This is the accepted version of the journal article: Bénard, Léo; Dubois, Jérôme; Heusener, Michael; [et al.]. «Asymptotics of twisted Alexander polynomials and hyperbolic volume». Indiana Uni- versity mathematics journal, Vol. 71, Issue 3 (2022), p. 1155-1207. DOI 10.1512/iumj.2022.71.8937 This version is available at https://ddd.uab.cat/record/275428 under the terms of the license ASYMPTOTICS OF TWISTED ALEXANDER POLYNOMIALS AND HYPERBOLIC VOLUME LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI Abstract. For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we prove this asymptotic behavior for cusped hyperbolic manifolds of finite volume. The proof relies on results of Müller, and Menal-Ferrer and the last author. Using the uniformity of the convergence, we also deduce a similar asymptotic result for the Mahler measures of those polynomials. 1. Introduction Alexander polynomials are Laurent polynomials associated to a free abelian cover of a three-dimensional manifold M . Classically M is a knot or link exte- rior in the three-dimensional sphere S 3 , and the associated free abelian cover is the maximal one. The Alexander polynomial carries only metabelian infor- mation on the fundamental group, and the idea behind the twisted Alexander polynomial is to associate a Laurent polynomial invariant to a free abelian cover of M together with a linear representation of the fundamental group π1 (M ). Twisted Alexander polynomials of knots have been defined by Lin [23] and Wada [44]. Kitano [21] showed that they are Reidemeister torsions, generalizing Milnor’s theorem on the (untwisted) Alexander polynomial [30]. In this article we are interested in orientable, hyperbolic three-dimensional manifolds of finite volume since such a manifold has a natural representation of its fundamental group into PSL2 (C), the hyperbolic holonomy, that is unique up to conjugation. The holonomy representation lifts to SL2 (C) and a lift is unique up to multiplication with a representation into the center of SL2 (C) (see [8]). The corresponding twisted Alexander polynomial has been considered, among others, by Dunfield, Friedl and Jackson in [10]. Here we compose the lift of the holonomy representation with the irreducible representation of SL2 (C) in SLn (C), the (n − 1)-th symmetric power, and study its asymptotic behavior as n tends to +∞. Examples of twisted Alexander polynomials associated to a lift of the holo- nomy representation into SL2 (C) can be found in [10]. More examples, also for the n-th symmetric power are presented in http://dunfield.info/torsion. Date: June 9, 2022. 2010 Mathematics Subject Classification. 57M50, 57M25. 1 2 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI For illustration we have included in the Appendix A the twisted polynomials for the figure-eight knot associated for the n-th symmetric power for n ≤ 9 (following our normalisation which is different from [14]). Before considering non-compact, orientable, hyperbolic three-manifolds of fi- nite volume in general, we discuss first the case of a hyperbolic knot complement S 3 \ K. Let ρn : π1 (S 3 \ K) → SLn (C) be the composition of a lift of the holo- nomy with the (n − 1)-th symmetric power SL2 (C) → SLn (C). Let ∆ρKn denote the Alexander polynomial of K twisted by ρn , which equals Wada’s definition for n even, but it is Wada’s polynomial divided by (t − 1) when n is odd, so that its evaluation at t = 1 does not vanish. The set of unit complex numbers is denoted by S1 = {ζ ∈ C | |ζ| = 1}. The following is a particular case of the main result of this paper. Theorem 1.1. For any ζ ∈ S1 , log |∆ρKn (ζ)| 1 lim = vol(S 3 \ K) n→∞ n2 4π uniformly on ζ. For a knot exterior there are two lifts ρ of the holonomy, Theorem 1.1 holds true for both choices of lift. Based on Yamaguchi’s work ([48]), Goda showed in [14] that |∆ρKn (1)| is equal to the Reidemeister torsions with coefficients in ρn and so Theorem 1.1 generalizes results of Müller [35] and of Menal-Ferrer and the last author [29]. Theorem 1.1 is a particular case of Theorem 1.6 below. To extend the def- inition of twisted Alexander polynomial to general cusped manifolds (Defini- tion 2.6), we need to make some assumptions, that are always satisfied for hyperbolic knot exteriors. Let M be an orientable, non-compact, connected, finite volume hyperbolic three-manifold. It admits a compactification M by adding l ≥ 1 peripheral tori, one for each end: ∂M = T12 ⊔ · · · ⊔ Tl2 . Let α : π1 (M ) ↠ Zr be an epimorphism. In this paper we require the two following hypotheses, that are always satis- fied in the case of a knot complement. Assumption 1.2. For each peripheral torus Ti2 , α(π1 (Ti2 )) ∼ = Z. This assumption is a necessary condition for the acyclicity (over the field of rational fractions) of the Zr -cover associated with α, so that the twisted Alexan- der polynomials do not vanish. Assumption 1.2 holds true for the abelianization map of a knot in a homology sphere, or more generally for the abelianization map of a link in a homology sphere having the property that the linking number of pairwise different components vanish. Furthermore, for any cusped, oriented, hyperbolic 3-manifold M , there exists an epimorphism α : π1 (M ) ↠ Z satisfy- ing Assumption 1.2 (compose the abelianization map with a generic surjection of H1 (M, Z) onto Z). ASYMPTOTICS OF TWISTED POLYNOMIALS 3 Let li ∈ π1 (Ti2 ) be a generator of ker(α|π1 (T 2 ) ), we say that li is a longitude i for α, and we use the terminology α-longitude. We choose ρ : π1 (M ) → SL2 (C) a lift of the hyperbolic holonomy satisfying the following: Assumption 1.3. The lift of the holonomy ρ : π1 (M ) → SL2 (C) satisfies tr(ρ(li )) = −2. for each α-longitude li , i = 1, . . . , l. For a knot exterior in a homology sphere and the abelianization map, As- sumption 1.3 is satisfied for every lift of the holonomy. More generally, it is also satisfied for every lift of the holonomy for a link exterior in a homology sphere with the property that the linking number of pairwise different components vanish. In general, for any M and α satisfying Assumption 1.2, there exists at least one lift of the holonomy satisfying Assumption 1.3 [29, Proposition 3.2]. In terms of spin structures, this is the condition for a spin structure to extend along certain Dehn fillings which we will consider in Section 3. For n ≥ 2, recall that the unique n-dimensional irreducible holomorphic rep- resentation of SL2 (C) is given by the natural action on the (n−1)-th symmetric power of C2 . We denote it by Symn−1 : SL2 (C) → SLn (C). For a lift of the holonomy representation ρ : π1 (M ) → SL2 (C), we denote the composition with the (n − 1)-th symmetric power by ρ Symn−1 ρn : π1 (M ) − → SL2 (C) −−−−−→ SLn (C). Remark 1.4. This convention follows the notation of [29], but it differs from [35], it is shifted by 1. To construct the twisted Alexander polynomial, we consider the polynomial representation associated to α: ᾱ : π1 (M ) → C[t±1 ±1 1 , . . . , tr ] α1 (γ) αr (γ) γ 7→ t1 · · · tr where α = (α1 , . . . , αr ) are the components of α. We define the twisted Alexan- der polynomial ∆α,n M in Definition 2.6 as the inverse of the Reidemeister torsion of the pair (M, ᾱ ⊗ ρn ), after removing some factors (tβ1 1 · · · tβr r − 1) when n is odd (one factor for each peripheral torus). It is a Laurent polynomial with variables t±1 ±1 ±1 1 , . . . , tr defined up to sign and up to multiplicative factors ti . Remark 1.5. We stress out the fact that, for ζ1 , . . . , ζr ∈ S1 , only the modulus |∆α,n M (ζ1 , . . . , ζr )| is well defined. The main result of this paper is: Theorem 1.6. Let M be an oriented, hyperbolic three-manifold of finite vol- ume. We suppose that α : π1 (M ) ↠ Zr is an epimorphism and that ρ : π1 (M ) → SL2 (C) is a lift of the holonomy representation. If M is closed, or if the Assumptions 1.2 and 1.3 are satisfied, then for any ζ1 , . . . , ζr ∈ S1 , log |∆α,n M (ζ1 , . . . , ζr )| vol(M ) lim = n→∞ n2 4π 4 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI uniformly on the ζ1 , . . . , ζr . Remark 1.7. Assumptions 1.2 and 1.3 are vacuous for closed manifolds. The logarithmic Mahler measure of a Laurent polynomial P (t1 , . . . , tr ) ∈ C[t±1 ±1 1 , . . . , tr ] is defined as Z 2π Z 2π 1 m(P ) = · · · log |P (eiθ1 , . . . , eiθr )| dθ1 · · · dθr . (2π)r 0 0 With Theorem 1.6, as the convergence is uniform on ζ1 , . . . , ζr ∈ S1 , we also prove: Theorem 1.8. Under Assumptions 1.2 and 1.3, m(∆α,n M ) vol(M ) lim 2 = . n→∞ n 4π Assume now that M is fibered over the circle and let α : π1 (M ) → Z be induced by the fibration M → S1 . We chose a representative ∆α,n M (t) so that ∆α,n M (t) ∈ C[t] (it is a polynomial in one variable) and ∆α,n M (0) ̸ = 0. Corollary 1.9. We have 1 X 1 lim log |λ| = vol(M ), n→∞ n2 2π λ∈Spec(∆α,n M ) where Spec(∆α,n α,n M ) = {λ ∈ C | ∆M (λ) = 0}. Moreover, the maximum of the modulus of the roots grows at least exponen- tially with n. Proof. If p(t) = a di=1 (t − αi ) ∈ C[t] is a polynomial without zeros on the unit Q circle and p(0) ̸= 0, then Jensen’s formula implies that d Z 2π X 1 log |p(eiθ )| dθ = m(p) .  log |a| + log max(|αi |, 1) = 2π 0 i=1 We aim to apply this formula to the normalized polynomial ∆α,n M (t). From the fiberness of M it follows that ∆α,n M (0) = ±1 ̸ = 0. Moreover, ∆ α,n M (t) is symmet- ric which implies for all λ ∈ C that ∆M (λ) = 0 if and only if ∆α,n ∗ α,n M (1/λ) = 0. Finally, by Theorem 1.11 the Alexander polynomial has no root on the unit circle. Hence we can apply the above formula for ∆α,nM (t), and we obtain: log |λ| = 2 m(∆α,n X M ), λ∈Spec(∆α,n M ) and the first statement follows directly from Theorem 1.8. Finally notice that deg ∆α,n M (t) is linear on n, so the second statement follows directly from the first. □ Remark 1.10. When M is not fibered, it may happen that ∆α,n M is not monic and we must take into account |∆α,n M (0)| in Jensen’s formula. ASYMPTOTICS OF TWISTED POLYNOMIALS 5 Given ζ1 , . . . , ζr ∈ S1 , we compose α : π1 (M ) ↠ Zr with the homomorphism Zr → S1 (n1 , . . . , nr ) 7→ ζ1n1 · · · ζrnr and we denote the composition by χ : π1 (M ) → S1 . Namely, we evaluate ᾱ at tj = ζj : if α = (α1 , . . . , αr ) are the components of α, then (1) χ : π1 (M ) → S1 α (γ) γ 7→ ζ1 1 · · · ζrαr (γ) . In fact, Theorem 1.6 is a theorem on Reidemeister torsions, as |∆α,n M (ζ1 , . . . , ζr )| is the inverse of the modulus of the Reidemeister torsion of M twisted by the representation χ ⊗ ρn (in some cases perhaps up to some factor independent of n or after the choice of basis in homology), see Section 2. The definition of twisted Alexander polynomial as a Reidemeister torsion requires a vanishing theorem in cohomology, Theorem 2.3. Its proof mimics the classical vanishing theorem on L2 -cohomology of Matsushima–Murakami, as we explain in Appendix C . As a direct consequence of this vanishing theorem, we obtain that the twisted Alexander polynomials have no roots on the unit circle: Theorem 1.11. Under Assumptions 1.2 and 1.3, for any ζ1 , . . . , ζr ∈ S1 , ∆α,n M (ζ1 , . . . , ζr ) ̸= 0. We apply this theorem to study the dynamics of a pseudo-Anosov diffeo- morphism on the variety of representations. Let Σ be a compact orientable surface, possibly with boundary and with negative Euler characteristic. For a pseudo-Anosov diffeomorphism ϕ : Σ → Σ, consider its action on the rela- tive variety of (conjugacy classes of) representations ϕ∗ : R(Σ, ∂Σ, SLn (C)) → R(Σ, ∂Σ, SLn (C)). The mapping torus M (ϕ) is a hyperbolic manifold of fi- nite volume and its holonomy restricts to a representation of π1 (Σ) in SL2 (C) whose conjugacy class is fixed by ϕ∗ . In particular the conjugacy class of the composition [ρn ] = [Symn−1 ◦hol|π1 (Σ) ] in R(Σ, ∂Σ, SLn (C)) is fixed by ϕ∗ . In Appendix D we prove: Theorem 1.12. The tangent map of ϕ∗ at [ρn ] on R(Σ, ∂Σ, SLn (C)) has no eigenvalues of norm one. For n = 2 and ∂Σ = ∅, this was proved by M. Kapovich in [19]. Kapovich answered thereby a question of McMullen [26], namely, the point [ρ2 ] is a hy- perbolic fixed point of ϕ∗ . The relation with the rest of the paper comes from the formula (Proposi- tion D.3)  n−1 Y α,2k+1 det (dϕ∗ )[ρn ] − t Id = ∆M (ϕ) (t), k=1 where α : π1 (M (ϕ)) ↠ Z is induced by the natural fibration of the mapping torus over S1 with fiber Σ. Then the result follows from Theorem 1.11. 