F Dal'bo

M Peigné

J C Picaud

A Sambusetti

Volume growth and rigidity of negatively curved manifolds of finite volume

We study the asymptotic behaviour of the volume growth function of simply connected, Riemannian manifolds X of strictly negative curvature admitting a nonuniform lattice Γ. If X is asymptotically 1/4-pinched, we prove that Γ is divergent, with finite Bowen-Margulis measure, and that the volume growth of balls B(x, R) in X is asymptotically equivalent to a purely exponential function c(x)e ω(X)R , where ω(X) is the volume entropy of X. This generalizes Margulis' celebrated theorem for negatively curved spaces with compact quotients. A crucial step for this is a finite-volume version of the entropy-rigidity characterization of constant curvature spaces: any finite volume n-manifold with sectional curvature -b 2 ≤ k(X) ≤ -1 and volume entropy equal to (n -1) is hyperbolic. In contrast, we show that for spaces admitting lattices which are not 1/4-pinched, depending on the critical exponent of the parabolic subgroups and on the finiteness of the Bowen-Margulis measure, the growth function can be exponential, lower-exponential or even upper-exponential.

Introduction

Let X be a complete, simply connected manifold with strictly negative curvature. In the sixties, G. Margulis [START_REF] Margulis | Certain applications of ergodic theory to the investigation of manifolds of negative curvature Funkcional[END_REF], using measure theory on the foliations of the Anosov system defined by the geodesic flow, showed that if Γ is a uniform lattice of X (i.e. a torsionless, discrete group of isometries such that X = Γ\X is compact), then the orbital function of Γ is asymptotically equivalent to a purely exponential function:

v Γ (x, y, R) = #{γ ∈ Γ | d(x, γy) < R} ∼ c Γ (x, y)e δ(Γ)R
where δ(Γ) = lim R→∞ R -1 v γ (x, x, R) is the critical exponent of Γ, and ∼ means that the quotient tends to 1 when R → ∞. By integration over fundamental domains, one then obtains an asymptotic equivalence for the volume growth function of X: v X (x, R) = volB(x, R) ∼ m(x)e δ(Γ)R .

It is well-known that the exponent δ(Γ) equals, for uniform lattices, the volume entropy ω(X) = lim sup 1 R ln v X (x, R) of the manifold X; the function m(x), depending on the center of the ball, is the Margulis function of X.

Since then, this result has been generalized in different directions. Notably, G. Knieper showed in [START_REF] Knieper | On the asymptotic geometry of nonpositively curved manifolds[END_REF] that the volume growth function of a Hadamard space X (a complete, simply connected manifolds with nonpositive curvature) admitting uniform lattices is purely exponential, provided that X has rank one, that is:

v X (x, R) ≍ e ω(X)R
where f ≍ g means that 1/A < f (R)/g(R) < A for some positive A, when R ≫ 0. In general, he showed that v X (x, R) ≍ R d-1 2 e ω(X)R for rank d manifolds; however, as far as the authors are aware, it is still unknown whether there exists a Margulis function for Hadamard manifolds of rank 1 with uniform lattices, i.e. a function m(x) such that v X (x, R) ∼ m(x)e ω(X)R , even in the case of surfaces. Another remarkable case is that of asymptotically harmonic manifolds of strictly negative curvature, where the strong asymptotic homogeneity implies the existence of a Margulis function, even without compact quotients, cp. [START_REF] Castillon | On asymptotically harmonic manifolds of negative curvature[END_REF].

In another direction, it seems natural to ask what happens for a Hadamard space X of negative curvature admitting nonuniform lattices Γ (i.e. vol(Γ\X) < ∞): is v X purely exponential and, more precisely, does X admit a Margulis function? Let us emphasize that if X also admits a uniform lattice then X is a symmetric space of rank one (by [START_REF] Eberlein | Geometry of nonpositively curved manifolds[END_REF], Corollary 9.2.2); therefore, we are interested in spaces which do not have uniform lattices, i.e. the universal covering of finite volume, negatively curved manifolds which are not locally symmetric.

It is worth to stress here that the orbital function of Γ is closely related to the volume growth function of X, but it generally has, even for lattices, a different asymptotic behaviour than v X (x, R). The weak equivalence v Γ (x, R) ≍ e δ(Γ)R is known for convexcocompact discrete subgroups of isometries of H n since [START_REF] Sullivan | The density at infinity of a discrete group of hyperbolic motions[END_REF], [START_REF] Patterson | The limit set of a Fuchsian group[END_REF], by Patterson-Sullivan theory. A precise asymptotic equivalence fo v Γ was proved by T. Roblin [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] in a very general setting. Namely, he proved that for any nonelementary group of isometries Γ of a CAT(-1) space X with non-arithmetic length spectrum1 and X = Γ\X, one has:

(a) v Γ (x, y, R) ∼ c Γ (x, y)e δ(Γ)R if the Bowen-Margulis measure of U X is finite;

(b) v Γ (x, y, R) = o(R)e δ(Γ)R , where o(R) is infinitesimal, otherwise.

Thus, the behaviour of v Γ (x, R) strongly depends on the finiteness of the Bowen-Margulis measure µ BM ; also, the asymptotic constant can be expressed in terms of µ BM and of the family of Patterson-Sullivan measures (µ x ) of Γ, as c Γ (x, y) = µx µy µ BM . A useful criterion ensuring that µ BM (U X) < ∞, hence a precise asymptotics for v Γ (x, R), is the following (see [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF]) Finiteness Criterion. Let Γ be a divergent, geometrically finite group, X = Γ\X. We have µ BM (U X) < ∞ if and only if for every maximal parabolic subgroup P of Γ p∈P d(x, px)e -δ(Γ)d(x,px) < +∞.

(1)

On the other hand, any convergent group Γ exhibits a behaviour as in (b), since it certainly has infinite Bowen-Margulis measure (by Poincaré recurrence, µ BM (U X) < ∞ implies that the geodesic flow is totally conservative, and this is equivalent to divergence, by Hopf-Tsuji-Sullivan's theorem). Notice that, whereas uniform lattices always are divergent and with finite Bowen-Margulis measure, for nonuniform lattices Γ divergence and condition [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF] in general may fail. Namely, this can happen only in case Γ has a "very large" parabolic subgroup P , that is such that δ(P ) = δ(Γ): we will call exotic such a lattice Γ, and we will say that such a P is a dominant parabolic subgroup. Convergent, exotic lattices are constructed by the authors in [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF]; also, one can find in [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF] some original counting results for the orbital function of Γ in infinite Bowen-Margulis measure, more precise than (b).

However, as we shall see, the volume growth function v X has a wilder behaviour than v Γ . In [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF] we proved that for nonuniform lattices in pinched, negatively curved spaces X, the functions v Γ and v X can have different exponential growth rates, i.e. ω(X) = δ(Γ). In the Example 6.2 we will see that the function v X might as well have different superior and inferior exponential growth rates ω ± (X) (notice, in contrast, that δ(Γ) always is a true limit). Nevertheless, δ(Γ) still encodes a lot of information on the manifold X even if Γ is non-uniform. The first result we prove in this paper is a generalization of a volume-entropy characterization of constant curvature spaces, due to G. Knieper when the quotient Γ\X is compact (cp. [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF]; see also [START_REF] Bonk | Rigidity for quasi-Möbius group actions[END_REF] for an analogue in case of convex-cocompact lattices): Theorem 1.1 Let X be a Hadamard manifold with curvature -b 2 ≤ K X ≤ -a 2 < 0 and Γ a nonuniform lattice of X. If δ(Γ) = (n -1)a, i.e. if it equals the volume entropy of the space H n a with constant curvature -a 2 , then X has constant curvature -a 2 . The volume-entropy characterization of constant curvature (and locally symmetric) metrics has a long history and has been declined in many different ways so far, for uniform lattices or convex-cocompact representations (beyond [START_REF] Knieper | Spherical means on compact Riemannian manifolds of negative curvature[END_REF] and [START_REF] Bonk | Rigidity for quasi-Möbius group actions[END_REF], see also [START_REF] Hamenstädt | Entropy-rigidity of locally symmetric spaces of negative curvature[END_REF], [START_REF] Bourdon | Structure conforme au bord et flot géodśique d'un CAT(?1)-espace[END_REF], [START_REF] Izeki | Limit sets of Kleinian groups and conformally flat Riemannian manifolds[END_REF], [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF], [START_REF] Besson | Lemme de Schwarz réel et applications géométriques[END_REF], [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF]). To prove Theorem 1.1, we use the barycenter method initiated by Besson-Courtois-Gallot in [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]- [START_REF] Besson | Lemme de Schwarz réel et applications géométriques[END_REF]. There exist finite-volume versions of [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]& [START_REF] Besson | Lemme de Schwarz réel et applications géométriques[END_REF], given by Boland-Connell-Souto [START_REF] Boland | Volume rigidity for finite volume manifolds[END_REF] and Storm [START_REF] Storm | The minimal entropy conjecture for nonuniform rank one lattices[END_REF]: these two works, together, imply that if a Hadamard manifold X with curvature K X ≤ -1 has a quotient of finite volume X = Γ\X and ω(X) = n -1, then it is hyperbolic, provided that one knows that X is homotopically equivalent to a finite-volume, hyperbolic manifold X0 . In contrast, notice that in Theorem 1.1 no supplementary topological assumption on the quotient manifold X is made. Also, notice that if we drop the assumption K X ≥ -b 2 , the manifold X might as well be of infinite type (i.e. with infinitely generated fundamental group, or even without any cusp, see examples in [START_REF] Nguyen Phan | On finite volume, negatively curved manifolds[END_REF]), hence not even homotopically equivalent to a finite-volume, hyperbolic manifold.

The second result of the paper concerns the Bowen-Margulis measure and an aymptote for the volume growth function of 1 Theorem 1.2 Let X be a Hadamard space with curvature -b 2 ≤ K X ≤ -a 2 , and let Γ be a nonuniform lattice of X. If X = Γ\X has asymptotically 1/4-pinched curvature (that is, for any ǫ > 0, the metric satisfies -k

2 + ≤ K X ≤ -k 2 -with k 2 + ≤ 4k 2 -+ ǫ on each cuspidal end outside some compact set Cǫ ⊂ X), then: (i) Γ is divergent; (ii) the Bowen-Margulis measure µ BM of U X is finite; (iii) ω + (X) = ω -(X) = δ(Γ); (iv) there exists a function m(x) ∈ L 1 ( X) such that v X (x, R) ∼ m(x)e δ(Γ)R , where m(x) is the lift of m to X.
From the divergence of Γ, it then follows that the geodesic flow of any asymptotically 1 4 -pinched, negatively curved manifold is ergodic and totally conservative w.r. to µ BM , by the celebrated Hopf-Tsuji-Sullivan Theorem (see [START_REF] Sullivan | The density at infinity of a discrete group of hyperbolic motions[END_REF], [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]). Condition (iv) also implies that volume equidistributes on large spheres, i.e. the volume v ∆ X (x, R) of annuli in X of thickness ∆ satisfies the asymptotics v ∆ X (x, R) ∼ 2m(x) sinh(∆δ(Γ))e ω(X)R . Notice that the above theorem also covers the classical case of noncompact symmetric spaces of rank one (where the proof of the divergence and the asymptotics is direct).

