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Bulletin of the London Mathematical Society Advance Access originally published online on October 3, 2008
Bulletin of the London Mathematical Society 2008 40(6):1065-1069; doi:10.1112/blms/bdn089
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© 2008 London Mathematical Society

Pro-p groups of positive deficiency

Jonathan A. Hillman

School of Mathematics and Statistics
University of Sydney
Sydney, NSW 2006
Australia
jonh@maths.usyd.edu.au

Alexander Schmidt

NWF-I Mathematik
Universität Regensburg
D-93040 Regensburg
Germany

Received 28 February 2008. Revision received 9 July 2008.

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 Zp, with F a finitely generated free pro-p group.

2000 Mathematics Subject Classification 20E18.


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