ZETA FUNCTIONS IN HIGHER TEICHM ¨ULLER THEORY

. In this note we introduce zeta functions and L -functions for discrete and faithful representations of surface groups in PSL( d, R ), for d ≥ 3. These are natural generalizations of the well known classical Selberg zeta function and L -function for Fuchsian groups, corresponding to the case d = 2. We show that these complex functions have meromorphic extensions to the entire complex plane C .


Introduction
Zeta functions and L-functions play a central role in number theory, geometry and dynamical systems.An important example is the Selberg zeta function associated to a compact hyperbolic surface.A hyperbolic metric on a compact surface is determined by a representation of its fundamental group into the rank-1 Lie group PSL(2, R).In this article, we address the natural question of how to extend the above approach to zeta functions Z(s, ρ) naturally associated to appropriate representations ρ into higher rank Lie groups, which are the basis of the relatively new area of higher Teichmüller theory.Here, when defining the zeta function, we replace the length of closed geodesics with the logarithm of the spectral radius of the image of (conjugacy classes of) group elements under the representation.A suitable class of representations to study are those known as projective Anosov (see section 2).
A simplified form of one of our main results is the following.
Theorem 1.1.Let Γ be the fundamental group of a compact oriented surface of genus at least 2 and let ρ : Γ → SL(d, R) (d ≥ 3) be a projective Anosov representation.Then Z(s, ρ) converges for Re(s) sufficiently large, and extends to a meromorphic function in the entire complex plane.
This appears in a more precise form as Theorem 2.15, which is the analogue of the classical result that Selberg zeta functions have meromorphic extensions to the entire complex plane.
The original Selberg zeta function may also be considered dynamically as a zeta function associated to the geodesic flow over the surface.In this context, it is common to also consider the closely related Ruelle zeta function.Both these functions may be studied via a body of tools and techniques known as Thermodynamic Formalism.In particular, this leads to results about extension beyond their half-plane of convergence.
We now briefly outline the contents of the rest of the paper.In the next section, we recall the Selberg and Ruelle zeta functions and introduce analogous zeta functions associated to projective Anosov representations.In section 3, we discuss surface groups and their symbolic coding.We associate to this a dynamical system, namely a subshift of finite type and explain how the induced action on projective space allows us to introduce a function defined on the shift that encodes the weightings d ρ (g).In section 4, we use an approach due to Ruelle [28], to obtain results on extending our zeta functions as meromorphic functions in the entire complex plane by writing them in terms of determinants of transfer operators.In section 5, we discuss the location of zeros and poles for the zeta functions and obtain an error estimate on the associated counting function.This is based on the ideas originally introduced by Dolgopyat [11] for estimating iterates of transfer operators.A final section considers the more general case of L-functions associated to unitary representations.Some of the technical results we require in the paper (Lemma 3.3, Lemma 3.8 and Lemma 5.4) originally appeared in the preprint [25], and we include their proofs for completeness.
We would like to take this opportunity to thank the anonymous referee for their careful reading of the paper and many constructive comments.In particularly, we are grateful to them for correcting an error in the original version of section 5.

Representations and zeta functions
2.1.Selberg and Ruelle zeta functions for PSL(2, R).In 1956, Selberg introduced a zeta function associated to the fundamental group Γ of a compact oriented surface V of genus g ≥ 2 or, more precisely, to representations of such groups as Fuchsian groups, i.e. discrete subgroups of PSL(2, R).
The surface V admits a family of hyperbolic metrics (metrics of constant curvature −1) which are parametrised by the Teichmüller space T (V ).Let us consider a particularly convenient viewpoint using representations.We recall that T (V ) can be identified with one of the two connected components of Hom(Γ, PSL(2, R))/PSL(2, R) which consist of discrete and faithful representations ρ : Γ → PSL(2, R). (Here PSL(2, R) acts on Hom(Γ, PSL(2, R)) by conjugation.)If ρ ∈ T (V ) then one recovers the hyperbolic metric by realising V as H 2 /ρ(Γ), where H 2 is the Poincaré upper half plane and where PSL(2, R) acts by Möbius transformations as orientation preserving isometries of H 2 .(We also note that, since Γ is torsion free, ρ : Γ → PSL(2, R) can be lifted to a representation ρ : Γ → SL(2, R) [10].To simplify notation, we will use ρ(g) to denote the matrix in SL(2, R) in the hope that the context will make the usage clear.)Another connected component T ′ (V ) corresponds to reversing the orientation.
The Selberg zeta function is a function of a single complex variable s ∈ C, formally defined by where the second product is over the set P of all primitive conjugacy classes [g] of elements g ∈ Γ\{1}.(TheSelberg zeta function is more usually denoted by Z but we wish to reserve this for a related but slightly different function defined below.)A conjugacy class is called primitive if it does not contain an element of the form g n 0 , for g 0 ∈ Γ and n > 1.The real number ℓ ρ (g) is twice the logarithm of the largest eigenvalue for ρ(g) (or, equivalently, ℓ ρ (g) is the length of the unique closed geodesic on V in the free homotopy class corresponding to [g]).The product converges to a non-zero analytic function for Re(s) > 1 and Selberg showed the following fundamental result.
Theorem 2.1 (Selberg).The Selberg zeta function S(s, ρ) extends to an entire function of order 2 and has a simple zero at s = 1.
An account of this theorem is contained in Hejhal's book [16], where it appears as Theorem 4.11 and Theorem 4.25.The result was obtained independently by Randol [27].
Theorem 2.2.The Ruelle zeta function R(s, ρ) has a meromorphic extension to C with a simple pole at s = 1.Moreover, R(s, ρ) can be written as a ratio of two entire functions of order 2.
In [28], Ruelle gave an alternative proof of Theorem 2.2 which will be the basis of our approach when we define and study analogues of these zeta functions in higher rank Teichmüller spaces.
We now want to consider a natural generalization of these definitions and results.