One may wonder about the meaning (and necessity) of the 1 4 -pinching condition. This turns out to be an asymptotic, geometrical condition on the influence and wildness of maximal parabolic subgroups of Γ associated to the cusps of X = Γ\X. Parabolic groups, being elementary, do not necessarily have a critical exponent which can be interpreted as a true limit; rather, for a parabolic group of isometries P of X, one can consider the limits

δ + (P ) = lim sup R→∞ 1 R ln v P (x, R) δ -(P ) = lim inf R→∞ 1 R ln v P (x, R)
and the critical exponent δ(P ) of the Poincaré series of P coincides with δ + (P ). Accordingly, we say that a lattice Γ is sparse if it has a maximal parabolic subgroup P such that δ + (P ) > 2δ -(P ) (conversely, we will say that Γ is parabolically 1 2 -pinched if it is not sparse). Such parabolic groups in Γ are precisely associated to cusps whose growth can wildly change and this can globally influence the growth function of X. Namely, we can prove: Theorem 1.3 Let X be a Hadamard manifold with pinched, negative curvature -b 2 ≤ K X ≤ -a 2 < 0. If X has a nonuniform lattice Γ which is neither exotic nor sparse, then Γ is divergent and with finite Bowen-Margulis measure; moreover, v X ≍ v Γ and X has a Margulis function m(x), whose projection is L 1 on X = Γ\X. Theorem 1.2 is a particular case of Theorem 1.3, as (using the volume-entropy characterization 1.1 of constant curvature spaces) we can show that any lattice in a negatively curved, 1 4 -pinched space is neither exotic nor sparse. The last part of the paper is devoted to studying sparse and exotic lattices, and the following result shows that Theorem 1.3 is the best that we can expect for Hadamard spaces with quotients of finite volume: Theorem 1.4 Let X be a Hadamard manifold with pinched negative curvature -b 2 ≤ K X ≤ -a 2 < 0 admitting a nonuniform lattice Γ.

(i) If Γ is exotic and the dominant subgroups P satisfy δ(Γ) = δ + (P ) < 2δ -(P ), then both v X and v Γ are purely exponential or lower-exponential, with the same exponential growth rate ω(X) = δ(Γ). Namely

• either µ BM < ∞, and then v X is purely exponential and X has a Margulis function;

• or µ BM = ∞, and in this case v X is lower-esponential.

The two cases can actually occur, cp. Examples 6.3(a)&(b).

(ii) If Γ is exotic and a dominant subgroup • µ BM < ∞, with v Γ purely exponential and v X upper-exponential;

P satisfies δ(Γ) = δ + (P ) = 2δ -(P ), then ω(X) = δ(Γ) but in general v X ≍ v Γ ,
• µ BM = ∞, with v Γ lower-exponential and v X upper-exponential.

By lower-(respectively, upper-) exponential we mean a function f with exponential growth rate ω = lim sup R→∞ 1 R ln f (R), but such that lim inf R→∞ f (R)/e ωR = 0 (resp. lim sup R→∞ f (R)/e ωR = +∞).

We shall see that all these examples can be obtained as lattices in ( 14ǫ)-pinched spaces, for arbitrary ǫ > 0, which shows the optimality of the 1 4 -pinching condition. On the other hand, if Γ is sparse, one can even have ω + (X) > ω -(X) > δ(Γ), and the Example 6.2 shows that virtually any asymptotic behaviour for v X can occur. Thus, the case of exotic lattices with a parabolic subgroup such that δ + (P ) = 2δ -(P ) can be seen as the critical threshold where a transition happens, from functions v Γ , v X with same asymptotic behaviour to functions with even different exponential growth rate.

Notations.

Given two functions f, g : R+ → R+, we will systematically write f C ≺ g for R > R0 (or g C ≻ f ) if there exists C > 0 such that f (R) ≤ Cg(R) for these values of R. We say that f and g are weakly asymptotically equivalent and write f C ≍ g when g C ≺ f C ≺ g for R ≫ 0; we will simply write f ≍ g and f ≺ g when the constants C and R0 are unessential. We say that f and g are asymptotically equivalent and write f ∼ g when limR→+∞ f (R)/g(R) = 1. We define the upper and lower exponential growth rates of a function f respectively as:

ω + (f ) = lim sup R→+∞ R -1 ln f (R) and ω -(f ) = ω(f ) = lim inf R→+∞ R -1 ln f (R)
and we simply write ω(f ) when the two limits coincide. Also, we will say that f is purely exponential if f ≍ e ω(f )R , and that f is lower-exponential (resp. upper-exponential) when lim infR→+∞

f (R)) e ω(f )R = 0 (resp. lim sup R→+∞ f (R) e ω(f )R = +∞).
Finally, if f and g are two real functions, we will use the notation f * ∆ g for the discrete convolution of f and g with gauge ∆, defined by (f * ∆ g)(R) = h+k=⌊R/∆⌋ h,k≥1 f (h∆)g(k∆). We notice here that, for nondecreasing functions f and g, this is weakly equivalent to the usual convolution, namely

∆ • (f * ∆ g) (R -∆) ≤ (f * g) (R) = R 0 f (t)g(R -t)dt ≤ 2∆ • (f * ∆ g) (R + 2∆).

Growth of parabolic subgroups and of lattices modulo parabolic subgroups

Throughout all the paper, unless otherwise stated, X will be a Hadamard space of dimension n, with pinched negative sectional curvature -b 2 ≤ K X ≤ -a 2 < 0.

For x, y ∈ X and ξ ∈ X(∞), we will denote [x, y] (resp. [x, ξ]) the geodesic segment from x to y (resp. the ray from x to ξ). We will repeatedly make use of the following, classical result in strictly negative curvature: there exists ǫ(a, ϑ) = 1 |a| log( 2 1-cos ϑ ) such that any geodesic triangle xyz in X making angle ϑ = ∠ z (x, y) at z satisfies:

d(x, y) ≥ d(x, z) + d(z, x) -ǫ(a, ϑ).
(

) 2
Let b ξ (x, y) = lim z→ξ d(x, z)d(z, y) be the Busemann function centered at ξ. The level set ∂H ξ (x) = {y | b ξ (x, y) = 0} (resp. the suplevel set H ξ (x) = {y | b ξ (x, y) ≥ 0} is the horosphere (resp. the horoball) with center ξ and passing through x. From (2) we easily deduce the following: Lemma 2.1 For any d > 0, there exists

ǫ 1 = ǫ 1 (a, d) ≥ ǫ(a, π
2 ) with the following property: given two disjoint horoballs

H 1 , H 2 at distance d = d(H 1 , H 2 ) = d(z 1 , z 2 ) with z i ∈ ∂H i , then for any x ∈ H 1 and y ∈ H 2 we have d(x, z 1 ) + d(z 1 , z 2 ) + d(z 2 , y) -ǫ 1 (a, d) ≤ d(x, y) ≤ d(x, z 1 ) + d(z 1 , z 2 ) + d(z 2 , y).
Proof. As K X ≤ -a 2 and horoballs are convex, for any y ∈ H 2 the angle ϑ(y) = ∠ z 1 z 2 , y satisfies tan ϑ(y) ≤ 1 sinh(d/|a|) (cp. for instance [START_REF] Sambusetti | Asymptotic properties of coverings in negative curvature[END_REF], Prop.8). Then, we have

∠ z 1 x, y ≥ π 2 -ϑ(y) ≥ ϑ(d) with ϑ(d) > 0 for d = 0, hence, by (2), d(x, y) ≥ d(x, z 1 ) + d(z 1 y) -ǫ(a, ϑ(d)) ≥ d(x, z 1 ) + d(z 1 , z 2 ) + d(z 2 , y) -ǫ 1 (a, d) for ǫ 1 (a, d) = ǫ(a, ϑ(d)) + ǫ(a, π 2 ).2
Let d ξ denote the horospherical distance between two points on a same horosphere centered at ξ. If ψ ξ,t : X → X denotes the radial flow in the direction of ξ, we define:

t ξ (x, y) = inf{t > 0 |d ξ (ψ ξ,t+∆ (x), ψ ξ,t (y)) < 1} if b ξ (x, y) = ∆ ≥ 0; inf{t > 0 |d ξ (ψ ξ,t (x), ψ ξ,t-∆ (y)) < 1} if b ξ (x, y) = ∆ < 0. ( 3
)
If y is closer to ξ than x, let x ∆ = [x, ξ[∩∂H ξ (y): then, t ξ (x, y) represents the minimal time we need to apply the radial flow ψ ξ,t to the points x ∆ and y until they are at horospherical distance less than 1. Using (2) and the lower curvature bound K X ≥ -b 2 , we obtain in [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF] the following estimate, which is also crucial in our computations:

Approximation Lemma 2.2 There exists ǫ 0 = ǫ 0 (a, b) ≥ ǫ(a, π 2
) such that for all x, y ∈ X and ξ ∈ X(∞) we have:

2t ξ (x, y) + |b ξ (x, y)| -ǫ 0 ≤ d(x, y) ≤ 2t ξ (x, y) + |b ξ (x, y)| + ǫ 0
In this section we give estimates for the growth of annuli in a parabolic subgroup and in quotients of a lattice by a parabolic subgroup, which will be used later. So, let us fix some notations. We let

A ∆ (x, R) = B x, R + ∆ 2 \ B x, R -∆ 2
be the annulus of radius R and thickness ∆ around x. For G acting on X, we will consider the orbital functions

v G (x, y, R) = # (B(x, R) ∩ Gy) v ∆ G (x, y, R) = # A ∆ (x, R) ∩ Gy and we set v G (x, R) = v G (x, x, R), v ∆ G (x, R) = v ∆ G (x, x, R) and v ∆ G (x, R) = ∅ for ∆ < 0.
We will also need to consider the growth function of coset spaces, endowed with the natural quotient metric: if H < G, we define

d x (g 1 H, g 2 H) := d(g 1 Hx, g 2 Hx) and v G/H (x, R) := #{gH | |gH| x = d x (H, gH) < R} v ∆ G/H (x, R) = v G/H x, R + ∆ 2 -v G/H x, R - ∆ 2 .
We will use analogous notations for the growth functions of balls and annuli in the spaces of left and double cosets H\G, H\G/H with the metrics

d x (Hg 1 , Hg 2 ) := d(Hg 1 x, Hg 2 x) = |g -1 1 Hg 2 | x d x (Hg 1 H, Hg 2 H) := d(Hg 1 Hx, Hg 2 Hx) = |g -1 1 Hg 2 H| x .
The growth of the orbital function of a bounded parabolic group P is best expressed by introducing the horospherical area function. Let us recall the necessary definitions: Definitions 2.3 Let P be a bounded parabolic group of X fixing ξ ∈ X(∞): that is, acting cocompactly on X(∞) -{ξ} (as well as on every horosphere ∂H centered at ξ). Given x ∈ X, let D(P, x) be a Dirichlet domain centered at x for the action of P on X; that is, a convex fundamental domain contained in the closed subset

D(P, x) = {y ∈ X | d(x, y) ≤ d(px, y) for all p ∈ P } We set S x = D(P, x) ∩ ∂H ξ (x) and C x = D(P, x) ∩ H ξ (x)
, and denote by S x (∞) the trace at infinity of D(P, x), minus ξ; these are, respectively, fundamental domains for the actions of P on ∂H ξ (x), H(x) and X(∞)-{ξ}. The horospherical area function of P is the function

A P (x, R) = vol [P \ψ ξ,R (∂H ξ (x))] = vol [ψ ξ,R (S x )]
where the vol is the Riemannian measure of horospheres. We also define the cuspidal function of P , which is the function

F P (x, R) = vol [B(x, R) ∩ H ξ (x)]
that is, the volume of the intersection of a ball centered at x and the horoball centered at ξ and passing through x. Notice that the functions A P (x, R), F P (x, R) only depend on the choice of the initial horosphere ∂H ξ (x).

