Zeta functions in higher Teichmüller theory

Abstract

In this note we introduce zeta functions and L-functions for discrete and faithful representations of surface groups in $\textrm{PSL}(d, \mathbb R)$, for $d \ge 3$. 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 $\mathbb C$.

1 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 $\textrm{PSL}(2,\mathbb R)$. In this article, we address the natural question of how to extend the above approach to zeta functions $Z(s,\rho )$ naturally associated to appropriate representations $\rho$ 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 Sect. 2).

A simplified form of one of our main results is the following.

Theorem 1.1

Let $\Gamma$ be the fundamental group of a compact oriented surface of genus at least 2 and let $\rho : \Gamma \rightarrow \textrm{SL}(d, \mathbb R)$ ($d \ge 3$) be a projective Anosov representation. Then $Z(s, \rho )$ 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 Sect. 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 $d_\rho (g)$. In Sect. 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 Sect. 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 (Lemmas 3.3, 3.8 and 5.4) originally appeared in the preprint [24], 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 Sect. 5.

2 Representations and zeta functions

2.1 Selberg and Ruelle zeta functions for $\textrm{PSL}(2, \mathbb R)$

In 1956, Selberg [31] introduced a zeta function associated to the fundamental group $\Gamma$ of a compact oriented surface V of genus $g \ge 2$ or, more precisely, to representations of such groups as Fuchsian groups, i.e. discrete subgroups of $\textrm{PSL}(2,\mathbb R)$.

The surface V admits a family of hyperbolic metrics (metrics of constant curvature $-1$) which are parametrised by the Teichmüller space $\mathcal T(V)$. Let us consider a particularly convenient viewpoint using representations. We recall that $\mathcal T(V)$ can be identified with one of the two connected components of

$$\begin{aligned} \textrm{Hom}(\Gamma ,\textrm{PSL}(2, \mathbb R))/\textrm{PSL}(2, \mathbb R) \end{aligned}$$

which consist of discrete and faithful representations $\rho : \Gamma \rightarrow \textrm{PSL}(2, \mathbb R)$. (Here $\textrm{PSL}(2,\mathbb R)$ acts on $\textrm{Hom}(\Gamma ,\textrm{PSL}(2, \mathbb R))$ by conjugation.) If $\rho \in \mathcal T(V)$ then one recovers the hyperbolic metric by realising V as $\mathbb H^2/\rho (\Gamma )$, where $\mathbb H^2$ is the Poincaré upper half plane and where $\textrm{PSL}(2, \mathbb R)$ acts by Möbius transformations as orientation preserving isometries of $\mathbb H^2$. (We also note that, since $\Gamma$ is torsion free, $\rho : \Gamma \rightarrow \textrm{PSL}(2,\mathrm R)$ can be lifted to a representation $\widetilde{\rho }: \Gamma \rightarrow \textrm{SL}(2,\mathbb R)$ [10]. To simplify notation, we will use $\rho (g)$ to denote the matrix in $\textrm{SL}(2,\mathbb R)$ in the hope that the context will make the usage clear.) Another connected component $\mathcal T'(V)$ corresponds to reversing the orientation.

The Selberg zeta function is a function of a single complex variable $s\in \mathbb C$, formally defined by

$$\begin{aligned} S(s, \rho ) = \prod _{n=0}^\infty \prod _{[g] \in \mathcal P} \left( 1 - e^{-(s+n) \ell _\rho (g)}\right) , \end{aligned}$$

(1.1)

where the second product is over the set $\mathcal P$ of all primitive conjugacy classes [g] of elements $g \in \Gamma {\setminus } \{1\}$.(The Selberg 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 $g_0^n$, for $g_0 \in \Gamma$ and $n >1$. The real number $\ell _\rho (g)$ is twice the logarithm of the largest eigenvalue for $\rho (g)$ (or, equivalently, $\ell _\rho (g)$ 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 $\textrm{Re}(s) > 1$ and Selberg showed the following fundamental result.

Theorem 2.1

(Selberg) The Selberg zeta function $S(s,\rho )$ 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].

One may also, following Ruelle (cf. [28]), consider the related zeta function

$$\begin{aligned} R(s,\rho ) = \prod _{[g] \in \mathcal P} \left( 1-e^{-s\ell _\rho (g)}\right) ^{-1}. \end{aligned}$$

(1.2)

Since $R(s,\rho ) = S(s+1,\rho )/S(s,\rho )$, one immediately obtains the following result from Theorem 2.1.

Theorem 2.2

The Ruelle zeta function $R(s,\rho )$ has a meromorphic extension to $\mathbb C$ with a simple pole at $s=1$. Moreover, $R(s,\rho )$ 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, 17], has received considerable attention, see, for example, the surveys [6, 20, 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 $\Gamma$ in the higher rank group $\textrm{PSL}(d, \mathbb R)$ (for $d \ge 3$). The most elementary class of such representations are the so-called Fuchsian representations, obtained directly from representations into $\textrm{PSL}(2,\mathbb R)$.

Example 2.3

(Fuchsian representations) It is well known that there is an irreducible representation $\widetilde{\tau }: \textrm{SL}(2, \mathbb R) \rightarrow \textrm{SL}(d, \mathbb R)$, unique up to conjugation. This has an explicit construction, which we briefly recall. Let $\mathcal S_d$ denote the d-dimensional vector space of homogeneous polynomials in 2 variables of degree $\le d-1$. We can choose a basis for $\mathcal S_d$ of the form

$$\begin{aligned} x^{d-1}, x^{d-2}y, \cdots , x^{d-i-1} y^i, \cdots , x y^{d-2}, y^{d-1}. \end{aligned}$$

For

$$\begin{aligned} g = \left( \begin{matrix} a &{}\,\,\, b \\ c&{}\,\,\,d \end{matrix} \right) \in \textrm{SL}(2, \mathbb R), \end{aligned}$$

we can define $\widetilde{\tau }(g) \in \textrm{SL}(d,\mathbb R)$ by specifying its action on the basis elements as follows:

$$\begin{aligned} \widetilde{\tau }(g): x^{d-i-1} y^i \mapsto (ax + cy)^{d-i-1} (bx + dy)^i. \end{aligned}$$

(It is an easy calculation to check that $\det \widetilde{\tau }(g) =1$.) Furthermore, $\widetilde{\tau }$ factors to an irreducible representation $\tau : \textrm{PSL}(2, \mathbb R) \rightarrow \textrm{PSL}(d, \mathbb R)$. Representations $\rho : \Gamma \rightarrow \textrm{PSL}(d,\mathbb R)$ of the special form $\rho = \tau \circ \rho _0$ where $\rho _0 \in \mathcal T(V)$ (or $\mathcal T'(V)$), are called Fuchsian representations.

We can now define the representations we wish to study.

