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Schottky groups and counting. (Groupes de Schottky et comptage.) (French) Zbl 1087.22010
The author proves a counting result for certain discrete subgroups $$\Gamma$$ of linear semisimple Lie groups $$G$$. The group $$\Gamma$$ is a Schottky group in the sense of Y. Benoist [Geom. Funct. Anal. 7, 1–47 (1997; Zbl 0947.22003)], a free subgroup of $$G$$ obtained by a procedure generalizing the classical construction of Schottky.
The counting can be described in geometric terms as follows. Let $$X$$ be the symmetric space of $$G$$. In the paper $$X$$ is supposed to be of non-compact type. For $$x\in X$$ and $$a\geq 0$$ put $$N_\Gamma(x,a)=\text{card}\{\gamma\in\Gamma\mid d(x,\gamma x)\leq a\}$$. If $$X$$ has strictly negative curvature the behavior of this quantity is well understood. For general $$X$$ the main result of the paper is that there is a constant $$\tau$$ such that for every $$x\in X$$ there is a constant $$C>0$$ such that $$N_\Gamma(x,a)\sim_{a\to\infty} C e^{\tau a}$$. This result can be translated into group theoretical terms, using the Cartan decomposition. This theorem is in contrast to a counting result of A. Eskin and C. McMullen [Duke Math. J. 71, 181–209 (1993; Zbl 0798.11025)] for lattices, where the quantity in question behaves asymptotically like $$a^{\frac{r-1}2}\cdot e^{\tau a}$$, where $$r$$ is the rank of $$X$$. Quint’s result is thus in keeping with the philosophy that Schottky groups behave like rank $$1$$ lattices, not like lattices in $$G$$.
I will describe a few steps of the proof. A technical result, proved in Chapter 2, gives a multiplicative formula for the dominant eigenvalue of the product of proximal linear maps. The author uses this to prove a density result for the asymptotic cone in the sense of Benoist of a Zariski dense subsemigroup $$\Gamma$$ of $$G$$. This is applied to verify a technical assumption in a result of Dolgopyat. Dolgopyat’s theorem gives control over the norm of certain Ruelle operators. Here the author makes use of his paper [J.-F. Quint, Ergodic Theory Dyn. Syst. 23, 249–272 (2003; Zbl 1037.22024)]. In Chapter 5 a Tauberian theorem, proved in Chapter 6, is used to finish the proof. For the proof of the Tauberian theorem a difficulty is caused by the fact that the norm on the Cartan subalgebra corresponding to the Riemannian metric on $$X$$ is given by a quadratic form and not the absolute value of a linear form on the Weyl chamber. One might ask the question if similar results hold also for norm-like metrics in the sense of [H. Abels and G. A. Margulis, Coarsely geodesic metrics on reductive groups. Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 163–183 (2004)].

##### MSC:
 22E40 Discrete subgroups of Lie groups 53C35 Differential geometry of symmetric spaces 37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
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