Françoise Dal'bo-Milonet

Marc Peigné

Jean-Claude Picaud

Andrea Sambusetti

Françoise Dal'bo

Convergence and Counting in Infinite Measure

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Introduction

Let Γ be a Kleinian group. i.e. a discrete, torsionless group of isometries of a Hadamard space X of negative, pinched curvature -B 2 ≤ K X ≤ -A 2 < 0, with quotient X = Γ\X. This paper is concerned with two mutually related problems :

1) The description of the distribution of the orbits of Γ on X, namely of fine asymptotic properties of the orbital function : v Γ (x, y; R) := {γ ∈ Γ/d(x, γ • y) ≤ R} for x, y ∈ X. This has been the subject of many investigations since Margulis' [START_REF] Margulis | Certain applications of ergodic theory to the investigation of manifolds of negative curvature[END_REF] (see Roblin's book [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] and Babillot's report on [START_REF] Babillot | Rigidité, groupe fondamental et dynamique[END_REF] for a clear overview). The motivations to understand the behavior of the orbital function are numerous : for instance, a simple but important invariant is its exponential growth rate

δ Γ = lim sup R→∞ 1 R log(v Γ (x, y; R))
which has a major dynamical significance, since it coincides with the topological entropy of the geodesic flow when X is compact, and is related to many interesting rigidity results and characterization of locally symmetric spaces, cp. [START_REF] Hamenstädt | Entropy-rigidity of locally symmetric spaces of negative curvature[END_REF], [START_REF] Bonk | Rigidity for quasi-Möbius group actions[END_REF], [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF].

2) The pointwise behavior of the Poincaré series associated with Γ :

P Γ (x, y, s) := γ∈Γ e -sd(x,γ•x) , x, y ∈ X for and s = δ Γ , which coincides with its exponent of convergence. The group Γ is said to be convergent if P Γ (x, y, δ Γ ) < ∞, and divergent otherwise. Divergence can also be understood in terms of dynamics as, by Hopf-Tsuju-Sullivan theorem, it is equivalent to ergodicity and total conservativity of the geodesic flow with respect to the Bowen-Margulis measure on the unit tangent bundle U X (see again [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] for a complete account).

The regularity of the asymptotic behavior of v Γ , in full generality, is well expressed in Roblin's results, which trace back to Margulis' work in the compact case : Theorem 1.1 (Margulis [27] -Roblin [START_REF] Roblin | Sur la fonction orbitale des groupes discrets en courbure négative[END_REF], [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]). Let X be a Hadamard manifold with pinched negative curvature and Γ a non elementary, discrete subgroup of isometries of X with non-arithmetic length spectrum 1 : (i) the exponential growth rate δ Γ is a true limit ;

(ii) if ||m Γ || < ∞, then v Γ (x, y; R) ∼ ||µx||.||µy|| δΓ||mΓ|| e δΓR ; (iii) if ||m Γ || = ∞ then v Γ (x, y; R) = o(e δΓR ),
where (µ x ) x∈X denotes the family of Patterson conformal densities of Γ, and m Γ the Bowen-Margulis measure on U X. Theorem 1.1 shows that the key assumption for a regular behavior of v Γ is that the Bowen-Margulis measure m Γ is finite. This condition is clearly satisfied for uniform lattices Γ (i.e. when X = X/Γ is compact), and more generally for groups Γ such that m Γ has compact support (e.g., convex cocompact groups), but it may fail for nonuniform lattices, that is when X = X/Γ has finite volume but is not compact.

The finiteness of m Γ has a nice geometrical description in the case of geometrically finite groups. Recall that any orbit Γ • x accumulates on a closed subset Λ Γ of the geometric boundary ∂X of X, called the limit set of Γ ; the group Γ (or the quotient manifold X) is said to be geometrically finite if Λ Γ decomposes in the set of radial limit points (the limit points ξ which are approached by orbit points in the M -neighborhood of any given ray issued from ξ, for some M < ∞) and the set of bounded parabolic points (those ξ fixed by some parabolic subgroup P acting cocompactly on ∂X \ {ξ}) ; for a complete study of geometrical finiteness in variable negative curvature see [START_REF] Bowditch | Geometrical finiteness with variable negative curvature Duke Math[END_REF] and Proposition 1.10 in [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF], and for a description of their topology at infinity see [START_REF] Dal'bo | On the horoboundary and the geometry of rays of negatively curved manifolds[END_REF]. A finite-volume manifold X is a particular case of geometrically finite manifold : it can be decomposed into a compact set and finitely many cusps Ci , i.e. topological ends of X of finite volume which are quotients of a horoball H ξi centered at a bounded parabolic point ξ i ∈ ∂X by a maximal parabolic subgroup P i ⊂ Γ fixing ξ i .

The principle ruling the regularity of the orbital function v Γ of nonuniform lattices, as pointed out in [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF] and in the following papers of the authors [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negative curvature manifolds[END_REF], [START_REF] Dal'bo | On the growth of quotients of Kleinian groups[END_REF], [START_REF] Dal'bo | Volume growth and rigidity of negatively curved manifolds of finite volume[END_REF], is that the orbital functions v Pi (defined in a similar way as v Γ ) capture the relevant information on the wildness of the metric inside the cusps, which in turn may imply ||m Γ || = ∞ and the irregularity of v Γ . In this regard, distinctive properties of the group Γ and of its maximal parabolic subgroups P i are their type (convergent or divergent, as defined above) and the critical gap property (CGP), i.e. if δ Pi < δ Γ for all i. Actually, in [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF] it is proved that, for geometrically finite groups Γ, the divergence of P i implies δ Pi < δ Γ , and that the critical gap property implies that the group Γ is divergent with ||m Γ || < ∞. On the other hand there exist geometrically finite groups Γ which do not satisfy the CGP : we call such groups exotic, and we say that a cusp is dominant if it is associated to a parabolic subgroup P with δ P = δ Γ . Geometrically finite, exotic groups may as well be convergent or divergent : in the first case, they always have ||m Γ || = ∞ (by Poincaré recurrence and Hopf-Tsuju-Sullivan theorem, as ||m Γ || < ∞ implies total conservativity) ; in the second case, in [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF] it is proved that the finiteness of m Γ depends on the convergence of the special series p∈Pi d(x, px)e -δΓd(x,px) < +∞.

The main aim of this paper is to present examples of lattices Γ for which v Γ has an irregular asymptotic behavior. According to our discussion, we will then focus on exotic, non-uniform lattices. The convergence property of exotic lattices is an interesting question on its own : while uniform lattices (as well as convex-cocompact groups) always are divergent, the only known examples of convergent groups, to the best of our knowledge, are given in [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF] and have infinite covolume. The first result of the paper is to show that both convergent and divergent exotic lattices do exist. Actually, in Section 3, by a variation of the construction in [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF] we obtain : Theorem 1.2. For any N ≥ 2, there exist N -dimensional, finite volume manifolds of pinched negative curvature whose fundamental group Γ is (exotic and) convergent.

Constructing exotic, divergent lattices is more subtle. We prove in Section 5 : Theorem 1.3. There exist non compact finite area surfaces of pinched negative curvature whose group Γ is exotic and divergent.

We stress the fact that the examples of Theorem 1.2 have infinite Bowen-Margulis measure ; on the other hand, the surfaces of Theorem 1.3 can have finite or infinite Bowen-Margulis measure, according to the chosen behaviour of the metric in the cusps. Moreover, we believe that the assumption on the dimension in Theorem 1.3 is just technical, but at present we are not able to construct similar examples in dimension N ≥ 3.

Finally, in Section 6 we address the initial question about how irregular the orbital function can be, giving estimates for the orbital function of a large family of exotic lattices with infinite Bowen-Margulis measure : Theorem 1.4. Let κ ∈]1/2, 1[. There exist non compact finite area surfaces with pinched negative curvature whose fundamental group Γ satisfies the following asymptotic property : for any

x, y ∈ X v Γ (x, y; R) C e δΓR R 1-κ L(R) as R → +∞
for some slowly varying function2 L : R + → R + and some constant C = C Γ,x,y > 0.

A general, but more technical, result on the orbital function of divergent, exotic lattices is given in Theorem 6.1. As far as we know, except for some sharp asymptotic formulas established by Pollicott and Sharp [START_REF] Pollicott | Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature[END_REF] for the orbital function of normal subgroups Γ of a cocompact Kleinian group (hence, groups which are far from being geometrically finite or with finite Bowen-Margulis measure), these are the only examples of such precise asymptotic behavior for the orbital function of Kleinian groups with infinite Bowen-Margulis measure.

Remark. This work should be considered as a companion paper to [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negative curvature manifolds[END_REF]& [START_REF] Dal'bo | Volume growth and rigidity of negatively curved manifolds of finite volume[END_REF], where we study the asymptotic properties of the integral version of v Γ , i.e. the growth function of X :

v X (x; R) := vol X (B(x, R)).
In [START_REF] Dal'bo | Volume growth and rigidity of negatively curved manifolds of finite volume[END_REF], we obtain optimal conditions on the geometry on the cusps in order that there exists a Margulis function, that is a Γ-invariant function c :

X → R + such that v X (x; R) ∼ c(x)e ω X R for R → +∞
where ω X is the exponential growth rate of the function v X (the integral analogue of δ Γ ). Notice that ω X can be different from δ Γ , also for lattices, as we showed in [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negative curvature manifolds[END_REF].

2. Geometry of negatively curved manifolds with finite volume 2.1. Landscape. Additionally to those given in the introduction, we present here notations and familiar results about negatively curved manifolds. Amongst good references we suggest [START_REF] Ballmann | Manifolds of nonpositive curvature Progress in Mathematics[END_REF], [START_REF] Eberlein | Geometry of Nonpositively curved manifolds University of[END_REF], [START_REF] Ballmann | Lectures on spaces of nonpositive curvature DMV Seminar[END_REF] and, more specifically related to this work, [START_REF] Heintze | On the geometry of horospheres[END_REF] and [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negative curvature manifolds[END_REF]. In the sequel, X = X/Γ is a N -dimensional complete connected Riemannian manifold with metric g whose sectional curvatures satisfy : -B 2 ≤ K X ≤ -A 2 < 0 for fixed constants A and B ; we will assume 0 < A ≤ 1 ≤ B since in most examples g will be obtained by perturbation of a hyperbolic one and the curvature will equal -1 on large subsets of X.

The family of normalized distance functions :

d(x 0 , x) -d(x, •) converges uniformly on compacts to the Busemann function B ξ (x 0 , •) for x → ξ ∈ ∂X.
The horoballs H ξ (resp. the horospheres ∂H ξ ) centered at ξ are the sup-level sets (resp. the level sets) of the function B ξ (x 0 , •). Given a horosphere ∂H ξ passing through a point x, we also set, for all t ∈ R,

H ξ (t) := {y/B ξ (x 0 , y) ≥ B ξ (x 0 , x) + t} resp. ∂H ξ (t) := {y/B ξ (x 0 , y) = B ξ (x 0 , x) + t}.
We will refer to t = B ξ (x 0 , y) -B ξ (x 0 , x) as to the height of y (or of the horosphere ∂H ξ (t)) in H ξ . Also, when no confusion is possible, we will drop the index ξ ∈ ∂X denoting the center of the horoball. Also, recall that the Busemann function satisfies the fundamental cocycle relation

B x (x, z) = B x (x, y) + B x (y, z)
which will be crucial in the following.