2.2.
Higher rank Teichmüller theory.In recent years there has been considerable interest in generalising results in classical Teichmüller theory to what is now often referred to as higher Teichmüller theory, involving representations of surface groups (and more general groups) in higher rank Lie groups.This point of view, which had its origins in the work of Goldman [14] and [17], has received considerable attention, see, for example, the surveys [5], [20] and [33].In this note we will address the natural problem of studying analogues of the Selberg and Ruelle zeta functions in the context of higher Teichmüller theory.
We wish to consider representations of Γ in the higher rank group PSL(d, R) (for d ≥ 3).The most elementary class of such representations are the socalled Fuchsian representations, obtained directly from representations into PSL(2, R).
Example 2.3 (Fuchsian representations).It is well known that there is an irreducible representation τ : SL(2, R) → SL(d, R), unique up to conjugation.This has an explicit construction, which we briefly recall.Let S d denote the d-dimensional vector space of homogeneous polynomials in 2 variables of degree ≤ d − 1.We can choose a basis for S d of the form we can define τ (g) ∈ SL(d, R) by specifying its action on the basis elements as follows: τ (g) : (It is an easy calculation to check that det τ (g) = 1.)Furthermore, τ factors to an irreducible representation τ : PSL(2, R) → PSL(d, R).Representations ρ : Γ → PSL(d, R) of the special form ρ = τ •ρ 0 where ρ 0 ∈ T (V ) (or T ′ (V )), are called Fuchsian representations.
We can now define the representations we wish to study.which contain Fuchsian representations.These are called Hitchin components.When d is odd there is a single Hitchin component but when d is even there are two Hitchin components (corresponding to T (V ) and T ′ (V ) having distinct images).
The study of representations in the Hitchin component has proved particularly fruitful.Hitchin studied the connected components as in Definition 2.4, proving that they were each diffeomorphic to R (2g−2)(d 2 −1) , where Γ is the fundamental group of a surface of genus g ≥ 2 [17], and further work of Goldman [14] began to describe their role in parametrising geometric structures on surfaces.Labourie [22] showed that the Hitchin components consist entirely of discrete and faithful representations.More recently the dynamics of the individual representations in the component have attracted attention (see, for example, the survey [20]).
We note that any ρ : Γ → PSL(d, R) in a Hitchin component may be lifted to a representation ρ : Γ → SL(d, R).This follows since a Fuchsian representation τ • ρ 0 into PSL(d, R) lifts to a representation τ • ρ 0 into SL(d, R) and, by Theorem 4.1 of [10], everything in the same connected component has the same lifting property.In what follows, we shall consider representations into SL(d, R), which we shall denote by ρ (but remembering that this is the lift of a representation into PSL(d, R)).
A consequence of this definition is that there is a well behaved map from the Gromov boundary of Γ to G k (R d ), the Grassmannian of k-planes in R k .Here, the Gromov boundary is defined as follows.A sequence (g n ) ∞ n=0 ∈ Γ Z + is a geodesic ray (in the Cayley graph of Γ) if |g n | = n and g n+1 g −1 n ∈ Γ 0 .(In particular, g 0 = 1.)Then the Gromov boundary of Γ, which we denote by ∂ ∞ Γ, is the space of equivalence classes of geodesic rays which remain a bounded distance apart.It can be equipped with a natural "visual" metric, where two points are close if one can choose representative rays which agree for a large initial segment (see Chapitre 7 of [13]).In our case, where Γ is the fundamental group of a compact surface of genus at least two, ∂ ∞ Γ is homeomorphic to the unit circle (by a quasisymmetric homeomorphism) [13].There is a natural action of Γ on ∂ ∞ Γ and each g ∈ Γ \ {1} has a unique attracting fixed point g + ∈ ∂ ∞ Γ and {g We now have the following result.
Lemma 2.6.If a representation ρ : Γ → SL(d, R) is k-Anosov then there exist Hölder continuous maps (3) there exist a > 0 and c > 0 such that for every geodesic ray where σ i denotes the ith singular value (k-dominated).
In this paper, we will deal with representations which are 1-Anosov.These representations are also called projective Anosov.Noting that G k (R d ) is the projective space RP d−1 , in this case we will only need the map ξ : ∂ ∞ Γ → RP d−1 corresponding to ξ (1) in the lemma above.In this case, we have the following contraction property, which, though not part of the original definitions in [22], [15], comes from the equivalent definition due to Kapovich, Leeb and Porti ( [18], Definition 6.45) and [19], or from part (3) of Lemma 2.6.
Lemma 2.7.There exist a > 0 and c > 0 such that for every geodesic ray Projective Anosov representations were originally introduced by Labourie (ii) There may be additional restrictions on Anosov representations of hyperbolic groups.For example, if Γ is a torsion-free hyperbolic group which has an Anosov representation into SL(3, R) then Γ is either a free group or a surface group ( [7], Theorem 1.1).

