Schottky groups and counting.
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 \textit{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 \textit{A. Eskin} and \textit{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 [\textit{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 [\textit{H. Abels} and \textit{G. A. Margulis}, Coarsely geodesic metrics on reductive groups. Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 163--183 (2004)].
scientific article; zbMATH DE number 2171513
scientific article