On some exotic Schottky groups

Abstract

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,

Convergence and Counting in Infinite Measure

“…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 [30], 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…”

“…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] :…”

“…An example of an exotic divergent discrete group has already been constructed in [30] : 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 [30] 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.…”

Transition de phase de fonctions orbitales pour des groupes de Schottky en courbure négative

“…Il ressort de [14] que le choix de a influe tout d'abord sur la valeur de l'exposant critique δ a,Γ de Γ, relativement à la métrique g a . En effet, lorsque a est suffisamment grand (disons a a 0 > 0), la surfaceX est de plus en plus « proche » d'une surface hyperbolique et l'exposant δ a,Γ devient strictement supérieur à 1 2 ; le groupe Γ est alors divergent d'après [5].…”

“…En effet, lorsque a est suffisamment grand (disons a a 0 > 0), la surfaceX est de plus en plus « proche » d'une surface hyperbolique et l'exposant δ a,Γ devient strictement supérieur à 1 2 ; le groupe Γ est alors divergent d'après [5]. Plus précisément, il est montré dans [14] qu'il existe une valeur a * ∈ ]0, a 0 [ du paramètre (2) Le choix du terme log log n s'impose dans l'expression (1.3) afin d'obtenir un groupe convergent. Celui du terme log L(log n) est inspiré des lois stables en calcul des probabilités ; il permet d'élargir la classe de métriques pour lesquelles existent des estimations asymptotiques précises de la fonction orbitale de Γ.…”