A proof of the Scott-Wiegold conjecture on free products of cyclic groups
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.
scientific article; zbMATH DE number 1787489
scientific article