2.4.
Zeta functions for projective Anosov representations.We now come to the main object of study in this note.Let Γ be the fundamental group of a compact oriented surface of genus g ≥ 2 and let ρ : Γ → SL(d, R) be a projective Anosov representation.We recall the following definition.Definition 2.9.A matrix is called proximal if it has a unique eigenvalue which is real and strictly maximal in modulus.
We then have the following, which is a consequence of the definition.Lemma 2.10.Let ρ : Γ → SL(d, R) be a projective Anosov representation.Then for each g ∈ Γ \ {1} the matrix ρ(g) is proximal.
We use this to associate a natural weight to each g ∈ Γ \ {1}.Let where λ(g) is the maximal eigenvalue of the matrix ρ(g).By Lemma 2.10, since ρ(g) ∈ SL(d, R), |λ(g)| > 1 and hence d ρ (g) > 0. Note that d ρ (g) only depends on the conjugacy class [g].We will generalize the definition of the Selberg and Ruelle zeta functions to these representations.Definition 2.11.Given a projective Anosov representation ρ : Γ → SL(d, R), we can associate a generalized Ruelle zeta function defined by and a generalized Selberg zeta function defined by for s ∈ C wherever the products converge.
In particular, from (1.2) and (1.3) we have the identities Remark 2.12.Note that, in the special case d = 2, with ρ ∈ T (V ) we have that d ρ (g) = ℓ ρ (g)/2.Therefore, ζ(s, ρ) = R(s/2, ρ).However, the relationship between Z(s, ρ) and S(s, ρ) is more complicated with To describe the half-plane of convergence of these new zeta functions, we recall the following definition.See [5], [6] or [25] for a further discussion, including the existence of the limit.Definition 2.13.The entropy h(ρ) of a projective Anosov representation ρ : Γ → SL(d, R) is defined to be the growth rate of the number of primitive conjugacy classes [g] ∈ P with d ρ (g) at most T as T → +∞, i.e., (The value of h(ρ) would be the same if we used all non-trivial conjugacy classes in the definition, rather than just the primitive ones.) It is easy to see from the definitions that Z(s, ρ) and ζ(s, ρ) converge to non-zero analytic functions provided Re(s) > h(ρ).In the original setting, with d = 2 and ρ ∈ T (V ), the entropy is always equal to 2 and so does not need to be explicitly introduced.(We remark that, in this case, h(ρ) = 2 is twice the topological entropy of the geodesic flow over , where ℓ ρ (g) is the length of the corresponding closed geodesic.)2.5.Meromorphic extensions of the zeta functions.Our main results on the zeta functions for projective Anosov representations can be viewed as generalizations of Theorem 2.2 and Theorem 2.1, respectively.We begin with the result for ζ(s, ρ), since, with our approach, the proof of this naturally comes first.
Theorem 2.14.Let ρ : Γ → SL(d, R) be a projective Anosov representation.Then ζ(s, ρ) extends to a meromorphic function in the entire complex plane and has a simple pole at s = h(ρ).Furthermore, ζ(s, ρ) may be written as the quotient of two entire functions of order at most d.
This will allow us to deduce the following result for Z(s, ρ).
Theorem 2.15.Let ρ : Γ → SL(d, R) be a projective Anosov representation.Then Z(s, ρ) is a meromorphic function to the entire complex plane and has a simple zero at s = h(ρ).Furthermore, Z(s, ρ) may be written as the quotient of two entire functions of order at most d + 1.
We will prove these results in section 4. We will also consider the zeros of the zeta functions in section 5.This leads to estimates on the error terms in the counting functions for the weights for the subclass of hyperconvex representations.
Remark 2.16.An alternative way of weighting the matrices, and hence of producing zeta functions, would be to take the difference between the logarithms of largest and smallest eigenvalues, i.e. dρ (g) = log λ(g)/λ(g −1 ) = log (λ d /λ 1 ) .
This may be reduced to the situation we consider by composing ρ with the Adjoint representation of SL(d, R) and so Theorem 2.14 and Theorem 2.15 will still apply.(We are grateful to the referee for this observation, which replaces our original more complicated argument.)

