Amenability, Critical Exponents of Subgroups and Growth of Closed Geodesics

Let $\Gamma$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $\Gamma_0$ has critical exponent equal to the critical exponent of $\Gamma$ if and only if $\Gamma / \Gamma_0$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $X / \Gamma$. These statements are analogues of classical results of Kesten for random walks on groups and of Brooks for the spectrum of the Laplacian on covers of Riemannian manifolds.


Introduction
Let X be a connected simply connected and complete Riemannian manifold with sectional curvatures bounded between two negative constants. We call such an X a pinched Hadamard manifold. Let Γ be a non-elementary and convex co-compact group of isometries of X (see section 2 for precise definitions) and let Γ 0 ⊳ Γ be a normal subgroup. We write G = Γ/Γ 0 . We define the critical exponent of Γ to be the abscissa of convergence of the Dirichlet series g∈Γ e −sdX (o,go) , for any choice of base point o ∈ X, and denote it by δ(Γ). The critical exponent δ(Γ 0 ) of Γ 0 is defined in the same way. The fact that Γ is non-elementary and convex co-compact means that δ(Γ) > 0 and it is clear that δ(Γ 0 ) ≤ δ(Γ). It is natural to ask when we have equality and our main result will give a precise answer to this question, which will depend only on G as an abstract group. (We will discuss the history of this and related problems in the next section.) Before stating our result, we introduce a related problem. Consider the quotient manifolds M = X/Γ and M = X/Γ 0 ; M is a regular cover of M with covering group G. Then M has a countably infinite set C(M ) of prime closed geodesics. For γ ∈ C(M ), we write l(γ) for its length.  [17,18].) If G is infinite then there are infinitely many closed geodesics on M with length a most a given T (since a single closed geodesic has infinitely many images under the action of G). However, we can obtain a finite quantity by considering the set Again we may ask when we have equality.
We remark that, while it is well known that h(M ) = δ(Γ), and both are equal to the topological entropy of the geodesic flow over X/Γ, it is not a priori clear whether or not h( M ) is equal to δ(Γ 0 ).
Our main result is the following.
Theorem 1.1. Let Γ be a convex co-compact group of isometries of a pinched Hadamard manifold X and let Γ 0 be a normal subgroup of Γ. Then the following are equivalent: This follows from Theorems 7.5 and 8.1 below, together with the following theorem of Roblin [22] (see also the expository account in [23]). Theorem 1.2 (Roblin [22]). Let Γ be a convex co-compact group of isometries of a pinched Hadamard manifold X and let Γ 0 be a normal subgroup of Γ. If Γ/Γ 0 is amenable then δ(Γ 0 ) = δ(Γ).
Actually, Roblin's Theorem applies it the more general situation where X is a CAT(−1) space. Remark 1.3. In the special case where X = H d+1 and δ(Γ) > d/2, the statement δ(Γ 0 ) = δ(Γ) if and only if G is amenable is a result of Brooks [3] (and holds when Γ is geometrically finite, which is a more general condition than convex cocompactness). We discuss this in more detail in the next section. If X = H d+1 and Γ is essentially free then Stadlbauer showed the same result holds without the assumption δ(Γ) > d/2 [26]. The class of essentially free groups includes all nonco-compact geometrically finite Fuchsian groups (i.e. discrete groups of isometries of H 2 ) and all Schottky groups.
We conclude the introduction by outlining the contents of the paper. In the next section, we define the key concepts associated to groups that are mentioned above and discuss some history of this and analogous problems. Our approach to Theorem 1.1 is via dynamics. More precisely, we consider the geodesic flow over M and M and a class of symbolic dynamical systems that model them. These symbolic systems belong to a class called countable state Markov shifts: we introduce these in section 3 and define a key quantity called the Gurevič pressure. We also mention recent results of Stadlbauer and Jaerisch that will be key to our analysis. In sections 4, 5 and 6, we consider the geodesic flows over M and M and discuss how they may be modeled by symbolic systems, particular using a skew product construction to record information about lifts to the cover. In section 7, we link various zeta functions defined by the closed geodesics to the Gurevič pressure and hence, using Stadlbauer's result, prove that the equality of h(M ) and h( M ) is equivalent to amenability of the covering group. Finally, in section 8, we obtain the corresponding result for critical exponents, using Jaerisch's result and some approximation arguments.

