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 . We obtain an explicit asymptotic for their orbital growth function. Namely, for any and any smooth slowly varying function , we construct -dimensional Hadamard manifolds of negative and pinched curvature, whose group of oriented isometries possesses convergent geometrically finite subgroups such that, as ,
for some depending on the base point .
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 , et nous obtenons une estimation précise du comportement asymptotique de la fonction orbitale de ce groupe. Plus précisément, pour tout et toute fonction à variations lentes , nous construisons des variétés de Hadamard de dimension dont le groupe des isométries qui préservent l’orientation possède des sous-groupes discrets et géométriquement finis tels que, lorsque ,
où est une constante strictement positive qui dépend du point .
Revised:
Accepted:
Published online:
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
@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}, pages = {1307--1340}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {3}, year = {2020}, doi = {10.5802/aif.3335}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3335/} }
TY - JOUR AU - Peigné, Marc AU - Tapie, Samuel AU - Vidotto, Pierre TI - Orbital counting for some convergent groups JO - Annales de l'Institut Fourier PY - 2020 SP - 1307 EP - 1340 VL - 70 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3335/ DO - 10.5802/aif.3335 LA - en ID - AIF_2020__70_3_1307_0 ER -
%0 Journal Article %A Peigné, Marc %A Tapie, Samuel %A Vidotto, Pierre %T Orbital counting for some convergent groups %J Annales de l'Institut Fourier %D 2020 %P 1307-1340 %V 70 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3335/ %R 10.5802/aif.3335 %G en %F 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/articles/10.5802/aif.3335/
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