Surface groups and symbolic dynamics
In this section we recall the ideas that are used to build a bridge between the geometry and the dynamics.
3.1.Action on projective space.As discussed above, the representation ρ : Γ → SL(d, R) induces an action of Γ on the projective space RP d−1 .More explicitly, for g ∈ Γ, we define ρ(g) : Since, by Lemma 2.10, ρ(g) has a real eigenvalue which is strictly maximal in modulus, the map ρ(g) has a unique attracting fixed point; we will denote this by ξ g ∈ RP d−1 .Let ξ : ∂ ∞ Γ → RP d−1 be the map introduced in the previous section.As an immediate consequence of Lemma 2.7, for each g ∈ Γ \ {1}, we have ξ(g + ) = ξ g , the fixed point for ρ(g).
We now define a compact subset of projective space which will be useful in the sequel.
Here ρ is a representation into SL(3, R) and Λ ⊂ RP 2 is the associated limit set.For g ∈ Γ \ {1}, the point ξ g is an attracting fixed point for ρ(g) : RP 2 →: RP 2 .Definition 3.1.We define the limit set to be the closure of the fixed points.

Since {g
Example 3.2 (Case d = 3).In this case the limit set Λ ⊂ RP 2 is diffeomorphic to a circle bounding a strictly convex region Ω when ρ a representation in the Hitchin component.In particular, in Lemma 2.6 we have that for η ∈ ∂ ∞ Γ the point ξ (1) (η) ∈ Λ and θ (1) (η) is tangent to the curve at that point.This is described in the earlier work of Choi and Goldman [9] (see also [22]) and applies to projective Anosov representations via the equivalence of the definitions shown in [21].
We can use the following simple lemma to relate the weight d ρ (g) to the action of ρ(g) on RP d−1 .
Proof.We can consider the linear action of ρ(g) on R d , then the fixed point corresponds to an eigenvector v and the result follows from a simple calculation using that the linear action of ρ(g) ∈ SL(d, R) preserves area in R d .More precisely, ξ g corresponds to an eigenvector v for the maximal eigenvalue λ(g), with |λ(g)| > 1, for the matrix ρ(g).We can assume without loss of generality that v = 1 and then for arbitrarily small δ > 0 we can consider a δ-neighbourhood of v which is the product of a (d − 1)-dimensional neighbourhood in RP d−1 and a δ-neighbourhood in the radial direction.The effect of the linear action of ρ(g) is to replace v by λ(g)v, and thus stretch the neighbourhood in the radial direction by a factor of |λ(g)|.Since ρ(g) has determinant one, the volume of the (d − 1)-dimensional neighbourhood contracts by |λ(g)| −1 .To calculate the effect of the projective action ρ(g), we need to rescale λ(g)v to have norm one, which corresponds to multiplication by the diagonal matrix diag(|λ(g)| −1 , . . ., |λ(g)| −1 ).In particular, the (d − 1)-dimensional neighbourhood in RP d−1 shrinks by a factor of approximately |λ(g)| −d , giving the result.
3.2.Surface groups and coding.We now want to introduce a natural coding for the group Γ which will allow us to analyse the projective action of Γ on Λ.This is motivated by the more familiar boundary coding associated to Fuchsian groups due to Bowen and Series [32].We take the same basic approach to introducing a dynamical perspective to the study of projective Anosov representations based on subshifts of finite type that was used in [25] (for representations in a Hitchin component).Since Γ is the fundamental group of a compact surface V with genus g ≥ 2, it has the presentation g } for the symmetrized generating set and | • | : Γ → Z + for the word length with respect to Γ 0 .
Such surface groups are particular examples of Gromov hyperbolic groups and as such they are strongly Markov groups in the sense of Ghys and de la Harpe [13], i.e. they can be encoded using a directed graph and an edge labelling by elements in Γ 0 .This approach allows allows us to relate the group to a dynamical system, namely a subshift of finite type, and thus to use the machinery of thermodynamic formalism to define a so called pressure function and, later, transfer operators, which may be used to analyse our zeta functions.In the particular case of surface groups, the coding follows directly from the work of Adler and Flatto [1] and Series [32] on coding the action on the boundary and the associated subshift of finite type is mixing.
We have the following result.