Background and History
Let X be a pinched Hadamard manifold, i.e. a connected simply connected complete Riemannian manifold such that its sectional curvatures lie in a interval [−κ 1 , −κ 2 ], for some κ 1 > κ 2 > 0. Associated to X is a well defined topological space ∂X called the Gromov boundary. This can be defined to be the set of equivalence classes of geodesic rays emanating for a fixed base point, where two rays are equivalent it their distance apart is bounded above. Let Γ be a group of isometries acting freely and properly discontinuously on X. We say that Γ is nonelementary if it is not a finite extension of a cyclic group. Fix o ∈ X. Then the orbit Γo = {go : g ∈ Γ} accumulates only on ∂X and we call the set of accumulation points L Γ the limit set of Γ; this is independent of the choice of o. Let C(Γ) denote the intersection of X with the convex hull (with respect to the metric on X) of L Γ . We say that Γ is convex co-compact if C(Γ)/Γ is compact.
A (countable) group G is amenable if ℓ ∞ (G, R) admits an invariant mean, i.e. that there exists a bounded linear functional ν : ℓ ∞ (G, R) → R such that, for all f ∈ ℓ ∞ (G, R), The concept was introduced by von Neumann in 1929. One sees immediately from this definition that finite groups are amenable by taking An alternative criterion for amenability was given by Følner [7]: G is amenable if and only if, for every ǫ > 0 and every finite set {g 1 . . . , g n } ⊂ G, there exists a finite set F ⊂ G such that #(F ∩ g i F ) ≥ (1 − ǫ)#F , i = 1, . . . , n. Using this criterion, it is easy to see that abelian groups are amenable and that, more generally, groups with subexponential growth are amenable [9]. Furthermore, since amenability is closed under extensions, solvable groups are amenable. In particular, there are amenable groups with exponential growth (e.g. lamplighter groups). On the other hand, a group containing the free group on two generators is not amenable and non-elementary Gromov hyperbolic groups (a class which includes the convex co-compact groups above) are not amenable.
There are numerous results that connect growth and spectral properties of groups and manifolds to amenability. The prototype is the following theorem of Kesten from 1959. Let G be a countable group and let p : G → R + be a symmetric probability distribution (i.e. g∈G p(g) = 1 and p(g −1 ) = p(g) for all g ∈ G) such that its support, supp(p), generates G. This defines a symmetric random walk on G with transition probabilities P (g, g ′ ) = p(g −1 g ′ ). If we define λ(G, P ) to be the ℓ 2 (G)-spectral radius of P then we have λ(G, P ) = lim sup n→∞ P n (g, g) 1/n = lim n→∞ P 2n (g, g) 1/2n , for any g ∈ G. It is clear that λ(G, P ) ≤ 1.
Note that, while λ(G, P ) depends on the probability p, whether or not it takes the value 1 depends only on the group G.
Subsequently, results inspired by Kesten's Theorem were obtained in a variety of other situations. In the setting of group theory, the most notable result is Grigorchuk's co-growth criterion [8] (see also Cohen [4]) for finitely generated groups. Recall that a finitely generated group G may be written as F/N , where F is a free group of rank k and N is a normal subgroup. If | · | denotes the word length on F with respect to a free generating set then lim n→∞ (#{x ∈ F : |x| = n}) 1/n = 2k −1. Subsequently, various extensions of this to graphs and (non-backtracking) random walks were obtained by Woess [27], Northshield [14,15] and Ortner and Woess [16].
In the setting of Riemannian manifolds, an analogue is provided by the following spectral result of Brooks. Let M be a complete Riemannian manifold which is of "finite topological type", i.e. that it is topologically the union of finitely many simplices, and let M be a regular covering of M with covering group G. Let λ 0 (M ) and λ 0 ( M ) denote the infimum of the spectrum of the Laplace-Beltrami operator on M and M , respectively; then λ 0 ( M ) ≥ λ 0 (M ). Brooks show that amenability of G implied equality and that, together with an additional condition, the converse holds. More precisely, he proved the following.
where the infimum is taken over co-dimension 1 submanifolds S that divide F \ K into an interior and an exterior. If λ 0 ( M ) = λ 0 (M ) the G is amenable.