6 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI Summary of the proof. Most of the paper is devoted to prove Theorem 1.6 for ζ1 , . . . , ζr ∈ eiπQ . For that purpose, we consider sequences of closed man- ifolds Mp/q obtained by Dehn filling, that converge geometrically to M by Thurston’s hyperbolic Dehn filling theorem. The assumption ζ1 , . . . , ζr ∈ eiπQ allows us to chose the Dehn fillings Mp/q so that the twist χ : π1 (M ) → S1 in (1) factors through π1 (Mp/q ). Then the strategy is to apply Müller’s theo- rem [35] to the asymptotic behavior of the torsion of closed Dehn fillings Mp/q . Even if in Müller’s paper there is no twist, as χ is unitary we can modify the proof of Müller’s theorem by considering Ruelle functions twisted by χ. The key ingredient is a proof of a version of Fried’s theorem (Theorem 4.2) relat- ing the value of the Ruelle zeta function with the Reidemeister torsion of any closed manifold. Then we prove our main theorem for ζ1 , . . . , ζr ∈ eiπQ by an- alyzing the behavior of those twisted Ruelle zeta functions and the arguments of Müller’s proof under limits of Dehn fillings, as in [29]. To conclude the proof for arbitrary ζ1 , . . . , ζr ∈ S1 , we establish an intermediary result (Corol- lary 6.6) where Dehn fillings do not appear anymore. As Corollary 6.6 holds for ζ1 , . . . , ζr ∈ eiπQ , we use continuity and a density argument to extend it to any unitary ζ1 , . . . , ζr ∈ S1 . Organization of the paper. In Section 2 we define (a normalized version of) the twisted Alexander polynomial ∆α,n M , in particular we establish the basic results in cohomology required for that, based on Appendix C and we prove Theorem 1.11. Section 3 is devoted to construct Dehn fillings that approx- imate M , so that the character in (1) extends to them, as we assume that ζ1 , . . . , ζr ∈ eiπQ . In Section 4 we prove a twisted version (Theorem 4.2) of Fried’s Theorem and some properties of the Ruelle zeta functions that will be used later. Essentially this section extends the results of [35] to the case where a unitary twist is added. Section 5 discusses the behavior of Reidemeister torsion and Ruelle zeta functions under sequences of approximating Dehn fillings, and the proof of the main theorem is completed in Section 6. The paper contains four appendices. In Appendix A we present for small values of n the polynomials ∆α,n 41 (t) for the canonical surjection α from the knot group to the integers. In Appendix B we recall the main properties of combinatorial torsion. The results in L2 -cohomology needed in Section 2 are established in Appendix C. Finally, in Appendix D we establish Theorem 1.12. Acknowledgements. The authors warmly thank Nicolas Bergeron for many enlightening conversations on related topics and also Shu Shen for indicating them that Müller’s proof of Fried’s theorem should generalize with an additional unitary twist. They also thank the anonymous referee for his/her careful reading and meaningful suggestions that contributed to improve the paper. This work was supported by the Université Clermont Auvergne (projet Emer- gence de l’I-SITE en 2019). A large part of this work has been conducted while L.B. was hosted by the University of Geneva, supported by the NCCR SwissMAP (Swiss National foundation). L.B. is partially funded by the RTG 2491 ”Fourier Analysis and Spectral Theory”. M.H. and J.P. have been funded ASYMPTOTICS OF TWISTED POLYNOMIALS 7 by the MEC grant MTM2015-66165-P and J.P. by the MEC through “Marı́a de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0445). 2. Reidemeister torsion and twisted Alexander polynomials In this section we define the twisted Alexander polynomial for a cusped hy- perbolic manifold M , equipped with an epimorphism α : π1 (M ) ↠ Zr satisfying Assumption 1.2, and a lift ρ of the holonomy satisfying Assumption 1.3. Before defining the polynomial from the Reidemeister torsion of the pair (M, ᾱ ⊗ ρn ), we need to consider homology and cohomology of (M, χ ⊗ ρn ), as the represen- tation χ ⊗ ρn is a specialization of ᾱ ⊗ ρn . In Subsection 2.1 we study (co)-homology of (M, χ ⊗ ρn ) and (M, ᾱ ⊗ ρn ). In Subsection 2.1.3 we consider the Reidemeister torsion of (M, χ ⊗ ρn ). Then we define the twisted Alexander polynomial in Subsection 2.2 from the Reide- meister torsion of (M, ᾱ ⊗ ρn ). We express evaluations of the twisted Alexander polynomial at unit complex numbers as Reidemeister torsions of the represen- tations χ ⊗ ρn and we prove Theorem 1.11. Preliminary constructions and results on homology, cohomology and Reide- meister torsion are gathered in Appendix B, where we also recall some prop- erties of Symn−1 . This section also relies on results on L2 -cohomology from Appendix C. 2.1. Cohomology of (M, χ ⊗ ρn ) and (M, ᾱ ⊗ ρn ). When χ is trivial, the results of this subsection on cohomology twisted by χ ⊗ ρn = ρn can be found in [29, Section 4]. In Corollary C.7 (in Appendix C) we prove that the inclusion ∂M ,→ M induces a monomorphism (2) 0 → H 1 (M , χ ⊗ ρn ) → H 1 (∂M , χ ⊗ ρn ). Thus to understand the cohomology of M we need to understand the cohomol- ogy of the peripheral tori Ti2 , i = 1, . . . , l, where ∂M = T12 ⊔ · · · ⊔ Tl2 is the decomposition in connected components. In particular l is the number of cusps of M . 2.1.1. Peripheral cohomology. Lemma 2.1. If Assumptions 1.2 and 1.3 hold, then for any peripheral torus Ti2 ( 0 if n even or χ(π1 (Ti2 )) ̸= {1}, (a) dimC H 0 (Ti2 , χ ⊗ ρn ) = 1 if n odd and χ(π1 (Ti2 )) = {1}. (b) dimC H 0 (Ti2 , χ ⊗ ρn ) = dimC H 2 (Ti2 , χ ⊗ ρn ) = 12 dimC H 1 (Ti2 , χ ⊗ ρn ). Proof. (a) To compute its dimension, we view H 0 (Ti2 , χ ⊗ ρn ) as the space of 2 invariants (Cn )χ⊗ρn (π1 (Ti )) . For any non-trivial element γ in π1 (Ti2 ) its image ρ(γ) by the holonomy is parabolic, with trace 2ϵγ , for some ϵγ = ±1. Hence χ(γ) ρn (γ) has only one eigenspace, with dimension one and eigenvalue χ(γ)ϵn−1 γ (see Remark B.7 in Appendix B). 8 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI By Assumptions 1.2 and 1.3, when n is even or when χ(π1 (Ti2 )) ̸= {1}, there is always an element γ ∈ π1 (Ti2 ) that satisfies χ(γ)(ϵγ )n−1 ̸= 1, thus dimC (H 0 (Ti2 , χ ⊗ ρn )) = 0. In case n is odd and χ(π1 (Ti2 )) = {1}, then χ(γ)(ϵγ )n−1 = 1 for every γ ∈ π1 (Ti2 ). (b) Poincaré duality induces a nondegenerate pairing, see Remark B.6: H 0 (Ti2 , χ ⊗ ρn ) × H 2 (Ti2 , χ ⊗ ρn ) → C. Notice that we use χ and its inverse χ (its complex conjugate). Hence by (a): dimC (H 0 (Ti2 , χ ⊗ ρn )) = dimC (H 0 (Ti2 , χ̄ ⊗ ρn )) = dimC (H 2 (Ti2 , χ ⊗ ρn )). Then the assertion follows from these computations and vanishing of the Euler characteristic of T 2 . □ To compute further cohomology groups, we discuss L2 -forms, in particu- lar de Rham cohomology. Let Eχ⊗ρn denote the flat bundle on M (or on any submanifold) twisted by the representation χ ⊗ ρn , see Appendix C. For each peripheral torus Ti2 , let Ti2 × [0, ∞) ⊂ M denote the cusp, which is an end of M , and consider the space of forms on the cusp valued on the bun- dle Eχ⊗ρn , Ω∗ (Ti2 × [0, ∞), Eχ⊗ρn ). It is equipped with a metric as in Ap- pendix C, in particular we may talk about L2 -forms, as forms with a finite norm. A cohomology class is called L2 if represented by an L2 -form, and the subspace of L2 -cohomology classes in H i (Tj2 × [0, +∞), Eχ⊗ρn ) is denoted by H i (Tj2 × [0, +∞), Eχ⊗ρn )L2 . Lemma 2.2. Assume that n is odd and that the restriction of the character χ(π1 (Tj2 )) is trivial. Then: (a) Every class in H i (Tj2 × [0, +∞), Eρn )L2 is represented by a form v ⊗ ω, 2 where ω is an i-form on T 2 and v ∈ (Cn )ρ(π1 (T )) . (b) dimC H 0 (T 2 × [0, +∞), Eρn )L2 = dimC H 1 (T 2 × [0, +∞), Eρn )L2 = 1 and H 2 (T 2 × [0, +∞), Eρn )L2 = 0. Proof. In [27, Lemma 3.3] the same statement is proved for the composition with the adjoint representation on the Lie algebra sln (C), Ad ◦ Symn−1 . Recall that 2 from Lemma 2.1 the space of invariants (Cn )ρn (π1 (Ti )) is one-dimensional, for n odd. Then the lemma follows from Clebsch-Gordan formula, see Equation (33) in Appendix B. □ 2.1.2. Cohomology of M. In this paragraph we prove the properties of the co- homology of M required for our definition of twisted Alexander polynomial. Theorem 2.3. Let M , ρ, and α satisfy Assumptions 1.2 and 1.3. (a) If n is even or if χ is non trivial on every peripheral subgroup, then H ∗ (M, χ⊗ ρn ) = 0. (b) If n is odd, then dimC H 1 (M ; χ ⊗ ρn ) = dimC H 2 (M ; χ ⊗ ρn ) is the number of peripheral subgroups to which the restriction of χ is trivial. Proof. For both (a) and (b), first notice that M has the homotopy type of a 2–complex, hence H i (M, χ ⊗ ρn ) = 0 for any i ≥ 3. In addition, the space of invariants H 0 (M, χ⊗ρn ) ∼ = (Cn )χ⊗ρn (π1 (M )) also vanishes since ρn is irreducible. ASYMPTOTICS OF TWISTED POLYNOMIALS 9 To prove (a), the vanishing of H 1 (M, χ ⊗ ρn ) is a consequence of Lemma 2.1 and the monomorphism in (2). We conclude that H 2 (M, χ ⊗ ρn ) = 0 because the Euler characteristic χ(M ) is zero. For (b) assume that n is odd. We use that H 1 (M, χ ⊗ ρn ) has no L2 -forms, by Theorem C.1, hence by Lemmas 2.1 and 2.2, the map H 1 (M , χ ⊗ ρn ) → H 1 (Ti2 , ρn ) has rank at most one if χ|π1 (T 2 ) is trivial, and 0 otherwise. Thus, if s is the i number of peripheral tori Ti2 where χ restricts trivially, by (2), dimC H 1 (M ; χ ⊗ ρn ) ≤ s. On the other hand, using duality twice (Poincaré and homology/cohomology) H 3 (M , ∂M ; χ ⊗ ρn ) = 0 and, by the long exact sequence of the pair, H 2 (M , χ ⊗ ρn ) ↠ H 2 (∂M , χ ⊗ ρn ) is a surjection. Hence by Lemma 2.1: dimC H 2 (M ; χ ⊗ ρn ) ≥ s. Finally, as χ(M ) = 0, dimC H 1 (M ; χ ⊗ ρn ) = dimC H 2 (M ; χ ⊗ ρn ) = s. □ We need to precise the bases for the cohomology groups. It is easier to describe them for the homology groups. For a torus Ti2 such that χ(π1 (Ti2 )) is trivial, if hi ∈ Cn is invariant by π1 (Ti2 ), then the class of hi ⊗ Ti2 is a well defined element in H2 (Ti2 , χ ⊗ ρn ), and so is hi ⊗ li in H1 (Ti2 , χ ⊗ ρn ). Lemma 2.4. Assume that χ is trivial precisely on π1 (T12 ), . . . , π1 (Ts2 ). Let hi ∈ Cn be non-zero and invariant by π1 (Ti2 ), for i = 1, . . . , s. Let i∗ denote the map induced by inclusion in homology. Then: (a) {i∗ (h1 ⊗ T12 ), . . . , i∗ (hs ⊗ Ts2 )} is a basis for H2 (M, χ ⊗ ρn ). (b) {i∗ (h1 ⊗ l1 ), . . . , i∗ (hs ⊗ ls )} is a basis for H1 (M, χ ⊗ ρn ). Proof. For i = 1, . . . , s, since χ is trivial on Ti2 , (a) follows from the isomor- phisms 2 H2 (Ti2 , ρn ) ∼ = H 0 (Ti2 , ρn ) ∼ = (Cn )ρn (π1 (T )) , and from the isomorphism i 0 → H2 (T12 , ρn ) ⊕ · · · ⊕ H2 (Ts2 , ρn ) → ∗ H2 (M , χ ⊗ ρn ) → 0 coming from the long exact sequence in homology. For (b) we claim first that hj ⊗ lj is non-zero in H1 (Tj2 , ρn ), for j = 1, . . . , s. We prove the claim by computing cellular homology explicitly. For this pur- pose, chose a cell decomposition of the torus with one 0-cell, one 2-cell and two 1-cells, that are loops, and assume that one of these loops represents lj . Fur- thermore, using the description of ρn (π1 (Tj2 )), a straightforward computation shows that hj ⊗ lj is not a boundary, see (7) below. Alternatively, as in the proof of Lemma 2.2, an equivalent statement is proved in [27, Lemma 3.4] for Ad ◦ Symn−1 , and our claim follows from Clebsch-Gordan formula (33). From the proof of Theorem 2.3 we have an injection i∗ 0 → H 1 (M, χ ⊗ ρn )) → H 1 (T12 , ρn ) ⊕ · · · ⊕ H 1 (Ts2 , ρn ) 10 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI and a surjection i H1 (T12 , ρn ) ⊕ · · · ⊕ H1 (Ts2 , ρn ) → ∗ H1 (M, χ ⊗ ρn ) → 0. We also have naturality with the pairing between homology and cohomology (see Appendix B): ⟨i∗ (−), −⟩ = ⟨−, i∗ (−)⟩ where the pairing on ∂M is understood to be the sum of pairings on each component Ti2 . Thus, by Poincaré duality, (3) ker(i∗ ) = im(i∗ )⊥ . 2 By Remark B.7, hj ∈ (Cn )ρn (π1 (Ti )) is isotropic for the ρn -invariant bilinear form. Hence by Lemma 2.2(a), the pairing between hj ⊗ lj and any L2 -class in H 1 (Tj2 × [0, ∞), ρn ) vanishes. Thus, by dimension considerations: ⊥ (4) ⟨h1 ⊗ l1 , . . . , hs ⊗ ls ⟩ = H 1 (∂M × [0, ∞), χ ⊗ ρn )L2 . Furthermore, by Theorem C.1: im(i∗ ) ∩ H 1 (∂M × [0, ∞), χ ⊗ ρn )L2 = 0. As dim im(i∗ ) = dim H 1 (∂M × [0, ∞), χ ⊗ ρn )L2 = 1 2 dim H 1 (∂M , χ ⊗ ρn ) = s, (5) im(i∗ ) ⊕ H 1 (∂M × [0, ∞), χ ⊗ ρn )L2 = H 1 (∂M , χ ⊗ ρn ). Finally, (5), (3) and (4) yield ker(i∗ ) ⊕ ⟨h1 ⊗ l1 , . . . , hs ⊗ ls ⟩ = H1 (∂M , χ ⊗ ρn ), in particular ⟨h1 ⊗l1 , . . . , hs ⊗ls ⟩∩ker(i∗ ) = 0. Thus {i∗ (h1 ⊗l1 ), . . . , i∗ (hs ⊗ls )} are linearly independent, hence a basis. □ When χ is trivial, Lemma 2.4 is [29, Proposition 4.6]. 