Definition 2.4

The natural generalization of the Teichmüller space are connected components of

$$\begin{aligned} \textrm{Hom}(\Gamma , \textrm{PSL}(d, \mathbb R))/\textrm{PSL}(d, \mathbb R) \end{aligned}$$

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 $\mathcal T(V)$ and $\mathcal T'(V)$ 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 $\mathbb R^{(2g-2)(d^2-1)}$, where $\Gamma$ is the fundamental group of a surface of genus $g \ge 2$ [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 $\rho : \Gamma \rightarrow \textrm{PSL}(d,\mathbb R)$ in a Hitchin component may be lifted to a representation $\widetilde{\rho }: \Gamma \rightarrow \textrm{SL}(d,\mathbb R)$. This follows since a Fuchsian representation $\tau \circ \rho _0$ into $\textrm{PSL}(d,\mathbb R)$ lifts to a representation $\widetilde{\tau }\circ \widetilde{\rho }_0$ into $\textrm{SL}(d,\mathbb R)$ 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 $\textrm{SL}(d,\mathbb R)$, which we shall denote by $\rho$ (but remembering that this is the lift of a representation into $\textrm{PSL}(d,\mathbb R)$).

2.3 Anosov representations

It is now convenient to consider a more general setting. Consider the case of discrete finitely generated torsion-free (word) hyperbolic group $\Gamma$. Let $\Gamma _0$ be a finite symmetric generating set and let $|\cdot |: \Gamma \rightarrow \mathbb Z^+$ be the word length with respect to $\Gamma _0$.

Definition 2.5

A discrete and faithful representation $\rho : \Gamma \rightarrow \textrm{SL}(d, \mathbb R)$ is called k-Anosov (where $1 \le k \le d/2$) if there exists $C, \mu > 0$ such that for each $g \in \Gamma {\setminus }\{1\}$ the eigenvalues $\lambda _i(\rho (g))$, $1\le i \le d$, for $\rho (g)$ ordered by modulus (i.e., $|\lambda _1(\rho (g))| \ge |\lambda _2(\rho (g))| \ge \cdots \ge |\lambda _d(\rho (g))|$) satisfy

$$\begin{aligned} \frac{\lambda _k(\rho (g))}{\lambda _{k+1}(\rho (g))} \ge C e^{|g|}. \end{aligned}$$

This particularly short definition is due to Kassel and Potrie [21] (see also [20]) but corresponds to earlier definitions, beginning with Labourie [22] and including Guichard and Wienhard [15].

A consequence of this definition is that there is a well behaved map from the Gromov boundary of $\Gamma$ to $\mathcal G_k( \mathbb R^{d})$, the Grassmannian of k-planes in $\mathbb R^k$. Here, the Gromov boundary is defined as follows. A sequence $(g_n)_{n=0}^\infty \in \Gamma ^{\mathbb Z^+}$ is a geodesic ray (in the Cayley graph of $\Gamma$) if $|g_n|=n$ and $g_{n+1}g_n^{-1} \in \Gamma _0$. (In particular, $g_0 = 1$.) Then the Gromov boundary of $\Gamma$, which we denote by $\partial _\infty \Gamma$, 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 $\Gamma$ is the fundamental group of a compact surface of genus at least two, $\partial _\infty \Gamma$ is homeomorphic to the unit circle (by a quasisymmetric homeomorphism) [13]. There is a natural action of $\Gamma$ on $\partial _\infty \Gamma$ and each $g \in \Gamma {\setminus } \{1\}$ has a unique attracting fixed point $g^+ \in \partial _\infty \Gamma$ and $\{g^+ \hbox {: } g \in \Gamma {\setminus } \{1\}\}$ is dense in $\partial _\infty \Gamma$.

We now have the following result.

Lemma 2.6

If a representation $\rho : \Gamma \rightarrow \textrm{SL}(d, \mathbb R)$ is k-Anosov then there exist Hölder continuous maps

$$\begin{aligned} \xi ^{(k)}: \partial _\infty \Gamma \rightarrow \mathcal G_k( \mathbb R^d) \hbox { and } \theta ^{(k)}: \partial _\infty \Gamma \rightarrow \mathcal G_{d-k}( \mathbb R^{d}) \end{aligned}$$

such that for $\eta , \eta ' \in \partial \Gamma _\infty$:

• (1) for $\eta \ne \eta '$ we have that $\xi ^{(k)}(\eta ) \oplus \theta ^{(k)}(\eta ') = \mathbb R^d$ (transversality);

• (2) $\xi ^{(k)}(g\eta ) = \rho (g)\theta ^{(k)}(\eta )$ for all $g \in \Gamma {\setminus } \{1\}$ (equivariance); and

• (3) there exist $a>0$ and $c>0$ such that for every geodesic ray $(g_n)_{n=0}^\infty$ corresponding to $\eta \in \partial _\infty \Gamma$ we have that

  $$\begin{aligned} \frac{\sigma _k(\rho (g_n))}{\sigma _{k+1}(\rho (g_n))} \ge ce^{an}, \end{aligned}$$

  where $\sigma _i$ denotes the ith singular value (k-dominated).

Parts (1) and (2) are standard and part (3) may be found in [4].

In this paper, we will deal with representations which are 1-Anosov. These representations are also called projective Anosov. Noting that $\mathcal G_k( \mathbb R^d)$ is the projective space $\mathbb R P^{d-1}$, in this case we will only need the map $\xi : \partial _\infty \Gamma \rightarrow \mathbb R P^{d-1}$ corresponding to $\xi ^{(1)}$ in the lemma above. In this case, we have the following contraction property, which, though not part of the original definitions in [15, 22], 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 $(g_n)_{n=0}^\infty$ corresponding to $\eta \in \partial _\infty \Gamma$ we have that the associated action $\widehat{\rho }(g_n): \mathbb R P^{d-1} \rightarrow \mathbb R P^{d-1}$ satisfies $\Vert D_{\xi (\eta )}\widehat{\rho }(g_n)\Vert \le c e^{-n a}$, for $n \ge 0$.

Projective Anosov representations were originally introduced by Labourie [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 $\textrm{SL}(d,\mathbb R)$ induces a projective Anosov representation into $\textrm{SL}\left( \left( {\begin{matrix} d \\ k \end{matrix}}\right) ,\mathbb R \right)$.

(ii) There may be additional restrictions on Anosov representations of hyperbolic groups. For example, if $\Gamma$ is a torsion-free hyperbolic group which has an Anosov representation into $\textrm{SL}(3,\mathbb R)$ then $\Gamma$ 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 $\Gamma$ be the fundamental group of a compact oriented surface of genus $g \ge 2$ and let $\rho : \Gamma \rightarrow \textrm{SL}(d,\mathbb R)$ 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 $\rho : \Gamma \rightarrow \textrm{SL}(d, \mathbb R)$ be a projective Anosov representation. Then for each $g \in \Gamma \setminus \{1\}$ the matrix $\rho (g)$ is proximal.

