Complete growth functions of hyperbolic groups
type: Item
instance of: scholarly article
title: Complete growth functions of hyperbolic groups
zbMATH Open document ID: 0880.20024
author: Tatiana Nagnibeda
author: R. I. Grigorchuk
published in: Inventiones Mathematicae
publication date: 1997-09-09T00:00:00Z
review text: Given a finitely generated group \(G\) and a finite system of generators \(S\) in \(G\), we study the formal power series \(F_{G,S}(z)=\sum_{g\in G}gz^{|g|}\) with coefficients in the group ring \(\mathbb{Z}[G]\), which we call the complete growth function of \(G\) with respect to \(S\). We discuss the problem of rationality of such power series, in particular for hyperbolic groups, and methods of computations. The complete growth functions of surface groups are explicitly computed. The coefficients of this power series may also be viewed as elements of the reduced \(C^*\)-algebra of the group. We establish a relation between the radius of convergence of the operator growth series and the growth rate of the group. The notion of complete growth series has sense for any formal language over a finite alphabet. An interesting example provided by groups is the complete growth series of the language of geodesics of \(G\) with respect to \(S\). We discuss its rationality and compute some examples.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F38
zbMATH DE Number: 1058850
zbMATH Keywords: finitely generated groups
zbMATH Keywords: finite systems of generators
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: complete growth functions
zbMATH Keywords: surface groups
zbMATH Keywords: reduced \(C^*\)-algebras
zbMATH Keywords: operator growth series
zbMATH Keywords: growth rates
zbMATH Keywords: geodesics
zbMATH Keywords: rationality
MaRDI profile type: Publication
full work available at URL: s002220050181
OpenAlex ID: W2073964378
DOI: 10.1007/S002220050181
description: scientific article; zbMATH DE number 1058850