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 .
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 .
Révisé le :
Accepté le :
Publié le :
Classification : 58F17, 58F20, 20H10
Mots clés : exposant de Poincaré, groupe convergent/divergent, fonction orbitale
@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'institut Fourier}, volume = {70}, number = {3}, year = {2020}, doi = {10.5802/aif.3335}, language = {en}, url = {https://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, Tome 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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