Ruelle Zeta Functions of Hyperbolic Manifolds and Reidemeister Torsion

This paper is concerned with the behavior of twisted Ruelle zeta functions of compact hyperbolic manifolds at the origin. Fried proved that for an orthogonal acyclic representation of the fundamental group of a compact hyperbolic manifold, the twisted Ruelle zeta function is holomorphic at s=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=0$$\end{document} and its value at s=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=0$$\end{document} equals the Reidemeister torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried’s result to arbitrary finite dimensional representations of the fundamental group. The Reidemeister torsion is replaced by the complex-valued combinatorial torsion introduced by Cappell and Miller.


Introduction
Let X be a d-dimensional closed, oriented hyperbolic manifold. Then there exists a discrete torsion free subgroup ⊂ SO 0 (d, 1) such that X = \H d , where H d = SO 0 (d, 1)/ SO(d) is the d-dimensional hyperbolic space. Every γ ∈ \ {e} is loxodromic and the -conjugacy class [γ ] corresponds to a unique closed geodesic τ γ . Let (γ ) denote the length of τ γ . A conjugacy class is called prime if γ is not a non-trivial power of some other element of . Let χ : → GL(V χ ) be a finite dimensional complex representation of and let s ∈ C. Then the Ruelle zeta function (1.1) The infinite product is absolutely convergent in a certain half plane Re(s) > C and admits a meromorphic extension to the entire complex plane [11,31]. The Ruelle zeta function is a dynamical zeta function associated to the geodesic flow on the unit sphere bundle S(X ) of X . There are formal analogies with the zeta functions in number theory such as the Artin L-function associated to a Galois representation. Analogues to the role of zeta functions in number theory, one expects that special values of the Ruelle zeta function provide a connection between the length spectrum of closed geodesics and geometric and topological invariants of the manifold. In [10], Fried has established such a connection. To explain his result we need to introduce some notation. Recall that a representation χ is called acyclic, if the cohomology H * (X , F χ ) of X with coefficients in the flat bundle F χ → X associated to χ vanishes. Let χ be an orthogonal acyclic representation. Then F χ is equipped with a canonical fiber metric which is compatible with the flat connection. Let k,χ be the Laplacian acting in the space k (X , F χ ) of F χ -valued k-forms. Regarded as operator in the space of L 2 -forms, it is essentially self-adjoint with a discrete spectrum Spec( k,χ ) consisting of eigenvalues λ of finite multiplicity m(λ). Let ζ k (s; χ) = λ∈Spec( k,χ ) m(λ)λ −s be the spectral zeta function of k,χ [30]. The series converges absolutely in the half plane Re(s) > d/2 and admits a meromorphic extension to the complex plane, which is holomorphic at s = 0. Then the Ray-Singer analytic torsion T RS (X , χ) ∈ R + is defined by log T RS (X , χ) : 2) [28]. Now we can state the result of Fried [10,Theorem 1]. He proved that for an acyclic unitary representation χ the Ruelle zeta function R(s, χ) is holomorphic at s = 0 and where ε = (−1) d−1 and the absolute value can be removed if d > 2. If χ is not acyclic, but still orthogonal, R(s, χ) may have a pole or zero at s = 0. Fried [10] has determined the order of R(s, χ) at s = 0 and the leading coefficient of the Laurent expansion around s = 0. Let b k (χ ) := dim H k (X , F χ ). Assume that d = 2n + 1. Put Then by [10,Theorem 3], the order of R(s, χ) at s = 0 is h and the leading term of the Laurent expansion of R(s, χ) at s = 0 is C(χ ) · T RS (X , χ) 2 s h , (1.4) where C(χ ) is a constant that depends on the Betti numbers b k (χ ). In [13, p. 66] Fried conjectured that (1.3) holds for all compact locally symmetric manifolds X