On some exotic Schottky groups

We construct a Cartan-Hadamard manifold with pinched negative curvature whose group of isometries possesses divergent discrete free subgroups with parabolic elements who do not satisfy the so-called"parabolic gap condition". This construction relies on the comparaison between the Poincar\'e series of these free groups and the potential of some transfer operator which appears naturally in this context.


Introduction
Throughout this paper, X will denote a complete and simply connected Riemannian manifold of dimension N ≥ 2 whose sectional curvature is bounded between two negative constants −B 2 ≤ −A 2 < 0. We denote by d the distance on X induced by the Riemannian metric and by ∂X the boundary at infinity ; the isometries of X act as conformal transformations on ∂X when it is endowed by the so-called Gromov-Bourdon metric.
A Kleinian group of X is a non elementary torsion free and discrete subgroup Γ of orientation preserving isometries of X ; this group Γ acts freely and properly discontinuously on X and the quotient manifold M := X/Γ has a fundamental group which can be identified with Γ. One says that Γ is a lattice when the Riemannian volume of X/Γ is finite.
The limit set Λ Γ of a Kleinian group Γ is the least non empty Γ-invariant subset of ∂X ; this is also the set of accumulation points of some (any) orbit Γ · x of x ∈ X. This set is of interest for further reasons ; in particular, if (φ t ) t denotes the geodesic flow on the unit tangent bundle T 1 (X/Γ) of X/Γ, its non-wandering set Ω Γ coincides with the projection on T 1 (X/Γ) of the set of unit tangent vectors on X whose points at infinity in both directions belong to Λ Γ .
Note that the convex-hull C(Λ Γ ) of Λ Γ is a Γ-invariant closed subset of X and that the projection of Ω Γ onto the manifold X/Γ is in fact equal to C(Λ Γ )/Γ.The group Γ is said convex cocompact when it acts co-compactly on C(Λ Γ ) and more generally geometrically finite when it acts like a lattice on some (any) ǫ-neigbourhood C ǫ (Λ Γ ) of C(Λ Γ ) (in otherwords, when vol(C ǫ (Λ Γ )/Γ) < +∞ for some (any) ǫ > 0).
It is shown in [1] that the existence and unicity of a measure of maximal entropy for the geodesic flow restricted to Ω Γ is equivalent to the finiteness of a natural invariant Radon measure on T 1 (X/Γ) with support Ω Γ , the so-called Patterson-Sullivan measure m Γ . In this paper, we construct examples of isometry groups Γ for which the restriction of the geodesic flow (φ t ) t to the set Ω Γ exhibits particular properties with respect to ergodic theory. In particular, for those groups, the Patterson-Sullivan measure may be infinite and the associated dynamical system (φ t , Ω Γ ) will thus have no measure of maximal entropy.
We now recall briefly the construction of the Patterson-Sullivan measure associated with a Kleinian group Γ. The critical exponent of Γ is the exponential growth of its orbital function defined by It does not depend on x ∈ X and coincides with the exponent of convergence of the Poincaré series of Γ defined by P Γ (s, x) := γ∈Γ e −sd (o,γ·o) ; this series converges if s > δ Γ and diverges when s < δ Γ . The group Γ is divergent when the Poincaré series diverges at the critical exponent ; otherwise Γ is convergent.
A construction due to Patterson in constant curvature provides a family of δ Γ -conformal measures σ = (σ x ) x∈X supported on the limit set Λ Γ . D. Sullivan showed also how to assign to σ an invariant measure for the geodesic flow (φ t ) t restricted to Ω Γ . This construction has been extended by several people to the situation of a variable curvature space X and an arbitrary Kleinian group Γ acting on it [K1], [Y]. 1 Marc Peigné LMPT, UMR 6083, Faculté des Sciences etTechniques, Parc de Grandmont, 37200 Tours -mail : peigne@lmpt.univ-tours.fr It is important to recall that the family of measures σ associated with Γ is unique if and only if Γ is divergent (see [Ro] for a complete statement). In this case, the corresponding (φ t ) t -invariant measure constructed by Sullivan depends only on Γ, it is the Patterson-Sullivan measure m Γ of Γ.
We review now some basic results concerning the finiteness of the measure m Γ . When Γ is convex-coccompact, this measure is of course finite since it is a Radon measure with compact support. The same property holds when Γ is a geometrically finite group acting on a locally symmetric space [Su2], [CI] ; nevertheless, there exist non-geometrically finite groups with finite Bowen-Margulis measure [P].
The situation is much more complicated in the general variable curvature case, even for geometrically finite groups, because of the existence of parabolic subgroups.
There exist in particular criteria which ensure that a geometrically finite group Γ is divergent, for instance when its Poincaré exponent δ Γ is strictly greater than the one of each of its parabolic subgroups [DOP, Théorème A]. This is the so-called parabolic gap condition (PGC), which is satisfied in particular when the parabolic subgroups of Γ are themselves divergent. Furthermore, the Patterson-Sullivan measure m Γ of a divergent geometrically finite group Γ is finite if and only if, for any parabolic subgroup P of Γ, one has where δ P denotes the critical exponent of P [DOP, Théorème B] ; this holds in particular when the critical gap property is satisfied and in this case, by the Poincaré recurrence theorem, the geodesic flow (φ t ) t is completely conservative with respect to m Γ .
When Γ is convergent, the Patterson-Sullivan measure m Γ is infinite and the geodesic flow (φ t ) t is completely dissipative with respect to m Γ . On may choose the metric in such a way there exist non elementary geometrically finite groups of convergent type ; in this case, the parabolic gap condition is not satisfied and the parabolic subgroups of Γ of maximal Poincaré exponent are convergent. In [DOP] an explicit construction of such groups is proposed.
As far as we know, there were no examples of geometrically finite groups Γ of divergent type which do not satisfy the critical gap property ; this contruction is really of interest because it gives examples of geometrically finite manifolds for which the geodesic flow is completely conservative with respect to m Γ but this measure is infinite. We have the Theorem 1.1. There exist Hadamard manifolds with pinched negative curvature whose group of isometries contains geometrically finite Schottky groups Γ of divergent type which do not satisfy the parabolic gap condition PGC. Furthermore, the Patterson-Sullivan measure m Γ may be finite or infinite.
The paper is organized as follows : Section §2 deals with the construction of convergent parabolic groups ; we recall in particular the results presented in [DOP]. Section §3 is devoted to the construction of Hadamard manifolds containing convergent parabolic elements and whose groupe of isometries is non elementary. In section §4 we construct Schottky groups with convergent parabolic factor and we explain how to choose the metric inside the corresponding cuspidal end to prove Theorem 1.1.
We fix here once and for all some notation about asymptotic behavior of functions : Notations 1.2. Let f, g be two functions from R + to R + . We shall write f c g (or simply f g) when f (R) ≤ cg(R) for some constant c > 0 and R large enough. The notation f c ≍ g (or simply f ≍ g) means f c g the Riemannian hyperbolic metric is given by dx 2 + dy 2 y 2 where dx 2 + dy 2 is the classical euclidean metric on R N −1 × R * + . We denote by i the origin (0, · · · , 0, 1) of H N and by . the euclidean norm in R N . Let p be a parabolic isometry of H N fixing ∞ ; its induces on R N −1 an euclidean isometry which can be decomposed as the product p = R p • T p = T p • R p of an affine rotation R p and a translation T p with vector of translation s p . By an elementary calculous in hyperbolic geometry, one may check that the sequence (d(i, p n · i) − 2 ln n s p ) n≥1 converges to 0. The Poincaré exponent of the group p is thus equal to 1 2 and p is divergent. More generally, for any parabolic subgroup P of the group of isometries of H N , the sequence (d(i, p · i) − 2 ln s p ) p converges to 0 as p → ∞ in P. By one of Bieberbach's theorems, the group P contains a finite index abelian subgroup Q which acts by translations on a subspace R k of R N −1 ; in other words, there exist k linearly independant vectors s 1 , · · · , s k and a finite set F ⊂ P such that any p ∈ P may be decomposed as p = p n1 s1 · · · p n k s k f with n 1 , · · · , n k ∈ Z and f ∈ F so that The Poincaré exponent of P is thus equal to k 2 and the group is divergent. All these calculous may be done in the following (less classical) model : using the natural diffeomorphism between H N and R N defined by (x, y) → (x, t) := (x, ln y) one may endow R N with the hyperbolic metric g hyp := e −2t dx 2 + dt 2 .
In this model, we fix the origin o = (0, · · · , 0) and the vertical lines {(x, t)/t ∈ R} are clearly geodesics. For any t ∈ R, we denote by H t the hyperplane {(x, t) : x ∈ R N −1 } ; this corresponds to the horosphere centered at +∞ and passing through (0, · · · , 0, t). For any x, y ∈ R N −1 , the distance between x t := (x, t) and y t := (y, t) for the metric e −2t dx 2 induced by g hyp on H t is equal to e −t x − y ; furthermore, if t is choosen in such a way that this distance is equal to 1 (namely t = ln x − y ), then the union of the 3 segments [x 0 , x t ], [x t , y t ] and [y t , y 0 ] lies at a bounded distance of the hyperbolic geodesic joigning x 0 and y 0 which readily implies that d(x 0 , y 0 ) − 2 ln x − y is bounded.
This crucial fact is the key to understand geometrically the estimations above ; it first appeared in [DOP] and allowed the authors to construct negatively curved manifolds with convergent parabolic subgroups, we recall in the following subsection this construction.
2.2. The metrics T a,u on R N . We consider on R N −1 × R a Riemannian metric of the form g = T 2 (t)dx 2 + dt 2 , where dx 2 is a fixed euclidean metric on R N −1 and T : R → R * + is a C ∞ non-increasing function. The group of isometries of g contains the isometries of R N −1 × R fixing the last coordinate. The sectionnal curvature at ( It is convenient to consider the non-decreasing function u : R * + → R satisfying the following implicit equation T (u(s)) = 1 s .
Then, the value of the curvature of g is : Note that g has negative curvature if and only if T is convex. For instance, we have seen in the previous subsection that for u(s) = log s one gets T (t) = e −t and obtains a model of the hyperbolic space of constant curvature −1.
As it was seen in [DOP], the function u is of interest since it gives precise estimates (up a bounded term) of the distance between points lying on the same horosphere H t := {(x, t) : x ∈ R N −1 } where t ∈ R is fixed. Namely, the distance between x t := (x, t) and y t := (y, t) for the metric T 2 (t)dx 2 induced by g on H t is equal to T (t) x − y ; for t = u( x − y ), this distance is thus equal to 1, and the union of the 3 segments [x 0 , x t ], [x t , y t ] and [y t , y 0 ] lies at a bounded distance of the hyperbolic geodesic joigning x 0 and y 0 (see [DOP], lemme 4) : this readily implies that d(x 0 , y 0 ) − 2u( x − y ) is bounded.
