A note on Lyndon properties in one-relator groups.
type: Item
instance of: scholarly article
title: A note on Lyndon properties in one-relator groups.
zbMATH Open document ID: 1228.20035
author: Benjamin Fine
author: Anja Rosenberger
author: Gerhard Rosenberger
published in: Results in Mathematics
publication date: 2011-06-22T00:00:00Z
review text: The authors call the property in a group \(G\) that \(x^2y^2z^2=1\) for \(x,y,z\in G\) implies that \(\langle x,y,z\rangle\) is cyclic `Lyndon's property'. Lyndon showed this property for free groups. Let \(F_3\) be a free group of rank 3 with basis \(\{x_1,x_2,x_3\}\). A word \(w=w(x_1,x_2,x_3)\) is called `regular' if there is no Nielsen transformation from \(\{x_1,x_2,x_3\}\) to a system \(\{y_1,y_2,y_3\}\) with \(w\in\langle y_1,y_2\rangle\). The word \(w\) is called `quadratic' if each \(x_i\) in \(w\) occurs exactly twice, each time as \(x_i\) or \(x_i^{-1}\). In the present paper it is shown: Theorem 1. Let \(G\) be the one-relator group \(G=\langle a,b,c,\dots;\;R^m\rangle\) with \(m\geq 3\) odd and \(R\) a cyclically reduced word, not a proper power in the free group on \(a,b,c,\dots\). Let \(w(x_1,x_2,x_3)\) be a regular quadratic word in the free group \(F\) on \(x_1,x_2,x_3\) and let \(\varphi\colon F\to G\) be a homomorphism from \(F\) into \(G\) with \(\varphi (x_i)=u_i\) for \(i=1,2,3\). If \(w(u_1,u_2,u_3)=1\) in \(G\) than the subgroup \(\langle u_1,u_2,u_3\rangle\) is cyclic. In particular, \(G\) satisfies the Lyndon property.
reviewed by: Stephan Rosebrock
Mathematics Subject Classification ID: 20F70
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E08
zbMATH DE Number: 5911446
zbMATH Keywords: equations over groups
zbMATH Keywords: quadratic equations
zbMATH Keywords: one-relator groups
zbMATH Keywords: Lyndon properties
zbMATH Keywords: free groups
zbMATH Keywords: regular words
zbMATH Keywords: Nielsen transformations
MaRDI profile type: Publication
full work available at URL: s00025-010-0092-7
OpenAlex ID: W2074412492
cites work: On generalised free products
cites work: The equation $x^py^q=z^r$ and groups that act freely on $Łambda $-trees
cites work: On Lyndon's equation in some Λ-free groups and HNN extensions
cites work: Q4664541
cites work: Commutators as Powers in Free Products of Groups
cites work: Q4226362
cites work: Quadratic properties in group amalgams
cites work: Q3123486
cites work: Q4866219
cites work: A classification of fully residually free groups of rank three or less
cites work: A classification of conjugately separated abelian, commutative transitive, and restricted Gromov one-relator groups.
cites work: Q4277427
cites work: Irreducible affine varieties over a free group. I: Irreducibility of quadratic equations and Nullstellensatz
cites work: Irreducible affine varieties over a free group. II: Systems in triangular quasi-quadratic form and description of residually free groups
cites work: Implicit function theorem over free groups.
cites work: Elementary theory of free non-abelian groups.
cites work: Hyperbolic groups and free constructions
cites work: Q4656957
cites work: Q4145882
cites work: Some results on one-relator groups
cites work: On cancellations in HNN-groups
cites work: \(\exists\)-free groups
cites work: Q3955567
cites work: Über alternierende Wörter in freien Produkten mit Amalgam
cites work: Sq-Universality in Hnn Groups and One Relator Groups
cites work: The isomorphism problem for hyperbolic groups. I
cites work: Diophantine geometry over groups. I: Makanin-Razborov diagrams
cites work: Diophantine geometry over groups. II: Completions, closures and formal solutions
cites work: Diophantine geometry over groups. III: Rigid and solid solutions.
cites work: Diophantine geometry over groups. IV: An iterative procedure for validation of a sentence.
DOI: 10.1007/S00025-010-0092-7
description: scientific article; zbMATH DE number 5911446