Weak forms of amenability for one-relator groups
type: Item
instance of: scholarly article
title: Weak forms of amenability for one-relator groups
zbMATH Open document ID: 0955.43002
author: Cédric Béguin
author: Tullio Ceccherini-Silberstein
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2001-02-20T00:00:00Z
review text: The authors discuss weak amenability on discrete groups \(G\). The authors call a group `\(a\)-\(T\)-moyennable' if it satisfies the condition that when there exists a sequence of normalised positive definite functions on \(G\), tending to 0 at infinity, the sequence converges pointwise to the constant function 1. This condition is not really an amenability condition, but mimics a property of all separable locally compact groups which follows from a result of P. Urysohn (circa 1924). The term \(a\)-\(T\)-moyennable was used by \textit{M. Gromov} [Asymptotic invariants of infinite groups, in: Geometric group theory, vol. 2 (1991; Zbl 0841.20039)] in the sense of essentially not having the property \(T\) of \textit{D. A. Kazhdan} [Funct. Anal. Appl. 1, 63-65 (1967; Zbl 0168.27602)]. \(G\) is called inner amenable if there is a finitely additive mean with total mass 1 on \(G -\{e\}\) which is invariant under inner automorphisms. \(G\) is usually assumed as having two generators and one relator, say \(G=\langle a,b:R\rangle\). Let \(N\) denote the normal subgroup generated by one of the generators which is assumed to appear with exponent sum 0. As in M. Dehn's Freiheitssatz [\textit{W. Magnus}, J. Reine Angew. Math. 163, 141-165 (1930; JFM 56.0134.03)], the graph of \(G\) is divided into `layers' of amalgams of freely generated subgroups and \(N\) expressed as the union of these. The authors use only the simplest situation [see Ch.~II.5 of \textit{B. Chandler} and \textit{W. Magnus}, The history of combinatorial group theory (Berlin etc. 1982; Zbl 0498.20001)] showing that \(N\) is freely generated whenever \(G\) has nontrivial centre. Then, by results quoted from \textit{P. Jolissaint, P. Julg} and \textit{A. Valette} in ``Locally compact groups with the Haagerup property'' (to appear), \(G/N\) is amenable and so also \(G\) is \(a\)-\(T\)-moyennable. Taking \(G\) to be an HNN extension [see \textit{G. Higman, B. H. Neumann} and \textit{H. Neumann}, J. Lond. Math. Soc. …
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 1423102
zbMATH Keywords: HNN extension
zbMATH Keywords: discrete group
zbMATH Keywords: generator
zbMATH Keywords: relator
zbMATH Keywords: amenability
zbMATH Keywords: Freiheitssatz
zbMATH Keywords: tree
zbMATH Keywords: fixed point
zbMATH Keywords: hyperbolic and elliptic automorphisms
zbMATH Keywords: JFM 56.0134.03
reviewed by: Aubrey Wulfsohn
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1423102