Amalgamated products and finitely presented groups
type: Item
instance of: scholarly article
title: Amalgamated products and finitely presented groups
zbMATH Open document ID: 0711.20015
DOI: 10.1007/BF02566605
author: Gilbert Baumslag
author: Peter B. Shalen
published in: Commentarii Mathematici Helvetici
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140193
review text: An amalgamated product \(A*_ CB\) is said to be non-trivial if the amalgamated subgroup C is not equal to either of the factors A, B. If, in addition, \(\max \{| A:C|,| B:C| \}>2\), then the amalgamated product is said to be proper. The authors first establish the following theorem (they point out that the theorem has been obtained independently by \textit{M. Bestvina} and \textit{M. Feighn} [unpublished]); If a finitely presented group G has a proper amalgamated product decomposition \(A*_ CB\), then it has a proper decomposition \(A'*_{C'}B'\) where \(A'\leq A\), \(B'\leq B\), \(C'\leq C\) and where \(A'\), \(B'\), \(C'\) are finitely generated (Theorem 1). They then make clever use of this result to obtain two further interesting theorems (Theorems 2, 3). Let \(G=<x_ 1,...,x_ m;R_ 1,...,R_ n>\) with \(m\geq 2\), and suppose that either m-n\(\geq 2\) (Theorem 2), or \(R_ 1,...,R_ n\) lie in the third derived subgroup of the free group on \(x_ 1,...,x_ m\) (Theorem 3). Then G has a proper amalgamated product decomposition in which the factors and amalgamated subgroup are finitely generated.
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 4168935
zbMATH Keywords: finitely presented group
zbMATH Keywords: amalgamated product decomposition
zbMATH Keywords: finitely generated
reviewed by: Stephen J. Pride
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4168935