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) <= 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) >= 1, then def (Gamma) = 1, cd G <= 2 and Gamma is virtually a direct product F x (p), with F a finitely generated free pro-p group.