The Cheeger-type condition in part (ii) holds if, for example, M is a convex co-compact quotient of the (d + 1)-dimensional real hyperbolic space H d+1 and λ 0 (M ) < d 2 /4.
More recently, Stadlabauer [26] and Jaerisch [10] have considered the relation between amenability and certain growth rates that occur in the study of skew product extensions of dynamical systems. It will be clear below that we are greatly indebted to their work in our analysis.

Countable State Markov Shifts and Gurevič Pressure
In this section we will define countable state Markov shifts and discuss some of their properties. Basic definitions and results are taken from chapter 7 of [12]. In the rest of the paper, we shall be concerned with finite state shifts and skew product extensions of these by a countable group, so we shall often specialise to these two cases.
Let S be a countable set, called the alphabet, and let A be a matrix, called the transition matrix, indexed by S × S with entries zero or one. We then define the space with the product topology induced by the discrete topology on S. This topology is compatible with the metric d(x, y) = 2 −n(x,y) , where n(x, y) = inf{n : x n = y n }, with n(x, y) = ∞ if x = y. If S is finite then Σ + A is compact. We say that A is locally finite if all its row and column sums are finite. Then Σ + A is locally compact if and only if A is locally finite. (The skew product extensions we consider have this latter property.) We define the (one-sided) countable state topological Markov shift σ : . This is a continuous map. We will say that σ is topologically transitive if it has a dense orbit and topologically mixing if, given non-empty open For A irreducible, set p ≥ 1 to be the greatest common divisor of periods of periodic orbits σ : Σ + A → Σ + A ; this p is called the period of A. We say that A is aperiodic if p = 1 or, equivalently, if there exists n ≥ 1 such that A n has all entries positive. Suppose that A is locally finite.
A is topologically mixing if and only if A is aperiodic. Suppose that A is irreducible but not aperiodic and fix i ∈ S. Then we may partition S into sets S l , l = 0, . . . , p − 1, defined by (This partition is independent of the choice of i.) For each l, let A l denote the restriction of A to S l × S l ; then σ : Σ + A l → Σ + A l+1 (mod p) and A p l is aperiodic. We say that an n-tuple w = (w 0 , . . . , w n−1 ) ∈ S n is an allowed word of length n if A(w j , w j+1 ) = 1 for j = 0, . . . , n − 2. We will write W n for the set of allowed words of length n. If w ∈ W n then we define the associated cylinder set [w] by We say that f is locally Hölder continuous if there exist 0 < θ < 1 and C ≥ 0 such that, for all n ≥ 1, V n (f ) ≤ Cθ n . (There is no requirement of V 0 (f ) and a locally Hölder f may be unbounded.) For n ≥ 1, we write Definition 3.1. Suppose that σ : Σ + A → Σ + A is topologically transitive and let f : Σ + A → R be a locally Hölder continuous function. Following Sarig [24], we define the Gurevič pressure, P G (σ, f ), of f to be where a ∈ S. (The definition is independent of the choice of a.) In [24], Sarig gives this definition in the case where σ : Σ + A → Σ + A is topologically mixing. However, the above decomposition of Σ + A = Σ + A0 ∪· · · ∪Σ + Ap−1 , with σ p topologically mixing on each component, together with the regularity of the function f , shows that the same definition may be made in the topologically transitive case.
We now specialise to the case where S is finite. In this situation, we call σ : Σ + A → Σ + A a (one-sided) subshift of finite type. The above definitions and results hold. If f : Σ + A → R is Hölder continuous then f is locally Hölder. Provided σ : is topologically transitive, the Gurevič pressure P G (σ, f ) agrees with the standard pressure P (σ, f ), defined by and if σ is topologically mixing then the lim sup may be replaced with a limit.
