Filtered ends of infinite covers and groups

Abstract

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 {K_n} of B by compact subsets, A - f^{-1}(K_n) "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 < {\tilde e}(G)(H) < \infty implies {\tilde e}(G)(K) = 1, where {\tilde e}(.)(.) is the number of filtered ends of a pair of groups in the sense of Kropholler and Roller.