 ω p +1 , n − ( p +1) ( a, ∂b ) ,ω p, n − p ( a, b ) = ( − p ω n − p,p ( b, a ) . Clearly, we have anti-symmetric and non-degenerate bilinear map [ ω p,n − p ] : H p ( C ∗ ) × H n − p ( C ∗ ) → R defined by [ ω p,n − p ]([ x ] , [ y ]) = ω p,n − p ( x, y ) . Suppose C ∗ is a real-symplectic chain complex of length 2 n, and that c p is abasis of C p , p = 0 , . . . , n. We say that the bases c p of C p and c n − p of C n − p are ω − compatible , if the matrix of ω p, n − p in bases c p , c n − p is the k × k identitymatrix I k × k when p = n and (cid:18) l × l I l × l − I l × l l × l (cid:19) when p = n, where k = dim R C p =dim R C n − p and 2 l = dim R C n . Let us introduce the following notation used throughout the paper. Let C ∗ be areal-symplectic chain complex and let h Cp , h C n − p be bases of H p ( C ∗ ) , H n − p ( C ∗ ) , H. ZEYBEK AND Y. S ¨OZEN respectively. Then, ∆ p, n − p ( C ∗ ) denotes the determinant of the matrix of the non-degenerate pairing [ ω p, n − p ] : H p ( C ∗ ) × H n − p ( C ∗ ) → R in bases h Cp , h C n − p . By an easy linear algebra argument, we have:
Lemma 2.3. ( [31] ) For a symplectic chain complex, there exist ω − compatible bases. ✷ Theorem 2.4. ( [27] ) Suppose C ∗ is a symplectic chain complex of length n, and c p , h Cp are bases of C p , H p ( C ∗ ) , respectively, p = 0 , . . . , n. Then, the followingformula is valid: T (cid:0) C ∗ , { c p } np =0 , { h Cp } np =0 (cid:1) = n − Y p =0 ∆ p, n − p ( C ∗ ) ( − p · q ∆ n,n ( C ∗ ) ( − n . ✷ We refer the reader to [27] for detailed proof and unexplained subjects. See also[28, 29, 30, 31] for further applications of Theorem 2.4.3.
Anosov Representations
Let Σ be a closed oriented Riemann surface of genus at least 2, h be a hyperbolicmetric on Σ, M = U T (Σ) be its unit tangent bundle of Σ and g t be the geodesicflow for the hyperbolic metric h . Since the geodesic flow on the unit tangent bundleof a negatively curved manifold is Anosov [16], g t is an Anosov flow. To be moreprecise, there is a g t -invariant splitting of the tangent bundle T Σ = E s ⊕ E u ⊕ E t . Here, • E t is a line bundle, which is tangent to the flow g t , • E u is expanding, namely, there are constant A > , α > t ∈ R and v ∈ E u , k D g t ( v ) k ≥ Aα t k v k , • E s is contracting, in other words, there are constants B > , ≤ β < t ∈ R and v ∈ E s , k D g t ( v ) k ≤ Bβ t k v k . Let e Σ be the universal covering of Σ, c M = U T ( e Σ) be the π (Σ)-cover of M andlet us also denote by g t the geodesic flow on c M .For a semi-simple Lie group G, let ( P + , P − ) be a pair of opposite parabolicsubgroups of G. We denote respectively the quotient spaces G/P + , G/P − , and G/L by F + , F − , and X , where L = P + ∩ P − . Note that considering the diagonal actionof G on F + × F − , X is the unique open G -orbit. This product structure inducestwo G-invariant distributions E + and E − on X . To be more precise, E + x = T x + F + and E − x = T x − F − , where x = ( x + , x − ) ∈ X ⊂ F + × F − . Clearly, any X -bundlecan be equipped with two distributions which will be denoted by the same letters E + and E − .Let ̺ : π (Σ) → G be a representation. By the diagonal action of π (Σ) , c M × X is a π (Σ)-space. Here, π (Σ) acts on c M as the deck transformation and the actionof π (Σ) on X by conjugation, via the representations ̺ . Thus, under this action X ̺ := c M × ̺ X = ( c M × X ) /π (Σ) . NOTE ON REIDEMEISTER TORSION OF G-ANOSOV REPRESENTATIONS 5
Note that the projection of c M × X onto c M descends to a map X ̺ → M , whichgives c M × ̺ X the structure of a flat X -bundle over M .Clearly, the geodesic flow g t can be lifted to a flow on c M × X by defining G t ( m, x ) := ( g t m, x ) . Let us also note that the resulting flow is invariant under the π (Σ) action.Therefore, it defines a flow on X ̺ , which we denote by the same symbol G t liftinggeodesic flow g t .We say that a representation π (Σ) → G is a ( P + , P − ) − Anosov, if the followingtwo conditions are satisfied:1) there is a section σ : M → X ̺ of the flat bundle X ̺ which is flat along theflow lines, namely, the restriction of σ to any geodesic leaf is flat,2) the lifted action of the geodesic flow g t on σ ∗ E + , σ ∗ E − is respectively expand-ing, contracting. More precisely, for some continuous family of norms on the fibersof the σ ∗ E + , σ ∗ E − bundles, the expanding-contracting properties are fulfilled.Let Rep Anosov ( π (Σ) , G ) be the set of all Anosov representations. It was provedin [19] by F. Labourie that Rep Anosov ( π (Σ) , G ) is open in Rep( π (Σ) , G ) . He alsoproved that every such representation 1-1, discerete, irreducible, and purely loxo-dromic. 4.