In the sequel, we will assume that the function u coincides with the function s → ln s on ]0, 1] ; in otherwords, the restriction to the set ]0, 1] of the corresponding function T u (t) satisfying (2) is equal to t → e −t . More generally, we will "enlarge " the area where T u (t) and e −t coincides to the domain R N −1 ×] − ∞, a] with a arbitrary, introducing the following Notation 2.1. Let a ∈ R and u : R * + → R be a C 2 non decreasing function such that • u(s) = ln s for any s ∈]0, 1] We endow R N −1 × R with the metric T 2 a,u (t)dx 2 + dt 2 , where T a,u is given by Note that this metric has constant curvature −1 on the domain R N −1 ×] − ∞, a].
2.3. On the existence of metrics with convergent parabolic groups. In this paragraph, we fix a ∈ R and endow R N −1 × R with the metric T 2 a,u (t)dx 2 + dt 2 where u(s) = ln s + α ln ln s for s large enough and some constant α > 0 ; in this case, the curvature varies, nevertheless one has lim s→∞ K(u(s)) = −1 and all derivatives of K(u(s)) tend to 0 as s → +∞ . We will first need the following Lemma 2.2. Fix κ ∈]0, 1[. For any α ≥ 0, there exists a constant s α ≥ 1 and a non decreasing C 2 function u α : R * + → R such that Proof. We first fix a C 2 non decreasing function φ : R → [0, α], which vanishes on R − and is equal to α on [1, +∞[. For any ǫ > 0, we consider the function v ǫ : [e, +∞[→ R defined by For any continuous function g : [e, +∞[ converging to 0 at infinity, one gets gφ ǫ → 0 uniformly on [e, +∞[ ; consequently, one obtains, as ǫ → 0 and uniformly on [e, +∞[ and one sets with s α := exp(exp(1/ǫ 0 )). We thus fix a, α ≥ 0 and endow R N = R N −1 × R with the metric T 2 a,uα (t)dx 2 + dt 2 where u α is given by Lemma 2.2. This metric has pinched negative curvature less than −κ 2 and constant negative curvature in the domain {(x, t) : t ≤ a}. Now, let P be a discrete group of isometries of R N −1 of rank k ∈ {1, · · · , N − 1}, i.e generated by k linearly independent translations p τ1 , · · · , p τ k in R N −1 . In order to simplify the notations,n = (n 1 , · · · , n k ) ∈ Z k will represent the translation of vector n 1 τ 1 +· · ·+n k τ k and |n| will denote its euclidean norm. These translations are also isometries of R N endowed with the metric T a,uα (t) 2 dx 2 + dt 2 given above and the corresponding Poincaré series of P is given by We have thus prove the Proposition 2.3. Let R N be endowed with the metric T 2 a,uα (t)dx 2 + dt 2 where u α is given by Lemma 2.2. If P is a discrete group of isometries of R N −1 of rank k, its critical Poincaré exponent is equal to k/2 ; furthermore, the group P is convergent if and only if α > 1.
Remark 2.4. One may also choose u is such a way that u −1 (t) = e t/2− √ t . If r = 1, the critical exponent of the associated Poincaré series is equal to 1 2 and the group P is also convergent ; this last example appears in [Sch], where some explicit results are given, in terms of the Poincaré series of the parabolic groups, which guarantee the equidistribution of the horocycles on geometrically finite negatively curved surfaces.

Weakly homogeneous Hadamard manifolds of type (a, u α )
In the previous section, we have endowed R N with a metric T a,u (t) 2 dx 2 + dt 2 ; unfortunately, in this construction, excepted for some particular choice of u, all the isometries fix the same point at infinity and the group Is(R N ) is thus elementary. We need now to construct an Hadamard manifold with a metric of this inhomogeneous type in the neighbourhood of some points at infinity but whose group of isometries is non elementary.
3.1. Metric of type (a, u) relatively to some group Γ and some horoball H. Consider first a non uniform lattice Γ of isometries of H N . The manifold M := H N /Γ has finite volume but is not compact ; it thus possesses finitely many cusp C 1 , · · · , C l , each cusp C i being isometric to the quotient of some horoball H i of H N (centered at a point ξ i ) by a Bieberbach group P i with rank N − 1. Each group P i also acts by isometries on R N −1 × R endowed with one of the metrics T a,u (t) 2 dx 2 + dt 2 given by Notation 2.1. Now, we endow R N −1 × R with one of these metrics T a,u (t) 2 dx 2 + dt 2 and choose a in such a way we may paste the quotient (R N −1 × [0, +∞[)/P 1 with M \ C 1 . The Riemannian manifold M remains negatively curved with finite volume. By construction, the group Γ acts isometrically on the universal covering X ≃ R N of M endowed with the lifted metric g a,u ; note that g a,u coincides with the metric T 2 a,u (t)dx 2 + dt 2 on the preimage by Γ of the cuspidal end C 1 (2) . All this discussion gives sense to the following definition : Definition 3.1. Fix a, α ≥ 0, let u α be the function given by Lemma 2.2 and (X, g) a negatively curved Hadamard manifold whose group of isometries contains a non uniform lattice Γ.
