Bieri–Groves FP_m conjecture for metabelian groups

Let

$$1\longrightarrow A\longrightarrow G\longrightarrow Q\longrightarrow 1$$

be the short exact sequence (1), where $A$ is regarded as a $\mathbb{Z}Q$-module, and let $m$ be a positive integer. The module $A$ is $m$-tame if every $m$-point subset of $\Sigma_A^c=S(Q)\setminus\Sigma_A$ lies in an open hemisphere of $S(Q)$; equivalently, $v_1+\cdots+v_m\neq 0$ for any $[v_1],\ldots,[v_m]\in\Sigma_A^c$. A group $G$ is of type $FP_m$ if it has a $\mathbb{Z}G$-projective resolution of the trivial module $\mathbb{Z}$ whose modules are finitely generated in dimensions at most $m$.

Bieri–Groves $FP_m$ conjecture. $G$ is of type $FP_m$ if and only if $A$ is $m$-tame as a $\mathbb{Z}Q$-module.

This conjecture extends the known equivalence between finite presentability and 2-tameness for metabelian groups. Both directions remain open for $m>2$, although several specific cases have been proved.