Main Theorems
Let Σ be a closed oriented Riemann surface with genus g ≥ , e Σ be the universalcovering of Σ . Let G ∈ { PSp(2 n, R )( n ≥ , PSO( n, n +1)( n ≥ , PSO( n, n +1)( n ≥ } and G be the corresponding Lie algebra with the non-degenerate Killing form B. For a representation ̺ : π (Σ) → G, consider the associated adjoint bundle E ̺ = e Σ × G / ∼ over Σ . Here, for all γ ∈ π ( S ) , ( γ · x, γ · t ) ∼ ( x, t ) , the action of γ in the first component as a deck transformation and in the second component asconjugation by ̺ ( γ ) . Suppose that K is a cell-decomposition of Σ so that the adjoint bundle E ̺ overΣ is trivial over each cell. Let e K be the lift of K to the universal covering e Σ ofΣ . Let Z [ π (Σ)] = { P pi =1 m i γ i ; m i ∈ Z , γ i ∈ π (Σ) , p ∈ N } be the integral groupring. Let C ∗ ( K ; G Ad ◦ ̺ ) = C ∗ ( e K ; Z ) ⊗ ̺ G ) = C ∗ ( e K ; Z ) ⊗ G / ∼ , where σ ⊗ t and allthe elements in orbit { γ · σ ⊗ γ · t ; γ ∈ π (Σ) } are identified, and where π (Σ) actson e Σ by the deck transformation and the action of π (Σ) on G is by conjugation.We have0 → C ( K ; G Ad ◦ ̺ ) ∂ ⊗ id −→ C ( K ; G Ad ◦ ̺ ) ∂ ⊗ id −→ C ( K ; G Ad ◦ ̺ ) → . Here, ∂ p is the usual boundary operator. Let H ∗ ( K ; G Ad ◦ ̺ ) denote the homologiesof the above chain complex. The cochains C ∗ ( K ; G Ad ◦ ̺ ) yield that H ∗ ( K ; G Ad ◦ ̺ ) . Here, C ∗ ( K ; G Ad ◦ ̺ ) denotes the set of Z [ π (Σ)]-module homomorphisms from C ∗ ( e K ; Z )to G . For more information, we refer the reader to [23, 27, 36], and the referencestherein.We say that ̺ : π (Σ) → G is purely loxodromic , if for every non-trivial γ ∈ π (Σ) , the eigenvalues of ̺ ( γ ) are real with multiplicity 1 . H. ZEYBEK AND Y. S ¨OZEN
Suppose that ̺ : π (Σ) → G is purely loxodromic. Consider chain complex0 → C ( K ; G Ad ◦ ̺ ) ∂ ⊗ id −→ C ( K ; G Ad ◦ ̺ ) ∂ ⊗ id −→ C ( K ; G Ad ◦ ̺ ) → . Let e pj be the p − cells of K which gives us a Z − basis for C p ( K ; Z ) . Fix a lift e e pj of e pj , j = 1 , . . . , m p . Then, c p = { e e pj } m p j =1 is a Z [ π (Σ)] − basis for C p ( e K ; Z ) . Let A = { a k } dim G k =1 be an R − basis of the semisimple Lie algebra G so that the matrix ofthe Killing form B is the diagonal matrix Diag ( p , . . . , , r − , . . . , − , where p + r =dim G . Such a basis is called B − orthonormal basis . Then, c p = c p ⊗ ̺ A is an R − basis for C p ( K ; G Ad ◦ ̺ ) and called a geometric basis for C p ( K ; G Ad ◦ ̺ ) . Let us assume that h p is an R -basis for H p ( K ; G Ad ◦ ̺ ) then T ( C ∗ ( K ; G Ad ◦ ̺ ) , { c p ⊗ ̺ A} p =0 , { h p } p =0 ) is called the Reidemeister torsion of the triple K, Ad ◦ ̺, and { h p } p =0 . The independence of the Reidemeister torsion of A , lifts e e pj , conjugacy class of ̺, and of the cell-decomposition follows by similar arguments given in [21], [27, Lemma1.4.2., Lemma 2.0.5.]. For the sake of completeness, the independence of A , lifts e e pj , and conjugacy class of ̺ will be explained below and for the independence ofthe cell-decomposition, the reader is referred to [27, Lemma 2.0.5.]. Proposition 4.1. T ( C ∗ ( K ; G Ad ◦ ̺ ) , { c p ⊗ ̺ A} p =0 , { h p } p =0 ) is independent of A , lifts e e pj , conjugacy class of ̺, and the cell-decomposition K. Proof.
