The Homological Invariants for Metabelian Groups of Finite Prüfer Rank: A Proof of the Σ^m-Conjecture

Abstract

We prove the homological part of the Σm-conjecture for metabelian groups G of finite Prüfer rank: if G is of type FPm then the complement of the higher homological invariant Σm(G; Z) introduced by R. Bieri and B. Renz [10] is given by the formula conv⩽m Σ1(G; Z)c = Σm(G; Z)c, where conv⩽m Σ1(G; Z)c is the union of the convex hulls of all subsets of at most m elements of Σ1 (G; Z)c ⊆ Hom(G; R).