The Σ2-conjecture for metabelian groups: the general case

The Bieri–Neumann–Strebel invariant Σm(G) of a group G is a certain subset of a sphere that contains information about finiteness properties of subgroups of G. In case of a metabelian group G the set Σ1(G) completely characterizes finite presentability and it is conjectured that it also contains complete information about the higher finiteness properties (FPm-conjecture). The Σm-conjecture states how the higher invariants are obtained from Σ1(G). In this paper we prove the Σ2-conjecture.