Let A ′ be another B − orthonormal basis of G . From change-base-formula(2.3) of Reidemeister torsion it follows that T ( C ∗ ( K ; G Ad ◦ ̺ ) , { c ′ p } p =0 , { h p } p =0 ) T ( C ∗ ( K ; G Ad ◦ ̺ ) , { c p } p =0 , { h p } p =0 ) = det( T ) − χ (Σ) . Here, c ′ p = c p ⊗ ̺ A ′ , T is the change-base-matrix from A ′ to A , and χ is the Eulercharacteristic.Note that since A , A ′ are B − orthonormal bases of G , det T is ± . The indepen-dence of the Reidemeister torsion from B − orthonormal basis A follows from thefact that the Euler-characteristic χ (Σ) of Σ is even.Next, let us fix γ ∈ π (Σ) . Assume c ′ p = { e e p · γ, e e p , . . . , e e pm p } is also a lift of { e p , . . . , e pm p } , where only another lift of e p is considered and the others are keptthe same. From the tensor product property it follows that e e p · γ ⊗ t = e e p ⊗ Ad ̺ ( γ ) ( t ) . By change-base-formula (2.3), we have T ( C ∗ ( K ; G Ad ◦ ̺ ) , { c ′ p } p =0 , { h p } p =0 ) T ( C ∗ ( K ; G Ad ◦ ̺ ) , { c p } p =0 , { h p } p =0 ) = det( A ) . Here, c p = c p ⊗ ̺ A , c ′ p = c ′ p ⊗ ̺ A , and A denotes the matrix of Ad ̺ ( γ ) : G → G with respect to basis A .To compute the determinant of the matrix of Ad ̺ ( γ ) , let us consider the basis B sp n ( R ) = E ii − E n + i,n + i , ≤ i ≤ n,E ij − E n + j,n + i , ≤ i = j ≤ n,E i,n + i , ≤ i ≤ n,E n + i,i , ≤ i ≤ n,E i,n + j + E j,n + i , ≤ i < j ≤ n,E n + i,j + E n + j,i , ≤ i < j ≤ n NOTE ON REIDEMEISTER TORSION OF G-ANOSOV REPRESENTATIONS 7 B so n,n ( R ) = E ij − E n + j,n + i , ≤ i = j ≤ n,E ii − E n + i,n + i , ≤ i ≤ n,E i,n + j − E j,n + i , ≤ i < j ≤ n,E n + i,j − E n + j,i , ≤ i < j ≤ n, and B so n,n +1 ( R ) = E ii − E n + i,n + i , ≤ i ≤ n + 1 ,E ,n + i +1 − E i +1 , , ≤ i ≤ n,E ,i +1 − E n + i +1 , , ≤ i ≤ n,E i +1 ,j +1 − E n + j +1 ,n + i +1 , ≤ i = j ≤ n,E i +1 ,n + j +1 − E j +1 ,n + i +1 , ≤ i = j ≤ n,E i + n +1 ,j +1 − E j + n +1 ,i +1 , ≤ j = i ≤ n of sp n ( R ) , so n,n ( R ) , and so n,n +1 ( R ) , respectively. Here, E ij denotes the matrixwith 1 in the ij entry and 0 elsewhere.By the assumption that ̺ is purely loxodromic, we have for each γ ∈ π (Σ) thereis Q = Q ( γ ) ∈ G such that Q̺ ( γ ) Q − = D = Diag( λ , . . . , λ m ) . Here, m is equalto 2 n for G ∈ { PSp(2 n, R ) , PSO( n, n ) } , and for G = PSO( n, n + 1) , m = 2 n + 1 . Note that by the spectral properties of such diagonalizable matrices we have D = Diag( λ , . . . , λ n , /λ , . . . , /λ n ) for G ∈ { PSp(2 n, R ) , PSO( n, n ) , } and D =Diag(1 , λ , . . . , λ n , /λ , . . . , /λ n ) for G = PSO( n, n + 1) . Note also that DE ij D − = λ i λ j E ij . From this it follows that the matrix of Ad D in the basis B sp n ( R ) is the diagonalmatrix with the diagonal entries , ≤ i ≤ n λ i λ j , ≤ i = j ≤ nλ i , ≤ i ≤ n λ i , ≤ i ≤ nλ i λ j , ≤ i < j ≤ n λ i , ≤ i ≤ n λ i λ j , ≤ i < j ≤ n, in the basis B so n,n ( R ) is the diagonal matrix with the diagonal entries λ i λ j , ≤ i = j ≤ n , ≤ i ≤ nλ i λ j , ≤ i < j ≤ n λ i λ j , ≤ i < j ≤ n, in the basis B so n,n +1 ( R ) is the diagonal matrix with the diagonal entries , ≤ i ≤ n + 1 λ i +1 , ≤ i ≤ n /λ i +1 , ≤ i ≤ n λ i +1 λ j +1 , ≤ i = j ≤ nλ i +1 λ j +1 , ≤ i < j ≤ n λ i +1 λ j +1 , ≤ j < i ≤ n. Note that the determinant of these diagonal matrices is 1. Hence, we proved theindependence of the Reidemeister torsion from the lifts.
