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

Abstract.

Let G be a finitely presented pro-${\cal C}$ group with discrete relations. We prove that the kernel of an epimorphism of G to $\hat{\Bbb Z}_{\cal C}$ is topologically finitely generated if G does not contain a free pro-${\cal C}$ 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 FP _m we define a homological invariant Σ^m(G) and prove that this invariant determines when a subgroup H of G that contains the commutator subgroup G′ is itself of type FP _m . This generalises work of J. King for Σ¹(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 Σ¹(G) and on the automorphism group of G.