Assume that X/Γ has one cusp C, let P be a maximal parabolic subgroup of Γ corresponding to this cusp, with fixed point ξ and let H be an horoball centered at ξ such that the γ · H, γ ∈ Γ, are disjoints or coincide.
One endows the manifold X with the metric g a,u defined by (1) g a,u has constant curvature −1 outside the set γ∈Γ γ · H (2) g a,u coincides with the metric T a,u (t) 2 dx 2 + dt 2 inside each horoball γ · H, γ ∈ Γ.
One says that the Riemannian manifold (X, g a,u ) has type (a, u) relatively to the group Γ and the horoball H. More generally, one says that (X, g) has type u when, for some a ∈ R, some lattice Γ and some horoball H, it has type (a, u) relatively to Γ and H.
Remark 3.2. If the metric g has type (a, u) relatively to Γ and H, the curvature remains equal to −1 in the stripe R N −1 × [0, a] ⊂ H. In the limit case "a = +∞", one refinds the hyperbolic metric of constant curvature −1.
By construction, the elements of Γ are isometries of (X, g a,u ). It is a classical fact that the group of isometries of H N is quite large since in particular it acts transitively on the hyperbolic space (and even on its unit tangent bundle). This property remains valid when X is symmetric, otherwise its isometry group is discrete ( [E], Corollary 9.2.2) ; consequently, if g has type (a, u α ) with α > 0, the group of isometries of (X, g) does not inherit this property of transitivity, it is is discrete and has Γ as finite index subgroup ( [E], Corollary 1.9.34).
We fix now one and for once the following Notation 3.3. From now on, we consider an Hadamard manifold X, with origin o, whose group of isometries contains a non uniform lattice Γ ; we fix a maximal parabolic subgroup P of Γ with fixed point ξ ∈ ∂X and an horoball H centered at ξ such that the horoballs γ · H, γ ∈ Γ, are disjoints or coincide. We fix α ≥ 0 and we assume that, for any a ≥ 0, the manifold X may be endowed with a metric g a := g a,uα of type (a, u α ) relatively to Γ and H, where u α is given by Lemma 2.2.
We denote by d a the corresponding distance on X.
Note that, by construction, the sectional curvature of g a is pinched between two non positive constants and is less than −κ 2 for somme constant κ > 0 which does not depend on a. Furthermore, using the fact that u α is non negative and uniformly continuous on [1, +∞[, one obtains the Property 3.4. For any a, a ′ and α ≥ 0, there exists a constant K = K a,a ′ ,α ≥ 1 with K a,a ′ ,α → 1 as a ′ → a such that Remark 3.5. Note that if a ′ > a, one has in fact g a ≥ g a ′ and so d a ≥ d a ′ . It will be used in the last section.
2 By the choice of C 1 , the horoballs γ · H 1 , γ ∈ Γ, are disjoint or coincide, they are also isometric to R N−1 ×R + endowed with the hyperbolic metric e −2t dx 2 +dt 2 ; another way to endow R N with the new metric ga,u is to replace inside each horoball γ · H 1 the hyperbolic metric with the restriction of T 2 a,u (t)dx 2 + dt 2 to the half space R N−1 × R + 3.2. On the metric structure of the boundary at infinity. In this paragrah we describe the metric structure of the boundary at infinity of X; we need first to consider the Busemann function B (a) · (·, ·) defined by : for any x ∈ ∂X and any p, q in X The Gromov product on ∂X, based at the origin o, between the points x and y in ∂X is defined by where z is any point on the geodesic (x, y) (note that the value of (x|y) (a) does not depend on z). By [Bou], the function is a distance on ∂X ; furthermore, the cocycle property satisfied by the Busemann functions readily implies that for any x, y ∈ ∂X and γ ∈ Γ In other words, γ acts on (∂X, D a ) as a conformal transformation with coefficient of conformality ) at the point x, since equality (5) may be rewrite (6) D a (γ · x, γ · y) = |γ ′ (x)| a |γ ′ (y)| a D a (x, y).
We will need to control the regularity with respect to a of the Busemann function x → B (a) x (o, z). By Property 3.4, the spaces (X, d 0 ) and (X, d a ) are quasi-isometric and, for any a 0 > 0, there exist a constant K 0 ≥ 1 such that Note that, by [GH], one also gets The corresponding distances D a on ∂X are thus Hölder equivalent ; more precisely, we have the Property 3.6. For any a 0 > 0, there exists a real ω 0 ∈]0, 1] such that, for all a ∈ [0, a 0 ], one gets The regularity of the Busemann function x → B (a) x (o, p) where p is a fixed point in X is given by the following Fact, which precises a result due to M. Bourdon.
Fact 3.7. [BP] Let E ⊂ ∂X and F ⊂ X two sets whose closure E and F in X ∪ ∂X are disjoint. Then the family of functions x → B (a) x (o, p), with p ∈ F , is equi-Lipschitz continuous on E with respect to D a .

Divergent Schottky groups without PGC
4.1. On the existence of convergent Schottky groups when α > 1. The fact that α > 1 ensures that any subgroup of P is convergent. In [DOP], it is proved that Γ possesses also non elementary subgroups of convergent type ; we first recall this construction and precise the statement.
