A proof of the Scott-Wiegold conjecture on free products of cyclic groups
type: Item
instance of: scholarly article
title: A proof of the Scott-Wiegold conjecture on free products of cyclic groups
zbMATH Open document ID: 1026.20019
DOI: 10.1016/S0022-4049(02)00042-7
author: James Howie
published in: Journal of Pure and Applied Algebra
publication date: 2002-08-22T00:00:00Z
review text: A one-relator product of groups is a group obtained from a free product of groups by adding one relation. In particular, a one-relator product of three finite cyclic groups has a presentation of the form \(\langle x,y,z\mid x^p=y^q=z^r=W(x,y,z)=1\rangle\), and the Scott-Wiegold conjecture states that such a group is nontrivial. The author uses a clever and original method to prove the conjecture. After some reductions, such as that \(p\), \(q\), and \(r\) are relatively prime and each has exponent sum \(1\) in \(W(x,y,z)\), a nontrivial representation is constructed into \(\text{SO}(3)\). The key idea is to use a degree argument for certain equivariant maps between spheres to show that the images of \(x\), \(y\), and \(z\) can be chosen so that \(W(x,y,z)\) represents to the trivial element. The author deduces some important consequences for surgery on \(3\)-manifolds. Suppose that a connected sum of \(n\) \(3\)-manifolds can be obtained by Dehn surgery on a knot in \(S^3\). The conjecture implies that at least \(n-2\) of them are integral homology spheres. Combined with some recent results from \(3\)-manifold theory, this shows that in fact \(n\leq 3\), and if \(n=3\) then two of the summands are lens spaces and the third is an integral homology sphere. A natural generalization of the Scott-Wiegold conjecture says that a free product of \(2m+1\) cyclic groups cannot be trivialized by adding \(m\) relations. The method does not generalize to this case in any obvious way.
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 1787489
zbMATH Keywords: free products of cyclic groups
zbMATH Keywords: Scott-Wiegold conjecture
zbMATH Keywords: one-relator products
zbMATH Keywords: representations
zbMATH Keywords: Dehn surgery
zbMATH Keywords: connected sums
zbMATH Keywords: lens spaces
zbMATH Keywords: presentations
Wikidata QID: Q123198083
MaRDI profile type: Publication
cites work: Groups with two More Generators Than Relators
cites work: Generalized triangle groups
cites work: On proper powers in free products and Dehn surgery
cites work: Q4273692
cites work: Groups with balanced presentations
cites work: One-Relator Quotients and Free Products of Cyclics
cites work: Constructing lens spaces by surgery on knots
cites work: Knot surgery and primeness
cites work: Dehn Surgery and Satellite Knots
cites work: Only integral Dehn surgeries can yield reducible manifolds
cites work: Dehn surgery on knots in solid tori creating essential annuli
cites work: Symmetric knots satisfy the cabling conjecture
cites work: A Finitely Generated Infinite Simple Group
cites work: Free subgroups in groups of small deficiency
cites work: Symmetric knots and the cabling conjecture
cites work: Q4317172
cites work: Q3987886
cites work: Band Sums of Links which Yield Composite Links. The Cabling Conjecture for Strongly Invertible Knots
cites work: The reducibility of surgered \(3\)-manifolds and homology \(3\)-spheres
cites work: Producing reducible 3-manifolds by surgery on a knot
cites work: Groups with one more generator than relators
cites work: Dehn fillings of 3-manifolds and non-persistent tori
cites work: Identities for Lie algebras of pro-\(p\) groups
cites work: Dehn surgery on arborescent knots
full work available at URL: s0022-4049(02)00042-7
OpenAlex ID: W1982626960
description: scientific article; zbMATH DE number 1787489