Orbital counting for some convergent groups  [ Comptage orbital pour certains groupes convergents ]
Annales de l'Institut Fourier, à paraître, 34 p.

Nous construisons des variétés géométriquement finies à courbure strictement négative pincée, dont le flot géodésique possède une mesure de Bowen-Margulis non ergodique infinie, et dont la série de Poincaré converge à l’exposant δ Γ , et nous obtenons une estimation précise du comportement asymptotique de la fonction orbitale de ce groupe. Plus précisément, pour tout α]1,2[ et toute fonction à variations lentes L:(0,+), nous construisons des variétés de Hadamard (X,g) de dimension N2 dont le groupe des isométries qui préservent l’orientation possède des sous-groupes discrets et géométriquement finis Γ tels que, lorsque R+,

NΓ(R):={γΓ|d(o,γ·o)R}CΓ(o)L(R)RαeδΓR,

C Γ (o) est une constante strictement positive qui dépend du point o.

We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen–Margulis measure and whose Poincaré series converges at the critical exponent δ Γ . We obtain an explicit asymptotic for their orbital growth function. Namely, for any α]1,2[ and any smooth slowly varying function L:(0,+), we construct N-dimensional Hadamard manifolds (X,g) of negative and pinched curvature, whose group of oriented isometries possesses convergent geometrically finite subgroups Γ such that, as R+,

NΓ(R):={γΓ|d(o,γ·o)R}CΓ(o)L(R)RαeδΓR,

for some C Γ (o)>0 depending on the base point o.

Reçu le : 2018-10-01
Révisé le : 2019-06-13
Accepté le : 2019-07-11
Classification : 58F17,  58F20,  20H10
Mots clés: exposant de Poincaré, groupe convergent/divergent, fonction orbitale
@unpublished{AIF_0__0_0_A19_0,
     author = {Peign\'e, Marc and Tapie, Samuel and Vidotto, Pierre},
     title = {Orbital counting for some convergent groups},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Peigné, Marc; Tapie, Samuel; Vidotto, Pierre. Orbital counting for some convergent groups. Annales de l'Institut Fourier, à paraître, 34 p.

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