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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)].

22E40 Discrete subgroups of Lie groups
53C35 Differential geometry of symmetric spaces
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
Full Text: DOI Numdam EuDML
[1] Mesures de patterson-Sullivan en rang supérieur, Geometric and functional analysis, 12, 776-809, (2002) · Zbl 1169.22300
[2] Precise counting results for closed orbits of Anosov flows, Annales Scientifiques de l’École Normale Supérieure, 33, 33-56, (2000) · Zbl 0992.37026
[3] Lalley’s theorem on periodic orbits of hyperbolic flows, ergodic theory and dynamical systems, 18, 17-39, (1998) · Zbl 0915.58074
[4] Propriétés asymptotiques des groupes linéaires, Geometric and functional analysis, 7, 1-47, (1997) · Zbl 0947.22003
[5] Propriétés asymptotiques des groupes linéaires (II), Advanced studies in pure mathematics, 26, 33-48, (2000) · Zbl 0960.22012
[6] Limit sets of groups of linear transformations, 62, 367-385, (2000) · Zbl 1115.37305
[7] Prevalence of rapid mixing in hyperbolic flows, Ergodic theory and dynamical systems, 18, 1097-1114, (1998) · Zbl 0918.58058
[8] Mixing, counting and equidistribution in Lie groups, Duke mathematical journal, 71, 181-209, (1993) · Zbl 0798.11025
[9] Differential geometry, Lie groups and symmetric spaces, 80, (1978), Academic Press, San Diego · Zbl 0451.53038
[10] Closed orbits in homology classes, Publications mathématiques de l’IHES, 71, 5-32, (1990) · Zbl 0728.58026
[11] Renewal theorems in symbolic dynamics, with application to geodesic flows, Noneuclidean tessellations and their fractal limits, Acta mathematica, 163, 1-55, (1989) · Zbl 0701.58021
[12] \({\Bbb R}\)-regular elements in Zariski dense subgroups, Oxford quaterly journal of mathematics, 45, 541-545, (1994) · Zbl 0828.22010
[13] Zeta functions and the periodic orbit structure for hyperbolic dynamics, 187-188, (1990), Société Mathématique de France, Paris · Zbl 0726.58003
[14] Linear actions of free groups, Annales de l’institut Fourier, 51, 131-150, (2001) · Zbl 0967.37016
[15] Asymptotic expansions for closed orbits in homology classes, GeometriæDedicata, 87, 123-160, (2001) · Zbl 1049.37021
[16] Divergence exponentielle des sous-groupes discrets en rang supérieur, Commentarii Mathematicii Helvetici, 77, 563-608, (2002) · Zbl 1010.22018
[17] L’indicateur de croissance des groupes de Schottky, Ergodic theory and dynamical systems, 23, 249-272, (2003) · Zbl 1037.22024
[18] Ergodicité et équidistribution en courbure négative, 95, (2003), Société Mathématique de France · Zbl 1056.37034
[19] The infinite word problem and limit sets in Fuchsian groups, Ergodic theory and dynamical systems, 1, 337-360, (1981) · Zbl 0483.30029
[20] Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, Journal f\" ur die reine und angewandte Mathematik, 247, 196-220, (1971) · Zbl 0227.20015
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