Remark 2.4 Well-known estimates of the differential of the radial flow (cp. [START_REF] Heintze | Geometry of horospheres[END_REF]) yield, when -

b 2 ≤ K X ≤ -a 2 < 0, e -bt v ≤ dψ ξ,t (v) ≤ e -at v (4)
Therefore we deduce that, for any ∆ > 0,

e -(n-1)b∆ ≤ A P (x, R + ∆) A P (x, R) ≤ e -(n-1)a∆ (5)
The following Propositions show how the horospherical area A P and the cuspidal function F P are related to the orbital function of P ; they refine and precise some estimates given in [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF] for v P (x, R). Proposition 2.5 Let P be a bounded parabolic group of X fixing ξ, with diam(S x ) ≤ d.

There exist C = C(n, a, b, d) and C ′ = C ′ (n, a, b, d; ∆) such that: v P (x, y, R) C ≍ A -1 P x, R + b ξ (x, y) 2 ∀R ≥ b ξ (x, y)+ R 0 (6) v ∆ P (x, y, R) C ′ ≍ A -1 P x, R + b ξ (x, y) 2 ∀R ≥ b ξ (x, y)+R 0 and ∀∆ > ∆ 0 (7)
for explicit constants R 0 and ∆ 0 only depending on n, a, b, d.

Proposition 2.6 Same assumptions as in Proposition 2.5. We have:

F P (x, R) C ≍ R 0 A P (x, t) A P x, R+t 2 dt ∀R ≥ R 0 (8)
Remark 2.7 More precisely, we will prove (and use later) that:

(i) v P (x, y, R) C ≺ A -1 P x, R+b ξ (x,y) 2
for all R > 0;

(ii) v ∆ P (x, y, R)

C ′ ≺ A -1 P x, R+b ξ (x,y) 2
for all ∆, R > 0;

(iv)

F P (x, R) C ≺ R 0 A P (x,t) A P (x, R+t 2 )
dt for all R > 0.

As a direct consequence of ( 8) and ( 6) we have (see also Corollary 3.5 in [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF]):

Corollary 2.8 Let P be a bounded parabolic group of X. Then:

δ -(P ) ≤ ω -(F P ) ≤ ω + (F P ) ≤ max{δ + (P ), 2(δ + (P ) -δ -(P ))} ( 9
)
Proof of Proposition 2.5.

Since v P (x, y, R) = v P (y, x, R) and A P (x, R) = A P (y, R -b ξ (x, y)), we can assume that t = b ξ (x, y) ≥ 0. If z ∈ ∂H ξ (y) and d(x, z) = R, we know by Lemma 2.2 that 2t ξ (x, z) + t -ǫ 0 ≤ d(x, z) ≤ 2t ξ (x, z) + t + ǫ 0 so |t ξ (x, z) -R-t 2 | ≤ ǫ 0 /2. We deduce that d ξ ψ ξ, R+t+ǫ 0 2 (x), ψ ξ, R-t+ǫ 0 2 (z) ≤ 1, so the set ψ ξ, R-t+ǫ 0 2 (B(x, R) ∩ ∂H ξ (y)
) is contained in the unitary ball B + of the horosphere

∂H ξ (x + ), centered at x + = ψ ξ, R+t+ǫ 0 2 (x). Similarly, if R > t + ǫ 0 then t ξ (x, z) > 0, so d ξ ψ ξ, R+t-ǫ 0 2 (x), ψ ξ, R-t-ǫ 0 2 (z) ≥ 1, and the set ψ ξ, R-t-ǫ 0 2 (B(x, R) ∩ ∂H ξ (y)) con-
tains the unitary ball B -of ∂H ξ (x -), centered at the point x -= ψ ξ, R+t-ǫ 0 2 (x). We know that, by Gauss' equation, the sectional curvature of horospheres of X is between a 2b 2 and 2b(ba) (see, for instance, [START_REF] Buser | Gromov's almost flat manifolds[END_REF], §1.4); therefore, there exist positive constants

v -= v -(a, b) and v + = v + (a, b) such that vol(B + ) < v + and vol(B -) > v -.
Now, let S y = ψ ξ,t (S x ) be the fundamental domain for the action of P on ∂H ξ (y) deduced from S x . There are at least v P (x, y, Rd) distinct fundamental domains pS y included in B(x, R) ∩ ∂H ξ (y); since the radial flow ψ ξ,t is equivariant with respect to the action of P on the horospheres centered at ξ, there are also at least

v P (x, y, R -d) distinct fundamental domains ψ ξ, R-t+ǫ 0 2 (pS y ) included in ψ ξ, R-t+ǫ 0 2 (B(x, R) ∩ ∂H ξ (y)).
We deduce that v P (x, y, R-d)•A P (x, R+t+ǫ 0 2 ) < v + and, by [START_REF] Boland | Volume rigidity for finite volume manifolds[END_REF], this gives v P (x, y, R)

C ≺ A -1 P (x, R+t
2 ) for all R ≥ 0. On the other hand, if R > t + ǫ 0 , we can cover the set B(x, R) ∩ ∂H ξ (y) with v P (x, y, R + d) distinct fundamental domains pS y ; then, again,

ψ ξ, R-t-ǫ 0 2 (B(x, R) ∩ ∂H ξ (y)) can be covered by v P (x, y, R + d) fundamental domains ψ ξ, R-t-ǫ 0 2 (pS y ) as well, hence we deduce that v P (x, y, R + d) • A P (x, R+t-ǫ 0 2 ) ≥ v -.
This implies that v P (x, y, R)

C ≻ A -1 P (x, R+t 2 ) for all R > t + R 0 , for R 0 = ǫ 0 + d and a constant C = C(n, a, b, d).
To prove the weak equivalence [START_REF] Bowditch | Geometrical finiteness with variable negative curvature[END_REF], we just write, for R + ∆ 2 > t + R 0 :

v ∆ P (x, y, R) = v ∆ P (x, y, R + ∆/2) -v ∆ P (x, y, R -∆/2) ≥ C -1 A P R+t+∆/2 2 - C A P R+t-∆/2 2 ≥ C -1 e (n-1)a ∆ 4 -Ce -(n-1)a ∆ 4 A P x, R+t 2 = 2 sinh 1 4 (n -1)a∆ -ln C • A -1 P R + t 2
again by [START_REF] Boland | Volume rigidity for finite volume manifolds[END_REF], if ∆ > ∆ 0 = 4 ln C (n-1)a . Reciprocally, we have for all R, ∆ > 0:

v ∆ P (x, y, R) ≤ v P (x, y, R + ∆ 2 ) ≤ C A P x, R+t+∆/2 2 ≤ C ′ (n, a, b, d; ∆) A P x, R+t 2 2
Proof of Proposition 2.6. We just integrate (6) over a fundamental domain C x for the action of P on H ξ (x):

F P (x, R) = p∈P vol[B(x, R) ∩ pC x ] = Cx p∈P 1 B(x,R) (pz) dz = Cx v P (x, y, R) dy
so, integrating over each slice ψ ξ,t (S x ) by the coarea formula, we obtain

R-R0 0 ψ ξ,t (Sx) A -1 P x, R + t 2 dt C ≺ F P (x, R) C ≺ R 0 ψ ξ,t (Sx) A -1 P x, R + t 2 dt
(the left inequality holding for R > R 0 ). By [START_REF] Boland | Volume rigidity for finite volume manifolds[END_REF], both sides are weakly equivalent to the integral

R 0 A P (x, t) A P (x, R+t 2 )
dt, up to a multiplicative constant c = c(n, a, b, d).2

Remark 2.9 Thus, we see that the curvature bounds imply that v ∆ P (x, R) ≍ v P (x, R) for ∆ and R large enough. This also holds in general for non-elementary groups Γ with finite Bowen-Margulis measure, as in this case v

∆ Γ (x, R) ∼ 2 µx 2 µ BM sinh[ ∆ 2 δ(Γ)
] by Roblin's asymptotics. On the other hand, it is unclear whether the weak equivalence v ∆ Γ ≍ v Γ holds for non-elementary lattices Γ, when µ BM = ∞.

In the next section we will also need estimates for the growth of annuli in the spaces of left and right cosets of a lattice Γ of X, modulo a bounded parabolic subgroup P . Notice that, if P fixes ξ ∈ X(∞), the function v P \Γ (x, R) counts the number of points γx ∈ Γx falling in the Dirichlet domain D(P, x) of P with d(x, γx) < R; on the other hand, the function v Γ/P (x, R) counts the number of horoballs γH ξ (x) at distance (almost) less than R from x. It is remarkable that, even if these functions count geometrically distinct objects, they are weakly asymptotically equivalent, as the following Proposition will show. Actually, let H ξ be a horoball centered at the parabolic fixed point ξ of P < Γ; we call depth(H ξ ) the minimal distance min Γ\{e} d(H ξ , γH ξ ). Then, for S x defined as in Definition 2.3 we have: Proposition 2.10 Let Γ be a torsionless, non-elementary, discrete group of isometries of X, let P a bounded parabolic subgroup of Γ, and let x ∈ X be fixed. Assume that max{diam(S x ), 1/depth(H ξ (x))} ≤ d, and let ℓ be the minimal displacement d(x, γx) of the elements γ ∈ Γ whose domains of attraction

U ± (γ, x) = {y | d(γ ±1 x, y) ≤ d(x, y)} are included

in the Dirichlet domain D(P, x).

There exists a constant δ 0 = δ 0 (a, d) such that, for all ∆, R > 0: (x, R);

(iv) 1 4 v ∆-δ 0 -4ℓ Γ (x, R) ≤ v ∆ P \Γ/P (x, R) ≤ v ∆ Γ (x, R).
Notice that (iv) strenghtens a result of S. Hersonsky and F. Paulin on the number of rational lines with depth smaller than R (cp. [START_REF] Hersonsky | Counting orbits in coverings of negatively curved manifolds and Hausdorff dimension of cusps excursions[END_REF] Theorem 1.2, where the authors furthermore assume the condition δ P < δ Γ ). Actually, let H ξ be the largest horosphere centered at ξ non intersecting any other γH ξ for γ = e, and recall that the depth of a geodesic c = (ξ, γξ) is defined as the length of the maximal subsegment ĉ ⊂ c outside ΓH ξ . The double coset space P \ (Γ-P ) /P can be identified with the set of oriented geodesics (ξ, γξ) of X with γ ∈ Γ-P . Then, if x ∈ ∂H ξ , the counting function v P \(Γ-P )/P (x, R) corresponds to the number of geodesics of X = Γ\X which travel a time R outside the cusp C = P \H ξ , before entering and definitely staying (in the future and in the past) in C.

