F R Éd

Polyxeni Spilioti
email: spilioti@math.au.dk

ON THE SPECTRUM OF TWISTED LAPLACIANS AND THE TEICHM ÜLLER REPRESENTATION

Given a compact hyperbolic surface X = Γ\H 2 and a linear nonunitary representation : Γ → GL(V ), we investigate the spectrum of the twisted Laplacian ∆ acting on sections of the associated flat vector bundle E → X. We show that this non self-adjoint operator has its spectrum inside a parabola related to a critical exponent δ of the representation. In the case where is of Teichmüller type we then exhibit explicit parabolic regions determined by another constant δ 0 with an asymptotic spectral density which improves Weyl's law by a power factor. Both δ and δ 0 are uniquely determined by the so-called "Manhattan curve" related to the representation .

Introduction and main results

The Laplace-Beltrami operator is a key object in Riemannian geometry: its spectrum encodes some fine properties of the geometry of the manifold and the dynamics of the geodesic flow. This operator is self-adjoint and has discrete spectrum on compact manifolds. It has natural generalizations to vector bundles and remains self-adjoint if the vector bundle comes equipped with a natural hermitian metric. In particular, if this vector bundle is flat, the holonomy of a loop depends only on its homotopy class and there is an associated natural unitary representation of the fundamental group. Conversely, any unitary finite dimensional representation of the fundamental group of a compact manifold give rise to a flat vector bundle on which a natural self-adjoint Bochner-Laplace operator lives. There are however natural situations where non-unitary representations of the fundamental group lead to consider an associated non-self-adjoint Laplacian, whose discrete spectrum remains rather mysterious. In this paper, we will study non-self-adjoint Bochner-Laplace operators associated to linear, non-unitary representations of surface groups. There is a very rich litterature on this subject which leads to a wide class of non-self-adjoint Laplacians for which, apart from the general results pertaining to elliptic operators, almost nothing is known.

More precisely, let H 2 denote the hyperbolic plane and let X = Γ\H 2 be a compact hyperbolic surface obtained by quotienting H 2 by a discrete co-compact subgroup of PSL 2 (R). Given a linear representation : Γ → GL(V ), where V is a complex d-dimensional vector space, we can easily build a flat vector Bundle E over X by considering the quotient of H 2 × V by the Γ-action defined by γ.(z, v) : = (γ(z), (γ)v). Smooth sections of E are then identified with elements

F ∈ C ∞ (H 2 , V ) satisfying ∀ γ ∈ Γ, F (γz) = (γ)F (z).
Let ∆ be the usual hyperbolic Laplacian acting diagonally on smooth -equivariant sections, then this operator can be identified with the twisted Bochner-Laplace operator ∆ = -tr((∇ E ) 2 ),

where ∇ E is a flat connection on E . Choosing a hermitian metric on V , one can define L 2 (X, E ) as the space of square integrable sections of E . The operator ∆ is in general not self-adjoint. However, it has a self-adjoint principal symbol and hence it has nice spectral properties ( [START_REF] Aleksandrovich | Pseudodifferential operators and spectral theory[END_REF]). It is an elliptic operator and, because of the compactness of the base X, it has a unique closed extension with dense domain in L 2 (X, E ), with a compact resolvent. Its spectrum is therefore discrete and moreover is contained in a translate of a positive cone C ⊂ C with R + ⊂ C ( [START_REF] Aleksandrovich | Pseudodifferential operators and spectral theory[END_REF], [START_REF] Müller | A Selberg trace formula for non-unitary twists[END_REF]Lemma 2.1]). In addition, the spectrum of the twisted Laplacian satisfies a Weyl law, as showed by Müller in a pioneering work [START_REF] Müller | A Selberg trace formula for non-unitary twists[END_REF]. He also proved a general trace formula for integral operators induced by the twisted Laplacian, using Paley-Wiener test functions. This work generalizes Selberg's theory in a non-self-adjoint setting for all locally symmetric space of split rank one. The higher rank case is considered by Shen, where he used the Müller's Selberg trace formula to study the Ruelle zeta function and its relation to the complex analytic torsion ( [START_REF] Shen | Complex Valued Analytic Torsion and Dynamical Zeta Function on Locally Symmetric Spaces[END_REF]). In general, the twisted Laplacians are defined as coupling operators to a flat vector bundle ([12, Section 2]). One can define then also the twisted Hodge Laplacian, acting on the space of vector-valued differential forms. Consequently, there exists an extension of the usual notion of the analytic torsion ( [START_REF] Daniel | R-torsion and the laplacian on riemannian manifolds[END_REF]) to the case of non self-adjoint Hodge Laplacians. This is the Cappell-Miller torsion ( [START_REF] Sylvain | Complex-valued analytic torsion for flat bundles and for holomorphic bundles with (1, 1) connections[END_REF]) and the refined analytic torsion of Braverman and Kappeler ( [START_REF] Braverman | Refined analytic torsion as an element of the determinant line[END_REF], [START_REF] Braverman | Refined analytic torsion[END_REF]).

Let now P denote the set of primitive conjugacy classes in Γ and if γ ∈ P, let (γ) be the length of the associated primitive closed geodesic on X. We define the critical exponent δ of the representation by

δ( ) := inf    s > 0 : γ∈P,k≥1 |tr( (γ k ))|e -sk (γ) < ∞    .
A simple argument using the fact that the word length and the displacement function (with respect to hyperbolic distance in H 2 ) are equivalent shows that there exist M, C > 0 and a set of representatives {γ} of conjugacy classes such that, |tr( (γ))| ≤ M e C (γ) , hence δ( ) is finite. Since is assumed to be non unitary, we will have in general δ( ) > 1. For certain type of representations , this critical exponent is related to the celebrated Manhattan curve, see below for details.

We then define the parabolic regions C σ ⊂ C for all σ ∈ (1/2, +∞) by

C σ := z = x + iy ∈ C : x ≥ σ(1 -σ) + y 2 (1 -2σ) 2 .
We will also use the notation P σ := ∂C σ to denote the parabola given by the equation {x + iy : x = σ(1 -σ) + y 2

(1-2σ) 2 }. In the degenerate case where σ = 1/2, we set P σ = C σ = [1/4, +∞). Our first result is the following.

Theorem 1.1. Let ∆ be the twisted Laplacian defined as above, then one always have Sp(∆ ) ⊂ C δ( ) .

We will show in §3 that this inclusion is in some sense sharp: there exists families of linear representations such that ∆ satisfies inf

λ∈Sp(∆ ) Re(λ) = δ(1 -δ).
It is tempting to believe that one always has this identity, but this is not true: in §3 we exhibit a family of linear representations for which δ(1 -δ) ∈ Sp(∆ ).

The proof of Theorem 1.1 is standard and follows from the already known properties of the twisted Selberg zeta functions and a Heat trace argument. As proved by Muller in [START_REF] Müller | A Selberg trace formula for non-unitary twists[END_REF], the twisted Laplacian satisfies a Weyl law: as T → +∞,

#{λ ∈ Sp(∆ ) : |λ| ≤ T } = dim(V )Vol(X) 4π T + o(T ).
We prove a much more precise fact for some specific type of representations which we call "of Teichmüller type". Given a fixed hyperbolic, compact surface X = Γ\H 2 , we will view the Teichmüller space T (X) as the space of faithfull discrete representations ρ : Γ → PSL 2 (R), quotiented by conjugations in PSL 2 (R). We denote the hyperbolic distance on H 2 by d(z, w), we will use the upper half-plane model H 2 = {Im(z) > 0} in the definition below. Definition 1.2. A linear representation : Γ → GL(V ) is said to be of Teichmüller type if there exist a norm . 0 on GL(V ), a representation ρ ∈ T (X) and constants C, β > 0 such that for all γ ∈ Γ,

(γ) 0 ≤ Ce βd(i,ρ(γ)i) , (2) C -1 e β (ρ(γ)) ≤ |tr( (γ))| ≤ Ce β (ρ(γ)) . (1)
Let : Γ → GL(V ) be a representation of Teichmüller type, as defined above. Clearly we have

δ( ) = inf    s > 0 : γ,k e βk ρ (γ)-sk (γ) < ∞    ,
where ρ : Γ → PSL 2 (R) is a Teichmüller deformation of Γ and ρ (γ) stands for (ρ(γ)). Following the work of Burger [START_REF] Burger | Intersection, the Manhattan curve, and Patterson-Sullivan theory in rank 2[END_REF], let M ρ be the convex region defined by

M ρ :=    (a, b) ∈ R 2 : γ,k e -ak ρ(γ)-bk (γ) < ∞    .
Then, the boundary of M ρ is a convex graph C which contains the points (0, 1) and (1, 0), called "the Manhattan curve". This curve is a straight line if and only if Γ and ρ(Γ) are conjugated in PSL 2 (R), see [START_REF] Burger | Intersection, the Manhattan curve, and Patterson-Sullivan theory in rank 2[END_REF][START_REF] Sharp | The Manhattan curve and the correlation of length spectra on hyperbolic surfaces[END_REF]. It was shown by Sharp in [START_REF] Sharp | The Manhattan curve and the correlation of length spectra on hyperbolic surfaces[END_REF] that this curve is actually real-analytic. In our setting, δ( ) is the unique real number such that (-β, δ( )) belongs to C, see the figure below.

The Manhattan curve C and the exponent δ( ).

The main result of this paper is the following improved Weyl law.

Theorem 1.3. Assume that is a a linear representation of Teichmüller type. There exists δ/2 < δ 0 < δ such that for all δ 0 < σ ≤ δ, one can find (σ) > 0 such that as r → +∞, we have

M σ (r) := #{λ ∈ Sp(∆ ) \ C σ : r ≤ |λ| ≤ r + √ r} = O(r 1/2-(σ) ).
Moreover, similarly as δ, the constant δ 0 is uniquely determined by β and the Manhattan curve: if (-2β, η) ∈ C, then δ 0 = η/2.

We have the following corollary.

Corollary 1.4. For all σ > δ 0 , as T → +∞ we have

N σ (T ) := #{λ ∈ Sp(∆ ) \ C σ : |λ| ≤ T } = O(T 1-(σ) ),
which beats the Weyl law by a polynomial factor. As a consequence, Weyl law actually holds in C σ for all σ > δ 0 : as T → +∞,

#{λ ∈ Sp(∆ ) ∩ C σ : |λ| ≤ T } = dim(V )Vol(X) 4π T + o(T ).
Proof. Consider the sequence (n p ) p≥0 defined by n 0 := 1 and n p+1 = n p + √ n p for all p ≥ 1. It is a standard fact that as p → +∞ we have

n p = p 2 4 (1 + o(1)).
We fix σ > δ 0 , let T >> 1 and choose the smallest integer p such that T ≤ n p+1 .

We have p = O( √ T ), while n p < T . We then write

N σ (T ) ≤ O(1) + p k=0 M σ (n k ) ≤ O(1) + (p + 1) max 0≤k≤p M σ (n k ) = O T 1 2 + 1 2 -(σ) ,
and the proof is done.

The above theorem therefore shows that in the "high frequency" regime when we have Re(λ) → +∞, the bulk of the spectrum lies in a smaller parabolic region C δ0 .

Spectrum of ∆ with P δ and Pσ , the bulk of the spectrum being inside Cσ . Some explicit bounds on δ 0 involving the "geodesic strech" can be derived, see §6.