H. ZEYBEK AND Y. S ¨OZEN
By the fact that the twisted chains and cochains for conjugate representations areisomorphic, we also have the independence of Reidemeister torsion from conjugacyclass of ̺. This is the end of proof Proposition 4.1. (cid:3)
Let us continue with the following well known result which will be used in theproof of our main theorem (Theorem 4.3). For the sake of completeness, we willalso give the proof of this auxilary result in details.
Lemma 4.2.
Let f : V × V → R be a non-degenarate, anti-symmetric bilinearmap on the real vector space V of dimension n. Let { v , . . . , v n } be a basis of V and let { v , . . . , v n } be the corresponding dual basis, namely v i ( v j ) = δ ij . Let f ∗ : V ∗ × V ∗ → R be the dual bilinear map of f, which is defined by f ∗ ( v i , v j ) := f ( v i , v j ) . If G ( f ; { v , . . . , v n } ) denotes the Gram matrix of f in the basis { v , . . . , v n } , then G ( f ∗ ; { v , . . . , v n } ) G ( f ; { v , . . . , v n } ) T = I n × n . Here, I n × n is the n × n identitymatrix and “ T ” denotes the transpose of a matrix.Proof. Let us first note that there is a symplectic basis { e , . . . , e n } of V so that theGram matrix G ( f ; { e , . . . , e n , e n +1 , . . . , e n } ) and G ( f ∗ ; { e , . . . , e n , e n +1 , . . . , e n } )are both equal to (cid:18) n × n I n × n − I n × n n × n (cid:19) . Thus, we have G ( f ∗ ; { e , . . . , e n , e n +1 , . . . , e n } ) G ( f ; { e , . . . , e n , e n +1 , . . . , e n } ) T = I n × n . If L is the change-base matrix from basis { e , . . . , e n } to { v , . . . , v n } of V andif M is the change-base matrix from basis { e , . . . , e n } to { v , . . . , v n } of V ∗ , thenclearly we have LM T = I n × n . Note also that G ( f ; { v , . . . , v n , v n +1 , . . . , v n } ) = L T G ( f ; { e , . . . , e n , e n +1 , . . . , e n } ) L,G ( f ∗ ; { v , . . . , v n , v n +1 , . . . , v n } ) = M T G ( f ∗ ; { e , . . . , e n , e n +1 , . . . , e n } ) M. From these it follows that G ( f ∗ ; { v , . . . , v n } ) G ( f ; { v , . . . , v n } ) T = I n × n . This finishes the proof of Lemma 4.2. (cid:3)
Theorem 4.3.
Assume that Σ is a closed orientable surface of genus g ≥ and ̺ : π (Σ) → G is an irreducible, purely loxodromic representation. Assume also K is a cell-decomposition of Σ , c p is the geometric bases of C p ( K ; G Ad ◦ ̺ ) , p = 0 , , , and h is a basis for H (Σ; G Ad ◦ ̺ ) . Then, the following formula is valid: T ( C ∗ ( K ; G Ad ◦ ̺ ) , { c p } p =0 , { , h , } ) = p det Ω ω B , where ω B : H (Σ; G Ad ◦ ̺ ) × H (Σ; G Ad ◦ ̺ ) ⌣ B −→ H (Σ; R ) R Σ −→ R is the Atiyah-Bott-Goldman symplectic form for the Lie group G, Ω ω B is the matrix of ω B in the basis h , and where h is the Poincar´e dual basis of H (Σ; G Ad ◦ ̺ ) corresponding to h of H (Σ; G Ad ◦ ̺ ) . Proof.