Proposition 4.1. There exist Schottky subgroups G of Γ and a 0 ≥ 0 such that • G has Poincaré exponent 1 2 and is convergent on (X, g 0 ) • G has Poincaré exponent > 1 2 and is divergent on (X, g a ) for a ≥ a 0 . Note that the group G necessarily contains a parabolic element, otherwise it would be convex co-compact and thus of divergent type, whatever metric g a endows X. Proof. We first work in constant negative curvature −1 and fix a parabolic isometry p ∈ P. Since Γ is non elementary, there exists an hyperbolic isometry q ∈ Γ whose fixed points are distinct from the one of p. If necessary, one may shrink the horoball H in such a way that the projection of the axis of h on the manifold M = H N /Γ remains outside the cuspidal end C ≃ H/P ; in others words, one may fix o on the axis of q and assume that for any n ∈ Z * the geodesic segments [o, q n · o] lie outside the set γ∈Γ γ · H (so that in the area where the curvature is constant when X will be endowed with the metric g a ).
By the dynamic of the elements of H N there exist two compact sets U p and U q in X ∪∂X as follows : (1) U p is a neigbourhood of the fixed point ξ p of p ; (2) U q is a neigbourhood of the fixed points ξ + q and ξ − q of q ; (3) there exists θ > 0 such that for any x ∈ U p and y ∈ U q the angle x o y is greater than θ ; (4) for all k ∈ Z * one has q k (X ∪ ∂X) \ U q ⊂ U q and p k (X ∪ ∂X) \ U p ⊂ U p .
By the Klein's tennis table lemma, the group p, q generated by p and q is free. Therefore each element γ ∈ p, q , γ = Id, may be decomposed in a unique way as a product α n1 1 α n2 2 . . . α n k k with α i ∈ {p, q}, n i ∈ Z * and α i = α i+1 ; the integer k is the length of γ and α k is its last letter.
Let us now endow X with the metric g 0 = g 0,uα . If x ∈ U p and y ∈ U q , the path which is the disjoint union of the geodesic ray (x, o] and [o, y) is a quasi-geodesic in (X, g 0 ); therefore there exists a constant C > 0 which only depends on the sets U p and U q and on the bounds on the curvature -that is, on the choice of the function u α -such that

It follows that
Recall that d 0 (o, p m · o) = 2 ln m + 2α ln ln |m| + a bounded term ; since α > 1, the series m∈Z * e −sd0(o,p m ·o) converges at its critical exponent δ p = 1 2 . We may now replace q by a sufficient large power q k in order to get It comes out that the critical exponent of the group G generated by p and h := q k is less than 1 2 and that P G ( 1 2 ) < +∞ ; since p ∈ G, one also gets δ G ≥ δ p = 1 2 . Finally δ G = 1 2 and G is convergent, with respect to the metric d 0 on X.
Let us now prove that for a large enough the group G is divergent on (X, d a ). By the triangular inequality one first gets Recall first that, when the curvature is constant and equal to −1 (that is to say "a = +∞" in the definition of g a ), the quantity d H N (o, p m · o) − 2 log |m| is bounded, so the parabolic group p is divergent with critical exponent 1 2 . There thus exists ǫ 0 > 0 such that, for ǫ ∈]0, ǫ 0 ], one gets which proves that the critical exponent of G is strictly greater than 1 2 for a = +∞. The same property holds in fact for finite but large enough values of a. Indeed, there exists m a ≥ 1, with m a → +∞ as a → +∞, such that the geodesic segments [o, p m · o] for −m a ≤ m ≤ m a remain inside the stripe R N −1 × [0, a] ⊂ H corresponding to the cuspidal end of type (a, α) ; since g a has curvature −1 in this stripe (see Remark 3.2), the quantities There thus exists a 0 > 0 such that, for a ≥ a 0 one gets g∈G ǫ −( 1 2 +ǫ)da(o,g·o) = +∞. This last inequality implies that δ G > 1 2 when a ≥ a 0 ; by ( [DOP], Proposition 1), the group G is thus divergent since δ p = 1 2 . We now want to check that there exists some a ∈]0, a 0 [ such that the group G is divergent with δ G = 1 2 when X is endowed with the metric g a ; to prove this, we need to compare the Poincaré series P G (s) with the potential of some Ruelle operator L a,s associated with G that we introduce in the following paragraph.
From now on, we fix a Schottky group G = p, h satisfying the conclusions of Proposition 4.1 and subsets U p and U h in X ∪ ∂X satisfying conditions (1), (2), (3) and (4) above.

Spectral radius of the Ruelle operator and Poincaré exponent.
We introduce the family (L a,s ) (a,s) of Ruelle operators associated with G = p, h defined formally by : for any a ∈ [0, a 0 ], s ≥ 0, x ∈ ∂X and any bounded Borel function φ : ∂X → R The sequence (p n · o) n accumulates at ξ p . So, for any x ∈ U h the angle at o of the triangle x o p n · o is greater than θ/2 for n large enough and the sequence (B (a) x (p −n ·o, o)−d a (o, p n · o)) n is bounded uniformly in x ∈ U h and a ≥ 0. In the same way, the sequence (B (a) ) n is bounded uniformly in x ∈ U p and a ≥ 0. It readily implies that L a,s φ(x) is finite when s ≥ max(δ p , δ h ) = 1 2 and that it acts on the space L ∞ (∂X) of bounded Borel functions on ∂X.
