On profinite groups with finite abelianizations

Profinite groups with finite p-abelianizations arise in various contexts: group theory, number theory and geometry. Using Ph. Furtw?ngler’s transfer vanishing theorem it will be proved that a finitely generated profinite group Ĝ with this property satisfies 〚Ĝ〛) = 0 (Thm. A). As a consequence one finds that a hereditarily just-infinite non-virtually cyclic pro-p group has only one end (Cor. B). Applied to 3-dimensional Poincaré duality groups, Theorem A yields a generalization of A. Reznikov’s theorem on 3-dimensional co-compact hyperbolic lattices violating W. Thurston’s conjecture (Thm. C).