Exponential divergence of discrete subgroups of higher rank.
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 \textit{S.~J.~Patterson}, Acta Math. 136, 241--273 (1976; Zbl 0336.30005); \textit{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 \textit{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 \textit{H.~Abels, G. A.~Margulis}, and \textit{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).
scientific article; zbMATH DE number 1837407