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We initiate the study of profinite groups of non-negative deficiency. Theprincipal focus of the paper is to show that the existence of a finitelygenerated normal subgroup of infinite index in a profinite group $G$ ofnon-negative deficiency gives rather strong consequences for the structure of$G$. To make this precise we introduce the notion of $p$-deficiency ($p$ aprime) for a profinite group $G$. This concept is more useful in the study ofprofinite groups then the notion of deficiency. We prove that if the$p$-deficiency of $G$ is positive and $N$ is a finitely generated normalsubgroup such that the $p$-Sylow subgroup of $G/N$ is infinite and $p$ dividesthe order of $N$ then we have $\cd_p(G)=2$, $\cd_p(N)=1$ and $\vcd_p(G/N)=1$for the cohomological $p$-dimensions; moreover either the $p$-Sylow subgroup of$G/N$ is virtually cyclic or the $p$-Sylow subgroup of $N$ is cyclic. Aprofinite Poincar\'e duality group $G$ of dimension 3 at a prime $p$($PD^3$-group) has deficiency 0. In this case we show that for $N$ and $p$ asabove either $N$ is $PD^1$ at $p$ and $G/N$ is virtually $PD^2$ at $p$ or $N$is $PD^2$ at $p$ and $G/N$ is virtually $PD^1$ at $p$. In particular if $G$ ispro-$p$ then either $N$ is infinite cyclic and $G/N$ is virtually Demushkin or$N$ is Demushkin and $G/N$ is virtually infinite cyclic. We apply this resultsto deduce structural information on the profinite completions of ascendingHNN-extensions of free groups. We also give some implications of our theory tothe congruence kernels of certain arithmetic groups.

Normal Subgroups of Profinite Groups of Non-negative Deficiency