By the invariance of the Cartan-Killing form B of G under conjugation, for k = 0 , , , we have the non-degenerate form, the Kronecker pairing, < · , · > : C k ( K ; G Ad ◦ ̺ ) × C k ( K ; G Ad ◦ ̺ ) → R defined by B ( t, θ ( σ )) , θ ∈ C k ( K ; G Ad ◦ ̺ ) , σ ⊗ ̺ t ∈ C k ( K ; G Ad ◦ ̺ ) . It is extended to < · , · > : H k (Σ; G Ad ◦ ̺ ) × H k (Σ; G Ad ◦ ̺ ) → R . NOTE ON REIDEMEISTER TORSION OF G-ANOSOV REPRESENTATIONS 9
Invariance of B under the conjugation and the non-degeneracy of B yield thecup product ⌣ B : C k ( K ; G Ad ◦ ̺ ) × C ℓ ( K ; G Ad ◦ ̺ ) → C k + ℓ ( K ; R )defined by ( θ k ⌣ B θ ℓ )( σ k + ℓ ) = B ( θ k (( σ k + ℓ ) front )) , θ ℓ (( σ k + ℓ ) back ) . Clearly, ⌣ B hasthe extension ⌣ B : H k (Σ; G Ad ◦ ̺ ) × H ℓ (Σ; G Ad ◦ ̺ ) → H k + ℓ (Σ; R ) . Let us denote by K ′ the dual cell-decomposition of Σ corresponding to the celldecomposition K. Suppose that cells σ ∈ K, σ ′ ∈ K ′ meet at most once and alsothe diameter of each cell is less than, say, half of the injectivity radius of Σ . By thefact that the Reidemeister torsion is invariant under subdivision, this assumptionis not loss of generality. Let c ′ p be the basis of C p ( f K ′ ; Z ) corresponding to the basis c p of C p ( e K ; Z ) , and let c ′ p = c ′ p ⊗ ̺ A be the corresponding basis for C p ( K ′ ; G Ad ◦ ̺ ) . We have the intersection form( · , · ) k, − k : C k ( K ; G Ad ◦ ̺ ) × C − k ( K ′ ; G Ad ◦ ̺ ) → R (4.1)defined by ( σ ⊗ t , σ ⊗ t ) k, − k = X γ ∈ π (Σ) σ . ( γ · σ ) B ( t , γ · t ) , where “ . ” is the intersection number pairing. Note that ( · , · ) k, − k are ∂ − compatiblebecause the intersection number pairing “ . ” is compatible with the usual boundaryoperator in the sense ( ∂α ) .β = ( − | α | α. ( ∂β ) , where | α | is the dimension of thecell α. Since the intersection number form “ . ” is anti-symmetric and B is invariantunder adjoint action, then ( · , · ) k, − k is anti-symmetric.By the independence of the twisted homologies from the cell-decomposition, weget the non-degenerate anti-symmetric form( · , · ) k, − k : H k (Σ; G Ad ◦ ̺ ) × H − k (Σ; G Ad ◦ ̺ ) → R . (4.2)Combining the isomorphisms induced by the Kronecker pairing and the intersectionform, we obtain the Poincar´e duality isomorphismsPD : H k (Σ; G Ad ◦ ̺ ) ∼ = H − k (Σ; G Ad ◦ ̺ ) ∗ ∼ = H − k (Σ; G Ad ◦ ̺ ) . Thus, for k = 0 , , , we have the following commutative diagram H − k (Σ; G