Pierre-Ambroise Lacourt
email: pierre-ambroise.lacourt@univ-fcomte.fr

François Courvoisier

Jassem Safioui

Souleymane Diallo

Romain Martinenghi

Luca Furfaro

Maxime Jacquot

Jean-Marc Merolla

Luc Froehly

Laurent Larger

Femtosecond laser preforming of millimeterscale whispering gallery mode resonant disks from crystalline substrate

Millimeter-sized whispering gallery mode resonators produced from bulk crystalline substrates show great potential as optoelectronics components but are still delicate to produce. We report here on an improved manufacturing technique based on femtosecond laser ablation, which allowed us to obtain state-of-the-art performance (quality factor around 10 9 ) in calcium fluoride with only half the processing time required before. Our results are supported by optical profilometric and cavity ringdown measurements, and could be extended to a variety of other substrates.

Introduction

Whispering gallery mode resonators (WGMR) have received extensive interest in the past decades, as they provide a wide family of compact, rugged and versatile photonics components. They can be implemented in a wide range of substrates, either amorphous, crystalline or integrated, at scales ranging from micrometric to centimetric. This wide variety in size and shape, as well as their excellent behavior in both linear and non-linear regimes, has made them suitable for many applications, including narrow-linewidth opto-electronic oscillators, integrated Sagnac gyrometers, quantum optics and optical Kerr combs generation [START_REF] Vahala | Optical microcavities[END_REF][START_REF] Matsko | Review of applications of whisperinggallery mode resonators in photonics and nonlinear optics[END_REF][START_REF] Ilchenko | Optical resonators with whispering-gallery modes-part II: applications[END_REF][START_REF] Chiasera | Spherical whispering-gallery-mode microresonators[END_REF][START_REF] Strekalov | Nonlinear and quantum optics with whispering gallery resonators[END_REF][START_REF] Reynolds | Fluorescent and lasing whispering gallery mode microresonators for sensing applications[END_REF].

Among the available substrates, fluoride crystals such as magnesium, calcium, barium and lithium fluorides (MgF2, CaF2, BaF2 and LiF) have attracted particular attention. Indeed, they offer a good combination of mechanical and optical properties: medium to high MOHS hardness, fairly good thermal stability, high transparency at telecom wavelengths, and insensitivity to water vapor absorption. Thus very low intrinsic loss can be achieved after polishing, enabling high quality factors (Qi = Δf/f0 above 10 8 ). Also, their low refractive index, inferior to that of standard silica fibers, facilitates coupling since fiber tapers can be employed. Moreover, these substrates can support second-and third-order optical nonlinearities, allowing such effects as frequency generation and Kerr, Raman or Brillouin effects [START_REF] Ilchenko | Nonlinear optics and crystalline whispering gallery mode cavities[END_REF].

Fabrication of microdisks or microrods using laser processing techniques have been reported before in amorphous substrates [START_REF] Del'haye | Laser-machined ultra-high-Q microrod resonators for nonlinear optics[END_REF]; however crystalline disks in the millimeter range have so far been produced mostly via mechanical grinding and polishing, which presents severe limitations. Indeed, since some force has to be applied to the substrate, the only possible shape is that of a single, isolated edge. The range of achievable bevel angles is also restricted, particularly with softer materials (MOHS hardness 4 for CaF2). As a result, to prevent chipping or breaking, the bevel angle needs to be kept well above 20°. Consequently, the transverse curve radius of the polished edge remains high (tens of microns) which is detrimental when single transverse mode operation is desired. Also, deviations from the ideal shape (eccentricity, ellipticity) are likely to occur as a reason of the unavoidable resulting mechanical stress. These defects, as well as disk breaking, typically occur in the early stages of the process, when the general shape of the WGMR is produced from a blank and a significant amount of material needs to be removed.

As a result, laser processing is a candidate of choice to improve the processing of crystalline substrates, as it is contactless and thus removes the drawbacks induced by mechanical stress. Femtosecond laser machining in particular has shown great potential, as it provides drastically reduced side effects such as heating and shock wave induced damage when compared to either thermal (CO2) or nanosecond (YAG) processing. Moreover, submicron precision over a wide range of materials has been demonstrated. However, though water-assisted femtosecond cutting of crystalline substrates has been reported [START_REF] Wang | Fabrication of high-Q microresonators in dielectric materials using a femtosecond laser: Principle and applications[END_REF], it has only been applied to micro-toroïds yet, and relies on CO2 laser annealing to provide both for surface shaping (rounding) and polishing, which is difficult to implement for larger disks. In this communication, we report on the femtosecond laser cutting and shaping of CaF2 WGMRs 12 mm in diameter. After mechanical polishing, an intrinsic Q-factor of 9.1 10 8 has been obtained, with significantly reduced processing times.

Fabrication process

A general schematic of the machining setup is represented in Fig. 1. The source we used is a commercially available Amplitude Tangerine amplified femtosecond laser. For the purpose of our setup, the beam was fixed and the sample was moved along five degrees of freedom (3D translation, plus pitch and yaw) using Aerotech PRO-LM stages with an accuracy of ±1 µm. The pulses were focused on the target using a f' = 50 mm lens. The source delivered 280 fs pulses at 1030 µm wavelength in single shot bursts, with a pulse energy of 100 µJ at a 5 kHz repetition rate. First, the 12 mm-diameter disks were cut from 2", 1 mm thick commercial CaF2 wafers (Korth Kristalle GmbH) using the same setup (see Fig. 1.a). A 1.1 mm through-hole was also drilled at this stage to allow the gluing of a handling stub. The disk preforming was then realized in two sequential steps as illustrated in Fig. 1.b,c. First, the disk was tilted at the desired angle of 50°. The disk was then rotated while the beam progressively ablated the substrate in a lathelike manner. Machining was stopped when half the disk thickness had been almost reached, minus a safety margin of a few microns. Then, the disk was translated as shown in Fig. 1.c so that machining of the second edge could be performed in a similar fashion. During each step, regular microscope observation was performed to control progress and ensure machining quality, using the in-situ imaging system.

The overall geometry (disk diameter, eccentricity and bevel angle) was measured using optical microscopy. The mechanical characterization results are summarized in Fig. 2, with a photographic image of the disk after preforming but before polishing (2.a) as well as a geometric profile of the disk bevel obtained via microscopic scanning (Nikon proprietary software, 2.b). This reconstructed profile showed that the bevel angle was well defined and symmetric, with a diedric half-angle of 39.9°. The disk circularity was also found satisfactory, with a radius of 604 µm and an ellipticity inferior to 1 µm. It should be noted that diffraction effects produce slightly tilted edges with an angle of 3.1° with respect to disk normal. This defect was however inconsequential as it was compensated during the following steps. As expected from preliminary work the machined surfaces exhibited significant diffusion under microscopic observation, consistent with > 100 nm surface roughness. This was in accord with measurements performed via white-light vertical scanning interferometric profilometry during our preliminary tests on CaF2 wafers.

Once both disk edges were obtained, subsequent mechanical polishing was performed following the procedure reported in Ref. [START_REF] Coillet | Microwave Photonics Systems Based on Whispering-gallery-mode Resonators[END_REF]. An abrasive compound (aluminum oxide or diamond powder suspension for lower grains) was deposited on polishing tissue, which was held in place by a metallic guide. The guide was translated at low speed (1 cm/minute typical) while the disk was rotated at high velocity (2500 to 5000 RPM) by an air-spindle motor. Progress was checked on site at regular intervals with a binocular microscope, especially before moving to the next abrasive grade. An important aspect is that most of the excess material having been removed via laser ablation, this procedure could begin with a significantly lower grain size (1 µm instead of 29 µm), thus reducing the number of polishing steps and hence the processing time before reaching optics-grade surface roughness. As a summary, representative durations are given in Table 1. Table 1 -Breakdown of polishing times per abrasive grade, with and without femtosecond laser preforming. Please note that each change in grain size requires between 15 and 30 minutes of cleaning and setting up, which has not been taken into account in the total polishing time.

Grade Without fs preforming

Characterization and results

The final surface roughness was measured using a contactless optical profilometer operating under white light vertical-scanning interferometry. A 40X magnification Mirau-type objective lens and a conventional phase shifting algorithm were used. The 3D reconstructed image of the disk after optical-grade polishing is shown in Figure 2.c, over a 30x123 µm² area. The raw data was then processed to account for the disk curvature along both axes. The residual surface roughness was then extracted from the 3D image and found to be 4.4 nm RMS, which is compatible with the high quality factors sought after.

To assess the optical and resonant properties of the disk, 1550 nm-wavelength light from a Koheras Adjustik (NKT Photonics) tunable laser was coupled to the disk using a locally produced fiber taper. By scanning the input laser frequency around the resonance, cavity ringdown was observed and recorded, allowing us to extract the intrinsic and external quality factors via numerical fitting [START_REF] Savchenkov | Optical resonators with ten million finesse[END_REF][START_REF] Trébaol | Ringing phenomenon in coupled cavities: Application to modal coupling in whispering-gallery-mode resonators[END_REF]. Fig. 3 represents our cavity ringdown results, the red curve being the measured signal and the blue curve the fitting curve corresponding to an intrinsic quality factor of Qi = 9.1 10 8 .