The computable examples of §3 show that δ 0 is sharp.

In the case when is a one-dimensional non-unitary representation, Anantharaman [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF] proves a similar result as Theorem 1.3. Indeed, if we have

(γ) = e γ ω ,
where ω is a harmonic real-valued 1-form on X = Γ\H 2 , and γ ∈ H 1 (X, Z) is the homology class determined by γ. The spectrum of ∆ acting on sections of the line bundle E is the same as the spectrum of

∆ ω f := ∆ X f + 2ω(∇f ),
acting on functions on X. This spectral problem can therefore be understood in terms of a generalized stationary damped wave equation on X, and the semiclassical technique of Sjöstrand [START_REF] Sjöstrand | Asymptotic distribution of eigenfrequencies for damped wave equations[END_REF] can be refined as done in [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF] to obtain the one-dimensional analog of Theorem 1.3. Unfortunately, this one-dimensional technique cannot be applied for higher dimensional representations. Hence our method is fairly different, but our result can be viewed as a generalization of [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF] to the higher dimensional representations case. It is also interesting to observe that in [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF], independently from [START_REF] Müller | A Selberg trace formula for non-unitary twists[END_REF], a trace formula for ∆ ρ is derived and utilized, by an analytic continuation argument of the "usual" Selberg trace formula for a unitary representation.

The paper is organized as follows. In §2, we recall the known facts on twisted Laplacians, including the heat trace formula and the main result on the Selberg zeta function from [START_REF] Frahm | Twisted Ruelle zeta function at zero for compact hyperbolic surfaces[END_REF]. We then prove easily Theorem 1.1. In §3, we exhibit representations of Teichmüller type and compute the spectrum for several examples: depending on the parity of the dimension of the representation space, these spectra lie on parabolas and are related to (self-adjoint) Dirac and Laplace spectra. In §4, we recall the so-called Bowen-series coding for co-compact Fuchsian groups and relate δ to the topological pressure by using some basic Teichmüller theory and thermodynamical formalism. The proof of Theorem 1.3 really starts in §6, where we relate the counting problem to an estimate from above of certain Hilbert-Schmidt norms for a family of transfer operators. These transfer operators act on a family of (vector valued) Bergman spaces of holomorphic functions and rather delicate off-diagonal estimates for Bergman kernels of shrinking domains are needed: they are proved in the Appendix. We finally analyse δ 0 and prove some explicit bounds via Thermodynamical formalism. At the technical level, the core of the proof of Theorem 1.3 bears some similarities with [START_REF] Jakobson | Resonances and density bounds for convex co-compact congruence subgroups of SL 2 (Z)[END_REF][START_REF] Naud | Density and location of resonances for convex co-compact hyperbolic surfaces[END_REF], but there are some serious additional difficulties and differences which make it interesting: the non-unitarity of the representation involved, and the need to use more general complex domains (hence the non-trivial Bounds on Bergman kernels).

Acknowledgements. We thank Julien Marché and Andres Sambarino for several discussions around the examples of §3 and pointing out several possible generalizations. We also thank Maxime Wolff for pointing out references on Thurston's asymetric distance.

Twisted Selberg zeta functions and the main parabola

In this section we prove Theorem 1.1, which is completely general. We define first the twisted Bochner-Laplace operator as introduced in [28, Section 4]. This operator acts on sections of a vector bundle over X. It is an elliptic operator which is in general not self-adjoint. Nevertheless, it has a self-adjoint principle symbol (with respect to the choice of a metric) and hence it has qualitatively similar spectral properties.

Let be a finite-dimensional, complex representation

: Γ → GL(V )
of Γ. Denote by E = V × Γ H 2 → X the associated flat vector bundle over X, equipped with a flat connection ∇ E . We recall the construction of the twisted Bochner-Laplace operator ∆ , acting on smooth sections of E . The second covariant derivative (∇ E ) 2 is defined by

(∇ E ) 2 V,W := ∇ E V ∇ E W -∇ E ∇ LC V W
, where V, W ∈ C ∞ (X, T X), T X is the tangent bundle of X, and ∇ LC denotes the Levi-Civita connection on T X. The twisted Bochner-Laplace operator ∆ is defined to be the corresponding connection Laplacian on E , i.e., the negative of the trace of the second covariant derivative:

∆ := -tr (∇ E ) 2 .
(

Locally, this operator is described as follows.

We consider an open subset U of

X such that E | U is trivial, i.e., E | U ∼ = U × C m , where m = rank(E ) = dim V .
Let e 1 , . . . , e m be any basis of flat sections of

E | U . Then, each φ ∈ C ∞ (U, E | U ) can be written as φ = m i=1 φ i ⊗ e i , where φ i ∈ C ∞ (U ), i = 1, . . . , m. Then, ∇ E Y φ = m i=1 ∇ Y φ i ⊗ e i , (Y ∈ C ∞ (X, T X)). (2)
The local expression above is independent of the choice of the basis of flat sections of E | U , since the transition maps comparing flat sections are constant. By (1) and (2), the twisted Bochner-Laplace operator acting on

C ∞ (U, E χ | U ) is given by ∆ φ = m i=1 (∆φ i ) ⊗ e i , (3)
where ∆ denotes the usual Laplace-Beltrami operator on X for the hyperbolic metric. Let now E be the pullback to X = H 2 of E . Then,

E ∼ = X × V , and C ∞ ( X, E ) ∼ = C ∞ ( X) ⊗ V . (4)
With respect to the isomorphism (4), it follows from (3) that the lift ∆ of ∆ to E takes the form

∆ = ∆ ⊗ Id V , (5)
where ∆ is the Laplace-Beltrami operator on H 2 . If we choose a Hermitian metric on E , then ∆ acts in L 2 (X, E ) with domain C ∞ (X, E ). However, it is not a formally self-adjoint operator in general. By (3), ∆ has principal symbol

σ ∆ (x, ξ) = ξ 2 x Id (E )x (x ∈ X, ξ ∈ T * x X).
Hence, ∆ is an elliptic, second order differential operator with the following spectral properties: its spectrum is discrete and contained in a translate of a positive cone C ⊂ C such that R + ⊂ C. This fact follows from classical spectral theory of elliptic operators, under the assumption of the compactness of the manifold. We refer the reader to [START_REF] Aleksandrovich | Pseudodifferential operators and spectral theory[END_REF], and also [START_REF] Müller | A Selberg trace formula for non-unitary twists[END_REF]Lemma 2.1]. From now on, for notational simplicity, we will denote this Laplacian by ∆ .

The semi-group e -t∆ is well defined for t > 0 and trace class on L 2 (X, E ). The following "heat trace formula" [START_REF] Frahm | Twisted Ruelle zeta function at zero for compact hyperbolic surfaces[END_REF] holds:

tr(e -t∆ ) = µ∈Sp(∆ ) e -tµ (6)
= 1 4π 2 dim(V )Vol(X) R e -t(λ 2 +1/4) λπ tanh(λπ)dλ + 1 2π γ∈P,k≥1 tr( (γ k )) (γ) 2 sinh(k (γ)/2) e -t/4 e -(k (γ)) 2 4t
.

By working with this heat trace one can then deduce a "resolvent trace formula" [START_REF] Frahm | Twisted Ruelle zeta function at zero for compact hyperbolic surfaces[END_REF] and obtain the following fact on the twisted Selberg zeta function.

Theorem 2.1. For Re(s) large enough, set

Z Γ (s, ) := k≥0 γ∈P det I -(γ)e -(s+k) (γ) .
Then Z Γ (s, ) has a holomorphic extension to C. Its zeros set is the divisor (with multiplicities)

j {1/2 ± ir j } ∪ ∞ k=0 (2g -2)dim(V )(2k + 1){-k},
where {1/4 + r 2 j } is the spectrum of ∆ and eigenvalues are repeated according to their multiplicity. In addition the following functional equation holds for all s ∈ C \ Z,

Z Γ (s, ) = Z Γ (1 -s, )exp dim(V )Vol(X) s-1/2 0 r tan(πr)dr .
Remark 2.2. By Theorem 2.1, the twisted Selberg zeta function Z Γ (s, ) has a holomorphic extension to C. Indeed, by considering the "resolvent trace formula" [19, eq. (4.21)], we have that the logarithmic derivative L(s, ) of Z Γ (s, ) has simple poles with positive residues. This follows from Proposition 4.2.4 and Proposition 4.2.5 in [START_REF] Frahm | Twisted Ruelle zeta function at zero for compact hyperbolic surfaces[END_REF]. Still, we can not conclude more information about the order of the zeros of Z Γ (s, ), since the "spectral" and the "trivial" zeros could overlap.

2.1. Proof of Theorem 1.1. For Re(s) large, we have by a direct calculation

Z Γ (s, ) Z Γ (s, ) = γ∈P,k≥1 (γ) 1 -e -k (γ) tr( (γ k ))e -sk (γ) .
By definition of δ( ), we know that the right hand side converges absolutely for Re(s) > δ, which shows that Z Γ (s, ) has no zeros in the half-plane {Re(s) > δ}.

Using the functional equation, we deduce that non-trivial zeros of Z Γ (s, ) are in the strip {1 -δ ≤ Re(s) ≤ δ}. Given s j (1 -s j ) ∈ Sp(∆ ) with s j = 1/2 ± ir j , then we know that s j must be a zero of Z Γ (s, ): therefore we get that

Sp(∆ ) ⊂ Z -∪ C δ .
We have to discard the possible negative integer eigenvalues that are below δ(1-δ).

To this end we go back to the heat trace formula [START_REF] Bers | Uniformization, moduli, and Kleinian groups[END_REF] in the regime t → +∞. Let us denote by N (x) the counting function

N (x) := k (γ)≤x |tr( (γ k ))|.
By definition of δ, we have the crude bound valid for all > 0 as x → ∞:

N (x) = O (e (δ+ )x ).
Set α := inf µ∈Sp(∆ ) Re(µ). By discreteness of the spectrum, we know that there are finitely many eigenvalues µ such that Re(µ) = δ, and therefore a simple argument involving Lebesgue dominated convergence shows that there exists C 0 > 0 such that for all t large,

C 0 e -tα ≤ | j e -tµj | = |tr(e -t∆ )|.
Using the heat trace formula we get for t large

C 0 e -tα ≤ O(e -t/4 ) + O   e -t/4 γ∈P,k≥1 |tr( (γ k ))| (γ) 2 sinh(k (γ)/2) e -(k (γ)) 2 4t   .
Furthermore, we have

S(t) := γ∈P,k≥1 |tr( (γ k ))| (γ) 2 sinh(k (γ)/2) e -(k (γ)) 2 4t = O ∞ 0 xe -x/2 e -x 2 4t dN (x) .
A Stieltjes integration by parts combined with the crude bound on N (x) shows that for all > 0 we have :

S(t) = O ∞ 0 e (δ+ -1/2)x e -x 2 4t dx .
It is a standard calculation (Laplace transform of a Gaussian) to show that as a → ∞ we have

∞ 0 e av e -v 2 dv = e a 2 /4 ( √ π + o(1)).
Using this asymptotic and a change of variable show that for all > 0 and all t large,

S(t) = O √ te (-δ(1-δ)+1/4+ )t ,
which implies that as t → +∞,

C 0 e -αt ≤ O √ te (-δ(1-δ)+ )t
and therefore α ≥ δ(1 -δ). The proof is done.

2.2.