2. A function L(t) is said to be "slowly varying" or "of slow growth" if it is positive, measurable and L(λt)/L(t) → 1 as t → +∞ for every λ > 0.

An origin x 0 ∈ X being fixed, the Gromov product between x, y ∈ ∂X ∼ = S n-1 , x = y, is defined as

(x|y) x0 = B x (x 0 , z) + B y (x 0 , z) 2
where z is any point on the geodesic (x, y) joining x to y ; then, for any 0 < κ 2 ≤ A 2 the expression D(x, y) = e -κ(x|y)x 0 defines a distance on ∂X, cp [START_REF] Bourdon | Structure conforme au bord et flot géodésique d'un CAT(-1)-espace[END_REF]. Recall that for any γ ∈ Γ one gets

(1)

D(γ • x), γ • y) = e -κ 2 Bx(γ -1 •x0,x0) e -κ 2 By(γ -1 •x0,x0) D(x, y).
In other words, the isometry γ acts on (∂X, D) as a conformal transformation with coefficient of conformality |γ (x)| = e -κBx(γ -1 •x0,x0) at x, since equality (1) may be rewritten

(2) D(γ • x), γ • y) = |γ (x)||γ (y)| D(x, y).
Recall that Γ is a torsion free nonuniform lattice acting on X by hyperbolic or parabolic isometries. For any ξ ∈ ∂X, denote by (ψ ξ,t ) t≥0 the radial semi-flow defined as follows : for any x ∈ X, the point ψ ξ,t (x) lies on the geodesic ray [x, ξ) at distance t from x. By classical comparison theorems on Jacobi fields (see for instance [START_REF] Heintze | On the geometry of horospheres[END_REF]), the differential of ψ ξ,t : ∂H ξ → ∂H ξ (t) satisfies e -Bt ||v|| ≤ ||dψ ξ,t (v)|| ≤ e -At ||v|| for any t ≥ 0 and any vector v in the tangent space T (∂H ξ ) ; consequently, if µ t is the Riemannian measure induced on ∂H ξ (t) by the metric of X, we have, for any Borel set

F ⊂ ∂H ξ e -B(N -1)t µ 0 (F ) ≤ µ t (ψ ξ,t (F )) = F |Jac(ψ ξ,t )(x)|dµ 0 (x) ≤ e -A(N -1)t µ 0 (F ).
As X = Γ\X is non compact and vol( X) < ∞, the manifold X can be decomposed into a disjoint union of a relatively compact subset K and finitely many cusps C1 , ..., Cl , each of which is a quotient of a horoball H ξi , centered at some boundary point ξ i , by a maximal parabolic subgroup P i . As a consequence of Margulis' lemma, we can choose the family (H ξi ) 1≤i≤l so that any two Γ-translates of the H ξi are either disjoint or coincide (cp. [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF], Proposition 1.10) ; we call these H ξi a fundamental system of horospheres for X. Accordingly, the Dirichlet domain D of Γ centered at the base point x 0 can be decomposed into a disjoint union

D = K ∪ C 1 ∪ • • • ∪ C l ,
where K is a convex, relatively compact set containing x 0 in its interior and projecting to K, and the C i are connected fundamental domains for the action of P i on H ξi , projecting to Ci . We let S i = D ∩ ∂H ξi be the corresponding, relatively compact fundamental domain for the action of P i on ∂H ξi , so that

C i = D ∩ H ξi S i × R + .
Fixing an end C, and omitting in what follows the index i, let µ t be the Riemannian measure induced by the Riemannian metric on the horosphere ∂H ξ (t) corresponding to C. In [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negative curvature manifolds[END_REF] we defined the horospherical area function associated with the cusp C as :

A(t) = µ t (P \∂H ξ (t)) = µ t (ψ ξ,t (S)).
This function depends on the choice of the initial horosphere ∂H ξ for the end C, and the following result shows that this dependance is unessential for our counting problem : Proposition 2.1. [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negative curvature manifolds[END_REF] There exists a constant c = c(A, B, diam(S)) such that

v P (R) c 1 A( R 2 )
. This weak equivalence is the key to relate the irregularity of the metric in the cusp to the irregular asymptotic behaviour of the orbital function of P . The second crucial step of our work will then be to describe precisely the contribution of v P in the asymptotic behavior of v Γ assuming δ P = δ Γ .

Cuspidal geometry.

The strategy to construct examples with irregular orbital functions as in Theorems 1.2, 1.3 and 1.4 is to perturb in a suitable manner the metric of a finite volume hyperbolic manifold Γ\H N in a cuspidal end P \H. If H = {y/B ξ (x 0 , y) ≥ t 0 }, the hyperbolic metric writes on H ∂H × R + ≡ R N × R + as g = e -(t-t0) dx 2 + dt 2 in the horospherical coordinates y = (x, t), where dx 2 denotes the induced flat Riemannian metric of ∂H and t = B ξ (x 0 , y) -t 0 . We will consider a new metric g in P \H ξ whose lift to H writes, in the same coordinates, as

g = τ 2 (t)dx 2 + dt 2 .
We extend this metric by Γ-invariance to ΓH and produce a new Hadamard space (X, g) with quotient X = Γ\X. The new manifold X has again finite volume, provided that

+∞ 0 e -ωτ (N -1)t τ N -1 (t) dt < ∞.
Also, notice that the sectional curvatures at (x, t) are given by K [START_REF] Bishop | Geometry of Manifolds AMS Chelsea Pubishing[END_REF]). In sections 4, 5 and 6 we will apply this strategy to specific analytic profiles τ associated with a function τ , and depending on additional real parameters a, b, η, defined as follows. For any convex function τ : R → R + with +∞ 0 τ n-1 (t)dt < ∞ and satisfying the conditions

(x,t) ( ∂ ∂xi , ∂ ∂xj ) = -τ τ 2 and K (x,t) ( ∂ ∂xi , ∂ ∂t ) = -τ τ (see
(3) ∀ t ≤ t 0 τ (t) = e -(t-t0) (4) A < τ /τ < B (5) ω τ = lim sup t→+∞ | ln(τ (t))| t < B we will set, a ≥ t 0 τ a (t) = e -a τ (t -a) for t ∈ R.
This profile defines a manifold X with a cusp C which is hyperbolic at height less than a and then has (renormalized) profile equal to τ . Moreover, given parameters a ≥ t 0 , b ≥ 0 and η ∈]0, A[, a straightforward calculus proves the existence of a profile τ a,b,η such that

         A -η < τ a,b,η (t)/τ a,b,η (t) < B + η for all t, τ a,b,η (t) = e -t for t ≤ a, τ a,b,η (t) = e -ωτ t for t ∈ [∆ + a, ∆ + a + b], τ a,b,η (t) = e -(2∆+a+b) τ (t -(2∆ + a + b)) for t ≥ 2∆ + a + b.
for some constant ∆ = ∆(A, B, η) 0. The function τ a,b,η defines a cusp C which is hyperbolic till height a, with constant curvature -ω 2 τ in a band of width b at height ∆ + a, and which then has asymptotic (suitable renormalized) profile τ after height 2∆ + a + b.

Proof of Theorem 1.2 : construction of convergent lattices

An example of manifold of negative curvature with infinite volume and whose fundamental group is convergent is due to Dal'Bo-Otal-Peigné [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF]. We propose in this section a variation of their argument to produce a convergent nonuniform lattice. We will consider a finite volume hyperbolic manifold Γ\H N with one cuspidal end P \H and deform the metric in this end as explained before to obtain a new Hadamard space (X, g) such that the quotient X = Γ\X has finite volume and a dominant cusp C = P \H with a convergent parabolic group P , whose exponent δ P : -is greater than the Poincaré exponent of Γ acting on H N , that is N -1 ; -equals the Poincaré exponent δ Γ of Γ corresponding to the new metric. For this, we choose τ satisfying the conditions (3), ( 4), [START_REF] Bishop | Geometry of Manifolds AMS Chelsea Pubishing[END_REF]

with δ P = (N -1)ωτ 2 > (N -1)
and +∞ 0 e -ωτ (N -1)t τ (t) N -1 dt < ∞, and we consider the profile τ a for some a > 0 to be precised. Remark that the first condition can be satisfied only if ω τ > 2 which requires B 2 /A 2 > 4. We will denote by d the distance on (X, g) corresponding to this new metric, and by d 0 the hyperbolic distance. We emphasize that the perturbation of the metric will not change neither the algebraic structure of the groups Γ and P , nor the horospheres H(t) (which are only modified in size and not as subsets of H n ) and their radial flow ; however, the orbital function has different behavior before and after perturbation. Now, we need to introduce a natural decomposition of geodesic segments according to their excursions in the cusp, which will enable us to encode elements of Γ by sequences of parabolic elements travelling far in the cusp and elements staying in a fixed, compact subset. We use the same notations as in 2.1 for the compact subset K, the fundamental horosphere H = H ξ1 of X, and the Borel fundamental domain S be for the action of P on ∂H.

Let h > 0 : for every γ ∈ Γ, the geodesic segment [x 0 , γ • x 0 ] intersects r = r(γ) disjoint translates g.H(h) (with the convention r = 0 if the intersection with g∈Γ g.H(h) is empty). In case r ≥ 1, denote by z - 1 , • • • , z - r (resp. z + 1 , • • • , z + r ) the hitting (resp. exit) points of the oriented geodesic segment [x 0 , γ • x 0 ] with translates of H(h) in this order. Hence we get [x0, γ • x0] ∩ g∈Γ g • H(h) = [z - 1 , z1 + ] ∪ • • • ∪ [z - r , z + r ].
Accordingly, when r ≥ 1 we can define the points y -

1 , y + 1 • • • y - r , y + r on [x 0 , g • x 0 ] such that, for any 1 ≤ i ≤ r, the geodesic segment [y - i , y + i ] is the connected component of [x 0 , γ • x 0 ] ∩ g∈Γ g • H containing [z - i , z + i ].
We also set y + 0 = x 0 and y - r+1 = γ • x 0 . With these notations, there exist uniquely determined isometries

g 1 , • • • , g r ∈ Γ and p 1 , • • • , p r ∈ P such that y - 1 ∈ g 1 • S, y + 1 ∈ g 1 p 1 • S, • • • , y + r ∈ g 1 p 1 • • • g r p r • S. Finally, we define g r+1 by the relation γ = g 1 p 1 • • • g r p r g r+1
which we call the horospherical decomposition of γ at height h. Notice that this decomposition depends only on the initial hyperbolic metric. We also set

x + 0 = x 0 , x - r+1 = γ • x 0 and x - i = g 1 p 1 • • • g i • x 0 , x + i = g 1 p 1 • • • g i p i • x 0 for 1 ≤ i ≤ r.
We then have :

