Filtered ends of infinite covers and groups

AbstractLet f:A→B be a covering map. We say that A has e filtered ends with respect to f (or B) if, for some filtration {Kn} of B by compact subsets, A−f−1(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 that ẽ(G,K)=1, where ẽ(⋅,⋅) is the number of filtered ends of a pair of groups in the sense of Kropholler and Roller