Homological Invariants for pro- p Grou ps and Some Finitely Presented pro-${cal C}$ Grou ps

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 FP_m 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 FP_m. 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.Both authors are partially supported by CNPq, Brazil.