Schreier theorem on groups which split over free abelian groups

Abstract

Let G be either a free product with amalgamation A *c B or an HNN group A*c, where C is isomorphic to a free abelian group of finite rank. Suppose that both A and B have no nontrivial, finitely generated, normal subgroups of infinite indices. We show that if G contains a finitely generated normal subgroup N which is neither contained in C nor free, then the index of N in G is finite. Further, as an application of this result, we show that the fundamental group of a torus sum of 3-manifolds MI and M2, the interiors of which admit hyperbolic structures, have no nontrivial, finitely generated, nonfree, normal subgroup of infinite index if each of MI and M2 has at least one nontorus boundary.