Lemma 3.4.We can associate to (Γ, Γ 0 ) (i) a directed graph G = (V, E) with a distinguished vertex * ; and (ii) an edge labelling ω : E → Γ 0 , such that (1) no edge terminates at * ; (2) there is at most one directed edge joining each ordered pair of vertices; Furthermore, the subgraph obtained be deleting the vertex * has the aperiodicity property that there exists N ≥ 1 such that, given any two v, v ′ ∈ V \{ * }, there is a directed path of length N from v to v ′ .
Remark 3.5.Part (4) of Lemma 3.4 plays a subtle, but important, role in allowing us to relate the dynamical theory of zeta functions to conjugacy classes in Γ.In order to study other groups Γ than surface groups one would need to find a replacement for this property.
3.3.Symbolic dynamics.We now introduce the symbolic coding for the limit set Λ.This is particularly useful in introducing a partition of Λ which gives an effective way to reduce the group action to the orbits of a single transformation.
We can associate to the directed graph G = (V, E) a subshift of finite type Σ, as follows.Let E * ⊂ E be the set of directed edges that start at * .The set of states in Σ is E \ E * and a transition e → e ′ is allowed if and only if e ′ follows e in G.More precisely, if there are k edges in E \ E * then we define a k × k matrix A by A(e, e ′ ) = 1 if e ′ follows e in the directed graph and define (We note that this definition does not involve the edge labels on E.) The set Σ is a compact space with respect to the metric The shift map is the local homeomorphism σ : Σ → Σ defined by (σx) n = x n+1 .By the final statement in Lemma 3.4, A is aperiodic (i.e.there exists N ≥ 1 such that A N has all entries positive) and, equivalently, the shift σ : Σ → Σ is mixing (i.e. for all open non-empty U, V ⊂ Σ, there exists There is a natural surjective Hölder continuous map π : Σ → ∂ ∞ Γ defined by letting π((x n ) ∞ n=0 ) be the equivalence classes of the geodesic ray ( Definition 3.6.We define a map κ : However, the shift σ : Σ → Σ only encodes information about Γ as an abstract group.In order to keep track of the additional information given by the representation of Γ in SL(d, R) and its action on RP d−1 we need to introduce a Hölder continuous function r : Σ → R. Definition 3.7.We can associate a map r : Σ → R defined by r(x) = − 1 d log det(D κ(x) ρ(g x 0 )) (i.e. the Jacobian of the derivative of the projective action), where g x 0 = ω(x 0 ) is the generator corresponding to the first term in There is a natural one-to-one correspondence between non-trivial conjugacy classes in Γ and periodic orbits for the shift σ, by part (4) of Lemma 3.4.
We now have the following simple but key result.
Lemma 3.8.The function r : Σ → R is Hölder continuous, and if σ n x = x is a periodic point corresponding to the conjugacy class [g] of an element g ∈ Γ then r n (x) = d ρ (g).
Proof.The Hölder continuity of r follows immediately from the Hölder continuity of ξ, which is given by Lemma 2.7.The second statement follows from the equivariance and the observation κ(σx) = ρ(g x 0 )κ(x).Moreover, that the periodic point x has an image κ(x)(= ξ g ) which is fixed by ρ(g) and the result follows from Lemma 3.3.

3.4.
Semi-flows and entropy.In this subsection we give an alternative definition of the entropy h(ρ) of a representation, as defined in the introduction, which is useful when taking the symbolic viewpoint.We can associate to σ : Σ → Σ and the positive function r : Σ → R + the space where (x, r(x)) ∼ (σx, 0), and the semi-flow σ r t : Σ r → Σ r defined by σ r t (x, u) = (x, u + t), for t ≥ 0 (respecting the identifications).
The topological entropy h(σ r ) for the semi-flow is defined to be the entropy of the time one map.It is a standard result that an alternative way of defining h(σ r ) is as the growth rate where P(σ r ) denotes the set of prime periodic orbits of σ r and l(τ ) denotes the least period of the orbit τ [23].
Lemma 3.9.If ρ is a projective Anosov representation then the entropy of the representation is equal to the topological entropy of the semi-flow σ f .
Proof.We observe that there is a one-one correspondence between the periodic orbits for the semi-flow σ r and for the shift σ.By Lemma 3.8 and Definition 2.13, we see that h(ρ) = h(σ r ).

