The Σ2-conjecture for metabelian groups: the general case

AbstractThe 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

Subdirect products of groups and the n-(n+1)-(n+2) Conjecture

We analyse the subgroup structure of direct products of groups. Earlier work on this topic has revealed that higher finiteness properties play a crucial role in determining which groups appear as subgroups of direct products of free groups or limit groups. Here, we seek to relate the finiteness properties of a subgroup to the way it is embedded in the ambient product. To this end we formulate a conjecture on finiteness properties of fibre products of groups. We present different approaches to this conjecture, proving a general result on finite generation of homology groups of fibre products and, for certain special cases, results on the stronger finiteness properties F_n and FP_n.Comment: 32 page