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 H1(G, ℤ [H\G]). We also consider some generalizations of this theorem.
Kropholler, P. H., & Roller, M. A. (1996). Remarks on a theorem of Swarup on ends of pairs of groups. Journal of Pure and Applied Algebra, 109(1), 107–110. https://doi.org/10.1016/0022-4049(95)00080-1