Pro‐p groups of positive deficiency - Hillman, Jonathan; Schmidt, Alexander (Bulletin of the London Mathematical Society, 2008)
Let Γ be a finitely presentable pro?p group with a nontrivial, finitely generated closed normal subgroup N of infinite index. Then def (Γ) ? 1, and if def (Γ) = 1 then Γ is 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 Γ is nontrivial and def (Γ) ? 1, then def (Γ) = 1, cd G ? 2 and Γ is virtually a direct product F ? ?p, with F a finitely generated free pro?p group.