Schottky groups and counting.
type: Item
instance of: scholarly article
title: Schottky groups and counting.
zbMATH Open document ID: 1087.22010
author: Jean-François Quint
publication date: 2005-05-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116195
review text: 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. Sys…
reviewed by: Herbert Abels
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 37D35
zbMATH DE Number: 2171513
zbMATH Keywords: Lie groups
zbMATH Keywords: discrete subgroups
zbMATH Keywords: higher rank geometry
zbMATH Keywords: thermo-dynamical formalism
MaRDI profile type: Publication
OpenAlex ID: W2477557444
cites work: Patterson-Sullivan measures in higher rank
cites work: Precise counting results for closed orbits of Anosov flows
cites work: Lalley's theorem on periodic orbits of hyperbolic flows
cites work: Asymptotic properties of linear groups
cites work: Q4497333
cites work: Limit sets of groups of linear transformations.
cites work: Prevalence of rapid mixing in hyperbolic flows
cites work: Mixing, counting, and equidistribution in Lie groups
cites work: Q3897665
cites work: Closed orbits in homology classes
cites work: Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits
cites work: R-REGULAR ELEMENTS IN ZARISKIDENSE SUBGROUPS
cites work: Q3999409
cites work: Linear actions of free groups
cites work: Asymptotic expansions for closed orbits in homology classes.
cites work: Exponential divergence of discrete subgroups of higher rank.
cites work: L'indicateur de croissance des groupes de Schottky
cites work: Ergodicité et équidistribution en courbure négative
cites work: The infinite word problem and limit sets in Fuchsian groups
cites work: Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque.
DOI: 10.5802/AIF.2102
published in: Annales de l’institut Fourier
Recommended article: Counting abelian group structures
Recommended article: On Schurity of Finite Abelian Groups
Recommended article: Counting homomorphisms onto finite solvable groups.
Recommended article: COUNTING CONJUGACY CLASSES
Recommended article: Counting the Subgroups of Some Finite Groups
Recommended article: Counting problems and semisimple groups
Recommended article: Schröder and group theory
Recommended article: Counting Schur rings over cyclic groups
Recommended article: Counting congruence subgroups.
Recommended article: Counting Certain Pairings in Arbitrary Groups
description: scientific article; zbMATH DE number 2171513