Discussion

As can be seen in Fig. 2, the geometrical and mechanical properties of the preformed disks are satisfactory. The bevel angle is well defined and accurately maintained over the whole disk circumference. Ellipticity and eccentricity are also significantly lower than what was observed in our anterior results as reported in Ref. [START_REF] Coillet | Microwave Photonics Systems Based on Whispering-gallery-mode Resonators[END_REF]. More importantly, the processing time was drastically reduced: preforming took roughly one hour instead of three. Polishing was also obtained more easily and rapidly without reduction in surface quality. Altogether, we estimate that the total processing time (from bulk wafer to fully characterized component) is halved. Cavity ringdown measurements confirms that the intrinsic quality factor of the newly produced disks is comparable to previous art, slightly under 10 9 .

We believe that optimization of the machining process could further improve performance, in particular in reducing initial surface roughness, which has a direct impact on the mechanical polishing process. Also, even though our proof-of-concept work was focused on CaF2, we expect this process to adapt fairly well to most substrates, after suitable optimization. Further progress could take WGMRs one step closer to pre-industrial applications.

Conclusion

Femtosecond laser machining of whispering gallery mode resonators from bulk substrates is shown to be a valid alternative to mechanical grinding and polishing. Even though fine-grain polishing is still required to reach optical grade surface finish, femtosecond laser processing significantly reduces production time while maintaining state-of-the-art performance and flexibility. Further work includes optimization of the production scheme as well as portability to other crystalline substrates.

Fig. 1 -

 1 Fig. 1 -Schematic of the machining procedure. (a) Cutting of blank disk from 2" wafer; (b) machining of first bevel angle; (c) machining of second angle. The arrows illustrate the degrees of freedom actively employed during each step. Please note that the beam focusing angle is exaggerated for clarity.

Fig. 2 -

 2 Fig. 2 -Mechanical characterization of preformed disk: (a) photographic image of disk glued to holding stub; (b) microscopic profile reconstruction before polishing; (c) optical profilometry measurement of disk surface after polishing. Measured rugosity after curvature compensation is 4.4 nm over the 20x123µm² observation window.

Fig. 3 -

 3 Fig. 3 -Cavity ring-down measurement of disk resonant properties. Red : measured signal, blue : numerical fitting. The intrinsic quality factor used for the fitting curve was Qi = 9.1 10 8 .

Funding.

This work was funded by the French Agence Nationale de la Recherche (ISITE project FC18028.FEM.IS "Sypher", SMARTLIGHT (ANR-21-ESRE-004), the Banque Publique d'Investissement (project DOS 0074771/00 "Corioli"), and EIPHI Graduate School ANR-17-EURE-0002.

Data Availability.

Data underlying the results presented in this paper are not publicly available at this time but may be obtained form the authors upon reasonable request.

Disclosures.

The authors declare no conflict of interest.


Jean Raimbault
email: jean.raimbault@math.univ-toulouse.fr

TWISTED L 2 -TORSION ON THE CHARACTER VARIETY

Keywords: 2010 Mathematics Subject Classification. Main 57M27; secondary 57M50 Hyperbolic 3-manifold, L 2 -torsion, character variety

We define a twisted L 2 -torsion on the character variety of a 3manifold M and study some of its properties. In the case where M is hyperbolic of finite volume, we prove that the L 2 -torsion is a real analytic function on a neighborhood of any lift of the holonomy representation.

Introduction

Let M be a compact oriented 3-manifold with µ toroidal boundary components, π = π 1 (M ) its fundamental group, and assume that its interior admits a complete hyperbolic metric. Then the Riemannian volume Vol(M ) given by this metric is finite and it is a topological invariant of M by the Mostow-Prasad rigidity theorem. The complete hyperbolic structure also gives rise to a holonomy representation from π to the isometry group PSL 2 (C) of hyperbolic 3-space H 3 (see for example [20, pages 139-152]). The latter lifts to a finite set of representations ρ 0 : π → SL 2 (C); in this paper we will usually pick one of these arbitrarily and call it the holonomy representation. In general the lifts are not isomorphic to each other as representations of π but this does not matter for our purposes here.

Another topological invariant of M is the combinatorial L 2 -torsion τ (2) (M ). It is defined in a manner similar to the classical combinatorial torsions but using a chain complex for the universal cover with the action of the fundamental group, and the Fuglede-Kadison determinants of equivariant operators. For hyperbolic manifolds it can be computed using analytic means, by a result of Wolfgang Lück and Thomas Schick [START_REF] Lück | L 2 -torsion of hyperbolic manifolds of finite volume[END_REF]. It follows that τ (2) (M ) = e -Vol(M ) 6π .

Given a linear representation of the fundamental group of M we can use it to twist the chain complex and for the holonomy representation of a hyperbolic 3-manifold the result of Lück-Schick has been extended in the PhD thesis of Benjamin Waßerman (see [START_REF] Waßermann | An L 2 Cheeger-Müller theorem on compact manifolds with boundary[END_REF] where the corresponding result is announced). In this paper we want to check that the L 2 -torsion τ (2) (M, ρ) twisted by an SL 2 (C)-representation ρ is well-defined, at least near the holonomy. We study the properties of the function ρ → τ (2) (M, ρ) defined on the SL 2 (C)-character variety. We will discuss the extension of the formula in terms of volume below.

1.1. Well-definedness and regularity of the L 2 -torsion function. For hyperbolic knot complements, such a twisted L 2 -torsion was defined in Weiping Li's and Weiping Zhang's paper [START_REF] Li | Twisted L 2 -Alexander-Conway invariants for knots[END_REF]. However, the non-triviality of this invariant was never addressed in this reference, as it is not established there that the relevant complexes are of determinant-class. We remedy to this and observe that in fact the argument used to check well-definedness also implies the following result.

Theorem A. The twisted L 2 -torsion function τ (2) (M, ρ) : X(M ) → R ≥0 is real-analytic on an open neighborhood U of a holonomy character [ρ 0 ] of M in its character variety X(M ). This result fits within a broader program. In general, the question of defining L 2torsions for twisted L 2 -chain complexes, and studying its continuity in the twisting representation, is asked by Lück in [START_REF] Lück | Twisting L 2 -invariants with finite-dimensional representations[END_REF] (see e.g. Problem 10.11 there). It has been answered positively by Yi Liu [START_REF] Liu | Degree of L 2 -Alexander torsion for 3-manifolds[END_REF] in the "abelian" case: for coefficients coming from 1-dimensional representations. Our result gives a partial answer to this problem in a non-abelian setting.

Theorem A is local in character, that is we do not know an explicit description of the locus U . Our argument is not well-suited for this purpose, as we use a spectral gap condition to check the determinant-class condition (see Lemma 5.3) and we establish the latter through a continuity argument. It would be interesting to prove this spectral gap directly for all holonomies of cone-manifold structures; ideally the theorem would be proven to hold on the whole Dehn surgery space (see [START_REF] Hodgson | Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery[END_REF]). On the rest of the character variety we see no reason why there should be such a spectral gap and we see no other way to check the determinant-class condition, outside of the dense subset of characters with values in Q, where the determinant class holds by well-known arguments (see [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF]Chapter 13])

Let us say a few more words on the proof of Theorem A. As we said above the main ingredient is to establish a spectral gap for the relevant operators. By a standard continuity argument we immediately deduce this from the result for the holonomy representation which is essentially established by Nicolas Bergeron and Akshay Venkatesh in [START_REF] Bergeron | The asymptotic growth of torsion homology for arithmetic groups[END_REF] (under the name of "strong acyclicity"). The analyticity of τ (2) (M, ρ) on this neighbourhood then follows from a general result, Lemma 3.6, which seems standard but we could not find in the literature, and from the regularity of the twisted Laplacians on the character variety, as operator-valued functions.

1.2. The L 2 -torsion function and geometric invariants. In view of the results of Lück-Schick and their extension it is natural to ask whether there is a relation between this kind of twisted L 2 -torsion and the volume function Vol defined on X(M ) (see also [12, p. 486] and [10, p. 248]).

Using the relation between the twisted L 2 -torsion and the volume for the family of hyperbolic manifolds obtained by Dehn surgery on M and a surgery formula, one might expect an expression of the L 2 -torsion function as the volume function plus a sum of lengths of the lengths functions L i , i = 1, . . . µ appearing in [START_REF] Walter | Volumes of hyperbolic three-manifolds[END_REF] which for our purposes can be defined as follows: if ρ is the holonomy of a cone-manifold structure on a Dehn filling of M then L i (ρ) equals the length of the core curve of the i-th cusp.

