THE HOMOLOGICAL INVARIANTS FOR METABELIAN-GROUPS OF FINITE PRUFER RANK - A PROOF OF THE SIGMA(M)-CONJECTURE

We prove the homological part of the Sigma(m)-conjecture for metabelia n groups G of finite Prufer rank: if G is of type FPm then the complem ent of the higher homological invariant Sigma(m)(G;Z) introduced by R. Bieri and B. Rent [10] is given by the formula conv(less than or equa l to m)Sigma(1)(G;Z)(c) = Sigma(m)(G;Z)(c), where conv(less than or eq ual to m)Sigma(1)(G;Z)(c) is the union of the convex hulls of all subs ets of at most m elements of Sigma(1)(G;Z)(c) subset of or equal to Ho m(G;R).