Lemma 3.1. Let γ = g 1 p 1 • • • p r g r+1 be the horospherical decomposition of γ at height h : (i) for every i ∈ {1, • • • , r + 1} the geodesic segments [x + i-1 , x - i ] and [x 0 , g i • x 0 ] lie outside the set g∈Γ g.H(c), for c = diam(K) ; (ii) for every i ∈ {1, • • • , r} the geodesic segments [x 0 , p i •x 0 ] have length greater than 2(h-c) and lie outside the set g∈Γ g.H(h -c). Proof. Assume r ≥ 1 and fix 1 ≤ i ≤ r + 1. By construction, each geodesic segment [y + i-1 , y - i ] lies outside g∈Γ g.H. Then, each segment [x + i-1 , x - i ] lies outside g∈Γ g.H(c) since d(x + i-1 , y + i-1 ) and d(x - i , y - i ) are both smaller than c. Since [x + i-1 , x - i ] = g 1 p 1 • • • g i-1 p i-1 • [x 0 , g i •x 0 ]
and the set g∈Γ g.H(c) is Γ-invariant, the same holds for the segment [x 0 , g i •x 0 ]. To prove statement [START_REF] Babillot | Homologie des géodésiques fermées sur des variétés hyperboliques avec bouts cuspidaux[END_REF], notice that the segment [y - i , y + i ] intersects the set g∈Γ g.H(h) and has endpoints in ∂H, so that d(y - i , y + i ) ≥ 2h ; one concludes using the facts that d(x - i , y - i ) and d(x + i , y + i ) are both smaller than c and that [x - i ,

x + i ] = g 1 p 1 • • • p i-1 g i • [x 0 , p i • x 0 ]. 2
Moreover, the distance function is almost additive, with respect to this decomposition : Lemma 3.2. There exists a constant C ≥ 0 such that for all γ ∈ Γ with horospherical decomposition γ = g 1 p 1 • • • g r p r g r+1 at height h :

d(x 0 , γ • x 0 ) ≥ r+1 i=1 d(x 0 , g i • x 0 ) + r i=1 d(x 0 , p i • x 0 ) -rC.
Proof. With the above notations one gets, for C := 4diam(K)

d(x0, γ • x0) = r+1 i=1 d(y + i-1 , y - i ) + r i=1 d(y - i , y + i ) ≥ r+1 i=1 d(x + i-1 , x - i ) + r i=1 d(x - i , x + i ) -4rdiam(K) = r+1 i=1 d(x0, gi • x0) + r i=1 d(x0, pi • x0) -rC. 2
Now assume a > max(h, diam(K)) and let

P a = {p ∈ P/d 0 (x 0 , p • x 0 ) ≥ 2a} Γ a = {γ ∈ Γ/[x 0 , γ • x 0 ] ∩ g.H(a) = ∅ for all g ∈ Γ} for t ≥ 0. Let γ ∈ Γ with horospherical decomposition γ = g 1 p 1 • • • g r p r g r+1 at height h. By Lemma 3.1, the geodesic segments [x 0 , g i • x 0 ], 1 ≤ i ≤ r + 1, stay outside the perturbed set g∈Γ g • H(a), so that g i ∈ Γ a and d(x 0 , g i • x 0 ) = d 0 (x 0 , g i • x 0 )
. Consequently, the Poincaré series of Γ for the perturbed metric is

PΓ(x0, δ) ≤ 1 + Γa\{Id} e -δd(x 0 ,γ•x 0 ) + p∈Pa e -δd(x 0 ,p•x 0 ) + r≥1 g 1 ,••• ,g r+1 ∈Γa p 1 ,••• ,pr ∈Pa e -δd(x 0 ,g 1 p 1 •••gr pr g r+1 •x 0 )
≤ γ∈Γ e -δd 0 (x 0 ,γ•x 0 ) + p∈Pa e -δd(x 0 ,p•x 0 ) + r≥1 e Cδ g∈Γa e -δd(x 0 ,g•x 0 ) p∈Pa e -δd(x 0 ,p•x 0 ) r ≤ γ∈Γ e -δd 0 (x 0 ,γ•x 0 ) + p∈Pa e -δd(x 0 ,p•x 0 ) + r≥1 e Cδ γ∈Γ e -δd 0 (x 0 ,γ•x 0 ) p∈Pa e -δd(x 0 ,p•x 0 ) r For δ = δ P the term γ∈Γ e -δd0(x0,γ•x0) < +∞ since δ P > N -1. On the other hand, the group P being convergent with respect to the new metric g, we deduce that p∈Pa e -δ P d(x0),p•x0)) → 0 as a → +∞ ; we may thus choose a large enough so that

e Cδ P γ∈Γ e -δ P d0(x0,γ•x0) p∈Pa e -δ P d(x0,p•x0) < 1 which readily implies P Γ (x 0 , δ P ) < +∞, hence δ P ≥ δ Γ . As P is a subgroup of Γ, this implies that δ P = δ Γ , hence Γ is a convergent group. 2

Critical gap property versus divergence

In this section we start constructing a hyperbolic lattice Γ of H 2 which is generated by suitable parabolic isometries, so that the resulting surface Γ\H 2 has finite volume. In the Poincaré model for the hyperbolic plane, we choose r ≥ 2 and 2r boundary points

ξ 0 = η 0 , ξ 1 , η 1 , • • • , ξ r , η r = ξ 0 of S 1 = ∂B 2 in
cyclic order, and consider (uniquely determined) parabolic isometries p 1 , • • • , p r such that for 1 ≤ i ≤ r we have p i • ξ i = ξ i and p i • η i-1 = η i . We remark that all η i belong to the Γ orbit of the point η 0 , which is a parabolic fixed point of the isometry

p 0 := p r p r-1 • • • p 1 . Property 4.1. The group Γ = p 1 , • • • , p r is a free non abelian group over p 1 , • • • , p r .
The quotient Γ\B 2 is a finite surface with r + 1 cuspidal ends, with a cusp C i for each parabolic subgroup P i = p i for i = 1, .., r, and another cusp C 0 corresponding to the parabolic subgroup P 0 = p 0 fixing ξ 0 .

Each element γ ∈ Γ \ {Id} can be written in a unique way as a word with letters in the alphabet

A := {p ±1 1 , • • • , p ±1 r } ; namely, one gets (6) γ = p 1 j1 • • • p n jn with p 1 j1 , • • • , p n jn ∈ A
, n ≥ 1 and with adjacent letters which are not inverse to each other. Such expression with respect to the natural (but not canonical) choice of the alphabet A is called a coding of elements of Γ. We will call j 1 is the first index of γ, denoted by i γ ; similarly j n the last index and is denoted by l γ .

4.1.

A new coding for elements of Γ. We code here the elements of Γ by blocks, with some admissibility rules to be precised. This new coding is designed to obtain a contraction property for an operator that will be introduced and studied in the next sections whose restriction to some suitable space of functions present remarkable spectral properties. We first rewrite the decomposition (6) as follows

(7) γ = p 1 i1 p 2 i2 • • • p m im with m ≥ 1, 1 , • • • , m ∈ Z * and i j = i j+1 for 1 ≤ j < m.
When all the j , 1 ≤ j ≤ m, belong to {±1}, one says that γ is a level 1 word ; the set of such words is denoted by W 1 .

Then, we select all the j , 1 ≤ j ≤ m, with | j | ≥ 2 and write γ as

(8) γ = p l0 j0 Q 1 p l1 j1 Q 2 • • • p l k-1 j k-1 Q k p l k j k
where :

• k ≥ 0 • |l 1 |, • • • , |l k-1 | ≥ 2, • |l 0 |, |l k | = 1,
• each Q j is either empty or a level 1 word, with i Qj = j i-1 and l Qj = j i . The decomposition by blocks (8) is still unique ; it only uses letters from the new alphabet

B := P 1 ∪ • • • ∪ P r ∪ W 1
where P i := {p n i /|n| ≥ 2} for 1 ≤ i ≤ r, possibly with p l0 i0 = 1 or p lr ir = 1. We will call blocks the letters of this new alphabet, and say that a word β 1 • • • β m in the alphabet B is admissible if the last letter of any block β i is different from the first one of β i+1 for 1 ≤ i ≤ m -1. So, any γ ∈ Γ \ {Id} can be written as a finite, admissible word β 1 • • • β m on B ; the ordered sequence of the β i 's is called the B-decomposition of γ and the number m of blocks is denoted by |γ| B . Finally, we denote by Σ B the set of all finite admissible words with respect to B.

4.2.

A new metric in the cusps. We consider a fundamental system of horoballs

H 0 , H 1 , H 2 , • • • , H r centered respectively at the parabolic points ξ 0 , ξ 1 , ξ 2 , • • • , ξ r
and such that all the horoballs γ • H i , for γ ∈ Γ, 0 ≤ i ≤ r, are disjoint or coincide, as in section 2.1. Then, we modify the hyperbolic metric in the cuspidal ends Ci = P i \H i as follows. We choose positive constants a 0 , a 1 , • • • , a r-1 , a r , b and η, functions τ 0 , τ 1 , • • • , τ r-1 and τ r as in section 2.2 such that ω τr = max(ω τ0 , • • • , ω τr ) > 1 and we prescribe the profile τ i,ai for the i-th cusp Ci for 0 ≤ i ≤ r -1, and the profile

Ping-Pong by blocks.

For any 1 ≤ i ≤ r, we consider the sub-arcs

I i := [η i-1 , η i ] and I i := [p -1 i • η i-1 , p i • η i ] of S 1 containing ξ i .
There exists a ping-pong dynamic between these intervals : namely, for any block β ∈ B, we have

• if β ∈ W 1 , then β • I i ⊂ I i β for any i = l β • if β ∈ P l with l = i β = l β , then β • I i ⊂ I l for any i = l.
Moreover, for any γ with B-decomposition β 1 • • • β m , we define a compact subset K γ ⊂ S 1 as follows :

• K γ = ∪ i =lγ I i , if β m ∈ W 1 • K γ = ∪ i =lγ I i , if β m ∈ P l with l = l γ .
Then, using the fact that the closure of the sets I i and ∂X \ I i are disjoint, one gets :

Lemma 4.3. There exists a constant C = C(A, η) > 0 such that d b (x 0 , γ • x 0 ) -C ≤ B x (γ -1 • x 0 , x 0 ) ≤ d b (x 0 , γ • x 0 )
for any γ ∈ Γ and any x ∈ K γ .

This lemma implies in particular the following contraction property (see Prop.

β = (β n ) n≥1 such that the concatenation γ * β = (b 1 • • • b m β 1 • • • β i • • • ) is admissible. We also set J γ := Cl{ π(γ * β) | β ∈ K γ },
that is the closure of points corresponding to admissible sequences obtained by concatenation with the B-decomposition of γ.

