The growth of linear groups
type: Item
instance of: scholarly article
title: The growth of linear groups
zbMATH Open document ID: 0892.20023
author: Yehuda Shalom
published in: Journal of Algebra
publication date: 1998-06-03T00:00:00Z
review text: In this nice short paper, the author gives a fairly simple proof of the growth dichotomy for finitely generated linear groups over fields of characteristic zero, namely that each such group has either polynomial or exponential growth. The proof avoids using the Tits alternative (every finitely generated linear group is virtually soluble or contains a free group of rank two). Indeed, the author claims that his main theorem (which follows easily from Tits' result) can replace the Tits alternative in many contexts (e.g.\ in Gromov's proof that every finitely generated group of polynomial growth is virtually nilpotent). The main theorem of the paper says that a finitely generated linear (in characteristic zero) and amenable group is virtually soluble (i.e., it has a soluble subgroup of finite index). From this the growth dichotomy readily follows since any group of subexponential growth is amenable and the dichotomy holds for every virtually soluble group by a theorem due to Wolf and Milnor.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1115988
zbMATH Keywords: finitely generated linear groups
zbMATH Keywords: polynomial growth
zbMATH Keywords: exponential growth
zbMATH Keywords: growth dichotomy
zbMATH Keywords: Tits alternative
zbMATH Keywords: subgroups of finite index
reviewed by: Michael Stoll
MaRDI profile type: Publication
OpenAlex ID: W2014506781
cites work: A Note on Borel's Density Theorem
cites work: Groups of polynomial growth and expanding maps. Appendix by Jacques Tits
cites work: Produits de matrices aléatoires et applications aux propriétés géometriques des sous-groupes du groupe linéaire
cites work: Q4040376
cites work: Q3358790
cites work: Free subgroups in linear groups
cites work: Kazhdan groups acting on compact manifolds
DOI: 10.1006/JABR.1997.7186
description: scientific article; zbMATH DE number 1115988