Ad ◦ ̺ ) × H k (Σ; G Ad ◦ ̺ ) ⌣ B −→ H (Σ; R ) x PD x PD (cid:9) x H k (Σ; G Ad ◦ ̺ ) × H − k (Σ; G Ad ◦ ̺ ) ( , ) k, − k −→ R . Here, the isomorphism R → H (Σ; R ) sends 1 to the fundamental class of H (Σ; R ) . By the irreducibility of ̺, we have H (Σ; G Ad ◦ ̺ ) , H (Σ; G Ad ◦ ̺ ) , H (Σ; G Ad ◦ ̺ ) , and H (Σ; G Ad ◦ ̺ ) are all zero. Hence, H (Σ; G Ad ◦ ̺ ) × H (Σ; G Ad ◦ ̺ ) ⌣ B −→ H (Σ; R ) x PD x PD (cid:9) x H (Σ; G Ad ◦ ̺ ) × H (Σ; G Ad ◦ ̺ ) ( , ) , −→ R . (4.3)Recall that ω B : H (Σ; G Ad ◦ ̺ ) × H (Σ; G Ad ◦ ̺ ) ⌣ B −→ H (Σ; R ) R Σ −→ R is called theAtiyah-Bott-Goldman symplectic form for the Lie group G. Note also that from equation (4.3), ω B is nothing but the dual of the intersection pairing ( , ) , . Let C p = C p ( K ; G Ad ◦ ̺ ) , C ′ p = C p ( K ′ ; G Ad ◦ ̺ ) , and D p = C ∗ ⊕ C ′∗ . Considerthe intersection form (4.1), define it on C p × C − p and C ′ p × C ′ − p as 0 . Let ω p, − p : D p × D − p → R be defined by using ( · , · ) p, − p . Then, D ∗ becomes asymplectic chain complex. Note that ω , : H ( D ∗ ) × H ( D ∗ ) → R is equal to (cid:18) · , · ) , − ( · , · ) , (cid:19) and ( · , · ) , is the intersection form (4.2) for k = 1 . Then,from Lemma 2.2, Theorem 2.4, independence of the Reidemeister torsion from thecell-decomposition of Σ , and the fact that D ∗ is a symplectic chain complex itfollows that T ( D ∗ , { c p ⊕ c ′ p } p =0 , { ⊕ , h ⊕ h , ⊕ } ) = q ∆ , ( D ∗ ) ( − . (4.4)Since the intersection form (4.2) for k = 1 is non-degenerate, then equation (4.4)becomes T ( D ∗ , { c p ⊕ c ′ p } p =0 , { ⊕ , h ⊕ h , ⊕ } ) = ∆ , ( C ∗ ) ( − . (4.5)Let us consider the short-exact sequence0 → C ∗ ֒ → D ∗ = C ∗ ⊕ C ′∗ ։ C ′∗ → . Here, C ∗ ֒ → D ∗ denotes the inclusion, D ∗ ։ C ′∗ denotes the projection. Clearly,the bases c p of C p , c p ⊕ c ′ p of D ∗ , and c ′ p of C ′∗ are compatible. By Lemma 2.2 andthe independence of the Reidemeister torsion from the cell-decomposition of Σ , wehave T ( D ∗ , { c p ⊕ c ′ p } p =0 , { ⊕ , h ⊕ h , ⊕ } ) = (cid:0) T ( C ∗ , { c p } p =0 , { , h , } ) (cid:1) . (4.6)Thus, combining equations (4.5) and (4.6), we obtain T ( C ∗ , { c p } p =0 , { , h , } ) = q ∆ , ( C ∗ ) ( − . (4.7)The fact that ⌣ B is the dual of the intersection pairing ( , ) , and Lemma 4.2yield that T ( C ∗ , { c p } p =0 , { , h , } ) = p det Ω ω B . (4.8)This concludes the proof of Theorem 4.3. (cid:3) Corollary 4.4.