Extensions of Zeta Functions
We now turn to the results on the meromorphic extensions of zeta functions.The proof of Theorem 2.14 is based on the approach of Ruelle in [28].We then use Theorem 2.14 to prove the corresponding result for Z(s, ρ) in Theorem 2.15.
4.1.Partitions.By the construction in the previous section we have associated to the representation ρ a Hölder continuous map κ : Σ → RP d−1 which intertwines the shift map on Σ with the projective action, i.e. κ(σx) = ρ(ω(x 0 ))κ(x).In particular, we use κ to define a particular partition P i = κ([i]) of the limit set Λ ⊂ RP d−1 , where We then have that (1) Λ = k i=1 P i , and (2) P i ∩ P j consists of at most one point, for distinct i = j.ζ(s, ρ) is based on the study of a transfer operator defined on analytic functions.In particular, we can realise the symbolic maps Σ ⊃ [j] ∋ x → ix ∈ [i] ⊂ Σ (where A(i, j) = 1) as local real analytic contractions

The analytic extension of the zeta function
on the disjoint union of the sets.
Figure 2. The contractions ψ i : P j → P i satisfy ψ i (U j ) ⊂ U i for the neighbourhoods of the complexifications (suggested in the figure).

Transfer operators.
We can consider a Banach space of analytic functions on a neighbourhood of the disjoint union of the partition k i=1 P i .In particular, let us fix neighbourhoods U i ⊃ P i in the complexification of RP d−1 , chosen suitably small as will be indicated below.
This allows us to define a convenient Banach space of analytic functions.
Definition 4.1.We define B 0 to be the space of bounded analytic functions We can also define a natural family of bounded linear operators on this space.Definition 4.2.For s ∈ C, the transfer operator where det(Dψ i (•)) s is understood as the complexification of the real analytic function.
In order for the operator to be well defined we need to show that we can choose the neighbourhoods U i so that whenever A(i, j) = 1 (1 ≤ i, j ≤ k) then ψ i (U j ) ⊂ U i .Since, by construction, ψ(P j ) ⊂ P i , it suffices to show that for any η ∈ P i we have D η ψ < 1.However, associating to η those geodesics (γ) arising from the coding (in Lemma 3.4) we can apply Lemma 2.7.If ae −c < 1 then we are done.Otherwise we can choose n ≥ 2 sufficiently large so that ae −cn and use the standard recoding of Σ by replacing symbols by words of length n.
The advantage of being able to work in the Banach space of bounded analytic functions is that the operators have far better spectral properties than, say, Banach spaces of C 1 functions or Hölder continuous functions.More precisely, the transfer operators are compact operators and, furthermore, satisfy the stronger property of nuclearity.This follows directly from the work of Ruelle, based on the ideas of Grothendieck.Lemma 4.3 (Nuclearity: after Grothendieck and Ruelle).The transfer operators L s : B 0 → B 0 are nuclear.More precisely, we can find Proof.This can be deduced from [28].
We can also consider the natural generalisation of these operators to the space of analytic j-forms on k i=1 U i (for j = 1, . . ., d−1).More precisely, let B j denote the Banach space of analytic j-forms on k i=1 U i whose coefficients are bounded.Then the definition of L s naturally extends to define transfer operators L s,j : B j → B j .The analogue of Lemma 4.3 holds for these operators.