By a similar argument, for any k ≥ 1, the quantities n are bounded uniformly in m 1 , n 1 , · · · , m k , n k ∈ Z * and x ∈ U p , y ∈ U h . One thus gets is equal to the spectral radius ρ ∞ (L a,s ) of L a,s on L ∞ (∂X). We have thus established the Fact 4.2. The Poincaré series P G (s) and the potential k≥1 |L k a,s 1| ∞ diverge or converge simultaneously. In particular, if δ a denotes the Poincaré exponent of G for the metric g a , one gets Consequently, since G satisfies the conclusions of Proposition 4.1, one gets • the series k≥1 |L 2k 0,1/2 1| ∞ converges ; • for a 0 large enough, the series k≥1 |L 2k a0,1/2 1| ∞ diverges which implies in particular ρ ∞ (L 0,1/2 ) ≤ 1 and ρ ∞ (L a0,1/2 ) ≥ 1.
We will prove that for some value a * ∈]0, a 0 [ one gets ρ ∞ (L a * ,1/2 ) = 1 ; the unicity of a * will be specified in the last section.
We first need to control the regularity of the function a → ρ ∞ (L a0,1/2 ). It will be quite simple to check that the function a → L 0,1/2 is continuous from R + to the space of bounded operators on L ∞ (∂X) ; unfortunately, the function L → ρ ∞ (L) is in general only lower semi-continuous. In the case of the family of Ruelle operators we consider here, this function will be in fact continuous, because of the very special form of the spectrum in this situation.
x (γ −1 o, o) Furthermore, δ a will denote the Poincaré exponent of G with respect to the metric g a and, for γ ∈ {p, h} and n ∈ Z * , the "weight" function w a (γ n , .) is defined by ∈Uγ e −δaba(γ n ,x) With these notations, the Ruelle operator L a introduced in the previous paragraph may be expressed as follows : for any φ ∈ L ∞ (∂X) and any x ∈ ∂X, The iterates of L a are given by where G(k) is the set of γ = α n1 1 · · · α n k k ∈ G of length k (with α i+1 = α ±1 i ) and w a (γ, x) = 1 x / ∈Uα k e −δGba(γ,x) when γ has last letter α k ; observe that we have the following "multiplicative cocycle property" : We will see that the L a , a ≥ 0, act on the space C(∂X) of real valued continuous functions on ∂X and that the map a → L a is continuous. Nevertheless, the function a → ρ ∞ (a) is only lower semi-continuous in general and may present discontinuities. The main idea to avoid this difficulty is to introduce a Banach space on which the L a act quasicompactly.
In the sequel, we will consider the restriction of the L a to some subspace of C(∂X) of Hölder continuous functions. The spaces (L a,ω (∂X), |.|) are C-Banach space and the identity map from (L a,ω (∂X), |.| a,ω ) to (C(∂X), |.| ∞ ) are compact.
We now want to prove that each operator L a acts on L a,ω (∂X); in fact, we need a stronger result, i.e that each L a , for 0 ≤ a ≤ a 0 , acts on L ω (∂X). It will be a direct consequence of the following : Lemma 4.5. There exists ω 0 ∈]0, 1[ such that for any ω ∈]0, ω 0 ], any γ ∈ {p, h}, any a ∈ [1, a 0 ] and any n ∈ Z * , the function w a (γ n , .) belongs to L ω (∂X); furthermore, the sequence e δGda(o,γ n ·o) |w a (γ n , .)| ω n is bounded.
Proof. The cluster points of the sequence (γ n · o) n belong to U γ . Since the curvature is pinched, the quantity b a (γ n , x))−d(o, γ n ·o) is bounded uniformly in n ∈ Z * , x ∈ ∂X \U γ and a ∈ [0, a 0 ] ; so is the sequence e δGda(o,γ n ·o) |w a (γ n , .)| ∞ n . In order to control the ω-Hölder coefficient of w a (γ n , .), we use Fact 3.7 : the functions x → B (a) x (o, γ −n ·o) are equi-Lipschitz continuous on ∂X \U α with respect to D a , since once again the cluster points of the sequence (γ n · o) n belong to U γ . More precisely, the sequence e δGda(o,γ n ·o) |w a (γ n , .)| ω n is bounded for any a ∈ [0, a 0 ] and for ω given by inequality (9) .
From now and for once , we fix ω 0 ∈]0, 1[ satisfying the conclusion of the above Lemma. We know that, for a ∈ [0, a 0 ], the operator L a acts on L ω (∂X) whenever ω ∈]0, ω 0 ] ; let ρ ω (a) denote the spectral radius of L a on L ω (∂X). We have the Proposition 4.6. For any ω ∈]0, ω 0 ] and a ∈ [1, a 0 ], one gets is a simple eigenvalue of the operator L a acting on L ω (∂X) and the associated eigenfunction is non negative on ∂X.
Furthermore, the operator L a is quasi-compact on L ω (∂X) : there exists r < 1 such that the essential spectral radius of L a on L ω (∂X) is less than rρ ω (a).
In particular the eigenvalue ρ ω (a) is isolated in the spectrum of L a , it is simple and the corresponding eigenfunction is non-negative.
