Sets of primitive elements in a free group.
type: Item
instance of: scholarly article
title: Sets of primitive elements in a free group.
zbMATH Open document ID: 1266.20038
published in: Journal of Algebra
publication date: 2012-11-09T00:00:00Z
review text: Let \(F_n\) be the free group on \(n\) generators and \(X=\{x_1,\dots,x_{n}\}\) a standard basis for \(F_n\). A set \(U=\{u_1,u_2,\dots,u_m\}\), \(m\leq n\), is called a primitive set in \(F_n\) if it can be extended to a basis of \(F_n\). In this paper the authors study primitive sets from three classical viewpoints. 1. A method of Whitehead -- A 3-manifold \(M\) which is homeomorphic to the connected sum of \(n\) copies \(S^1\times S^2\) has its fundamental group isomorphic to \(F_n\). One defines (see for details the authors [J. Group Theory 13, No. 4, 601-611 (2010; Zbl 1206.20025)]) a certain set \(\Sigma=\{S_1,S_2,\dots,S_n\}\) of \(n\) disjoint 2-spheres in \(M\) to be the standard sphere basis for \(M\) which is in some sense dual to the standard basis for \(F_n\). A collection \(\tau=\{T_1,T_2,\dots,T_m\}\) of disjoint 2-spheres embedded in \(M\) is said to be a sphere basis if \(m=n\), and there is a diffeomorphism of \(M\) which sends \(S_j\) to \(T_j\). The collection \(\tau=\{T_1,T_2,\dots,T_m\}\) is said to be a primitive sphere set if \(m\leq n\) and there is a diffeomorphism of \(M\) which maps \(T_j\) onto \(S_j\). This is equivalent to the existence of spheres \(\{T_{m+1},\dots,T_n\}\) such that \(\{T_1,T_2,\dots,T_m,T_{m+1},\dots,T_n\}\) is a sphere basis. Let \(M\), \(\Sigma\) and \(F_n\) as above. Theorem. Let \(U=\{u_1,u_2,\dots,u_m\}\), \(m\leq n\), be a set of distinct reduced words in \(F_n\). Then \(U\) is a primitive set if and only if there exists closed paths \(\{\alpha_1,\alpha_2,\dots,\alpha_m\}\) in \(M\) and 2-spheres \(\{T_1,T_2,\dots,T_m\}\) in \(M\) such that: (1) the word obtained by the intersection of \(\alpha_j\) with \(\Sigma\) is precisely \(u_j\); (2) \(a_i\cap T_i\) is a singleton; (3) \(a_i\cap T_j=\emptyset\) if \(i\neq j\). 2. A method of Nielsen -- Here the authors, using Nielsen transformations, prove that the preimage of a primitive set contains a primitive set. Theorem. Let \(G\) be a free group of rank \(m\) with b…
reviewed by: Dimitrios Varsos
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 6104539
zbMATH Keywords: free groups
zbMATH Keywords: primitive elements
zbMATH Keywords: Nielsen transformations
zbMATH Keywords: Whitehead automorphisms
zbMATH Keywords: Stallings method
author: A. H. Clifford
author: Richard Z. Goldstein
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.01.033
OpenAlex ID: W2012538830
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DOI: 10.1016/J.JALGEBRA.2012.01.033
description: scientific article; zbMATH DE number 6104539