We now consider skew product extensions of a shift of finite type σ : Σ + A → Σ + A , which we will assume to be topologically mixing. Let G be a countable group and let ψ : Σ + A → G be a function depending only on two co-ordinates, ψ(x) = ψ(x 0 , x 1 ). (One may consider more general ψ but this set-up suffices for our needs.) This data defines a skew product extension σ : and ψ(i, j) = g −1 h and A((i, g), (j, h)) = 0 otherwise. Clearly, A is locally finite and so the topological transitivity and topological mixing of σ are equivalent to A being irreducible and aperiodic, respectively.
Let f : Σ + A → R be Hölder continuous and define f : ; then f is locally Hölder continuous and its Gurevič pressure P G ( σ, f ) is defined. In fact, it is easy to see that, due to the mixing of σ, It is clear that P G ( σ, f ) ≤ P (σ, f ) and it is interesting to ask when equality holds. Stadlbauer has shown this depends only on the amenability of the group G, provided the skew product and the function f satisfy appropriate symmetry conditions, which we now describe.
Suppose there is a fixed point free involution κ : S → S such that A(κj, κi) = A(i, j), for all i, j ∈ S. We say that the skew product σ : The following is the main result of Stadlbauer [26], restricted to the case where the base is a (finite state) subshift of finite type. (More generally, Stadlbauer considers skew product expansions of countable state Markov shifts.) A × G be a transitive symmetric skew-product extension of a mixing subshift of finite type σ : Σ + A → Σ + A by a countable group G. Let f : Σ + A → R be a weakly symmetric Hölder continuous function and define f : Remark 3.4. In [26], Stadlbauer considers skew products with ψ depending on only one co-ordinate. However, replacing S by W 2 , one can easily recover the above formulation.
Another characterisation of P (σ, f ) is as the logarithm of the spectral radius of the the transfer operator L f : Jaerisch has recently shown a result analogous to Proposition 3.3, where the Gurevič pressure is replaced by the logarithm of the spectra radius of a transfer operator associated to σ and f , acting on a suitably chosen Banach space. We state this result more precisely below. (In fact, Jaerisch's result holds without the symmetry assumptions described above but we will not need this greater generality.) Fix a Hölder continuous function f : Σ + A → R. Then there is a unique σ-invariant probability measure µ f , called the equilibrium state for f , with the property that P (σ, f ) = h µ f (σ) + f dµ f , where h µ f (σ) denotes the measure theoretic entropy of σ with respect to µ f . Following [26], define a Banach space and · ∞ is the usual essential supremum norm on L ∞ (Σ + A , µ f ). Writing Proposition 3.5 (Jaerisch [10]). Let σ : Σ + A × G → Σ + A × G be a transitive skewproduct extension of a mixing subshift of finite type σ : We will use both Proposition 3.3 and Proposition 3.5 in subsequent arguments. We end this section by discussing two-sided subshifts of finite type and suspended flows over them. Given a finite alphabet S and transition matrix A, we define n=0 ∈ S Z : A(x n , x n+1 ) = 1 ∀n ∈ Z and the (two-sided) shift of finite type σ : Σ A → Σ A by (σx) n = x n+1 . As before, we give Σ A with the product topology induced by the discrete topology on S and this is compatible with the metric d(x, y) = 2 −n(x,y) , where n(x, y) = inf{|n| : x n = y n }, with n(x, y) = ∞ if x = y. Then Σ A is compact and σ is a homeomorphism. There is an obvious one-to-one correspondence between the periodic points of σ : Σ A → Σ A and σ : Σ + A → Σ + A . Furthermore, we may pass from Hölder functions on Σ A to Hölder functions on Σ A is such a way that sums around periodic orbits are preserved. More precisely, we have the following lemma, due originally to Sinai. We may also define suspended flows over σ : Σ A → Σ A . Given a strictly positive continuous function r : Σ A → R + , we define the r-suspension space 0). The suspended flow σ r t : X r A → X r A is defined by σ r t (x, s) = (x, s+t) modulo the identifications. Clearly, there is a natural one-to-one correspondence between periodic orbits for σ r t : Σ r A → Σ r A and periodic orbits for σ : Σ A → Σ A , and a σ r -periodic orbit is prime if and only if the corresponding σ-periodic orbit is prime. Furthermore, if γ is a closed σ r -orbit corresponding to the closed σ-orbit {x, σx, . . . , σ n−1 x} then the period of γ is equal to r n (x).