Since every Anosov representation is 1-1, discerete, irreducible,and purely loxodromic [19] , then Theorem 4.3 also holds for Anosov representations. Application:A Volume element on some Hitchin components
For a closed oriented Riemann surface Σ with genus g > G, let us denote by Hom( π (Σ) , G ) the set of all homomorphisms from thefundamental group π (Σ) of Σ to G. Let us consider the orbit space Hom( π (Σ) , G ) /G, where the action of G onHom( π (Σ) , G ) by conjugation i.e. g · ̺ ( γ ) = g̺ ( γ ) g − , for g ∈ G, ̺ ∈ Hom( π (Σ) , G ) , and γ ∈ π (Σ) . It is well known that this is a real analytic variety. Moreover, foralgebraic G, Hom( π (Σ) , G ) /G is also algebraic. This orbit space is not necessarilyHausdorff (cf., e.g. [10]) but the space Rep( π (Σ) , G ) = Hom + ( π (Σ) , G ) /G of allreductive representations of π (Σ) in G is Hausdorff. A reductive representation isthe one that once composed with adjoint representation of G on its Lie algebra G is a sum of irreducible representations. NOTE ON REIDEMEISTER TORSION OF G-ANOSOV REPRESENTATIONS 11
Teichm¨uller space
Teich(Σ) of Σ is the space of isotopy classes of complex struc-tures on Σ . A complex structure on Σ is a homotopy equivalence of a homeomor-phism f : Σ → S. Here, S is a Riemann surface, and two such homeomorphisms f : Σ → S, f ′ : Σ → S ′ are said to be equivalent, if there exists a conformaldiffeomorphism g : S → S ′ so that ( f ′ ) − ◦ g ◦ f is isotopic to the identity map onΣ . One can lift a complex structure on Σ to a complex structure on the universalcovering e Σ of Σ . By the Uniformization Theorem, e Σ is biholomorphic to the upperhalf-plane H ⊂ C . It is well known that each biholomorphic homeomorphism of H is of the form f ( z ) = ( az + b ) / ( cz + d ) with a, b, c, d ∈ R , ad − bc = 1 . This yields a dis-crete, faithful homomorphism from π (Σ) to PSL(2 , R ) . This homomorphism is alsowell defined up to conjugation by the orientation preserving isometries of H . Thus,one can identify Teich(Σ) with the
Fricke space , i.e. the set Rep df ( π (Σ) , PSL(2 , R ))of discrete faithful representations from π (Σ) to PSL(2 , R ) . Fricke space is a connected component of Rep( π (Σ) , PSL(2 , R )) . Openness fol-lows from [35], closedness from [6, 24], and connectedness from the UniformizationTheorem together with the identification of Teich(Σ) as a cell.For a finite cover G of PSL(2 , R ) , W. Goldman investigated the connected compo-nents of the representation space Hom( π (Σ) , G ) /G [11]. He proved that there exist4 g − π (Σ) , PSL(2 , R )) / PSL(2 , R ) . There exist twohomeomorphic components, called Teichm¨uller spaces, which are homeomorphic to R | χ (Σ) | dim PSL(2 , R ) . For a split real form G of a semi-simple Lie group, N. Hitchin investigated theconnected components of Rep( π (Σ) , G ) in [15] by using techniques of Higgs bundle.He proved that there exists an interesting connected component not detected bycharacteristic classes. He called it as Teichm¨uller component but it is called now
Hitchin component .A Hitchin component
Rep
Hitchin ( π (Σ) , G ) of Rep( π (Σ) , G ) is the connectedcomponent containing Fuchsian representations, i.e. representations of the form ̺ ◦ ı, where ̺ : π (Σ) → PSL(2 , R ) is Fuchsian, ı : PSL(2 , R ) → G is the representationcorresponding to the 3 − dimensional principal subgroup of B. Kostant [17]. For G =PSp(2 n, R ) , ı denotes the 2 n − dimensional irreducible representation correspondingto symmetric power Sym n − ( R ) . This enables one to identify the Fricke space and thus Teich(Σ) by a subset ofRep( π (Σ) , G ) . N. Hitchin proved in [15] that each Hitchin component is homoeo-morphic to a ball of dimension (6 g −
6) dim G. Recall that it was proved by F.Labourie in [19] that the set Rep
Hitchin ( π (Σ) , G ) of Hitchin representations is asubset of Rep Anosov ( π (Σ) , G ) . Applying Theorem 4.3 and Corollary 4.4, we have the following result.
Corollary 5.1.
Let Σ be a closed orientable surface of genus g ≥ and ̺ be in Rep
Hitchin ( π (Σ) , G ) . Let K be a cell-decomposition of Σ , c p be the geometric basesof C p ( K ; G Ad ◦ ̺ ) , p = 0 , , , and h is a basis for H (Σ; G Ad ◦ ̺ ) . Then, we have T ( C ∗ ( K ; G Ad ◦ ̺ ) , { c p } p =0 , { , h , } ) = p det Ω ω B . Moreover, it is a volume element on the Hitchin component
Rep
Hitchin ( π (Σ) , G ) . Here, G is one of { PSp (2 n, R )( n ≥ , PSO ( n, n + 1) n ≥ , PSO ( n, n + 1)( n ≥ } and G is the corresponding Lie algebra with the non-degenerate Killing form B and ω B : H (Σ; G Ad ◦ ̺ ) × H (Σ; G Ad ◦ ̺ ) ⌣ B −→ H (Σ; R ) R Σ −→ R is the Atiyah-Bott-Goldmansymplectic form for G. Proof.