As indicated previously, this coding by blocks is of interest since the classical shift operator on Σ + B induces locally, exponentially expanding maps T n on ∂ 0 X ; the map T , described for instance in details in [START_REF] Dal | Groupes du Ping-Pong et géodésiques fermées en courbure -1[END_REF], has countably many inverse branches, each of them acting by contraction on some subset of ∂X. Namely, we consider on Σ + B the natural shift θ defined by θ(

β) := (β k+1 ) k≥1 , ∀ β = (β k ) k≥1 ∈ Σ + B
This map induces a transformation T : ∂ 0 X → ∂ 0 X via the coding π ; moreover, T can be extended to the whole ∂X by setting, for any γ with B-decomposition

γ = γ 1 γ 2 • • • γ n T (γ.ξ i ) := γ 2 • • • γ n .ξ i
and T (ξ i ) := ξ i for i = 0, ..., n. Then, for every block β ∈ B, the restriction of T to J β is the action by β -1 ; by the dynamic described above, the inverse branches of the map T have the following property Property 4.6. There exist 0 < r < 1 and a constant C > 0 such that, for any γ ∈ Γ with |γ| B = k and for x, y ∈ K γ we have

D 0 (γ • x, γ • y) ≤ Cr k D 0 (x, y).
Correspondingly, for |γ| B = k and x, y ∈ J γ , we have the following expanding property :

D 0 (T k x, T k y) ≥ 1 Cr k D 0 (x, y)
This property is crucial for the investigation of the spectral properties of the transfer operator, which will be introduced in the next section.

Existence of divergent exotic lattices

This section is devoted to prove Theorem 1.3. An example of an exotic divergent discrete group has already been constructed in [START_REF] Peigné | On some exotic kleinians groups[END_REF] : it is a Schottky group Γ generated by both parabolic and hyperbolic isometries, with LΓ = S 1 and, consequently, the quotient manifold X = Γ\X has infinite volume. The strategy in [START_REF] Peigné | On some exotic kleinians groups[END_REF] to obtain both exoticity and divergence is the following : one starts from a free, convergent Schottky group satisfying a ping-pong dynamics, obtained by perturbation of a hyperbolic metric in one cusp ; then, it is proved that the group becomes divergent if the perturbation of the metric is pushed far away in the cusp finally, it is shown that the divergent group can be made exotic (without losing the divergence property) by pushing the perturbation of the metric at a suitable height, by a continuity argument which is consequence of a careful description of the spectrum of a transfer operator naturally associated to Γ. Here, we adapt this approach to obtain a discrete group with finite covolume.

We start from the surface X = Γ\X with r + 1 cusps described in 4.2, with a dominant cusp Cr = P r \H r and make Γ convergent by choosing a 0 , ..., a r 0, as in Theorem 1.2. Besides a different coding by blocks (due to the generators which are all parabolic) which gives a slightly different expression for the transfer operator associated to Γ, the main difficulty here is to show that Γ can be made divergent. This cannot be achieved now by simply pushing the perturbation far away in the cusp, since in our case δ Pr is strictly greater than the critical exponent of the subset of elements staying in the compact, non-perturbed part ; to obtain the divergence we rather modify Cr with a profile τ r,ar,b which equals the profile of a cusp with constant curvature metric -ω 2 τr on a sufficiently large band of width b 0. The, we use the spectral properties of the tranfer operator to show that there exists a suitable value b = b * for which the lattice becomes simultaneuosly exotic and divergent.

5.1. On the spectrum of transfer operators. From the analytic point of view, the action of the geodesic flow is encoded by a transfer operator associated to the transformation T described above. Namely, for any Borel bounded functions g : ∂X → R we define

L z ϕ(x) =
T y=x e -zr(y) ϕ(y), x ∈ ∂X for a function r to be defined, depending on the metric g a0,••• ,ar,b , in particular on the width b of the band in the cusp C r where the curvature is -ω 2 τ . We will need to understand precisely the dependance on b when b ∈ [0, b 0 ], so we stress the dependance on b writing L b,z and r b . The function r b we consider is given by

r b : y → B β -1 •y (β -1 • x 0 , x 0 ) = B x (β -1 • x 0 , x 0 )
where B x is the Busemann function with respect to the metric g = g a0,••• ,ar,b . The idea behind the choice of r b is to relate the norm |L m b,s 1| ∞ to the Poincaré series of Γ ; on the other hand, the Busemann functions are good approximations of the distance functions d b by 4.3, and are more adapted to express the iterates of L b,s , by the cocycle property. To explicitly express the set T -1 (x) we remark that, the alphabet B being countable, the pre-images of x ∈ ∂X by T are the points y = β • x, for those blocks β ∈ B such that x belongs to K β , that is :

• if x ∈ I i then β ∈ ∪ j =i P j or β ∈ W 1 with l β = i,
• if x ∈ I i \ I i then β ∈ ∪ j =i P j . This motivates the introduction of weight functions w b,z (γ, •) : ∂X → C, which are defined for z ∈ C and γ ∈ Γ as w b,z (γ, x) = 1 Kγ (x)e -zr b (γ,x)

We can now make explicit the definition of transfer operators : for b ∈ [0, b 0 ], z ∈ C and any bounded Borel function ϕ : ∂X → C we set

(9) (L b,z ϕ)(x) = β∈B w b,z (β, x)ϕ(β • x) = β∈B 1 K β (x)e -zBx(β -1 •x0,x0) ϕ(β • x), ∀x ∈ ∂X
The operator L b,z is well defined when Re(z) ≥ δ and acts on the space C(∂X) of C-valued continuous functions on ∂X endowed with the norm | • | ∞ of the uniform convergence ; however, to obtain a quasi-compact operator with good spectral properties, we will consider its restriction to a subspace L α ⊂ C(∂X) of Hölder continuous functions with respect to D 0 , for α given by Lemma 4.2. Namely we let

L α := {ϕ ∈ C(∂X) : ϕ = |ϕ| ∞ + [ϕ] α < +∞}
C = C(z) > 0 such that for any γ ∈ Γ w b,z (γ, •) ≤ Ce -Re(z)d b (x0,γ•x0) .
Observe now that the w z (γ, •) satisfy the following cocycle relation : if the B-decomposition of γ = γ 1 γ 2 is given by the simple concatenation

γ 1 * γ 2 of the γ i , i.e. |γ 1 γ 2 | B = |γ 1 | B +|γ 2 | B , then w b,z (γ 1 γ 2 , x) = w b,z (γ 1 , γ 2 • x) • w b,z (γ 2 , x).
This equality leads to the following simple expression of the iterates of the transfer operators : for any n ≥ 1 and x ∈ ∂X

(10) (L k b,z ϕ)(x) = γ∈Γ |γ| B =n w b,z (γ, x)ϕ(γ • x) = γ∈Γ |γ| B =k 1 Kγ (x)e -zBx(γ -1 •x0,x0) ϕ(γ • x).
Following [START_REF] Peigné | On some exotic kleinians groups[END_REF] and using the contraction property in Lemma 4.4, we first describe the spectrum of the operators L b,z for real values of the parameter z greater than or equal to δ := ω τ /2 : Proposition 5.2. For any b ≥ 0 and s ≥ δ = ω τ /2, the operator L b,s acts both on (C(∂X), | • | ∞ ) and (L α , • ), with respective spectral radius ρ ∞ (L b,s ) and ρ α (L b,s ). Furthermore, the operator L b,s is quasi-compact 3 on L α , and :

(1) • ρ α (L b,s ) = ρ ∞ (L b,s ), • ρ α (L b,s ) is a simple, isolated eigenvalue of L b,s ,
• the eigenfunction h b,s associated with ρ α (L b,s ) is non negative on ∂X,

(2) for any s ≥ δ, the map b → L b,s is continuous from R + to the space of continuous linear operators on L α ,

(

) the function s → ρ ∞ (L b,s ) is decreasing on [δ, +∞[. Remark. 5.3. Let ρ b,s = ρ ∞ (L b,s ). 3
Then, L b,s h b,s = ρ b,s h b,s , the function h b,s being unique up to a multiplicative constant. By duality, there also exists a unique probability measure σ b,s on ∂X such that σ b,s L b,s = ρ b,s σ b,s ; the function h b,s becomes uniquely determined imposing the condition σ b,s (h b,s ) = 1, which we will assume from now on.

3. In other words its essential spectral radius on this space is less than ρα(L b,s ) 5.2. From convergence to divergence : proof of Theorem 1.3. Combining expression [START_REF] Bourdon | Structure conforme au bord et flot géodésique d'un CAT(-1)-espace[END_REF] with Lemma 4.3, one gets for any s, b > 0 and k ≥ 0

(11) |L k b,s 1| ∞ γ∈Γ |γ| B =k exp(-sd b (x 0 , γ • x 0 )).
Consequently, the Poincaré series P Γ (s) of Γ relatively to d b and the series k≥0

|L k b,s 1| ∞ converge or diverge simultaneously. Following [START_REF] Peigné | On some exotic kleinians groups[END_REF], we see that the function s → ρ ∞ (L b,s ) is strictly decreasing on R + ; the Poincaré exponent of Γ relatively to d b is then equal to

δ Γ = sup s ≥ 0 : ρ ∞ (L b,s ) ≥ 1 = inf s ≥ 0 : ρ ∞ (L b,s ) ≤ 1 .
This expression of the Poincaré exponent in terms of the spectral radius of the transfer operator will be useful to prove Theorem 1.3. We first get the Proof. By the previous section and the choice of the profiles τ i , 0 ≤ i ≤ r, we may fix the constants a 0 , • • • , a r large enough, in order that the group Γ acting on (X, g a0,••• ,ar,0 ) is a convergent lattice with exponent δ = ωτ 2 . Considering just the contribution of words containing powers of p r , by the triangle inequality we get