2.1.3. Reidemeister torsion. We use the convention of [31] and [43] for Rei- demeister torsion, so that it is compatible with the standard convention for analytic torsion but it is the reciprocal to the twisted Alexander polynomial. See Appendix B. As a consequence of Theorem 2.3, we have that | tor(M, χ ⊗ ρn )| ∈ R>0 is well defined when n is even or when n is odd and the restriction of χ to every peripheral torus is nontrivial. The absolute value in | tor(M, χ ⊗ ρn )| is needed, because χ introduces an indeterminacy of the argument, more precisely tor(M, χ ⊗ ρn ) is only defined up to multiplication by a unit complex number. In the non-acyclic case we shall consider | tor(M, χ ⊗ ρn ; b1 , b2 )|, where b1 and b2 are the basis of the homology provided by Lemma 2.4. Notice that | tor(M, χ⊗ρn ; b1 , b2 )| is independent on the vectors hi in Lemma 2.4. This follows since (Cn )π1 (Ti ) is one-dimensional and | tor(M, χ ⊗ ρn ; b1 , b2 )| does not change if we replace the vector hi in Lemma 2.4 by a multiple, since in the formula for the torsion the additional factors cancel out. ASYMPTOTICS OF TWISTED POLYNOMIALS 11 2.2. Twisted Alexander polynomials. In this subsection we introduce the twisted Alexander polynomial for a finite volume 3-manifold M (connected and orientable) and an epimorphism α : π1 (M ) ↠ Zr as in the introduction. We define it as the inverse of a Reidemeister torsion of ᾱ ⊗ ρn , where ᾱ(γ) = α (γ) α (γ) t1 1 · · · tr r , ∀γ ∈ π1 (M ). We use the notation C[t±1 ] = C[t±1 ±1 1 , . . . , tr ] for the ring of Laurent polyno- mials and C(t) = C(t1 , . . . , tr ) for its field of fractions. Lemma 2.5. If Assumptions 1.2 and 1.3 hold, then H ∗ (M, ᾱ⊗ρn ) = H∗ (M, ᾱ⊗ ρn ) = 0. Proof. Choose ζ1 , . . . ζr ∈ S1 generic so that the corresponding homomorphism χ : π1 (M ) → S1 in (1) has non trivial restriction on each peripheral subgroup π1 (Ti2 ). Then by Theorem 2.3 we have H ∗ (M, χ ⊗ ρn ) = 0. Using combina- torial cohomology, notice that the matrices used to compute H ∗ (M, χ ⊗ ρn ) are the evaluation at (t1 , . . . , tr ) = (ζ1 , . . . , ζr ) of the matrices used to compute H ∗ (M, ᾱ⊗ρn ). In addition, the C(t)-rank of a matrix with coefficients in C[t±1 ] is larger than or equal to its C-rank after evaluation at (t1 , . . . , tr ) = (ζ1 , . . . , ζr ). Thus, by acyclicity of χ ⊗ ρn , the C-rank of the matrices used to compute coho- mology is maximal, hence the C(t)-rank of these matrices before evaluation at (t1 , . . . , tr ) = (ζ1 , . . . , ζr ) is also maximal, and therefore ᾱ ⊗ ρn is acyclic. □ For each peripheral torus Ti2 chose mi so that π1 (Ti2 ) = ⟨li , mi ⟩ ∼ = Z2 , where r li is an α-longitude. Writing α(mi ) = (α1 (mi ), . . . , αr (mi )) ∈ Z , we denote α (mi ) tα(mi ) = ᾱ(mi ) = t1 1 · · · tαr r (mi ) . Definition 2.6. The twisted Alexander polynomial of (M, ᾱ ⊗ ρn ) is  1  for n even, tor(M, ᾱ ⊗ ρn )  α,n ∆M (t1 , . . . , tr ) := 1   for n odd. tor(M, ᾱ ⊗ ρn )(tα(m1 ) − 1) · · · (tα(ml ) − 1) It is an element of C(t) = C(t1 , . . . , tr ), a quotient of polynomials in the variables t1 , . . . , tr , defined up to sign and up to multiplication by monomials tn1 1 · · · tnr r . In Corollary 2.9 we prove that it is a Laurent poynomial, an element of C[t±1 ] = C[t±1 ±1 1 , . . . , tr ]. Remark 2.7. For even dimensional representations, this is the same as Wada’s polynomial [44], using Kitano’s Theorem [21]. For odd dimensional representa- tions, it is a normalization of the latter. We view C[t±1 ]n ∼= C[t±1 ] ⊗ Cn as a π1 (M )-module via ᾱ ⊗ ρn , and denote it by C[t ]ᾱ⊗ρn . For the definition of the order of a C[t±1 ]-module, see [43]. ±1 n Lemma 2.8. (a) Up to units in C[t±1 ]: 1 = orderC[t±1 ] H1 (M, C[t±1 ]nᾱ⊗ρn ) . tor(M, ᾱ ⊗ ρn ) (b) For n odd, orderC[t±1 ] H1 (M, C[t±1 ]nᾱ⊗ρn ) ∈ (tα(m1 ) −1) · · · (tα(ml ) −1)C[t±1 ]. 12 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI Corollary 2.9. The twisted Alexander polynomial is a Laurent polynomial: ∆α,n ±1 M ∈ C[t ]. Before proving Lemma 2.8 we need the following lemma: Lemma 2.10. Assume that n is odd. (a) For each peripheral torus Tj2 , H1 (Tj2 , C[t±1 ]nᾱ⊗ρn ) ∼ = C[t±1 ]/(tα(mj ) − 1). In addition, it is generated by the image of hj ⊗ lj via the natural map (Cn )ρn (lj ) ⊗ H1 (Sj1 ; Z) → H1 (Tj2 , C[t±1 ]nᾱ⊗ρn ), where (Cn )ρn (lj ) is the (1-dimensional) subspace invariant by ρn (lj ), 0 ̸= hj ∈ (Cn )ρn (lj ) , and Sj1 is a circle representing lj . (b) The inclusion induces a monomorphism H1 (∂M , C[t±1 ]nᾱ⊗ρn ) ,→ H1 (M , C[t±1 ]nᾱ⊗ρn ). Proof. To prove (a) consider Sj1 ×R → Tj2 the infinite cyclic covering correspond- ing to ᾱ with deck transformation group Z generated by τ : Sj1 × R → Sj1 × R. There is a long exact sequence in homology [32, 38] τ −1 · · · → Hi (Sj1 × R, C[t±1 ]nᾱ⊗ρn ) −→ ∗ Hi (Sj1 × R, C[t±1 ]nᾱ⊗ρn ) → Hi (Tj2 , C[t±1 ]nᾱ⊗ρn ) → · · · As ᾱ(π1 (Sj1 )) = 1, Hi (Sj1 × R, C[t±1 ]nᾱ⊗ρn ) ∼ = C[t±1 ] ⊗ Hi (Sj1 , Cnρn ) and the ±1 action of τ∗ on C[t ] corresponds to multiplication by tα(mj ) . Furthermore Hi (Sj1 , Cnρn ) = 0 for i ̸= 0, 1 and H1 (Sj1 , Cnρn ) ∼ = H 0 (Sj1 , Cnρn ) ∼ = (Cn )ρn (lj ) . Then (a) follows from these considerations. In the proof of (b), we need two preliminary steps. The first one is to show that (6) H1 (∂M , C[t±1 ]n ) ⊗χ C → H1 (M , C[t±1 ]n ) ⊗χ C is injective for every χ ∈ Hom(Zr , S1 ). By Lemma 2.4 (b) and Assertion (a) in this lemma, the composition H1 (∂M , C[t±1 ]nᾱ⊗ρn ) ⊗χ C → H1 (M , C[t±1 ]nᾱ⊗ρn ) ⊗χ C → H1 (M , χ ⊗ ρn ) is an isomorphism, hence (6) is a monomorphism. The second step is to show that, if we consider the algebraic subvariety Vj ⊂ (C∗ )r defined by tα(mj ) = 1, then Vj ∩ (S1 )r is Zariski dense in Vj . To prove this, by the action of SL(r, Z) we may assume α(mj ) = (k, 0, . . . , 0) for some k ∈ Z \ {0}. Then the subvariety Vj ⊂ (C∗ )r is defined by tk1 = 1 and the assertion is clear. After these preliminary claims, we prove (b): let a ∈ ker(H1 (∂M , C[t±1 ]nᾱ⊗ρn ) → H1 (M , C[t±1 ]nᾱ⊗ρn )) be an element of the kernel, consider aj ∈ H1 (Tj2 , C[t±1 ]nᾱ⊗ρn ) ∼ = C[t±1 ]/(tα(mj ) − 1) ASYMPTOTICS OF TWISTED POLYNOMIALS 13 its projection to the j-th component. Viewing C[t±1 ]/(tα(mj ) −1) as the function ring of Vj ⊂ (C∗ )r , injectivity of (6) implies that aj vanishes when evaluated on Vj ∩ (S1 )r . By Zariski density of Vj ∩ (S1 )r in Vj , aj vanishes on the whole Vj , hence aj = 0. □ Proof of Lemma 2.8. By Turaev [43] we have an equality (up to units in C[t±1 ]): 1 order(H1 (M, C[t±1 ]nᾱ⊗ρn )) = tor(M, ᾱ ⊗ ρn ) order(H0 (M, C[t±1 ]nᾱ⊗ρn )) order(H2 (M, C[t±1 ]nᾱ⊗ρn )) As the manifold M has the simple homotopy type of a 2-dimensional com- plex, we have that order(H2 (M, C[t±1 ]nᾱ⊗ρn )) = 1. Hence it suffices to prove that order(H0 (M, C[t±1 ]nᾱ⊗ρn )) = 1. So, looking for a contradiction, assume order(H0 (M, C[t±1 ]nᾱ⊗ρn )) ̸= 1 and pick λ = (λ1 , . . . , λr ) ∈ (C∗ )r a root of the polynomial order(H0 (M, C[t±1 ]nᾱ⊗ρn )) (this polynomial defines a hypersur- face in Cr , and since the order is defined up to factors t±1 ∗ r i , it intersects (C ) non trivially). Evaluation at (t1 , . . . , tr ) = (λ1 , . . . , λr ) defines a morphism α (γ) α (γ) Λ : π1 (M ) → C∗ , by Λ(γ) = λ1 1 · · · λr r . Since λ is a root of the poly- nomial order(H0 (M, C[t±1 ]nᾱ⊗ρn )), H0 (M, Λ ⊗ ρn ) ̸= 0. This means that Cn has non-zero coinvariants for the action of Λ ⊗ ρn . By duality, Cn has non- zero invariants by the action of Λ ⊗ ρn , in particular Cn has a proper subspace preserved by ρn . By Zariski density of the holonomy representation, this con- tradicts irreducibility of Symn−1 . This proves (a). For (b) construct the twisted chain complex from a CW–complex K, with |K| = M : C∗ (K, C[t±1 ]nᾱ⊗ρn ) = C[t±1 ]n ⊗ᾱ⊗ρn C∗ (K, e Z). We may assume furthermore that there are 2-cells of K, denoted by e2i , repre- senting Ti2 and 1-cells denoted by e1i representing li . Chose respective lifts ẽ2i of e2i , and ẽ1i of e1i , in the universal cover Ke that correspond to the same con- nected component of the lift of the peripheral torus Ti2 in the universal covering. Moreover, chose ẽ1i to be adjacent to ẽ2i , so that (7) ∂ẽ2 = (mi − 1)ẽ1 + (1 − li )f˜1 i i i for some other 1-cell fi1 . Notice that ⟨mi , li ⟩ ∼ = π1 (Ti2 ), with ᾱ(li ) = 1 and ᾱ(mi ) = tα(mi ) . Chose also hi ∈ Cn a non-zero element invariant by ρn (π1 (Ti2 )). Let L∗ ⊂ C∗ (K, C[t±1 ]nᾱ⊗ρn ) be the C[t±1 ]-subcomplex generated by the ele- ments hi ⊗ ẽji , j = 1, 2, i = 1, . . . , l. By the choice of lifts: (8) ∂(hi ⊗ ẽ2i ) = (tα(mi ) − 1)hi ⊗ ẽ1i , up to sign and up to powers of tα(mi ) , and ∂(hi ⊗ ẽ1i ) = 0 since e1i represents li . Hence L∗ is a subcomplex of C∗ (K, ᾱ ⊗ ρn ). Moreover it follows from (8) that this complex L∗ is also acyclic as a complex of C(t)-vector spaces. We have a short exact sequence of acyclic complexes of C(t)-vector spaces: (9) 0 → L∗ → C∗ (K, C[t±1 ]nᾱ⊗ρn ) → C∗ (K, C[t±1 ]nᾱ⊗ρn )/L∗ → 0. Furthermore we can construct geometric bases à la Milnor for C∗ (K, C[t±1 ]nᾱ⊗ρn ) that include the elements hi ⊗ ẽji , Definition B.2. Thus there are compatible 14 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI geometric bases in the sequence and the multiplicativity formula for the torsion [31, Theorem 3.2] provides the equality: tor(M, ᾱ ⊗ ρn ) = tor(L∗ ) tor(C∗ (K, C[t±1 ]nᾱ⊗ρn )/L∗ ). From (8) the contribution of tor(L∗ ) is (tm1 − 1) · · · (tml − 1). Finally, we show that the zeroth and second homology groups of C∗ (K, C[t±1 ]nᾱ⊗ρn )/L∗ vanish (hence its torsion is the inverse of a Laurent polynomial). For that purpose, notice that from Lemma 2.10 (a) and (8) we have a natural isomorphism H1 (L∗ ) ∼ = H1 (∂M, C[t±1 ]n ). ᾱ⊗ρn Hence by Lemma 2.10 (b) we have an injection induced by inclusion H1 (L∗ ) ,→ H1 (M, C[t±1 ]nᾱ⊗ρn ). Using this monomorphism, the long exact sequence in homology corresponding to (9), and the vanishing of Hi (M, C[t±1 ]nᾱ⊗ρn ) for i = 0, 2, it follows that zeroth and second homology groups of C∗ (K, C[t±1 ]nᾱ⊗ρn )/L∗ also vanish. □ Proposition 2.11. For n even, 1 |∆α,n M (ζ1 , . . . , ζr )| = . | tor(M, χ ⊗ ρn )| For n odd, 1 1 |∆α,n Y M (ζ1 , . . . , ζr )| = . | tor(M, χ ⊗ ρn ; b1 , b2 )| |ζ α(mi ) − 1| ζ α(mi ) ̸=1 In the proposition, b2 and b1 are the basis in homology of Lemma 2.4, accord- ing to the components where χ(π1 (Ti2 )) is trivial, α(mi ) ∈ Zr is a generator of α (m ) α (m ) the image of α(π1 (Ti2 )). We use the notation ζ α(mi ) = ζ1 1 i · · · ζr r i . The product in the odd case runs on the components where χ(π1 (Ti2 )) is non trivial. Proof. In the acyclic case (when n is even or when χ is non-trivial on each peripheral subgroup) the proposition follows from naturality, cf. [31, § 6]. The proof of the non-acyclic case is very similar to the proof of Lemma 2.8 b), but the subcomplex L∗ is only constructed from the peripheral tori for which the restriction of χ is trivial. Namely, assume that n is odd and that χ is trivial precisely on π1 (T12 ), . . . , π1 (Ts2 ). Then, choosing a CW-complex K as in the proof of Lemma 2.8, we take L∗ to be the subcomplex of C∗ (K, χ⊗ρn ) generated by elements hi ⊗ ẽji , j = 1, 2, i = 1, . . . , s. In this case the boundary operator is zero in L∗ , and a geometric basis for L∗ is precisely a lift of the basis b1 and b2 for H∗ (M, χ ⊗ ρn ), by Lemma 2.4. From the defining short exact sequence and the previous consideration, it follows that C∗ (K, χ ⊗ ρn )/L∗ is acyclic and its torsion equals | tor(M, χ ⊗ ρn ; b1 , b2 )|. Then the lemma follows from naturality applied to C∗ (K, C[t±1 ]n )/L′∗ , where L′∗ is the subcomplex C[t±1 ]-generated by the same elements as L∗ , hi ⊗ ẽji , j = 1, 2, i = 1, . . . , s. □ For ρ3 = Sym2 ◦ρ = Ad ◦ρ and χ trivial, Proposition 2.11 has been proved by Yamaguchi in [48] for knot exteriors and by Dubois and Yamaguchi in [9] in a more general setting. ASYMPTOTICS OF TWISTED POLYNOMIALS 15 Proof of Theorem 1.11. Proposition 2.11 expresses the evaluation of the twisted Alexander polynomial at unitary complex numbers as a Reidemeister torsion, which is an element of C∗ , up to multiplication by a unit complex number. In particular it is not zero. □ 3. Dehn fillings and rational twists In this section we consider Dehn fillings on M and sequences of those fillings that converge geometrically to M . In the first subsection we discuss compati- bility conditions with the twist and with the lift of the holonomy (equivalently the spin structure). In particular we restrict to rational Dehn fillings. In the second subsection we discuss surgery formulas for the torsion. Definition 3.1. A unitary twist χ : π1 (M ) → S1 ⊂ C is called rational if it takes values in e2πi Q . Assumption 3.2. In this section we assume that: (a) χ is rational, (b) χ restricted to each peripheral subgroup π1 (Tj2 ) ⊂ π1 (M ) is nontrivial. Assumption 3.2 (a) implies that χ induces a twist of certain Dehn fillings, as we will explain in the next subsection. Assumption 3.2 (b) allows to simplify the formulas when n is odd. The results in this section could be generalized without Assumption (b) by considering bases in homology, but this simpler version is sufficient for applying them in Section 6. 