On the bottom of the spectrum. From Theorem 1.1, we know that inf

µ∈Sp(∆ ) Re(µ) ≥ δ(1 -δ).
In §3, we will give examples for which this inequality is strict. There is however a large class of representations for which the equality holds true. We prove the following.

Proposition 2.3. Assume that for all γ ∈ Γ we have tr( (γ)) ≥ 0. Then we have

inf µ∈Sp(∆ ) Re(µ) = δ(1 -δ).
Proof. Assume that the inequality is strict, then we know that the meromorphic function on

C s → Z Γ (s, ) Z Γ (s, )
has no pole at s = δ, i.e. is analytic at s = δ. On the other hand we have for all Re(s) > δ

Z Γ (s, ) Z Γ (s, ) = γ∈P,k≥1 (γ) 1 -e -k (γ) tr( (γ k ))e -sk (γ) ,
which is a Dirichlet series of the type n a n e -sλn with a n ≥ 0, whose abcissa of absolute convergence is precisely δ. A classical theorem of Landau [START_REF] Landau | Handbuch der Lehre von der Verteilung der Primzahlen[END_REF] tells us that s = δ must be an analytic singularity, a contradiction.

We remark that if is a representation of Teichmüller type, then we must have δ( ) > 1, as depicted on the Manhattan curve. Therefore δ(1 -δ) < 0 which means that ∆ has indeed negative eigenvalues under the hypothesis of positivity of the character tr( (γ)).

If the character is real-valued, then from the heat trace formula one deduces that for all t > 0 we have j e -tµj = j e -tµj , which implies that Sp(∆ ) = Sp(∆ ), i.e., the spectrum is symmetric with respect to the x-axis. Alternatively, we can recover this fact from the identity

Z Γ (s, ) = Z Γ (s, ),
which follows from the equality (valid for Re(s) large)

Z Γ (s, ) = exp   - γ n≥1 tr( (γ n )) n e -s (γ n ) 1 -e -(γ n )   ,
and uniqueness of analytic continuation.

Examples of linear representations of surface groups

In this section, we give examples of representations : Γ → GL(V ) which are of Teichmüller type. For some of them the spectrum of ∆ can be explicitely computed in term of Laplace or Dirac spectrum on the base surface X. We then make some remarks about the bottom of the spectrum of ∆ for these specific representations.

Y 0 = 0 1 0 0 , Y 1 = 0 0 1 0 , Y 2 = 1 0 0 -1 .
The adjoint representation Ad : G → GL(V ) is defined by Ad(g)(X) = gXg -1 , and leaves invariant the killing form on V given by B(X, Y ) = tr(XY ).

Notice that Ad(-I) = I, hence Ad induces a representation

r a : PSL 2 (R) → GL(V ).
In the basis (Y 0 , Y 1 , Y 2 ), the associated quadratic form to B is

B(U, U ) = 2u 2 2 + 2u 0 u 1 for U = u 0 Y 0 + u 1 Y 1 + u 2 Y 2 ,
which is a quadratic form of signature [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF][START_REF] Adler | Geodesic flows, interval maps, and symbolic dynamics[END_REF]. This means that in a suitable basis (X 0 , X 1 , X 2 ), this quadratic form can be expressed as

B(X, X) = -x 2 0 + x 2 1 + x 2 2 , for X = x 0 X 0 + x 1 X 1 + x 2 X 2 . The Hyperboloid model of the hyperbolic plane H 2 is then H 2 := {X ∈ V : B(X, X) = -1 and x 0 > 0}.
It is a standard fact that Ad induces an isomorphism between P SL 2 (R) and SO + (2, 1) which is the group of direct isometries of the hyperboloid model. Notice that hyperbolic distance in the hyperboloid model is given by cosh

(d(U, V )) = -B(U, V ).
In particular, if Ad(g) is given in the basis (X 0 , X 1 , X 2 ) by the matrix

M =   a b c d e f g h i   ,
then the fact that M and M -1 preserve the killing form imply that we have the formula

M 2 = 4a 2 -1 = 4 cosh 2 d(Ad(g)o, o) -1, where o = (1, 0, 0) ∈ H 2 . Given an hyperbolic element γ ∈ PSL 2 (R), we can always conjugate it to γ = e (γ)/2 0 0 e -(γ/2 , so that a simple computation in the basis (Y 0 , Y 1 , Y 2 ) yields Ad( γ)Y 0 = e (γ) Y 0 , Ad( γ)Y 1 = e -(γ) Y 1 , Ad( γ)Y 2 = Y 2 , (7) tr
(Ad(γ)) = tr(Ad( γ)) = 1 + e (γ) + e -(γ) . (8) All the above computations show that for all ρ ∈ T (Γ), r a • ρ : Γ → SO + (2, 1) is indeed a Teichmüller type representation with β = 1.
In the specific case = r a •id, then we have by the prime orbit theorem δ( ) = 2. The spectrum of ∆ is actually computable. Indeed, using formulas [START_REF] Bolte | The Selberg trace formula for Dirac operators[END_REF], we can express the twisted Selberg zeta function as follows. For all Re(s) large, we have

Z Γ (s, ) = k∈N γ∈P det I -(γ)e -(s+k) (γ) = k∈N γ∈P 1 -e -(s+k-1) (γ) 1 -e -(s+k+1) (γ) 1 -e -(s+k) (γ) = Z Γ (s -1)Z Γ (s + 1)Z Γ (s),
where Z Γ (s) is the non-twisted Selberg zeta function of X = Γ\H 2 . The divisor of this zeta function is (with multiplicities)

{1/2 ± it j } ∪ ∞ k=0 (2g -2)(2k + 1){-k},
where the spectrum of the Laplacian ∆ on the base surface X is given by Sp(∆) = {1/4 + t 2 j , j ∈ N}, see for example Hejhal [START_REF] Hejhal | The Selberg trace formula for PSL[END_REF]. We point out that 1 is a simple zero of Z Γ (s). The divisor of Z Γ (s, ) is therefore

1 2 ± it j ∪ 3 2 ± it j ∪ - 1 2 ± it j ∪ 4(2g -2){0} ∪(2g -2){1} ∪ ∞ k=1 (2g -2)3(2k + 1){-k}.
By comparing it with the known divisor of Z Γ (s, ) from Theorem 2.1, we deduce that the bottom of the spectrum of ∆ is therefore -2 = δ(1 -δ), and is a simple eigenvalue. All but finitely many eigenvalues of ∆ are included in C 3/2 . Notice that we have in this specific example Sp(∆ ) ⊂ R ∪ P 3/2 .

Spectrum of ∆ with P δ = ∂C δ and P 3/2 = ∂C 3/2 . where γ is such that π( γ) = γ. Clearly r χ (γ) does not depend on the choice of γ and it is straightforward to check that this is a homomorphism. The existence of such a χ is non trivial and corresponds to the existence of spin structures on the riemann surface X, there are 2 2g such choices, where g is the genus of X, see [START_REF] Michael | Riemann surfaces and spin structures[END_REF].

Spin structures and SL

To such a spin structure one can associate a complex line bundle T 1/2 on X which is a square root of the canonical line bundle T (bundle of holomorphic 1-forms on X), i.e. T 1/2 ⊗ T 1/2 = T . Spinors (sections of T 1/2 ) correspond, from the point of view of automorphic forms, to smooth functions f : H 2 → C such that for all γ ∈ Γ,

f (γz) = χ( γ) cz + d |cz + d| f (z),
where γ = a b c d . The space of such automorphic forms is denoted by S(1).

The Laplacian ∆ 1 is given in the Poincaré model by

∆ 1 = -y 2 ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + iy ∂ ∂x ,
and leaves invariant S(1), see Hejhal [START_REF] Hejhal | The Selberg trace formula for PSL[END_REF], chapter 4. This operator ∆ 1 , is essentially self-adjoint, has a positive discrete spectrum starting at 1/4, see [START_REF] Hejhal | The Selberg trace formula for PSL[END_REF] and references herein. Following Roelcke, Hejhal has developped a Selberg trace formula for S(1), and there is a natural associated Selberg zeta function Z Γ (s, χ) which is defined as follows for Re(s) > 1:

Z Γ (s, χ) = γ∈P k≥0 1 -χ( γ)e -(s+k) (γ) ,
where γ ∈ SL 2 (R) is such that π( γ) = γ and tr( γ) > 0. It is important to stress that χ is not a character of Γ and that the sign changes given by χ( γ) depend on the homology class of a lift of the closed geodesic γ to the unit tangent bundle T 1 X. By a trace formula argument, see [START_REF] Sarnak | Determinants of Laplacians[END_REF], we know that Z Γ (s, χ) has a holomorphic extension to C and its trivial zeros are located at s = -1 2 -n, n ∈ N with multiplicities (2n + 2)(2g -2). The non-trivial zeros are given by s j = 1 2 ± ir j , where the spectrum of ∆ 1 is {1/4 + r 2 j , j ∈ N}. Since the bottom of the spectrum of ∆ 1 starts at 1/4, all non-trivial zeros of Z Γ (s, χ) actually lie on the line {Re(s) = 1/2}.

There is also an interpretation of r j as spectrum of a genuine Dirac operator acting on a rank 2 vector bundle, see [START_REF] Bolte | The Selberg trace formula for Dirac operators[END_REF]. As in the previous example, it is definitely possible to express the spectrum of ∆ rχ in terms of the spectrum of ∆ 1 . Indeed, we have for all γ ∈ Γ,

r χ (γ) = χ( γ) γ,
where γ can be chosen with tr( γ) > 0 so that γ is conjugate in SL 2 (R) to

e (γ)/2 0 0 e -(γ)/2 .
A similar computation as before shows that for Re(s) large we have

Z Γ (s, r χ ) = Z Γ (s + 1/2, χ)Z Γ (s -1/2, χ).
The divisor of Z Γ (s, r χ ) is therefore

{1 ± ir j } ∪ {±ir j } ∪ ∞ k=0 (2g -2)2(2k + 1){-k}.
By comparing with the known divisor from Theorem 2.1, we therefore conclude that the spectrum of ∆ rχ is included in the parabola P 0 , see picture below.

Spectrum of ∆r χ and Dirac spectrum on P 0 .

By the prime orbit theorem, we know that δ(r χ ) = 3/2, and we point out that δ(1 -δ) = -3/4 while the bottom of the spectrum of ∆ rχ is 0 and has multiplicity 2dim(h 0 (X, E 1/2 )), where h 0 (X, E 1/2 ) denotes the space of holomorphic sections of E 1/2 , see [START_REF] Hejhal | The Selberg trace formula for PSL[END_REF], chapter 4. Given a Teichmüller representation ρ : Γ → PSL 2 (R), we can choose a spin structure for ρ(Γ) and obtain a more general linear representation

= r χ • ρ : Γ → SL 2 (R).
It is straightforward to check that this is a representation of Teichmüller type with β = 1/2, since for all

γ a b c d ∈ PSL 2 (R),
the following formula holds in the Poincaré half-plane model:

a 2 + b 2 + c 2 + d 2 = 2 cosh(d(i, γi)).
3.3. The Fuchsian representation in SL n+1 (R). Let H n denote the space of bivariate homogeneous polynomials of degree n, that is

H n = Span{P j n (x, y) := x j y n-j , j = 0, . . . , n}. Any element M = a b c d ∈ SL 2 (R) acts linearly on H n by r n (M )Q := Q(ax + by, cx + dy),
and r n defines a representation

r n : SL 2 (R) → SL n+1 (R). If n is even then r n (-I) = r n (I) and therefore r n induces a natural representation of PSL 2 (R) into SL n+1 (R).
If n is odd, we need to use a spin structure on X = Γ\H 2 given by a character χ : Γ → {±1} such that χ(-I) = -1 and we define for all γ ∈ Γ such that π( γ) = γ by r n (γ) := (χ( γ)) n r n ( γ).

