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.

This article shows that the Cremona group of the plane is compactly presented. To do this, we prove that it is a generalized amalgamated product of three of its algebraic subgroups (automorphisms of the plane and Hirzebruch surfaces) divided by one relation.

This article shows that the Cremona group of the plane is compactly presented. To do this we prove that it is a generalised amalgamated product of three of its algebraic subgroups (automorphisms of the plane and Hirzebruch surfaces) divided by one relation.