Semistability at infinity, simple connectivity at infinity and normal subgroups

In this paper we prove the following two theorems about the behavior of the fundamental group near ∞ for certain group extensions. m 1. If N ⩽ A ⩽ G are groups with A one-ended, finitely presented and of infinite index in G, G finitely presented, and N a non-locally finite normal subgroup of G, then G is semistable at ∞. s a technical result that should provide a useful tool in conjunction with semistability results already in the literature. m 2. If 1 → H → G → K → 1 is a short exact sequence of infinite finitely generated groups with G finitely presented, K one-ended and H contained in a finitely presented subgroup L of infinite index in G then G is simply connected at ∞. s a generalization of a theorem of B. Jackson. Several applications of this result are discussed in the Introduction. y we construct a negative solution to a long standing problem in the form of a group extension 1 → H → G → K → 1 where H is a one-ended finitely generated group, G is the non-simply connected at ∞ group (Zn ∗ Z) × (Zn ∗ Z) and K is the ((n − 2)-connected at ∞) group Zn.