and acyclic orthogonal bundles over S(X ). This conjecture was recently proved by Shen [29]. Let χ be a unitary acyclic representation of . Let τ (X , χ) be the Reidemeister torsion [25,28]. It is defined in terms of a smooth triangulation of X . However, it is independent of the particular C ∞ -triangulation. Since χ is acyclic, τ (X , χ) is a topological invariant, i.e., it does not depend on the metrics on X and in F ρ . By [8,24] we have T RS (X , χ) = τ (X , χ). Assume that d is odd. Then (1.3) can be restated as (1.5) This provides an interesting relation between the length spectrum of closed geodesics and a secondary topological invariant. Another class of interesting representations arises in the following way. Let G := SO 0 (d, 1). Let ρ be a finite dimensional complex or real representation of G. Then ρ| is a finite dimensional representation of . In general, ρ| is not an orthogonal representation. However, the flat vector bundle F ρ associated with ρ| can be equipped with a canonical fiber metric which allows the use of methods of harmonic analysis to study the Laplace operators k,ρ . Put The behavior of R(s, ρ) at s = 0 has been studied by Wotzke [35]. Let θ : G → G be the Cartan involution of G with respect to K = SO(d). Let ρ θ := ρ • θ . Also denote by T RS (X , ρ) the analytic torsion of X with respect to ρ| and an admissible metric in F ρ . Assume that ρ is irreducible. If ρ ρ θ , then the cohomology H * (X , F ρ ) vanishes [2, Chapt. VII, Theorem 6.7]. Moreover, in this case Wotzke [35] has proved that R(s, ρ) is holomorphic at s = 0 and (1.6) If ρ ∼ = ρ θ , then R(s, ρ) may have a zero or a pole at s = 0. Wotzke [35] has also determined the order of R(s, ρ) at s = 0 and the coefficient of the leading term of the Laurent expansion of R(s, ρ) at s = 0. As in (1.4) the main contribution to the coefficient is the analytic torsion. Let τ (X , ρ) be the Reidemeister torsion [25] of X with respect to ρ| . Assume that ρ is irreducible and ρ ρ θ . As mentioned above, the cohomology H * (X , F ρ ) vanishes in this case and therefore, τ (X , ρ) is independent of the metrics on X and in F ρ . By [ This is another interesting relation between the length spectrum of X and topological invariants of X . For arithmetic subgroups ⊂ G, representations of G with -invariant lattices in the corresponding representation space exist. See [1,20].
The main purpose of this paper is to extend the above results about the behavior of the Ruelle zeta function at s = 0 to every finite dimensional representation χ of . To this end we use a complex version T C (X , χ) of the analytic torsion, which was introduced by Cappell and Miller [7] and which is closely related to the refined analytic torsion of Braverman and Kappeler [3]. It is defined in terms of the flat Laplacians k,χ , k = 0, . . . , d, which are obtained by coupling the Laplacian k on k-forms to the flat bundle F χ (see Sect. 2 for its definition). In general, the flat Laplacian k,χ is not self-adjoint. However, its principal symbol equals the principal symbol of a Laplace type operator. Therefore, it has good spectral properties which allows to carry over most of the results from the self-adjoint case. The Cappell-Miller torsion T C (X , χ) is defined as an element of the determinant line For an acyclic representation T C (X , χ) is a complex number and where T C (X , χ) is the Ray-Singer analytic torsion with respect to any choice of a fiber metric in F χ . Since χ is acyclic, T RS (X , χ) is independent of the choice of the metric in F χ . Let V k 0 be the generalized eigenspace of k,χ , k = 0, . . . , d, with generalized eigenvalue 0. Let d * , χ be the coupling of the co-differential d * χ : * (X ) → * (X ) to the flat bundle F χ . Then (V * 0 , d χ , d * , χ ) is a double complex in the sense of [7, §6]. Let (1.10) be its torsion [7, §6]. We note that T C (X , χ) and T 0 (X , χ) are both non-zero elements of the determinant line det H * (X , F χ ) ⊗ (det H * (X , F χ )) * . Hence there exists λ ∈ C with T C (X , χ) = λT 0 (X , χ). Set Put Furthermore, let d = 2n + 1 and put Then our main result is the following theorem. (1.14) As above there is a combinatorial formula. Let τ comb (X , χ) ∈ det H * (X , F χ )⊗ det H * (X , F * χ ) be the combinatorial torsion defined by Cappell and Miller [7,Sect. 9], which is defined in terms of a triangulation of X , but is independent of the choice of the triangulation. An equivalent definition is as follows. For a unimodular complex representation ρ of , one can define a complex valued Reidemeister torsion τ C (X , ρ) ∈ det H * (X , F ρ ) [9, §3]. See also Sect. 6.1. Now let χ * be the contragredient representation to χ . The representation χ ⊕ χ * is unimodular and therefore, the complex Reidemeister torsion τ C (X , χ ⊕ χ * ) ∈ det H * (X , F χ ⊕ F * χ ) is well defined. Then by [5] it follows that with respect to the canonical isomorphism we have τ comb (X , χ) = ±τ C (X , χ ⊕ χ * If χ is acyclic, then T C (X , χ), T 0 (X , χ) and τ comb (X , χ) are complex numbers and on the right hand side of (1.14) and (1.15) appear quotients of complex numbers. Now we apply Theorem 1.1 to representations of which are restrictions of representations of G. Denote by Rep(G) the space of finite dimensional complex representations of G. Then we have Corollary 1.2 Let ρ ∈ Rep(G) be irreducible and assume that ρ ρ θ . Then R(s, ρ) is holomorphic at s = 0 and (1.16) Using (1.6) and (1.9), it follows that The order h of R(s, ρ m ) at s = 0 is zero. Thus by (1.12) we have h 1 = 2h 0 . Note that ρ m is acyclic. Let k,ρ m be the usual Laplacian in k (X , F ρ m ) with respect to the admissible metric in F ρ m . Then for m ∈ N even we have ker k,ρ m = 0, ker k,ρ m = 0, k = 0, . . . , (1.19) which shows that for acyclic representations χ , in general, the flat Laplacian k,χ need not be invertible. Again we can replace T C (X , ρ) in (1.16) by the combinatorial torsion τ comb (X , ρ).
Since G is a connected semi-simple Lie group and ρ a representation of G, it follows from [25,Lemma 4.3] that ρ is actually a representation in SL(n, C). Thus ρ| and ρ * | are unimodular. Moreover, if ρ ρ θ , we have H * (X , In dimensions > 3 this is not known.
Using Theorem 1.1, we get This proposition was first proved by Spilioti [33] using the odd signature operator [3]. She also discusses the relation with the refined analytic torsion. As above we can also express R(0, χ) in terms of the combinatorial torsion and by (1.21) we have This agrees with (1.7).

Coupling Differential Operators to a Flat Bundle
We recall a construction of the flat extension of a differential operator introduced in [7]. Let X be a smooth manifold and E 1 and E 2 complex vector bundles over X . Let be a differential operator. Let F → X be a flat vector bundle. Then there is a canonically operator Let s 1 , . . . , s k be another local frame field of flat sections of F| U . Then , and it follows that the transition functions f i j are constant. Since D is linear, (D F | U )(ϕ) is independent of the choice of the local frame field of flat sections and therefore, D F is globally well defined. Let σ (D) be the principal symbol of D. Then the principal symbol σ ( Thus if D is elliptic, then D F is also an elliptic differential operator. As an example consider a Riemannian manifold X and the Laplace operator p on p-forms. Let F be a flat bundle over X . Denote by p (X , F) the space of smooth F-valued p-forms, i.e., p (X , F) = C ∞ (X , p T * (X ) ⊗ F). By the construction above we obtain the flat Laplacian p,F : p (X , F) → p (X , F). If the flat bundle is fixed, we will denote the flat Laplacian simply by p . The flat Laplacian can be also described as the usual Laplacian. Let d F : p−1 (X , F) → p (X , F) be the exterior