We use this to associate a natural weight to each $g \in \Gamma {\setminus } \{1\}$. Let

$$\begin{aligned} d_{\rho }(g) = \log |\lambda (g)|, \end{aligned}$$

where $\lambda (g)$ is the maximal eigenvalue of the matrix $\rho (g)$. By Lemma 2.10, since $\rho (g) \in \textrm{SL}(d,\mathbb R)$, $|\lambda (g)| >1$ and hence $d_\rho (g) >0$. Note that $d_{\rho }(g)$ 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 $\rho : \Gamma \rightarrow \textrm{SL}(d, \mathbb R)$, we can associate a generalized Ruelle zeta function defined by

$$\begin{aligned} \zeta (s,\rho ) = \prod _{[g] \in \mathcal P} \left( 1-e^{-sd_\rho (g)}\right) ^{-1}, \end{aligned}$$

(1.2)

and a generalized Selberg zeta function defined by

$$\begin{aligned} Z(s, \rho ) = \prod _{n=0}^\infty \prod _{[g] \in \mathcal P} \left( 1- e^{-(s+n) d_{\rho }(g)} \right) , \end{aligned}$$

(1.3)

for $s \in \mathbb C$ wherever the products converge.

In particular, from (1.2) and (1.3) we have the identities

$$\begin{aligned} \zeta (s,\rho ) = \frac{Z(s+1,\rho )}{Z(s,\rho )}. \end{aligned}$$

(1.4)

Remark 2.12

Note that, in the special case $d=2$, with $\rho \in \mathcal T(V)$ we have that $d_\rho (g) = \ell _\rho (g)/2$. Therefore, $\zeta (s,\rho ) = R(s/2,\rho )$. However, the relationship between $Z(s,\rho )$ and $S(s,\rho )$ is more complicated with

$$\begin{aligned} Z(s,\rho ) = S(s/2,\rho ) S(s/2 + 1/2,\rho ). \end{aligned}$$

To describe the half-plane of convergence of these new zeta functions, we recall the following definition. See [5, 6] or [24] for a further discussion, including the existence of the limit.

Definition 2.13

The entropy $h(\rho )$ of a projective Anosov representation $\rho : \Gamma \rightarrow \textrm{SL}(d, \mathbb R)$ is defined to be the growth rate of the number of primitive conjugacy classes $[g] \in \mathcal P$ with $d_{\rho }(g)$ at most T as $T \rightarrow +\infty$, i.e.,

$$\begin{aligned} h(\rho ) = \lim _{T \rightarrow +\infty } \frac{1}{T} \log \left( \#\{[g] \in \mathcal P \hbox {: } d_{\rho }(g) \le T \}\right) . \end{aligned}$$

(1.5)

(The value of $h(\rho )$ 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 $Z(s, \rho )$ and $\zeta (s,\rho )$ converge to non-zero analytic functions provided $\textrm{Re}(s) > h({\rho })$. In the original setting, with $d=2$ and $\rho \in \mathcal T(V)$, the entropy is always equal to 2 and so does not need to be explicitly introduced. (We remark that, in this case, $h(\rho )=2$ is twice the topological entropy of the geodesic flow over $\mathbb H^2/\rho (\Gamma )$ because $d_\rho (g) = \ell _\rho (g)/2$, where $\ell _\rho (g)$ 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 $\zeta (s, \rho )$, since, with our approach, the proof of this naturally comes first.

Theorem 2.14

Let $\rho : \Gamma \rightarrow \textrm{SL}(d, \mathbb R)$ be a projective Anosov representation. Then $\zeta (s,\rho )$ extends to a meromorphic function in the entire complex plane and has a simple pole at $s=h(\rho )$. Furthermore, $\zeta (s,\rho )$ 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 $Z(s, \rho )$.

Theorem 2.15

Let $\rho : \Gamma \rightarrow \textrm{SL}(d, \mathbb R)$ be a projective Anosov representation. Then $Z(s,\rho )$ is a meromorphic function to the entire complex plane and has a simple zero at $s=h(\rho )$. Furthermore, $Z(s,\rho )$ may be written as the quotient of two entire functions of order at most $d+1$.

We will prove these results in Sect. 4. We will also consider the zeros of the zeta functions in Sect. 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.

$$\begin{aligned} \bar{d}_\rho (g) = \log \left| \lambda (g)/\lambda (g^{-1})\right| = \log \left( \lambda _d/\lambda _1\right) . \end{aligned}$$

This may be reduced to the situation we consider by composing $\rho$ with the Adjoint representation of $\textrm{SL}(d,\mathbb R)$ and so Theorems 2.14 and 2.15 will still apply. (We are grateful to the referee for this observation, which replaces our original more complicated argument.)

3 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.

Fig. 1

Here $\rho$ is a representation into $\textrm{SL}(3,\mathbb R)$ and $\Lambda \subset \mathbb R P^2$ is the associated limit set. For $g \in \Gamma \setminus \{1\}$, the point $\xi _g$ is an attracting fixed point for $\widehat{\rho }(g): \mathbb R P^2 \rightarrow : \mathbb R P^2$

3.1 Action on projective space

As discussed above, the representation $\rho : \Gamma \rightarrow \textrm{SL}(d,\mathbb R)$ induces an action of $\Gamma$ on the projective space $\mathbb R P^{d-1}$. More explicitly, for $g \in \Gamma$, we define $\widehat{\rho }(g): \mathbb R P^{d-1} \rightarrow \mathbb R P^{d-1}$ by

$$\begin{aligned} \widehat{\rho }(g)[v] = \frac{\rho (g)v}{\Vert \rho (g)v\Vert _2}, \end{aligned}$$

where $v \in \mathbb R^d \setminus \{0\}$ is a representative element.

Suppose $g \in \Gamma \setminus \{1\}$. Since, by Lemma 2.10, $\rho (g)$ has a real eigenvalue which is strictly maximal in modulus, the map $\widehat{\rho }(g)$ has a unique attracting fixed point; we will denote this by $\xi _g \in \mathbb R P^{d-1}$. Let $\xi : \partial _\infty \Gamma \rightarrow \mathbb R P^{d-1}$ be the map introduced in the previous section. As an immediate consequence of Lemma 2.7, for each $g \in \Gamma \setminus \{1\}$, we have $\xi (g^+) = \xi _g$, the fixed point for $\widehat{\rho }(g)$.

We now define a compact subset of projective space which will be useful in the sequel (Fig. 1).

Definition 3.1

We define the limit set

$$\begin{aligned} \Lambda := \overline{\{\xi _g \hbox {: } g \in \Gamma \setminus \{1\}\}} \subset \mathbb R P^{d-1} \end{aligned}$$

to be the closure of the fixed points.

Since $\{g^+ \hbox {: } g \in \Gamma \setminus \{1\}\}$ is dense in $\partial _\infty \Gamma$, immediately follows that $\Lambda = \xi (\partial _\infty \Gamma )$.

Example 3.2

(Case $d=3$) In this case the limit set $\Lambda \subset \mathbb R P^2$ is diffeomorphic to a circle bounding a strictly convex region $\Omega$ when $\rho$ a representation in the Hitchin component. In particular, in Lemma 2.6 we have that for $\eta \in \partial _\infty \Gamma$ the point $\xi ^{(1)}(\eta ) \in \Lambda$ and $\theta ^{(1)}(\eta )$ 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 $d_\rho (g)$ to the action of $\widehat{\rho }(g)$ on $\mathbb R P^{d-1}$.

