Abstract. We study the growth and divergence of quotients of Kleinian groups G (i.e. discrete, torsionless groups of isometries of a Cartan-Hadamard manifold with pinched negative curvature). Namely, we give general criteria ensuring the divergence of a quotient groupḠ of G and the 'critical gap property' δḠ < δ G . As a corollary, we prove that every geometrically finite Kleinian group satisfying the parabolic gap condition (i.e. δ P < δ G for every parabolic subgroup P of G) is growth tight. These quotient groups naturally act on non-simply connected quotients of a Cartan-Hadamard manifold, so the classical arguments of Patterson-Sullivan theory are not available here; this forces us to adopt a more elementary approach, yielding as by-product a new elementary proof of the classical results of divergence for geometrically finite groups in the simply connected case. We construct some examples of quotients of Kleinian groups and discuss the optimality of our results.

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 quotientX = 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 :This has been the subject of many investigations since Margulis' [27] (see Roblin's book [33] and Babillot's report on [1] 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 ratewhich has a major dynamical significance, since it coincides with the topological entropy of the geodesic flow whenX is compact, and is related to many interesting rigidity results and characterization of locally symmetric spaces, cp.[23], [9], [6].2) The pointwise behavior of the Poincaré series associated with Γ :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 UX (see again [33] 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 : [33]). 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 ;where (µ x ) x∈X denotes the family of Patterson conformal densities of Γ, and m Γ the Bowen-Margulis measure on UX.Here, f ∼ g means that f (t)/g(t) → 1 when t → ∞ ; for c ≥ 1, we will write f c ≍ g when 1 c ≤ f (t)/g(t) ≤ c for t ≫ 0 (or simply f ≍g when the constant c is not specified). The best asymptotic regularity to be expected is the existence of an equivalent, as in (ii) ; an explicit computation of the second term in the asymptotic development of v Γ is a difficult question for locally symmetric spaces (and almost a hopeless question in the general Riemannian setting).1. This means that the set L(X) = {ℓ(γ) ; γ ∈ Γ} of lengths of all closed geodesics ofX = Γ\X is not contained in a discrete subgroup of R.

International audienceWe study the relation between the exponential growth rate of volume in a pinched negatively curved manifold and the critical exponent of its lattices. These objects have a long and interesting story and are closely related to the geometry and the dynamical properties of the geodesic flow of the manifold

We show that every nontrivial free product, different from the infinite dihedral group, is growth tight with respect to any algebraic distance: that is, its exponential growth rate is strictly greater than the corresponding growth rate of any of its proper quotients. A similar property holds for the amalgamated product of residually finite groups over a finite subgroup. As a consequence, we provide examples of finitely generated groups of uniform exponential growth whose minimal growth is not realized by any generating set. 2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. -Nous montrons que tout produit libre non trivial, différent du groupe diédral infini, est à croissance forte par rapport à n'importe quelle distance algébrique : son taux de croissance exponentielle est toujours strictement supérieur à celui d'un quotient propre quelconque. On prouve une propriété similaire pour les produits amalgamés de groupes résiduellement finis sur un sous-groupe fini. Comme application de ce résultat, nous présentons une classe de groupes à croissance exponentielle uniforme, dont la croissance minimale n'est réalisée par aucune partie génératrice finie. 2002 Éditions scientifiques et médicales Elsevier SAS N k=1 2n(2n − 1) k−1 , so that Ent(F n , S n ) = log(2n − 1).