Proof. Fix x, y ∈ ∂X ∩ U p ; one gets Note that in these sums, it is sufficient to consider the γ ∈ G(k) with last letter α k = p. For such γ the quantity b a (γ, x) is greater than d a (o, γ · o) − C for some constant C which depends only on the angle θ 0 and the bounds on the curvature ; in particular b a (γ, x) ≥ 1 for all but finitely many γ ∈ G with last letter = p. It readily follows that lim inf uniformly in x ∈ U p . In other words, there exists 0 < r < 1 and C > 0 such that |γ ′ (x)| a ≤ Cr k for any k ≥ 1, x ∈ U p and γ ∈ G(k) with last letter = p. The same argument works when x, y ∈ ∂X ∩ U h . We thus obtain the inequality w a (γ, .)] ω . Note that L a is a non-negative operator, so that the quantity lim sup k |L k a 1| 1/k ∞ is equal to the spectral radius ρ ∞ (a) of L a on C(Λ). Using a version due to H. Hennion of the Ionescu-Tulcea-Marinescu's theorem concerning quasi-compact operators, one may conclude that the essential spectral radius of L a on L ω (∂X) is less than rρ ∞ (a) ; in other words, the spectral values of L a with modulus ≥ rρ ∞ (a) are isolated eigenvalues with finite multiplicity in the spectrum of L a . This implies in particular that ρ ω (a) = ρ ∞ (a). The inequality ρ ω (a) ≥ ρ ∞ (a) is obvious since the function 1 belongs to L ω (∂X). Furthermore, the strict inequality would imply the existence of a function φ ∈ L a such that L a φ = λφ for some λ ∈ C of modulus > ρ ∞ (a) ; this would give |λ||φ| ≤ L a |φ| so that |λ| ≤ ρ ∞ (a), which is a contradiction.
It remains to control the value ρ ω (a) in the spectrum of L a . The operator L a being non negative and compact on L ω (∂X), its spectral radius ρ ω (a) is an eigenvalue with associated eigenfunction φ a ≥ 0.
Assume that φ a vanishes at x 0 ∈ ∂X and let g ∈ {p, h} such that x 0 ∈ U g ; the equality L a φ a (x 0 ) = ρ ω (a)φ a (x 0 ) implies that φ a (γ · x 0 ) = 0 for any γ ∈ G with last letter = g. By minimality of the action of G on ∂X one thus has φ a = 0 on ∂X. Consequently, the function φ a is non negative.
Let us now check that ρ ω (a) is a simple eigenvalue of L a . Consider the operator P defined formaly by P (f ) = 1 ρ ω (a)φ a L a (f φ a ) ; this operator is well defined on L ω (∂X) since φ a does not vanish, it is non negative, quasi-compact with spectral radius 1 and Markovian (that is to say P 1 = 1). If f ∈ L ω (∂X) satisfies the equality P f = f one considers a point x 0 ∈ ∂X such that |f (x 0 )| = |f | ∞ and g ∈ {p, h} such that x 0 ∈ U g . An argument of convexity applied to the inequality P |f | ≤ |f | readily implies |f (x 0 )| = |f (γ · x 0 )| for any γ ∈ G with last letter = g ; by minimality of the action of G on ∂X it follows that the modulus of f is constant on ∂X. Applying again an argument of convexity and the minimality of the action of G on its limit set, one proves that f is in fact constant on ∂X ; it follows that Cφ a is the eigenspace associated with ρ ω (a) on L ω (∂X).

4.4.
Regularity of the function a → L a . In this section we will establish the following Proposition 4.7. For any 0 < ω < ω 0 , the function a → L a is continuous from [1, a 0 ] to the space of continuous linear operators on (L ω (∂X), |.| ω ).
Since the axis of h lies in the area of X where the curvature is −1, the quantity d a (o, h n · o) − |n|l h , where l h denotes the hyperbolic lenght of the closed geodesic associated with h, is bounded uniformly in a ∈ [0, a 0 ] and n ∈ Z * ; the same holds for the quantity d a (p n · o, o) − d 0 (p n · o, o). Consequently |w a ′ (γ n , .) − w a (γ n , .)| ≤ C ′ e −δGd0(γ n o,o) |e −δG(b a ′ (γ n ,.)−ba(γ n ,.)) − 1| and we have to study the regularity of the function a → b a (γ n , x), for any point x / ∈ U γ . By inequalities (8), one gets (y|z) a ′ → (y|z) a as a ′ → a, when (y|z) a remains bounded. There are thus two cases to consider: • We first consider the case γ = p. For any n ∈ Z * let y n be the point in ∂X such that o belongs to the geodesic ray [p n · o, y n ) (for the metric g a ) ; this ray is in fact a quasi-geodesic for any a ′ ∈ [0, a 0 ], so the point o belongs to some bounded neigbourhood of the geodesic ray (for g a ′ ) from p n · o to x n ( note that inf n∈Z * D 0 (y n , ξ p ) > 0 by convexity of the horospheres). For any n ∈ Z * and a ′ ∈ [0, a 0 ] one gets Since p n · x → ξ p as |n| → +∞ and inf n∈Z * D 0 (y n , ξ p ) > 0, one gets (p n · x|y n ) a ′ → (p n · x|y n ) a as a ′ → a, uniformly in n ∈ Z * and x / ∈ U p . In the same way, since inf n∈Z * D 0 (y n , ξ p ) > 0, the sequence (p −n · y n ) n converges to ξ p as |n| → +∞ so that (x|p −n · y n ) a ′ → (x|p −n · y n ) a uniformly in n ∈ Z * and x / ∈ U p . Atlast one has b a ′ (p n , p −n · as a ′ → a, uniformly in n ∈ N * .
