Schreier theorem on groups which split over free abelian groups

Let G G be either a free product with amalgamation A ∗ C B A *_C B or an HNN group A ∗ C , A *_C, where C C is isomorphic to a free abelian group of finite rank. Suppose that both A A and B B have no nontrivial, finitely generated, normal subgroups of infinite indices. We show that if G G contains a finitely generated normal subgroup N N which is neither contained in C C nor free, then the index of N N in G G is finite. Further, as an application of this result, we show that the fundamental group of a torus sum of 3 3 -manifolds M 1 M_1 and M 2 M_2 , the interiors of which admit hyperbolic structures, have no nontrivial, finitely generated, nonfree, normal subgroup of infinite index if each of M 1 M_1 and M 2 M_2 has at least one nontorus boundary.