derivative defined as above. Let : p (X ) → n− p (X ) denote the Hodge -operator. Then the flat extension d * , Then d * , If we choose a Hermitian fiber metric on F, we can define the usual Laplace operator which is formally self-adjoint. Now note that d * , Hence the principal symbol σ ( F )(x, ξ) of F is given by More generally, let E → X be a Hermitian vector bundle over X . Let ∇ be a covariant derivative in E which is compatible with the Hermitian metric. We denote by C ∞ (X , E) the space of smooth sections of E. Let E = ∇ * ∇ be the Bochner-Laplace operator associated to the connection ∇ and the Hermitian fiber metric. Then E is a second order elliptic differential operator. Its leading symbol σ ( E ) : π * E → π * E, where π is the projection of T * X , is given by Let F → X be a flat vector bundle and the coupling of E to F. Then the principal symbol of E⊗F is given by

Regularized Determinants and Analytic Torsion
Let E be as above. Let be an elliptic second order differential operator which is a perturbation of E by a first order differential operator, i.e., is a first order differential operator. This implies that P is an elliptic second order differential operator with leading symbol σ (P)(x, ξ) given by Though P is not self-adjoint in general, it still has nice spectral properties [30, Chapt. I, §8]. We recall the basic facts. For I ⊂ [0, 2π ] let The following lemma describes the structure of the spectrum of P. For λ ∈ C\spec(P) let R λ (P) := (P −λ Id) −1 be the resolvent. Given λ 0 ∈ spec(P), let λ 0 be a small circle around λ 0 which contains no other points of spec(P). Put Then λ 0 is the projection onto the root subspace V λ 0 . This is a finite-dimensional subspace of C ∞ (X , E) which is invariant under P and there exists N ∈ N such that which is invariant under the closureP of P in L 2 and the restriction of (P − λ 0 I) to V λ 0 has a bounded inverse. The algebraic multiplicity m(λ 0 ) of λ 0 is defined as such that the restriction of P to V k has a unique eigenvalue λ k , for each k there exists N k ∈ N such that (P − λ k I) N k V k = 0, and |λ k | → ∞. In general, the sum (3.5) is not a sum of mutually orthogonal subspaces. See [23, Sect. 2] for details. Assume that P is invertible. Recall that an angle θ ∈ [0, 2π) is called an Agmon angle for P, if there exists ε > 0 such that spec(P) ∩ [θ−ε,θ+ε] = ∅. (3.7) The zeta function admits a meromorphic extension to the entire complex plane which is holomorphic at s = 0 [30, Theorem 13.1]. Let R θ := {ρe iθ : ρ ∈ R + }. Denote by log θ (λ) the branch of the logarithm in C \ R θ with θ < Im log θ < θ + 2π . We enumerate the eigenvalues of P such that By Lidskii's theorem [16,Theorem 8.4] if follows that for Re(s) > d/2 we have where (λ k ) −s θ = e −s log θ (λ k ) . We will need a different description of the zeta function in terms of the heat operator e −t P , which can be defined using the functional calculus developed in [23,Sect. 2] by where ⊂ C is the same contour as in [ for t ≥ 1. Since spec(P) is contained in the half plane Re(s) > 0, we can choose the Agmon angle as θ = π . Using the asymptotic expansion of Tr(e −t P ) as t → 0, it follows from (3.8) and (3.11) that Then the regularized determinant of P is defined by (3.12) As shown in [3, 3.10], det θ (P) is independent of θ . Therefore we will denote the regularized determinant simply by det(P).

W. Müller
Assume that the vector bundle E is Z/2Z-graded, i.e., E = E + ⊕ E − and P preserves the grading, i.e., assume that with respect to the decomposition Then we define the graded determinant det gr (P) of P by det gr (P) = det(P + ) det(P − ) . (3.13) Next we introduce the analytic torsion defined in terms of the non-self-adjoint operators p,χ . We use the definition given in [7,Sect. 8]. Recall that the principal symbol of p,χ is given by (2.3). Therefore, p,χ satisfies the assumptions of Sect. 3.