3.1. Compatible Dehn fillings. For each peripheral torus Ti2 we have an α- longitude, namely an element li ∈ π1 (Ti2 ) that generates the kernel of α|π1 (T 2 ) , i by Assumption 1.2. We fix a basis for the fundamental group of the peripheral group that contains this element: ⟨mi , li ⟩ = π1 (Ti2 ) ∼ = Z2 . As trace(ρ(li )) = −2 (Assumption 1.3), we may chose trace(ρ(mi )) = +2, after replacing mi by mi li if needed. Once we have fixed the mi and li , given pairs of coprime integers pi , qi , the Dehn filling with filling meridians pi mi + qi li is denoted by Mp1 /q1 ,...,pl /ql . To simplify notation we write Mp/q := Mp1 /q1 ,...,pl /ql . The inclusion map is denoted by i : M → Mp/q , it induces an epimorphism i∗ : π1 (M ) → π1 (Mp/q ). Another convention is that (p, q) → ∞ means that p2i + qi2 → +∞ for i = 1, . . . , l. By Thurston’s hyperbolic Dehn filling theorem, when p2i + qi2 is sufficiently large for each i = 1. . . . , l, then Mp/q is hyperbolic. The (conjugacy class of the) holonomy of Mp/q composed with i∗ converges to the (conjugacy class of the) holonomy of M in the set of conjugacy classes of such representations Hom(π1 (M ), PSL(2, C))/ PSL(2, C). As we work with representations in SL(2, C), we need to impose compatibility conditions on the Dehn filling to get the same conclusion for the lifts. We shall also impose conditions so that the rational twist χ factors through i∗ to a twist of Mp/q . 16 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI Definition 3.3. The Dehn filling Mp/q = Mp1 /q1 ,...,p1 /ql is called compatible with χ and ρ if, for each i = 1, . . . , l: (1) χ(mpi i ) = 1, and (2) qi ≡ 1 mod 2. Since χ(li ) = 1 by Assumption 1.2, Condition (1) in the definition amounts to say that the twist of M factors through π1 (Mp/q ). As we assume χ rational, this is achieved by taking pi ∈ order(χ(mi ))Z. Condition (2) in the definition is explained by the following lemma. Lemma 3.4. For an infinite family of compatible Dehn fillings Mp/q such that (p, q) → ∞, there exists a lift of the holonomy ρp/q of Mp/q in SL(2, C) such that lim [ρp/q ◦ i∗ ] = [ρ] (p,q)→∞ in Hom(π1 (M ), SL(2, C))/ SL(2, C), where ρ is the lift of the holonomy of the complete structure on M . Proof. As we chose mi so that trace(ρ(mi )) = +2, Condition (2) in Defini- tion 3.3 means trace(ρ(pi mi + qi li )) = −2. Using the natural bijection between spin structures and lifts of the holonomy, this is precisely the condition required for a spin structure on M to extend to M p/q , see [29] for instance. In terms of lifts of representations, this is the compatibility condition for the lifts of the de- formation for the holonomy in Thurston’s hyperbolic Dehn filling theorem. □ We shall use the following notation: ρp/q n := Sym n−1 ◦ρp/q : π1 (Mp/q ) → SLn (C), ϱp/q p/q n := ρn ◦ i∗ : π1 (M ) → SLn (C). p/q Thus for large p2i + qi2 , i = 1, . . . , l, [ϱn ] lies in a neighborhood of [ρn ] in Hom(π1 (M ), SLn (C))/ SLn (C). 3.2. Dehn filling formula. We shall only consider compatible Dehn fillings. In particular χ factors through π1 (Mp/q ). Since Mp/q is closed, Corollary C.7 p/q p/q yields that H ∗ (Mp/q , χ ⊗ ρn ) vanishes. However H ∗ (M, χ ⊗ ϱn ) does not need to vanish, we have: Lemma 3.5. If Assumption 3.2 holds for χ, then for sufficiently large p2i + qi2 , i = 1, . . . , l, we have: H ∗ (M, χ ⊗ ϱp/q n ) = 0. Proof. We first claim that the inclusion ∂M → M induces a monomorphism (10) 0 → H ∗ (M, χ ⊗ ϱp/q ∗ p/q n ) → H (∂M , χ ⊗ ϱn ). The proof of the claim is analogous to Theorem 2.3, but simpler, as we no not need a theorem in L2 -cohomology but Mayer-Vietoris exact sequence. More precisely, we apply Mayer-Vietoris to the pair (M , V ), where V is the union of the attached solid tori in the Dehn filling, so that V ∩ M = ∂M and V ∪ M = ASYMPTOTICS OF TWISTED POLYNOMIALS 17 p/q Mp/q . As H ∗ (Mp/q , χ ⊗ ρn ) = 0, Mayer-Vietoris long exact sequence yields an isomorphism induced by inclusion maps: p/q ∼ (11) H ∗ (M ; χ ⊗ ϱp/q ∗ ∗ p/q n ) ⊕ H (V ; χ ⊗ ϱn ) = H (∂M ; χ ⊗ ϱn ), which proves that (10) is a monomorphism. Using this, to prove the lemma we show the vanishing of the cohomology of each peripheral 2-tori Tj2 . For this purpose, we use that p/q ∼ n (χ⊗ϱn )(π1 (Tj2 )) H 0 (Tj2 , χ ⊗ ϱp/q n ) = (C ) p/q 2 p/q where (Cn )(χ⊗ϱn )(π1 (Tj )) denotes the subspace invariant by (χ ⊗ ϱn )(π1 (Tj2 )). If γj denotes the soul of the j-th attached solid torus Vj , then, after conjugation,  λ(γ )/2  ρp/q (γj ) = e j 0 (12) −λ(γj )/2 0 e where λ(γj ) is the complex length, λ(γj ) = ℓ(γj ) + i θ(γj ), and ℓ(γj ) > 0 is the (real) length of the geodesic γj in Mp/q . Therefore, as we assume that χ is non trivial on every peripheral subgroup: p/q p/q )(π1 (Tj2 )) (Cn )(χ⊗ϱn = (Cn )χ(γj )ρn (γj ) = 0. p/q As H 0 (Tj2 , χ ⊗ ϱn ) = 0, from Poincaré duality and χ(Tj2 ) = 0 we deduce the p/q vanishing of the cohomology groups in every dimension, H ∗ (Tj2 , χ ⊗ ϱn ) = 0, which concludes the lemma. □ By applying Milnor’s formula on the torsion of a Mayer-Vietoris sequence, we have (further details on the proof can be found in [29, Lemmas 5.7 and 5.11]): Proposition 3.6. Under Assumption 3.2 for χ: l n−1 Y Y λ(γj ) tor(Mp/q , χ ⊗ ρp/q p/q n ) = tor(M, χ ⊗ ϱn ) (e 2 (n−1−2k) χ(mj ) − 1). j=1 k=0 where the complex length λ(γj ) is defined in (12). We want to consider sequences of admissible Dehn fillings Mp/q such that (p, q) → ∞, hence by Lemma 3.4 ϱp/q → ρ. By naturality of the torsions: Lemma 3.7. Under Assumption 3.2 for χ: lim tor(M, χ ⊗ ϱp/q n ) = tor(M, χ ⊗ ρn ). (p,q)→∞ As mentioned at the beginning of the section, if we did not assume non- triviality of χ on each peripheral torus, a generalization of Proposition 3.6 and Lemma 3.7 would also holds, but one would need to consider bases in homology. 18 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI 4. Ruelle zeta functions and Fried’s theorem In this section we establish the analytic tools that will be used in Section 6 to prove our version (Theorem 6.1) of a result of Müller, relating the logarithm of the Reidemeister torsion of a closed hyperbolic 3-manifold, its volume and some geometric quantity expressed as a function on the lengths of the geodesics. The new point in our Theorem 6.1 is the appearance of a unitary twist χ. In this section we fix a closed hyperbolic 3-manifold N with fundamental group π1 (N ) = Γ and a lift of its holonomy ρ : Γ → SL2 (C). For any natural integer n, we denote by ρn : Γ → SLn (C) the (n − 1)-th symmetric power of the holonomy ρ. Finally, we fix a group homomorphism χ : Γ → S1 (sometimes called the twist). The two statements we need to prove in this section are the following, the definitions of the two different Ruelle zeta functions come right after in Subsec- tion 4.1. Proposition 4.1. For any integer k, the unitary Ruelle zeta function Rχ,k extends meromorphically to the whole complex plane. Moreover, it satisfies the functional equation: |Rχ,k (s)| = e4 Vol(N )s/π |Rχ,−k (−s)|. Theorem 4.2 (Fried’s Theorem). The Ruelle zeta function Rχ⊗ρn extends meromorphically to C, it is holomorphic at s = 0 and |Rχ⊗ρn (0)| = | tor(N, χ ⊗ ρn )|2 . 4.1. Ruelle zeta functions. In this paragraph we introduce two Ruelle zeta functions Rχ⊗ρn and Rχ,k (we call the latter the unitary Ruelle zeta function). Those functions are defined as in [35], except for the appearance of the twist χ. Since the twist has modulus one, it does not modify the arguments from [35] showing the convergence of the defining series, hence we will not discuss this point. On the other hand, the unitary Ruelle zeta function, as well as Selberg zeta functions with a twist that we do not define here, are extensively studied in Bunke’s and Olbrich’s book [6]. The basics of the study of those functions are due to Fried in his seminal article [12]. Any closed geodesic ϕ in the manifold N corresponds uniquely to a conjugacy class [γ] in the set [Γ] of conjugacy classes in the fundamental group of N . A prime element in Γ is an element that cannot be written as a non-trivial power of another one. We will use the notation PC(N ) for the set of conjugacy classes [γ] of prime elements γ of Γ (equivalently, the set of prime closed geodesics in N ). Any γ is mapped by the holonomy representation to a conjugate of the  λ(γ)/2 e 0 matrix −λ(γ)/2 where λ(γ) = ℓ(γ) + iθ(γ) is the complex length of 0 e the geodesic ϕ represented by γ. In particular ℓ(γ) > 0 denotes the length of γ and the parameter θ(γ) is determined modulo 4πiZ (4π instead of 2π because we take care of the spin structure, equivalently, the lift to SL(2, C)). ASYMPTOTICS OF TWISTED POLYNOMIALS 19 Definition 4.3. For s ∈ C with Re(s) > 2, the Ruelle zeta function is Y det Id −χ(γ)ρn (γ)e−sℓ(γ) .  Rχ⊗ρn (s) = [γ]∈PC(N ) The unitary Ruelle zeta function is Y k Rχ,k (s) = (1 − χ(γ)e 2 iθ(γ) e−sℓ(γ) ). [γ]∈PC(N ) k The term unitary comes from the unitary representation γ 7→ e 2 iθ(γ) . The relation between both functions is the following, whose proof is a term- wise direct computation, see [35, (3.14)] for a proof. Lemma 4.4. For s ∈ C with Re(s) > 2, n Y n Rχ⊗ρn (s) = Rχ,n−2k (s − ( − k)). 2 k=0 In particular meromorphic continuation of the former follows from meromor- phic continuation of the latter, which is the content of the first statement of Proposition 4.1. The proof of Proposition 4.1 is exactly the same as [35, Proposition 3.2]: one uses a similar functional equation for a Selberg zeta function Zχ,k proved in [6, Section 3.3.2] (see also [35, Section 4], though there is no twist χ in the latter reference). Then one expresses the unitary Ruelle zeta function Rχ,k as a product of Selberg zeta functions, and the proposition follows from the study of Selberg zeta functions carried out in [6]. 4.2. Fried’s theorem. In this subsection we prove Theorem 4.2. Again, it follows the lines of [35], which itself reproduces the proof contained in Wotzke’s PhD thesis [46]. One needs some care to introduce our twist χ, and we will describe the global picture of the proof, stressing out the points where the twist plays a role. 4.2.1. Analytic torsion. The first step is to replace the Reidemeister torsion by the analytic torsion: the representation χ ⊗ ρn , despite non-unitary, allows to equip the flat bundle Eχ⊗ρn = Γ\(H3 × Cn ) with a canonical hermitian metric, as we explain in Appendix C. Using the Hodge star on N and this hermitian metric on Eχ⊗ρn , one defines the Hodge-Laplace operator ∆pχ,n acting on the space Ωp (N, Eχ⊗ρn ) of p-forms on N with coefficients in Eχ⊗ρn . p Since N is compact, for any t > 0 the heat operator e−t∆χ,n is of trace class. p The asymptotic behavior for small and large t of the heat trace Tr e−t∆χ,n is local in N , in the sense that they are properties of the Laplacian acting on the universal cover H3 where the twist χ does not comes into play and it follows from the usual arguments that the expression   Z ∞ 3 1 d  1 X p log T (N, χ ⊗ ρn ) = ts−1 (−1)p p Tr(e−t∆χ,n )dt 2 ds Γ(s) 0 p=1 s=0 20 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI defines an invariant of (N, χ ⊗ ρn ) called the analytic torsion. In the rest of the section, the following alternating sum of heat traces will be denoted by 3 X p (13) Kχ,n (t) = (−1)p p Tr(e−t∆χ,n ). p=1 The representation ρn has determinant one, and as χ has module one, χ ⊗ ρn is unimodular, hence we can apply Cheeger-Müller theorem, in the version of [33] (see also [4]): Theorem 4.5 (Cheeger-Müller). For N closed, the combinatorial and analytic torsions coincide, that is |tor(N, χ ⊗ ρn )| = T (N, Eχ⊗ρn ). Hence to prove our Theorem 4.2 we are led to relate the analytic torsion with the evaluation at 0 of the Ruelle zeta function. 