Pro-p groups of positive deficiency

Abstract

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