The surgery formula only computes the L 2 -torsion of some intermediate cover of M , and we do not see how to extract enough information on the L 2 -torsion of the universal cover from the latter. However, the Lück-Schick result and continuity of τ (2) (M, ρ) imply that log τ (2) (M, ρ) = -11 12π Vol(ρ) + o(1) near the holonomy representation ρ 0 . A corollary of the analyticity of the L 2 -torsion proved in Theorem A and of [START_REF] Walter | Volumes of hyperbolic three-manifolds[END_REF]Equation (3)] is the following first-order approximation:

(1) log τ (2) (M, ρ) = - 11 12π Vol(ρ) + 11 24 µ i=1 L i (ρ) + O(max L i (ρ) 2 ).
This can be deduced as follows. First, τ (2) (M, ρ 0 ) = exp(-11 12π Vol(M )) as follows from the Cheeger-Müller theorem of Wasserman [START_REF] Waßermann | An L 2 Cheeger-Müller theorem on compact manifolds with boundary[END_REF] and the computation of analytic L 2 -torsion by Bergeron-Venkatesh [START_REF] Bergeron | The asymptotic growth of torsion homology for arithmetic groups[END_REF]. Now the L 2 i are local analytic parameters near the holonomy and the volume function admits a Puiseux development of valuation 1/2 and the expansion in ( 1) is the only one that is differentiable at ρ 0 (this is the contents of [START_REF] Walter | Volumes of hyperbolic three-manifolds[END_REF]Equation (3)]) which the expansion for τ (2) (M, ρ) must be by Theorem A.

We do not know a closed formula for the twisted L 2 -torsion outside of the holonomy representation and so we are not even able to determine whether it is nonconstant in a neighbourhood of the holonomy representation. Computing higherorder terms in (1) would be an interesting related question. Note that it follows from our Proposition 3.4 (1) that τ (2) (ρ) = e -1 3π Vol(M ) for unitary representations ρ, hence the torsion is not constant on the whole character variety in general.

1.3. L 2 -torsion function for non-hyperbolic manifolds. Our definition of the twisted L 2 -torsion is rather general, and it can be considered on the character variety of any 3-manifold. The main point is to ensure that this invariant is not trivial (i.e. zero). To this purpose one needs to check whether some delicate conditions hold, namely that the complex under consideration is weakly acyclic and of determinant class. It is achieved in several cases in this paper. Aside of hyperbolic manifolds, we consider the case where N is a Seifert fibered manifold. We prove: Proposition 1.1. Let N be a compact Seifert manifold. Let ρ be an irreducible representation of π 1 (N ). Then the complex C * (N, ρ) is weakly L 2 -acyclic and of determinant class, and τ (2) (N ; ρ) = 1.

From Proposition 1.1 we deduce a JSJ formula for the twisted L 2 torsion: Theorem B. Let M be a compact aspherical 3-manifold and N 1 , . . . , N r be the hyperbolic components in the JSJ decomposition of M . Let ρ : π 1 (M ) → SL 2 (C) be a representation such that for any i, the complex C

(2) * (N i , ρ| π1(Ni) ) is L 2 -acyclic and of determinant class and that for any Seifert piece N ⊂ M , the restriction ρ| π1(N ) is irreducible. We have

τ (2) (M ; ρ) = r i=1 τ (2) (N i ; ρ).
In particular if M is a graph manifold then τ (2) (M ; ρ) = 1 for any representation ρ that restricts to an irreducible representation on each Seifert fibered piece.

The hypotheses in the theorem are somewhat unnecessary but give a cleaner statement:

(1) we do not know whether it can happen that a hyperbolic piece is not of determinant class; on the other hand we will see that Seifert pieces always are. (2) If the representation is reducible on a Seifert piece we can extract a formula from the proof which involves further factors (see Remark 4.4 below). (3) In principle it is possible to give a similar formula for the twisted L 2 -torsion of any 3-manifold by decomposing it into prime manifolds, which are either elliptical (for which the torsion is a classical Franz-Reidemeister torsion), S 2 × S 1 (for which the computation is easy) or aspherical (for which the above result applies). We also relate our invariant with its abelian cousin studied in [START_REF] Dubois | The L 2 -Alexander torsion of 3-manifolds[END_REF] by Jérôme Dubois, Stefan Friedl and Wolfgang Lück, and treat the case of unitary representations in Proposition 3.4.

We note that in general, when M has a nontrivial JSJ decomposition it is not clear whether there are representations to SL 2 (C) which satisfy the hypotheses of the theorem, or indeed any representations with infinite image.

Organisation of the paper. Section 2 contains preliminary facts on character varieties of hyperbolic manifolds. In Section 3 we recall the general theory of L 2invariants, define the twisted L 2 torsion and state various technical results. In Section 4 we prove Theorem B and Proposition 1.1. Finally, in Section 5 we prove Theorem A.

Acknowledgments. We thank Thomas Schick for some useful suggestions on the topic, and we are grateful to the anonymous referees for their interesting remarks and suggestions. L.B. is partially funded by the RTG 2491 "Fourier Analysis and Spectral Theory".

Preliminaries on character varieties and volume function

In this section we collect facts about the SL 2 (C)-character variety of a hyperbolic 3-manifold. In Subsection 2.1 we define the character variety in general, in Subsection 2.2 we specialize to the case of hyperbolic manifolds.

2.1. Character varieties. We refer to [START_REF] Sikora | Character varieties[END_REF] for a survey of character varieties in general. If Γ is a finitely generated group and G a complex affine algebraic group the representation variety R(Γ, G) is the set Hom(Γ, G), which has the structure of an affine variety, whose defining equations are given by the relations defining Γ and by the determinant one condition. The group G acts by conjugation on R(Γ, G); if G is reductive then the categorical quotient R(Γ, G) / / G is a well-defined affine algebraic variety. It is called the G-character variety of Γ and denoted by X(Γ, G). For a representation ρ ∈ R(Γ, G) we will denote its image in X(Γ, G) by [ρ].

In the sequel we will only consider the case G = SL 2 (C). The character variety X(Γ, SL 2 (C)) is not homeomorphic in general to the topological quotient R(Γ, SL 2 (C))/ SL 2 (C); however there is a SL 2 (C)-invariant open subset for which the two quotients coincide as we now describe. Recall that a representation is irreducible if it does not admit a proper nonzero invariant subspace. Let R * (Γ, SL 2 (C)) be the subset of irreducible representations, which is a Zariski-open subset of the space R(Γ, SL 2 (C)). Orbits of SL 2 (C) on R * (Γ, SL 2 (C)) are closed and so its image

X * (Γ, SL 2 (C)) in X(Γ, SL 2 (C)) is homeomorphic to R * (Γ, SL 2 (C))/ SL 2 (C).
2.2. Character varieties of hyperbolic 3-manifolds. Now we restrict to the case of interest to us of studying X(π, SL 2 (C)) where π = π 1 (M ) is the fundamental group of an orientable hyperbolic manifold with µ cusps. We will use the notations

R(M ) = R(π, SL 2 (C)), X(M ) = X(π, SL 2 (C)).
The complete hyperbolic structure of M corresponds to a holonomy representation ρ 0 : π → PSL 2 (C) such that M = ρ 0 (π)\H 3 . It is known (see [START_REF] Culler | Lifting representations to covering groups[END_REF]) that this representation lifts to a representation ρ 0 : π 1 (M ) → SL 2 (C) which we will also call the holonomy representation. There is in general no unique choice for ρ 0 , we assume that an arbitrary one has been made. It is an irreducible representation and we call its image [ρ 0 ] in X(M ) the holonomy character.

We will need a way to work directly with representations rather than characters, for which the following well-known lemma (see [START_REF] Porti | Torsion de Reidemeister pour les variétés hyperboliques[END_REF]Proposition 3.2]) will be useful.

Lemma 2.1. There exists a section of the projection R(M ) → X(M ) which is real-analytic on a neighbourhood of the holonomy character [ρ 0 ].

The L 2 -torsion

We introduce the twisted L 2 -torsion τ (2) (M, ρ) for unimodular representations of 3-manifold groups. Twisted L 2 -torsions are considered in a general context in [START_REF] Lück | Twisting L 2 -invariants with finite-dimensional representations[END_REF] and we use this and the book [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF] as a reference. In Subsection 3.1 we recall without proof all the results needed to make the paper self-contained. In Subsection 3.2 we use this to define τ (2) (M, ρ). In Subsection 3.3 we give some useful properties and some simple examples. Finally, in Subsection 3.4 we prove a regularity lemma for the determinant which we will use in Section 5.