4.2.2. Selberg trace formula. The second ingredient is the Selberg trace formula. Again [Γ] denotes the set of conjugacy classes in Γ = π1 (N ). From the identification N = Γ\H3 it follows that forms in Ωp (N, Eχ⊗ρn ) are exactly the equivariant p-forms on H3 with value in Cn , and the Laplace operator ∆pχ,n can be seen as the restriction of the Laplace operator ∆e pn acting on Cn -valued forms on H3 . Using invariance properties of the Laplace operator, and denoting G = SL2 (C), it follows that the heat trace can be decomposed as a series Z −t∆pχ,n X (14) Tr e = χ(γ) hpn (t, g −1 γg)dg γ∈Γ Γ\G ep where hpn (t, g) is the trace of the kernel of the heat operator e−t∆n that we compose with the regular representation R(g) to get an endomorphism of the fiber of Eχ⊗ρn -valued p-forms at the class of the point g in H3 = SL2 (C)/ SU(2). It turns out that one can compute the heat trace on the ground manifold N with the heat trace on its universal cover, summing up on all translated fundamental domains in a tiling of H3 . A reference for a proof of (14) is [34, (5.7)]. Inserting (14) in (13), one obtains X Z (15) Kχ,n (t) = χ(γ) kn (t, g −1 γg)dg γ∈Γ Γ\G where kn (t) = 3p=1 (−1)p p hpn (t). P In fact, one can regroup the summands by conjugacy classes and compute explicitly the integrals in the right hand side of (15). This is performed in [35, Section 7] and the result is [(7.10), loc. cit.], note that here the twist χ is just ASYMPTOTICS OF TWISTED POLYNOMIALS 21 added in each summand: XZ (16) Kχ,n (t) = Vol(N ) Pk (iλ)Θk,λ (kn (t))dλ k∈Z R Z X χ(γ)ℓ(γ) X −kθ(γ)/2 + e Θk,λ (kn (t))e−ℓ(γ)λ dλ, 2πD(γ)nΓ (γ) R [γ]̸=1 k∈Z where • ℓ(γ) and θ(γ) are the real and the imaginary part of the complex length λ(γ) of γ. 2 • Pk is the Plancherel polynomial Pk (z) = 4π1 2 ( k4 − z 2 ). • Θk,λ is the character of the corresponding representation πk,λ of G called the unitary principal series, see [35, Section 2.4]. • The integer nΓ (γ) is defined as being maximal for the property that n (γ) there exists γ0 ∈ Γ such that γ = γ0 Γ . • D(γ) is the Weyl denominator of γ: D(γ) = (1 − e−(ℓ(γ)+iθ(γ)) )(1 − e−(ℓ(γ)−iθ(γ)) ). It turns out that the series indexed by k in (16) are indeed finite sums, since there are only finitely many indices k such that Θk,λ (kn (t)) is not zero. Precisely, there are exactly four indices kj , j = 1, . . . , 4, such that Θkj ,λ (kn (t)) ̸= 0. They are computed in [35, Section 7]: k1 = n + 1, k2 = −k1 , k3 = −n − 3, k4 = −k3 , and one obtains (see [35, (7.17)]): 4 2 (−1)j e−tλj X (17) Kn (t) = j=1   Z k θ(γ)/2 −ℓ(γ)2 /(4t) 2 X χ(γ)ℓ(γ)e j e × Vol(N ) e−tλ Pkj (iλ)dλ +  R D(γ)n Γ (γ) (4πt)1/2 [γ]̸=1 where the corresponding λj are computed in [35, (3.31)]: n+3 n+1 λ1 = , λ2 = −λ1 , λ3 = − , λ4 = −λ3 . 2 2 4.2.3. Auxiliary operators. At this stage it is not clear yet that the expan- sion (17) of the sum of heat traces Kχ,n (t) that defines the analytic torsion can be related with the evaluation at 0 of the Ruelle zeta function Rχ⊗ρn . The idea of Müller and Wotzke is to use some auxiliary operators ∆χ,n (j), j = 1, . . . , 4, and to show that each of the four summands in (17) can indeed be written as the (super-)trace of one of this operator acting on sections of some graded vector bundles. Those operators ∆χ,n (j) are defined in [35, (6.15)] as a combination of some Casimir type operators studied in [6, Section 1.1.3]. Again, Müller does not include the case of a non-trivial unitary twist χ that we need, but all the basic work is done in [6] for those Casimir operators with a twist. 22 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI Applying the same strategy that in Subsection 4.2.2 and expressing the trace of those operators with the Selberg trace formula yields: 1 2 (18) Trs (e−t∆χ,n (j) ) = e−tλj  2  Z 2 X χ(γ)ℓ(γ)eki θ(γ)/2 e−ℓ(γ)2 /(4t) × Vol(N ) e−tλ Pkj (iλ)dλ +  R D(γ)nΓ (γ) (4πt)1/2 [γ]̸=1 so that inserting (18) in (17) gives 4 1X (19) Kχ,n (t) = (−1)j Trs (e−t∆χ,n (j) ). 2 j=1 We need to check that those operators have trivial kernels, it is done without a twist in [35, Lemma 7.2]: Lemma 4.6. For any j = 1, . . . , 4, we have ∆χ,n (j) > 0. Proof. We will show that it is a consequence of the vanishing of the kernel of the Hodge Laplacians ∆pχ,n , see Theorem C.1. First, we can express the Hodge Laplacian as a direct sum of Bochner-Laplace operators on H3 (see [36, (5.7)]) that are denoted with a tilde. (recall that Symn is (n + 1)-dimensional): ep = M ∆ e ν + (Symn (Ω) − ν(ΩK )) Id . ∆ n+1 ν∈K b p [νn+1 : ν]̸=0 Here K = SU(2), the representations ν of K in the sum are those that appear in p the decomposition into irreducible representations of the restriction νn+1 to K of the action of G on C n+1 -valued p-forms through Sym . The scalars Symn (Ω) n and ν(ΩK ) are corresponding eigenvalues of the Casimir operator, the first one is computed in [35, (6.16)]. From this equation and the fact that for any p, one has ∆pχ,n+1 > 0, we deduce that the operator Aχ,ν defined as Aχ,ν = ∆χ,ν − ν(ΩK ) Id (see [35, (4.7)] for the notations) is bounded below by − Symn (Ω) for any ν such that p [νn+1 : ν] ̸= 0. Inserting that in the definition of ∆χ,n (j) given in [35, (6.15)] it follows that ∆χ,n (j) > 0, as claimed. □ Lemma 4.6 ensures that we can apply the Mellin transform to (19), one gets: Z ∞ Z ∞ 4 1 1 X ts−1 Kχ,n (t)dt = ts−1 (−1)j Trs (e−t∆χ,n (j) )dt Γ(s) 0 2Γ(s) 0 j=1 and taking the derivative at s = 0 gives 4 j+1 Y (20) T (N, Eχ⊗ϱn )4 = detgr (∆χ,n (j))(−1) . j=1 ASYMPTOTICS OF TWISTED POLYNOMIALS 23 4.2.4. A determinant formula. The relation between the right hand side of (20) and the Ruelle zeta function is given by the following proposition: Proposition 4.7. Denote by ρn : Γ → SLn (C) the complex conjugate of ρn , we have 4 4n Vol(N )s Y j+1 Rχ⊗ρn (s)Rχ⊗ρn (s) = e− π detgr (s2 − 2λj s + ∆χ,n (j))(−1) . j=1 This proposition is stated and proved as [35, Proposition 6.2] without the twist χ, nevertheless its proof relies on two facts that are known to be true with the twist inserted: • A determinant formula for the Selberg zeta function ([6, Theorem 3.19]). • The expression of the Ruelle zeta function as a product of Selberg zeta functions, whose proof [35, Proposition 3.5] works identically in our situation. With those two points, Müller’s proof generalizes verbatim to our statement. Proof of Theorem 4.2. It follows from Lemma 4.6 that we can take the limit as s goes to 0 in Proposition 4.7, we obtain 4 j+1 Y (21) Rχ⊗ρn (0)Rχ⊗ρn (0) = detgr (∆χ,n (j))(−1) j=1 Note that in the product on [γ], the conjugacy classes of both γ and of γ −1 are taken into account. Since ρ(γ) is conjugated in SL2 (C) to ρ(γ −1 ) and using χ(γ −1 ) = χ(γ), one has det(1 − χ(γ)ρn (γ)) = det(1 − χ(γ −1 )ρn (γ −1 )). Exchanging the role of γ and γ −1 in the product defining Rχ⊗ρn we obtain: Rχ⊗ρn (0) = Rχ⊗ρn (0) hence the left hand side of (21) is equal to |Rχ⊗ρn (0)|2 . Inserting this in (20) yields T (N, Eχ⊗ϱn )2 = |Rχ⊗ρn (0)| and Theorem 4.2 is proved. □ 5. Approximation by Dehn fillings In this section we describe the geometric convergence of Dehn fillings Mp/q to M , focusing on the behavior of geodesics and their role in Ruelle functions. 5.1. Geometric convergence. For a sequence of compatible Dehn fillings Mp/q such that (p, q) → ∞, not only we have convergence of representations p/q [ϱn ] → [ρn ], but we have also pointed bi-Lipschitz convergence, see Thurston’s notes [42] or [15] for an overview, and [2, Theorem E.5.1] for this precise state- ment: Theorem 5.1 (Thurston). Given ϵ > 0 and δ > 0, there exists C > 0 such that, if p2j + qj2 > C(ϵ, δ) for i = 1, . . . , l, then there is a (1 + ϵ)-bi-Lipschitz [δ,+∞) homeomorphism of the δ-thick parts M [δ,+∞) → Mp/q . 24 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI The δ-thick part of N is defined as N [δ,+∞) = {x ∈ N | inj(x) ≥ δ}, where inj(x) denotes the injectivity radius of x. p /q p /q Let γ1 1 1 , . . . , γl l l denote the souls of the filling solid tori of Mp/q , whose length converges to zero as (p, q) → ∞. p /q p /q Proposition 5.2. Except for γ1 1 1 , . . . , γl l l , all primitive closed geodesics of Mp/q must intersect the δ-thick part, provided that 0 < δ < δ0 for a δ0 > 0 depending only on M . Proof. By Margulis lemma, using the thin-thick decomposition (see [42] again) and taking δ0 > 0 less than half the length of the shortest geodesic of M , M \M [δ,+∞) is the union of cusp neighborhoods. Therefore, by choosing δ0 even [δ,+∞) less, by Theorem 5.1 Mp/q \ Mp/q is the union of Margulis tubes around the p /q p /q geodesics γ1 1 1 , . . . , γl l l . Then the proposition holds true because Margulis tubes contain no closed geodesics other than their souls. □ As the diameter of M [δ,+∞) goes to infinity when δ → 0, by Proposition 5.2 and geometric convergence we have: Proposition 5.3. For given L > 0 generic (so that L is not the length of any geodesic in M ) there exists a constant C(L) > 0 such that if p2j + qj2 > C(L), for i = 1 . . . , l, then the inclusion induces a bijection: p /qi ±1 {[γ] ∈ PC(M ) | ℓ(γ) ≤ L} ←→ {[γ] ∈ PC(Mp/q ) | ℓ(γ) ≤ L, γ ̸= (γi i ) }. As the length of the γi converges to zero, the inclusion induces a bijection: 1 {[γ] ∈ PC(M ) | ℓ(γ) ≤ L} ←→ {[γ] ∈ PC(Mp/q ) | ≤ ℓ(γ) ≤ L}. L Furthermore, the length and the holonomy of each geodesic in PC(M ) is the limit of length and holonomy of the corresponding geodesic in Mp/q as (p, q) → ∞. See [29, Section 6.3 and 6.4] for a detailed proof, for instance. Another consequence of bi-Lipschitz convergence is a uniform estimate on the growth of geodesics. We next quote Lemma 6.3 from [29], based on [7]: Lemma 5.4. Let X be a complete hyperbolic 3-manifold. For a compact domain K ⊂ X, #{[γ] ∈ PC(X) | γ ∩ K ̸= ∅, ℓ(γ) ≤ L} ≤ Ce2L , with C = π e8 diam(K) / vol(K). This is not the best estimate, for instance the estimate #{[γ] ∈ PC(N ) | ℓ(γ) ≤ L} < Ce2L /2L. due to Margulis [24] is better, see also [7], but Lemma 5.4 provides a uniform bound for the family of Dehn fillings. From Proposition 5.2 and Theorem 5.1, [δ,+∞) by taking K = M [δ,+∞) or K = Mp/q , Lemma 5.4 yields: ASYMPTOTICS OF TWISTED POLYNOMIALS 25 Lemma 5.5. There is a uniform C such that #{[γ] ∈ PC(X) | ℓ(γ) ≤ t} ≤ Ce2t , for X = M and X = Mp/q . 5.2. Estimates for Ruelle functions. We want to apply the results of the previous subsection to find uniform estimates on Ruelle functions for the Dehn fillings. We start with two elementary inequalities: (22) For z ∈ C, |z| < 1, log |1 − z| ≤ log(1 − |z|) . (23) For z ∈ C, |z| < 1/2, log |1 − z| ≤ 4|z|. To prove (22) apply logarithms to 1 1 − |z| ≤ |1 − z| ≤ 1 + |z| ≤ 1 − |z| and take into account that log(1 − |z|) < 0. Inequality (23) is then straightfor- ward. The next lemma reformulates the key calculus required for analysis of Ruelle functions, without using the formalism of measures of [29]. Lemma 5.6. For ϵ > 0 there exists C ′ (ϵ) such that, if s > 2 + ϵ and L ≥ 1, then X log |1 − χ(γ)e−s ℓ(γ) | ≤ C ′ (ϵ) eL(2+ϵ−s) [γ]∈PC(X) ℓ(γ)>L for X = Mp/q or X = M , where C ′ (ϵ) is uniform on X and the unitary twist χ. Proof. We omit the subscript [γ] ∈ PC(X) from the sums, which is always understood in the summations along the proof, and is combined with restrictions on the length of the geodesics. First, by (23) X X log |1 − χ(γ)e−s ℓ(γ) | ≤ 4 e−s ℓ(γ) . ℓ(γ)>L ℓ(γ)>L We divide the set PC(X) according to lengths. Set lj = (1 + 2j ϵ)L. Then by using Lemma 5.5: X ∞ X X ∞ X −s ℓ(γ) −s ℓ(γ) (24) e ≤ e ≤ C e2lj+1 e−s lj ℓ(γ)>L j=0 lj <ℓ(γ)≤lj+1 j=0 Since 2lj+1 − s lj = (2 + ϵ − s)L + j(− 2s + 1)ϵ L, the bound in (24) can be explicitly computed: ∞ X eL(2+ϵ−s) C C e2lj+1 e−s lj = C ≤ j=0 1−e (−s/2+1)ϵ L 1 − e−ϵ2 /2 and we are done. □ 26 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI The following bound is used in the proof of the theorem on the asymptotic behavior. Lemma 5.7. For a closed hyperbolic three-manifold N there exists a constant C(N ) depending only on N such that ∞ X (25) log |Rχ,−k (k/2)| ≤ C(N ). k=5 Proof. For each k ≥ 5 split log |Rχ,−k (k/2)| into two summations: X log |Rχ,−k (k/2)| ≤ log |1 − χ(γ)e−kλ(γ)/2 | [γ]∈PC(N ) ℓ(γ)≤1 X + log |1 − χ(γ)e−kλ(γ)/2 | . [γ]∈PC(N ) ℓ(γ)>1 We bound the contribution of the first summation. There exists k0 (depending on the length of the shortest geodesic of N ) such that for each k > k0 we have for all [γ] ∈ PC(N ): 1 |χ(γ)e−kλ(γ)/2 | = |e−kℓ(γ)/2 | < . 2 By (23) we obtain for all [γ] ∈ PC(N ): log |1 − χ(γ)e−kλ(γ)/2 | ≤ 4e−kℓ(γ)/2 . As the number of geodesics of length ≤ 1 is finite, the contribution of the summation indexed by ℓ(γ) ≤ 1 in the left-hand side of (25) is bounded (by finitely many geometric series, starting from k0 ). For the summation of geodesics [γ] with ℓ(γ) > 1, we use Lemma 5.6 (that we stated for Dehn fillings but applies to any closed hyperbolic manifold if we do not require uniformity on the manifold). As k ≥ 5, this yields again a bound by a geometric series. □ Remark 5.8. In Lemma 5.7 we do not have uniformity on the Dehn fillings because of short geodesics (k0 depends on the length of the shortest geodesic in N ). We will get rid of short geodesics by Dehn filling formulas in the next section (see Lemma 6.3). Notice that we do have uniformity on the twist χ. 6. Asymptotic behavior of torsions I this section we prove Theorem 1.6 and Theorem 1.8 from the introduction. 