Quasiaspherical knots with infinity many ends (Q789067)

A smooth n-knot K in \(S^{n+2}\) is quasiaspherical if the universal cover of its complement has trivial homology in dimension \(n+1\). Theorem: a knot is quasiaspherical if and only if the fundamental group of its complement cannot be written as free product with amalgamation over a finite group, with a meridian of the knot contained in one of the factors. The ''only if'' part of this result is due to \textit{G. A. Swarup} [J. Pure Appl. Algebra 6, 291-296 (1975; Zbl 0323.57011)]; the sufficiency was previously announced by the present authors, but details are given in this paper. Since from the work of Stallings K will have infinitely many ends if and only if its group splits either as a free product with amalgamation or as an HNN-extension over a finite subgroup, it follows that a knot which is not aspherical has infinitely many ends. The converse however is here shown to be false. An example is given of a 2-knot whose group splits as an HNN-extension over a finite subgroup, but the meridian is represented by the stable letter of the extension ensuring that it cannot also be split as a free product with amalgamation in the way required by the theorem.