Then, we set

Γ = { γ | γP ∈ Γ/P } Γ 0 = {γ 0 | γ 0 ∈ Γ, γ 0 ξ ∈ S x (∞)} ∪ {e}.
We have bijections Γ ∼ = Γ/P and Γ 0 ∼ = P \Γ, as S x (∞) is a fundamental domain. Moreover, every γ 0 ∈ Γ 0 almost minimizes the distance to x in its right coset P γ 0 . Actually, for all γ ∈ Γ set z(γ) = (ξ, γξ) ∩ ∂H ξ (x) and z ′ (γ) = (ξ, γξ) ∩ γ∂H ξ (x); then, for all p ∈ P we have, by Lemma 2.1

d(x, pγ 0 x) ≥ d(x, pz(γ)) + d(pz(γ), pz ′ (γ)) + d(pz ′ (γ), pγ 0 x) -ǫ 1 (a, d) ≥ d(x, γ 0 x) -c (10) as d(H ξ (x), pγ 0 H ξ (x)) = d(pz(γ), pz ′ (γ)), for c = 2d + ǫ 1 (a, d).
We will now define a bijection between pointed metric spaces i : (P \Γ, x 0 ) → (Γ/P, x 0 ) which almost-preserves the distance to their base point x 0 = P (with respect to their quotient distances | • | x = d x (P, •) as seen at the beginning of the section), as follows. For every γ ∈ Γ we can write γ = γp γ , for uniquely determined γ ∈ Γ and p γ ∈ P ; given a right coset P γ, we take γ 0 ∈ Γ 0 representing P γ and then set i(P γ) := p γ 0 γ 0 P . The map i is surjective. Actually, given γP , we take p ∈ P such that pγξ ∈ S x (∞), so that P γ = P γ 0 , for γ 0 = pγ ∈ Γ 0 ; then, we write γ 0 = γ 0 p γ 0 , and we deduce that i(P γ) = i(P γ 0 ) = i(P γ 0 p -1 ) = p -1 γ 0 P = p -1 γ 0 p -1 γ 0 P = γP . We now check that i is injective. Given γ 0 = γ 0 p γ 0 and γ ′ 0 = γ ′ 0 p γ ′ 0 in Γ 0 representing two right cosets P γ and P γ ′ , assume that p γ 0 γ 0 P = p γ ′ 0 γ ′ 0 P . Then, γ 0 ξ = p γ ′ 0 ξ for p = p -1 γ 0 p γ ′ 0 ∈ P , which yields p γ 0 = p γ ′ 0 as γ 0 ξ, γ ′ 0 ξ ∈ S x (∞) and S x (∞) is a fundamental domain for the left action of P ; so, γ 0 P = γ ′ 0 P , which implies that γ 0 = γ ′ 0 too (as Γ is a section of Γ/P ). Therefore, P γ = P γ 0 = P γ 0 p γ 0 = P γ ′ 0 p γ ′ 0 = P γ ′ 0 = P γ ′ . To show that i almost preserves | | x , we notice that, given a class P γ and writing its representative in Γ 0 as γ 0 = γ 0 p γ 0 , we have [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF] and by Lemma 2.1

|P γ| x ≤ |γ 0 | x ≤ d(x, γ 0 x) + d( γ 0 x, γ 0 p γ 0 x) = | γ 0 | x + |p γ 0 | x while, by
|P γ| x ≥ |γ 0 | x -c ≥ d(x, z ′ (γ 0 )) + d(z ′ (γ 0 ), γ 0 p γ 0 ) -ǫ 1 (a, d) -c ≥ | γ 0 | x + |p γ 0 | x -2c as d(z ′ (γ 0 ), γ 0 x) < d. On the other hand |i(P γ)| x = |p γ 0 γ 0 P | x ≤ d(x, p γ 0 x) + d(p γ 0 x, p γ 0 γ 0 P x) = |p γ 0 | x + | γ 0 | x while, as z(p γ 0 γ 0 ) = p γ 0 z( γ 0 ) and z ′ (p γ 0 γ 0 ) = p γ 0 z ′ ( γ 0 ), we get by Lemma 2.1 |i(P γ)| x ≥ d(x, p γ 0 z( γ 0 )) + d(p γ 0 z( γ 0 ), p γ 0 γ 0 P x) -ǫ 1 (a, d) ≥ |p γ 0 | x + | γ 0 | x -c.
This shows that |P γ| xc ≤ |i(P γ)| x ≤ |P γ| x + 2c. We then immediately deduce that v P \Γ (x, R -2c) ≤ v Γ/P (x, R) ≤ v P \Γ (x, R + c), as well as (i) for δ 0 = 4c. The proof of the left-hand inequality in (ii) is a variation for annuli of a trick due to Roblin, cp. [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]. Actually, as LP LΓ, we can choose a γ ∈ Γ with d(x, γx) = ℓ and such that the domains of attraction U ± (γ, x) are included in the domain D(P, x). Let v D(P,x) (x, R) be the number of points of the orbit Γx falling in D(P, x) ∩ B(x, R).

We have:

v ∆ Γ (x, R) ≤ v ∆ D(P,x) (x, R) + v ∆+2ℓ D(P,x) (x, R) ≤ 2v ∆+2ℓ D(P,x) (x, R)
since, for γx ∈ A ∆ (x, R), either γx ∈ D(P, x), or γγx ∈ D(P, x) and γγx ∈ A ∆+2ℓ (x, R).

As the points of P falling in D(P, x) minimize the distance to x modulo the left action of P , we also have v ∆+2ℓ D(P,x) (x, R) = v ∆+2ℓ P \Γ (x, R), which proves (ii). Assertion (iii) follows directly from (i) and (ii). To show (iv), we need to estimate the number of classes γP modulo the left action of P , that is the elements of Γ such that γx belongs to the fundamental domain D(P, x). We choose an element γ ∈ Γ with U ± (γ, x) ⊂ D(P, x) as before, and apply again Roblin's trick to the classes γP . The set Γx can be parted in two disjoint subsets: the subset Γ 1 := Γ ∩ D(P, x), and the subset Γ 2 := Γ ∩ D(P, x) c , whose elements γ then satisfy γ γ ∈ D(P, x) and |γ γ|

x ≤ | γ| x + ℓ. Then v ∆ Γ/P (x, R) = v ∆ Γ 1 (x, R) + v ∆ Γ 2
(x, R) ≤ 2v ∆+2ℓ P \Γ/P (x, R) and we conclude by (iii).2

3 Orbit-counting estimates for lattices

In this section we give estimates of the orbital function v Γ (x, y, R) and of v X (R) in terms of the orbital function of the parabolic subgroups P i and the associated cuspidal functions F P i of Γ. These estimates will be used in §4 and §6; they stem from an accurate dissection of large balls in compact and horospherical parts, assuming that ambient space X admits a nonuniform lattice action.

Let Γ be a lattice of X. The quotient manifold X = Γ\X is geometrically finite, and we have the following classical results due to B. Bowditch [START_REF] Bowditch | Geometrical finiteness with variable negative curvature[END_REF] concerning the structure of the limit set Γ and of X:

(a) L(Γ) = X(∞) and it is the disjoint union of the radial limit set L rad (Γ) with finitely many orbits L bp Γ = Γξ 1 ∪ . . . ∪ Γξ m of bounded parabolic fixed points; this means that each ξ i ∈ L bp G is the fixed point of some maximal bounded parabolic subgroup P i of Γ;

(b) (Margulis' lemma) there exist closed horoballs H ξ 1 , . . . , H ξm centered respectively at ξ 1 , . . . , ξ m , such that gH ξ i ∩ H ξ j = ∅ for all 1 ≤ i, j ≤ m and all γ ∈ Γ -P i ;

(c) X can be decomposed into a disjoint union of a compact set K and finitely many "cusps" C1 , ..., Cm : each Ci is isometric to the quotient of H ξ i by the maximal bounded parabolic group P i ⊂ Γ. We refer to K and to C = ∪ i Ci as to the compact core and the cuspidal part of X.

Throughout this section, we fix x ∈ X and we consider a Dirichlet domain D(Γ, x) centered at x; this is a convex fundamental subset, and we may assume that D contains the geodesic rays [x, ξ i [. Accordingly, setting S i = D∩∂H ξ i and C i = D∩H ξ i ≃ S i ×R + , the fundamental domain D can be decomposed into a disjoint union

D = K ∪ C 1 ∪ • • • ∪ C m
where K is a convex, relatively compact set containing x in its interior (projecting to a subset K in X), while C i and S i are, respectively, connected fundamental domains for the action of P i on H ξ i and ∂H ξ i (projecting respectively to subsets Ci , Si of X).

Finally, as LP i = {ξ i }, for every 1 ≤ i ≤ m we can find an element γ i ∈ Γ, with ℓ i = d(x, γ i x), which is in Schottky position with P i relatively to x, i.e. such that the domains of attraction U ± (γ i ) = {y | d(γ ±1 i x, y) ≤ d(x, y)} are included in the Dirichlet domain D(P i , x), as in Proposition 2.10.

For the following, we will then set d = max{diam(K), diam(S i ), 1/depth(H ξ i ), ℓ i } ≥ ǫ 0 . Proposition 3.1 (Counting Formula) Assume that x, y ∈ X project respectively to the compact core K and to a cuspidal end Ci of X = Γ\X. There exists C ′′ = C(n, a, b, d) such that:

[v Γ (x, •) * v P i (x, y, •)](R-D 0 ) C ′′ ≺ v Γ (x, y, R) C ′′ ≺ [v Γ (x, •) * v P i (x, y, •)](R+D 0 ) ∀R ≥ 0
for a constant D 0 only depending on n, a, b, d.

Proof. We will write, as usual, |γ| x = d(x, γx) and |γP | x = d(x, γP x), and choose a constant ∆ > max{R 0 , ∆ 0 , 2δ 0 + 4d}, where R 0 , ∆ 0 , δ 0 are the constants of Propositions 2.5 and 2.10. We first show that

B(x, R) ∩ Γy ⊂ N k = 1 γ ∈ Γ, |γ| ≤ k∆ B (γx, (N -k)∆) ∩ (γP i )y (11)
for N = ⌊ R ∆ ⌋ + 2. Actually, let γy ∈ B(x, R) ∩ γH ξ i and set ȳi = [x, γξ] ∩ γ∂H ξ i . By using the action of the group γP i γ -1 on γH ξ i , we can find γ = γp, with p ∈ P i , such that ȳi ∈ γC i . Since the angle ∠ ȳi (x, γy) at ȳi is greater than π 2 , we have:

d(x, γy) ≤ d(x, ȳi ) + d(ȳ i , γy) ≤ d(x, γy) + ǫ 0 < R + ǫ 0 with |γ| ≤ d(x, ȳi ) + d < R + d + ǫ 0 ≤ N ∆. Then, if k∆ ≤ |γ| < (k + 1)∆, we deduce d(γx, γy) ≤ d(ȳ i , γy) + d ≤ R + ǫ 0 -d(x, γx) + 2d < (N -k)∆ which shows that γy = γp -1 y ∈ B(γx, (N -k)∆) ∩ (γP i )y = γ [B(x, (N -k)∆) ∩ P i y].
Thus, we obtain:

v Γ (x, y, R) ≤ N k=1 v Γ (x, k∆) • v P i (x, y, (N -k)∆) ≺ v Γ * v P i (R + 2∆)
This proves the right hand side of our inequality. The left hand is more delicate, as we need to dissect the ball B(x, R) in disjoint annuli. So, consider the set Γ i of minimal representatives of Γ/P i as in the proof of Proposition 2.10. We have:

A 4∆ (x, R) ∩ Γy ⊃ N k=0 γ ∈ Γ i k∆ -∆ 2 ≤ | γ| < k∆ + ∆ 2 A ∆ ( γx, (N -k)∆) ∩ ( γP i )y (12)
for N = ⌊ R ∆ ⌋ + 1. In fact, given γy = γp i y ∈ A ∆ ( γx, (N -k)∆) with γx ∈ A ∆ (x, k∆) we have again 12) is a disjoint union, as the annuli with the same center do not intersect by definition, while for γ = γ ′ the orbits γP i y and γ ′ P i y lie on different horospheres γH i = γ ′ H i , which are disjoint by Margulis' Lemma. From [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF] and by Proposition 2.10 we deduce that for all R > 0 it holds:

N ∆ -2∆ ≤ | γ| + d( γx, γy) -2d -ǫ 0 ≤ d(x, γy) ≤ | γ| + d( γx, γy) < N ∆ + ∆ as ∆ > 2d + ǫ 0 , hence γy ∈ A 4∆ (x, R). Notice that (
v 4∆ Γ (x, y, R) ≥ 1 2 N k=0 v ∆/2 Γ (x, k∆) • v ∆ P i (x, y, (N -k)∆) (13)
as ∆ > 2ℓ i . Now, we set h i = b ξ i (x, y) and we sum (13) over annuli of radii R n = n∆, and we get:

v Γ (x, y, R) ≥ 1 4 ⌊ R ∆ ⌋-2 n=0 v 4∆ Γ (x, y, n∆) ≻ ⌊ R ∆ ⌋-1 k=0   ⌊ R ∆ ⌋-1 n≥k v ∆/2 Γ (x, (n -k)∆)   • v ∆ Pi (x, y, k∆) ≥ ≥ ⌊ R ∆ ⌋-1 k≥ h i ∆ +1 v Γ (x, R -(k + 2)∆) • v ∆ Pi (x, y, k∆) C ′ ≻ ⌊ R ∆ ⌋-1 k= h i ∆ +1 v Γ (x, R -(k + 2)∆) A Pi x, k∆+hi 2 (14) as v ∆ P i (x, y, k∆) ≻ A -1 P i x, k∆+h i 2 if k∆ ≥ h i + ∆ > h i + R 0 by Proposition 2.5.
Using again Proposition 2.5 and ( 5), it is easily verified that the expression in ( 14) is greater than the continuous convolution v Γ (x, •) * v P i (x, y, •) (R + 4∆), up to a multiplicative constant CC ′ ∆. This ends the proof, by taking D 0 = 4∆.2

The Counting Formula enables us to reduce the estimate of the growth function v X to a group-theoretical calculus, that is to the estimate of a the convolution of v Γ with the cuspidal functions F P i of maximal parabolic subgroups P i of Γ:

Proposition 3.2 (Volume Formula)

There exists a constant C ′′′ = C ′′′ (n, a, b, d, vol(K)), such that:

v Γ (x, •) * i F Pi (x, •) (R-2D 0 ) C ′′′ ≺ v X (x, R) C ′′′ ≺ v Γ (x, •) * i F Pi (x, •) (R+2D 0 ) ∀R ≥ 0 (15)
for D 0 = D 0 (n, a, b, d) as in Proposition 3.1.

Proof. Let h i = d(x, H ξ i ); we may assume that the constants R 0 , D 0 of Propositions 2.5 and 3.1 satisfy

D 0 ≫ d ≥ diam(K) ≥ h i ≫ R 0 . Now call S i (h) = ψ ξ i ,h [S i ]; integrating v Γ (x,
y, R) over the fundamental domain D yields, by Proposition 3.1:

v X (x, R + 2D 0 ) = D v Γ (x, y, R + 2D 0 )dy = K v Γ (x, y, R + 2D 0 )dy + m i=1 Ci v Γ (x, y, R + 2D 0 )dy C ′′ ≻ m i=1 R+D0 2hi v Γ (x, R + 2D 0 -t) t-hi hi Si(h)
v Pi (x, y, t) dy dh dt which then gives by Propositions 2.5 and 2.6, as

h = b ξ i (x, y) ≤ t -h i < t -R 0 , v X (x, R + 2D 0 ) C ′′ ≻ R+D0 2hi v Γ (x, R + 2D 0 -t) m i=1 t-hi hi A Pi (x, h) A Pi x, t+h 2 dh dt ≥ R+D0-2hi 0 v Γ (x, R -t) m i=1 t 0 A Pi (x, s + h i ) A Pi x, t+s+3hi 2
ds dt ≻ R 0 v Γ (x, R -t) m i=1 F Pi (x, t)dt.
Reciprocally, we have v Γ (x, R -D 0 ) ≤ v Γ (x, y, R) ≤ v Γ (x, R + D 0 ) so again by Proposition 3.1 and Remarks 2.7 we obtain

v X (x, R -2D 0 ) C ′′ ≺ vol(K) • v Γ (x, R -D 0 ) + m i=1 Ci R-2D0 0 v Γ (x, t)v Pi (x, y, R -t)dt dy C ′′′ ≺ v Γ (x, R -D 0 ) + R-2D0 0 v Γ (x, t) m i=1 R-t 0 A Pi (x, h) A Pi x, R-t+h 2 dh dt as v P i (x, y, R -t) = 0 for R -t < b ξ i (x, y) = h. This proves the converse inequality, since v Γ (x, R -D 0 ) ≺ v Γ (x, R -D 0 )F P i (R 0 ) ≤ 1 D 0 -R 0 R-R 0 R-D 0 v Γ (x, t)F P i (x, R -t)dt.2
As a consequence of the Volume Formula and of Corollary 2.8, we deduce 2 : Corollary 3.3 If F P i are the cuspidal functions of the parabolic subgroups of Γ:

(i) ω + (X) = max{δ(Γ), ω + (F P 1 ), ..., ω + (F Pm )}. (ii) ω + (X) = ω -(X) = δ(Γ) if Γ is 1 2 -parabolically pinched.

Margulis function for regular lattices

In this section we assume that Γ is a lattice which is neither sparse nor exotic. We need to recall a general criterion for the divergence of the Poincaré series of Γ, which can be found in [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF], [START_REF] Dal'bo | On the growth of quotients of Kleinian groups[END_REF]:

Divergence Criterion. Let Γ be a geometrically finite group: if δ + (P ) < δ(Γ) for every parabolic subgroup P of Γ, then Γ is divergent.

Proof of Theorem 1.3. Let Γ be a nonuniform lattice of X which is neither sparse nor exotic. As Γ is not exotic, it satisfies the gap property δ(P ) < δ(Γ) for all parabolic subgroups; by the Divergence and Finiteness Criterion we deduce that the group is divergent and that

µ BM (U X) < ∞. Therefore v Γ (x, R) c Γ (x)
≍ e δ(Γ)R is purely exponential (for some c Γ (x) depending on Γ, x). We will now show that X has a Margulis function. Let D be the fundamental domain for Γ and P i the maximal parabolic subgroups fixing ξ i as at the beginning of §3: we call w(x, y, R) = v X (x, R)e -δ(Γ)R , so that have

v X (x, R) e δ(Γ)R = D v Γ (x, y, R) e δ(Γ)R dy = K w(x, y, R)dy + m i=1 Ci w(x, y, R)dy (16)
We know that v Γ (x, y, R) ≤ v Γ (x, R + d) ≤ c Γ (x)e δ(Γ)R for y ∈ K, so we can pass to the limit for R → ∞ under the integral sign in the first term. For the integrals over the cusps, we have:

w(x, y, R) C ′′ ≺ [v Γ (x, •) * v Pi (x, y, •)](R+D 0 ) e δ(Γ)R cΓ(x) ≺ ∞ b ξ i (x,y) e -δ(Γ)t A Pi x, b ξ i (x,y)+t 2 dt = w(x, y)
2 Part (i) of this corollary already appears in [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF], where an upper estimate for vX is proved. Notice that in [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF] we erroneously stated that also ω -(X) = max{δ(Γ), ω -(FP 1 ), ..., ω -(FP m )}; an explicit counterexample to this is given in Example 6.2.

Notice that the dominating function w(x, y) is finite as δ + (P i ) < δ(Γ). We will now show that w(x, y) ∈ L 1 (C i ). With the same notations h i = d(x, H ξ i ) and S i (h) = ψ ξ i ,h (S i ) as before, we have for all i:

Ci w(x, y)dy = ∞ hi y∈Si(h)   ∞ b ξ i (x,y) e -δ(Γ)t A Pi x, b ξ i (x,y)+t 2 dt   dydh = ∞ hi ∞ h e -δ(Γ)t A Pi (h) A Pi x, h+t 2 dtdh = ∞ hi e -δ(Γ)t t hi A Pi (h) A Pi x, h+t 2 dh dt C ≺ ∞ hi e -δ(Γ)t F Pi (t)dt (17)
which converges, as Γ is not sparse and so ω + (F P i ) ≤ δ + (P i ) < δ(Γ), by Corollary 2.8.

We therefore obtain from ( 16), by dominated convergence, using Roblin's asymptotics

lim R→+∞ v X (x, R) e δ(Γ)R = µ x µ BM D µ y dy =: m(x) < +∞.
Notice that m(x) defines an L 1 -function on X = Γ\X, as its integral over D is finite.2

Proof of Theorem 1.4(i). We assume now that X has a nonuniform lattice Γ which is exotic, with the dominant parabolic subgroups P i , for i = 1, ..., d,

satisfying δ := δ(Γ) = δ + (P i ) ≤ 2(δ -(P i ) -ǫ),
for some ǫ > 0. When µ BM (U X) < ∞, the same lines of the above proof apply: v Γ (x, R) ≍ c Γ (x)e δR is purely exponential, and for the same functions w(x, y, R), w(x, y) we again obtain [START_REF] Hersonsky | Counting orbits in coverings of negatively curved manifolds and Hausdorff dimension of cusps excursions[END_REF]; but we need some more work to deduce that, for the dominant cusps P i , the integral of e -δt F P i (t)dt converges. So, for every dominant subgroup P i , we write v P i (x, t) = o i (t)e δt , for some subexponential functions o i (t); so, A P i (x, t) ≍ e -2δt /o i (2t) for t ≥ R 0 . As Γ is exotic, the dominant parabolic subgroups P i are convergent: actually, for any divergent subgroup Γ 0 < Γ with limit set LΓ 0 LΓ one has δ(Γ 0 ) < δ(Γ) (see [START_REF] Peigné | Groupes du ping-pong et géodésiques fermées en courbure -1[END_REF]). Therefore, the Poincaré series of P i gives, for ∆ > ∆ 0 ≫ 0

∞ > p∈Pi e -δd(x,px) ≻ k≥1 v ∆ Pi (x, k∆) e δk ≍ ∞ ∆ o i (t)dt
by Proposition 2.5, so the functions o i (t) are integrable. This shows that

w(x, y) = ∞ b ξ i (x,y) e -δt A Pi x, b ξ i (x,y)+t 2 dt = e δb ξ i (x,y) ∞ b ξ i (x,y) o i (h + t)dt < ∞
Moreover, as every dominant P i is strictly 1 2 -pinched, we have

v P i (x, t) ≻ e 1 2 (
δ+ǫ)t for some ǫ > 0, that is A P i (x, t) ≺ e -(δ+ǫ)t for all t > 0. Then Proposition 2.6 yields

F Pi (R) ≍ R 0 A Pi (s) A Pi ( s+R 2 ) ds ≺ e δR R 0 e -ǫs o i (s + R)ds for R ≫ 0 (18)
hence [START_REF] Hersonsky | Counting orbits in coverings of negatively curved manifolds and Hausdorff dimension of cusps excursions[END_REF] gives in this case:

Ci w(x, y)dy C ≺ ∞ hi e -δ(Γ)t F Pi (t)dt ≍ ∞ hi t 0 e -ǫs o i (s + t)ds dt ≤ ∞ 0 e -ǫs ∞ s o i (s + t)dt ds
which converges, since o i is integrable. We can therefore pass to the limit for R → ∞ under the integral in [START_REF] Hamenstädt | Entropy-rigidity of locally symmetric spaces of negative curvature[END_REF], obtaining the asymptotics for v X (x, R) as before.