6.1. Müller’s theorem for closed Dehn filling. We give first the proof of Müller’s theorem for the Dehn fillings Mp/q . We follow [35], just with the minor change of the rational unitary twist χ: ASYMPTOTICS OF TWISTED POLYNOMIALS 27 Theorem 6.1 (Müller). For χ rational, Mp/q a compatible Dehn filling and m ≥ 3: m−1 tor(Mp/q , χ ⊗ ρ2m ) X 1 log = log |Rχ,−2k−1 (k + 12 )| − Vol(Mp/q )(m2 − 4), tor(Mp/q , χ ⊗ ρ4 ) π k=2 m tor(Mp/q , χ ⊗ ρ2m+1 ) X 1 log = log |Rχ,−2k (k)| − Vol(Mp/q )(m − 2)(m + 3). tor(Mp/q , χ ⊗ ρ5 ) π k=3 Proof. We prove the odd-dimensional case, the even dimensional case is similar. Observe first that, by Lemma 4.4, 2m Y Rχ⊗ρ2m+1 (s) = Rχ,2m−2k (s − (m − k)) k=0 m Y = Rχ,0 (s) Rχ,2k (s − k)Rχ,−2k (s + k) k=1 Ym (26) = Rχ⊗ρ5 (s) Rχ,2k (s − k)Rχ,−2k (s + k) k=3 Then, taking s = 0 in (26) and by Theorem 4.2: m Y (27) T (Mp/q , Eχ⊗ρ2m+1 )2 = T (Mp/q , Eχ⊗ρ5 )2 |Rχ,2k (−k)||Rχ,−2k (k)|. k=3 Next recall from Proposition 4.1 that |Rχ,2k (−k)| = |Rχ,−2k (k)|e−4k Vol(Mp/q )/π , then (27) becomes m m T (Mp/q , Eχ⊗ρ2m+1 ) 2 X X log = − Vol(N ) k+ log |Rχ,−2k (k)| T (Mp/q , Eχ⊗ρ5 ) π k=3 k=3 and the statement follows from Cheeger–Müller Theorem, Thm. 4.5. □ This theorem holds for any closed oriented hyperbolic 3-manifold, not only for Dehn fillings. Combined with Lemma 5.7 it yields a twisted version of Müller’s theorem: Corollary 6.2 ([35]). Let N be a closed hyperbolic, oriented 3-manifold. Let χ : π1 (N ) → S1 be a homomorphism. Then: log | tor(N, χ ⊗ ρn )| Vol(N ) lim 2 =− . n→∞ n 4π Note that the corollary above is also a consequence of the more general work of Bismut–Ma–Zhang in [3]. 28 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI 6.2. Proof of the main theorem. Assume that the twist χ is rational and that χ restricted to every peripheral subgroup of M is nontrivial, until Lemma 6.7. For an admissible Dehn filling Mp/q , let A = {(γp1 /q1 )±1 , . . . , (γpl /ql )±1 } denote the set of oriented souls of the filling tori, namely the l short geodesics added for the Dehn filling, with both orientations (hence A has cardinality 2l). Define, for k ≥ 5, p/q X Bχ,k = log |1 − χ(γ)e−kλp/q (γ)/2 |. [γ]∈PC(Mp/q )−A The convergence of this series follows from Margulis’ estimate ([24]), because k/2 ≥ 5/2 > 2. We discuss below in Lemma 6.4 further properties of this series. p/q p/q p/q Recall that ϱn = ρn ◦ i∗ , where ρn is the symmetric power of the lift of the holonomy of Mp/q and i∗ : π1 (M ) → π1 (Mp/q ) is induced by inclusion. Lemma 6.3. Given a rational twist χ of M that is nontrivial on each peripheral torus, for any integer m ≥ 3: p/q l  m−1 tor(M, χ ⊗ ϱ2m ) m2 − 4  X 2 X p/q log p/q = − ℓ(γ p /q i i ) + Vol(M p/q ) + Bχ,2k+1 tor(M, χ ⊗ ϱ4 ) 2 π i=1 k=2 and p/q l m tor(M, χ ⊗ ϱ2m+1 ) X 2  X p/q log p/q = − (m−2)(m−3) 2 ℓ(γpi /qi )+ Vol(Mp/q ) + Bχ,2k tor(M, χ ⊗ ϱ5 ) π i=1 k=3 Proof. We discuss the even case, 2m, and assume for simplicity that there is only one cusp, l = 1. Set λ = λ(γp1 /q1 ) and ζ = χ(m1 ) the image by the twist of any meridian of the boundary torus. By Proposition 3.6: p/q 2m−1 tor(Mp/q , χ ⊗ ρ2m ) X log p/q = log |e(2m−1−2k)λ/2 ζ − 1| tor(M, χ ⊗ ϱ2m ) k=0 m−1 X 1 1 = log |(e(k+ 2 )λ ζ − 1)(e−(k+ 2 )λ ζ − 1)|. k=0 Thus p/q m−1 tor(Mp/q , χ ⊗ ρ2m ) tor(M, χ ⊗ ϱp/q 4 ) X 1 1 log p/q p/q = log |(e(k+ 2 )λ ζ−1)(e−(k+ 2 )λ ζ−1)|. tor(Mp/q , χ ⊗ ρ4 ) tor(M, χ ⊗ ϱ2m ) k=2 With Theorem 6.1 it yields (28) p/q m−1 tor(M, χ ⊗ ϱ2m ) X p/q 1 1 log p/q = log |Rχ,−2k−1 (k + )| − Vol(Mp/q )(m2 − 4) tor(M, χ ⊗ ϱ4 ) 2 π k=2 m−1 X 1 1 − log |(e(k+ 2 )λ ζ − 1)(e−(k+ 2 )λ ζ − 1)|, k=2 ASYMPTOTICS OF TWISTED POLYNOMIALS 29 p/q where Rχ,−2k−1 denotes the twisted Ruelle zeta function of Mp/q . By definition p/q of Bk : p/q p/q 1 1 (29) log |Rχ,−2k−1 (k+ 21 )| = log |Bχ,2k+1 |+log |(1−ζe−(k+ 2 )λ )(1−ζe−(k+ 2 )λ )|. To combine (28) and (29), we use: 1 1 1 (1 − ζe−(k+ 2 )λ )(1 − ζe−(k+ 2 )λ ) 1 − ζe−(k+ 2 )λ (30) 1 1 = 1 (1 − ζe−(k+ 2 )λ )(1 − ζe(k+ 2 )λ ) 1 − ζe(k+ 2 )λ 1 1 1 − ζe−(k+ 2 )λ 1 = e−λ( 2 +k) = e−ℓ(λ)(k+ 2 ) . −(k+ 12 )λ ζe −1 The lemma follows from (28), (29) and (30). □ Define, for k ≥ 5, a unitary Ruelle function on the cusped manifold M : Y Rχ,−k (k/2) = (1 − χ(γ)e−kλ(γ)/2 ). γ∈PC(M ) Lemma 6.4. For k ≥ 5: (a) The series X X log |1 − χ(γ)e−kλ(γ)/2 | and log |1 − e−kℓ(γ)/2 | γ∈PC(M ) γ∈PC(M ) converge uniformly. (b) There exists a constant C > 0, uniform in χ, such that ∞ X ∞ X log |Rχ,−2k (k)| ≤ C, and log |Rχ,−2k−1 (k + 21 )| ≤ C. k=3 k=3 p/q (c) The series Bχ,k also converges uniformly, uniformly on (p, q) and the twist χ. In addition, p/q lim Bχ,k = Rχ,−k (k/2) (p,q)→∞ uniformly on the twist χ. In the lemma, uniformity on (p, q) or on χ means that the series can be bounded term-wise in absolute value by a convergent series, independently on (p, q) or/and on χ. Proof. Assertion (a) follows from Margulis bound on geodesic length growth, using inequalities (22) and (23). Assertion (b) has the very same proof as Lemma 5.7. For (c), we get uniformity on (p, q) from Lemma 5.5 and the fact that the sum does not include any of the short geodesics of the set A = p/q p/q {(γ1 )±1 , . . . , (γn )±1 }; hence there is a uniform lower bound away from zero p/q on the length of the geodesics that appear in the sum of Bχ,k , and from this, with Proposition 5.3 and Lemma 5.5, we get uniformity. Finally, the limit follows also from Proposition 5.3. □ 30 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI Proposition 6.5. For a rational twist that is nontrivial on each peripheral torus, and m ≥ 3, m tor(M, χ ⊗ ρ2m+1 ) X 1 log = log |Rχ,−2k (k)| − Vol(M )(m − 2)(m + 3) tor(M, χ ⊗ ρ5 ) π k=3 and m tor(M, χ ⊗ ρ2m ) X 1 log = log |Rχ,−2k−1 (k + 12 )| − Vol(M )(m − 2)(m + 2) tor(M, χ ⊗ ρ4 ) π k=2 Proof. We take limits on Lemma 6.3 when (p, q) → ∞. On the left hand side of the formula in Lemma 6.3, we apply Lemma 3.7. On the right hand side, we apply that Vol(Mp/q ) → Vol(M ), that ℓ(γpi /qi ) → 0 [42, 15], and Lemma 6.4. □ Using Propositions 2.11 and 6.5, we get: Corollary 6.6. Assume that ζ = (ζ1 , . . . , ζr ) ∈ (S1 )r satisfies that ζj ∈ eπiQ , for j = 1, . . . , r, and ζ α(mk ) ̸= 1, for k = 1, . . . , l. Then: (a) for 2m even: m ∆α,2m M (ζ) 1 X log = Vol(M )(m − 2)(m + 2) − log |Rχ,−2k−1 (k + 21 )| , ∆α,4 M (ζ) π k=2 (b) for 2m + 1 odd: m ∆α,2m+1 M (ζ) 1 X log = Vol(M )(m − 2)(m + 3) − log |Rχ,−2k (k)|. ∆α,5 M (ζ) π k=3 The proof of Theorem 1.6 for χ rational and non trivial on peripheral sub- groups follows from Corollary 6.6 and Lemma 6.4 (b). Next we remove the hypothesis on the twist χ. Lemma 6.7. Corollary 6.6 holds for any ζ = (ζ1 , . . . , ζr ) ∈ (S1 )r , without any assumption on rationality of ζ1 , . . . , ζr . Proof. The proof is a density argument, using continuity of the terms that appear in Corollary 6.6, that we need to justify. By Theorem 1.11, we know that ∆α,n α,n M (ζ) does not vanish, hence log |∆M (ζ)| 1 is continuous on ζ = (ζ1 , . . . , ζr ) ∈ S , for n ≥ 2. For the continuity of Rχ,−k (k/2), k ≥ 5, as in the proof of Lemma 5.7 we split again the series log |Rχ,−k (k/2)| in two: a finite sum indexed by geodesics of length < L and a series indexed by geodesics of length > L. The finite sum is continuous on ζ, so we need to chose L so that the series indexed by geodesics of length > L is arbitrarily small, uniformly on ζ. More precisely, by (22), for each [γ] ∈ PC(M ) |1 − χ(γ)e−kλ(γ)/2 | ≤ |1 − e−kℓ(γ)/2 |. ASYMPTOTICS OF TWISTED POLYNOMIALS 31 As the series [γ]∈PC(M ) log |1 − e−kℓ(γ)/2 | converges (Lemma 6.4), for every P ε > 0 there exists L = L(ε) > 0 such that X log |1 − χ(γ)e−kλ(γ)/2 | < ε, [γ]∈PC(M ) ℓ(γ)>L uniformly on χ. As PC(M ) has finitely many elements of length ≤ L, continuity of Rχ,−k (k/2) on χ is clear. □ Proof of Theorems 1.6 and 1.8. By Lemma 6.7, equations of Corollary 6.6 hold true for any ζ. Moreover, the bound on the series in the right-hand side given by Lemma 6.4 (b) is uniform in χ, so that we obtain Theorem 1.6 by dividing by (2m)2 in case (a), or by (2m + 1)2 in case (b), and by letting m tend to infinity. This proves Theorem 1.6, and Theorem 1.8 follows in the same way, again using uniformity on ζ of the bound from Lemma 6.4. □ Appendix A. Some twisted Alexander polynomials for the figure-eight knot We computed the twisted Alexander polynomials ∆ρ41n for the figure-eight knot for n up to 10. n ∆ρ41n (t) 2 t2 − 4 t + 1 3 t2 − 5 t + 1 2 4 t2 − 4 t + 1 5 t4 − 9 t3 + 44 t2 − 9 t + 1 6 t6 − 12 t5 + 156 t4 − 388 t3 + 156 t2 − 12 t + 1   7 t2 − 6 t + 1 t4 − 7 t3 + 372 t2 − 7 t + 1 8 t8 − 16 t7 + 1045 t6 − 15808 t5 + 39160 t4 − 15808 t3 + 1045 t2 − 16 t + 1   9 t6 + 9 t5 + 2403 t4 − 15690 t3 + 2403 t2 + 9 t + 1 t2 − 26 t + 1 10 t10 − 20 t9 + 3630 t8 − 350916 t7 + 5322645 t6 − 13176480 t5 +5322645 t4 − 350916 t3 + 3630 t2 − 20 t + 1 Appendix B. Review on combinatorial torsion The goal of this appendix is to review basic properties of combinatorial tor- sion. We restrict to compact orientable three-dimensional manifolds N , possibly with boundary. To simplify notation, we write Γ = π1 (N ). We also fix a field F of characteristic 0, and a representation ρ : Γ → GLn (F). 32 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI B.1. Twisted chain complexes. Fix a CW-complex structure K on N . The complex of chains on the universal covering K e is the free Z-module on the cells of K, e equipped with the usual boundary operator, and it is denoted by C∗ (K, Z). It has an action of Γ = π1 (N ) that turns it into a left Z[Γ]-module. e The group Γ acts on Fn via ρ on the left, and for the tensor product Γ acts on Fn on the right using inverses: any γ ∈ Γ maps v ∈ Fn to ρ(γ −1 )(v). We write n n ρ F and Fρ to emphazise the left and right Z[Γ]-module structures, respectively. The twisted chain and cochain complexes are defined as: (31) C∗ (K, ρ) = Fnρ ⊗Γ C∗ (K, e Z), (32) C ∗ (K, ρ) = HomΓ (C∗ (K, e Z),ρ Fn ). Those are complexes and co-complexes of finite-dimensional vector spaces, and the corresponding homology and cohomology groups are denoted by H∗ (K, ρ) and H ∗ (K, ρ). B.2. Geometric bases. For a cell ee ∈ K, e denotes the free Z[Γ]-module e Z[Γ]e of rank one on its Γ-orbit (i.e. the free module on all lifts of a given cell e in K). Lemma B.1. We have natural isomorphisms of F-vector spaces: e, Fn ) → Fn HomΓ (Z[Γ]e Fn ⊗Γ Z[Γ]e e → Fn θ 7→ θ(e e) v ⊗ ee 7→ v The proof is straightforward. Chose {v1 , . . . , vn } a basis for Fn and let {ei1 , . . . eiji } be the set of i-dimensional cells of K. For each cell eij chose a lift eeij to K. e Then {vk ⊗ eei }i,j,k is an F- j basis for Ci (K, ρ). Similarly {(e eij )∗ ⊗ vl }i,j,l is an F-basis for C i (K, ρ) where eij )∗ ⊗ vl (γe  i (e ek ) = ρ(γ)vl δjk . Definition B.2. We call this basis a geometric basis for C∗ (K, ρ), respectively for C ∗ (K, ρ). B.3. Combinatorial torsion. Recall the definition of torsion of a complex of finite dimensional F-vector spaces C∗ with bases {ci }i for the chain complexes and bases {hi }i for the homology groups, following for instance [31]. For that purpose we consider the space of boundaries Bi = im(∂ : Ci+1 → Ci ), the space of cycles Zi = ker(∂ : Ci → Ci−1 ) and the homology Hi = Zi /Bi . We chose bi an F-basis for Bi . Using the exact sequences 0 → Bi → Zi → Hi → 0, 0 → Zi → Ci → Bi−1 → 0 we lift bi it to a subset ebi of Ci+1 , and hi to a subset e hi of Ci , so that ebi−1 ∪ e hi ∪bi is an F-basis for Ci . We denote [ebi−1 ∪ e hi ∪ bi : ci ] the determinant of the matrix i ei−1 e i i which takes c to b ∪ h ∪ b (in the colomns of the matrix are the coordinates of ebi−1 ∪ e hi ∪ bi with respect to ci ). Then we define 3 i Y i i tor(C∗ , {c }i , {h }i ) = hi ∪ bi : ci ](−1) ∈ F∗ [ebi−1 ∪ e i=0 ASYMPTOTICS OF TWISTED POLYNOMIALS 33 If we have defined a geometric basis {ci }i as in Definition B.2, then the torsion is: e ρ), {ci }i , {hi }i ) ∈ F∗ / ± det ρ(Γ) . tor(N, ρ, {hi }i ) = tor(C∗ (K,  It is straightforward to check that it is well defined (see [31] and [38]). Topo- logical invariance follows from uniqueness of triangulations on three-manifolds. B.4. Duality homology-cohomology. We aim to define the torsion from the cohomological point of view. Let V be a finite dimensional F-vector space, and let ρ : Γ → GL(V ) be a representation. The contravariant representation or dual representation ρ∗ : Γ → GL(V ∗ ) is defined by ρ∗ (γ)(f ) = f ◦ ρ(γ −1 ). Lemma B.3. The representations ρ and ρ∗ are equivalent if and only if there exists a non-degenerate bilinear form B : V ⊗ V → F which is Γ-invariant. If we choose a basis in V and its dual basis in V ∗ , we obtain matrix repre- sentations ρ, ρ∗ : Γ → GLn (F), and they are related by ρ∗ (γ) = ρ(γ −1 )t . Notice that (ρ∗ )∗ = ρ. Example B.4. For any representation ρ : Γ → SL2 (F), the module V = F2 has a skew-symmetric non-degenerate bilinear form defined by the determinant. Namely, the vectors (x1 , x2 ) and (y1 , y2 ) ∈ F2 are mapped to   x1 y1 det  . x2 y2 In view of Lemma B.3, ρ∗ and ρ are equivalent. More concretely, for any matrix A ∈ SL2 (F) we have = (A−1 )t . 