The homological invariant Sigma^1(G), where G is afinitely generated metabelian group, was used to classify finitely presentable metabelian groups. The description of such groups was based in the geometric properties of Sigma^1(G). The FPm-Conjecture relates geometric properties of this invariant with the homological type FPm of G. The FP3-Conjecture was proved when G is a split extension of abelian groups. Algebric and geometric methods were used in this proof. The proposal of this work is to simplify the proof of FP3-Conjecture for discrete groups and if it is possible to generalize for dimension 4 (thus proving the FP4-Conjecture in the split extension case).
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