Lemma 3.3

If $g \in \Gamma \setminus \{1\}$ then

$$\begin{aligned} d_\rho (g) = -\frac{1}{d} \log \det (D_{\xi _g} \widehat{\rho }(g)). \end{aligned}$$

Proof

We can consider the linear action of $\rho (g)$ on $\mathbb R^d$, then the fixed point corresponds to an eigenvector v and the result follows from a simple calculation using that the linear action of $\rho (g) \in \textrm{SL}(d, \mathbb R)$ preserves area in $\mathbb R^d$. More precisely, $\xi _g$ corresponds to an eigenvector v for the maximal eigenvalue $\lambda (g)$, with $|\lambda (g)| > 1$, for the matrix $\rho (g)$. We can assume without loss of generality that $\Vert v\Vert =1$ and then for arbitrarily small $\delta >0$ we can consider a $\delta$-neighbourhood of v which is the product of a $(d-1)$-dimensional neighbourhood in $\mathbb R P^{d-1}$ and a $\delta$-neighbourhood in the radial direction. The effect of the linear action of $\rho (g)$ is to replace v by $\lambda (g) v$, and thus stretch the neighbourhood in the radial direction by a factor of $|\lambda (g)|$. Since $\rho (g)$ has determinant one, the volume of the $(d-1)$-dimensional neighbourhood contracts by $|\lambda (g)|^{-1}$. To calculate the effect of the projective action $\widehat{\rho }(g)$, we need to rescale $\lambda (g) v$ to have norm one, which corresponds to multiplication by the diagonal matrix $\textrm{diag}(|\lambda (g)|^{-1},\ldots ,|\lambda (g)|^{-1})$. In particular, the $(d-1)$-dimensional neighbourhood in $\mathbb R P^{d-1}$ shrinks by a factor of approximately $|\lambda (g)|^{-d}$, giving the result. $\square$

3.2 Surface groups and coding

We now want to introduce a natural coding for the group $\Gamma$ which will allow us to analyse the projective action of $\Gamma$ on $\Lambda$. 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 [24] (for representations in a Hitchin component).

Since $\Gamma$ is the fundamental group of a compact surface V with genus $g \ge 2$, it has the presentation

$$\begin{aligned} \Gamma = \left\langle a_1,\ldots ,a_g, b_1,\ldots ,b_g \hbox {: } \prod _{i=1}^g[a_i, b_i] = 1\right\rangle . \end{aligned}$$

We write $\Gamma _0 =\{a_1^{\pm 1}, \cdots , a_g^{\pm 1}, b_1^{\pm 1}, \cdots , b_g^{\pm 1} \}$ for the symmetrized generating set and $|\cdot |: \Gamma \rightarrow \mathbb Z^+$ for the word length with respect to $\Gamma _0$.

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 $\Gamma _0$. 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 $(\Gamma ,\Gamma _0)$

• (i) a directed graph $\mathcal G = (\mathcal V,\mathcal E)$ with a distinguished vertex $*$; and

• (ii) an edge labelling $\omega : \mathcal E \rightarrow \Gamma _0$,

such that

• (1) no edge terminates at $*$;

• (2) there is at most one directed edge joining each ordered pair of vertices;

• (3) the map from the set finite paths in the graph starting at $*$ to $\Gamma \setminus \{1\}$ defined by

  $$\begin{aligned} (e_1, \ldots , e_n) \mapsto \omega (e_1)\cdots \omega (e_n) \end{aligned}$$

  is a bijection and $|\omega (e_1)\cdots \omega (e_n)|=n$;

• (4) the map from closed paths in $\mathcal G$ to conjugacy classes to $\Gamma$ induced by $\omega$ is a bijection and for such a closed path $(e_1, \ldots , e_n,e_1)$, n is the minimum word length in the conjugacy class of $\omega (e_1)\cdots \omega (e_n)$.

Furthermore, the subgraph obtained be deleting the vertex $*$ has the aperiodicity property that there exists $N \ge 1$ such that, given any two $v,v' \in \mathcal V \setminus \{*\}$, 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 $\Gamma$. In order to study other groups $\Gamma$ 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 $\Lambda$. This is particularly useful in introducing a partition of $\Lambda$ which gives an effective way to reduce the group action to the orbits of a single transformation.

We can associate to the directed graph $\mathcal G = (\mathcal V,\mathcal E)$ a subshift of finite type $\Sigma$, as follows. Let $\mathcal E^* \subset \mathcal E$ be the set of directed edges that start at $*$. The set of states in $\Sigma$ is $\mathcal E {\setminus } \mathcal E^*$ and a transition $e \rightarrow e'$ is allowed if and only if $e'$ follows e in $\mathcal G$. More precisely, if there are k edges in $\mathcal E \setminus \mathcal E^*$ then we define a $k \times k$ matrix A by $A(e,e') =1$ if $e'$ follows e in the directed graph and define

$$\begin{aligned} \Sigma = \{x = (x_n) \in \{1, \cdots , k\}^{\mathbb Z^+} \hbox {: } A(x_n, x_{n+1})=1, n \ge 0\}. \end{aligned}$$

(We note that this definition does not involve the edge labels on $\mathcal E$.) The set $\Sigma$ is a compact space with respect to the metric

$$\begin{aligned} d(x,y) = \sum _{n=0}^\infty \frac{1- \delta (x_n, y_n)}{2^n}. \end{aligned}$$

The shift map is the local homeomorphism $\sigma : \Sigma \rightarrow \Sigma$ defined by $(\sigma x)_n = x_{n+1}$. By the final statement in Lemma 3.4, A is aperiodic (i.e. there exists $N \ge 1$ such that $A^N$ has all entries positive) and, equivalently, the shift $\sigma : \Sigma \rightarrow \Sigma$ is mixing (i.e. for all open non-empty $U,V \subset \Sigma$, there exists $N \ge 1$ such that $\sigma ^{-n}(U) \cap V \ne \varnothing$ for all $n \ge N$).

There is a natural surjective Hölder continuous map $\pi : \Sigma \rightarrow \partial _\infty \Gamma$ defined by letting $\pi ((x_n)_{n=0}^\infty )$ be the equivalence classes of the geodesic ray $(g_n)_{n=0}^\infty$ in $\partial _\infty \Gamma$, where $g_n = \omega (x_0) \omega (x_1) \cdots \omega (x_n)$.

Definition 3.6

We define a map $\kappa : \Sigma \rightarrow \Lambda \subset \mathbb R P^{d-1}$ by $\kappa = \xi \circ \pi$.

However, the shift $\sigma : \Sigma \rightarrow \Sigma$ only encodes information about $\Gamma$ as an abstract group. In order to keep track of the additional information given by the representation of $\Gamma$ in $\textrm{SL}(d, \mathbb R)$ and its action on $\mathbb RP^{d-1}$ we need to introduce a Hölder continuous function $r: \Sigma \rightarrow \mathbb R$.

