Pro-p groups of positive deficiency

Abstract

Let Γbe a finitely presentable pro-p group with a nontrivial finitely generated closed normal subgroup N of infinite index. Then def(Γ)\leq 1, and if def(Γ)=1 then Γis a pro-p duality group of dimension 2, N is a free pro-p group and Γ/N is virtually free. In particular, if the centre of Γis nontrivial and def(Γ)\geq 1, then def(Γ)=1, cd G \leq 2 and Γis virtually a direct product F \times Z_p, with F a finitely generated free pro-p group.

final version, to appear in Bull. LMS