3.1. L 2 -invariants. Let π be a discrete group and Cπ its group ring. The reduced von Neumann algebra N π is the commutant of the left action of Cπ on 2 (π) in the algebra of bounded operators on the Hilbert space 2 (π) (see [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF]Definition 1.1]). A finitely generated Hilbert N π-module is a quotient of some 2 (π) n by a closed N π-invariant subspace ([14, Definition 1.5]). There is a functor Λ from the category of based free Cπ-modules to the category of Hilbert N π-modules ([15, p. 727]), it can be defined as the L 2 -completion of M with respect to the basis g • b where b is an element of the Cπ-basis of M and g ∈ π. In the sequel we will often make no distinction between M and ΛM or f and Λf .

For V a Hilbert N π-module we denote by B N π (V ) the set of bounded (resp. bounded with a bounded inverse) N π-equivariant operators on V . The subset such operators with a bounded inverse is denoted by GL N π (V ). We also denote by tr N π the usual trace on

B N π (V ) if V is of finite type (see [14, Definitions 1.2, 1.8]).
For the definition of the L 2 -torsion we need the definition of the Fuglede-Kadison determinant det N π (A) of a morphism A between two Hilbert N π-modules. If A is a positive self-adjoint bounded operator in B N π (V ) with trivial kernel then the operator log(A) is well-defined-but not bounded in general. If log(A) is in B N π (V ) its Fuglede-Kadison determinant of A can be computed by the formula:

(2) det N π (A) = exp(tr N π log(A)). The operator log(A) is in B N π (V ) if and only if A ∈ GL N π (V )
, which is the main case we will consider in the sequel. The Fuglede-Kadison determinant is defined in general, and the equality above holds formally (see [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF]Definition 3.11]). When the trace is -∞ the determinant is zero. If det N π (A) > 0 then the operator A is said to be of determinant class. Being of determinant class depends only on the behaviour at 0 of the spectral density function of A; the simplest case is when its Novikov-Shubin invariants (see [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF]Definitions 2.8,2.16]) are positive.

Remark 3.1. It is a difficult question to know whether a morphism is of determinant class. A notable exception occur when π is virtually abelian, in this case all morphisms between finitely generated Cπ-modules are of determinant class, as follows from [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF]Example 3.13] and [START_REF] Everest | Heights of polynomials and entropy in algebraic dynamics[END_REF]Lemma 1.8].

For many groups π, including all residually finite groups, so in particular al 3manifold groups, if f is given by a matrix over the integral group ring Zπ, then it is of determinant class (see [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF]Chapter 13]). So if ρ is defined over Q or even Q we have τ (2) (M, ρ) = 0. However when π is nonabelian and the representation has transcendental coefficients it is essentially completely open to give general criteria.

A complex of Hilbert

N π-modules (C * , d * ) is L 2 -acyclic if its reduced L 2 -homology ([14, Definition 1.17]) vanishes, that is if Im(d p ) is dense in ker(d p-1
). In this situation the combinatorial Laplace operators

(2) (C * ) ∈]0, +∞[ is defined by (4) τ (2) (C * ) 2 = n p=0 det N π (∆ p ) (-1) p p .
We extend the definition of L 2 -torsion to all complexes by taking the convention that τ (2)

(C * ) = 0 if C * is not of determinant class.
3.2. Twisted L 2 -torsion for unimodular representations. Let M be a finite CW-complex, π its fundamental group and ρ : π → GL(E) a finite-dimensional unimodular representation. There is a twisted chain complex of C(π)-modules

C * ( M , ρ) = C * ( M ) ⊗ C E
where any element γ of the group π acts by γ ⊗ ρ(γ) on C(π) ⊗ C E by the rightregular representation on the first factor times ρ acting on the left on the second one, this is denoted by η E C * ( M ) in [START_REF] Lück | Twisting L 2 -invariants with finite-dimensional representations[END_REF]. We chose a Cπ-basis for C * ( M ), that is, an orientation and a lift to the universal cover for each cell. We chose a basis for E as well, we get a complex of free based Cπ-modules. If its completion ΛC * ( M , ρ), which we will rather denote by C

(2) * (M, N π ⊗ ρ) in the rest of this paper, is L 2 -acyclic then we define the twisted L 2 -torsion of (M, ρ) by: ( 5)

τ (2) (M, ρ) = τ (2) (C (2) * (M, N π ⊗ ρ)).
Since the representation ρ is unimodular, it follows immediately from [15, Theorem 6.7(1), Lemma 3.2(1)] that ( 5) does not depend on on the choices of bases for C * ( M ) and E.

We are interested in the twisted L 2 -torsion as defined above for 3-manifolds; however in some occurences we will need the L 2 -torsion for more general covers. The twisted L 2 -torsion of a cover M → M associated to a surjective morphism π → Λ is defined exactly as above, using the chain complexes C * ( M ) in place of C * ( M ) and replacing the group ring and von Neumann algebra Cπ, N π by CΛ, N Λ respectively. In this case we will denote the complex by C

(2) * (M, N Λ ⊗ ρ), and the L 2 -torsion by τ (2) (M, N Λ ⊗ ρ).

(2) (M, ρ) = τ (2) (M, ρ ).
The next well-known lemma (see for example [4, Lemma 3.1]) will be useful to compute the L 2 -torsion of Seifert manifolds.

Lemma 3.3. Let C * = 0 → Cπ k A -→ Cπ k+l B -→ Cπ l → 0.
Let A ∈ M k,k (Cπ) and B ∈ M l,l (Cπ) be the square matrices obtained from the k first lines of A and l last columns of B respectively. If A , B are acyclic and of determinant class then so is C * and moreover

τ (2) (C * ) = det N π (A ) -1 det N π (B ).
Proof. Consider the diagram

E *   C *   D * 0 ----→ 0 ----→ Cπ l B ----→ Cπ l ----→ 0     0 ----→ Cπ k A ----→ Cπ k+l B ----→ Cπ l ----→ 0     0 ----→ Cπ k A ----→ Cπ k ----→ 0 ----→
0 where vertical arrows in the middle are given by natural inclusions and projections. This gives a short exact sequence of complexes 0 → E * → C * → D * → 0 and it follows from additivity of L 2 -torsion [14, Theorem 3.35( 1)] (the terms on the right in this equation vanish because of acyclicity and determinants of inclusions and projections are 1) that C * is acyclic, of determinant class and

τ (2) (C * ) = τ (2) (D * ) • τ (2) (E * ).
Since the torsion of a complex with a single nonzero differential d is equal to det N π (d) (-1) p where p is the degree of d, the result follows.

In some very special cases the computation of the L 2 -torsion is immediate.

Proposition 3.4. (1) If M is hyperbolic and the representation ρ : π 1 (M ) → SU(2) ⊂ SL 2 (C) is unitary then τ (2) (M, ρ) = τ (2) (M ) 2 = exp(-Vol(M )/3π) (2) Assume that H 1 (M ) = Z, and let φ : π 1 (M ) → Z be a choice of an abelianization map. If ρ : π 1 (M ) → SL 2 (C) is reducible, conjugated to ρ(γ) = λ φ(γ) * 0 λ -φ(γ)
for some λ ∈ C * then

τ (2) (M, ρ) = τ (2) (M, φ)(|λ|)τ (2) (M, φ)(|λ| -1 )
where the abelian-twisted L 2 -torsion τ (2) (M, φ) is defined in [START_REF] Dubois | The L 2 -Alexander torsion of 3-manifolds[END_REF].

Proof. Point ( 1) is a consequence of [15, Theorem 4.1 (4)] For (2), observe that using Lemma 3.2 one can assume that ρ has the triangular form above. It induces an exact sequence of π-modules

0 → 2 (π) → 2 (π) 2 → 2 (π) → 0
where the π-actions are given by the representations λ φ , ρ, λ -φ respectively. Now we can apply [START_REF] Lück | Twisting L 2 -invariants with finite-dimensional representations[END_REF]Lemma 3.3] and conclude since we have the equality τ (2) (M, -φ)(|λ|) = τ (2) (M, φ)(|λ| -1 ).

3.4.

Regularity under spectral gap. Let V be a Hilbert space, B(V ) the space of bounded linear operators on V . On B(V ) we will use the operator norm defined by

|T | = sup v =1 T v .
Let S(V ) ⊂ B(V ) the subspace of self-adjoint operators. By the spectral theorem any self-adjoint operator T in S(V ) has a spectrum σ(T ) which is a closed subset of R. We will use the following lemma.

Lemma 3.5. The function S(V ) → R defined by T → inf σ(T ) is continuous.

Proof. It also follows from the spectral theorem that

(6) inf σ(T ) = inf v∈V \{0} T v, v v 2 = inf v =1 T v, v Since T ∈ S(V ) is bounded we have that inf σ(T ) = λ > -∞. Let ε > 0. Then for any v ∈ V with v = 1 we have T v, v ≥ λ. If |S -T | < ε then for all such v we have Sv, v ≥ T v, v -Sv -T v
by Cauchy-Schwarz and the right-hand side is then at least λ -ε. Together with [START_REF] Everest | Heights of polynomials and entropy in algebraic dynamics[END_REF] this proves that σ is lower-continuous.