Coverings and Geodesic Flows
As in the introduction, we shall write M = X/Γ, M = X/Γ 0 and G = Γ/Γ 0 . There is a natural dynamical system related to the geometry of M , namely the geodesic flow on the unit-tangent bundle where · x is the norm induced by the Riemannian structure on T x M . For future reference, we write p : SM → M for the projection. The geodesic flow φ t : SM → SM is defined as follows. Given (x, v) ∈ SM , there is a unique unit-speed geodesic γ : R → M with γ(0) = x andγ(0) = v. We then define φ t (x, v) = (γ(t),γ(t)). The non-wandering set Ω(φ) ⊂ SM is defined to be the set of points x ∈ SM with the property that for every open neighbourhood U of x, there exists t > 0 such that φ t (U ) ∩ U = ∅. It can be characterised as the set of vectors tangent to The restriction of the geodesic flow to its non-wandering set, φ t : Ω(φ) → Ω(φ), is an example of a hyperbolic flow. A C 1 flow φ t : Ω → Ω is hyperbolic if (1) there is a continuous Dφ-invariant splitting of the tangent bundle where E 0 is the line bundle tangent to the flow and where there exists constants C, c > 0 such that There is a natural one-to-one correspondence between (prime) periodic orbits for φ t : Ω(φ) → Ω(φ) and (prime) closed geodesics on M , with the least period being equal to the length of the closed geodesic. We will typically write γ for either a closed geodesic or a periodic orbit and allow the context to distinguish them. We will write l(γ) for the length (period) of γ and use these numbers to define a function of a complex variable ζ φ (s), the zeta function, by whenever the product converges. In fact, the product converges for Re(s) > h := h(M ) > 0 (defining a non-zero analytic function in this half-plane) and has a simple pole at s = h [17]. This number h is also characterised as the topological entropy of φ.

Markov Sections and Symbolic Dynamics
A particularly useful aspect of hyperbolic flows is that they admit a description by finite state symbolic dynamics. We shall outline this construction below.
Given ǫ > 0, we define the (strong) local stable manifold W s ǫ (x) and (strong) local unstable manifold W u ǫ (x) for a point x ∈ SM by Provided ǫ > 0 is sufficiently small, these sets are diffeomorphic to (dim M − 1)dimensional embedded disks. If x and y are sufficiently close then there is a unique t ∈ [−ǫ, ǫ] such that W s ǫ (x) ∩ W u ǫ (φ t (y)) = ∅ and, furthermore, this intersection consists of a single point denoted [x, y]. This pairing [·, ·] is called the local product structure.
Let D 1 , . . . , D k be a family of co-dimension one disks that form a local cross section to the flow and let P denote the Poincaré map between them. For each i = 1, . . . , k, let T i ⊂ int(D i ) ∩ Ω(φ) be sets which are chosen to be rectangles in the sense that whenever x, y ∈ T i then [x, y] ∈ T i and proper (i.e. T i = int(T i ) for each i). (Here and subsequently, the interiors are taken relative to D i .) We then say that T 1 , . . . , T k are Markov sections for the flow if The local product structure on SM induces a local product structure, also denoted [·, ·] on transverse sections by projecting along flow lines. The rectangles T i may be chosen so that T i = [U i , S i ], where U i and S i are closed subsets of local unstable and stable manifolds, respectively. Associated to this, we have projection maps ρ u i : T i → U i and ρ s i : T i → S i . Proposition 5.1 (Bowen [2]). For all ǫ > 0, the flow has Markov sections T 1 , . . . , T k such that diam(T i ) < ǫ, for i = 1, . . . , k and such that These sections may be chosen to reflect the time-reversal symmetry of the geodesic flow. The Markov sections allow us to relate φ t : Ω(φ) → Ω(φ) to a suspended flow over a mixing subshift of finite type.