γ∈Γ e -δd b (x0,γ•x0) ≥ k≥1 Q 1 ,••• ,Q k ∈W 1 |l1|,••• ,|lm|≥2 e -δd b (x0,p l 1 r Q1•••p l k r Q k •x0) ≥ k≥1   |l|≥2 e -δd b (x0,p l r •x0) Q∈W1 e -δd b (x0,Q•x0)   k .
When |l| is large enough, say |l| ≥ n ar ≥ 1, the geodesic segments [x 0 , p l • x 0 ] intersect the "horospherical band" H ξr (a r , a r + b) := H ξr (a r ) \ H ξr (a r + b) in which the curvature is -ω 2 τ ; more precisely, for any b > 0, there exists n b ≥ n ar such that : for any n ar ≤ l ≤ n b , two points such that there exist integers 1 ≤ n ar < n b When |l| is large enough, say |l| ≥ n ar ≥ 2, the geodesic segments [x 0 , p l • x 0 ] intersect the horoball H ξr (a r ) ; furthermore, for any b > 0, there exists

n b ≥ n ar , with n b → +∞ as b → +∞, such that [x 0 , p l • x 0 ] ∩ H ξr (a r + b) = ∅ when n ar ≤ |l| ≤ b so that the geodesic segment [x 0 , p l • x 0 ] ∩ H ξr (a r )
remains in a subset of X where the curvature is -ω 2 τ . Now, for such l, the lenth of the segment [x 0 , p l • x 0 ], evaluated either with respect to the metric g b or to the hyperbolic metric of curvature -ω 2 τ , equals the length of [x 0 , p l • x 0 ] ∩ H ξr (a r ) up to 2a r + c, for some c > 0 depending only on the bounds of the curvature ; this yields, for

n ar ≤ |l| ≤ n b d b (x 0 , p l • x 0 ) ≥ d 0 (x 0 , p l • x 0 ) ω τ -2(2a r + c) ≥ 2 ω τ ln |l| -2(2a r + c) so that, for δ = ωτ 2 : |l|≥2 e -δd b (x0,p l •x0) ≥ n b |l|=na r e -δd b (x0,p l •x0) n b |l|=na r 1 |l| 2/ωτ
(L 0,δ ) = ρ ∞ (L 0,δ ) ≤ 1 and ρ α (L b0,δ ) = ρ ∞ (L b0,δ ) ≥ 1 ; thus, there exists b * ∈ [0, b 0 ] such that ρ w (L b * ,δ ) = ρ ∞ (L b * ,δ ) = 1. Since the function s → ρ ∞ (L b * ,s ) is strictly decreasing on [δ, +∞[, one gets ρ ∞ (L b * ,s ) < 1 as soon as s > δ.
For such values of s, the Poincaré series P Γ (s) of Γ relatively to the metric g b * thus converges, this implies that its Poincaré exponent δ Γ,b * is ≤ δ ; we have in fact δ Γ,b * = δ since δ pr = δ and p r ∈ Γ. Finally, the eigenfunction h b * ,δ of L b * ,δ associated with ρ α (L b * ,δ ) being non negative on ∂X, one gets h b * ,δ 1 and so

k≥0 |L k b * ,δ 1| ∞ k≥0 |L k b * ,δ h b * ,δ | ∞ = k≥0 |h b * ,δ | ∞ = ∞
which implies, by [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms Lecture Notes in Mathematics[END_REF], that Γ is divergent with respect to the metric g b * . 2

Counting for some divergent exotic lattices

We consider here the group Γ = p 1 , • • • , p r , r ≥ 2, where the p i , 1 ≤ i ≤ r, satisfy the conditions given in Property 4.1 ; we endow X with a metric of the form g = g a0,••• ,ar,b * as defined in subsection 4.2, where profiles τ 0 , τ 1 , • • • , τ r satisfy conditions (3), (4), ( 5) and constants a 0 , a 1 , a 2 , • • • , a r and b * are suitably chosen in such a way

• H 0 : the group Γ is exotic and divergent with respect to g, with Poincaré exponent

δ = δ Γ = ω 2 with ω = max(ω τ0 , • • • , ω τr ) > 2. • H 1 : the exponential growth of the set {β ∈ W 1 /d(x 0 , β • x 0 ) ≤ R} as R → +∞ is strictly less than δ.
• H 2 : there exist κ ∈]1/2, 1[ and a slowly varying function ( 4) L such that

(12) p∈Pr/d(x0,p•x0)>t e -δd(x0,p•o) t→+∞ ∼ L(t) t κ .
• H 3 : the groups P l , 0 ≤ l ≤ r -1, satisfy the condition

(13) p∈P l /d(x0,p•x0)>t e -δd(x0,γ•o) = o L(t) t κ .
Since b * is now fixed, we will omit it it the sequel and set d = d b * . Under those hypotheses, the subgroup P r corresponding to the cusp C r has a dominant influence on the behavior of the orbital function of Γ. Hypothesis H2 is inspired by probability theory, it corresponds to a heavy tail conditions satisfied by random walks which have been intensively investigated [START_REF] Erickson | Strong renewal theorems with infinite mean[END_REF]. Theorem 1.4 will be a direct consequence of the following one : Theorem 6.1. Let Γ be a finite area Fuchsian group satisfying Property 4.1 and let us endow X with a metric g such that hypotheses H 0 , H 1 , H 2 and H 3 hold. Then, for any 1 ≤ j ≤ r, any fixed x j / ∈ I j , there exists C j > 0 such that

(14) {γ ∈ Γ j /B xj (γ -1 • x 0 , x 0 ) ≤ R} R→+∞ ∼ C j e δΓR R 1-κ L(R)
.

where Γ j denotes the set of γ ∈ Γ with last letter j.

Indeed, since x j / ∈ I j is fixed, there exists c(x j ) > 0 such that

d(x 0 , γ • x 0 ) -c(x j ) ≤ B xj (γ -1 • x 0 , x 0 ) ≤ d(x 0 , γ • x 0 )
for any γ ∈ Γ j and ( 14) thus implies

{γ ∈ Γ j /d(x 0 , γ • x 0 ) ≤ R} R→+∞ e δΓR R 1-κ L(R)
.

Theorem 1.4 follows summing over j ∈ {1, • • • , r}.

Let us make some remarks :

(1) We have seen in the previous section how one may choose profiles τ 0 , • • • , τ r and parameters a 0 , • • • , a r and b * in such a way hypothesis H 0 holds.

(2) Hypothesis H 1 holds when ω := max(ω τ0 , • • • , ω τr ) > 2, a 0 = +∞ and the constants a 1 , • • • , a n are chosen large enough in such a way all the geodesic paths [x 0 , β•x 0 ], β ∈ W 1 , remain in a closed subset of X where the curvature is -1 ; the exponential growth of {β ∈ W 1 /d(x 0 , β • x 0 ) ≤ R} as R → +∞ is less or equal to 1 and then strictly less than δ = ω/2.

(3) Hypothesis H 2 holds in particular when

d(x 0 , p n r • x 0 ) = 2 ln n + 2(1 + κ)(ln ln n + O(n))
ω for some ω > 2. This equality involves only the asymptotic geometry on the cusp C r as it is equivalent to prescribe a profile. Hence it is compatible with any choice of the parameter b. The critical exponent of P r is thus δ = ω/2 and one gets, as t → +∞, By Theorem B in [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF], it readily follows that the Bowen-Margulis measure associated with Γ is infinite.

(5) Observe at last that hypothesis H 3 is satisfied in particular when the gap property δ P l < δ hold for any 0 ≤ l ≤ r -1.

We present in this section the main steps of the proof of Theorem 6.1 ; we refer to [START_REF] Dal'bo | Groupes du ping-pong et géodésiques fermées en courbure -1[END_REF] for details. For any R ≥ 0, let us denote W j (R, •) the measure on R defined by : for any Borel non negative function ψ : R → R

W j (R, ψ) := γ∈Γj e -δBx j (γ -1 •x0,x0) ψ(B xj (γ -1 • x 0 , x 0 ) -R).
One gets 0 ≤ W j (R, ψ) < +∞ when ψ has a compact support in R since the group Γ is discrete, furthermore r j=1 W j (R, ψ) = e -δR v Γ (R) when ψ(t) = e δt 1 t≤0 . If one proves that for any non negative and continuous function ψ with compact support and such that

R + ψ(x)dx > 0 (15) W j (R, ψ) R→+∞ ∼ C j R 1-κ L(R) R ψ(x)dx,
this convergence will also hold for non negative functions with compact support in R and whose discontinuity set has 0 Lebesgue measure ; Theorem 6.1 follows since v Γ (R) = e δR r j=1 n≥0 W j (R, e δt 1 ]-(n+1),-n] (t)). From now on, we fix a continuous function ψ : R → R + with compact support ; one gets, for 1 ≤ j ≤ r fixed

W j (R, ψ) = k≥0 γ∈Γj /|γ| B =k e -δBx j (γ -1 •x0,x0) ψ(B xj (γ -1 • x 0 , x 0 ) -R) .
Notice that for γ ∈ Γ j with B-decomposition

γ = β 1 • • • β k , one gets B xj (γ -1 • x 0 , x 0 ) = r(β 1 • • • β k • x j ) + r(β 2 • • • β k • x j ) + • • • + r(β k • x j ) = r(y) + r(T • y) + • • • + r(T k-1 • y) = S k r(y)
where one denotes y := γ • x j , so that ( 16)

W j (R, ψ) = k≥0 y∈∂X/T k •y=xj e -δS k r(y) ψ(S k r(y) -R).
By a classical argument in probability theory due to Stone (see for instance [START_REF] Erickson | Strong renewal theorems with infinite mean[END_REF]), it suffices to check that the convergence [START_REF] Dal'bo | On the horoboundary and the geometry of rays of negatively curved manifolds[END_REF] holds when ψ has a C ∞ Fourier transform with compact support : indeed, the test function ψ may vary in the set H of functions of the form ψ(x) = e itx ψ 0 (x) where ψ 0 is an integrable and strictly positive function on R whose Fourier transform is C ∞ with compact support. When ψ ∈ H, one can use the inversion Fourier formula and write ψ(x) = 1 2π R e -itx ψ(t)dt, so that, for any 0 < s < 1

W j (s, R, ψ) := k≥0 s k y∈∂X/T k •y=xj e -δS k r(y) ψ(S k r(y) -R) = k≥0 s k y∈∂X/T k •y=xj 1 2π R e itR e -(δ+it)S k r(y) ψ(t)dt = 1 2π R e itR ψ(t) k≥0 s k y∈∂X/T k •y=xj e -(δ+it)S k r(y) dt = 1 2π R e itR ψ(t) k≥0 s k L k δ+it 1(x j ) dt = 1 2π R e itR ψ(t) (I -sL δ+it ) -1 1(x j )dt (17)
where, for any z ∈ C, the operator L z is the transfer operator associated to the function r formally defined in the previous section by : for any Borel bounded function ϕ : ∂X → C and any x ∈ ∂X L z ϕ(x) := y∈∂X/T •y=x e -zr(y) ϕ(y).

We know that the operators L z , Re(z) ≥ δ, are bounded and quasi-compact on the space L ω (∂X) of Hölder continuous function on (∂X, D 0 ). The subsection 6.1 is devoted to the control of the peripherical spectrum of L δ+it on L ω . In subsection 6.2, we describe the local expansion of the dominant eigenvalue. Atlast we achieve the proof using arguments coming from renewal theory (subsection 6.3).

6.1. The spectrum of L δ+it on L ω . First we need to control the spectrum of L z when z = δ + it, t ∈ R. By Lemma 5.1, the operators L z are bounded on L ω when Re(z) ≥ δ ; the spectral radius of L z will be denoted ρ ω (z) throughout this section. In the following Proposition, we describe its spectrum on L ω when Re(z) = δ. Proposition 6.2. There exist 0 > 0 and ρ 0 ∈]0, 1[ such that, for any t ∈ R with modulus less than 0 , the spectral radius ρ ω (δ + it) of L δ+it is > ρ 0 and the operator L δ+it has a unique eigenvalue λ t of modulus ρ(δ + it), which is simple and closed to 1, the rest of the spectrum being included in a disc of radius ρ 0 .

Furthermore, for any A > 0, there exists ρ A ∈]0, 1[ such that ρ ω (δ + it) ≤ ρ A for any t ∈ R such that 0 ≤ |t| ≤ A. Notation 6.3. We denote σ the unique probability measure on ∂X such that σL δ = σ and h the element of L ω such that L δ h = h and σ(h) = 1.

