Growth of conjugacy classes in Gromov hyperbolic groups.
type: Item
instance of: scholarly article
title: Growth of conjugacy classes in Gromov hyperbolic groups.
zbMATH Open document ID: 1042.20031
author: Michel Coornaert
author: Gerhard Knieper
published in: Geometric and Functional Analysis. GAFA
publication date: 2002-01-01T00:00:00Z
review text: Consider a group \(\Gamma\) acting by isometries on a metric space \(X\). Given an element \(\gamma\in\Gamma\), the number \(\ell(\gamma)=\inf_{x\in X}d(x,\gamma x)\) only depends on the conjugacy class \([\gamma]\) of \(\gamma\) in \(\Gamma\). \(\ell(\gamma)\) is called the length of the conjugacy class \([\gamma]\). For all \(t\geq 0\), let \[ P(t)=\text{card}\{[\gamma]\mid\gamma\in\Gamma\text{ is primitive and }\ell(\gamma)\leq t\} \] denote the number of conjugacy classes of primitive elements in \(\Gamma\) having length at most \(t\). An element \(\gamma\in\Gamma\) is said to be primitive if it cannot be written as a power \(\gamma=\alpha^n\), where \(\alpha\in\Gamma\) and \(n\geq 2\). An asymptotic estimate for \(P(t)\), as \(t\) tends to infinity, has been given by \textit{G. A. Margulis} [Funct. Anal. Appl. 3, 335-336 (1969); translation from Funkts. Anal. Prilozh. 3, No. 4, 89-90 (1969; Zbl 0207.20305)] in the case when \(X\) is the universal cover of a closed Riemannian manifold \(M\) with negative sectional curvature and \(\Gamma=\pi_1M\). Margulis obtained the estimate \[ P(t)\sim\tfrac{e^{ht}}{ht} \] as \(t\to\infty\), where \(h>0\) is a constant depending only on \(X\) (the notation \(f(t)\sim g(t)\) means that \(f(t)/g(t)\to 1\)). The constant \(h\) is equal to the topological entropy of the geodesic flow on the unit tangent bundle of \(M\). It is also equal to the `volume entropy' of \(X\), i.e. one has, for all \(x\in X\), \[ h=\lim_{R\to\infty}\tfrac{\log\text{vol\;}B(x,R)}{R}=\lim_{r\to \infty}\tfrac{\log\text{card\;}\Gamma_R(x)}R \] where \(B(x,R)\) denotes the closed ball of radius \(R\) centered at \(x\) and \[ \Gamma_R(x)=\{\gamma\in\Gamma\mid d(x,\gamma x)\leq R\}. \] A generalization of Margulis' result for rank 1 manifolds has been obtained by the second author of the paper under review [Geom. Funct. Anal. 7, No. 4, 755-782 (1997; Zbl 0896.53033)] as following. Let \(X\) be the universal cover of the closed rank 1 manifold \(M\) and let \(\…
reviewed by: Alireza Abdollahi
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 1890450
zbMATH Keywords: conjugacy classes
zbMATH Keywords: Gromov hyperbolic groups
zbMATH Keywords: primitive elements
MaRDI profile type: Publication
full work available at URL: s00039-002-8254-8
OpenAlex ID: W1981030382
DOI: 10.1007/S00039-002-8254-8
description: scientific article; zbMATH DE number 1890450