The Skew Product Extension
In this section we will describe a skew product extension of (the one-sided version of) the shift of finite type introduced above, which will serve to encode information about how orbits on SM lift to S M , and relate this construction to the result of Stadlbauer, Proposition 3.3, stated above.
Choose ǫ 0 > 0 sufficiently small that every open ball in SM with diameter less than ǫ 0 is simply connected. Let U ⊂ SM be such an open ball. Then π −1 (U ) = g∈G g · U , where U is a connected component of π −1 (U ). Since we can choose the Markov sections T i to have arbitrarily small diameters, for each i = 1, . . . , k, we can choose U i be an open ball of diameter less than ǫ 0 containing T i . As above, we may and we may assume that this decomposition is chosen with ι( T i ) = T κi , where ι is the time-reversing involution ι : S M → S M given by ι(x, v) = (x, −v).
We will use the notation Notice that each lifted section g · T i is transverse to the flow φ t : S M → S M . We write write P : T → T for the Poincaré map.
Proof. We will begin be proving the existence and uniqueness of g. Let x 1 , x 2 ∈ T i ∩ P −1 (T j ) and let c 1 and c 2 be the φ-orbit segments from c 1 (0) = x 1 and c 2 (0) = x 2 to c 1 (1) = P(x 1 ) and c 2 (1) = P(x 2 ). We will show that there is a unique g ∈ G such that the unique lifts of c 1 and c 2 that begin in T i both have terminal points in g · T j . The statement for all h ∈ G will follow by translating by the isometry h ∈ G. Let c 1 and c 2 be lifts of c 1 and c 2 with c 1 (0), c 2 (0) ∈ T i . Having chosen the Markov partition to have sufficiently small diameters and the flow times between rectangles to be sufficiently small, there is an open ball U ⊂ SM of diameter less than ǫ 0 containing T i , T j , c 1 and c 2 . Let U be the connected component of π −1 (U ) containing c 1 . Then U ∩ T i = ∅ and U ∩ g · T j = ∅ and so T i ∪ g · T j ⊆ U . It follows that c 2 is entirely contained in U and so we must have c 2 (1) ∈ g · T j as required.
For the final part, we note that ι( c 1 ) is an orbit segment from g · T κj to T κi . It follows from the uniqueness in the previous that g(κj, κi) = g(i, j) −1 .
We use the preceding lemma to define a skew product extension of the onesided shift of finite type σ : Σ + A → Σ + A . Define ψ : Σ + A → G (depending on two co-ordinates) by ψ(x) = ψ(x 0 , x 1 ) = g(x 0 , x 1 ), where g = g(x 0 , x 1 ) is the unique element of G given by Lemma 6.1. Then the skew product σ : Furthermore, part (2) of Lemma 6.1 shows that the skew product extension is symmetric (with respect to the involution κ), i.e. that ψ(κj, κi) = ψ(i, j) −1 . We shall apply Proposition 3.3 to the skew product extension σ : To do this, we need to establish that two further conditions are satisfied: that σ is transitive and that r is weakly symmetric. We start with transitivity. Lemma 6.2. If G is not equal to π 1 (M ) then the map σ : Proof. If G is not equal to π 1 (M ) then the geodesic flow φ t : S M → S M is transitive. A proof is given in [5] (page 94) for the case where X = H 2 but the argument clearly generalizes. (See also [6] for the case of variable curvature when Γ is co-compact.) Let x ∈ S M be a point with dense φ-orbit. Without loss of generality x ∈ T and then { P n x} ∞ n=−∞ is dense in T . Suppose that A((i j , g j ), (i j+1 , g j+1 )) = 1, where A is the transition matrix for Σ + A × G, for j = 0, . . . , n. Then is non-empty and open in T . (Here int(g j · T ij ) is taken with respect to the codimension one disk containing g j · T ij .) Since x has dense P-orbit, P m x ∈ U for some m ∈ Z. Then P m+j (x) ∈ int(g j · T ij ) for j = 0, . . . , n. By definition, this