Proof of Proposition 6.2 This is exactly the same proof that the one presented in [START_REF] Babillot | Homologie des géodésiques fermées sur des variétés hyperboliques avec bouts cuspidaux[END_REF](Proposition 2.2) and [START_REF] Dal'bo | Groupes du ping-pong et géodésiques fermées en courbure -1[END_REF] : the operators L δ+it are quasi-compact on L ω and it is sufficient to control their peripherical spectrum. When t is closed to 0, we use the perturbation theory to conclude that the spectrum of L δ+it is closed to the one of L δ : it is thus necessary to prove that the map t → L δ+it is continuous on R. The following Lemma is devoted to precise the type of continuity of this function. Lemma 6.4. Under the hypotheses H 1 , H 2 and H 3 , there exists a constant C > 0 such that

L δ+it -L δ+it ≤ C|t -t| κ L 1 |t -t|
Proof. We will use the following classical fact ( [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF] p.272 and [START_REF] Erickson | Strong renewal theorems with infinite mean[END_REF] Lemmas 1 & 2) : Lemma 6.5. Let µ be a probability measure on R + , set F µ (t) := µ[0, t] and m(t) := t 0 (1 -F µ (s))ds and assume that there exist κ ∈ R and a slowly varying function L such that 1 -F µ (t) ∼ L(t) t κ as t → +∞. One thus gets

(t) = 1 -e -iπ κ 2 Γ(1 -κ)t κ L 1 t (1 + o(t)). (19) μ
Noticing that m(t) :

= t 0 xµ(dx) = m(t) -t(1 -F µ (t)) one also gets m(t) ∼ κm(t) as t → +∞ ; furthermore, decomposing +∞ 0 |e itx -1| µ(dx) as [0,1/t] |e itx -1| µ(dx) + ]1/t,+∞[ |e itx -1| µ(dx)
and applying the previous estimations, one gets, for any t ∈ R +∞ 0 |e itx -1| µ(dx) t κ L(1/t).

We now apply [START_REF] Dal'bo | Volume growth and rigidity of negatively curved manifolds of finite volume[END_REF] with the probability measures µ l , 1 ≤ l ≤ r, on R + defined by µ l := c l p∈P l δ d(x0,p•x0) where c l > 0 is some normalizing constant. As a direct consequence, under hypotheses H 1 , H 2 and H 3 , one gets (up to a modification of L by multiplicative constant) [START_REF] Eberlein | Geometry of Nonpositively curved manifolds University of[END_REF] similarly, for 1 ≤ l < r, one has ( 22)

p∈Pr/d(x0,p•x0)≤t d(x 0 , p • x 0 )e -δd(x0,p•x0) t→+∞ ∼ t 1-κ 1 -κ L(t).
p∈P l |e itd(x0,p•x0) -1| × e -δd(x0,p•x0) = t κ L(1/t)o(t).
In the same way, by lemma 4.3, for any x ∈ ∂X \ I r , one gets ( 23)

p∈Pr/r(p•x)≤t e -δr(p•x)) L(t) t κ , p∈Pr/r(p•x))>t r(p • x)e -δr(p•x) t 1-κ L(t), (24)
p∈Pr |e itr(p•x) -1| × e -δr(p•x) t κ L(1/t) and, for 1 ≤ l ≤ r -1 and x ∈ ∂X \ I l ( 25
) p∈P l |e itr(p•x) -1| × e -δr(p•x) = t κ L(1/t)o(t).
Similarly, since the exponential growth of the set {β ∈ W 1 /d(x 0 , β • x 0 ) ≤ R} is < δ, we also have, for any 1 ≤ l ≤ r and x ∈ ∪ l =l I l (26)

β∈W 1 l β =l |e itr(β•x) -1| × e -δr(β•x) t κ L(1/t)o(t).
Noticing that for any β ∈ B and

x ∈ K β |w δ+it (β, x) -w δ+it (β, x)| ≤ |e i(t-t )•x)) -1| × e -δr(β•x)
one readily gets, combining the above estimations ( 23) to (26) all together

(27) L δ+it -L δ+it ∞ ≤ β∈B |w δ+it (β, •) -w δ+it (β, •)| ∞ |t -t | κ L 1 |t -t | .
Now, for any β ∈ B and x, y ∈ K β , one gets

w δ+it (β, x) -w δ+it (β, x) -w δ+it (β, y) -w δ+it (β, y) ≤ e -δr(β•x) × e i(t-t )r(β•x) -1 -e i(t-t )r(β•y) -1 + e -δr(β•x) -e -δr(β•y) × e i(t-t )r(β•y) -1 ≤ e -δr(β•x) × e i(t-t )(r(β•x)-r(β•y)) -1 + e -δr(β•y) e δ(r(β•y)-r(β•x)) -1 × e i(t-t )r(β•y) -1 e -δr(β•x) [r • β] × |t -t | + e -δr(β•y) [r • β] × e i(t-t )r(β•y) -1 D 0 (x, y) so that, as above β∈B [w δ+it (β, •) -w δ+it (β, •)] |t -t | κ L 1 |t -t |
. We achieve the proof of the Lemma combining this last inequality with (27). 2 6.2. On the local expansion of the dominant eigenvalue λ t . We explicit here the local expansion near 0 of the map t → λ t : Proposition 6.6. Under the hypotheses H 0 -H 4 , there exists C Γ > 0 such that (28)

λ t = 1 -C Γ e -iπ κ 2 t κ L(1/t) (1 + o(t)).
Proof. Recall first that 1 is a simple eigenvalue of L δ with L δ h = h and σ(h) = 1 ; since t → L δ+it is continuous on R, for t closed to 0 there exists a function h t ∈ L ω such that L δ+it h t = λ t h t , this function being unique if we impose the normalization condition σ(h t ) = 1. The maps t → λ t and t → h t have the same type of continuity than t → L δ+it and one gets the identity

λ t = σ (L t h t ) = σ (L t h) + σ (L δ+it -L δ )(h t -h)) .
By the previous subsection, the second term of this last expression is t κ L(1/t)

2

. It remains to precise the local behavior of the map t → σ (L t h) ; one gets

σ (L t h) = 1 + r l=0 σ l with σ 0 := β∈W1 K β h(β • x))e -δr(β•x) (e -itr(β•x) -1)σ(dx) and, for 1 ≤ l ≤ r σ l := β∈ P l K β h(β • x))e -δr(β•x) (e -itr(β•x) -1)σ(dx).
By (26), the terms σ l , l = r, are of the form t κ L(1/t)o(t). To control the term σ r , one sets ∆(n, x) := r(p n r • x) -d(x 0 , p n r • x 0 ) for any x ∈ ∂X \ I r and n ∈ Z. The following lemma readily implies that the quantity ∆(n, x) tends to -(x|ξ r ) x0 as |n| → +∞. Lemma 6.7. For any parabolic group P :=< p > with fixed point ξ, we have

B x (p ±n • x 0 , x 0 ) = d b (x 0 , p n • x 0 ) -2(ξ|x) x0 + x (n)
with lim n→+∞ x (n) = 0, the convergence being uniform on compact sets of ∂X \ {ξ}.

Proof. Let (x m ) be a sequence of elements of X converging to x. We have One gets ∂X\Ir h(p n r • x))e -δ∆(n,x) σ(dx) > 0 for any |n| ≥ 2 ; by [START_REF] Dal'bo | Groupes du ping-pong et géodésiques fermées en courbure -1[END_REF], there exists c > 0 such that

σ r1 = -ce -iπ κ 2 Γ(1 -κ)t κ L 1 t (1 + o(t)).
On the other hand Equality (28) follows immediately. 2 6.3. Renewal theory and proof of Theorem 6.1. For technical reasons (see for instance [START_REF] Erickson | Strong renewal theorems with infinite mean[END_REF]) which will appear in the control of the term W Notice that, when ψ is a continuous function with compact support in R + , the terms ψ(-S k r(y) -R) of these sums vanish for R large enough, so that W j (R, ψ) = W j (R, ψ) and W j (s, R, ψ) = W j (s, R, ψ) in this case. By [START_REF] Dal'bo | On the growth of quotients of Kleinian groups[END_REF], for any 0 < s < 1, one gets, Fix 0 > 0 and let ρ(t) be a symmetric C ∞ -function on R which is equal to 1 on a [-0 , 0 ] and which vanishes outside [-2 0 , 2 0 ] ; one thus decomposes W j (s, R, ψ) as

|σ r2 | ≤ |t|
W j (s, R, ψ) = W (1)
j (s, R, ψ) + W

(2) j (s, R, ψ) + W The functions t → λ t has the same regularity as t → L δ+it ; by lemma 6.4 one can thus check that ψ 1 : t → ψ(t)(1 -ρ(t)) (I -L δ+it ) -1 1(x j ) + (I -L δ-it ) -1 1(x j ) and ψ 2 : t → ψ(t)ρ(t) (I -L δ+it ) -1 1(x j ) -σ(∂X \ I j )h(x j ) 1 -λ t + (I -L δ-it ) -1 1(x j ) -σ(∂X \ I j )h(x j ) 1 -λ -t satisfy the inequality |ψ k (s) -ψ k (t)| |s -t| κ L 1 |s-t| , k = 1, 2. This yields some information on the speed of convergence to 0 of their Fourier transform : indeed, for any θ < κ, there exists C θ > 0 such that ( 29) W

(1)

j (R, ψ) ≤ C θ R θ
and W

(2)

j (R, ψ) ≤ C θ R θ
(which readily implies lim R→+∞ R 1-κ L(R) W

(1)

j (R, ψ) = lim R→+∞ R 1-κ L(R) W (2)
j (R, ψ) = 0 as soon as θ ∈]1 -κ, κ[, which is possible since κ > 1/2). On the other hand, by section 5 in [START_REF] Erickson | Strong renewal theorems with infinite mean[END_REF], one gets for 1/2 < κ < 1 lim R→+∞ R 1-κ L(R) W

2 ,

 2 1 (t)dt < ∞ and the end C = P \H is a new cusp ; we call the function τ the analytic profile of the cusp C. The horospherical area function A associated with the profile τ satisfies A τ N -1 ; by Proposition 2.1, it implies that : (a) the parabolic group P has critical exponent δ P = (N -1)ωτ for ω τ := lim sup R→+∞ 1 R | ln(τ (R))|, (b) P is convergent if and only if

≤

  τ r,ar,b on the last. dominant cusp C r = C. This yields a new surface X = (B 2 , g a0,••• ,ar,b ), with quotient X = Γ\X of finite area. Since the metric on X depends, in particular, on the value of the parameter b, which will play a crucial role in what follows, we shall denote the induced distances on X, ∂X ∼ = S 1 and the conformal factor respectively by d b , D b and | • | b ; on the other hand, we shall omit the index b in the Busemann function and in the Gromov product, to simplify notations. The dependence of D b on the parameter b is described by the following lemma, whose proof can be found in [30] : Lemma 4.2. Let b 0 > 0 be fixed. There exists c ≥ 1 and α ∈]0, 1] such that the family of distances (D b ) 0≤b≤b0 , are Hölder equivalent ; namely for all b ∈ [0, b 0 ] we have D b ≤ cD α 0 .