From the fact that ̺ belongs to Rep Hitchin ( π (Σ) , G ) it follows that it is di-crete, faithfull, irreducible, and purely loxodromic (cf. [2, 8, 19]). The irreducibilityyields that H (Σ , G Ad ◦ ̺ ) and H (Σ , G Ad ◦ ̺ ) are both zero. Hence, by Theorem 4.3,we obtain T ( C ∗ ( K ; G Ad ◦ ̺ ) , { c p } p =0 , { , h , } ) = p det Ω ω B . It is well known that H (Σ , G Ad ◦ ̺ ) , H (Σ , G Ad ◦ ̺ ) can be identified respectively withthe tangent space T ̺ Rep
Hitchin ( π (Σ) , G ) , cotangent space T ∗ ̺ Rep
Hitchin ( π (Σ) , G )of Rep Hitchin ( π (Σ) , G ) (cf., e.g. [10]). Recall also that Reidemeister torsion T ( A ∗ )of a general chain complex A ∗ of length n belongs to ⊗ np =0 (det( H p ( A ∗ ))) ( − p +1 ([27, 36]). Here, det( H p ( A ∗ )) is the top exterior power V dim R H p ( A ∗ ) H p ( A ∗ ) of H p ( A ∗ ) and det( H p ( A ∗ )) − is the dual of det( H p ( A ∗ )) . Thus, we get a volumeelement on the Hitchin component Rep
Hitchin ( π (Σ) , G ) of Rep( π (Σ) , G ) . (cid:3) Let us note that since Teich(Σ) ⊂ Rep
Hitchin ( π (Σ) , G ) , Corollary 5.1 is also validfor Teich(Σ) representations.For the isomorphism T ̺ Teich(Σ) ∼ = H (Σ; G Ad ◦ ̺ ) , in [10], Goldman proved that ω PSL(2 , R ) : H (Σ; G Ad ◦ ̺ ) × H (Σ; G Ad ◦ ̺ ) → R and Weil-Petersson 2 − form differonly by a constant multiple. More precisely, ω WP = − ω PSL(2 , R ) . Bonahon parametrized the Teichm¨uller space of Σ by using a maximal geodesiclamination λ on Σ [1]. Geodesic laminations are generalizations of deformationclasses of simple closed curves on Σ . More precisely, a geodesic lamination λ onthe surface Σ is by definition a closed subset of Σ which can be decomposed intofamily of disjoint simple geodesics, possibly infinite, called its leaves. The geodesiclamination is maximal if it is maximal with respect to inclusion; this is equivalentto the property that the complement Σ − λ is union of finitely many triangles withvertices at infinity.The real-analytical parametrization given by Bonahon identifies Teich(Σ) to anopen convex cone in the vector space H ( λ, R ) of all transverse cocycles for λ. Inparticular, at each ̺ ∈ Teich(Σ) , the tangent space T ̺ Teich(Σ) is now identifiedwith H ( λ, R ) , which is a real vector space of dimension 3 | χ (Σ) | . A transverse cocycle σ for λ on Σ is a real-valued function on the set of all arcs k transverse to (the leaves) of λ with the following properties: • σ is finitely additive, i.e. σ ( k ) = σ ( k ) + σ ( k ) , whenever the arc k trans-verse to λ is decomposed into two subarcs k , k with disjoint interiors, • σ is invariant under the homotopy of arcs transverse to λ, i.e. σ ( k ) = σ ( k ′ )whenever the transverse arc k is deformed to arc k ′ by a family of arcswhich are all transverse to the leaves of λ. NOTE ON REIDEMEISTER TORSION OF G-ANOSOV REPRESENTATIONS 13 e incoming s e right s e left s s=switchtie The space H ( λ, R ) has also anti-symmetric bilinear form, namely the Thurstonsymplectic form ω Thurston . Let λ be a maximal geodesic lamination on Σ and Φ bea fattened train-track carrying the maximal geodesic lamination.The Thurston symplectic form is the anti-symmetric bilinear form ω Thurston : H ( λ ; R ) × H ( λ ; R ) → R defined by ω Thurston ( σ , σ ) = 12 X s det (cid:20) σ ( e left s ) σ ( e right s ) σ ( e left s ) σ ( e right s ) (cid:21) , where σ i ( e ) ∈ R is the weight associated to the edge e by the transverse cocycle σ i . Note that, ω Thurston is actually independent of the train-track Φ . It is proved in [26] that up to a multiplicative constant, ω Thurston is the same as ω PSL(2 , R ) , and hence is in the same equivalence class of ω WP . More precisely, for theidentification T ̺ Teich(Σ) ∼ = H ( λ ; R ) , the following is valid ω PSL(2 , R ) = 2 ω Thurston . As a final word on this study, Reidemeister torsion of ̺ ∈ Teich(Σ) can beexpressed in terms of ω Thurston . References [1] F. Bonahon,
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Hacettepe University, Department of Mathematics, 06800 Ankara, Turkey
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