NORMAL SUBGROUPS OF PROFINITE GROUPS OF FINITE COHOMOLOGICAL DIMENSION

Abstract

A profinite group G of finite cohomological dimension with (topologically) finitely generated closed normal subgroup N is studied. If G is pro-p and N is either free as a pro-p group or a Poincaré group of dimension 2 or analytic pro-p, it is shown that G/N has virtually finite cohomological dimension cd(G)−cd(N). Some other cases when G/N has virtually finite cohomological dimension are also considered. If G is profinite, the case of N projective or the profinite completion of the fundamental group of a compact surface is considered.