The ∑²-conjecture for metabelian groups: The general case

The Bieri-Neumann-Strebel invariant ∑^m (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 ∑¹ (G) completely characterizes finite presentability and it is conjectured that it also contains complete information about the higher finiteness properties (FP_m-conjecture). The ∑^m-conjecture states how the higher invariants are obtained from ∑¹ (G). In this paper we prove the ∑²-conjecture.