Note that r n now makes sense regardless of the parity of n and defines a linear representation r n : Γ → SL n+1 (R).

An easy calculation shows that if γ ∈ Γ, then we have

tr( r n (γ)) = (χ( γ)) n n j=0 e (2j-n) (γ)/2 . ( 9
)
Using the fact that the matrix coefficients of r n (M ) are homogeneous polynomials of degree n in a, b, c, d, a simple compactness argument shows that for any matrix norm . , one can find C > 0 such that for all γ ∈ Γ,

r n (γ) ≤ C γ n , (10)
where

γ a b c d and γ 2 = a 2 + b 2 + c 2 + d 2 . If ρ : Γ → PSL 2 (R)
is a Teichmüller representation, then clearly [START_REF] Braverman | Refined analytic torsion[END_REF] and ( 9) both show that = r n • ρ is a representation of Teichmüller type with β = n/2. Notice that the trace formula ( 9) also shows that for n = 1, 2 these representations are equivalent to the ones previously defined.

By the same type of calculations as previously performed, one can deduce in the case ρ = Id that we have the identities:

Z Γ (s, ) = n j=0 Z Γ s -j + n 2 if n is even, Z Γ (s, ) = n j=0 Z Γ s -j + n 2 , χ if n is odd.
As before, this leads to explicit expressions for the spectrum of ∆ in terms of Dirac or Laplace spectrum, depending on the parity of n, which perfectly generalizes to arbitrary dimensions the results obtained before. It is possible to complicate the picture even more by choosing p representations 1 , . . . , p such that

ds 2 = 4dzdz (1 -|z| 2 ) 2 .
We will view the co-compact group Γ as a subgroup of PSU(1, 1) i.e., the set of Mobius transforms of the form

γ(z) = az + b bz + a ,
where |a| 2 -|b| 2 = 1. The action of Γ on S 1 can be "coded" by a single expanding, piecewise analytic map on S 1 often referred as a Bowen-Series map associated to Γ. The construction is not unique and depends on a choice of a suitable fundamental domain with polygonal geodesic boundary and associated side-pairing isometries which generate Γ. See [START_REF] Bowen | Markov maps associated with Fuchsian groups[END_REF][START_REF] Series | Geometrical methods of symbolic coding[END_REF][START_REF] Adler | Geodesic flows, interval maps, and symbolic dynamics[END_REF], and also [START_REF] Pollicott | Some applications of thermodynamic formalism to manifolds with constant negative curvature[END_REF][START_REF] Pollicott | A remarkable formula for the determinant of the Laplacian[END_REF]. More precisely, we have the following facts, which summarize the main points proved in the above cited papers.

Proposition 4.1.

(1) There exists a finite set of generators and their inverses G ⊂ Γ.

(2) There exists a finite covering S1 = ∪ k j=1 I j by k closed intervals 1 I j such that Int(I j ) ∩ Int(I i ) = ∅ for all i = j.

(3) There exists a piecewise analytic map T : S 1 → S 1 such that for all j = 1, . . . , k, T | Int(Ij ) = γ j ∈ G and T is Markov: for each I j , T (I j ) is a finite union of intervals I i . (4) The map T is eventually expanding i.e. , there exists D > 1 such that

inf x∈S 1 |(T 2 ) (x)| ≥ D. (5)
The map T is topologically mixing: if A is the k × k matrix defined by A(i, j) = 1 if T (I i ) ⊃ I j , A(i, j) = 0 otherwise, then there exists p 0 > 1 such that A p0 has all its entries positive.

We need more definitions. If T n x = x, i.e. x is a periodic point for T , we denote by N (x) its primitive period i.e., the smallest integer N ≥ 1 such that T N x = x. Obviously, N (x) divides n. Given an element γ ∈ Γ \ {Id}, we will denote by |γ| its word length with respect to the set of generators G which is

|γ| = min{N : γ = γ 1 . . . γ N with γ j ∈ G}.
An admissible word α = (α 1 , . . . , α n ) ∈ {1, . . . , k} n is a word (of length n = |α|) such that A(α i , α i+1 ) = 1 for all i = 1, . . . , n -1. If α is an admissible word, we set

g α := γ -1 α1 • . . . • γ -1
αn . For all j such that A(α n , j) = 1, g α maps I j in I α1 , and we have for all x ∈ I j , T n (g α x) = x i.e. g α is an inverse branch of T n with word length |g α | = n. The set of all inverse branches of T n corresponds to the set of mobius maps {g α : |α| = n, α admissible}. Each inverse branch g α has a unique attracting fixed point denoted by x - gα ∈ I α1 .

The main interest in the above coding is the following property.

Proposition 4.2. With at most finitely many exceptions2 , there is a bijection between:

• Primitive conjugacy classes {γ} in Γ with word length n and translation length (γ). • Primitive periodic orbits {x, T x, . . . , T n-1 x} with period n and |(T n ) (x)| = e (γ) . • Primitive inverse branches g α with |α| = n and |g α (x - gα )| = e -(γ) . 4.2. Topological pressure and Mostow's map. Le I := k i=1 I i be the disjoint union of the intervals I i . If 0 < r < 1 we denote by C r (I) the space of piecewise r-Hölder functions on each I i , with respect to the usual angular distance d S 1 (x, y) on S 1 . Given ϕ ∈ C r (I) for some 0 < α < 1, we denote by P (ϕ) the topological pressure, defined by

P (ϕ) = sup µ h µ (T ) + S1 ϕdµ ,
where the supremum runs over all T -invariant probability measures and h µ (T ) is the measure theoretic entropy of µ. This supremum is realized by a unique probability measure µ ϕ called the equilibrium state of the potential ϕ, which is mixing, see [START_REF] Parry | Zeta functions and the periodic orbit structure of hyperbolic dynamics[END_REF] for example. Moreover, by [START_REF] Parry | Zeta functions and the periodic orbit structure of hyperbolic dynamics[END_REF] we have the following limit expression for the topological pressure:

e P (ϕ) = lim n→+∞ T n x=x e ϕ (n) (x) 1/n
, where the sum runs over all n-periodic points of T and

ϕ (n) (x) = ϕ(x) + ϕ(T x) + . . . + ϕ(T n-1 x).
In the latter, we will denote by τ (x) := log |T (x)| the distortion function for the Bowen-Series map T . By Proposition 4.2, we know that whenever T n x = x is a primitive periodic orbit, we have

τ (n) (x) = log |(T n ) (x)| = (γ),
where γ ∈ P is in correspondence with the primitive periodic orbit {x, T x, . . . , T n-1 x}.

Alternatively, if x - α is the unique attracting fixed point of an inverse branch γ = g α , then we have log

|(g α ) (x - α )| = -τ (n) (x - α ) = -(γ)
. Given a Teichmüller representation ρ : Γ → PSU(1, 1), we would like to express the distorted translation lengths ρ (γ) := (ρ(γ)) via a suitable cocycle over the dynamical system T : S 1 → S 1 . The main result of this section is as follows.

Proposition 4.3. For each Teichmüller representation ρ, there exists 0 < r ≤ 1 and a Hölder function ψ ∈ C r (I) such that for all γ ∈ P with (γ) = τ (n) (x), with T n x = x, we have

ρ (γ) = ψ (n) (x).
Morevover, if : Γ → GL(V ) is a Teichmüller like representation associated to ρ with exponent β, the characteristic exponent δ( ) coincides with the unique zero of the map σ → P (βψ -στ ).

Before we can give a proof, we need to recall a few facts. The Busemann cocycle C ξ (z, w) is defined (given z, w ∈ D and ξ ∈ ∂D) by

C ξ (z, w) = lim t→∞ (d(z, ξ t ) -d(w, ξ t )),
where ξ t is a parametrized geodesic line whose end point as t → ∞ is ξ ∈ S 1 . We summarize below the main useful properties of the Busemann cocycle in the disc model, valid for all z, w, y ∈ D and ξ ∈ S 1 .

•

C ξ (z, w) = -C ξ (w, z). • C ξ (z, y) = C ξ (z, w) + C ξ (w, y). • C ξ (z, w) = log (1-|w| 2 )|z-ξ| 2 (1-|z| 2 )|w-ξ| 2 . • For all γ ∈ PSU(1, 1), we have C ξ (γz, γw) = C γ -1 ξ (z, w).
• For all γ ∈ PSU(1, 1), we have

e C ξ (0,γ -1 0) = |γ(0) -γ(ξ)| 2 1 -|γ(0)| 2 = |γ (ξ)|.
Basic Teichmüller theory, see [START_REF] Bers | Uniformization, moduli, and Kleinian groups[END_REF] pages 267-268, tells us that there exists a quasiconformal homeomorphism F ρ : D → D such that for all γ ∈ Γ, we have

ρ(γ) = F • γ • F -1 .
Moreover, F ρ extends as a homeomorphism of ∂D which is by Mori's theorem [START_REF] Mori | On an absolute constant in the theory of quasi-conformal mappings[END_REF] Hölder continuous. This boundary map F , often called Mostow's map, conjugates in a Hölder regular way the action of Γ and ρ(Γ) on the boundary S 1 . Notice that if x + γ , x - γ are the repelling and attracting fixed points of γ ∈ Γ, then the repelling and attracting fixed points of ρ(γ) are F (x + γ ) and F (x - γ ). We then define ψ as follows: for all x ∈ I j , set

ψ(x) := C F (x) (0, ρ(γ j ) -1 0).
It is immediate to check that ψ is indeed piecewise Hölder on I. Let x ∈ S 1 be such that T n x = x and x ∈ I i1 , T x ∈ I i2 ,..., T n-1 x ∈ I in so that

T n x = γ in • . . . • γ i1 x = γ(x), with x = x + γ . Writing ψ (n) (x) = ψ(x) + . . . + ψ(T n-1 x) = C F (x) (0, ρ(γ i1 ) -1 0) + C F (γi 1 x) (0, ρ(γ i2 ) -1 0) + . . . + C F (γi n-1 ...γi 1 x) (0, ρ(γ in ) -1 0),
and using the fact that ρ(γ) • F = F • γ, and the invariance property of the Busemann function by isometries, we get

ψ (n) (x) = C F (x) (0, ρ(γ i1 ) -1 0) + C F (x) (ρ(γ i1 ) -1 0, ρ(γ i2 γ i1 ) -1 0) + . . . +C F (x) (ρ(γ in-1 . . . γ i1 ) -1 0, ρ(γ in . . . γ i1 ) -1 0).
The cocycle property then yields

ψ (n) (x) = C F (x) (0, ρ(γ in . . . γ i1 ) -1 0) = log |ρ(γ) (F (x))| = ρ (γ),
and the first claim is proved. By the very definition of representations of Teichmüller type, we know that

δ( ) = inf    s > 0 : γ∈P,k≥1 e βk ρ (γ)-sk (γ) < ∞    .
We can use Proposition 4.2 and consider the sum

S n (s) := n≥1 T n x=x 1 N (x) e βψ (n) (x)-sτ (n) (x) .
Using Fubini we can write formally

S n (s) = n≥1 d|n 1 d T d x=x N (x)=d e n d (βψ (d) (x)-sτ (d) (x)) = k≥1 d≥1 1 d T d x=x N (x)=d e k(βψ (d) (x)-sτ (d) (x)) = k≥1 γ∈P
e βk ρ (γ)-sk (γ) + k≥1 g∈{g1,...,gm}⊂P e βk ρ(g)-sk (g) .

where g 1 , . . . , g m are finitely many primitive conjugacy classes that are overcounted by the Bowen-series coding. Clearly by the properties of the topological pressure, S n (s) converges if and only if P (βψ -sτ ) < 0. In the above sum, the extra terms do not change the nature of the convergence, therefore δ( ) = inf{s : P (βψ-sτ ) < 0}. The topological pressure σ → P (βψ-στ ) := f (σ) is actually a real analytic function ( [START_REF] Parry | Zeta functions and the periodic orbit structure of hyperbolic dynamics[END_REF]) which, as follows directly from the variational definition, is strictly decreasing and satisfies lim σ→+∞ f (σ) = -∞ while lim σ→-∞ f (σ) = +∞. As a conclusion, f (σ) has a unique zero which coincides with δ( ).

