In [BM11], an ergodic theoremà la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. A quick corollary is the irreducibility of the associated unitary representation. These results are generalized [Boy15] to the context of convex cocompact groups of isometries of a CAT(-1) space, using Theorem 4.1.1 of [Rob03], with the hypothesis of non arithmeticity of the spectrum. We prove all the analog results in the case of the free group Fr of rank r even if Fr is not the fundamental group of a closed manifold, and may have an arithmetic spectrum. X u := {x ∈ X | u is a prefix of ⌊x⌋} B u := {ξ ∈ B | u is a prefix of ξ}

No abstract

We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group Γ := SL 2 (A), where A := F q [X, X −1 ] is the ring of Laurent polynomials with coefficients in F q , endowed with the length function coming from a natural action of Γ on a product of two trees, to show that is has the radial rapid decay (RRD) property and doesn't have the rapid decay (RD) property. The criterion also applies to irreducible lattices in semisimple Lie groups with finite center endowed with a length function defined with the help of a Finsler metric. These examples answer a question asked by Chatterji and moreover show that, unlike the RD property, the RRD property isn't inherited by open subgroups. Contents 12 5.1. Structure of the proof 12 5.2. The Koopman representation on the product of the boundaries 12 5.3. Estimates on the growth of Γ 13 5.4. Estimates on the Harish-Chandra function 15 5.5. Comparing the decay of the Harish-Chandra function with the volume growth in the group SL 2 (F q [X, X −1 ]) 16 6. Proof of Corollary 2 17 6.1. RD and amenable subgroups of exponential growth 17 6.2. The lamplighter subgroup 17 References 18

We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group Γ := SL 2 (A), where A := Fq[X, X −1 ] is the ring of Laurent polynomials with coefficients in Fq, endowed with the length function coming from a natural action of Γ on a product of two trees, and show that it has the radial rapid decay (RRD) property and doesn't have the rapid decay (RD) property. We show that the criterion also applies to all irreducible lattices (uniform or not) in semisimple Lie groups with finite center endowed with a length function defined with the help of a Finsler metric. When the rank is greater or equal to two and the lattice is non-uniform, the lattice has RRD but not RD. These examples answer a question asked by Chatterji and moreover show that, unlike the RD property, the RRD property isn't inherited by open subgroups.Résumé. -Nous établissons un nouveau critère dynamique entraînant la propriété de décroissance rapide radiale. Nous explicitons le cas particulier du groupe Γ := SL 2 (A), où A := Fq[X, X −1 ] est l'anneau des polynômes de Laurent à coefficients dans le corps fini Fq, muni d'une fonction longueur provenant d'une action naturelle de Γ sur le produit de deux arbres. Nous prouvons que pour cette fonction longueur, ce groupe vérifie la propriété de décroissance rapide radiale (RRD), mais ne vérifie pas la propriété de décroissance rapide (RD). Nous prouvons aussi que notre critère s'applique à tout réseau irréductible (uniforme ou non), de tout groupe de Lie semi-simple à centre fini, muni d'une certaine fonction longueur définie à l'aide d'une métrique de Finsler. Lorsque le rang réel est supérieur ou égal à deux et que le réseau n'est pas uniforme, le réseau vérifie la propriété RRD, mais pas la propriété RD. Ces exemples répondent à une question de Chatterji et montrent que, contrairement à la propriété RD, la propriété RRD n'est pas héréditaire par passage à un sous-groupe ouvert.