On the other hand there exists a w ∈ V , w = 1 such that T w, w ≤ λ + ε. For S as above we have Sw, w ≤ T w, w + Sw -T w ≤ λ + 2ε so σ(S) ≤ λ + 2ε. With [START_REF] Everest | Heights of polynomials and entropy in algebraic dynamics[END_REF] this proves that σ is upper-continous.

Let S >0 (V ) be the set of bounded self-adjoint operators A ∈ S(V ) such that inf σ(A) > 0, and let S >0 N π (V ) = S >0 (V ) ∩ B N π (V ). By Lemma 3.5, the set

S >0 N π (V ) is an open subset in S N π (V ).
It is an open question to determine exactly the domain of continuity in B N π (V ) of the Fuglede-Kadison determinant. However it is a general principle that it is "as regular as possible" on the open subset GL N π (V ); we will use an instance of this valid in the real-analytic category.

The space B N π (V ) is a Banach space for the norm operator, as a norm-closed subspace of B(V ). If U ⊂ R n is an open subset and E a Banach space we say that a function A : U → E is real-analytic if it admits an expression as a convergent power series in the neighbourhood of every x ∈ U .

We will use the habitual notations: for an n-tuple of integers α = (α 1 , . . . , α n ) and x ∈ R n , let |α| = i α i and

x α = x α1 1 • • • x αn n .
The former real-analyticity condition can be conveniently formulated as follows: for every x ∈ U there exists a sequence (a α ) ∈ E N n and r 0 > 0 such that

α a α E r |α| 0 < +∞ and for any v ∈ R n such that |v i | < r 0 for 1 ≤ i ≤ n (that is, v ∞ < r 0 ), we have A(x + v) = α∈N m a α v α .
Lemma 3.6. Let V be a Hilbert-N π-module of finite type, U an open subset of R n and A : U → S >0 N π (V ) a real-analytic map. Then the map x → det N π A(x) is a real-analytic function on U .

Proof. We see from (2) that we need to prove that the function x → tr N π log(A(x)) is real-analytic. This will follow immediately from the two following points: i) If is a continuous linear form on a Banach space B and x → T (x) a real-analytic map U → B then x → (T (x)) is real-analytic; ii) If I ⊂ R, f is a real-analytic function with a power series expansion which converges on I, and σ(A(x)) ⊂ I for all x ∈ U then x → f (A(x)) is realanalytic. Note that f = log satisfies this assumption for any I ⊂ ]0, +∞[.

Let us prove i): write α∈N n T α v α a convergent power series with a positive radius of convergence for T , then by continuity of the series α (T α )v α converges as well, and it equals the image of the former by the linear map on the domain of convergence. Now we prove ii): let f (t) = k c k t k be a power series expression for f : I ⊂ R → R, let x ∈ U and let α∈N n a α v α be the power series expansion for A(x + v), which converges for v ∞ ≤ r 1 for some r 1 > 0. We denote M = α∈N n a α B r |α| 1 . Assuming a 0 = 0 for convenience, we want to show that the formal series

k≥0 c k α∈N n a α v α k is convergent on a neighbourhood of v = 0. Let r 0 be the radius of convergence for k c k t k , then for v ∞ < r 1 r 0 /M we have that α∈N n a α B |v α | < r 0 by convexity. For those v the series k |a k | α∈N n a α B |v α |
k converges; in other words the power series k≥0 a k α∈N n a α v α k is absolutely convergent on a neighbourhood of 0.

Seifert manifolds and JSJ decomposition

In this subsection we prove Theorem B from the introduction, whose statement we recall here. Note that a similar statement, with a much simpler proof, also holds for torsions with unitary coefficients. Theorem 4.1. Let M be a compact aspherical 3-manifold and N 1 , . . . , N r be the hyperbolic components in the JSJ decomposition of M . Let ρ : π 1 (M ) → SL 2 (C) be a representation such that for any Seifert piece N ⊂ M , the restriction ρ| π1(N ) is irreducible, and for every hyperbolic piece C * (N i , N π 1 (N i ) ⊗ ρ| π1(Ni) ) is determinantclass. Then we have τ (2)

(M ; ρ) = r i=1 τ (2) (N i ; ρ).
In particular if M is Seifert or more generally a graph manifold then τ (2) (M ; ρ) = 1.

Proof. Since M is aspherical it is in particular irreducible and we can perform a JSJdecomposition of M . Its JSJ-components are either hyperbolic or Seifert fibered manifolds. The theorem then follows immediately from the following two claims:

(1) If N 1 , N 2 are compact with toric boundary, N is a gluing of N 1 , N 2 along a collection of incompressible boundary components then

τ (2) (N, ρ) = τ (2) (N 1 , ρ| π1(N1) )τ (2) (N 2 , ρ| π1(N2) ).
(2) For any ρ : π 1 (N ) → SL 2 (C), if N is Seifert (compact with toric boundary components) then the complex C

(2) * (N, N π 1 (N )⊗ρ| π1(N ) ) is L 2 -acyclic and of determinant class. Moreover, if ρ is an irreducible representation of π 1 (N ) then τ (2) (N ; ρ) = 1. Let N = N 1 ∪ N 2 as in the statement of Claim (1) and T 1 , . . . , T r the boundary tori along which N 1 and N 2 are glued to each other. Let π = π 1 (N ), for X = N 1 , N 2 , T j we denote by C

(2) * (X, ρ) the twisted complex of Hilbert N π-modules associated to the preimage of X in the universal cover N . Since the T j are incompressible, this preimage is a disjoint union of copies of X. By the restriction law [15, Theorem 6.7( 6)] we have that

τ (2) (C (2) * (X, ρ)) = τ (2) (X, ρ| π1(X) )
for those X. As we have an exact sequence

0 → r j=1 C (2) * (T j , ρ) → C (2) * (N 1 , ρ) ⊕ C (2) * (N 2 , ρ) → C
(2) * (N, ρ) → 0 it follows by using the gluing formula [15, Theorem 6.7(3)] that

τ (2) (N, ρ) = τ (2) (N 1 , ρ| π1(N1) ) • τ (2) (N 2 , ρ| π1(N2) ) r j=1 τ (2) (T j , ρ| π1(Tj ) )
.

In Lemma 4.2 below we show that the L 2 -torsion of a 2-torus with coefficients in any local system is 1 so τ (2) (T j , ρ| π1(Tj ) ) = 1 for all j and we finally obtain the formula in Claim (1). Claim ( 2) is a subcase of the more general Proposition 4.3 that we prove below. The arguments are essentially lifted from [START_REF] Kitano | Reidemeister torsion of Seifert fibered spaces for SL(2; C)-representations[END_REF] and adapted to the L 2 -setting.

We now prove a lemma on tori used in the proof above and which we will also need to deal with Seifert manifolds. It could be deduced from general Poincaré duality but we prefer to give a more explicit proof. Lemma 4.2. Let ρ : Z 2 → SL(V ) be any unimodular representation, Λ an infinite group and Z 2 → Λ a surjective morphism. Then the complex C

(2) * (T 2 , N Λ ⊗ ρ) is L 2 -acyclic and of determinant class, and its torsion τ (2)

(T 2 , N Λ ⊗ ρ) = 1.
Proof. The determinant class condition is always satisfied (see Remark 3.1). So we need to compute the torsion and show it equals 1.

First we reduce to the case where ρ is semisimple. Let m, be generators for π 1 (T 2 ). As ρ(m) and ρ( ) commute they are trigonalizable with Jordan blocks of the same shape. It follows that there exists a sequence g n ∈ SL(V ) such that ρ n ( ) := g n ρ( )g -1 n and ρ n (m) := g n ρ(m)g -1 n converge to a pair of commuting semisimple elements ρ ∞ ( ), ρ ∞ (m) ∈ SL(V ). By Lemma 3.2 we have τ (2) (T 2 , ρ) = τ (2) (T 2 , ρ n ). It follows from the formula for the Fuglede-Kadison determinant over abelian groups that ρ → τ (2)

(T 2 , ρ) is continuous. So τ (2) (T 2 , ρ) = τ (2) (T 2 , ρ ∞ ).
We can now assume ρ to be semisimple, so we can decompose ρ ∼ = dim(V ) j=1

χ j where χ j are 1-dimensional. It follows that τ (2) (T 2 , ρ) = j τ (2) (T 2 , χ j ) so we can just assume that ρ is 1-dimensional. Now the complex whose L 2 -torsion we have to compute is just

0 → 2 (Z 2 ) d1 -→ 2 (Z 2 ) 2 d0 -→ 2 (Z 2 ) → 0
with the boundary operators given by the following matrices over the group ring:

d 1 = 1 -⊗ ρ( ) 1 -m ⊗ ρ(m) d 0 = m ⊗ ρ(m) -1 ⊗ ρ( ) -1 .
It is immediate that the complex is L 2 -acyclic, and it follows from Lemma 3.3 that the torsion is equal to det

N Λ (1 -⊗ ρ( )) -1 det N Λ (1 -⊗ ρ( )) = 1.
Proposition 4.3. Let N be a compact Seifert manifold. Let ρ be an irreducible representation of π 1 (N ). Then the complex C * (N, ρ) is weakly L 2 -acyclic and of determinant class, and τ (2) (N ; ρ) = 1.

