Homological Invariants for pro-p Groups and Some Finitely Presented pro- ${\cal C}$ Groups

Abstract

Let G be a finitely presented pro- group with discrete relations. We prove that the kernel of an epimorphism of G to is topologically finitely generated if G does not contain a free pro- group of rank 2. In the case of pro-p groups the result is due to J. Wilson and E. Zelmanov and does not require that the relations are discrete ([15], [17]). For a pro-p group G of type FPm we define a homological invariant m(G) and prove that this invariant determines when a subgroup H of G that contains the commutator subgroup G is itself of type FPm. This generalises work of J. King for 1(G) in the case when G is metabelian [9]. Both parts of the paper are linked via two conjectures for finitely presented pro-p groups G without free non-cyclic pro-p subgroups. The conjectures suggest that the above conditions on G impose some restrictions on 1(G) and on the automorphism group of G