implies that the σ-orbit of (ϑ(π(x)), g 0 ) ∈ Σ + A × G (where ϑ(π(x)) is identified with a point in the one-sided shift) passes through the (arbitrary) cylinder [(i 0 , g 0 ), . . . , (i n , g n )] and is thus dense in Σ + A × G. Therefore, σ : Σ + A × G → Σ + A × G is transitive. Let r : Σ A → R be the Hölder continuous function defined by Proposition 5.3. By Lemma 3.6, there is a Hölder continuous function on Σ + A , which we will abuse notation by continuing to call r, with the same sums around periodic orbits. Lemma 6.3. For any ξ ∈ R, the function −ξr : Σ + A → R is weakly symmetric. Proof. It suffices to show that r is weakly symmetric. Since σ : Σ + A → Σ + A is mixing, there exists N ≥ 0 such that, for each length n cylinder [z] = [z 0 , . . . , z n−1 ], we may find a periodic point x ∈ [z] of period n + N . Writing x = (x 0 , . . . , x n+N −1 , x 0 , . . .), we set κx = (κx n+N −1 , . . . , κx 0 , κx n+N −1 , . . .). Clearly, σ N (κx) is a periodic point of period n + N and σ N (κx) ∈ [κz]. Furthermore, r n+N (x) = l(γ), for some φperiodic orbit γ and where ιγ is the time-reversed periodic orbit corresponding to γ. We therefore have for some constant C > 0.
We end the section by noting the following result. Lemma 6.4 ([17]). The entropy h is the unique real number for which P (σ, −hr) = 0.

Zeta functions
In this section we shall prove that the equality of h = h(M ) and h( M ) is equivalent to amenability of G. To do this, we need to relate the growth of periodic φ-orbits which have periodic lifts to the Gurevič pressure. We begin by characterising the lifting property in terms of the skew product. Proof. We will treat ϑ(x) as the initial point on γ. Let γ be the lift of γ which starts in T x0 . By Lemma 6.1, γ ends in ψ n (x) · T x0 and is thus periodic if and only if ψ n (x) = 1 G .
Next we define a zeta function, analogous to ζ φ (s), associated to C by This has abscissa of convergence h := h( M ). A similar function may be defined for the set F prime periodic orbits of the suspended flow which correspond via ϑ to periodic φ-orbits in C: We also write It is this last function that will be directly related to Gurevič pressure.  Thus, Z(ξ) converges if P G ( σ, −ξr) < 0 and diverges if P G ( σ, −ξr) > 0.
We may now prove our main result for closed geodesics. Proof. By Lemma 6.4, we have P (σ, −hr) = 0 and, by Proposition 3.3, P G ( σ, −hr) < P (σ, −hr) unless G is amenable, in which case equality holds. Hence, if G is amenable then P ( σ, −hr) = 0 and so h = h. On the other hand, if G is not amenable then P G ( σ, −ξr) = 0 for some ξ < h and so, by Corollary 7.3 and Lemma 7.4, h < h.

Critical exponents
To complete the proof of Theorem 1.1, we will now turn our attention to the critical exponents δ(Γ) and δ(Γ 0 ). We will prove the following.
As noted in the introduction, δ(Γ) = h(M ) = h and, to be consistent with earlier sections, we will simply denote this quantity by h. We will first prove a result for the shift of finite type σ : Σ + A → Σ + A and its skew product extension and then use approximation arguments to deduce Theorem 8.  Proof. The growth rate of N (i, x, T ) is equal to the abscissa of convergence on the Dirichlet series which is finite, an easy argument shows that the map s → φ(s) is continuous. We may also write where L −s r is the transfer operator for the skew product σ. Noting that 1 [i]×1G ∈ H ∞ , we have If G is not amenable then, by Proposition 3.5, ρ ∞ (−h r) < e P (σ,−hr) = 1. By the continuity of φ, there exists ǫ > 0 such that for s ∈ (h − ǫ, h + ǫ), φ(s) < 1. In particular, there exists s 0 < h such that η(s 0 ) converges and so the abscissa of convergence of η(s) is strictly less than h.