0 1  0 −1  −1 0 A 1 0 The pairing V ∗ ⊗ V → F induces a perfect pairing of complexes ⟨ , ⟩ : Ci (K, ρ) ⊗ C i (K, ρ∗ ) → F, defined by: ⟨v ⊗ ẽ, θ⟩ = θ(ẽ)(v), e v ∈ V and θ ∈ homΓ (C∗ (K), where ẽ is a cell of K, e V ∗ ). It is easy to check that it is well defined, non-degenerate and that it is compatible with the boundaries and coboundaries: ⟨∂· , ·⟩ = ±⟨· , δ·⟩ where the sign depends only on the dimension. Hence in its turn it induces a non-degenerate Kronecker pairing between homology and cohomology Hi (K, ρ) × H i (K, ρ∗ ) → F . Now we can relate the torsion in homology with the torsion in cohomology. We denote by B i , Z i and H i the coboundary, cocycle and cohomology spaces, i respectively. In addition, we take b̄i basis for B i that we lift to ēb in C i−1 . We define the torsion of a cocomplex with bases in cohomology hi as: 3 ēi ∪ b̄i : c̄i ](−1)i+1 ∈ F∗ Y ∗ i i tor(C , {c̄ }i , {h̄ }i ) = [ebi+1 ∪ h i=0 34 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI To relate torsion in homology and cohomology, notice that the geometric basis ci of Ci (K, ρ) and c̄i of C i (K, ρ∗ ) can be chosen to be dual. Then the matrices of the boundary operators with respect to those basis are transpose to the matrices of the respective coboundary operators. From this, we have: Proposition B.5. If the basis hi for Hi (K, ρ) and the basis h̄i for H i (K, ρ∗ ) are dual for each i, then tor(Ci (K, ρ), {ci }, {hi }) = tor(C i (K, ρ∗ ), {c̄i }, {h̄i }). Remark B.6. We shall also use Poincaré duality with twisted coefficients, see for instance [18, 38]. For N a compact orientable manifold: H i (N ; ρ)∗ ∼ = H dim N −i (N, ∂N ; ρ∗ ) B.5. The representations we are interested in the paper. Here we list the representations we use in the paper. We describe in which space they are defined. Since the torsion lies by definition in F/ ± det ρ(Γ), we need to understand det ρ(Γ). We start with a representation ρ : Γ → SL2 (C), and we put ρn := Symn−1 ◦ ρ. (1) For the representation ρn : Γ → SLn (C), the torsion is well defined up to sign, as det ρ(Γ) = {1}. Recall that Cn has a non-degenerate Symn−1 - invariant bilinear form which is symmetric for n odd and antisymmetric for n even. By irreducibility, the form is unique up to scalar. For n = 2 this form is the determinant (see Example B.4). For general n, it is the symmetrization of this bilinear form on C2 , an explicit formula is given in Lemma 3.1.4 in [41]. Thus ρ∗ ∼= ρn . n (2) For the representation α ⊗ ρn : Γ → GLn (C(t1 , . . . , tr )), the torsion is well defined up to sign and multiplication by monomials tm = tm mr 1 · · · tr . 1 There is no Γ-invariant bilinear form on C(t1 , . . . , tr )n , and hence (α ⊗ ρn )∗ and α ⊗ ρn are not equivalent (for non-trivial α), nevertheless (α ⊗ ρn )∗ = α−1 ⊗ ρ∗ ∼= α−1 ⊗ ρn . n (3) In the case of the representation χ ⊗ ρn : Γ → GLn (C) for a character χ : Γ → S1 ⊂ C, only the modulus of the torsion is well defined. There is no Γ-invariant bilinear form on Cn , and (χ ⊗ ρn )∗ = χ ⊗ ρ∗ ∼ = χ ⊗ ρn . n By the classical duality theorems of Franz [11] and Milnor [30] we have tor(N, α ⊗ ρn ) = ±tm tor(N, α−1 ⊗ ρn ) for some multiplicative factor ±tm , and | tor(N, χ ⊗ ρn )| = | tor(N, χ̄ ⊗ ρn )|. Remark B.7. Let us recall some basic facts about the irreducible representa- tion Symn−1 : SL2 (C) → SLn (C). For details we refer to Springer’s book [41, Section 3.1]. • The representation Symn−1 factors through PSL2 (C) for n odd. ASYMPTOTICS OF TWISTED POLYNOMIALS 35 • The space Cn has a non-degenerate Symn−1 -invariant bilinear form, that is symmetric for n odd and antisymmetric for n even. By irre- ducibility, this form is unique up to scalar. An explicit formula is given in Lemma 3.1.4 in [41] (for n = 2 see Example B.4). In higher dimen- sions the Symn−1 -invariant bilinear form is the symmetrization of the determinant. • The image of a non-trivial unipotent element in SL2 (C) is a regular unipotent element in SLn (C), i.e. it is conjugate to an upper-triangular matrix which has only ones on the diagonal and a single block in the Jordan-Hölder form. It follows that the image of a parabolic element g ∈ SL2 (C), with trace ϵg 2, for some ϵg = ±1 has a unique eigenspace, of dimension one and with eigenvalue ϵn−1g . Moreover, this 1-dimensional eigenspace is an isotropic subspace of Cn . • We also use Clebsch-Gordan formula: (33) Ad ◦ Symn−1 ∼ = Sym2(n−1) ⊕ Sym2(n−2) ⊕ · · · ⊕ Sym4 ⊕ Sym2 (see for instance [41, Exercise 3.2.4]) Appendix C. Vanishing of L2 -cohomology The goal of this appendix is to show that the classical vanishing theorems in cohomology à la Matsushima-Murakami [25] apply to our situation with a twist χ. C.1. Review on L2 -forms on hyperbolic manifolds. In this appendix M is an oriented hyperbolic three-manifold (possibly of infinite volume) and χ : π1 (M ) → S1 a unitary character, possibly trivial. In the rest of the paper we assume that M has finite volume, but not in this appendix. Let ρ : π1 (M ) → SL2 (C) be a lift of the holonomy, and Symn−1 : SL2 (C) → SLn (C) be the n-dimensional holomorphic irreducible representation. The composition is denoted by ρn = Symn−1 ◦ρ : π1 (M ) → SLn (C). We consider the flat vector bundle Cn → Eχ⊗ρn → M with total space: Eχ⊗ρn = Cn ×χ⊗ρn M f. We describe the hermitian metric on the bundle (i.e. on each fibre). View the universal covering M f as the quotient SL(2, C)/SU(2), start with a hermit- n ian product on C invariant by the action of the compact group SU(2), and translate it along SL(2, C)/SU(2) ∼= H3 , via Symn−1 . This hermitian product is compatible with the action of ρn by construction, but also with the action of χ, because hermitian products are invariant by multiplication by unit complex numbers. Thus it induces a non-flat hermitian metric on the bundle Eχ⊗ρn . 36 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI We consider Ωp (M, Eχ⊗ρn ) the space of p-forms valued in Eχ⊗ρn , namely smooth sections of the bundle Eχ⊗ρn ⊗ Ωp (M ). The Riemannian metric on T M and the hermitian metric on the fibres yield a Hodge star operator ∗, a codifferential δ, and a Laplacian ∆pχ,n on Ωp (M, Eχ⊗ρn ). They also provide a hermitian product on p-forms: Z (ϕ, ψ) = ϕ ∧ ∗ψ ∀ϕ, ψ ∈ Ωpc (M, Eχ⊗ρn ), M where ∧ denotes the exterior product on forms in Ω∗ (M ) combined with the her- mitian product on Eχ⊗ρn , and Ωpc (M, Eχ⊗ρn ) the space of compactly supported forms in Ωp (M, Eχ⊗ρn ) Pointwise we use the Riemannian metric on M and the hermitian product on the bundle to define a hermitian product on forms ⟨ϕ, ψ⟩x at any x ∈ M , so that (ϕ ∧ ∗ψ)x = ⟨ϕ, ψ⟩x dvol for every x ∈ M. De Rham cohomology of the cocomplex (Ω∗ (M, Eχ⊗ρn ), d) is denoted by H (M, Eχ⊗ρn ); it is isomorphic to the simplicial cohomology H ∗ (M ; χ ⊗ ρn ). ∗ The aim of this appendix is to prove that every closed form with finite norm is exact: Theorem C.1. For any form ω ∈ Ω1 (M, Eχ⊗ρn ) satisfying (ω, ω) < ∞, if dω = 0 then there exists η ∈ Ω0 (M, Eχ⊗ρn ) such that ω = dη. Theorem C.1 is in fact a theorem on vanishing of L2 -cohomology and it is a version of a theorem of Garland [13]. This theorem is proved in Subsection C.2. C.2. Proof of the theorem. The proof is based on the following theorem of Andreotti-Vesentini [1] and Garland [13]: uniform ellipticity implies that closed forms of finite norm are exact. Theorem C.2 (Thm 3.22 in [13]). If there exists a constant c > 0 such that for every form ω ∈ Ω1 (M, Eχ⊗ρn ) with compact support (34) (d ω, d ω) + (δ ω, δ ω) ≥ c(ω, ω), then Theorem C.1 holds. Inequality (34) is called uniform ellipticity because, for forms ω with compact support, it is equivalent to (∆1χ,n ω, ω) ≥ c(ω, ω). In order to prove uniform ellipticity, we use the formalism of Matsushima and Murakami [25], as in [13, 39, 17, 28, 40]. Since H3 ∼ = SL(2, C)/SU(2), from the decomposition sl(2, C) = su(2) ⊕ isu(2), orthogonal for the real Killing form, there is a natural identification ϕx of the tangent space at each point Tx H3 with isu(2). Let E = Cn ×H3 be the trivial bundle, equipped with the natural flat connec- tion, with covariant derivative ∇v and the standard differential d : Ωp (H3 , E) → Ωp+1 (H3 , E). Following [25] we define a new covariant derivative e v = ∇v − symn−1 (ϕx (v)), ∇ ∀v ∈ Tx H3 , ASYMPTOTICS OF TWISTED POLYNOMIALS 37 where symn−1 : sl2 (C) → sln (C) is the representation of the Lie algebra associ- ated to Symn−1 . The corresponding connection D : Ω0 (H3 , E) → Ω1 (H3 , E) is given by Ds(v) = d s(v) − symn−1 (ϕx (v))(s) for every section s ∈ Ω0 (H3 , E) and every tangent vector v ∈ Tx H3 . By con- struction, D is a connection: D(f s) = f Ds + sd f for any function f and any section s, and it can be checked that it is metric: d⟨s1 , s2 ⟩x = Ds1 ∧ s2 + s1 ∧ Ds2 , where x 7→ ⟨s1 , s2 ⟩x is a function on H3 . The connection D is introduced in [25] as induced from the natural connection associated to the principal bundle on SLn (C) → SL2 (C), corresponding to the representation Symn−1 . Now, for any frame {e1 , e2 , e3 } of H3 , let {ω 1 , ω 2 , ω 3 } denote its dual coframe. As d = 3j=1 ω j ∧ ∇ej [47, (6.19)] we have: P Proposition C.3. [25] On Ω∗ (H3 , E) we have d=D+T and δ = D∗ + T ∗ , where X X (35) D= ωj ∧ ∇ ee , j T = ω j ∧ symn−1 (ϕx (ej )), j j X X ∗ ∗ (36) D =− i(ej )∇ ee , j T = i(ej ) symn−1 (ϕx (ej )). j j Up to now these operators are defined on the bundle E on H3 , and we want to descend them to the bundle on M twisted by χ ⊗ ρn . Notice that the operators and Proposition C.3 are found in the literature without the twist, so we need to justify why it works in our situation. We may view the definitions and Proposition C.3 on the trivial bundle E = Cn × H3 → H3 as being equivariant for the action of π1 (M ) via the representation ρn . On the other hand, these formulas are C-linear, so they are equivariant for the action via χ ⊗ ρn : Remark C.4. Proposition C.3 holds true on Ω∗ (M, Eχ⊗ρ ). Proposition C.5. [25] There is a Weitzenböck formula: ∆ = dδ + δd = D D∗ + D∗ D + T T ∗ + T ∗ T = ∆D + H where ∆D = D D∗ + D∗ D and H = T T ∗ + T ∗ T . Moreover, for any form ω ∈ Ω1 (M, Eχ⊗ρn ) with compact support: (37) (dω̃, dω̃) + (δ ω̃, δ ω̃) = (Dω̃, Dω̃) + (D∗ ω̃, D∗ ω̃) + (H ω̃, ω̃). A proof of both propositions can be found in [25] and also in [17, 28, 40]. The Weitzenböck formula requires the identity D T ∗ + T D∗ + D∗ T + T ∗ D = 0 on forms with compact support. Identity (37) is proved from Stokes theorem, cf. [17, Equation (5)]. On the other hand: 38 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI Proposition C.6. [25] For any ω ∈ Ω1 (M, Eχ⊗ρn ), pointwise, ⟨Hω, ω⟩x ≥ cn ⟨ω, ω⟩x , at every point x ∈ M, for a uniform constant cn that depends only on Symn−1 . See also [39, 28] for a proof. Thus uniform ellipticity holds for any form in ω ∈ Ω1 (M, Eχ⊗ρn ) with compact support; so Theorem C.2 applies. Next we assume that the orientable hyperbolic manifold M has finite topol- ogy, that it has a compactification M that consists in adding surfaces, as it has finitely many ends that are topologically products. Every element in the kernel of H 1 (M ; χ ⊗ ρn ) → H 1 (∂M ; χ ⊗ ρn ) is represented by a differential form with compact support in M , in particular it has finite norm. Thus, by Theorem C.1: Corollary C.7. We have an injection induced by inclusion: H 1 (M ; χ ⊗ ρn ) ,→ H 1 (∂M ; χ ⊗ ρn ). Appendix D. Dynamics of pseudo-Anosov diffeomorphisms on the variety of characters Let Σ be a compact orientable surface, possibly with boundary, connected and with negative Euler characteristic, and let ϕ: Σ → Σ be a pseudo-Anosov diffeomorphism. Note that ϕ does not necessarily act by the identity on ∂Σ, but it may permute the boundary components. Consider the mapping torus of ϕ: M (ϕ) = Σ × [0, 1]/(x, 1) ∼ (ϕ(x), 0). Its fundamental group is a semi-direct product (38) π1 (M (ϕ)) ∼ = π1 (Σ) ⋊ Z ∼ = ⟨π1 (Σ), τ | τ γτ −1 = ϕ# (γ) for all γ ∈ π(Σ)⟩ . Here ϕ# : π1 (Σ) → π1 (Σ) denotes the isomorphism