On normal subgroups of an amalgamated product of groups with applications to knot theory

TL;DR

This paper establishes criteria for normal subgroups in amalgamated groups to be finitely generated and applies these to determine when multilinks in homology 3-spheres fiber, linking group theory with knot theory.

Contribution

It provides new necessary and sufficient conditions for finitely generated normal subgroups in amalgamated products and connects these to fibering properties of multilinks in 3-spheres.

Findings

Normal subgroups are finitely generated under specific conditions.

Multilinks fiber if and only if all splice components fiber.

Application of Stallings' theorem to knot theory contexts.

Abstract

In this paper, we give some necessary and sufficient conditions for a normal subgroup of an amalgamated product of groups to be finitely generated. We apply these conditions together with Stallings' fibering theorem to prove that an irreducible multilink in a homology 3-sphere fibers if and only if each of its multilink splice components fibers.