Let r > 0 be such that Re(λ) = r for all generalized eigenvalues λ of p,χ . Let p,r be the spectral projection on the span of the generalized eigenvectors with eigenvalues with real part less than r . Let p,χ ,r := (1− p,r ) p,χ . Let S( p, χ, r ) be the set of all nonzero generalized eigenvalues with real part less than r . Furthermore, let V p 0 be the generalized eigenspace of p,χ with generalized eigenvalues 0. Then (V * 0 , d, d * , ) is double complex in the sense of [7]. Let T 0 (X , χ) ∈ (det H * (X , F χ )) ⊗ (det H * (X , F χ )) * (3.14) be the torsion of the double complex. Then the Cappell-Miller torsion is defined by where m(λ) denotes the algebraic multiplicity of λ. Let k,0 be the spectral projection on the generalized eigenspace of k,χ with generalized eigenvalue 0. Let If we choose an Agmon angle we can also write If χ is acyclic, i.e., H * (X , E χ ) = 0, then T 0 (X , χ) and T C (X , χ) are complex numbers.

Twisted Ruelle Zeta Functions
In this section we consider compact oriented hyperbolic manifolds of odd dimension d = 2n + 1 and we recall some basic properties of Ruelle type zeta functions.
To begin with we fix some notation. Let G = SO 0 (d, 1) and K = SO(d). Then  (g, a). Let α ∈ a * be the unique positive root of (g, a). Let a + := {H ∈ a : α(H ) > 0}. Put A + := exp(a + ). Let g C := g ⊗ C and denote by Z(g C ) the center of the universal enveloping algebra of g C .
Let ⊂ G be a discrete, torsion free, co-compact subgroup. Then acts fixed point free on H d . The quotient X = \H d is a closed, oriented hyperbolic manifold and each such manifold is of this form. Given γ ∈ , we denote by [γ ] the -conjugacy class of γ . The set of all conjugacy classes of will be denoted by C( ). Let γ = 1. Then there exist g ∈ G, m γ ∈ M, and a γ ∈ A + such that gγ g −1 = m γ a γ .  In order to verify that the product converges in some half plane, we first recall that there exists C > 0 such that for all R > 0 we have [6, (1.31)]. We also need the following auxiliary lemma. For the proof see [31,Lemma 3.3]. It follows from (4.4) and Lemma 4.1 that the product on the right hand side of (4.3) converges absolutely and uniformly in some halfplane Re(s) > C (see [31,Prop. 3.5]). Furthermore, R(s, χ) admits a meromorphic extensions to the entire complex plane [31] and satisfies the following functional equation [32]: For unitary representations χ , these results were proved by Bunke and Olbrich [6]. The main technical tool is the Selberg trace formula. For the extension to the non-unitary case the Selberg trace formula is replaced by a Selberg trace formula for non-unitary twists, developed in [23]. The proofs are similar except that on has to deal with non-self-adjoint operators.
There are also expressions of the zeta functions in terms of determinants of certain elliptic operators. To explain the formulas we need to recall the definition of the relevant differential operators. Given τ ∈ K , let E τ → X be the homogeneous vector bundle associated to τ and let E τ := \ E τ be the corresponding locally homogeneous vector bundle over X . Denote by C ∞ (X , E τ ) the space of smooth sections of E τ . There is a canonical isomorphism [21, §1]. Let ∈ Z(g C ) be the Casimir element and denote by R the right regular representation of G in C ∞ ( \G). Then R ( ) acts on the right hand side of (4.7) and via this isomorphism, defines an operator in C ∞ (X , E τ ). We denote the operator induced by −R ( ) by A τ . Denote by ∇ τ the canonical connection in E τ and let ∇ τ be the induced connection in E τ . Let τ := (∇ τ ) * ∇ τ be the associated Bochner-Laplace operator acting in C ∞ (X , E τ ) . Let K ∈ Z(k C ) be the Casimir element of K . Assume that τ is irreducible. Let λ τ := τ ( K ) denote the Casimir eigenvalue of τ . Then we have (4.8) [21, §1]. Thus A τ is a formally self-adjoint second order elliptic differential operator. Let F χ → X be the flat vector bundle defined by χ . Let be the coupling of A τ to F χ . Denote by R(K ) and R(M) the representation rings of K and M, respectively. Let i : M → K be the inclusion and i * : R(K ) → R(M) the induced map of the representation rings. The Weyl group W (A) acts on R(M) in the canonical way. Let R ± (M) denote the ±1-eigenspaces of the non-trivial element w ∈ W (A). Let σ ∈ R(M). It follows from the proof of Proposition 1.1 in [6] that there exist m τ (σ ) ∈ {−1, 0, 1}, depending on τ ∈ K , which are equal to zero except for finitely many τ ∈ K , such that (4.9) if σ ∈ R + (M), and (4.11) Then E(σ ) has a grading defined by the sign of m τ (σ ). Let σ ∈ M. Denote by ν σ the highest weight of σ . Let b be the standard Cartan subalgebra of m [26,Sect. 2]. Let m be the half-sum of positive roots of (m C , b C ) and a the half-sum of positive roots of (g C , a C ). Put (4.13) We note that σ p is irreducible except when p = n, in which case σ n is the direct sum of the two spin representations σ + n , σ − n . Moreover, σ ∓ n = wσ ± n . Thus σ n = σ + n + wσ + n and (4.14) Recall that A χ (σ p ) acts in the space of sections of a graded vector bundle. Then by [32,Prop. 1.7] we have the following determinant formula.
