# zbMATH — the first resource for mathematics

Exponential divergence of discrete subgroups of higher rank. (Divergence exponentielle des sous-groupes discrets en rang supérieur.) (French) Zbl 1010.22018
The main object of the paper under review is a Zariski dense discrete subgroup $$\Gamma$$ of a connected real semisimple Lie group $$G$$ with finite centre. The author is interested in asymptotic properties of $$\Gamma$$. If the $$\mathbb{R}$$-rank of $$G$$ equals 1, an important rôle is played by the real number $\limsup\limits_{a\to\infty}\left(\frac{1}{a}\log (\#\{\gamma\in\Gamma |d(x,\gamma x)\leq a\})\right)$ which is the convergence exponent of the Dirichlet series $$\sum_{\gamma\in\Gamma}e^{-td(x,\gamma x)}$$ ($$t\in\mathbb{R}$$) [see S. J. Patterson, Acta Math. 136, 241–273 (1976; Zbl 0336.30005); D. Sullivan, Publ. Math., Inst. Hautes Étud. Sci. 50, 171–202 (1979; Zbl 0439.30034)]. The goal of the paper is to extend the study of exponential divergence of discrete subgroups to the higher rank case. If $$\mu\colon G\to\mathfrak a^+$$ is the projection corresponding to the Cartan decomposition $$G=K(\exp\mathfrak a^+)K$$, the set $$\mu (\Gamma)$$ (the limit cone) is convex with nonempty interior part [see Y. Benoist, Geom. Funct. Anal. 7, 1–47 (1997; Zbl 0947.22003)]. For any open cone $$C$$ of a Cartan subspace $$\mathfrak a$$, $$\tau_C$$ denotes the convergence exponent of the Dirichlet series $$\sum_{\gamma\in\Gamma ,\mu (\gamma)\in C}e^{-t\|\mu (\gamma)\|}$$ where $$t\in\mathbb{R}$$ and $$\|\cdot\|$$ stands for a norm on $$\mathfrak a$$ invariant under the Weyl group. For $$x\in\mathfrak a$$ set $$\psi (x)=\|x\|\inf\tau _C$$ where the infimum is taken over the set of open cones $$C$$ in $$\mathfrak a$$ containing $$x$$. The homogeneous function $$\psi$$ does not depend on the choice of a norm, and the convergence exponent of the Dirichlet series $$\sum_{\gamma\in\Gamma}e^{-t\|\mu (\gamma)\|}$$ equals $$\sup\limits_{x\in\mathfrak a\backslash\{0\}}\frac{\psi (x)}{\|x\|}$$. Let $$\rho$$ denote the sum of the roots of $$\mathfrak a$$ multiplied by the dimension of their weight spaces in $$\mathfrak g$$. The main result of the paper says that the function $$\psi$$ is estimated from above by $$\rho$$, concave and semicontinuous from above, and the set $$\{x\in\mathfrak a |\psi (x)>-\infty\}$$ coincides with the limit cone of $$\Gamma$$; moreover, $$\psi$$ is positive on the limit cone and strictly positive on its interior part. The main tool in the proof of these results is a “generic product” in $$\Gamma$$ which is studied using results of H. Abels, G. A. Margulis, and G. A. Soifer [Israel J. Math. 91, 1–30 (1995; Zbl 0933.22015)] on proximal linear maps. Analogues of the main results hold in the case where the ground field is any locally compact valued field (the Bruhat–Tits theory provides necessary tools).

##### MSC:
 22E40 Discrete subgroups of Lie groups 53C35 Differential geometry of symmetric spaces
Full Text: