Two generator subgroups of free products with commuting subgroups
type: Item
instance of: scholarly article
title: Two generator subgroups of free products with commuting subgroups
zbMATH Open document ID: 1007.20022
DOI: 10.1016/S0022-4049(01)00152-9
author: Benjamin Fine
author: Gerhard Rosenberger
author: Richard Weidmann
published in: Journal of Pure and Applied Algebra
publication date: 2002-08-22T00:00:00Z
review text: 2-generator subgroups of products with commuting subgroups are studied. The concept of a free product with commuting subgroups is a natural intermediate construction between a free product (the commuting subgroups are trivial) and a direct product (the commuting subgroups equal the factors), just like a graph group lives between a free and a free Abelian group. Automorphisms of graph groups were studied and, using geometric techniques, the centralizer problem was solved by \textit{H. Servatius} [J. Algebra 126, No. 1, 34-60 (1989; Zbl 0682.20022)]. Here the geometric techniques developed by \textit{I. Kapovich} and \textit{R. Weidmann} [Arch. Math 73, No. 3, 172-181 (1999; Zbl 0988.20012)] are used to show that a 2-generator subgroup of a product of free groups with commuting subgroups is either free, or of the form \(\langle a,b\mid[a^p,b^q]\rangle\) for some natural exponents \(p\) and \(q\), or it is the fundamental group of a graph of cyclic groups were the underlying graph is either homeomorphic to a segment or a circle.
reviewed by: Brigitte Servatius
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1787470
zbMATH Keywords: free products with commuting subgroups
zbMATH Keywords: fundamental groups
zbMATH Keywords: graphs of groups
zbMATH Keywords: 2-generator subgroups
MaRDI profile type: Publication
cites work: Q4040880
cites work: Q3351592
cites work: On the conjugacy problem in a free product with commuting subgroups
cites work: One-Relator Quotients and Free Products of Cyclics
cites work: Algorithms and geometry for graph products of groups
cites work: Two-generated groups acting on trees
cites work: The Subgroups of a Free Product of Two Groups with an Amalgamated Subgroup
cites work: Subgroups of HNN Groups and Groups with one Defining Relation
cites work: Q5513066
cites work: The geometry of higman‐neumann‐neumann extensions
cites work: On cancellations in HNN-groups
cites work: Q3341041
cites work: THE NIELSEN METHOD FOR GROUPS ACTING ON TREES
cites work: Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam
description: scientific article; zbMATH DE number 1787470