By the Lemma of Kuga, this operator corresponds to the Laplacian k on k (X ). Let k,χ be the coupling of k to F χ . Then by (5.1) we get Inserting this equality into the alternating product of graded determinants on the right hand side of (4.15), we obtain Applying the determinant formula (4.15) together with (5.3), we finally get

(5.4)
Let h k be the dimension of the generalized eigenspace of k,χ with eigenvalue zero.
Proof Let : p (X , F χ ) → d− p (X , F χ ) be the extension of the Hodge -star operator, which acts locally as (ω ⊗ f ) = ( ω) ⊗ f , where ω is a usual p-form and f a local section of F χ . Since p = d− p , it follows from the definition of the Laplacians coupled to F χ that p,χ = d− p,χ . It follows that for every k ∈ N we have ( p,χ ) k = ( d− p,χ ) k .This proves the lemma.
Denote by h the order of the singularity of R(s, χ) at s = 0. Using (5.4) and Lemma 5.1 it follows that Let ( k,χ ) be the operator defined by (3.16). We note that for s ∈ C, |s| 1, there is a common Agmon angle for the operator ( k,χ ) + s(s + 2(n − p)). Therefore, in order to study the limit of det(( k,χ ) + s(s + 2(n − p))) as s → 0, we can use one and the same Agmon angle.

Acyclic Representations
In this section we assume that χ is acyclic. Then T C (X , χ), T 0 (X , χ) and τ comb (X , χ) are complex numbers and the right hand side of (1.14) is the quotient of the two complex numbers. Besides the Cappell-Miller torsion we need another version of a complex analytic torsion for arbitrary flat vector bundles F χ . This is the refined analytic torsion T ran (X , χ) ∈ det(H * (X , F χ )) introduced by Braverman and Kappeler [4]. The definition is based on the consideration of the odd signature operator B χ [3, 2.1]. It is defined as follows. Let be the chirality operator defined by Let ∇ χ be the flat connection in F χ . Then the odd signature operator is defined as It leaves the even subspace ev (X , F χ ) invariant. Let B ev,χ be the restriction of B χ to ev (X , F χ ). Then T ran (X , χ) ∈ det(H * (X , F χ )) is defined in terms of B ev,χ . If χ is acyclic, then T ran (X , χ) is a complex number. In Combining (6.2) and (6.3), we obtain (1.9).

Restriction of Representations of the Underlying Lie Group
The first case that we consider are representations which are restrictions to of representations of G.