On the other hand, if µ BM (U X) = ∞, then v Γ (x, R) = o Γ (R)e δR is lower-exponential, and by [START_REF] Heintze | Geometry of horospheres[END_REF] we have

F P i (x, R) = f i (R)e δR with f i (R) =
R 0 e -ǫs o i (s + R)ds for the dominant cusps, and f i (R) ≺ e -ǫR , with ǫ > 0, for the others; in both cases, f i ∈ L 1 , since the functions o i (t) are subexponential. Proposition 3.2 then gives, for any arbitrarily small ε ′ > 0

v X (x, R) e δR ≺ 1 e δR R 0 v Γ (x, t) i F Pi (R -t)dt ≺ R 0 o Γ (t) i f i (R -t)dt ≤ i f i 1 • sup t> R 2 o Γ (t) + o Γ ∞ • i R R/2 f i (t)dt ≤ ε ′ • i f i 1 + o Γ ∞ provided that R ≫ 0, since o Γ (t) is infinitesimal and the f i are integrable. This shows that v X (x, R) is lower-exponential too.2 Remark 4.1 We have seen that, if µ BM (U X) = ∞, then v Γ (x, R) = o Γ (R)e δR and v X (x, R) = o X (R)e δR , with o Γ , o X infinitesimal, and F P i (x, R)=f i (R)e δR with f i ∈ L 1 ; so, o Γ 1 ≺ o X 1 ≤ ∞ 0 v X (x, R) e δR dR ≺ ∞ 0 R 0 o Γ (t) i f i (R -t)dtdR ≤ o Γ 1 • i f i 1
and we can say that o Γ is L 1 if and only if o X is.

Entropy rigidity and 1 4 -pinched manifolds

This section is devoted to the proof of the rigidity Theorem 1.1 and Theorem 1.2. We prove it for a = 1, as the general case follows from this by applying an homothety. The proof is through the method of barycenter, initiated by Besson-Courtois-Gallot [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF], [START_REF] Besson | Lemme de Schwarz réel et applications géométriques[END_REF], and follows the lines of [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF] (Theorem 1.6, holding for compact quotients). The main difficulty in the finite volume case is to show that the map produced by the barycenter method is proper: we recall the main steps of the construction, referring the reader to [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF] for estimates which are now well established, and we focus on the new estimates necessary to prove properness.

Let X = Γ\X, fix a point x 0 ∈ X and call for short b ξ (x) = b ξ (x, x 0 ). The function b ξ is strictly convex if K X ≤ -1 < 0, since for every point y we have

Hess y b ξ ≥ g y -(db ξ ) y ⊗ (db ξ ) y (19)
where g denotes the metric tensor of X; moreover, it is folklore that if the equality holds in [START_REF] Izeki | Limit sets of Kleinian groups and conformally flat Riemannian manifolds[END_REF] at every point y and for every direction ξ, then the sectional curvature is constant, and X is isometric to the hyperbolic space H n . The idea of the proof is to show that the condition δ(Γ) = n -1 forces the equality in [START_REF] Izeki | Limit sets of Kleinian groups and conformally flat Riemannian manifolds[END_REF].

Recall that, for every measure µ on X(∞) whose support is not reduced to one point, we can consider its barycenter, denoted bar[µ], that is the unique point of minimum of the function y → B µ (y) = X(∞) e b ξ (y) dµ(ξ) (which is C 2 and strictly convex function, as b ξ (y) is). If supp(µ) is not a single point, it is easy to see that lim y→X(∞) B µ (y) = +∞, cp. [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF]. Consider now the map F : X → X defined by

F (x) = bar e -b ξ (x) µ x = argmin y → X(∞) e b ξ (y,x) dµ x (ξ)
where (µ x ) is the family of Patterson-Sullivan measures associated with the lattice Γ.

We briefly recall the main properties of the family (µ x ), cp. [START_REF] Sullivan | The density at infinity of a discrete group of hyperbolic motions[END_REF], [START_REF] Patterson | The limit set of a Fuchsian group[END_REF]:

(i) they are absolutely continuous w.r. to each other, and dµ x ′ dµx (ξ) = e -δ(Γ)b ξ (x ′ ,x) ; (ii) µ x (γ -1 A) = µ γx (A) for every isometry γ of X and every Borel set A ⊂ X(∞);

(iii) if Γ is a lattice, then the support of µ x is the whole boundary X(∞).

In [START_REF] Courtois | Critical exponents and rigidity in negative curvature[END_REF] it is proved that the map F satisfies the following properties: a. F is equivariant with respect to the action of Γ, i.e. F (γx) = γF (x); b. F is C 2 , with Jacobian

|Jac x F | ≤ δ(Γ) + 1 n n • det -1 (k x ) (20)
where k x (u, v) is the bilinear form on T x X defined as

k x (u, v) = X(∞) e b ξ (F (x),x)) (db ξ ) 2 F (x) + Hess F (x) b ξ (u, v) dµ x (ξ) X(∞) e b ξ (F (x),x)) dµ x (ξ) (21)
Notice that the eigenvalues of k x are all greater or equal than 1, by [START_REF] Izeki | Limit sets of Kleinian groups and conformally flat Riemannian manifolds[END_REF].

Property (a) stems from the equivariance (i) of the family of Patterson-Sullivan measures with respect to the action of Γ, and from the cocycle formula for the Busemann function: b ξ (x 0 , x) + b ξ (x, y) = b ξ (x 0 , y). Property (b) comes from the fact that the Busemann function is C 2 on Hadamard manifolds, and is proved by direct computation, which does not use cocompactness. By equivariance, the map F defines a quotient map F : X → X, which is homotopic to the identity through the homotopy

Ft (x) = bar e -b ξ (x) (tµ x + (1 -t)λ x ) mod Γ
where λ x is the visual measure from x. Actually, the map F t = bar e -b ξ (x) (tµ x +(1-t)λ x ) passes to the quotient since it is still Γ-equivariant; moreover, we have bar e -b ξ (x) λ x = x as for all v ∈ T x X:

dB e -b ξ (x) λx x (v) = X(∞) (db ξ ) x (v)e b ξ (x) e -b ξ (x) dλ x (ξ) = UxX g x (u, v)du = 0.
We now prove that:

Proposition 5.1 The homotopy map Ft is proper.

Assuming for a moment Proposition 5.1, the proof of Theorem 1.1 follows by the degree formula: since F is properly homotopic to the identity, it has degree one, so

vol( X) = X F * dv g ≤ X |Jac x F |dv g ≤ δ(Γ) + 1 n n • X det -1 (k x )dv g ≤ δ(Γ) + 1 δ(H n ) + 1 n •vol( X)
as det(k x ) ≥ 1 everywhere. So, if δ(Γ) = δ(H n ) = n -1, we deduce that det(k x ) = 1 everywhere and k = g, hence the equality in the equation ( 19) holds for every y = F (x) and ξ. Since F is surjective, this shows that X has constant curvature -1.2

To show that the map Ft is proper, we need some precise estimates on the Patterson-Sullivan measures of a lattice. For x ∈ X and ζ ∈ X(∞), let xζ(t) be the geodesic ray from x to ξ; we define the "spherical cap" V ζ (x, R) ⊂ X(∞) as the set of points at infinity ξ whose projection on xζ falls between xζ(R) and ζ.

The following estimates are proved in [START_REF] Peigné | Autour de l'exposant de Poincaré d'un groupe Kleinien[END_REF]: Lemma 5.2 Let Γ be a nonuniform lattice of X, with curvature k X ≤ -a 2 . Let x 0 ∈ X, and let

D = K ∪ C 1 ∪ • • • ∪ C m a
decomposition of the Dirichlet domain of Γ centered at x 0 as in Sect. §3, corresponding to maximal, bounded parabolic subgroups P 1 , ..., P m with fixed points ξ 1 , ..., ξ m . There exists a constant c such that for every

ζ ∈ X(∞), if x R = x 0 ζ(R) ∈ γC i and r = b ξ i (x 0 , γ -1 x R ) we have: (a) µ x 0 (V ζ (x 0 , R)) c ≻ e -δ(Γ)(R-r) • p ∈ P i d(x 0 , px 0 ) > 2r + c ′ e -δ(Γ)d(x 0 ,px 0 ) , if γξ i ∈ V ζ (x 0 , R) (b) µ x 0 (V ζ (x 0 , R)) c ≻ e -δ(Γ)(R+r) • v P i (2r -c ′ )
otherwise.

Proof of Proposition 5.1. We denote by z the projection of points z ∈ X to X. Call δ = δ(Γ) and µ t

x = e -b ξ (x) (tµ x + (1t)λ x ); we need to show that if t k → t 0 and if xk → ∞ in X, then ȳk = Ft k (x k ) = bar[µ t k

x k ] goes to infinity too. Assume by contradiction that the points ȳk stay in a compact subset of X: so (up to a subsequence) xk , ȳk lift to points x k , y k such that y k → y 0 ∈ X and d(y

0 , x k ) = d(ȳ 0 , xk ) = R k → ∞.
By the cocycle relation b ξ (y, x) = b ξ (y, y 0 ) + b ξ (y 0 , x) and by the density formula for the Patterson-Sullivan measures dµx dµy 0 (ξ) = e -b ξ (x,y 0 ) , we have

(dB µ t x ) y (v) = X(∞) e b ξ (y,x) (db ξ ) y (v)dµ t x = t X(∞) e b ξ (y,y0) e (δ+1)b ξ (y0,x) (db ξ ) y (v) dµ y0 + (1 -t) X(∞) e b ξ (y,x) (db ξ ) y (v) dλ x (22)
We will now estimate the two terms in the above formula, and show that (dB µ t k x k

) y k does not vanish for R k ≫ 0, a contradiction. So, let ζ k be the endpoints of the geodesic rays y 0 x k (t), and let

v k = (∇b ζ k ) y k . Also, consider the caps V ζ k (y 0 , R k ) and V ζ k (y 0 , R k /2).
Let us first consider the contributions of the integrals in [START_REF] Margulis | Certain applications of ergodic theory to the investigation of manifolds of negative curvature Funkcional[END_REF] over

X \ V ζ k (y 0 , R k /2). If ξ ∈ X(∞) \ V ζ k (y 0 , R k /2), the projection P of ξ over y 0 ζ k falls closer to y 0 than to x k , hence b ξ (y 0 , x k ) ≤ 2ǫ 0 by (2); moreover, b ξ (y k , y 0 ) ≤ d(y k , y 0 ) → 0, so the first integral on X \ V ζ k (y 0 , R k /2) for x = x k , y = y k and v = v k gives: X\V ζ k (y0, R k 2 ) e b ξ (y k ,y0) e (δ+1)b ξ (y0,x k ) (db ξ ) y k (v k ) dµ y0 < 2e 2ǫ0(δ+1) µ y0 for k ≫ 0. Analogously, the second integral on X \ V ζ k (y 0 , R k /2) yields X\V ζ k (y0, R k 2 ) e b ξ (y k ,x k ) (db ξ ) y k (v k ) dλ x k < 2e 2ǫ0 vol(S n-1 )
for k ≫ 0, since |b ξ (y k , x k )-b ξ (y 0 , x k )| < d(y k , y 0 ). So, these contributions are bounded. We now compute the contributions of the integrals over

