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).
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Weigel, T. On profinite groups with finite abelianizations. Sel. math., New ser. 13, 175 (2007). https://doi.org/10.1007/s00029-007-0035-7
Published:
DOI: https://doi.org/10.1007/s00029-007-0035-7