Let us now return to the geometric setting of M = X/Γ and M = X/Γ 0 . For o ∈ X, let x and x denote the projections to M and M , respectively, so, in particular, π( x) = x. We have where N Γ0 (T ) = #{g ∈ Γ 0 : d X (o, go) ≤ T } and this counting function is equal to the number of geodesic loops of length at most T based at x ∈ M . Equivalently, N Γ0 (T ) is equal to the number of geodesic loops of length at most T based at x ∈ M which each lift to a closed loop based at x. In terms of the geodesic flow, N Γ0 (T ) is equal to the number of φ-orbit segments of length at most T which start and end in the fibre S x M and which each lift to a φ-orbit segment which starts and ends in the fibre S x M .
If τ is a φ-orbit segment from S x M to itself then it has a lift to S M running from S x M to S g· x M , for some well-defined g ∈ G, and we write τ = g.
We now use an approach adapted from [19] and [20]. The idea is to approximate N Γ0 (T ) by a function that counts orbit segments between pieces of unstable and stable manifolds (and which lift in an appropriate way). These latter counting functions may be related to the symbolic counting functions N (i, x, T ). 8.1. First approximation. For each i = 1, . . . , k, let P i denote the parallelepiped Then we have k i=1 P i = Ω(φ). The projection maps ρ u i : T i → U i and ρ s i : T i → S i (defined in section 5) induce projection maps R u i : P i → U i and R s i : P i → S i by R u i (φ t y) = ρ u i (y) and R s i (φ t y) = ρ s i (y), respectively, for y ∈ T i . Let P (x) = {P i : P i ∩ S x M = ∅}. We define the unions S x = Pi∈P (x) R s i (P i ) and U x = Pi∈P (x) R u i (P i ).
Lemma 8.3. Given ǫ > 0, we can choose the Markov sections {T i , . . . , T k } such that there is a bijection between (1) φ-orbit segments τ beginning and ending in S x M (of length l(τ )); and (2) φ-orbit segments τ ′ beginning in U x and ending in S x (of length l(τ ′ )), satisfying τ = τ ′ and |l(τ ) − l(τ ′ )| ≤ ǫ, apart possibly for a set of orbit segments intersecting the boundaries of the parallelepipeds. This exceptional set of orbit segments has growth rate bounded by h ′ < h.
We will write N (U x , S x , T ) for the number of φ-orbit segments τ from U x to S x with τ = 1 G and l(τ ) ≤ T . It then follows from the lemma that and hence that lim sup Ideally, we might hope that U x and S x are themselves each single pieces of stable and unstable manifold, in which case we could proceed directly to the final part of the proof, but, in general, we will need to make further approximations. 8.2. Second approximation. We may replace the subshift of finite type σ : Σ A → Σ A which models φ with a new subshift, σ : Σ A (N ) → Σ A (N ) , whose symbols are the set W 2N +1 of allowed words of length 2N + 1, for N ≥ 1. Associated to this is a finer set of Markov sections {T i : i ∈ W 2N +1 }, where, for i = (i −N , . . . , i −1 , i 0 , i 1 , . . . , i N ) ∈ W 2N +1 , Furthermore, we also obtain finer partitions {U i : i ∈ W 2N +1 } and {S i : i ∈ W 2N +1 } of the sets k i=1 U i and k i=1 S i , respectively. Lemma 8.4. We can choose N sufficiently large and index sets I 1 , Let N (U i , S j , T ) denote the number of φ-orbit segments τ from U i to S j such that l(τ ) ≤ T and τ = 1 G . Then the lemma gives us that for at least one pair (i, j) ∈ I 2 × J 2 . Next, we note that N (U i , S j , T ) is equal to ∞ n=1 #{w ∈ W n (i, x) : r n (wx) ≤ T and ψ n (wx) = 1 G }, for some x ∈ [j], up to an error which has growth rate at most h ′ < h. (We are now working in the one-sided version of the finer subshift, σ : Σ + A (N ) → Σ + A (N ) but, in particular, Lemma 8.2 still holds.) We are now if a position to complete the proof of Theorem 8.1. Suppose that δ(Γ 0 ) = h Then, by the approximation arguments given above, there exists i ∈ W 2N +1 and x ∈ Σ + A (N ) , for some N ≥ 1, such that lim sup Applying Lemma 8.2, we have that G is amenable.