Some technical bounds

For all the admissible words α of length n such that A(α n , i) = 1, we denote by C α∨i the "cylinder set" associated to the word α ∨ i of length n + 1 obtained by concatenation of α and i:

C α∨i := g α (I i ).
For all n ≥ 0, cylinder sets form a "partition" of S 1 :

S 1 = k i=1 |α|=n, A(α,i)=1
C α∨i , and whenever

α ∨ i = β ∨ j, we have Int(C α∨i ) ∩ Int(C β∨j ) = ∅.
From the fact that the Bowen-Series map is eventually expanding, we know that there exist C 0 > 0 and 0 < θ 0 ≤ θ 1 < 1 such that for all α, i with A(α, i) = 1, for all x ∈ I i , we have

C -1 0 θ n 0 ≤ |g α (x)| ≤ C 0 θ n 1 . (11)
Given γ ∈ PSU(1, 1), we can always choose a smooth lift γ : R → R such that for all θ ∈ R, γ(e iθ ) = e i γ(θ) . Using the fact that γ is a Mobius map of the form

γ(z) = az + b az + b with |a| 2 -|b| 2 = 1, we can deduce that ∀ θ ∈ R, γ (θ) = |γ (e iθ )|. (12)
Combining ( 11) and ( 12) we get that for all α, i with A(α, i) = 1 and |α| = n, we have diam

S 1 (C α∨i ) ≥ C 0 θ n 0 , (13)
for some uniform constant C 0 > 0. We are now ready to prove the following fact.

Lemma 5.1. (Separation) There exist C 1 > 0 and 0 < θ 2 < 1 such that for all n and all words α = β with |α| = |β| = n, for all i such that A(α n , i) = A(β n , i) = 1, we have for all [START_REF] Climenhaga | Positive entropy equilibrium states[END_REF] we have dist S 1 (C α∨i , C β∨i ) ≥ C 0 θ n 0 , and the claim is proved since both chordal and angular distance are equivalent on S 1 . We can therefore restrict to the case when C α∨i ∩ C β∨i = ∅, in which case this intersection is a single point. Let us set

x ∈ I i |g α (x) -g β (x)| ≥ C 1 θ n 2 . Proof. First observe that if C α∨i ∩ C β∨i = ∅, then by
C α∨i = {e iθ : a ≤ θ ≤ b}, C β∨i = {e iθ : b ≤ θ ≤ c},
where a < b < c and set also

I i = {e iθ : ω 1 ≤ θ ≤ ω 2 },
with ω 1 < ω 2 . We lift both maps g α , g β so that g α (e iθ ) = e i gα(θ) , g β (e iθ ) = e i g β (θ) , and choose those lifts3 such that g

α ([ω 1 , ω 2 ]) = [a, b] while g β ([ω 1 , ω 2 ]) = [b, c]. For all ω 1 ≤ θ ≤ ω 2 , we have g β (θ) -g α (θ) ≥ θ ω1 |g β (ϕ)|dϕ ≥ (θ -ω 1 )C -1 0 θ n 0 ,
while we also have

g β (θ) -g α (θ) ≥ (b -a) - θ ω1 |g β (ϕ)|dϕ ≥ (b -a) -(θ -ω 1 )C 0 θ n 1 .
Hence for all ω 1 ≤ θ ≤ ω 2 we get

| g β (θ) -g α (θ)| ≥ G 0 (θ) := max (θ -ω 1 )C -1 0 θ n 0 , (b -a) -(θ -ω 1 )C 0 θ n 1 .
A simple calculation then shows that for all ω 1 ≤ θ ≤ ω 2 and all n large we have

G 0 (θ) ≥ (b -a)C -1 0 θ n 0 C -1 0 θ n 0 + C 0 θ n 1 ≥ C 0 C -1 0 θ 2n 0 C -1 0 θ n 0 + C 0 θ n 1 ≥ C 1 θ 2 0 θ 1 n ,
and the proof is done.

We point out that we have used here in an essential way the fact that the Mobius maps g α , g β are orientation preserving and their derivatives enjoy exponential bounds.

We need another technical fact. Lemma 5.2. (Norm Bound) Let : Γ → GL(V ) be a linear representation of Teichmüller type, with parameter β and associated Teichmüller representation ρ. Then for any matrix norm . on GL(V ), there exists C 2 > 0 such that for all n and all admissible word α with length n we have

-1 (g α ) ≤ C 2 e β ρ (gα) = C 2 e βψ (n) (x + gα ) . Proof. Let g α = γ -1 α1 •. . .•γ -1
αn be an inverse branch ot T n associated to an admissible word α. Let x - gα and x + gα denote the attracting and repulsing fixed points of g α which are distinct since Γ is a co-compact group. By construction 4 of Bowen-Series maps [START_REF] Bowen | Markov maps associated with Fuchsian groups[END_REF][START_REF] Adler | Geodesic flows, interval maps, and symbolic dynamics[END_REF], we have x - gα ∈ I α1 , while x + gα ∈ I j where I j ∩ I αn = ∅. In particular d S 1 (x - gα , x + gα ) ≥ min 1≤i≤k (diam S 1 (I i )). Since Mostow's map F ρ : S 1 → S 1 is a continuous homomorphism, an immediate compactness argument shows that there exists η > 0 such that for all g α

d S 1 (x + ρ(gα) , x - ρ(gα) ) = d S 1 (F ρ (x + gα ), F ρ (x - gα) ) ≥ η.
Given an hyperbolic isometry γ, in the disc model, the hyperbolic distance from the origin 0 to the axis A γ of γ is uniquely determined by the angular distance between the two fixed points x + γ , x - γ ∈ S 1 . We therefore deduce that there exists M > 0 such that for all n, for all inverse branch g α we have

d(A gα , 0) ≤ M.
Using property 1) of representations of Teichmüller type, we can write for all g α , -1 (g α ) 1 ≤ Ce βd(ρ(g -1 α )0,0) = Ce βd(0,ρ(gα)0) .

By the triangle inequality, choosing a point z ∈ A ρ(gα) , we have

d(0, ρ(g α )0) ≤ d(0, z) + d(z, ρ(g α )z) + d(ρ(g α )z, ρ(g α )0),
and thus d(0, ρ(g α )0) ≤ 2d(A ρ(gα) , 0) + ρ (g α ). We can now conclude that -1 (g α ) 1 ≤ Ce 2βM +β ρ (gα) , and the proof is done.

4 See for example [START_REF] Adler | Geodesic flows, interval maps, and symbolic dynamics[END_REF], Thm 3.1, Thm 3.4 and Def 6.1. Both "left" and "right" Bowen-Series map work. See also [START_REF] Pollicott | A remarkable formula for the determinant of the Laplacian[END_REF] for a picture of the side pairings and mapping properties.

6.

Fredholm determinants and holomorphic function spaces 6.1. Using zeta functions. Following Ruelle [START_REF] Ruelle | Zeta-functions for expanding maps and Anosov flows[END_REF] and Pollicott [START_REF] Pollicott | Some applications of thermodynamic formalism to manifolds with constant negative curvature[END_REF], since each isometry γ i : I i → S 1 is real-analytic, and because of the contraction properties, one can find for each i an open set C ⊃ U i ⊃ I i such that whenever A(i, j) = 1,

γ -1 i (U j ) ⊂ U i .
Here γ -1 i : U j → U i is simply understood as the Mobius map γ -1 j (which is holomorphic on a neighborhhod of S 1 ) acting on the complex open set U j . Since the mappings γ -1 i : U j → U i are holomorphic contractions, it is not surprising that they have strong spectral properties when acting on "favorable function spaces". For a precise definition of "favorable function spaces" and possible choices we recommend to read [START_REF] Bandtlow | On the Ruelle eigenvalue sequence[END_REF].

Recall that V denotes the representation space of : Γ → GL(V), which is endowed with a Hermitian metric ., . V , with associated norm . V . In our case, we will use Bergman spaces H 2 (U i ) defined as

H 2 (U i , V ) := f : U i → V : f is holomorphic and Ui f (z) 2 V dm(z) < ∞ ,
where m denotes Lebesgue measure on C. Each space H 2 (U i , V ) is a Hilbert space endowed with the L 2 -norm. We now set

H 2 := k i=1 H 2 (U i , V )
And define a transfer operator L s, :

H 2 → H 2 by ∀ z ∈ U j , L s, (F )(z) := A(i,j)=1 [(γ -1 i ) ] s (z) -1 (γ -1 i )F • γ -1 i (z),
where [(γ -1 i ) ](z) is defined by analytic continuation to U j of |(γ -1 i ) |, defined on I j . The complex power [(γ -1 i ) ] s (z) is then defined by taking a suitable holomorphic branch of the logarithm such that the restriction to I j coincides with the usual complex power of a real number, see the next subsection for details. The n-th iterate L n s, acts as follows:

∀ z ∈ U j , L n s, (F )(z) := A(αn ,j)=1 |α|=n [g α ] s (z) -1 (g α )F • g α (z),
where the sum runs over all admissible words of length n. By the general facts on Holomorphic contraction systems [START_REF] Bandtlow | On the Ruelle eigenvalue sequence[END_REF], we know that L s, is a compact trace class operator when acting on H 2 . In addition, the trace is given by

tr(L n s, ) = k j=1 A(αn ,j)=1 |α|=n tr( -1 (g α )) |g α | s (x - gα ) 1 -g α (x - gα )
.

Lefschetz type formulas in this context date back to Ruelle [START_REF] Ruelle | Zeta-functions for expanding maps and Anosov flows[END_REF], see also [START_REF] Pollicott | Some applications of thermodynamic formalism to manifolds with constant negative curvature[END_REF]. For a proof in the vector valued case, including non-unitary representations, we refer the reader to [START_REF] Fedosova | Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy[END_REF]. A direct consequence of this trace formula combined with Proposition 4.2 is the following key fact: for all s ∈ C, we have det(I -L s, ) = Z Γ (s, ) k≥0 g1,...,gm det(I -(g j )e -(s+k) (gj ) ), where {g 1 , . . . , g m } ⊂ P is a finite set of primitive conjugacy classes that are "overcounted" by the Bowen-Series coding, see [START_REF] Pollicott | A remarkable formula for the determinant of the Laplacian[END_REF][START_REF] Pollicott | Some applications of thermodynamic formalism to manifolds with constant negative curvature[END_REF]. In particular, zeros of Z Γ (s, ) are a subset of zeros of the Fredholm determinant det(I -L s, ). Notice also that for all n ≥ 1, zeros of det(I -L s, ) are themselves, with multiplicity, a subset of the zero set of det(I -L n s, ), by using the product formula for trace class operators: det(I -L n s, ) = det(I -L s, ) det(I + L s, + . . . + L n-1 s, ) (see [START_REF] Simon | Trace ideals and their applications[END_REF]). The main result of this paper (improved Weyl law) will follow from the next statement. Theorem 6.1. Let δ 0 be the unique zero of the map σ → P (2βψ -2στ ), where the weights ψ and τ are defined as in §4. For all σ 0 > δ 0 , there exist constants κ > 0, M 0 > 0 and η 0 , T 0 > 0 such that for n(T

) = 2[κ log T ], for all s = σ + it with σ ≥ σ 0 and T 0 ≤ |t| ≤ T , log | det(I -L n(T ) s, )| ≤ M 0 T 1-η0 .
First, we explain why we have δ 0 < δ, the fact that δ 0 > δ/2 will follow from the bounds proved in the last subsection (see geodesic stretch and bounds on δ 0 ). By the theory of equilibrium states for Markov expanding maps (or subshifts of finite type), we know that

P (2βψ -2δ 0 τ ) = 0 = h µ0 (T ) + 2β ψdµ 0 -2δ 0 τ dµ 0 ,
where µ 0 is a T -invariant probability measure associated to the Hölder potential 2βψ -2δ 0 τ . This equilibrium state µ 0 has positive entropy h µ0 (T ) > 0 by a general fact on equilibrium states, see [START_REF] Climenhaga | Positive entropy equilibrium states[END_REF]. Therefore, we have

0 < 2 h µ0 (T ) + β ψdµ 0 -δ 0 τ dµ 0 ≤ 2 sup µ h µ (T ) + β ψdµ -δ 0 τ dµ = 2P (βψ -δ 0 τ ).
Hence we get P (βψ -δ 0 τ ) > 0 which implies that δ 0 < δ. It is not difficult to see that δ 0 is positive. For more refined explicit bounds on δ 0 involving δ and the "geodesic stretch", see the end of this section.

Let us now show how this bound implies Theorem 1.3. First observe that if λ = s(1 -s) with s = σ + it satisfies λ ∈ C δ \ C σ0 then we can assume that

σ 0 ≤ σ ≤ δ .
Using the fact that asymptotically as |t| → ∞ we have |λ| = |t|+O(1), we deduce that #{λ ∈ Sp(∆ ) :

λ ∈ C δ \ C σ0 and r ≤ |λ| ≤ r + √ r} ≤ #{Z Γ (s, ) = 0 : σ 0 ≤ σ ≤ δ and √ r -M ≤ |t| ≤ √ r + M },
for some constant M > 0. To bound from above the number of zeros of Z Γ (s, ) in this region as r → +∞, we use Jensen's formula as below.

Lemma 6.2. Let f be a holomorphic function on the open disc D(w, R), and assume that f (w) = 0. let N f (r) denote the number of zeros of f in the closed disc D(w, r). For all r < r < R, we have

N f ( r) ≤ 1 log(r/ r) 1 2π 2π 0 log |f (w + re iθ )|dθ -log |f (w)| .
For a reference, one can look for example at [START_REF] Ransford | Potential theory in the complex plane[END_REF], Corollary 4.5.2. Let us set

R ± 0 := {s = σ + it ∈ C : σ 0 ≤ σ ≤ δ and ± √ r -M ≤ ±t ≤ ± √ r + M }.
We treat the case of R + 0 , the case of R - 0 being similar. Choose > 0 so that δ 0 < σ 0 -< σ 0 . Set R = |A -σ 0 -iM | + /3 and R = |A -σ 0 -iM | + 2 /3. By fixing A large enough, we can ensure that for all r large enough we have

R + 0 ⊂ D(A + i √ r, R) ⊂ D(A + i √ r, R ) ⊂ {Re(s) ≥ σ 0 -}.
In order to apply Jensen's formula to det(I -L n(T ) s,

) with

T = √ r + R , we need to control from below log | det(I -L n(T ) A+i √
r, )|. By the Lefschetz trace formula, we have for large Re(s)

det(I -L n(T ) s, ) = exp - ∞ p=1 1 p tr(L pn(T ) s,
) , with tr(L pn(T ) s,

) = k j=1 A(α |α| ,j)=1 |α|=pn(T ) tr( -1 (g α )) |g α | s (x - gα ) 1 -g α (x - gα )
.

Using property (2) from representations of Teichmüller type, we have

|tr(L pn(T ) s, )| ≤ C T pn(T ) x=x
e βψ (pn(T )) (x)-Re(s)τ (pn(T )) (x) .

In particular, if Re(s) = A is large enough, then the topological pressure P (βψ-Aτ ) is negative and we get a positive lower bound on | det(I -L

n(T ) s,
)| which is uniform in r. Applying the above Lemma on the disc

D(A + i √ r, R ) to the holomorphic function det(I -L n(T ) s, ) we get #({Z Γ (s, ) = 0} ∩ R + 0 ) = O(T 1-η0 ) + O(1)
, and thus

#{λ ∈ Sp(∆ ) : λ ∈ C δ \ C σ0 and r ≤ |λ| ≤ r + √ r} = O(T 1-η0 ) = O(r 1/2-η0/2
), which is the bound of Theorem 1.3. 6.2. Refined function space and proof of Theorem 6.1. We will need in the sequel to refine the function space H 2 . Let h > 0 denote a small parameter. For each j = 1, . . . , k set U j (h) := I j + D(0, h), where D(0, h) is the open euclidean disc of radius h centered at 0 in C. Then U j (h) is a simply connected open subset of C. It is clear thar for all h > 0 small enough we have for all j = 1, . . . , k, U j (h) ⊂ U j . We then set

H 2 h := k i=1 H 2 (U i (h), V ),
endowed with the L 2 norm. These h-dependent families of Bergman spaces have been first used in the work of Guillopé-Lin-Zworski [START_REF] Guillopé | The Selberg zeta function for convex cocompact Schottky groups[END_REF] for resonances of Schottky manifolds, where they considered families of complex balls of radius h. In our analysis we will instead have to consider tubular neigborhoods of intervals.

Proposition 6.3.

(1) There exist C > 0 and 0 < θ 3 < 1 such that for all i = 1, . . . , k, for all admissible word α with |α| = n and A(α n , i) = 1, we have sup

z∈Ui |g α (z)| ≤ C 3 θ n 3 .
(2) There exists n 0 ≥ 1 such that for all h > 0 small enough and all n ≥ n 0 , L n s, is a well defined compact trace class operator on H 2 h . In addition, if |α| = n ≥ n 0 and g α maps

U i (h) into U α1 (h), we have for all z ∈ U i (h), dist(g α (z), ∂U α1 (h)) ≥ h 2 .
Moreover, the traces (hence the Fredholm determinant) do not depend on h.

(3) There exists B 0 > 0 such that for all admissible word α with |α| = n and indices i, j with A(α n , i) = A(α n , j) = 1, we have for all z ∈ U i and w ∈ U j

g α (z) g α (w) ≤ B 0 .
(4) There exists B 1 (σ 0 ) > 0 such that for all 0 ≤ Re(s) ≤ M 0 , for all i = 1, . . . , k, for all admissible word α with |α| = n and A(α

n , i) = 1, we have for h ≤ |Im(s)| -1 and z ∈ U i (h), |[g α ] s (z)| ≤ B 1 |g α (z)| Re(s) .
Proof. These estimates are rather standard for expanding Markov maps. Nevertheless we give some details. The claim now follows from a simple iteration argument.

(2) Let U i (h) be defined as above and pick an inverse branch g α with word length n, such that A(α n , i) = 1. Let z ∈ U i (h), and choose z ∈ I i such that |z -z| < h. By 1), we have |g α (z) -g α ( z)| ≤ Cθ n 3 h, and since g α ( z) ∈ I α1 , we get that g α (z) ∈ U α1 (h) and dist(g α (z), ∂U α1 (h)) ≥ h/2, as long as Cθ n 3 ≤ 1/2. Therefore the operator L n s, is well defined on H 2 h for all n ≥ n 0 . Because it acts by uniform contractions on holomorphic functions, it is a trace class operator. The fact that traces do not depend on h is because they are expressed only in term of fixed points which belong to the intervals I j 's. By analytic continuation, the determinant is also independent of h.

(3) Follows directly from 1) after taking logarithms and applying the chain rule. (4) This is similar to [START_REF] Naud | Density and location of resonances for convex co-compact hyperbolic surfaces[END_REF][START_REF] Jakobson | Resonances and density bounds for convex co-compact congruence subgroups of SL 2 (Z)[END_REF], however since we work with the disc model, there are some slight differences that we need to address. Given γ ∈ Γ with

γ(z) = az + b az + b ,
we have for all z ∈ S 1 ,

|γ (z)| = 1 (az + b)(az -1 + b) ,
so that the analytic continuation [γ ](z) is given by the same formula (which is holomorphic in a neighborhood of S 1 ). Given an inverse branch g α with |α| = n and A(α n , i) = 1, z = re iθ ∈ U i (h), we write

[g α ](re iθ ) = 1 |ae iθ + b| 2 + E(a, b, r, θ)
,

where

E(a, b, r, θ) = bae -iθ 1 r -1 + bae iθ (r -1).
It is immediate to check that for all h small enough we have

|E(a, b, r, θ)| ≤ 3|ab|h. Observe now that |E(a, b, r, θ)| |ae iθ + b| 2 ≤ 3h | b a | |e iθ + b a | 2 .
Since -b a is the center of the isometric circle of g α and z = re iθ ∈ U i (h), this quantity is bounded uniformly as

|E(a, b, r, θ)| |ae iθ + b| 2 ≤ Ch. (14)
In particular, from ( 14), we get that for all h small enough, for all z ∈ U i (h)

|[g α ](z)| ≤ A 1 |g α (z)|,
where A 1 > 0 is some uniform constant. Going back to [g α ] s (z), we have

[g α ] s (z) = e sL([g α ](z)) ,
where L(w) is the principal holomorphic determination of the logarithm defined on C \ R -given by L(w) = log |w| + iArg(w), where Arg(w) ∈ (-π, +π) is the principal argument of w. Writing

|[g α ] s (z)| = e Re(s) log |[g α ](z)| e -Im(s)Arg([g α ](z)) ,
we use [START_REF] Dolgopyat | On decay of correlations in Anosov flows[END_REF] to bound the argument as

|Arg([g α ](z))| ≤ A 2 h. It is now clear that if |Im(s)| ≤ h -1 and 0 ≤ Re(s) ≤ M 0 , then we get |[g α ] s (z)| ≤ A 3 (M 0 )|g α (z)| Re(s) ,
for some A 3 (M 0 ) > 0 and the proof is done. Before we can give a proof of Theorem 6.1, we need to recall a few facts on Bergman kernels which are essential at the technical level. If Ω is a bounded open set in C, then the Hilbert space H 2 (Ω, C) has a reproducing kernel i.e., for all f ∈ H 2 (Ω, C), we have

f (z) = Ω B Ω (z, w)f (w)dm(w),
where B Ω (z, w) is a smooth kernel on Ω × Ω, called the Bergman kernel of Ω. Given a Hilbert basis (φ l ) l∈N of H 2 (Ω, C), we have for all z, w ∈ Ω, l∈N φ l (z)φ l (w) = B Ω (z, w), the convergence being uniform on every compact subset of Ω × Ω, see for example Krantz [START_REF] Steven | Geometric analysis of the Bergman kernel and metric[END_REF]Chapter 1]. In the latter, we will work on narrow domains Ω = U i (h) with h → 0. We will need to use the following bounds on kernels B Ui(h) (z, w) which are proved in the Appendix (Proposition A.1).

