Filtered Ends of Infinite Covers and Groups

Let f : A → B be a covering map. We say A has e filtered ends with respect to f (or B) if for some filtration {Kn} of B by compact subsets, A − f(Kn) “eventually” has e components. The main theorem states that if Y is a (suitable) free H-space, if K < H has infinite index, and if Y has a positive finite number of filtered ends with respect to H\Y , then Y has one filtered end with respect to K\Y . This implies that if G is a finitely generated group and K < H < G are subgroups each having infinite index in the next, then 0 < ẽ(G, H) < ∞ implies ẽ(G, K) = 1, where ẽ(·, ·) is the number of filtered ends of a pair of groups in the sense of Kropholler and Roller.