Definition 3.7

We can associate a map $r: \Sigma \rightarrow \mathbb R$ defined by

$$\begin{aligned} r(x) = -\frac{1}{d} \log \det (D_{\kappa (x)}\widehat{\rho }(g_{x_0})) \end{aligned}$$

(i.e. the Jacobian of the derivative of the projective action), where $g_{x_0} = \omega (x_0)$ is the generator corresponding to the first term in $x = (x_n)_{n=0}^\infty \in \Sigma$.

There is a natural one-to-one correspondence between non-trivial conjugacy classes in $\Gamma$ and periodic orbits for the shift $\sigma$, by part (4) of Lemma 3.4.

We now have the following simple but key result.

Lemma 3.8

The function $r: \Sigma \rightarrow \mathbb R$ is Hölder continuous, and if $\sigma ^nx =x$ is a periodic point corresponding to the conjugacy class [g] of an element $g \in \Gamma$ then $r^n(x) = d_\rho (g)$.

Proof

The Hölder continuity of r follows immediately from the Hölder continuity of $\xi$, which is given by Lemma 2.7. The second statement follows from the equivariance and the observation $\kappa (\sigma x)= \rho (g_{x_0}) \kappa (x)$. Moreover, that the periodic point x has an image $\kappa (x) (= \xi _g)$ which is fixed by $\widehat{\rho }(g)$ and the result follows from Lemma 3.3. $\square$

3.4 Semi-flows and entropy

In this subsection we give an alternative definition of the entropy $h(\rho )$ of a representation, as defined in the introduction, which is useful when taking the symbolic viewpoint.

We can associate to $\sigma : \Sigma \rightarrow \Sigma$ and the positive function $r: \Sigma \rightarrow \mathbb R^+$ the space

$$\begin{aligned} \Sigma ^r = \{(x, u) \in \Sigma \times \mathbb R \hbox {: } 0 \le u \le r(x)\}/\sim , \end{aligned}$$

where $(x,r(x)) \sim (\sigma x, 0)$, and the semi-flow $\sigma ^r_t: \Sigma ^r \rightarrow \Sigma ^r$ defined by $\sigma ^r_t(x,u) = (x, u+t)$, for $t \ge 0$ (respecting the identifications).

The topological entropy $h(\sigma ^r)$ 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 $h(\sigma ^r)$ is as the growth rate

$$\begin{aligned} h(\sigma ^r) = \lim _{T \rightarrow +\infty } \frac{1}{T} \log \#\{\tau \in \mathcal P(\sigma ^r) \hbox {: } l(\tau ) \le T\}, \end{aligned}$$

where $\mathcal P(\sigma ^r)$ denotes the set of prime periodic orbits of $\sigma ^r$ and $l(\tau )$ denotes the least period of the orbit $\tau$ [23].

Lemma 3.9

If $\rho$ is a projective Anosov representation then the entropy of the representation is equal to the topological entropy of the semi-flow $\sigma ^f$.

Proof

We observe that there is a one-one correspondence between the periodic orbits for the semi-flow $\sigma ^r$ and for the shift $\sigma$. By Lemma 3.8 and Definition 2.13, we see that $h(\rho ) =h(\sigma ^r)$. $\square$

4 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 $Z(s, \rho )$ in Theorem 2.15.

4.1 Partitions

By the construction in the previous section we have associated to the representation $\rho$ a Hölder continuous map $\kappa : \Sigma \rightarrow \mathbb R P^{d-1}$ which intertwines the shift map on $\Sigma$ with the projective action, i.e. $\kappa (\sigma x) = \widehat{\rho }(\omega (x_0)) \kappa (x)$. In particular, we use $\kappa$ to define a particular partition $P_i = \kappa ([i])$ of the limit set $\Lambda \subset \mathbb RP^{d-1}$, where

$$\begin{aligned}{}[i] = \{x \in \Sigma \hbox {: } x_0=i\}, \quad 1 \le i \le k. \end{aligned}$$

We then have that

• (1) $\Lambda = \bigcup _{i=1}^k P_i$, and

• (2) $P_i \cap P_j$ consists of at most one point, for distinct $i \ne j$.

The analytic extension of the zeta function $\zeta (s,\rho )$ is based on the study of a transfer operator defined on analytic functions. In particular, we can realise the symbolic maps $\Sigma \supset [j] \ni x \mapsto ix \in [i] \subset \Sigma$ (where $A(i,j)=1$) as local real analytic contractions

$$\begin{aligned} \psi _i: \coprod _{A(i,j)=1} P_j \rightarrow P_i \end{aligned}$$

on the disjoint union of the sets (Fig. 2).

Fig. 2

The contractions $\psi _i: P_j \rightarrow P_i$ satisfy $\psi _i(U_j) \subset U_i$ for the neighbourhoods of the complexifications (suggested in the figure)

4.2 Transfer operators

We can consider a Banach space of analytic functions on a neighbourhood of the disjoint union of the partition $\coprod _{i=1}^k P_i$. In particular, let us fix neighbourhoods $U_i \supset P_i$ in the complexification of $\mathbb R P^{d-1}$, 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 $f: \coprod _{i=1}^k U_i \rightarrow \mathbb C$ with the supremum norm

$$\begin{aligned} \Vert f \Vert = \max _i \sup _{z\in U_i} |f(z)|, \end{aligned}$$

i.e., $B_0 = \left\{ f: \coprod _{i=1}^k U_i \rightarrow \mathbb C \ \textrm{ analytic}\hbox {: } \Vert f\Vert < \infty \right\}$.

We can also define a natural family of bounded linear operators on this space.

Definition 4.2

For $s \in \mathbb C$, the transfer operator $\mathcal L_s: B_0 \rightarrow B_0$ is defined by

$$\begin{aligned} \mathcal L_s f(z) = \sum _{A(i,j)=1} \det (D\psi _i(z))^s f(\psi _iz), \hbox { for } z \in U_j, \end{aligned}$$

where $\det (D\psi _i (\cdot ))^s$ 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 $A(i,j)=1$ ($1 \le i,j \le k$) then $\overline{\psi _i(U_j)} \subset U_i$. Since, by construction, $\psi (P_j) \subset P_i$, it suffices to show that for any $\eta \in P_i$ we have $\Vert D_\eta \psi \Vert < 1$. However, associating to $\eta$ those geodesics $(\gamma )$ 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 $n\ge 2$ sufficiently large so that $ae^{-cn}$ and use the standard recoding of $\Sigma$ 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 $\mathcal L_s: B_0 \rightarrow B_0$ are nuclear. More precisely, we can find

• (1) $v_m \in B_0$ with $\Vert v_m\Vert _{B_0}=1$,

• (2) $l_m \in B_0^*$ with $\Vert l_m\Vert _{B_0^*}=1$, and

• (3) $\lambda _m \in \mathbb C$ with $|\lambda _m| = O(e^{-cm})$, for some $c>0$,

such that

$$\begin{aligned} \mathcal L_s f= \sum _{m=1}^\infty \lambda _m l_m(f) v_m. \end{aligned}$$

Proof

This can be deduced from [28]. $\square$

We can also consider the natural generalisation of these operators to the space of analytic j-forms on $\coprod _{i=1}^k U_i$ (for $j=1,\ldots , d-1$). More precisely, let $B_j$ denote the Banach space of analytic j-forms on $\coprod _{i=1}^k U_i$ whose coefficients are bounded. Then the definition of $\mathcal L_s$ naturally extends to define transfer operators $\mathcal L_{s,j}: B_j \rightarrow B_j$. The analogue of Lemma 4.3 holds for these operators.

