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We use Kashiwara crystal basis theory to associate a random walk W to each irreducible representation V of a simple Lie algebra. This is achieved by endowing the crystal attached to V with a (possibly non uniform) probability distribution compatible with its weight graduation. We then prove that the generalized Pitmann transform defined in [1] for similar random walks with uniform distributions yields yet a Markov chain. When the representation is minuscule, and the associated random walk has a drift in the Weyl chamber, we establish that this Markov chain has the same law as W conditionned to never exit the cone of dominant weights. For the defining representation V of gl n , we notably recover the main result of [19]. At the heart of our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice. This theorem also have applications in representation theory since it permits to precise the behavior of some outer multiplicities for large dominant weights.

Abstract. -We consider a large class of non compact hyperbolic manifolds M = H n /Γ with cusps and we prove that the winding process (Yt) generated by a closed 1-form supported on a neighborhood of a cusp C, satisfies a limit theorem, with an asymptotic stable law and a renormalising factor depending only on the rank of the cusp C and the Poincaré exponent δ of Γ. No assumption on the value of δ is required and this theorem generalises previous results due to Y.

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Summary -Let Γ be a non-elementary Kleinian group acting on a Cartan-Hadamard manifold X ; denote by Λ(Γ) the non-wandering set of the geodesic flow (φ t ) acting on the unit tangent bundle T 1 (X/Γ). When Γ is convex cocompact (i.e. Λ(Γ) is compact), the restriction of (φ t ) to Λ(Γ) is an Axiom A flow : therefore, by a theorem of Bowen-Ruelle, there exists a unique invariant measure on Λ(Γ) which has maximal entropy. In this paper, we study the case of an arbitrary Kleinian group Γ. We show that there exists a measure of maximal entropy for the restriction of (φ t ) to Λ(Γ) if and only if the Patterson-Sullivan measure is finite ; furthermore when this measure is finite, it is the unique measure of maximal entropy.By a theorem of Handel-Kitchens, the supremum of the measure-theoretic entropies equals the infimum of the entropies of the distances d on Λ(X) ; when Γ is geometrically finite, we show that this infimum is achieved by the Riemannian distance d on Λ(X). Classification A.M.S :Primary 37C40, 37D40, 37B40, , 37D35 Secondary 28A50Dans cet article,X désignera une variété riemannienne complète simplement connexe, dont les courbures sectionnelles sont comprises entre deux constantes négatives −α 2 et −β 2 (0 < α ≤ β). Un groupe Kleinien sera un groupe discret d'isométries deX, non-élémentaire (i.e. qui ne possède pas de sous-groupes abéliens d'indice fini) et sans torsion : un groupe Kleinien Γ opère sans points fixes surX et on note X =X/Γ la variété riemannienne quotient. L'ensemble non-errant du flot géodésique (φ t ) agissant sur le fibré unitaire T 1 X est noté Λ(Γ) : c'est sur cet ensemble que se concentre la dynamique intéressante du flot.Nous allons relier certains invariants de la restriction de (φ t )à Λ(Γ)à l'exposant critique du groupe Γ, un invariant de Γ. Ce nombre δ(Γ) est défini comme l'exposant critique de la série, dite de Poincaré,

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.

We contruct a Cartan-Hadamard manifold with pinched negative curvature whose group of isometries possesses divergent discrete free subgroups with parabolic elements who do not satisfy the so-called "parabolic gap condition" introduced in [DOP]. This construction relies on the comparaison between the Poincaré series of these free groups and the potential of some transfer operator which appears naturally in this context. c f. Analogously, we whall write f c ∼ g (or simply f ∼ g) when |f (R) − g(R)| ≤ c for some constant c > 0 and R large enough.2. On the existence of convergent parabolic groups 2.1. The real hyperbolic space. We first consider the real hyperbolic space of dimension N ≥ 2, identified to the upper half-space H N := R N −1 × R * + . In this model,

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.