Jonathan A. Hillman and Alexander Schmidt: Pro-p groups of positive deficiency

       Authors: Jonathan A. Hillman and Alexander Schmidt
       Title: Pro-p groups of positive deficiency
       Year: 2008
       Pages: 6
       In: Bull. London Math. Soc., 40 (2008), 1065-1069

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	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, cd G ≤ 2 and Γ is virtually a direct product F × Z_p, with F a finitely generated free pro-p group.

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, cd G ≤ 2 and Γ is virtually a direct product F × Z_p, with F a finitely generated free pro-p group.

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