Orbital counting for some convergent groups
Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1307-1340.

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.

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.

Received:
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Accepted:
Published online:
DOI: 10.5802/aif.3335
Classification: 58F17, 58F20, 20H10
Keywords: Poincaré exponent, convergent/divergent groups, orbital function
Mot clés : exposant de Poincaré, groupe convergent/divergent, fonction orbitale

Peigné, Marc 1; Tapie, Samuel 2; Vidotto, Pierre 2

1 Institut Denis Poisson, UMR 7013 Université de Tours Université d’Orléans CNRS 37200 Tours (France)
2 Laboratoire Jean Leray 2 rue de la Houssinière BP92208 44322 Nantes Cedex 3 (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Peigné, Marc; Tapie, Samuel; Vidotto, Pierre. Orbital counting for some convergent groups. Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1307-1340. doi : 10.5802/aif.3335. https://aif.centre-mersenne.org/articles/10.5802/aif.3335/

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