, Volume 13, Issue 1, pp 175-181

On profinite groups with finite abelianizations

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Abstract.

Profinite groups with finite p-abelianizations arise in various contexts: group theory, number theory and geometry. Using Ph. Furtwängler’s transfer vanishing theorem it will be proved that a finitely generated profinite group Ĝ with this property satisfies $H^1(\hat{G}, {\mathbb{F}}_{p}$ 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).