Remarks on a theorem of Swarup on ends of pairs of groups

Abstract

We give an algebraic proof of a recent theorem of Swarup, which states that if H is a subgroup of infinite index in a finitely generated group G and if e ( G, N) = 1 for all subgroups N▷H with H N ≅ Z , then e(G, H) = 1 + rank H 1 (G, Z[H/G]) . We also consider some generalizations of this theorem.