  We initiate the study of profinite groups of non-negative deficiency. The
principal focus 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 $G$. This concept is more useful in the study of
profinite groups then the notion of deficiency. 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 $G/N$ is infinite and $p$ divides
the 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. A
profinite 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 $p$ as
above 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 $G$ is
pro-$p$ 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 results
to deduce structural information on the profinite completions of ascending
HNN-extensions of free groups. We also give some implications of our theory to
the congruence kernels of certain arithmetic groups.
