Filtered ends of infinite covers and groups

Let f:A→Bf:A→B be a covering map. We say that AA has ee filtered ends with respect to ff (or BB) if, for some filtration {Kn}{Kn} of BB by compact subsets, A−f−1(Kn)A−f−1(Kn) “eventually” has ee components. The main theorem states that if YY is a (suitable) free HH-space, if K