Let ρ : G → GL(V ρ ) be a finite dimensional real ( resp. complex) representation of G. Denote by F ρ → X the flat vector bundle associated to ρ| . Let E ρ → G/K be the homogeneous vector bundle associated to ρ| K . By [18,Part I,Prop. 3.3] there is a canonical isomorphism F ρ ∼ = \ E ρ . (6.4) Let g = k ⊕ p be the Cartan decomposition of g. By [18, Part I, Lemma 3.1], there exists an inner product ·, · in V ρ such that Such an inner product is called admissible. It is unique up to scaling. Fix an admissible inner product. Since ρ| K is unitary with respect to this inner product, it induces a metric in \ E τ and by (6.4) also in F ρ . Denote by T RS (X , ρ) the Ray-Singer analytic torsion of (X , F ρ ) with respect to the metric on X and the metric in Next we briefly recall the definition of the complex Reidemeister torsion [9]. Let V be C-vector space of dimension m. Let v = (v 1 , . . . , v m ) and w = (w 1 , . . . , w m ) be two basis of V . Let T = (t i j ) be the matrix of the change of basis from v to w, i.e., be a co-chain complex of finite dimensional complex vector spaces. We assume that C * is acyclic. Let Z q = ker(δ q ) and B q := Im(δ q−1 ) ⊂ C q . Let c q be a preferred base of C q . Choose a basis b q for B q , q = 0, . . . , n, and letb q+1 be an independent set in C q such that δ q (b q+1 ) = b q+1 . Then (b q ,b q+1 ) is a basis of C q and [b q ,b q+1 /c q ] depends only on b q and b q+1 . Therefore, we denote it by [b q , b q+1 /c q ]. Then the complex Reidemeister torsion τ C (C * ) ∈ C of the co-chain complex C * is defined by Let K be a C ∞ -triangulation of X and K the lift of K to a triangulation of the universal covering H d of X . Then C q ( K , C) is a module over the complex group algebra C[ ]. Now let ρ be an acyclic representation of in SL(N , C). Let be the twisted co-chain group and the corresponding co-chain complex. Then C * (K , ρ) is acyclic. Let e 1 , . . . , e r q be a preferred basis of C q (K , C) as a C[ ]-module consisting of the duals of lifts of q-simplexes and let v 1 , . . . , v N be a basis of C N . Then {e i ⊗ v j : i = 1, . . . , r q , j = 1, . . . , N } is a preferred basis of C q (K , ρ). Now consider the complex-valued Reidemeister torsion τ C (C * (K , ρ)). Since ρ is a representation in SL(N , C), a different choice of the preferred basis {e i } leads at most to a sign change of τ C (C * (K , ρ)).

Deformations of Acyclic Unitary Representations
Let Rep( , C m ) be the set of all m-dimensional complex representations of . It is well known that Rep( , C m ) has a natural structure of a complex algebraic variety [3, 13.6].   R(s, χ) is regular at s = 0 and from Theorem 1.1 we recover Fried's result [10] R(0, χ) = T RS (X , χ) 2 . (6.9) We equip Rep( , C m ) with the topology obtained from its structure as complex algebraic variety. The complement of the singular set is a complex manifold. Let W ⊂ Rep( , C m ) be the connected component of Rep( , C m ) which contains Rep u 0 ( , C m ). Let χ 0 ∈ W be a unitary acyclic representation and let E 0 be the associated flat vector bundle. By [15,Prop. 4.5] every vector bundles E χ , χ ∈ W , is isomorphic to E 0 . Thus the flat connection on E χ , which is induced by the trivial connection on X × C m , corresponds to a flat connection ∇ χ on E 0 . Now recall that where d * , Let ∇ 0 be the unitary flat connection on E 0 . Let C(E 0 ) denote the space of connections on E 0 . Recall that C(E 0 ) can be identified with 1 (X , End(E 0 )) by associating to a connection ∇ ∈ C(E 0 ) the 1-form ∇ − ∇ 0 ∈ 1 (X , End(E 0 )). We equip C(E 0 ) with the C 0 -topology defined by the sup-norm ω sup := max x∈X |ω(x)|, ω ∈ 1 (X , End(E 0 )), where | · | denotes the natural norm on 1 (T * X ) ⊗ E 0 . Since E 0 is acyclic, D 0 := ∇ 0 + ∇ * 0 is invertible. If ∇ χ − ∇ 0 1 it follows as in [3, Prop 6.8] that D χ is invertible and hence χ = (D χ ) 2 is invertible too. Thus we get Let χ ∈ V . Then we have h k = dim( k,χ ) = 0, k = 0, . . . , d, and therefore the order h of R(s, χ) at s = 0 vanishes. Also C(d, χ) = 1 and T 0 (X , χ) = 1. Thus by Theorem 1.1 we obtain Proposition 1.3.
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