Before starting the proof, let us recall some facts about oriented Seifert fibered spaces (see [START_REF] Jankins | Lectures on Seifert Maniolds[END_REF]). In a Seifert fibered space N with boundary a union of k tori, there is a finite number of exceptional fibers F 1 , . . . F r , each of which is given with a pair of integers (p i , q i ). We denote by T i a tubular neighborhood of a singular fiber F i , it is homeomorphic to a solid torus. The manifold N \ (T 1 ∪ . . . ∪ T r ) is an S 1 -bundle on a surface with k + r boundary components. The obstruction for this bundle to be trivial can be concentrated on the neighborhood of a single regular fiber F 0 with neighborhood T 0 , so that N \ (T 0 ∪ T 1 ∪ . . . ∪ T r ) is homeomorphic to the product N 0 = S g,k+r+1 × S 1 of a surface of genus g with k + r + 1 boundary components, and the fibration twists b times around T 0 .

If the surface S is orientable, we have the following presentation for the fundamental group of N :

(7) π 1 (N ) = a 1 , b 1 . . . , a g , b g , c 1 , . . . , c k , f 0 , . . . f r , h | hx = xh ∀x ∈ π 1 (N ), f pi i h qi = 1, [a 1 , b 1 ] • • • [a g , b g ]c 1 • • • c k f 0 . . . f r = 1
where the a i , b i are standard generators of the surface group, the c i represent the boundary curves of the surface which do not bound a fiber, and f i represents a curve on the surface such that f i × S 1 ∂T i . Additionally (p 0 , q 0 ) = (1, -b), and finally h denotes the class of the generic fiber.

Proof. Weak acyclicity and determinant class property follow from the explicit computation we will make for the complexes involved. Lemma 4.2 yields

τ (2) (∂T i , N π 1 (T i ) ⊗ ρ| π1(∂Ti) ) = 1.
Applying the multiplicativity formula for the L 2 -torsion we get

τ (2) (N, ρ) = τ (2) (N 0 , N π 1 (N 0 ) ⊗ ρ| π1(N0) ) r i=0 τ (2) (T i , ρ| π1(Ti) )
First we will prove that τ (2) (T i , ρ| π1(Ti) ) = 1 for any index i = 0, . . . r, and then that τ (2)

(N 0 , N π 1 (N ) ⊗ ρ| π1(N0) ) = 1.
Since ρ is irreducible and h is central in π 1 (N ), necessarily ρ(h) = ±Id, and it follows that for each i, the generator i of π 1 (T i ) has finite order through ρ.

In particular its eigenvalues λ i , λ -1 i are unitary complex numbers, and the operator 1

-i ⊗ ρ( i ) : 2 (π 1 (T i )) 2 → 2 (π 1 (T i )) 2 has Fuglede-Kadison determinant equal to 1. Since T i retracts on the circle i , its L 2 torsion is τ (2) (T i , ρ π1(Ti) ) = det N π1(Ti) (1 -i ⊗ ρ( i )) -1 = 1,
and the first assertion is proved. Now N 0 retracts onto 2-complex Y which is a product of a circle with a bouquet of 2g + k + r circles indexed by a 1 , b 1 . . . , a g , b g , c 1 , . . . , c k , f 1 , . . . f r (the generator f 0 does not appear thanks to the last relation in π 1 (N )).

By the same computation as in [9, Proof of Proposition 4.2] the differentials in the π 1 (N )-complex C * (Y, N π 1 (N ) ⊗ ρ| π1(Y ) ) are given by

A = (a 1 ⊗ ρ(a 1 ) -1 • • • b g ⊗ ρ(b g ) -1 c 1 ⊗ ρ(c 1 ) -1 . . . f r ⊗ ρ(f r ) -1 h ⊗ ρ(h) -1)
and

B =         1 -h ⊗ ρ(h) 0 • • • 0 . . . . . . . . . . . . . . . 0 • • • 1 -h ⊗ ρ(h) 1 -a 1 ⊗ ρ(a 1 ) 1 -b 1 ⊗ ρ(b 1 ) • • • 1 -f r ⊗ ρ(f r )        By Lemma 3.3 the L 2 -torsion of the complex 0 → L 2 (π 1 (N )) 2g+k+r B -→ L 2 (π 1 (N )) 2g+k+r+1 A -→ L 2 (π 1 (N )) → 0 is equal to det N π1(N ) (B ) det N π1(N ) (A ) -1 where B =    1 -h ⊗ ρ(h) 0 0 0 . . . 0 0 0 1 -h ⊗ ρ(h)   
is the (2g + k + r) × (2g + k + r) matrix obtained by keeping all but the last line of B and A = (1 -h ⊗ ρ(h)) is the matrix obtained by deleting all but the last column of A. Both of these Fuglede-Kadison determinants equal 1. So τ (2) (Y, N π 1 (N ) ⊗ ρ| π1(Y ) ) = 1, and by homotopy invariance [START_REF] Lück | Twisting L 2 -invariants with finite-dimensional representations[END_REF]Theorem 6.7(2)] it follows that τ (2) (N 0 , N π 1 (N ) ⊗ ρ| π1(N0) ) = 1 as well, and it proves the second assertion and finishes the proof of Proposition 4.3 in this case.

It remains to deal with the case where S is not orientable; this can be done following a similar scheme as above (see [START_REF] Kitano | Reidemeister torsion of Seifert fibered spaces for SL(2; C)-representations[END_REF] for the classical case). A simpler argument in our setting is to apply the orientable case to the double cover associated to π 1 (N ) → π 1 (S) → Z/2 and use multiplicativity of the twisted L 2 -torsion in covers [START_REF] Lück | Twisting L 2 -invariants with finite-dimensional representations[END_REF]Theorem 6.7(5)].

Remark 4.4. If ρ is not irreducible then the arguments above are still valid except that the image ρ(h) of the generic fiber can have any complex number as an eigenvalue, hence det N π1(N ) (1 -h ⊗ ρ(h)) is not necessarily equal to 1. In general, denoting by λ ρ an eigenvalue of ρ(h) of modulus ≥ 1, we have ( 8)

τ (2) (N, ρ) = |λ ρ | 2g+k+r-2- r i=1 1 p i
For example if M (p, q) is the complement of the (p, q)-torus knot we have

τ (2) (M (p, q), ρ) = |λ ρ | 1-1 p -1 q .
since in this case M (p, q) is a fibration on a disk with two singular fibers with singularities (p, 1) and (q, 1). Note that this can also be computed directly by a simpler method analogous to [START_REF] Dubois | Weighted l2-invariants and applications to knot theory[END_REF], using the following presentation of the knot group

π 1 (M (p, q)) = a, b | a p = b q .
Proof of Remark 4.4. With the same matrices as in the proof of Proposition 4.3, we just have to compute the determinants involved. First we carefully compute the core curves of the solid tori T i . For given index i = 0, . . . r, a meridian of the solid torus T i is given by the null-homotopic curve f pi i h qi , see [START_REF] Hodgson | Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery[END_REF]. Hence a longitude i is given by i = f ni i h mi for m i p i -n i q i = 1. Now the eigenvalues of ρ( i ) are λ ±1/pi ρ , and we obtain

det N π1(N ) (1 -i ⊗ ρ( i )) = |λ ρ | 1/pi , det N π1(N ) (B ) = |λ ρ | 2g+k+r , det N π1(N ) (A ) = |λ ρ |
and the result (8) follows from

τ 2) (N, ρ) = det N π1(N ) (B ) det N π1(N ) (A ) r i=0 det N π1(N ) (1 -i ⊗ ρ( i ) .
Finally, we note that for reducible representation ρ the element ρ( i [a i , b i ]) has trace ±2. In the case N has empty boundary (k = 0), if b = 0 in [START_REF] Hodgson | Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery[END_REF], then the eigenvalues of ρ(h) must lie on the unit circle.

5. Twisted L 2 torsion for hyperbolic manifolds.

In this section we conclude the proof of Theorem A, whose statement we recall here:

Theorem 5.1. The twisted L 2 -torsion function τ (2) (M, ρ) : X(M ) → R
is real-analytic on an analytic neighborhood U of any lift [ρ 0 ] of the holonomy representation of M in the character variety X(M ).

In Subsection 5.1 and 5.2 we give alternative definitions for the L 2 -torsion τ (2) (M, ρ), which allow us to work with operators on a fixed Hilbert space (Lemma 5.2) and for which we have a spectral gap (Lemma 5.3) that allows us to apply Lemma 3.6 to deduce Theorem A. We give the proof of Lemma 5.3 in Subsection 5.3, using comparison with the analytic L 2 -invariants and the spectral gap property of the holonomy representation established in [1, Lemma 4.1].