Lemma 4 . 4 . 4 . 4 .Lemma 4 . 5 .

 444445 2.2 in [3] for a detailed proof) : There exist a real number r ∈]0, 1[ and a constant C > 0 such that for any γ ∈ Γ with length |γ| B = k, one gets ∀x ∈ K γ |γ (x)| 0 ≤ Cr k . Coding for the limit points. An infinite word on the alphabet B, i.e. an infinite sequence β = (β n ) n≥1 of elements of B is called admissible if any finite subword β 1 • • • β k is admissible ; the set of such words is denoted by Σ + B . The contraction property 4.4 implies the following fundamental fact : For any β ∈ Σ + B , the sequence (β 1 • • • β n • x 0 ) n≥0 converges to some point π( β) ∈ ∂X ; the map π : Σ + B → ∂X is one-to-one, and its image π(Σ + B ) coincides with the subset ∂ 0 X := ∂X \ r i=0 Γ.ξ i . Notice that, if γ has B-decomposition b 1 • • • b m , then the subset K γ defined in 4.3 is the closure of the subset corresponding, via the coding map π, to the the infinite sequences

  where [ϕ] α := sup x,y∈∂X x =y |ϕ(x) -ϕ(y)| D α 0 (x, y) ; then, L b,z acts on L α because of the following Lemma (which is deduced by Lemma 4.2 as Lemma III.3 in [3]) : Lemma 5.1. Each weight w b,z (γ, •) belongs to L α and for any z ∈ C, there exists

Lemma 5 . 4 .

 54 Assume that the profiles τ 0 ,• • • , τ 1 , • • • , τ r = τ are convergent and satisfy the condition ω τ = max(ω τ0 , • • • , ω τr ) > 2.Then there exist non negative reals a 0 , • • • , a r and b 0 > 0 such that • the group Γ has exponent ωτ 2 and is convergent with respect to g a0,••• ,ar,0 ; • the group Γ has exponent > ωτ 2 and is divergent with respect to g a0,••• ,ar,b0 .

ωτ / 2 →> 1 for some s > ωτ 2 . 2

 2122 +∞ as b → +∞. So, there exists b 0 ≥ 0 such that   |l|≥2 e -δd b (x0,p l •x0) Q∈W1 e -δd b (x0,Q•x0)   > 1 as soon as b ≥ b 0 ; in particular, by monotone convergence type argument, one gets   |l|≥2 e -sd b 0 (x0,p l •x0) Q∈W1 e -sd b 0 (x0,Q•x0)   This ensures that δ Γ ≥ s > ωτ 2 = δ Pr = max{δ Pi | 0 ≤ i ≤ r}, and Γ is divergent with respect to the metric g a0,••• ,ar,b by the critical gap property recalled in the introduction. Proof of Theorem 1.3. Recall that δ = ω τ /2. Since ρ α (L b,δ ) is an eigenvalue of L b,δ which is isolated in the spectrum of L b,δ , the function b → ρ α (L b,δ ) has the same regularity as b → L b,δ . For b 0 given by Lemma 5.4, we have ρ α

( 4 )

 4 p∈Pr/d(x0,p•x0)>t e -δd(x0,p•o) = n∈N/d(x0,p n r •x0)>t e -δd(x0,p n r The condition κ ∈]1/2, 1[ readily implies p∈Pr d(x 0 , p • x 0 )e -δd(x0,p•x0) N ≥1 N p∈Pr/N <d(x0,p•x0)≤N +1 e -δd(x0,p•x0) N ≥1 p∈Pr/d(x0,p•x0)>N e -δd(x0,p•o) N ≥1 L(N ) N κ = +∞.

1 -e

 1 F µ (t)) m(t) = 1 -κ and the characteristic function μ(t) := +∞ 0 itx µ(dx) of µ has the following local expansion as t → 0 +

and ( 21 )

 21 p∈Pr |e itd(x0,p•x0) -1| × e -δd(x0,p•x0) t κ L(1/t);

B

  x (p ±n • x 0 , x 0 ) = lim m d(p ±n • x 0 , x m ) -d(x 0 , x m ) = d(p ±n • x 0 , x 0 ) -lim m d(x 0 , x m ) + d(p ±n • x 0 , x 0 ) -d(p ±n • x 0 , x m ) with lim n lim m d(x 0 , x m ) + d(p ±n • x 0 , x 0 ) -d(p ±n • x 0 , x m ) = 2 lim n (p ±n • x 0 |x) x0 = 2(ξ - h |x) x0and the conclusion follows as the Gromov product (p ±n • x 0 |x) x0 tends uniformly to (ξ|x) x0 on compacts of S 1 \ {ξ}. 2 We writeσ r = |n|≥2 e -δd(x0,p n r •x0) e -itd(x0,p n r •x0) -1 ∂X\Ir h(p n r • x))e -δ∆(n,x) σ(dx) + |n|≥2 e -δd(x0,p n r •x0) ∂X\Ir h(p n r • x))e -δ∆(n,x) e -itr(p n r •x) -e -itd(x0,p n r •x0) σ(dx) = σ r1 + σ r2 .

|n|≥2e

  -δd(x0,p n r •x0) ∂X\Ir h(p n r • x))e -δ∆(n,x) |r(p n r • x) -d(x 0 , p n r • x 0 )|σ(dx) = O(t).

( 3 )

 3 j (R, ψ), we need to symmetrize the quantities W j (R, ψ) and W j (s, R, ψ) settingW j (R, ψ) := k≥0 y∈∂X/T k •y=xj e -δS k r(y) ψ(S k r(y) -R) + ψ(-S k r(y) -R) and W j (s, R, ψ) := k≥0 s k y∈∂X/T k •y=xje -δS k r(y) ψ(S k r(y) -R) + ψ(-S k r(y) -R) .

W

  j (s, R, ψ) := k≥0 s k y∈∂X/T k •y=xj e -δS k r(y) ψ(S k r(y) -R) + ψ(-S k r(y) -R) = 1 2π R e itR ψ(t) (I -sL δ+it ) -1 -(I -sL δ-it ) -1 1(x j )dt.

1 - 1 -j

 11 itR ψ(t)(1 -ρ(t)) (I -rL δ+it ) -1 1(x j )dt + 1 2π R e itR ψ(t)(1 -ρ(t)) (I -rL δ-it ) -1 1(x j )dt, itR ψ(t)ρ(t) (I -rL δ+it ) -1 1(x j ) -σ(∂X \ I j )h(x j ) 1 -rλ t dt + 1 2π R e itR ψ(t)ρ(t) (I -rL δ-it ) -1 1(x j ) -σ(∂X \ I j )h(x j ) 1 -rλ -t dt and W (3) j (s, R, ψ) = σ(∂X \ I j )h(x j ) 2π R e itR ψ(t)ρ(t) 1 1 -rλ t -1 rλ -t dt.Using Proposition 6.6, letting s → 1, one getsW j (R, ψ) = W itR ψ(t)(1 -ρ(t)) (I -L δ+it ) -1 1(x j ) + (I -L δ-it ) -1 1(x j ) dt, W itR ψ(t)ρ(t) (I -L δ+it ) -1 1(x j ) -σ(∂X \ I j )h(x j ) 1 -λ t dt + 1 2π R e itR ψ(t)ρ(t) (I -L δ-it ) -1 1(x j ) -σ(∂X \ I j )h(x j ) (s, R, ψ) = σ(∂X \ I j )h(x j ) 2πR e itR ψ(t)ρ(t)Re 1 1 -λ t dt.

( 3 )

 3 j (R, ψ) = C j ψ(0) = C j R ψ(x)dx with C j = σ(∂X \ I j )h(x j ) sin πκ π; notice that the value h(x j ) is uniquely determined by the normalization σ(h) = 1 (see [26] Theorem 4 for a detailed argument). This achieves the proof of Theorem 6.1.2

GEOMETRY OF NEGATIVELY CURVED MANIFOLDS WITH FINITE VOLUME

A function L(t) is said to be "slowly varying" or "of slow growth" if it is positive, measurable and L(λt)/L(t) → 1 as t → +∞ for every λ > 0.


Jacques Pothier

Suffer Little Children: McCarthy's Lazarillos and the Ordeal of Mobility in the Southern Canon

The road has long been associated with the genre of the novel as it started in Spain.

Arguably the oldest short novel of the Spanish tradition, the anonymous La Vida de Lazarillo de Tormes is often described as the first realization of the picaresque novel, a genre that was one of the oldest modes of the western novel, later made famous by the adventures of Don Quixote. Lazarillo is a poor child, who has to put himself at the service of more powerful and slightly less miserable people to get by, and he is as astute as Tom Sawyer, Huckleberry Finn or Brer Rabbit at foiling the adults who want to take advantage of his assumed innocence.

When we find "the child" on the road with his father throughout Cormac McCarthy's latest novel The Road, he continues a pattern that has occurred before in earlier McCarthy novels, although through most of these novels young people "ride on," preferably South, as do the two protagonists of All the Pretty Horses, who, like later day Quixote and Sancho Panza, "took the road South such road as it was" (53).

Let us consider these young people in the broader perspective of the canon of American literature, preferably in its southern tradition. It may be argued that Huck Finn cannot be considered a southern character, but he certainly becomes a type of the child as a critique of the mores of southern society, which he famously seeks to run away from by choosing to go West to the territories before the good ladies "sivilize" him. "In the evening he saddled his horse and rode out west from the house." This is not Huck Finn, but John Grady Cole, on page 5 of All the Pretty Horses, in another picaresque novel, leading the American hero in a land of Hispanic culture that it stemmed from. The picaresque novel needs the road, and the road happens to be strongly related to the American migratory imaginary of the march West to the frontier, to new lands. Even in colonial times, early

Americans enthusiastically embraced the metaphor of John Bunyan's Pilgrim's Progress to deal with spiritual adventures that did not necessarily involve a change of locale. There is nothing as quintessentially American as to hit the road (or, in the Melvillian variant, taking to the sea-"The intense Pequod sailed on," as chapter 131 of Moby-Dick mysteriously starts) when one feels uneasy with the constraints of civilization.

Yet, come to think of it that may be where the South would take exception. There seems to be something about the South that is alien to the road. "Durn that road," Faulkner's Anse Bundren famously said: in his fumbling epistemology, he sensed it, when he surmised in a half-articulate way that was not uninfluenced by personal laziness:

"When [the Lord] aims for something to be always a-moving, He makes it long ways, like a road or a horse or a wagon, but when He aims for something to stay put, He makes it up-and-down ways, like a tree or a man. And so He never aimed for folks to live on a road, because which gets there first, I says, the road or the house? [...] Because if He'd a aimed for man to be always a-moving and going somewheres else, wouldn't He a put him longways on his belly, like a snake? It stands to reason He would." (Faulkner 24) Let us take a second example of discomfort with mobility: the tall convict in The Wild Palms expresses agony, pushed to its tall-tale absurdity: he is desperate to recover the confined comfort of the penitentiary from which he has been wrenched by the violence of the elements-but we recall that what sent him to jail was a symbolic dis/location-an attempted robbery in the manner of the dime-novel gangsters that he identified with. Like Lazarillo, he is forced to survive by wandering around, but the irony turns him again into the Quixote he once was when he believed too much in fictions as guidebooks for life: against his will, he becomes the knight-errant who rescues a pregnant dame.

