The homological dimension–Hirsch length conjecture for elementary amenable groups

Let $G$ be an elementary amenable group and let $k$ be a non-zero commutative ring. The homological dimension–Hirsch length conjecture. If $G$ has no $k$-torsion, then

$$\operatorname{hd}_{k}(G)=h(G).$$

Equivalently, finiteness of $\operatorname{hd}_{k}(G)$ should force equality with the Hirsch length in the elementary amenable case. The equality is established when $k$ is a field of characteristic zero and for several important subclasses, including constructible virtually soluble groups; the conjecture is open for soluble groups and hence for general elementary amenable groups.