no. 3
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:ℝ→(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):=♯{γ∈Γ|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:ℝ→(0,+∞), nous construisons des variétés de Hadamard (X,g) de dimension N≥2 dont le groupe des isométries qui préservent l’orientation possède des sous-groupes discrets et géométriquement finis Γ tels que, lorsque R→+∞,

NΓ(R):=♯{γ∈Γ|d(o,γ·o)≤R}∼CΓ(o)L(R)RαeδΓR,

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

Received: 2018-10-01
Revised: 2019-06-13
Accepted: 2019-07-11
Published online: 2020-06-26
DOI: https://doi.org/10.5802/aif.3335
Classification: 58F17,  58F20,  20H10
Keywords: Poincaré exponent, convergent/divergent groups, orbital function 
@article{AIF_2020__70_3_1307_0,
     author = {Peign\'e, Marc and Tapie, Samuel and Vidotto, Pierre},
     title = {Orbital counting for some convergent groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     pages = {1307-1340},
     doi = {10.5802/aif.3335},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_3_1307_0/}
}
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/item/AIF_2020__70_3_1307_0/

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