On exponential growth and uniformly exponential growth for groups.
type: Item
instance of: scholarly article
title: On exponential growth and uniformly exponential growth for groups.
zbMATH Open document ID: 1065.20054
author: John S. Wilson
published in: Inventiones Mathematicae
publication date: 2004-09-22T00:00:00Z
review text: The paper gives a negative answer to \textit{M. Gromov}'s question [Structures métriques pour les variétés riemanniennes. Textes Mathématiques, 1. Paris (1981; Zbl 0509.53034)], (Remark 5.2) posed in 1981. Gromov asked `if groups of exponential growth necessarily have uniform exponential growth'. Here are some background information and definitions: for the element \(g\) of the group \(G\) generated by the finite set \(S\) the word length \(l_S(g)\) is the smallest integer \(n\) such that \(g\) is a product of \(n\) elements of \(S\cup S^{-1}\). For each \(n\geq 0\) define \(\gamma_S(n)=|\{g \in G\mid l_S(g)\leq n\}|\). The limit \(e_S(G)=\lim_{n\to \infty}(\gamma_S(n))^{1/n}\) exists and is called the `exponential growth rate' of \(G\) with respect to \(S\). If it is greater than \(1\) for one generating set, then it is greater than \(1\) for all generating sets, and \(G\) is said to have `exponential growth'. Moreover, a group has `uniformly exponential growth' if \(\inf_Se_S(G)>1\), where the infimum is taken over all finite generating sets \(S\) in \(G\). It has been shown that Gromov's question has a positive answer for groups of certain types, such as hyperbolic groups, soluble groups, linear groups in characteristic \(0\), etc. (see [\textit{P. de la Harpe}, Geom. Dedicata 95, 1-17 (2002; Zbl 1025.20027)] for a survey). Nevertheless, a negative answer to Gromov's question follows from the following theorem and from the fact that a finitely generated group having non-Abelian free subgroups must have exponential growth. Theorem 1. There is a group \(G\) which has non-Abelian free subgroups and a sequence \((S_n)\) of generating sets such that \(\lim_{n\to\infty}e_{S_n}(G)=1\). Each generating set \(S_n\) has two elements, one of order 2 and the other of order~3. Let \(A_{31}\) be the alternating group of degree \(31\). The wreath product mentioned below is a `permutational' wreath product. A larger class of groups that do not have uniformly exponential growth i…
Mathematics Subject Classification ID: 20F69
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 2101714
zbMATH Keywords: growth functions
zbMATH Keywords: uniformly exponential growth
zbMATH Keywords: finitely generated groups
zbMATH Keywords: wreath products
zbMATH Keywords: free subgroups
reviewed by: Vahagn H. Mikaelian
MaRDI profile type: Publication
full work available at URL: s00222-003-0321-8
OpenAlex ID: W1976343050
DOI: 10.1007/S00222-003-0321-8
description: scientific article; zbMATH DE number 2101714