We construct non-uniform convergent lattices of pinched, negatively curved Hadamard spaces, in any dimension . These lattices are exotic, by which we mean that they have a maximal parabolic subgroup such that . We also give examples of divergent, non-uniform exotic lattices in dimension . Finally, we consider a particular class of such exotic lattices, with infinite Bowen–Margulis measure and whose cusps have a particular asymptotic profile (satisfying a “heavy tail condition”), and we give precise estimates of their orbital function; namely, we show that their orbital function is lower exponential with asymptotic behaviour , for a slowly varying function .
Nous construisons des réseaux non uniformes et convergents d’isométries d’une variété d’Hadamard à courbure strictement négative et pincée de dimension quelconque. Ces réseaux sont dits exotiques, au sens où ils possèdent des sous-groupes paraboliques maximaux d’exposant critique . Nous donnons aussi des examples explicites de réseaux exotiques non uniformes et divergents en dimension . Enfin, nous étudions une classe particulières de tels réseaux exotiques non uniformes et divergents dont la mesure de Bowen–Margulis est infinie et dont les « cusps » présentent un profile asymptotique particulier, satisfaisant une propriété de « queue lourde », et proposons une estimation précise du comportement asymptotique de leur fonction orbitale ; plus précisément, nous montrons que leur fonction orbitale croît de façon sous-exponentielle avec un comportement à l’infini de la forme , où est une fonction à variations lentes.
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Keywords: Poincaré exponent, convergent/divergent groups, Bowen–Margulis measure, orbital function
Mot clés : Exposant de Poincaré, groupe convergent/divergent, mesure de Bowen–Margulis, fonction orbitale
Dal’bo, Françoise 1; Peigné, Marc 2; Picaud, Jean-Claude 2; Sambusetti, Andrea 3
@article{AIF_2017__67_2_483_0, author = {Dal{\textquoteright}bo, Fran\c{c}oise and Peign\'e, Marc and Picaud, Jean-Claude and Sambusetti, Andrea}, title = {Convergence and {Counting} in {Infinite} {Measure}}, journal = {Annales de l'Institut Fourier}, pages = {483--520}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {67}, number = {2}, year = {2017}, doi = {10.5802/aif.3089}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3089/} }
TY - JOUR AU - Dal’bo, Françoise AU - Peigné, Marc AU - Picaud, Jean-Claude AU - Sambusetti, Andrea TI - Convergence and Counting in Infinite Measure JO - Annales de l'Institut Fourier PY - 2017 SP - 483 EP - 520 VL - 67 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3089/ DO - 10.5802/aif.3089 LA - en ID - AIF_2017__67_2_483_0 ER -
%0 Journal Article %A Dal’bo, Françoise %A Peigné, Marc %A Picaud, Jean-Claude %A Sambusetti, Andrea %T Convergence and Counting in Infinite Measure %J Annales de l'Institut Fourier %D 2017 %P 483-520 %V 67 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3089/ %R 10.5802/aif.3089 %G en %F AIF_2017__67_2_483_0
Dal’bo, Françoise; Peigné, Marc; Picaud, Jean-Claude; Sambusetti, Andrea. Convergence and Counting in Infinite Measure. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 483-520. doi : 10.5802/aif.3089. https://aif.centre-mersenne.org/articles/10.5802/aif.3089/
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