4.3 Traces and determinants

A particular consequence of Lemma 4.3 is that the operator $\mathcal L_s$ (and its powers $\mathcal L_s^n$) are trace class, i.e. the eigenvalues are summable: $\sum _{m=1}^\infty |\lambda _m| < \infty$.

To calculate the trace of $\mathcal L_s^n$, we need to consider compositions of the contractions $\psi _i$ corresponding to periodic points in $\Sigma$. Let $\underline{i} = (i_1,\ldots ,i_n)$ be a string of length $|\underline{i}|=n$ an allowed string such that $A(i_1, i_2) = \cdots = A(i_{n-1}, i_n)=1$. If, in addition, $A(i_n,i_1) = 1$ then we call $\underline{i}$ a periodic allowed string. We let

$$\begin{aligned} \psi _{\underline{i}}: \coprod _{A(i_n,j)=1} P_j \rightarrow P_{i_1} \end{aligned}$$

denote the composition $\psi _{\underline{i}} = \psi _{i_1} \circ \cdots \circ \psi _{i_n}$. Since $A(i_n,i_1)=1$, this restricts to a contraction $\psi _{\underline{i}}: P_{i_1} \rightarrow P_{i_1}$, which has a unique fixed point denoted $z_{\underline{i}}$. We now have the following result.

Lemma 4.4

The trace of $\mathcal L_s^n: B_0 \rightarrow B_0$ is given by

$$\begin{aligned} \hbox { tr}(\mathcal L_s^n) = \sum _{|\underline{i}|=n} \frac{\det (D\psi _{\underline{i}}(z_{\underline{i}}))^s}{ \det (I - D\psi _{\underline{i}}(z_{\underline{i}}))}, \end{aligned}$$

where the summation is taken over all periodic allowed strings of length n. Furthermore,

$$\begin{aligned} \hbox { tr}(\mathcal L_{s}^n) + \sum _{j=1}^d (-1)^j \hbox { tr}(\mathcal L_{s,j}^n) = \sum _{|\underline{i}|=n} \det (D\psi _{\underline{i}}(z_{\underline{i}}))^s. \end{aligned}$$

(4.1)

Proof

This can be deduced from the general setting in [28, pages 232–234] . $\square$

We want to associate to the transfer operators a function of a single complex variable s. We can write

$$\begin{aligned} \det (I - \mathcal L_s) = \exp \left( - \sum _{n=1}^\infty \frac{1}{n} \hbox { tr}(\mathcal L_s^n) \right) . \end{aligned}$$

Lemma 4.4 explains the connection with the fixed points of the contractions. Similarly, we can associate to the transfer operators $\mathcal L_{s,j}$ the functions

$$\begin{aligned} \det (I - \mathcal L_{s,j}) = \exp \left( - \sum _{n=1}^\infty \frac{1}{n} \hbox { tr}(\mathcal L_{s,j}^n) \right) \end{aligned}$$

for $1 \le j \le d-1$.

The next lemma shows that these functions extend analytically to the entire complex plane. To simplify notation, we shall write $\mathcal L_{s,0}=\mathcal L_{s}$.

Lemma 4.5

The functions $s \mapsto \det (I - \mathcal L_{s,j})$ ( $0 \le j \le d-1$) extend to entire analytic functions of $s\in \mathbb C$ 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

$$\begin{aligned} \det (I - \mathcal L_{s,j}) = 1 + \sum _{m=1}^\infty c_m(s) \end{aligned}$$

where

$$\begin{aligned} c_m(s) = O \left( e^{|s| Dm} e^{ -Em^{\left( 1+ \frac{1}{d-1} \right) } } \right) , \end{aligned}$$

for some $E>0$ and $D = \max _{j}\Vert D\phi _j\Vert$. In particular, by considering critical points of the real function $t\mapsto |s| D t -E t^{\left( 1+ \frac{1}{d-1} \right) }$ we see that the upper bound on the terms occurs at

$$\begin{aligned} m = \left[ \left( \frac{|s| D}{E(1 + \frac{1}{d-1})} \right) ^{d-1} \right] \hbox { or } \left[ \left( \frac{|s| D}{E(1 + \frac{1}{d-1})} \right) ^{d-1} \right] + 1. \end{aligned}$$

giving an upper bound on the terms of $O\left( \exp \left( |s|^d D^d/(E (1 + \frac{1}{d-1}))^{d-1} \right) \right)$. Moreover, it is easy to see that this gives an upper bound

$$\begin{aligned} |\det (I - \mathcal L_{s,j})| = O\left( e^{c|s|^{d}}\right) , \end{aligned}$$

for some $c>0$. In particular, we see that the order is given by

$$\begin{aligned} \limsup _{R \rightarrow +\infty } \frac{\log \left( \max _{|s| \le R} \log |\det (I-\mathcal L_{s,j})| \right) }{\log R} \le d. \end{aligned}$$

$\square$

Proof of Theorem 2.14

To complete the proof of Theorem 2.14, we can then use the identity (4.1) to write

$$\begin{aligned} \zeta (s, \rho ) = \frac{\prod _{j \textrm{odd} }\det (I- \mathcal L_{s,j})}{ \prod _{j \textrm{even}} \det (I- \mathcal L_{s,j}) }, \end{aligned}$$

from which we deduce that $\zeta (s, \rho )$ is a meromophic function which, by Lemma 4.5, may be written as the quotient of two entire functions of order $\le d$.

Proof of Theorem 2.15

Given $s_0 \in \mathbb C$, we can choose $m \in \mathbb Z$ with $\textrm{Re}(s_0) \ge m$. Setting $n = |m| + [h(\rho )] + 1$ we can use (1.4) to write

$$\begin{aligned} Z(s, \rho ) = \frac{Z(s+n, \rho )}{\zeta (s, \rho ) \zeta (s+1, \rho ) \cdots \zeta (s+n-1, \rho )}. \end{aligned}$$

(4.2)

For s in a sufficiently small neighbourhood of $s_0$ we have that $\textrm{Re}(s+n) > h(\rho )$ and therefore $Z(s+n, \rho )$ converges to an analytic function. Moreover, the denominator is meromorphic by Theorem 2.14 and thus using (4.2) we deduce that $Z(s, \rho )$ is meromorphic.