induced by ϕ. Notice that a different choice of τ would yield the composition of ϕ# with an inner auto- morphism. According to Thurston’s hyperbolization theorem, M (ϕ) admits a finite vol- ume and complete hyperbolic metric [37]. The holonomy of this hyperbolic structure lifts to a representation π1 (M (ϕ)) → SL2 (C) and its restriction to π1 (Σ) yields a representation ρ2 : π1 (Σ) → SL2 (C). The composition of ρ2 with Symn−1 will be denoted by ρn := Symn−1 ◦ ρ2 . Notice that, by (38), ρn and ϕ∗ (ρn ) := ρn ◦ ϕ# are conjugate by ρn (τ ). This implies that the equivalence class [ρn ] of ρn is a fixed point of the action of ϕ∗ on the character variety   R Σ, SLn (C) := Hom π1 (Σ), SLn (C) SLn (C) . For definitions and more details see [20, Section 4.3] or [16]. ASYMPTOTICS OF TWISTED POLYNOMIALS 39 In the case of a closed surface (∂Σ = ∅) and n = 2, M. Kapovich proved in [19] that [ρ] is a hyperbolic fixed point of ϕ∗ : R(Σ, SL2 (C)) → R(Σ, SL2 (C)), namely the tangent map (dϕ∗ )[ρ2 ] : T[ρ2 ] R(Σ, SL2 (C)) → T[ρ2 ] R(Σ, SL2 (C)) has no eigenvalues of modulus one. In the case of a surface with boundary this assertion is no longer true, as the trace functions of the peripheral elements are invariant under a power of ϕ∗ . This causes (dϕ∗ )[ρ2 ] to have 1 as an eigenvalue (see [38, Section 4.5] for n = 2 and a punctured torus). In order to generalize Kapovich’s result, for surfaces with boundary we con- sider the relative character variety. Let ∂Σ = ∂1 ⊔ · · · ⊔ ∂s be the decomposition in connected components. The relative character variety is  R Σ, ∂Σ, SLn (C) := {[ρ] ∈ R(Σ, SLn (C)) | ρ(∂i ) and ρn (∂i ) are similar} , with the convention that R(Σ, ∅, SLn (C)) = R(Σ, SLn (C)) for closed surfaces. The main result of this appendix is the following: Theorem D.1. Let Σ be a compact orientable surface, possibly with boundary, connected and with negative Euler characteristic, and let ϕ : Σ → Σ be a pseudo- Anosov diffeomorphism. Then the character [ρn ] is a hyperbolic fixed point of ϕ∗ , i.e. the tangent map (dϕ∗ )[ρn ] : T[ρn ] R(Σ, ∂Σ, SLn (C)) → T[ρn ] R(Σ, ∂Σ, SLn (C)) has no eigenvalues of modulus one. The proof is based on the cohomological interpretation of the tangent spaces to varieties of characters. Recall that by a result of A. Weil [45] there is a natural isomorphism T[ρ ] R(Σ, SLn (C)) ∼ n = H 1 (Σ, Ad ◦ρn ) and the tangent space of the relative character variety can be interpreted as a kernel (see for instance [16, Proposition 18]): T[ρ ] R(Σ, ∂Σ, SLn (C)) ∼ n = ker(H 1 (Σ, Ad ◦ρn ) → H 1 (∂Σ, Ad ◦ρn )). Moreover, the tangent map dϕ∗[ρn ] corresponds to the induced map ϕ∗ in coho- mology. There is a natural surjection α : π1 (M (ϕ)) ↠ Z, α(τ ) = 1, and α(γ) = 0 for γ ∈ π1 (Σ), induced by the fibration Σ → M (ϕ) → S 1 . Remark D.2. In Section 2.2 we have defined the twisted Alexander polynomial for manifolds with cusps, but the same definition applies to closed manifolds, without requiring any assumption on α (just non-triviality). Thus we can define a one variable twisted Alexander polynomial ∆α,nM (ϕ) (t) even for M (ϕ) closed. The main results of the paper hold true and are simpler to prove in the closed 40 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI case, without needing the approximation results of Sections 3 and 6, for in- stance. Proposition D.3. Up to multiplication by a factor ±tm : n−1 ∆α,2k+1 Y det((dϕ∗ )[ρn ] − t Id) = M (ϕ) (t). k=1 Proof of Theorem D.1. Apply Proposition D.3 and Theorem 1.11. □ Remark D.4. There is a natural C-valued symplectic form on R(Σ, ∂Σ, SLn (C)) [22, 16]. By naturality, this symplectic form is ϕ∗ -invariant, therefore: det((dϕ∗ )−1 ∗ [ρn ] − t Id) = det((dϕ )[ρn ] − t Id). Proposition D.3 is based on the following lemma: Lemma D.5. (a) When Σ is closed, tor(M (ϕ), ᾱ ⊗ Ad ◦ρn ) = det((dϕ∗ )[ρn ] − t Id)−1 . (b) When ∂Σ ̸= ∅, if σϕ denotes the permutation matrix on the components of ∂Σ, then tor(M (ϕ), ᾱ ⊗ Ad ◦ρn ) = det((dϕ∗ )[ρn ] − t Id)−1 det(σϕ − t Id)1−n . Remark D.6. As ∂Σ has s components, σϕ is a permutation matrix of size s × s, that decomposes into l cycles, where l is the number of components of ∂M (ϕ) = T12 ⊔· · ·⊔Tl2 . Furthermore, if ci is the order of the cycle corresponding to Ti2 , the i-th component of ∂M (ϕ), then α(π1 (Ti2 )) = ci Z, s = c1 + · · · + cl , and Yl s−l det(σϕ − t Id) = (−1) (1 − tci ). i=1 Proof of Lemma D.5. Let K be CW-complex with underlying space |K| = M (ϕ). Consider its lift K to Σ × R and its lift Ke to the universal covering. We work with the following chain complexes: C ∗ (K, ᾱ ⊗ Ad ◦ ρn ) = homπ (M (ϕ)) (C∗ (K, 1 e Z), C(t) ⊗ sln (C)), ∗ C (K, ᾱ ⊗ Ad ◦ ρn ) = homπ1 (Σ) (C∗ (K, e Z), C(t) ⊗ sln (C)). Choosing τ , a representative in π1 (M (ϕ)) of a generator of Z, it acts on C ∗ (K, ᾱ ⊗ Ad ◦ ρ) by θ 7→ α(τ )Adρn (τ ) ◦ θ ◦ τ −1 , ∀θ ∈ C ∗ (K, ᾱ ⊗ Ad ◦ ρn ). The action does not depend on the choice of the representative τ ∈ π1 (M (ϕ)) of the generator of Z. We have then a short exact sequence of complexes: (τ −1) 0 → C ∗ (K, ᾱ ⊗ Ad ◦ ρn ) → C ∗ (K, ᾱ ⊗ Ad ◦ ρn ) −→ C ∗ (K, ᾱ ⊗ Ad ◦ ρn ) → 0. It induces Wang’s exact sequence in cohomology. Hence, as H i (K, ᾱ⊗Ad◦ρn ) ∼ = H i (Σ, ᾱ ⊗ Ad ◦ ρn ) vanishes for every i ̸= 1, by Milnor’s theorem on torsion of exact sequences [31, Theorem 3.2]: (39) tor(M (ϕ), ᾱ ⊗ Ad ◦ρn ) = det(τ ∗,1 − Id)−1 , ASYMPTOTICS OF TWISTED POLYNOMIALS 41 where τ ∗,1 is the morphism on H 1 (Σ, ᾱ ⊗ Ad ◦ ρn ) induced by τ . As α is trivial on π1 (Σ): H 1 (Σ, ᾱ ⊗ Ad ◦ ρn ) ∼ = C(t) ⊗ H 1 (Σ, Ad ◦ ρn ) and τ ∗,1 corresponds via this isomorphism to t ⊗ ϕ∗,1 , where ϕ∗,1 is the action that ϕ induces on H 1 (Σ, Ad ◦ ρn ). Hence (40) det(τ ∗,1 − Id) = det(t ϕ∗,1 − Id) = tn det(ϕ∗,1 − t−1 Id). From (39) and (40) we get tor(M (ϕ), ᾱ ⊗ Ad ◦ρn ) = det(ϕ∗,1 − t−1 Id)−1 , up to a factor ±tm . Next we look at the action of (ϕ∗,1 )±1 to the exact sequence coming from the cohomology of the pair (Σ, ∂Σ) i∗ (41) 0 → ker(i∗ ) → H 1 (Σ, Ad ◦ρn ) → H 1 (∂Σ, Ad ◦ρn )) → 0. We claim that: (1) The action of (ϕ∗,1 )−1 on ker(i∗ ) corresponds to the action of dϕ∗ by the isomorphism ker(i∗ ) ∼ = T[ρn ] R(Σ, ∂Σ, SLn (C)). (n−1) (2) The action of (ϕ∗,1 ) on H 1 (∂Σ, Ad ◦ρn )) is equivalent to σϕ × · · · × σϕ . The proof of these claims and the product formula of torsions applied to the exact sequence (41) will complete the proof of Lemma D.5. Proof of Claim (1). We consider the action on the variety of representations ρ 7→ Adρn (τ −1 ) ◦ρ ◦ ϕ# so that ρn is fixed and induces the previous action on the space of conjugacy classes of representations. Next we recall Weil’s isomorphism between the tan- gent space and group cohomology: T[ρn ] R(Σ, SLn (C)) ∼ = H 1 (π1 (Σ), Ad ◦ρn ) (42) d d ρt (γ)ρ0 (γ −1 )|t=0   dt ρt |t=0 7 → γ 7→ dt where ρt is a path of representations, smooth on t ∈ (−ε, ε) and with ρ0 = ρn . Here H 1 (π1 (Σ), Ad ◦ρn ) = Z 1 /B 1 where Z 1 is the space of cocycles or crossed morphisms, i.e. maps θ : π1 (Σ) → sln (C) that satisfy θ(γ1 γ2 ) = θ(γ1 ) + Adρn (γ1 ) (θ(γ2 )) ∀γ1 , γ2 ∈ π1 (Σ), and B 1 is the space of inner cocycles, i.e. maps γ 7→ a − Adρn (γ) (a) for some a ∈ sln (C). Using Weil’s isomorphism (42), the action of dϕ∗ on H 1 (π1 (Σ), Ad ◦ρn ) using classes of cocycles is [θ] 7→ [Adρn (τ −1 ) ◦θ ◦ ϕ# ] ∀θ ∈ Z 1 . The claim follows using the isomorphism between group cohomology of π1 (Σ) and cohomology of the surface Σ, which is aspherical, and the naturality of the actions induced by ϕ on each cohomology group. This establishes the first claim. Proof of Claim (2). We use the isomorphism: r r H (∂i , Ad ◦ρn ) ∼ M M 1 1 (43) H (∂Σ, Ad ◦ρn ) = = H0 (∂i , Ad ◦ρn ). i=1 i=1 42 LÉO BÉNARD, JÉRÔME DUBOIS, MICHAEL HEUSENER, AND JOAN PORTI Namely, we decompose ∂Σ along connected components and use Poincaré du- ality on each of the circles. Next, we use the canonical isomorphism (44) H0 (∂i , Ad ◦ρn ) ∼ = sln (C)Ad(ρ (∂ )) ∼ = Cn−1 n i where sln (C)Ad(ρn (∂i )) = sln (C) ⊗π1 (∂i ) Z denotes the space of coinvariants. It is isomorphic to the quotient of sln (C) by the image of (Ad(ρn (∂i )) − Id) [5], and its dimension n − 1 has been computed for instance in [28]. Next we aim to understand the action of τ on these spaces. We view the chain complex for computing the homology of each ∂i as a subcomplex of C∗ (K, Ad ρn ) = sln (C) ⊗π (Σ) C∗ (K, 1 e Z), the complex that computes the homology of Σ × R, where K, K and K e are as in the beginning of the proof of the lemma. Here every γ ∈ π1 (Σ) acts on e Z) by deck transformations and on sln (C) by Ad(ρn (γ −1 )). The action C∗ (K, of τ maps the chain m ⊗ c (for m ∈ sln (C) and c ∈ C∗ (K, e Z)) to m ⊗ c 7→ m · τ −1 ⊗ τ c = Ad(τ )(m) ⊗ τ c (see [38]). Each component ∂i lifts to a union of lines in the universal covering Σ e whose stabilizer by the action of π1 (Σ) is precisely a representative of π1 (∂i ) in the conjugacy class. Then choosing ẽ0 a 0-cell of K e that projects to ∂i , the canonical isomorphism (44) is induced by the projection sln (C) ⊗ ẽ0 → sln (C) ⊗π1 (∂i ) Z = sln (C)Ad(ρn (∂i )) , where the choice of π1 (∂i ) corresponds to the stabilizer of the line in K e that contains ẽ0 . A different choice of ẽ0 would be γ ẽ0 for γ ∈ π1 (Σ), then the subgroup π1 (∂i ) should be replaced by γπ1 (∂i )γ −1 . This leads to the natural isomorphism between coinvariant subspaces sln (C)Ad(ρn (∂i )) ∼ = sln (C)Ad(ρn (γ∂i γ −1 )) m 7→ Adρn (γ) (m) Furthermore, the action of ϕ∗,1 on H 1 (Σ, Ad ◦ρn ) corresponds via (43) and (44) to sln (C)Ad(ρn (∂i )) → sln (C)Ad(ρn (ϕ# (∂i ))) m 7→ Adρn (τ ) (m) If ϕ defines a cycle of order ci > 0 on the component ∂i (in particular ϕci (∂i ) = ∂i ), then, viewing ∂i as an element of the fundamental group, there exists γ ∈ π1 (Σ) that conjugates ϕc#i (∂i ) = τ ci ∂i τ −ci and ∂i . Namely γτ ci commutes with ∂i in π1 (M (ϕ)), in fact both γτ ci and ∂i belong to the same peripheral subgroup π1 (Ti2 ) of π1 (M (ϕ)). Therefore sln (C)Ad(ρn (∂i )) = sln (C)Ad(ρn (⟨∂i ,γτ ci ⟩)) = sln (C)Ad(ρn (π1 (T 2 ))) i and Ad(ρn (γτ ci )) acts as the identity on sln (C)Ad(ρn (∂i )) , by Lemma D.7 below. In other words, ϕci acts trivially on H 1 (∂i , Ad ◦ρn ) ∼ = Cn−1 , and the second claim follows. □ ASYMPTOTICS OF TWISTED POLYNOMIALS 43 Lemma D.7. For any peripheral torus Ti2 and for any nontrivial γ ∈ π1 (Ti2 ), Ad(ρn (γ)) acts trivially on the space of coinvariants sln (C)Ad(ρn (π1 (T 2 ))) . i Proof. By construction, the action of Ad(ρn (γ)) on the space of invariants 2 sln (C)Ad(ρn (π1 (Ti ))) = {m ∈ sln (C) | Adρn (g) (m) = m, ∀g ∈ π1 (Ti2 )} is trivial. As the C-valued Killing form on sln (C) is Ad-invariant, the space of coinvariants is the quotient of sln (C) by the orthogonal of the invariants. Namely  2 ⊥ sln (C)Ad(ρn (π1 (Ti ))) = {m − Adρn (g) (m) | m ∈ sln (C), ∀g ∈ π1 (Ti2 )} and  ⊥ 2 sln (C)Ad(ρn (π1 (T 2 ))) = sln (C)/ sln (C)Ad(ρn (π1 (Ti ))) . i Therefore, the Killing form induces a pairing 2 sln (C)Ad(ρn (π1 (Ti ))) × sln (C)Ad(ρn (π1 (T 2 ))) → C. i As the Killing form is non-degenrate and Ad-invariant, this pairing is perfect and Ad(ρn (γ))-invariant. In particular, as the action of Ad(ρn (γ)) is trivial on the space of invariants, it is also trivial on the space of coinvariants. □ Proof of Proposition D.3. When ∂Σ = ∅ this is a direct consequence of the Clebsch-Gordan formula (33) and of Lemma D.5. When ∂Σ ̸= ∅, first notice that α always satisfies Assumption 1.2. Further- more, we do not need to care about Assumption 1.3 because Ad ◦ρn decomposes by (33) in a sum of odd dimensional representations and no spin structure is involved in the computation of the torsion in this case (however we could chose a lift of the holonomy satisfying this assumption). Finally, we need to discuss the term det(σϕ −t Id)n−1 . Following Remark D.6, we decompose σϕ into l disjoint cycles of order c1 , . . . , cl , respectively, with c1 + · · · + cl = s. Therefore l Y det(σϕ − t Id) = ± (tci − 1) . i=1 On the other hand, each cycle corresponds to a peripheral torus Ti2 of M (ϕ) and α(π1 (Ti2 )) = ci Z, for i = 1, . . . , l. 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Mathematisches Institut, Georg-August Universität, Bunsenstrasse 3-5, 37073 Göttingen, Germany Email address: leo.benard@mathematik.uni-goettingen.de Université Clermont Auvergne, CNRS, Laboratoire de Mathématiques Blaise Pascal, F-63000 Clermont-Ferrand, France Email address: jerome.dubois@uca.fr Email address: michael.heusener@uca.fr Departament de Matemàtiques, Universitat Autònoma de Barcelona and Cen- tre de Recerca Matemàtica (CRM) 08193 Cerdanyola del Vallès, Spain Email address: porti@mat.uab.cat