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

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DOI: 10.5802/aif.3089
Classification: 58F17, 58F20, 20H10
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

1 IRMAR, Université de Rennes-I Campus de Beaulieu 35042 Rennes Cedex (France)
2 LMPT, UMR 6083 Faculté des Sciences et Techniques Parc de Grandmont 37200 Tours (France)
3 Istituto di Matematica G. Castelnuovo Sapienza Università di Roma P.le Aldo Moro 5 00185 Roma (Italy)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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