Normal subgroups of profinite groups of non-negative deficiency

The principal goal of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group G of non-negative deficiency gives rather strong consequences for the structure of G. To make this precise we introduce the notion of p-deficiency (p a prime) for a profinite group G. We prove that if the p-deficiency of G is positive and N is a finitely generated normal subgroup such that the p  -Sylow subgroup of G/N$G/N$ is infinite and p divides the order of N   then we have _cdp(G)=2${\mathrm{cd}}_p(G)=2$, _cdp(N)=1${\mathrm{cd}}_p(N)=1$ and _vcdp(G/N)=1${\mathrm{vcd}}_p(G/N)=1$ for the cohomological p-dimensions; moreover either the p  -Sylow subgroup of G/N$G/N$ is virtually cyclic or the p-Sylow subgroup of N is cyclic. If G is a profinite Poincaré duality group of dimension 3 at a prime p   (PD³${\mathit{PD}}^3$-group at p) we show that for N and p as above either N   is PD¹${\mathit{PD}}^1$ at p   and G/N$G/N$ is virtually PD²${\mathit{PD}}^2$ at p or N   is PD²${\mathit{PD}}^2$ at p   and G/N$G/N$ is virtually PD¹${\mathit{PD}}^1$ at p.