Exponential divergence of discrete subgroups of higher rank.
type: Item
instance of: scholarly article
title: Exponential divergence of discrete subgroups of higher rank.
zbMATH Open document ID: 1010.22018
author: Jean-François Quint
published in: Commentarii Mathematici Helvetici
publication date: 2002-12-01T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1837407
zbMATH Keywords: Lie groups
zbMATH Keywords: discrete subgroups
zbMATH Keywords: higher rank geometries
zbMATH Keywords: real semisimple Lie group
zbMATH Keywords: Dirichlet series
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
full work available at URL: s00014-002-8352-0
OpenAlex ID: W2088645488
DOI: 10.1007/S00014-002-8352-0
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description: scientific article; zbMATH DE number 1837407