• There exists a constant M > 0 independent of h such that for all z, w ∈ U i (h) with dist(z, ∂U i (h)) ≥ h/2 and dist(w, ∂U i (h)) ≥ h/2, we have for all h > 0 small enough

|B Ui(h) (z, w)| ≤ M h -2 .
• If in addition we have |z -w| ≥ ch ν for some c > 0 and 0 < ν < 1, then we have for all h > 0 small enough

|B Ui(h) (z, w)| ≤ M h -1-ν .
We are now ready to prove Theorem 6.1. We assume that s = σ + it with σ ≥ σ 0 > δ 0 and |t| ≤ T . We set n(T ) = 2[κ log(T )] where T is large and κ > 0 will be adjusted later on. We will let L n(T ) s, act on the function space H 2 h with h = T -1 . By a standard bound on Fredholm determinants, see [START_REF] Simon | Trace ideals and their applications[END_REF], we know that

log | det(I -L n(T ) s, )| ≤ L n(T )/2 s, 2 HS ,
where . HS denotes the Hilbert-Schmidt norm, when acting on the function space H 2 h . We now need to compute and estimate this Hilbert-Schmidt norm. For each i = 1, . . . , k, let (φ (i) l ) l∈N be a Hilbert basis of U i (h). Let e 1 , . . . , e d be an orthonormal basis of the representation space V which is endowed with a hermitian inner product ., . V . Then the family

φ (i) l (z)e m l∈N, 1≤i≤k, 1≤m≤d
, is a Hilbert basis of H 2 h = i H 2 (U i (h), V ) and we have

L n(T )/2 s, 2
HS = i,m l L n(T )/2 s, (φ (i)
l e m ) 2 H 2 h , where φ (i)
l e m ∈ H 2 h is understood as being

φ (i) l (z)e m = φ (i) l (z)e m if z ∈ U i (h) 0 otherwise.
Setting for conveniency n(T ) = n(T )/2, we therefore have

L n(T ) s, 2 HS = l,m,i,j Uj (h) A(α n ,j)=1 A(β n ,j)=1 [g α ] s (z)[g β ] s (z)φ (i) l (g α z)φ (i) l (g β z) -1 (g α )e m , -1 (g β )e m V dm(z).
Using Fubini (justified by uniform convergence on compact sets for the infinite sum over l) we get

L n(T ) s, 2 HS = j Uj (h) A(α n ,j)=1 A(β n ,j)=1 tr( -1 (g α ) -1 (g β ) * )[g α ] s (z)[g β ] s (z)B U (h) (g α z, g β z)dm(z),
Where U (h) = k i=1 U i (h) and B U (h) (z, w) is the reproducing kernel of H 2 h . We then use the fact that (Cauchy-Schwartz inequality)

|tr( -1 (g α ) -1 (g β ) * )| ≤ -1 (g α ) HS -1 (g β ) HS ,
where . HS is the Hilbert-Schmidt norm on GL(V ) associated to the inner product on V . Using the norm bound from Lemma 5.2 combined with Proposition 6.3 (4), we get

L n(T ) s, 2 HS ≤ C× j Uj (h) A(α n ,j)=1 A(β n ,j)=1 e β( ρ (gα)+ ρ (g β )) |g α (z)| Re(s) |g β (z)| Re(s) |B U (h) (g α z, g β z)|dm(z).
We will now split the above sum into two parts S 1 (h) + S 2 (h), where

S 1 (h) = j Uj (h) A(α n ,j)=1 e 2β ρ (gα) |g α (z)| 2Re(s) |B U (h) (g α z, g α z)|dm(z),
which we call the "diagonal part" and

S 2 (h) = j Uj (h) α =β e β( ρ (gα)+ ρ (g β )) |g α (z)| Re(s) |g β (z)| Re(s) |B U (h) (g α z, g β z)|dm(z),
which we call the "off-diagonal" part. We first deal with the diagonal part. Using Proposition 6.3 (3), we have

S 1 (h) ≤ C j Uj (h) A(αn,j)=1 e 2β ρ (gα)-2σ (gα) |B U (h) (g α z, g α z)|dm(z).
Using the crude bound on Bergman kernels combined with Proposition 6.3 (2), we get uniformly in z

|B U (h) (g α z, g α z)| ≤ Ch -2
, and since we have m(U (h)) ≤ Ch (either by direct computation or by using a general result on volumes of tubular neighborhood of smooth curves) we obtain that for Re(s) = σ ≥ σ 0 , S 1 (h) ≤ Ch -1 j A(α n ,j)=1 e 2β ρ (gα)-2σ0 (gα) , where the words are of length n(T ) = [κ log T ]. Fixing > 0 small enough, this term can be controlled via the topological pressure for all n(T ) ≥ n 0 : S 1 (h) ≤ CT e n(T )(P (σ0)+ ) , where P (σ 0 ) is a shorthand for P (2βψ -2σ 0 τ ). Clearly, as long as σ 0 > δ 0 we have P (σ 0 ) < 0 and we get a polynomial gain of magnitude O(T -η0 ) for some η 0 > 0.

We now deal with the "off-diagonal sum", and this is where the "off-diagonal estimate" on Bergman kernels is critical. Observe that if z ∈ U j (h) and α, β are two distinct admissible words of length n(T ), then picking z ∈ I j with |z -z| < h, we have by Lemma 5.1 and Proposition 6.3 (1),

|g α (z) -g β (z)| ≥ C 1 θ n(T ) 2 -2C 3 θ n(T ) 3 h. Taking κ > 0 small enough so that θ n(T ) 2 ≥ √ h, we end up with |g α (z) -g β (z)| ≥ C 4 √ h,
for all z ∈ U j (h) and all h = T -1 small enough. Using the off-diagonal bound on Bergman kernels and Proposition 6.3 (3), we get

S 2 (h) ≤ Ch -1-1/2 j Uj (h) α =β
e β( ρ (gα)+ ρ (g β ))-σ0 (gα)-σ0 (g β ) dm(z).

Using the topological pressure formula, we get (again fixing > 0 and taking T large enough) S 2 (h) ≤ CT 1/2 e (2P (βψ-σ0τ )+ ) n(T ) .

On the other hand, if µ is a T ergodic measure, one can show using Birkhoff's ergodic theorem (see [START_REF] Naud | Entropy and decay of correlations for real analytic semi-flows[END_REF], Lemma 3.4) that there exists a sequence (x p ) p∈N of n pperiodic points such that

ψdµ τ dµ = lim p→∞ 1 np ψ (np) (x p ) 1 np τ (np) (x p )
.

But we do have for all p

1 np ψ (np) (x p ) 1 np τ (np) (x p ) ≤ I + (ρ),
and hence the equality is proved. Using [START_REF] Burger | Intersection, the Manhattan curve, and Patterson-Sullivan theory in rank 2[END_REF], we know that if I -(ρ) = I + (ρ) then they are both equal to 1 and the Manhattan curve is a straight line passing through the points (0, 1) and (1, 0), which implies that δ = 1 + β and δ 0 = 1/2 + β. The proof is done.

Remark 6.5. The quantities I -(ρ) and I + (ρ) are called "geodesic stretch" and are related to Thurston's asymetric distance: in particular, we have [START_REF] Papadopoulos | On Teichmüller's metric and Thurston's asymmetric metric on Teichmüller space[END_REF] log

I + (ρ) I -(ρ) ≤ d T (X, ρ(Γ)\H 2 ),
where d T (X 1 , X 2 ) denotes the usual Teichmüller distance related to quasi-conformal deformations. In particular, if the Teichmüller distance between X and the "deformed" surface ρ(Γ)\H 2 is small, then δ 0 is close to (δ + β)/2.

Final remarks

They are many ways in which this paper can be generalized. It raises also a lot of open questions. We list them below.

• In this paper we have essentially adressed the case of representations

: Γ → SL d (R),
where is obtained as a composition of Teichmüller and the irreducible representation. It would be interesting to consider a more general deformation of that case, which is called the "Hintchin component" in the litterature.

It is likely that one has to combine techniques from Labourie [START_REF] Labourie | Anosov flows, surface groups and curves in projective space[END_REF] and the transfer operator methods used in this paper to reach this goal. One could also look at the quasi-Fuchsian deformation of the surface group Γ into PSL 2 (C). We believe this should follow from our technique without major modifications.

• By looking at examples, it seems that on the parabola P δ , δ(1 -δ) is the only possible eigenvalue. Can one prove that fact in a general setting? We have shown that if the character of is positive, then δ(1 -δ) is indeed the bottom of the spectrum of ∆ . Is it a simple eigenvalue? Can one express the base eigenfunction via Patterson-Sullivan theory in higher rank by analogy with the case of geometrically finite hyperbolic manifolds? • Based on the examples, we believe that the spectrum of ∆ always has an essential spectral gap i.e., there exists > 0 such that Sp(∆ ) ∩ C δ \ C δ- is a finite set. Spectral gaps in the context of negative curvature are often derived from Dolgopyat's techniques [START_REF] Dolgopyat | On decay of correlations in Anosov flows[END_REF] and their generalizations to noncompact situations with unitary twists [START_REF] Oh | Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of SL 2 (Z)[END_REF][START_REF] Magee | Explicit spectral gaps for random covers of Riemann surfaces[END_REF][START_REF] Sarkar | Exponential mixing of frame flows for convex cocompact hyperbolic manifolds[END_REF]. Alternatively, when the "trapped set" is a smooth symplectic submanifold and the transverse dynamics are normally hyperbolic, Nonnenmacher and Zworski's microlocal approach [START_REF] Nonnenmacher | Decay of correlations for normally hyperbolic trapping[END_REF] provide an explicit essential spectral gap. In this context of non-unitary representations, Dolgopyat's arguments cannot be applied directly and the micro-local approach is also likely to fail due to the lack of contact/symplectic structure. • The computable examples also suggest that the spectrum of ∆ may have in some situations a "band structure", with the number of bands given by dim(V ). In [START_REF] Faure | Prequantum transfer operator for symplectic Anosov diffeomorphism[END_REF], for Pollicott-Ruelle resonances related to U (1)-extensions of Anosov maps, a band structure of the spectrum is rigourously established by Faure and Tsujii, together with some eigenvalue distribution results based on Sjöstrand's method [START_REF] Sjöstrand | Asymptotic distribution of eigenfrequencies for damped wave equations[END_REF] for damped wave equations. We also mention the relevant work of Dyatlov [START_REF] Dyatlov | Resonance projectors and asymptotics for r-normally hyperbolic trapped sets[END_REF] for the existence of band structures in quantum resonances in a setting where the trapped set is normally hyperbolic. It would be interesting to investigate to what extend the above mentioned works could be adapted to our problem. • By analogy with the theory of resonances for hyperbolic manifolds, it would be interesting to know if micro-local techniques used by Dyatlov in [START_REF] Dyatlov | Improved fractal Weyl bounds for hyperbolic manifolds (with an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich)[END_REF] could be adapted to improve our Weyl bounds on the peripheral spectrum of ∆ . The non-smooth nature of Teichmüller deformations (Mostow's map is only Hölder) is likely to be an obstacle. • There is a current work [START_REF] Fedosova | Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy[END_REF] on meromorphic continuation of Selberg zeta functions with non-unitary twists, for geometrically finite groups. Despite the fact that there is no available interpretation yet of zeros as spectra of a twisted Laplacian in this non-compact context, it would be interesting to know if our result can be extended to this framework. • Finally, one can consider hyperbolic manifolds of higher dimension, or more general, locally symmetric spaces X = Γ\Y of real rank one, where Y is a real, complex, or quaternionic hyperbolic space, or the hyperbolic Cayley plane. Considering a non-unitary representation of the lattice, the Selberg trace formula, for suitable integral operators, induced by the twisted Laplacian, should be available, and hence the meromorphic continuation of the twisted Selberg zeta function could follow. Consequently, it would be interesting to see if our results can be extended to this case.

