Convergence and Counting in Infinite Measure

Dal’bo, Françoise; Peigné, Marc; Picaud, Jean-Claude; Sambusetti, Andrea

Annales de l'Institut Fourier, Tome 67 (2017) no. 2, pp. 483-520.

Nous construisons des réseaux non uniformes et convergents $\Gamma$ d’isométries d’une variété d’Hadamard à courbure strictement négative et pincée de dimension $N\ge 2$ quelconque. Ces réseaux sont dits exotiques, au sens où ils possèdent des sous-groupes paraboliques maximaux $P<\Gamma$ d’exposant critique $\delta \left(P\right)=\delta \left(\Gamma \right)$. 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 $\asymp \frac{e^{{\delta }_{\Gamma }R}}{R^{1-\kappa }L\left(R\right)}$, où $L$ est une fonction à variations lentes.

We construct non-uniform convergent lattices $\Gamma$ of pinched, negatively curved Hadamard spaces, in any dimension $N\ge 2$. These lattices are exotic, by which we mean that they have a maximal parabolic subgroup $P<\Gamma$ such that $\delta \left(P\right)=\delta \left(\Gamma \right)$. 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 $\asymp \frac{e^{{\delta }_{\Gamma }R}}{R^{1-\kappa }L\left(R\right)}$, for a slowly varying function $L$.

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

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