Traces and determinants.
A particular consequence of Lemma 4.3 is that the operator L s (and its powers L n s ) are trace class, i.e. the eigenvalues are summable: ∞ m=1 |λ m | < ∞.To calculate the trace of L n s , we need to consider compositions of the contractions ψ i corresponding to periodic points in Σ.Let i = (i 1 , . . ., i n ) be a string of length |i| = n an allowed string such that If, in addition, A(i n , i 1 ) = 1 then we call i a periodic allowed string.We let ψ i : A(in,j)=1 , this restricts to a contraction ψ i : P i 1 → P i 1 , which has a unique fixed point denoted z i .We now have the following result.
Lemma 4.4.The trace of L n s : B 0 → B 0 is given by , where the summation is taken over all periodic allowed strings of length n.Furthermore, Proof.This can be deduced from the general setting in [28], pages 232-234.
We want to associate to the transfer operators a function of a single complex variable s.We can write Lemma 4.4 explains the connection with the fixed points of the contractions.Similarly, we can associate to the transfer operators L s,j the functions The next lemma shows that these functions extend analytically to the entire complex plane.To simplify notation, we shall write L s,0 = L s .
Lemma 4.5.The functions s → det(I − L s,j ) ( 0 ≤ j ≤ d − 1) extend to entire analytic functions of s ∈ C of order at most d.
Proof.This can be (essentially) deduced from the proof of Theorem 1(b) in [28], although there is a minor correction required, as described in [12].In particular, one can expand where for some E > 0 and D = max j Dφ j .In particular, by considering critical points of the real function t → |s|Dt − Et (1+ 1 d−1 ) we see that the upper bound on the terms occurs at For s in a sufficiently small neighbourhood of s 0 we have that Re(s + n) > h(ρ) and therefore Z(s + n, ρ) converges to an analytic function.Moreover, the denominator is meromorphic by Theorem 2.14 and thus using (4.2) we deduce that Z(s, ρ) is meromorphic.
To obtain the order bound in Theorem 2.15 consider |s| < m, say, and set n = m + [h(ρ)] + 2. By the proofs of Theorem 2.14 and Lemma 4. Remark 4.6 (Equidistribution).For each t > δ can define a probability measure on Λ by µ t = g∈Γ\{1} |JacD ξg ρ(g)| t δ ξg g∈Γ\{1} |JacD ξg ρ(g)| t and then as t tends to δ this converges to a probability measure µ supported on Λ.This follows easily from the properties of the transfer operator L s , where µ is related to the maximal eigenprojection of the operator L h(ρ) .

Zeros and poles of the zeta functions
Having established that the zeta functions ζ(s, ρ) and Z(s, ρ) have meromorphic extensions to C in Theorems 2.14 and 2.15, it is very natural to ask about the location of the zeros and poles of zeta function.
In the case of the classical Selberg zeta function for hyperbolic surfaces the following is well known.Recall from Theorem 2.1 that the Selberg zeta function for a hyperbolic surface is entire with a simple zero at s = 0. Furthermore, the following result is well known.
In this classical case it is even known that the zeros in the critical strip 0 ≤ Re(s) ≤ 1 lie on [0, 1] ∪ 1 2 + iR .This is a consequence of the way the zeta function can be extended using the trace formula and the interpretation of the zeros s n in terms of the eigenvalues λ n = s n (1 − s n ) of the Laplace-Beltrami operator.
We can generalize both Proposition 5.1 and Proposition 5.2 to the zeta functions for a special class of projective Anosov representations.We say that ρ This is an open condition and examples include representations in Hitchin components [22].Importantly, for our analysis, recent work of Pozzetti, Sambarino and Weinhard (Proposition 7.4 and Corollary 7.6 of [26]) and Zhang and Zimmer (Theorem 1.1 of [34]) showed that if ρ is a (1, 1, 2)hyperconvex representation then ξ 1 (∂ ∞ Γ) is a C 1 curve.Indeed, Zhang and Zimmer prove stronger regularity, which we state this as a lemma.
We will now obtain our results on the zeta functions.First we will consider the Ruelle-type zeta function ζ(s, ρ).Proof.That ζ(s, ρ) has a non-zero meromorphic extension to a half-plane Re(s) > h(ρ)−ǫ, for some ǫ > 0, is a standard result (Corollary 10.6 of [23]).The pole free strip is a consequence of the method of Dolgopyat [11] and the fact that, as discussed above, the limit set Λ ⊂ RP d−1 is a C 1 curve [26], [34].This method gives bounds on the iterates of the transfer operators L s regarded as operators L s : C . Following Proposition 7.4 of [2], the key ingredients required for the proof are that: We see that (i) holds by Lemma 3.3 and Lemma 5.3 and (ii) follows from Lemma 3.8 and Lemma 5.4.
We can apply results on bounding iterates of transfer operators: Corollary 2 and Corollary 3 of [11] and Proposition 4 of [24], to show that there exist constants σ 0 < h, C > 0 and 0 < β < 1 such that whenever s = σ + it and n = p[log |t|] + l, where p ≥ 0 and 0 ≤ l ≤ [log |t|] − 1, then By Lemma 2 of [24] (which, apart from the dependence on |t| in the bound, appears in [29]) we have the estimate that for any x j ∈ P j , j = 1, . . ., k, and any β 0 satisfying max{β, max j Dψ j ∞ } < β 0 < 1, there exists C > 0 such that where χ P j is the indicator function for P j and i, ψ i and z i are as in Lemma 4.4.We can use this estimate to get a bound on the logarithm of the absolute value of the zeta function of the form log |ζ(s, ρ)| = O (|t| α ) , for some α > 0, as in (2.3) of [24].
In light of (1.4), and the observation that Z(s+1, ρ) is uniformly bounded on the half-plane Re(s) > h(ρ) − ǫ, provided ǫ < 1, this gives the following generalization of Proposition 5.1.Although Theorem 5.5 is weaker than the full result known for the Selberg zeta function in the case of hyperbolic surfaces it nonetheless has analogous consequences for error terms in counting functions.To conclude the section, we address how the positions of the zeros influence asymptotic formulae.It was shown in [30] (following the approach in Chapter 6 of [23]) that π ρ (T ) satisfies the asymptotic formula π ρ (T ) ∼ e h(ρ)T h(ρ)T , as T → +∞.
Using Theorem 5.5, we can deduce a stronger result with an error term provided we replace the leading term with a logarithmic integral.As usual, we will write li(x) := We then have the following result.
Proof.Given Theorem 5.5, we can apply the classical proof from number theory (see Proposition 6 and Proposition 7 of [24]).
Remark 5.9.It is worth noting that the trick used in Remark 2.16 does not apply in the context of this section, since the adjoint representation of a (1, 1, 2)-hyperconvex representation is not necessarily itself hyperconvex.