V ζ k (y 0 , R k /2) \ V ζ k (y 0 , R k ). For all ξ ∈ V ζ k (y 0 , R k /2) we have that (∇b ξ ) y 0 • (∇b ζ k ) y 0 is close to 1, for R k ≫ 0; moreover, as |(∇b ξ ) y k •v k -(∇b ξ ) y 0 •(∇b ζ k ) y 0 | n→0 -→ 0, we deduce that (db ξ ) y k (v k ) > 1 2 on V ζ k (y 0 , R k /2)
for n ≫ 0, hence these contributions are positive. Finally, let us compute the contributions of these integrals on the caps V ζ k (y 0 , R k ). For ξ ∈ V ζ k (y 0 , R k ), consider the ray y 0 ξ(t) from y 0 to ξ, and the projection P (t) of y 0 ξ(t) on the geodesic y 0 ζ k . We have, again by Lemma (2)

b ξ (y 0 , x k ) ≥ lim t→∞ [d(y 0 , P (t)) + d(P (t), ξ(t))] -[d(ξ(t), P (t)) + d(P (t), x k )] -2ǫ 0 ≥ R k -2ǫ 0 therefore we deduce that, for k ≫ 0 V ζ k (y0,R k ) e b ξ (y k ,y0) e (δ+1)b ξ (y0,x k ) (db ξ ) y k (v k ) dµ y0 ≥ 1 2 t k e (δ+1)(R k -2ǫ0) µ y0 (V ζ k (y 0 , R k )) (23)
V ζ k (y0,R k ) e b ξ (y k ,x k ) (db ξ ) y k (v k ) dλ x k ≥ 1 4 (1 -t k )e (R k -d(y k ,y0)-2ǫ0) vol(S n-1 ) (24)
It is clear that this last integral goes to infinity when R k ≫ 0; we will now prove that the right-hand side of (23) also diverges for R k → ∞. This will conclude the proof, as it will show that dB µ t k

x k (v k ), being a convex combination of two positive diverging terms, does not vanish for k ≫ 0. So, let D = K ∪ C 1 ∪ • • • ∪ C m be a decomposition of the Dirichlet domain of Γ centered at y 0 as in Sect. §3, corresponding to maximal, bounded parabolic subgroups P 1 , ..., P m with fixed points ξ 1 , ..., ξ m . We know that xk belongs to some cusp of X, so x k ∈ γC i , for some γ, so let

r k = b ξ i (y 0 , γ -1 x k ) ≤ R k . If γξ i falls in V ζ k (y 0 , R k )
and ∆ ≫ 0, as K X ≥ -b 2 we can use Lemma 5.2 and Proposition 2.5 and deduce that there exist constants c, C ′ such that

e (δ+1)R k µ y0 (V ζ k (y 0 , R k )) c ≻ e (δ+1)R k e -δ(R k -r k ) • h> 2r k ∆ v ∆ Pi ((h + 1)∆)e -δh∆ C ′ ≻ e R k +δr k • h> 2r k ∆ e (δ -(Pi)-ǫ-δ)k∆ ≻ e R k • e (2δ -(Pi)-δ-2ǫ)r k
for arbitrarily small ǫ > 0. Since K X ≤ -1, we know that A P i (y 0 , R) ≺ e -(n-1)R , so δ -(P i ) ≥ n-1 2 by Proposition 2.5; as R k ≥ r k the integral in [START_REF] Nguyen Phan | On finite volume, negatively curved manifolds[END_REF] diverges in this case. If, on the other hand, γξ i ∈ V ζ k (y 0 , R k ), always by 5.2 and 2.5, we have:

e (δ+1)R k µ y 0 (V ζ k (y 0 , R k )) c ≻ e R k -δr k • v P (y 0 , 2r k -c) ≻ e R k +(n-1-δ)r k
which also diverges as δ ≤ n -1. This concludes the proof that the map Ft is proper.2 Lemma 6.1 Let b > a > 0, β > α > 0 and ǫ > 0 be given. There exist D = D(a, b, α, β, ǫ) > 1 and D ′ = D ′ (a, b, α, β) > 0 such that if [p, q], [r, s] are disjoint intervals satisfying r ≥ Dq and p ≥ D ′ , then there exist C 2 , convex and decreasing functions φ ǫ , ψ ǫ on [p, s] satisfying:

       ∀ t ∈ [p, q], φ ǫ (t) = t β e -bt ∀ t ∈ [r, s], φ ǫ (t) = t α e -at ∀ t ∈ [p, s], t β e -bt ≤ φ ǫ (t) ≤ t α e -at ∀ t ∈ [p, s], a 2 -ǫ ≤ φ ′′ ǫ (t) φǫ(t) ≤ b 2 + ǫ and        ∀ t ∈ [p, q], ψ ǫ (t) = t α e -at ∀ t ∈ [r, s], ψ ǫ (t) = t β e -bt ∀ t ∈ [p, s], t β e -bt ≤ ψ ǫ (t) ≤ t α e -at ∀ t ∈ [p, s], a 2 -ǫ ≤ ψ ′′ ǫ (t) ψǫ(t) ≤ b 2 + ǫ Example 6.2 Sparse lattices.
Sparse lattices satisfying ω + (X) > δ(Γ) were constructed by the authors in [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF]. Here, we modify that construction to show that, for spaces X admitting sparse lattices, one can have ω + (X) > ω -(X) > δ(Γ) (in contrast, notice that δ(Γ) always is a true limit); this shows in pwrticular that sparse lattices generally do not have a Margulis function.

We start from a hyperbolic surface X0 = X 0 \Γ of finite volume, homeomorphic to a 3-punctured sphere, and, for any arbitrary small ǫ > 0, we perturb the hyperbolic metric g 0 on one cusp C = P \H ξ (x) into a metric g ǫ by choosing an analytic profile T ǫ obscillating, on infinitely many horospherical bands, from e -t to e -bt . Namely, choose a = 1, b > 2 and ǫ > 0 arbitrarily small, and let D, D ′ be the constants given by Lemma 6.1. For M ≫ 1, we define a sequence of disjoint subintervals of [M 4n , M 4n+1 ]:

[p n , q n ] := [M 4n , 2M 4n ], [r n , s n ] := p n + M 4n+1 2 , q n + M 4n+1 2 such that r n ≥ Dq n , p n+1 ≥ Ds n , p 1 ≥ D ′ (we can choose any M ≥ max{4D -1, 3 √ D} in order that these conditions are satisfied). Notice that t+M 4n+1 2 ∈ [r n , s n ] for all t ∈ [p n , q n ]. Then, by Lemma 6.1, we consider a C 2 , decreasing function T ǫ (t) satisfying:

(i) T ǫ (t) = e -t for t ∈ [M 4n-2 , M 4n ] ∪ [p n , q n ], and T ǫ (t) = e -bt for t ∈ [r n , s n ]; (ii) e -bt ≤ T ǫ (t) ≤ e -t and -b 2ǫ ≤ T ′′ ǫ (t)/T ǫ (t) ≤ -1 + ǫ. Thus, the new analytic profile T ǫ (t) of C coincides with the profile of a usual hyperbolic cusp on [M 4n-2 , 2M 4n ], and with the profile of a cusp in curvature -b 2 on the bands [r n , s n ] ⊂ [M 4n , M 4n+1 ]. We have, with respect to the metric g ǫ : dt ≺ e R 2 [START_REF] Sambusetti | Asymptotic properties of coverings in negative curvature[END_REF] since

M 4n+4 ≥ t+R 2 ≥ M 4n+3 2 ≥ M 4n+2 ;
d. δ(Γ) is arbitrarily close to δ + (P ), let's say δ(Γ) ≤ b 2 + δ 2 , if we perturb the hyperbolic metric sufficiently far in the cusp C, i.e. if r 1 ≫ 0 (this is Proposition 5.1 in [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF]).

It follows that ω -(X) > δ(Γ). Actually, assume that v Γ (x, R) ≻ e (δ(Γ)-η)R , for arbitrarily small η. By Proposition 3.2 and (26), for any R ≫ 0, if M 4n+1 ≤ R < M 4n+5 v X (x, R + 2D 0 ) ≥ v Γ (x, •) * F P (x, •) (x, R) ≻ e (δ(Γ)-η)(R-M 4n+1 ) • e ( b 2 +δ)M 4n+1 by taking just the term v Γ (x, R-t)F P (x, t)) of the convolution with t closest to M 4n+1 , where F P (t) ≻ e ( b 2 +δ)t ; as M 4n+1 ≥ R/M 4 we get v X (x, R + 2∆) ≻ e (δ(Γ)-η+ δ/2+η M 4 )R which gives ω -(X) ≥ δ(Γ) + δ 2M 4 , η being arbitrary. Finally, we show that ω + (X) > ω -(X). In fact, the cusps different from C being hyperbolic, we have, always by Proposition 3.2, that ω + (X) = ω + (F P ) ≥ b 2 + δ. On the other hand, we know that ω + (F P ) ≤ max{δ + (P ), 2(δ + (P )δ -(P )} = b -1, by Corollary 2.8; thus, assuming F P (x, t) ≺ e (b-1+η)t , for arbitrarily small η, equation [START_REF] Sambusetti | Asymptotic properties of coverings in negative curvature[END_REF]

yields for R = M 4n+4 v X (x, R -2D 0 ) ≤ M 4n+3 0 v Γ (x, R -t) • F P (x, t)dt + R M 4n+3 v Γ (x, R -t) • F P (x, t)dt ≺ M 4n+3 0 e δ(Γ)(R-t) • e (b-1+η)t dt + R M 4n+3
e δ(Γ)(R-t) • e Examples 6.3 Exotic, strictly 1 2 -parabolically pinched lattices. We say that a lattice Γ is strictly 1 2 -parabolically pinched when every parabolic sugroup P < Γ satisfies the strict inequality δ + (P ) < 2δ -(P ). Let X = Γ\X as before; we show here that, for Γ exotic and strictly 1 2 -parabolically pinched, the following cases which appear in Theorem 1.4 do occur:

(a) µ BM (U X) = ∞ and v X is lower-exponential;

(b) µ BM (U X) < ∞ and v X is purely exponential. We start by an example of lattice satisfying (a). In [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF] the authors show how to construct convergent lattices, in pinched negative curvature and any dimension n; we will take n = 2 here by the sake of simplicity.

In those examples, the metric is hyperbolic everywhere but one cusp C, which has analytic profile T (t) = t β e bt for t ≥ t 0 ≫ 0, with β > 1 and b > 2. Therefore, there is just one dominant maximal parabolic subgroup P , with A P (x, t) ≍ T (t) ≍ e bt , and δ + (P ) = δ -(P ) = b 2 ; moreover, the subgroup P is convergent as

p∈P e -b 2 d(x,px) ≤ k≥0 v P (x, k)e -b 2 k ≍ ∞ 1 e -b 2 t A P (x, t 2 ) dt ≍ ∞ 1 e -b 2 t (t) β • e -b t 2 dt ≍ ∞ 1 t -β dt < ∞.
By decomposing the elements of Γ in geodesic segments which, alternatively, either go very deep in the cusp or stay in the hyperbolic part of X, we show in [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF] that Γ is convergent too, provided that t 0 ≫ 0. Then, Γ is exotic with infinite Bowen-Margulis measure, and v Γ (x, R) is lower-exponential by Roblin's asymptotics. By Theorem 1.4(i), the function v X is lower-exponential behaviour as well, with the same exponential growth rate.

  and X does not admit a Margulis function. Namely, there exist cases (Examples 6.4(a)&(b)) where:

≻ e b 2 R • M 4n e ( b 2 - 1 )pn ≥ e b 2 R • e 1 M ( b 2 - 1 )R ( 26 )

 221212126 a. δ + (P ) = b 2 and δ -(P ) = 1 2 , by (i) and (ii), because of Proposition 2.5; b. ω + (F P ) ≥ b 2 + δ for δ = 1 M ( b 2 -1) > 0, because for R = M 4n+1 F P (x, R) ≻ as p n /R = 1 M ; c. ω -(F P ) ≤ 1 2 if M > 2, as for R ∈ [M 4n+3 , M4n+4] we obtain:F P (x, R) ≺

1 2 1 M )R being b 2 ≤

 212 t dt ≺ e δ(Γ)R • e (b-1+η-δ(Γ))M 4n+3 ≤ e ( b 2 + δ 2 + (b/2+η-δ(Γ) ≤ b 2 + δ 2 and M 4n+3 = R M . Hence ω -(X) < b 2 + δ ≤ ω + (X), if M ≫ 0 and η small enough.

