Pro-p groups of positive deficiency

Jonathan A. Hillman and Alexander Schmidt

Abstract

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, cdΓ≤2 and Γ is virtually a direct product F×{Z}_p, with F a finitely generated free pro-p group.

Keywords: cohomological dimension, deficiency, pro-p group.

AMS Subject Classification: Primary 20E18.