Normal subgroups of profinite groups of non-negative deficiency

Abstract

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 is infinite and p divides the order of N then we have cdp(G)=2, cdp(N)=1 and vcdp(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. If G is a profinite Poincaré duality group of dimension 3 at a prime p (PD3-group at p) we show that for N and p as above either N is PD1 at p and G/N is virtually PD2 at p or N is PD2 at p and G/N is virtually PD1 at p.We apply this results to deduce structural information on the profinite completions of ascending HNN-extensions of free groups and 3-manifold groups. We prove that the arithmetic lattices in SL2(C) are cohomologically good and give some implications of our theory to their congruence kernels.