Combinatorial Laplacians. Lemma 5.2. There exists a graduated Hilbert

N π-module V = V 0 ⊕ • • • ⊕ V 3 and functions D p : R(M ) → Hom N π (V p , V p-1
), p = 1, . . . , 3, which are regular in a Zariski-neighbourhood of ρ 0 in R(M ) and such that for every ρ the complex

(V * , D * (ρ)) is isomorphic to C (2) * (M, N π ⊗ ρ). In particular, if we set A p = D * p D p + D p+1 D * p+1 then ρ → A p (ρ)
are analytic in a neighbourhood of ρ 0 and we have

(9) τ (2) (M, ρ) = 3 p=1 det N π (A p (ρ)) p(-1) p .
Proof. We set

L p = C p ( M ) ⊗ C C 2
where π acts by γ •(e⊗v) = (γe)⊗v. We choose an arbitrary Cπ-base B p for C p ( M ); then L p is isomorphic (as a Cπ-module) to C p ( M ) ⊗ C 2 with diagonal action (as introduced in 3.2) via the map

I p (ρ) : γe ⊗ v → γe ⊗ ρ(γ)v for e ∈ B, γ ∈ π.
Choosing any base of C 2 we get a basis of L p , and the completion V p = ΛL p is isomorphic to C

p (M, N π ⊗ ρ) (see also [START_REF] Lück | Twisting L 2 -invariants with finite-dimensional representations[END_REF]Lemma 1.1]). Let ∂ p be the differentials of C * (M, N π⊗ρ), which are given by ∂ p (e⊗v) = ∂ p e⊗v. The corresponding boundary maps of V * are given by D p (ρ) = I -1 p-1 (ρ)∂pI p (ρ). The coefficients of I p (ρ) in the Cπ-bases of L * , C * ( M ) ⊗ C 2 are rational functions of ρ because they depend only on the coefficients of a finite number of ρ(γ), so the first part of the lemma is proven.

Obviously A p = I -1 p ∆ p I p and (9) follows. 5.2. L 2 -torsion. (2) (M, ∂M, N π ⊗ ρ) is then defined by the exact sequence

0 → C * (2) (M, ∂M, N π ⊗ ρ) → C * (2) (M, N π ⊗ ρ) i * -→ C * (2) (∂M, N π ⊗ ρ| ∂M ) → 0. We denote by ∆ p rel : C p (2) (M, ∂M, N π⊗ρ) → C p (2) (M, ∂M, N π⊗ρ) the combinatorial Laplacians of the complex C * (2) (M, ∂M, N π ⊗ ρ).
The crucial point for us is then the following, which we prove in the next subsection.

Lemma 5.3. There exists λ 0 > 0 and a neighborhood U of the holonomy such that for all [ρ] ∈ U , for all p = 0, . . . 3, we have σ(∆ p rel (ρ)) ⊂ [λ 0 , +∞[. 5.2.2. Poincaré duality. We follow the references [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF] (see the proofs of Theorems 1.36(3) and 3.93(3)) and [15, Theorem 6.7 [START_REF] Hodgson | Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery[END_REF]]. There is a homotopy equivalence P * : C * ( M ) → C 3- * ( M , ∂ M ) between the untwisted chain and relative cochain complexes; a construction is given in the proof of Theorem 2.1 in [START_REF] Wall | Surgery on compact manifolds, volume 69 of Mathematical Surveys and Monographs[END_REF]. We can extend it by the identity to a homotopy equivalence

P * : C (2) * (M, N π ⊗ ρ) → C 3- *
(2) (M, ∂M, N π ⊗ ρ) between the completed twisted complexes. By [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF]Theorem 2.19] it follows that they have the same Novikov-Shubin invariants. In particular ∆ p (ρ) has a spectral gap for any ρ ∈ U (where U is the neighbourhood of ρ 0 given by Lemma 5.3).

Proof of Theorem A. The relative cochain complex C *

(2) (M, ∂M, N π ⊗ρ) is of determinant class for ρ in the neighbourhood U given by Lemma 5.3. By Poincaré duality (see preceding paragraph) we get that C

(2) * ( M , ρ) is of determinant class and that for ρ ∈ U , inf σ(∆ p (ρ)) > 0, and by Lemma 5.2 we get that (V * , D * (ρ)) are determinant-class and the A p (ρ) have a spectral gap for ρ ∈ U .

The hypotheses of Lemma 3.6 are thus satisfied by [ρ] → A p (ρ) where ρ belongs to the section of the map R(M ) → X(M ) given by Lemma 2.1. It follows that the functions [ρ] → det N π A p (ρ) are real-analytic in U and so is the L 2 -torsion by (9). 5.3. Proof of lemma 5.3.

5.3.1.

Preliminaries on analytic L 2 -invariants. In this subsection we define the twisted analytic Laplacian operators and derive some of their properties. For our purposes here we need only consider twisting by the holonomy representation ρ 0 but the discussion can be adapted to deal with any ρ.

We consider the trivial rank 2 bundle H 3 × C 2 with the π 1 (M )-action

γ • ( x, v) = (γ • x, ρ 0 (γ) v)
for any γ ∈ π 1 (M ), x ∈ H 3 , v ∈ C 2 , and the associated complex Ω * c (H 3 , C 2 ) of compactly supported, C 2 -valued forms on H 3 with the natural Γ-equivariant differential d p : Ω p (H 3 , C 2 ) → Ω p+1 (H 3 , C 2 ). On the bundle H 3 × C 2 we choose an arbitrary SL 2 (C)-invariant norm, which amounts to choosing a norm on C 2 above a base point .

The spaces Ω p c (H 3 , C 2 ) admit a Hermitian product associated with this metric on H 3 × C 2 . We will use a uniform notation for various spaces of differential forms associated with this inner product: if X is a manifold, a prefix R before Ω p (X, ρ) indicates that we consider the completion associated to ρ of the p-forms with coefficients in the trivial bundle and regularity R. For example L 2 Ω p stands for square-integrable forms, H l Ω p for Sobolev spaces, note that these Hilbert spaces and their N π-module structure depend on ρ. The norm on any RΩ p (X, ρ) will be denoted by • R . For the Sobolev spaces from here on, an integer l > dim M = 3 is fixed.

The Hodge Laplacian ∆ p on the space L 2 Ω p (H ) of compactly supported smooth forms extends to a unique essentialy self-adjoint operator on the completion L 2 Ω p (H 3 , ρ). This operator has a well-defined spectrum σ(∆ p ) and it follows from (6) then we have [START_REF] Li | Twisted L 2 -Alexander-Conway invariants for knots[END_REF]

inf σ(∆ p ) = min inf ω∈Ω p c (H 3 ,C 2 )\{0} d p ω 2 L 2 ω 2 L 2 , inf ω∈Ω p+1 c (H 3 ,C 2 )\{0} d p+1 ω 2 L 2 ω 2 L 2
.

The following result essentially follows from [1, Lemma 4.1]; we give a short justification as this reference nominally covers only compact locally symmetric spaces.

To be more specific: the statement of [1, Lemma 4.1] is given only for the discrete spectrum but the argument also applies to continuous spectrum. This is justified by the fact that the representations in the packets contributing to the spectrum on p-forms also satisfy the condition Hom K (∧ p p * ⊗ C 2 , •) = 0 which is the only point used to give a lower bound on the Casimir eigenvalue in loc. cit. As rigorously justifying this would take too much time and preliminaries we give a suboptimal bootstrap argument below. M (ρ M )) the Laplacian on p-forms with coefficients of the associated C 2 -bundle over M with the hermitian metric descended from that described above on H 3 . Now let M n be a tower of coverings of a fixed closed hyperbolic manifold such that the injectivity radius inj(M n ) goes to infinity. Let µ n be the spectral density mesure of ∆ p M (ρ M )); it is an atomic measure by the spectral theorem in Riemannian geometry and by the preceding paragraph it is supported on the interval [δ, +∞[ for all n. Now it follows from the Lemma in the proof of Theorem 4.5 ([1, p. 410]) that the measures µ n / Vol(M n ) weakly converge to the spectral measure µ of ∆ p (ρ 0 ).

More precisely, the lemma states (in our notation) that for fixed t > 0 we have that Mn tr(e -t∆ p Mn (ρ Mn ) -tr(e -t∆ p (ρ0) ) d Vol goes to 0. The integrals of the heat kernels can be rewritten using the spectral measures of ∆ p Mn (ρ Mn ) and ∆ p (ρ 0 ) and we get that for any t > 0 we have +∞ 0 e -tλ dµ n (λ)/ Vol(M n ) converges to +∞ 0 e -tλ dµ. The latter implies weak convergence of the measures since the functions λ → e -tλ are dense in uniform convergence on compact subsets. See also [START_REF] Donnelly | On the spectrum of towers[END_REF]Theorem 1.1] whose proof extends to our setting using [START_REF] Bergeron | The asymptotic growth of torsion homology for arithmetic groups[END_REF]Lemma 3.8].

