Pro-p groups of positive deficiency

TL;DR

This paper investigates finitely presentable pro-p groups with positive deficiency, establishing bounds on their structure and properties, including conditions for being a duality group, free, or virtually a product, based on their deficiency and normal subgroups.

Contribution

It characterizes the structure of pro-p groups with positive deficiency, linking deficiency to duality properties, free subgroups, and virtually product structures.

Findings

If def(Γ) ≥ 1, then def(Γ) = 1 and Γ is a pro-p duality group of dimension 2.

Normal subgroups of infinite index are free pro-p groups.

Groups with nontrivial center and def(Γ) ≥ 1 are virtually a direct product of a free pro-p group and Z_p.

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)\leq 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)\geq 1, then def(\Gamma)=1, cd G \leq 2 and \Gamma is virtually a direct product F \times Z_p, with F a finitely generated free pro-p group.