Convergence and Counting in Infinite Measure
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, p. 483-520

We construct non-uniform convergent lattices Γ of pinched, negatively curved Hadamard spaces, in any dimension N2. These lattices are exotic, by which we mean that they have a maximal parabolic subgroup P<Γ such that δ(P)=δ(Γ). We also give examples of divergent, non-uniform exotic lattices in dimension N=2. 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 e δ Γ R R 1-κ L(R), for a slowly varying function L.

Nous construisons des réseaux non uniformes et convergents Γ d’isométries d’une variété d’Hadamard à courbure strictement négative et pincée de dimension N2 quelconque. Ces réseaux sont dits exotiques, au sens où ils possèdent des sous-groupes paraboliques maximaux P<Γ d’exposant critique δ(P)=δ(Γ). Nous donnons aussi des examples explicites de réseaux exotiques non uniformes et divergents en dimension N=2. 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 e δ Γ R R 1-κ L(R), où L est une fonction à variations lentes.

Received : 2016-06-11
Revised : 2016-03-21
Accepted : 2016-06-14
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3089
Classification:  58F17,  58F20,  20H10
Keywords: Poincaré exponent, convergent/divergent groups, Bowen–Margulis measure, orbital function
@article{AIF_2017__67_2_483_0,
     author = {Dal'bo, Fran\c coise and Peign\'e, Marc and Picaud, Jean-Claude and Sambusetti, Andrea},
     title = {Convergence and Counting in Infinite Measure},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     pages = {483-520},
     doi = {10.5802/aif.3089},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_2017__67_2_483_0/

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