• Consider now the case when γ = h ; for any n ≥ 1, one gets On the other hand h n · x → x + as n → +∞ so that (h n ·x|h n ·ξ + h ) a → (x + |ξ + h ) a ; since ξ + h = x + , the Gromov product (x + |ξ + h ) a is equal to − log d a (o, (x + ξ + h )) up to a bounded term and the sequence ((h n ·x|h n ·ξ + h ) a ) n≥1 is bounded uniformly in a ∈ [0, a 0 ], x / ∈ U h and n ∈ N. It readily follows that (h n · x|h n · ξ + h ) a ′ → (h n · x|h n · ξ + h ) a as a ′ → a, for any n ≥ 1. A similar argument holds for n ≤ −1. Finally, b a ′ (h n , x)−b a (h n , x) → 0 uniformly in n ≥ 0 and x / ∈ U h and the lemma is proved for γ = h.
On the other hand, the function s → ρ ω (L a * ,s ) is strictly decreasing on R + . Fix s > δ p ; one has ρ ω (a * ) < 1 and the series P G (s) thus converges when X is endowed with the metric g a * . This proves that for the value a * of the parameter a the critical exponent of G is less than δ p ; since p ∈ G, one has in fact δ G = δ p .
Atlast, since φ a * ≍ 1, one has k≥1 |L 2k a * ,δ p 1| ∞ ≍ k≥1 |L 2k a * ,δ p φ a * | ∞ ; these two series diverge in fact because of the equality L a * ,δ p φ a * = φ a * . By the Fact 4.2, it follows that for the value a * of the parameter a, the series P G (δ G ) diverges. By criteria (1), one easily sees that m Γ is finite when α > 2 and infinite when α ∈]1, 2]. This achieves the Proof. of the Main Theorem.
4.6. Complement. A natural question is the one of unicity of the value a * of the parameter a such that the spectral radius ρ ∞ (a) of L a is equal to 1; this unicity is not necessary to prove the main Theorem but nevertheless it is of interest to describe for instance the behavior of the orbital function of G when a varies. It will be the subject of a forecoming work.
By the continuity of the function a → ρ ∞ (a), the unicity of a * is a direct consequence of the strict monotonicity of this function. We thus have to prove that ρ(L a ) < ρ(L a ′ ) for any a, a ′ in [0, a 0 ] such that a < a ′ . Note first that, for any fixed x ∈ X one gets ρ(L a ) = ρ ∞ (L a ) = lim and we have thus to check that there exists C > 0 and ρ := ρ(a, a ′ ) < 1 such that, for any n ≥ 1, one gets For any x ∈ ∂X and y, we will denote by H (a) x (y) the horoball (with respect to the metric g a ) centered at x and passing through y ; furthermore, for any x ∈ X we denote by ψ x,y (x) its projection (with respect to g a ) on the horosphere ∂H (a) x (y). In order to simplify the argument, one first assume that the two following conditions hold • (C 1 ) for any x ∈ U p ∩ ∂X the points h n · o, n ∈ Z * , lie outside the horoball H Fix k ≥ 1 and γ ∈ Γ 2k with last letter in h. Let us decompose γ into a 2k a 2k−1 · · · a 1 with a 2i = p mi and a 2i−1 = h ni for 1 ≤ i ≤ k ; set γ 0 := Id and γ j := a j · · · a 1 for 1 ≤ j ≤ 2k. We fix x ∈ U p ∩ ∂X ; by the ping-pong dynamic, there exists c > 0 independent of γ such The cocycle property of the Busemann function thus leads to the following and one may thus write, as in (13) (16) γ∈Γ 2k By the previous assumption, all the quantities B (a) x (γ −1 j+1 · o, γ −1 j · o) above are non negative and we want to compare them with a similar one involving g a ′ . For any x ∈ ∂X and x, y ∈ X, the quantity B x (x, y) is equal to the "signed" length ( for g a ) of [x, ψ x,y (x)] a , the geodesic segment (for g a ) joigning x and ψ x,y (x) ; in otherwords, with obvious notations, one gets B (a) x (x, y) = [x,ψx,y(x)]a dg a where the integral is non negative when x is outside H (a) x (y) and negative when it lies inside. Similarly, we introduce the quantity β x (x, y) defined by β x (x, y) = β (a,a ′ ) x (x, y) := [x,ψx,y(x)]a dg ′ a .
Proof. The first large inequality is a direct consequence of the Remark after Property 3.4, namely g a ′ ≤ g a . To prove the second one, we note that for any y ∈ U h and any n ∈ Z with |n| large enough, the geodesic segment [p n · o, ψ x,o (p n · o)] a inters sufficiently inside the horoball H centered at ξ p and in particular in the area where g a and g a ′ differ (ie g a ′ > g a ) ; consequently β y (p n · o, o) − B (a) y (p n · o, o) > 0. the existence of η(n) > 0 follows by an argument of continuity with respect to y.
• for any x ∈ U h ∩ ∂X, the points g · o, g ∈ P, lie outside the horoball H (a) x (o). Any γ in Γ 2k(2N +1) with last letter h may be decomposed into γ = a 2k · · · a 1 with a 2i ∈ P and a 2i−1 ∈ H for 1 ≤ i ≤ k ; the same argument as above, with obvious modifications, leads to the inequality