The L-function in higher rank
We can generalize the definition of the zeta functions to L-functions by incorporating additional information associated to a unitary representation U : Γ → U(N ).More precisely, we make the following definition.If R χ 0 is the trivial representation then we see that these L-functions reduce to the corresponding zeta functions, i.e.Z(s, ρ, χ 0 ) = Z(s, ρ) and ζ(s, ρ, χ 0 ) = ζ(s, ρ).In the particular case of Fuchsian representations then these definitions reduce to the familiar definitions of L-functions for Fuchsian groups [12].

Definition 2 . 4 .
The natural generalization of the Teichmüller space are connected components of Hom(Γ, PSL(d, R))/PSL(d, R) [22].Examples include the representations in the Hitchin component introduced above and also the Benoist representations introduced in[3].Remark 2.8.(i) By consider the kth-exterior power, a k-Anosov representation into SL(d, R) induces a projective Anosov representation into SL d k , R .
giving an upper bound on the terms of O exp |s| d D d /(E(1 + 1 d−1 )) d−1 .Moreover, it is easy to see that this gives an upper bound | det(I − L s,j )| = O e c|s| d , for some c > 0. In particular, we see that the order is given by lim sup R→+∞ log max |s|≤R log | det(I − L s,j )| log R ≤ d.Proof of Theorem 2.14.To complete the proof of Theorem 2.14, we can then use the identity (4.1) to write ζ(s, ρ) = j odd det(I − L s,j ) j even det(I − L s,j ) , from which we deduce that ζ(s, ρ) is a meromophic function which, by Lemma 4.5, may be written as the quotient of two entire functions of order ≤ d.Proof of Theorem 2.15.Given s 0 ∈ C, we can choose m ∈ Z with Re(s 0 ) ≥ m.Setting n = |m| + [h(ρ)] + 1 we can use (1.4) to write 5 we can write ζ(s, ρ) = f (s) g(s) , where for any ǫ > 0 we can choose C > 0 such that |f (s)|, |g(s)| ≤ exp(C|s| d+ǫ ) and thus using (4.2) we can writeZ(s, ρ) = Z(s + n, ρ) G(s) F (s) (4.3)whereF (s) = n−1 k=0 f (s + k) and G(s) = n−1 k=0 g(s + k) are entire functions.In particular, Z(s + n, ρ) is uniformly bounded and since log |F (s)| ≤ C m+1 k=0 |s + k| d+ǫ ≤ C(m + 2)(2m) d+ǫ the order of F (s) is bounded by lim m→+∞ log(C(m + 2)(2m) d+ǫ ) log m = d + 1 + ǫ,and similarly the order for G(s) is bounded by d + 1 + ǫ.Finally, since ǫ > 0 can be arbitrarily small, we can deduce from (4.3) that Z(s, ρ) is a ratio of two entire functions of order d + 1.

( 1 )Theorem 6 . 3 .
ζ(s, ρ, χ) has a meromorphic extension to the entire complex plane C; and(2) ζ(s, ρ, χ) has a simple pole at s = h(ρ) if and only if R χ is trivial.The following result generalizes Theorem 2.15 and has a similar proof.Let ρ : Γ → SL(d, R) be a projective Anosov representation and let R χ : Γ → U(N ) be a unitary representation.The generalized Lfunction Z(s, ρ, χ) has the following properties.