Third, the dichotomy between the house (not necessarily the plantation house) and the road (mostly West) can be found in Light in August between the Dark House where Joe Christmas will find the end of his tragic journey, the "street which was to run for fifteen years" (Faulkner 563), while Hightower lounges in his own dark house, reminiscing about his ancestor, and Lena Grove walks on, all the way from Alabama. West-not as an escape, but in the direction of dreams.

If one considers the stories by Flannery O'Connor, in the collection titled A Good

Man is Hard to Find, one sees how much of the tension in these plots lies in the hesitation between moving and not moving, the road and the house. Most of the stories pit an outsider, an intruder, a traveler, against the people who live on an estate, in a house. Even the stories that do not seem to be about this form of displacement can be considered according to this theme. "A Stroke of Good Fortune" is almost too emblematic: the focal character, Ruby, would love to "move" along to another location-a subdivision in suburbia would be nicebut in the whole story the reader sees her painstakingly moving upward in the apartment block she is stuck in-continuing the upward migration she had to be content with, up from the "pits"-Pitman where she was "raised." The ironic diversion from a horizontal dimension to the vertical dimension suggesting spiritual progress is never as obvious as in this storymaybe just a little bit too obvious.

But to return to the children. Take the eponymous story, "A Good Man Is Hard to Find," a title which invites to seeking, to hitting the road to find-with the suggestion that this quest might have spiritual dimensions, as it will. In the first paragraph, the home of this middle-class southern family is set on edge: the home is under threat of blowing apart in the centrifugal tension of all its members divided between different directions, wanting to take diverging roads on the way to a week's vacation. As all the family seems to be bent on going to Florida-which is South but admittedly not part of the "Dixie" South-, the grandmother would prefer to choose a different destination. On the highway, the children pay absolutely no attention to the scenery that the grandmother tries to interest them in, emphasizing the landmarks of the eternal South: Stone Mountain, a plantation graveyard here, a negro shack there, while the grandmother also pays acute attention to the dangers along the road: the Misfit of course, but also patrolmen in ambush for speeding drivers; or Red Sammy, the proprietor of a road-side diner, that she is anxious to paint as a reassuring "good man."

Through her own hubris, the whole journey takes a different course, when she convinces her son to veer off the main road onto a snake-like dirt road (the road as ensnaring snake again) which is supposed to lead to a vaguely remembered old plantation that never materializes.

Instead the road takes on the vicious quality that Anse Bundren has exposed. In a strange but suggestive sentence, the son, who is the driver, vents his impatience and seems to invite the house itself to get moving, pitting the agency of the house against the seeming agency of the snake-like road-Anse Bundren would say, "I told you so": "This place had better turn up in a minute," Bailey said, "or I'm going to turn around" (O'Connor 144). When the grandmother nervously realizes that she put the whole family on the wrong track, she lets her cat slip off the bag where she had concealed her, and causes an accident. The kids scramble out of the car and are, at last, thrilled, and scream "in a frenzy of delight." But of course the downside is that they do come across the most feared Misfit, who methodically organizes the deadly end to this journey. This is only the first story in which the kids hit the road to escape the confinement of social constructs, but in no way the last: in "The River" young Harry Ashfield escapes the See the child. He is pale and thin, he wears a thin and ragged linen shirt. He stokes the skullery fire. Outside lie dark turned fields with rags of snow and darker woods beyond that harbor yet a few last wolves. His folk are known for hewers of wood and drawers of water but in truth his father has been a schoolmaster. He lies in drink, he quotes from poets whose names are now lost. The boy crouches by the fire and watches him. (Blood Meridian 3) See the child: he is quite a sight indeed. If he escapes from the decay his father embodies, as an orphan from Dickens, as Huck does from old Finn, as O'Connor's children so often try to do, it is to adjust as well as he can, but certainly better than many who think children are just fragile creatures, to a world of lies and violence-a world very close to that of the Lazarillo of the Spanish tradition: he suffers, but he learns to make others suffer to fend for himself. On the same page, a few lines further down: "He watches, pale and unwashed. He can neither read nor write and in him broods already a taste for mindless violence." And then: "At fourteen he runs away." This is the age when American children (boys, at any rate) hit the road, if they follow the example of Huck Finn. Summing up the story of an American boy's life toward the end of the novel, the kid says "that he was an American and that he was a long way from the country of his birth and that he had no family and that he had traveled much and seen many things and had been at war and endured hardships" (Blood Meridian 315).

Young boys-fourteen to sixteen-are also central characters in the Border Trilogy, all the way to McCarthy's recent best-seller The Road, where the kid really becomes a young "boy" (except for No Country for Old Men, a novel that leaves the stage for older people 1 ). In both All the Pretty Horses and The Crossing, young teenagers escape the dull reality of a twentieth century American Southwest that has now no place for cow-boys, to cross into Mexico where the world of the vaqueros would seem to have preserved the values of a dream-like West: the Hispanic South is made the museum of what is left of the West. It is a South whose alien, exotic quality is underlined by the use of the Spanish language in retranscribed dialogue. As Kenneth Lincoln has noted, even the title of All the Pretty Horses posits the legendary South as the lost paradise of the young protagonists: the phrase conjures up the old slave lullaby of a black mammy singing a white child to sleep-an African American slave who could not take care of her baby because she was too busy taking care of her master's child: Hush-a-bye, don't you cry, go to sleep my little baby. When you wake you shall have all the pretty horses. Dapples and grays, pintos and bays, coach and six little horses. (Lincoln 102)

Suffering little children again. The Border Trilogy starts in nostalgia, and The Road finds another boy in a from which all human values or coordinates have been washed away-a sense that one could already have from the Border Trilogy-see p. 6 of the first novel: "He [the "boy" here is John Grady Cole] turned south along the old war trail and he rode out to the crest of a low rise and dismounted and dropped the reins and stood like a man come to the end of something." And a few pages further, the last words from his father:

The last thing his father said was that the country would never be the same.

People don't feel safe no more, he said. We're like the Comanches was two hundred years ago. We dont know what's goin to show up here come daylight. We dont even know what color they'll be. (Border Trilogy 26) Like Quixote, John Grady Cole takes to the road with a Sancho Panza figure, Rawlins. But in his travels in northern Mexico, the young boy learns something about life.

The father of his girl-friend, Alejandra, teases him about his idealism, and reveals to the reader that the Spanish tradition is not lost on the author: "Beware gentle knight. There is no greater monster than reason." He looked at John Grady and smiled […] That of course is the Spanish idea. You see. The idea of Quixote. But even Cervantes could not envision such a country as Mexico. (Border Trilogy 149)

Like the kid in Blood Meridian, John Grady visits a heart of darkness, a descent to the Hell of a Mexican jail. But early in the novel John Grady and Rawlings have exchanged thoughts about Judgment Day-whether they believe in it or not. This sense of the ultimate end as uncertain but terrible intensifies from his experience. Personal misery takes on a global dimension, and is revealed as bottomless: … as he lay there the agony in his heart was like a stake. He imagined the pain of the world to be like some formless parasitic being seeking out the warmth of human souls wherein to incubate and he thought he knew what made one liable to its visitations. What he had not known was that it was mindless and so had no way to know the limits of those souls and what he feared was that there might be no limits. (Border Trilogy 259) The philosophy he reaches resembles the lesson of the "judge" in Blood Meridian: "I will tell you. Only that man who has offered up himself entire to the blood of war, who has been to the floor of the pit and seen horror [in the sound] and learned at last that it speaks to his inmost heart, only that man can dance" (Blood Meridian 331). In All the Pretty Horses beauty is also extracted at a very high price, described in arithmetic terms: "He thought that in the beauty of the world were hid a secret. He thought that the world's heart beat at some terrible cost and that the world's pain and its beauty moved in a relationship of diverging equity and that in this headlong deficit the blood of multitudes might ultimately be exacted for the vision of a single flower" (Border Trilogy 286).

Toward the end of the novel John Grady, the dueña Alfonsina, the grandmother of the lover John Grady he had to break up with, tells him about the story of their family and derives from it a philosophy on the history of violence and the way blood cements the real human bonds-a metaphor that had also been the judge's in Blood Meridian (329):

That night in the garden here at my father's house Gustavo said to me that those who have suffered great pain of injury or loss are joined to one another with bonds of a special authority and so it has proved to be. The closest bonds we will ever know are bonds of grief. The deepest community one of sorrow. [...] Those whom life does not cure death will. [...] What is constant in history is greed and foolishness and a love of blood and this is a thing that even God-who knows all that can be knownseems powerless to change. (Border Trilogy 240-1) A message of sadness about the world the child is exposed to-certainly confirmed in The Road, with the pattern of the picaresque reduced to its bare essentials-an innocent boy, his older fellow traveler, an empty world with a few straggling individuals dotting an almost empty road, leading to the vague dream of a lost southern sea-shore, harking back to the exiled army of Xenophon's famous "Thalassa." The world of The Road is heavy with the devastating entropy that could be felt in earlier novels and now threatens human sensations and language-both the words and the syntactic structures, worn down to fragments: "the world shrinking down about a raw core of parsible entities. The names of things slowly following those things into oblivion. Colors. The names of birds. Things to eat. Finally the names of things one believed to be true" (The Road 75). The end of a journey is a blank, both a wasteland after the devastation of humanity repeating earlier apocalypses, and a fresh but tiny beginning that stems from the dark legacy of the human condition.

Works cited

Anonymous. La vida de Lazarillo de Tormes y de sus fortunas y adversidades. 1554. http://www.4olin.com/ (Jan. 15, 2012). Faulkner, William. As I Lay Dying; Light in August, in Novels 1930-1935. New

  figure, right from the first famous paragraph of the novel:

  York: Library of America, 1985. Lincoln, Kenneth. Cormac McCarthy: American Canticles. New York: Palgrave Macmillan, 2009. McCarthy, Cormac. Blood Meridian or the Evening Redness in the West. 1985. London, Picador, 1989. ----------. The Border Trilogy [All the Pretty Horses, 1992; The Crossing, 1993; Cities of the Plain, 1998]. London: Pan Macmillan, 2002. ----------. No Country for Old Men. 2005. New York: Vintage, 2007. ----------. The Road. New York: Knopf, 2006. O'Connor, Flannery. Collected Works. New York: Library of America, 1988.

Incidentally there is a hint of O'Connor's Misfit in the character of Chigurh, the maniac killer, that may be more perceptible in the treatment of the Coen brothers' film adaptation.