This means that the additive subgroup of R generated by the length of closed geodesics in X = Γ\X is dense in R; it is the case, for instance, if dim(X) =

2, or when Γ is a lattice.

-pinched spaces with lattices. This strongly relies on the above characterization and on a Counting Formula (Proposition 3.1), which enables us to reduce the computation of v X to the analytic profile of the cusps of X and v Γ (so, in the last instance, to T.Roblin's asymptotics (a)&(b)):

(i) v ∆-δ 0 P \Γ (x, R) ≤ v ∆ Γ/P (x, R) ≤ v ∆+δ 0 P \Γ (x, R); (ii) 1 2 v ∆-2ℓ Γ (x, R) ≤ v ∆ P \Γ (x, R) ≤ v ∆ Γ (x, R); (iii) 1 2 v ∆-δ 0 -2ℓ Γ (x, R) ≤ v ∆ Γ/P (x, R) ≤ v ∆+δ 0 Γ

Proof. The right-hand inequalities in (ii), (iii), (iv) are trivial. Let us prove (i). We first define two sections of the projections P \Γ ← Γ → Γ/P . Consider the fundamental domain S x (∞) for the action of P on X(∞)-{ξ} defined in 2.3, and choose for each γ ∈ Γ, a representative γ of γP which minimizes the distance to x.

We are now ready to prove Theorem 1.2 :

Proof of Theorem 1.2. Assume that Γ is a nonuniform lattice in a 1 4 -pinched negatively curved manifold X, i.e. -b 2 ≤ K X ≤ -a 2 with b 2 ≤ 4a 2 . If X = H n a , then clearly v X (x, R) ≍ v Γ (x, R) is purely exponential, X has a Margulis function, and Γ is divergent. Otherwise, let P i be the maximal parabolic subgroups of Γ, up to conjugacy. By the formulas [START_REF] Boland | Volume rigidity for finite volume manifolds[END_REF], we know that for all x ∈ X e -(n-1)bR ≺ A P i (x, R) ≺ e -(n-1)aR , so by Proposition 2.5 we have a(n -1)

for all P i . Thus, Γ is parabolically 1 2 -pinched. It follows from Corollary 3.3 that ω + (X) = ω -(X) = δ(Γ). Moreover, for all P i we have

the strict inequality following by the rigidity Theorem 1.1, since X = H n a . The same argument applies when X is only asymptotically 1 4 -pinched, by replacing -a 2 , -b 2 with the bounds -k 2 +ǫ ≤ K X ≤ -k 2 -+ ǫ on the cusps Ci . Then, Γ is also non-exotic, and we can conclude by Theorem 1.3 that Γ is divergent, with finite Bowen-Margulis measure, v X ≍ v Γ and X has a L 1 Margulis function m(x).2

Examples

In this section we show that all the cases presented in Theorem 1.4 do occurr, by providing examples of spaces X with exotic or sparse lattices Γ which do not admit a Margulis function, and with functions v Γ , v X having different behaviour.

If C = P \H ξ (o) is a cusp of X = Γ\X, we write the metric of X in horospherical coordinates on H ξ (o) ∼ = ∂H ξ (o)×R + as g = T (x, t) 2 dy 2 + dt 2 , for x ∈ ∂H ξ (o) and t = b ξ (o, •). We call the function T (x, t) the analytic profile of the cusp C. The horospherical area A P (x, t) is then obtained by integrating T n-1 (x, t) over a compact fundamental domain S for the action of P on ∂H ξ (o); thus, we have

for all x ∈ C (for a constant c depending on X and o). Also, notice that, in the particular case where T (y, t) = T (t), for points x, y belonging to a same horosphere H ξ we have by the Approximation Lemma 2.2

We will repeatedly make use of the following lemma, which is a easy modification of one proved in [START_REF] Dal'bo | On the growth of non-uniform lattices in pinched negatively curved manifolds[END_REF]: We now give an example for (b). This is more subtle, as we need to take a divergent, exotic lattice Γ: the existence of such lattices is established, in dimension 2, in [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF]. Again, the simplest example is homeomorphic to a three-punctured sphere, with three cusps, and hyperbolic metric outside one cusp C, which has analytic profile

with b > 2 and A, B, D ≫ 0. As before, we have one dominant and convergent maximal parabolic subgroup P , with δ

In [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF] it is proved that, according to the values of A and B, the behaviour of the group Γ is very different: it is convergent with critical exponent δ(Γ) = δ + (P ), for A ≫ 0 and B = 0, while it is divergent with δ(Γ) > δ + (P ) if B ≫ A. By perturbation theory of transfer operators, it is then proved that there exists a value of B for which Γ is divergent with δ(Γ) = δ + (P ) precisely. Thus, for this particular value of B, the lattice Γ we is exotic, and has finite Bowen-Margulis measure by the Finiteness Criterion, as

It follows that v X ≍ v Γ is purely exponential, by Theorem 1.4(i).

Examples 6.4 Exotic, exactly 1 2 -parabolically pinched lattices. We say that a lattice Γ is exactly 1 2 -parabolically pinched when it is 1 2 -parabolically pinched and has a parabolic sugroup P < Γ satisfisfying the quality δ + (P ) = 2δ -(P ). We show here that for an exotic and exactly 1 2 -parabolically pinched lattice Γ, the following cases can occur:

(a) µ BM (U X) < ∞, with v Γ purely exponential and v X upper-exponential;

(b) µ BM (U X) = ∞, with v Γ lower-exponential and v X upper-exponential.

We start by (a). Consider a surface with three cusps as in the Examples 6.3, now perturbing the hyperbolic metric on the cusp C to an analytic profile defined as follows. First, choose a sequence of disjoint subintervals of

and then define, for b > 1 and 0 < γ < 1

2 t for all t ≥ t 0 ≫ 0 (in order that the conditions of Lemma 6.1 are satisfied, it is enough to choose any 0 < µ < 1 4D and M > D).

As before, the profile T gives a divergent, exotic lattice Γ for a suitable value of B and A ≫ 0, with dominant parabolic subgroup P having δ + (P ) = b 2 = δ(Γ), and δ -(P ) = b 4 . The Bowen-Margulis measure of Γ is finite, as (28) also holds in this case; thus, v Γ is purely exponential. Let us now show that v X is upper exponential: for every R = M 2n+1 we have, by Proposition 3.2,

)

Producing examples for case (b) is more difficult; for this, we will need an exotic lattice Γ whose orbital function satisfies

R γ e δ(Γ)R . Lattices with lowerexponential growth and infinite Bowen-Margulis measure are investigated in [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF], where a refined counting result is proved, according to the behaviour of the profile functions of the cusps (the examples in [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF] are, as far as we know, the only precise estimates of the orbital function for groups with infinite Bowen-Margulis measure). Here we only give the necessary analytic profiles of the cusps in order to have a function v X which is exponential or upper-exponential, referring to [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF] for the precise estimate of v Γ . We again start from a hyperbolic surface X0 = X 0 \Γ with three cusps as in 6.3, and perturb now the metric on two cusps. We choose b > 2 and 1 + γ < β < 2 + γ, and define the profiles for C1 and C2 as

and T 2 (t) = e -t for t ≤ A t 1+γ e -bt for t ≫ A for the same sequence of intervals [p n , q n ], [r n , s n ] as in [START_REF] Sullivan | The density at infinity of a discrete group of hyperbolic motions[END_REF]. If P 1 , P 2 are the associated maximal parabolic subgroups, we have δ -(P 1 ) = b 4 and δ + (P 1 ) = b 2 , while δ + (P 2 ) = δ -(P 2 ) = b 2 by construction. It is easily verified that these parabolic subgroups are convergent as γ > 0. Again, pushing the perturbation far in the cusps (i.e. choosing A ≫ 0) and for a suitable value of B, the lattice Γ becomes exotic and divergent; it has two dominant cusps, it is exactly 1 2 -parabolically pinched, and has infinite Bowen-Margulis measure, because (as γ < 1)

Accordingly, v Γ is lower-exponential. In [START_REF] Dal'bo | Convergence and counting in infinite measure[END_REF] it is proved that the least convergent dominant parabolic subgroup determines the asymptotics of v Γ ; in this case, the parabolic subgroup P 1 converges faster than P 2 , and the chosen profile for C2 then gives

so that τ + (t) = t β on [r n , s n ] and τ -(t) = t on [p n , q n ], we compute as in case (a):

which is upper-exponential as β < 2 + γ.

Remark 6.5 Notice that in all these examples b can be chosen arbitrarily close to 2a = 2. Thus, by the last condition in Lemma 6.1, the analytic profiles give metrics with curvature -4a 2ǫ ≤ K X ≤ -a 2 , for arbitrarily small ǫ > 0.


Faming Zhang
email: faming.zhang@uni-rostock.de

Bikramjit Basu

Lianjun Wang

Izabel Fernanda Machado

Claude Estournès

Nanomaterials Processed by Spark Plasma Sintering

come

e spark plasma sintering (SPS), a variant of eld-assisted sintering (FAST) or pulsed electric current sintering (PECS), is a novel pressure assisted pulsed electric current sintering process, which utilizes ON-OFF DC pulse energizing. Due to the repeated application of an ON-OFF DC pulse voltage and current ow between powder particles, the spark discharges and the Joule heating (local high temperature state) are therefore dispersed to the overall specimen. e SPS process is based on the electrical spark discharge phenomenon and is a high e cient, energy saving technique with a high heating rate and a short holding time.

e problem of rapid grain growth of nanomaterials during conventional sintering can be inhibited to a larger extent by using the SPS technique.

e SPS can be used for diverse novel bulk material applications, but it is particularly suitable for the processing of nanomaterials. Despite such anticipated advantages, the optimization of the process window (heating rate-temperature-time) in SPS process is a challenging task. More importantly, the underlying mechanisms for superfast densi cation still remains to be explored.

In the above perspective, this special issue contains some peer-reviewed research papers, which address some of the exciting issues or illustrate new nanomaterials development using SPS. is special issue's papers are all very much of the "head up" variety. First D. V. Dudina and A. K. Mukherjee reviewed the reactive SPS on the successes and challenges of nanomaterial synthesis. Secondly, some work on bioceramics is covered, for example, antimony-doped tin oxide nanoceramics (J. Wu et al.). Additionally, we look to some works on the nanostructured metals, for example, nanostructured nickel (F. Naimi et al.) and FeAl alloys (S. Paris et al.). In the end, the SPS of nanocomposites is also introduced, for example, graphene/Bi 2 Te 3 thermoelectric materials (B. Liang et al.) and Ti-ZrO 2 functionally graded materials (M. Jayachandran et al.).

is eld of "nanomaterials processed by SPS" develops very fast. is special issue is just a tip of the iceberg, but we also can conjure up the whole thing through seeing a part of it.

We thank all the authors for their contribution to the literature on SPS.

Faming Zhang Bikramjit Basu Lianjun Wang Izabel Fernanda Machado

Claude Estournès