In particular, if ϕ is any continuous function supported in ] -δ, δ[ we have that 0 = ϕdµ n / Vol(M n ) → ϕdµ and so the latter also vanishes, which means that ∆ p (ρ 0 ) has no spectrum in ] -δ, δ[.

5.3.2.

Comparison of analytic and combinatorial invariants for the holonomy. The main step in the proof of Lemma 5.3 is now to prove the spectral gap for the representation ρ = ρ 0 . We compare the spectra of Laplace operators on the complex C *

(2) ( M , ∂ M , N π ⊗ ρ) and on the Sobolev complexes H l- * Ω * (H 3 , ρ) using Whitney maps. Our arguments are essentially those leading to [14, Lemma 2.71] which we adapt to the twisted setting. We will use the notation • 2 for the norm on the cochain complex C

(2) * ( M , ∂ M , N π ⊗ ρ), and by • x for the norm of the fiber above a point x in H 3 .

First we recall the definition of Whitney maps. Using the analoguous of Lemma 5.2 for cohomology, we use a different -although isomorphic-model for the cochain complex C * (M, N π ⊗ ρ). We see it as the completion of the complex

C * ( M ) ⊗ C 2 but with action γ • (f ⊗ v) = (γf ) ⊗ (ρ(γ)v) and differentials d(f ⊗ v) = (df ) ⊗ v. If we fix a Cπ-basis of C p ( M ) the map γf ⊗ v → γf ⊗ ρ(γ)v
gives an isomorphism of Cπ-complexes from our former model to this one. We choose a smooth partition of unity e c on M , indexed by vertices c of the triangulation and we lift it to M . The important property that we require is that e c has its support contained in the open simplices adjacent to c. Then, if f σ is the cochain with value 1 on σ and 0 on other simplices we define

W p (f σ ⊗ v) = p! c∈σ ω σ,c ⊗ v
where ω σ,c is defined by the usual formula: if c 0 , . . . , c p are the vertices of σ ordered according to orientation, ω σ,ci = (-1) i e ci • j =i de cj . The form W p f σ ⊗ v is supported in the open star of σ; in particular it is compactly supported in M \ ∂ M if σ is not contained in ∂M .

Thus restricting W defines a map

C * ( M , ∂ M ) ⊗ C 2 → C ∞ c Ω * (H 3 , C 2 )
, and a direct computation (see [23, p. 140]) shows that it is a chain map-note that in the model we use here, on both sides the differentials satisfy d(f ⊗ v) = df ⊗ v, d(ω ⊗ v) = dω ⊗ v so we can use this computation as it is for our twisted case. This map from a Cπ-module of finite rank extends to a bounded chain map Indeed, if this holds then for any φ the Rayleigh quotient of W p φ satisfies

d(W p φ) H l-p-1 W p φ H l-p = W p (dφ) H l-p-1 W p φ H l-p ≤ (c -1 C) dφ 2 φ 2
and by Lemma 5.4 and ( 10) it follows that we must have dφ 2 φ 2 ≥ c C δ. As (10) also applies to the combinatorial laplacian we get that inf σ(∆ p rel ) ≥ c C δ. This will prove Lemma 5.3 for the holonomy representation ρ 0 , after we show inequality [START_REF] Liu | Degree of L 2 -Alexander torsion for 3-manifolds[END_REF].

Since W is bounded, the content of ( 11) is the minoration c φ 2 ≤ W p φ H l-p , which we will now prove. First we note (see [START_REF] Schick | Notes for a minicourse at icmat, madrid[END_REF]Section 7]) that on the image of W p the Sobolev and L 2 -norms are equivalent, so it suffices to show that c φ L 2 ≤ W p φ L 2 . To do so, for a p-simplex σ in M = H 3 let U σ be the maximal open subset of H 3 on which c∈σ e c = 1 and no e c vanishes for c ∈ σ. This is nonempty, and we have U σ ∩ U τ = ∅ if τ = σ is another p-simplex. In addition, if σ = [c 0 , . . . , c p ], replacing every instance of e c0 in W p (f σ ⊗ v) with (1 -p i=1 e ci ) ⊗ v (which is valid on U σ by definition) we get that

W p (f σ ⊗ v) = (de c1 ∧ • • • ∧ de cp ) ⊗ v on U σ ,
in particular it does not vanish there and so the integral of its norm on U σ is strictly positive. As there are only finitely many p-simplexes modulo π and the maps e γc are π-equivariant we get that there exists a > 0 such that [START_REF] Lott | Heat kernels on covering spaces and topological invariants[END_REF] W p (f σ ⊗ v) L 2 (Uσ) ≥ a inf (2) ( M , ∂ M , N π ⊗ ρ). For any σ let γ σ the element of π such that γ -1 σ σ belongs to B. Then by definition f σ ⊗ v 2 = ρ(γ σ ) -1 v . As the U σ are relatively compact and B is finite, there is some a (independent of σ) such that inf x∈ Uσ v x > a v for all σ ∈ B. As x → v x is π-invariant, it follows that for any σ we have

inf x∈ Uσ v x > a ρ(γ -1 )v = a f σ ⊗ v 2 .
With [START_REF] Lott | Heat kernels on covering spaces and topological invariants[END_REF] we finally get that

W p (f σ ⊗ v) L 2 (Uσ) ≥ a f σ ⊗ v 2 .
for a = aa > 0 and all σ. Now if φ is an arbitrary relative cochain we can write it as φ = σ b σ f σ ⊗ v σ so that we have

φ 2 2 = σ b 2 σ f σ ⊗ v σ 2 2
As the U σ are disjoint we get that

W p (φ) 2 L 2 ≥ σ W p (φ) 2 L 2 (Uσ) = σ W p (f σ ⊗ v) 2 L 2 (Uσ) ≥ (a ) 2 σ b 2 σ f σ ⊗ v 2 2 = (a ) 2 φ 2 2
where the inequality on the second line follows from the fact that W p (f τ ⊗ v) vanishes identically on U σ whenever τ = σ, and that on the third from the previous inequality. The last line is the inequality we were after.

( 3 )

 3 ∆ p = d p d * p+1 + d p+1 d * p have zero kernel. They are always positive, and the complex (C * , d * ) is said to be of determinant class if all ∆ p are of determinant class. Then the L 2 -torsion τ

3. 3 .

 3 Properties and first examples. The following basic fact follows immediately from [15, Lemma 3.2 (1)]. Lemma 3.2. Let ρ, ρ : π 1 (M ) → SL 2 (C) be two conjugate representations. Then τ

5. 2 . 1 .

 21 L 2 -cochain complexes. We define in the same way as in 3.2 the L 2 -cochain complexes C *(2) (M, N π ⊗ ρ) as the completion of the complex C * ( M ) ⊗ C 2 with diagonal action and similarly for C *(2) (∂M, N π⊗ρ). The relative L 2 -cochain complex C *

Lemma 5 . 4 .

 54 There exists δ > 0 such that for the holonomy ρ 0 and 0 ≤ p ≤ 3 we have σ(∆ p (ρ 0 )) ⊂ [δ, +∞[. Proof. The standard representation of SL 2 (C) on C 2 satisfies the hypotheses of [1, Lemma 4.1]. Hence there exists a δ > 0 such that σ(∆ p M (ρ M ))) ≥ δ for any closed hyperbolic 3-manifold M , where ρ M is the holonomy representation of M and ∆ p

W : C * ( 2 )

 2 (M, ∂M, N π ⊗ ρ) → H l- * Ω * (H 3 , ρ).To prove our claim about the spectral gap of C *(2) ( M , ∂ M , ρ) it suffices to prove the following inequalities: there exists C, c > 0 such that for all p and any cochainφ ∈ C p (2) ( M , ∂ M , N π ⊗ ρ) we have (11) c φ 2 ≤ W p φ H l-p ≤ C φ 2 .

  Let • be the norm on C 2 , and B the Cπ-basis of C * ( M ), that were used to define the norm • 2 on C *

5. 3 . 3 .

 33 Conclusion of the proof of Lemma 5.3. By Lemma 5.2 and the isomorphisms V * → C (2) * ( M , N π ⊗ρ) → C * (2) ( M , ∂ M , N π ⊗ρ) the matrices of the operators ∆ p (ρ) have coefficients which vary continuously for [ρ] in their domain of definition D. In particular each map ∆ p rel : D → B(C p (2) ( M , ∂ M , N π ⊗ ρ)) are continuous for the operator norm. Lemma 5.3 then follows from Lemma 3.5 since it holds at [ρ 0 ] by 5.3.2 and Lemma 5.4.

  3 , ρ) is defined as follows: if d * p is the formal adjoint of d p then the operator defined by ∆ p = d * p d p + d p+1 d * p+1 on the space Ω p c (H 3 , C 2