To obtain the order bound in Theorem 2.15 consider $|s| < m$, say, and set $n= m + [h(\rho )]+2$. By the proofs of Theorem 2.14 and Lemma 4.5 we can write $\zeta (s, \rho )= \frac{f(s)}{g(s)}$, where for any $\epsilon >0$ we can choose $C > 0$ such that $|f(s)|, |g(s)| \le \exp (C |s|^{d+\epsilon })$ and thus using (4.2) we can write

$$\begin{aligned} Z(s, \rho )= Z(s+n, \rho ) \frac{G(s)}{F(s)} \end{aligned}$$

(4.3)

where $F(s) = \prod _{k=0}^{n-1}f(s+k)$ and $G(s) = \prod _{k=0}^{n-1}g(s+k)$ are entire functions. In particular, $Z(s+n, \rho )$ is uniformly bounded and since

$$\begin{aligned} \log |F(s)| \le C \sum _{k=0}^{m+1} |s+k|^{d+\epsilon } \le C (m+2) (2m)^{d+\epsilon } \end{aligned}$$

the order of F(s) is bounded by

$$\begin{aligned} \lim _{m\rightarrow +\infty }\frac{\log (C (m+2) (2m)^{d+\epsilon })}{\log m} = d+1+ \epsilon , \end{aligned}$$

and similarly the order for G(s) is bounded by $d+1 +\epsilon$. Finally, since $\epsilon > 0$ can be arbitrarily small, we can deduce from (4.3) that $Z(s, \rho )$ is a ratio of two entire functions of order $d+1$.

Remark 4.6

(Equidistribution) For each $t > \delta$ can define a probability measure on $\Lambda$ by

$$\begin{aligned} \mu _t = \frac{ \sum _{g \in \Gamma \setminus \{1\}} |\hbox { Jac} D_{\xi _g} \widehat{\rho }(g)|^t \delta _{\xi _g} }{ \sum _{g \in \Gamma \setminus \{1\}} |\hbox { Jac} D_{\xi _g} \widehat{\rho }(g)|^t } \end{aligned}$$

and then as t tends to $\delta$ this converges to a probability measure $\mu$ supported on $\Lambda$. This follows easily from the properties of the transfer operator $\mathcal L_s$, where $\mu$ is related to the maximal eigenprojection of the operator $\mathcal L_{h(\rho )}$.

5 Zeros and poles of the zeta functions

Having established that the zeta functions $\zeta (s, \rho )$ and $Z(s, \rho )$ have meromorphic extensions to $\mathbb C$ 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.

Proposition 5.1

([16]) There exists $\epsilon > 0$ such that $S(s,\rho )$ has no zeros in $\textrm{Re}(s) > 1 - \epsilon$ other than $s=1$.

In this classical case it is even known that the zeros in the critical strip $0 \le \textrm{Re}(s) \le 1$ lie on $[0,1] \cup \left( \frac{1}{2}+i \mathbb R\right)$. 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 $\lambda _n = s_n(1-s_n)$ of the Laplace-Beltrami operator.

Using the relation $R(s, \rho ) = S(s+1, \rho )/S(s,\rho )$ we can deduce the corresponding result for $R(s,\rho )$.

Proposition 5.2

There exists $\epsilon > 0$ such that $R(s,\rho )$ has no zeros or poles in $\textrm{Re}(s) > 1 - \epsilon$ other than a simple pole $s=1$.

We can generalize both Propositions 5.1 and 5.2 to the zeta functions for a special class of projective Anosov representations. We say that $\rho : \Gamma \rightarrow \textrm{SL}(d,\mathbb R)$ is (1, 2)-Anosov if it is 1-Anosov and 2-Anosov. A (1, 2)-Anosov representation is (1, 1, 2)-hyperconvex if for every triple of pairwise distinct points $x,y,z \in \partial _\infty \Gamma$ we have

$$\begin{aligned} \xi ^1(x) \oplus \xi ^1(y) \oplus \xi ^{d-2}(z) = \mathbb R^d. \end{aligned}$$

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 $\rho$ is a (1, 1, 2)-hyperconvex representation then $\xi ^1(\partial _\infty \Gamma )$ is a $C^1$ curve. Indeed, Zhang and Zimmer prove stronger regularity, which we state this as a lemma.

Lemma 5.3

(Zhang and Zimmer [34]) If $\rho : \Gamma \rightarrow \textrm{SL}(d,\mathbb R)$ is (1, 1, 2)-hyperconvex then the limit set $\Lambda = \xi ^1(\partial _\infty \Gamma )$ is a $C^{1+\alpha }$ curve, for some $\alpha >0$.

Another ingredient is the following “mixing condition” on the weights $d_\rho (g)$.

Lemma 5.4

(Carvajales [8, Lemma A.2] ) Let $\rho$ be a projective Anosov representation. Then there does not exist $a > 0$ such that $\{d_\rho (g) \hbox {: } g \in \Gamma {\setminus } \{1\}\} \subset a \mathbb Z$.

We will now obtain our results on the zeta functions. First we will consider the Ruelle-type zeta function $\zeta (s, \rho )$.

Theorem 5.5

Let $\rho : \Gamma \rightarrow \textrm{SL}(d,\mathbb R)$ be a (1, 2)-Anosov representation which is (1, 1, 2)-hyperconvex. Then there exists $\epsilon > 0$ such that $\zeta (s,\rho )$ has no zero or poles in $\textrm{Re}(s) > h(\rho ) - \epsilon$ other than $h(\rho )$. Moreover, there exists $\alpha > 0$ such that we can bound $\log |\zeta (s,\rho )| = O(|\textrm{Im}(s)|^\alpha )$ for s satisfying $h(\rho ) - \epsilon< \textrm{Re}(s) < h(\rho )$ and $|\textrm{Im}(s)| \ge 1$.

Proof

That $\zeta (s,\rho )$ has a non-zero meromorphic extension to a half-plane $\textrm{Re}(s) > h(\rho ) -\epsilon$, for some $\epsilon >0$, 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 $\Lambda \subset \mathbb R P^{d-1}$ is a $C^1$ curve [26, 34]. This method gives bounds on the iterates of the transfer operators $\mathcal L_s$ regarded as operators $\mathcal L_s: C^1\left( \coprod _{i=1}^k P_i, \mathbb C\right) \rightarrow C^1\left( \coprod _{i=1}^k P_i, \mathbb C\right)$. It is convenient to use the norm

$$\begin{aligned} \Vert h\Vert _{1,t} = {\left\{ \begin{array}{ll} \max \left\{ \Vert h\Vert _\infty , \frac{\Vert h'\Vert _\infty }{|t|} \right\} &{}\hbox { if } |t| \ge 1\\ \max \left\{ \Vert h\Vert _\infty , \Vert h'\Vert _\infty \right\} &{}\hbox { if } |t| < 1 \end{array}\right. } \end{aligned}$$

on $C^1\left( \coprod _{i=1}^k P_i, \mathbb C\right)$. Following Proposition 7.4 of [2], the key ingredients required for the proof are that:

• (i) $r: \coprod _{i=1}^k P_i \rightarrow \mathbb R$ is a $C^1$ function; and

• (ii) $r: \coprod _{i=1}^k P_i \rightarrow \mathbb R$ is not cohomologous to a constant, i.e. there is no $C^1$ function $u: \coprod _i P_i \rightarrow \mathbb R$ and constant $c \in \mathbb R$ such that $r = u \circ T - u + c$.