Appendix A. Universal bounds for Bergman kernels

The results below are based on standard facts on Bergman kernels for simply connected domains. However, because we couldn't find the exact statement needed in the classical bet, we include a detailed proof here. In particular, the "off-diagonal" estimate is not surprising but seems to be new in this setting. Let D denote the unit disc. Let Ω ⊂ C be a bounded, simply connected open set. If φ : Ω → D is a conformal representation, then one can endow Ω with a hyperbolic distance ρ Ω (x, y) by pulling back the hyperbolic distance ρ D (z, z ) on D:

ρ Ω (x, y) := ρ D (φ(x), φ(y)).

Notice that ρ D (z, z ) = inf For example, if Ω = D(a, r) is a euclidean disc centered at a and of radius r, then we have λ Ω (z) = 2r -1 1 -| z-a r | 2 . The Bergman Kernel B Ω (z, w) is the Schwarz kernel of the orthogonal projection B : L 2 (Ω) → H 2 (Ω), where H 2 (Ω) is as usual the Hilbert space of holomorphic L 2 functions on Ω. For the unit disc we have the well known formula [START_REF] Steven | Geometric analysis of the Bergman kernel and metric[END_REF]Chapter 1]:

B D (z, w) = 1 π(1 -zw) 2 .
The Bergman kernel has conformal invariance i.e., under the above notations, we have

B Ω (z, w) = φ (z)φ (w)
π(1 -φ(z)φ(w)) 2 .

Using the formula for the hyperbolic distance ρ D (u, v) in the disc

cosh 2 (ρ D (u, v)/2) = |1 -uv| 2 (1 -|u| 2 )(1 -|v| 2 )
,

we obtain the formula:

|B Ω (z, w)| = 1 4π

λ Ω (z)λ Ω (w) cosh 2 (ρ Ω (z, w)/2) ,

which expresses the kernel in terms of hyperbolic distance and density on Ω. This formula will be enough to prove the following estimate.

Proposition A.1. Assume that Ω is a bounded simply connected domain. We have the following facts.

(1) Assume that z, w ∈ Ω satisfy dist(z, ∂Ω) ≥ c , dist(w, ∂Ω) ≥ c , for some c > 0. Then there exists M (c) > 0 such that for all 1 ≥ > 0,

|B Ω (z, w)| ≤ M (c) -2 .

(2) Assume in addition that z, w ∈ Ω satisfy:

c < dist(z, ∂Ω) ≤ , c ≤ dist(w, ∂Ω) ≤ , |z -w| ≥ c α for some c > 0 and 0 < α < 1, then there exists M (c) > 0 such that for all 1 ≥ > 0, |B Ω (z, w)| ≤ M (c) -1-α .

The above constants M (c), M (c) do not depend on the domain Ω.

The first bound is "a diagonal estimate" and essentially optimal without additional assumptions on z, w. The second bound ("off-diagonal estimate") shows that provided z, w are sufficiently far away from each other, we can gain over the trivial diagonal bound. These bounds are universal: we do not assume anything on the domain Ω except that it is bounded and simply connected.

Proof. The hyperbolic density λ Ω satisfies the domain monotonicity: if D(z, r) ⊂ Ω then we have λ Ω (z) ≤ λ D(z,r) (z) = 2r -1 . Using Formula (15) we deduce that under the hypotheses of claim 1 we have for all 1 ≥ > 0,

|B Ω (z, w)| ≤ 1 4π λ Ω (z)λ Ω (w) ≤ 1 π dist(z, ∂Ω) -1 dist(w, ∂Ω) -1 ≤ c -2 π -2 ,
and the first bound is proved.

The second bound is more subtle: we need to bound from below the hyperbolic distance ρ Ω (z, w). To this end, we will use an inequality due to Beardon [START_REF] Beardon | The Apollonian metric of a domain in R n[END_REF]. Given x, y ∈ Ω, where Ω is again bounded and simply connected, set Going back to formula [START_REF] Dyatlov | Improved fractal Weyl bounds for hyperbolic manifolds (with an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich)[END_REF] and plugging the above estimate we get the desired upper bound (valid for all 0 < ≤ 0 (c) )

|B Ω (z, w)| ≤ c -2 -2 π e 1 2 | log c| 1-α .
The proof is done since for 1 ≥ > 0 (c) we can obviously use the first universal bound.

We point out that no assumptions on the regularity of ∂Ω or the convexity of Ω have been made.

3. 1 .

 1 The adjoint representation. Let G = SL 2 (R) and denote by V = g the Lie algebra of G, seen as a real vector space spanned by Y 0 , Y 1 , Y 2 , where

  2 (R)-lifts. Let us denote by π : SL 2 (R) → PSL 2 (R) the natural projection and set Γ = π -1 (Γ), which is a subgroup of SL 2 (R) with -I ∈ Γ. Given a character χ : Γ → {+1, -1}, with χ(-I) = -1, we can define a linear representation r χ : Γ → SL 2 (R) by setting r χ (γ) := χ( γ) γ,

j 1 .

 1 = r dj • ρ : Γ → SL dj (R), and then define := 1 ⊕ . . . ⊕ p : Γ → SL d1+...dp (R), where 1 ⊕ . . . ⊕ p acts on C d1 × . . . C dp by 1 ⊕ . . . ⊕ p (γ)(v 1 , . . . , v p ) := ( 1 (γ)v 1 , . . . , p (γ)v p ). When ρ = Id, the spectrum of ∆ can be computed as above and is composed of a mixture of Dirac and Laplace spectrum. 4. Bowen-Series/Adler-Flatto coding and topological pressure 4.Boundary coding. From now on, we will use the unit disc model D, with boundary S 1 = ∂D of the hyperbolic plane, endowed with the metric of constant negative curvature

  We can view Γ as a discrete subgroup of PSL 2 (C). Each γ ∈ Γ has an isometric circle C γ := {z ∈ C : |γ (z)| = 1}, with radius r γ and center z γ = γ -1 (∞). Since Γ is Fuchsian and discrete, the radii r γ can only accumulate at 0 and the centers x γ can only accumulate on S 1 . Notice that x + γ ∈ D γ := {z ∈ C : |γ (z)| ≥ 1}. As a consequence, there exists n 0 such that for all admissible word α with n = |α| ≥ n 0 , then D gα ∩ A(αn,j)=1 U j = ∅. In particular, if |α| = n 0 , sup z∈∪ A(αn,j)=1 Uj |g α (z)| < 1.

  (t)| 2 dt, where the infimum runs over all C 1 paths γ : [0, 1] → D such that γ(0) = z and γ(1) = z . The quantity λ D (z) := 2 1 -|z| 2 is called the density of the hyperbolic metric and we have under the conformal mapping φ the identity λ Ω (z) = 2|φ (z)| 1 -|φ(z)| 2 .

α 2 |

 2 Ω (x, y) := sup a,b∈∂Ω log |z -a||w -b| |z -b||w -a| .This pseudo-metric is called the Apollonian metric by Beardon in[START_REF] Beardon | The Apollonian metric of a domain in R n[END_REF], and satisfies the remarkable boundα Ω (x, y) ≤ 2ρ Ω (x, y). Choosing points a, b ∈ ∂Ω such that |z -a| = dist(z, ∂Ω), |w -b| = dist(w, ∂Ω),we obtain for 0 < ≤ 0 (c),α Ω (z, w) ≥ |log dist(z, ∂Ω) + log dist(w, ∂Ω) -log |z -b| -log |w -a|| ≥ 2| log | -| log |z -b| + log |w -a||.Using the triangle inequality we have for all 0 < ≤ 0 (c)|z -b| ≥ |z -w| -|w -b| ≥ c α -≥ c 2 α .Similarly, we obtain also for all 0 < ≤ 0 (c)|w -a| ≥ c 2 α ,so that we get| log |z -b| + log |w -a|| ≤ 2α| log | + | log(c)|.Hence for all 0 < ≤ 0 (c) we have (1 -α)| log | -1 log c| ≤ ρ Ω (z, w).

In the Adler and Flatto construction[START_REF] Adler | Geodesic flows, interval maps, and symbolic dynamics[END_REF], one can take k = 16g -8, where g is the genus of X.

The Boundary coding overcounts finitely many primitive geodesics, which corresponds to geodesics ending at boundary points of the I j , see[36,

[START_REF] Pollicott | Some applications of thermodynamic formalism to manifolds with constant negative curvature[END_REF] for more comments.

This is possible because these inverse branches are maps with degree 1.

We then choose again κ > 0 small enough so that e (2P (βψ-σ0τ )+ ) n(T ) = O(T 1/2-η0 ), for some η 0 > 0 and the proof is done.

It is definitely possible to optimize these bounds and get an explicit exponent η 0 in terms of all the constants involved, but this is likely to be non-optimal. 6.3. Geodesic stretch and bounds on δ 0 . In this subsection we derive some lower and upper bounds on δ 0 related to the geodesic stretch of the Teichmüller representation ρ. Given ρ : Γ → PSL 2 (R) a Teichmüller representation, we set

We prove the following fact.

Proposition 6.4. Using the above notations, we have

In particular if

Proof. We prove only the upper bound, the lower bound being similar. By the variational principle for the topological pressure we have

where the supremum is taken over all T -invariant probability measures. Since this supremum is realized by a unique ergodic (and mixing) equilibrium state, we can replace this supremum by a sup over all ergodic T -invariant measures. On the other hand, we have

and we get that

we have P (2βψ -2στ ) ≤ 0 and therefore δ 0 ≤ σ. To complete the proof of the upper bound, we just need to show that sup

First observe that if T n x = x is a primitive periodic point of T with τ (n) (x) = (γ) and ψ (n) (x) = ρ (γ), then by taking µ = 1 n n-1 j=0 D T j x , where D y stands for the Dirac measure at y, we have ψdµ τ dµ = ρ (γ) (γ) .

Since we know that for all γ ∈ P, there exists an n-periodic point of T such that both τ (n) (x) = (γ) and ψ (n) (x) = ρ (γ) hold, we deduce that sup µ ψdµ τ dµ ≥ I + (ρ).