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 [25], to show that there exist constants $\sigma _0 < h$, $C > 0$ and $0< \beta < 1$ such that whenever $s= \sigma + i t$ and $n = p [\log |t|] + l$, where $p \ge 0$ and $0 \le l \le [\log |t|]-1$, then

$$\begin{aligned} \Vert \mathcal L_{-sr}^n\Vert _{1,t} \le C \beta ^{p[\log |t|]} e^{l P(-\sigma r)} \end{aligned}$$

with respect to the norm $\Vert h\Vert _{1,t}$.

By Lemma 2 of [25] (which, apart from the dependence on |t| in the bound, appears in [29]) we have the estimate that for any $x_j \in P_j$, $j=1, \ldots , k$, and any $\beta _0$ satisfying $\max \{\beta , \max _j\Vert D\psi _j\Vert _\infty \}< \beta _0 < 1$, there exists $C > 0$ such that

$$\begin{aligned} \left| \sum _{|\underline{i}| =n} \det (D\psi _{\underline{i}}(z_{\underline{i}}))^s - \sum _{j=1}^k \mathcal {L}_s \chi _{P_j}(x_j) \right| \le C |t| n \beta _0^n, \quad \forall n \ge 1, \end{aligned}$$

where $\chi _{P_j}$ is the indicator function for $P_j$ and $\underline{i}$, $\psi _{\underline{i}}$ and $z_{\underline{i}}$ 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 $\log |\zeta (s, \rho )|= O\left( |t|^\alpha \right) ,$ for some $\alpha > 0$, as in (2.3) of [25]. $\square$

In light of (1.4), and the observation that $Z(s+1, \rho )$ is uniformly bounded on the half-plane $\textrm{Re}(s) > h(\rho ) - \epsilon$, provided $\epsilon < 1$, this gives the following generalization of Proposition 5.1.

Theorem 5.6

Let $\rho : \Gamma \rightarrow \textrm{SL}(d,\mathbb R)$ be a (1, 2)-Anosov representation which is (1, 1, 2)-hyperconvex. Then there exists $\epsilon > 0$ such that $Z(s,\rho )$ has no zero or poles in $\textrm{Re}(s) > h(\rho ) - \epsilon$ other than the zero at $s= h(\rho )$. Moreover, there exists $\alpha > 0$ such that we can bound $\log |Z(s,\rho )| = O(|\textrm{Im}(s)|^\alpha )$ for s satisfying $h(\rho ) - \epsilon< \textrm{Re}(s) < h(\rho )$ and $|\textrm{Im}(s)| \ge 1$.

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.

Definition 5.7

For $T>0$, we define

$$\begin{aligned} \pi _\rho (T) = \#\{[g] \in \mathcal P \hbox {: } d_\rho ([g]) \le T \}, \end{aligned}$$

the number of primitive conjugacy classes with weight at most T.

It was shown in [30] (following the approach in Chapter 6 of [23]) that $\pi _\rho (T)$ satisfies the asymptotic formula

$$\begin{aligned} \pi _\rho (T) \sim \frac{e^{h(\rho ) T}}{h(\rho )T}, \quad \hbox { as } T \rightarrow +\infty . \end{aligned}$$

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

$$\begin{aligned} \textrm{li}(x):= \int _2^x \frac{1}{\log u} du \sim \frac{x}{\log x}, \quad \textrm{as}\ x \rightarrow +\infty . \end{aligned}$$

We then have the following result.

Theorem 5.8

There exists $\epsilon > 0$ such that

$$\begin{aligned} \pi _\rho (T) = \textrm{li}(e^{h(\rho )T}) \left( 1 + O(e^{-\epsilon T})\right) . \end{aligned}$$

Proof

Given Theorem 5.5, we can apply the classical proof from number theory (see Proposition 6 and Proposition 7 of [25]). $\square$

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.

6 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 $U: \Gamma \rightarrow \textrm{U}(N)$. More precisely, we make the following definition.

Definition 6.1

We can associate to each projective Anosov representation $\rho : \Gamma \rightarrow \textrm{SL}(d, R)$ and to each unitary representation $R_\chi : \Gamma \rightarrow \textrm{U}(N)$ (with character $\chi =\textrm{Trace}(R_\chi )$), a Selberg type L-function formally defined by

$$\begin{aligned} Z(s,\rho ,\chi ) = \prod _{n=0}^\infty \prod _{[g] \in \mathcal P} \det \left( 1- e^{-(s+n) d_{\rho }(g)} R_\chi ([g]) \right) , \quad s \in \mathbb C, \end{aligned}$$

and a Ruelle type L-function formally defined by

$$\begin{aligned} \zeta (s,\rho ,\chi ) = \prod _{[g] \in \mathcal P} \det \left( 1- e^{-s d_{\rho }(g)} R_\chi ([g]) \right) ^{-1}, \quad s \in \mathbb C, \end{aligned}$$

where the products converge.

If $R_{\chi _0}$ is the trivial representation then we see that these L-functions reduce to the corresponding zeta functions, i.e. $Z(s, \rho , \chi _0) = Z(s,\rho )$ and $\zeta (s,\rho ,\chi _0) = \zeta (s,\rho )$. In the particular case of Fuchsian representations then these definitions reduce to the familiar definitions of L-functions for Fuchsian groups [12].

It is easy to see that $Z(s,\rho ,\chi )$ and $\zeta (s,\rho ,\chi )$ converge for $\textrm{Re}(s) > h({\rho })$. We can also observe directly from the definitions that

$$\begin{aligned} \zeta (s, \rho ,\chi ) = \frac{Z(s+1, \rho , \chi )}{Z(s, \rho , \chi )}. \end{aligned}$$

(5.1)

The following result generalizes Theorem 2.14 and has a similar proof.

Theorem 6.2

Let $\rho : \Gamma \rightarrow \textrm{SL}(d, \mathbb R)$ be a projective Anosov representation and let $R_\chi : \Gamma \rightarrow \textrm{U}(N)$ be a unitary representation. The Ruelle-type L-function $\zeta (s,\rho ,\chi )$ has the following properties.

• (1) $\zeta (s,\rho ,\chi )$ has a meromorphic extension to the entire complex plane $\mathbb C$; and

• (2) $\zeta (s,\rho ,\chi )$ has a simple pole at $s=h(\rho )$ if and only if $R_\chi$ is trivial.

The following result generalizes Theorem 2.15 and has a similar proof.

Theorem 6.3

Let $\rho : \Gamma \rightarrow \textrm{SL}(d, \mathbb R)$ be a projective Anosov representation and let $R_\chi : \Gamma \rightarrow \textrm{U}(N)$ be a unitary representation. The generalized L-function $Z(s, \rho , \chi )$ has the following properties.

• (1) $Z(s,\rho ,\chi )$ has a meromorphic extension to the entire complex plane $\mathbb C$; and

• (2) $Z(s,\rho ,\chi )$ has a simple zero at $s=h(\rho )$ if and only if $R_\chi$ is trivial.