The cohomological-dimension conjecture for elementary amenable groups

Let $k$ be a non-zero commutative ring and let $G$ be an elementary amenable group with no $k$-torsion. Let $h(G)$ denote its Hirsch length and let $\aleph_{\omega}$ be the first limit cardinal in the sequence of alephs.

Cohomological-dimension conjecture. The cohomological dimension $\operatorname{cd}_{k}(G)$ is finite if and only if

$$|G|<\aleph_{\omega}.$$

Moreover,

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

if $G$ is constructible,

$$\operatorname{cd}_{k}(G)=h(G)+1$$

if $G$ is countable but not constructible, and

$$\operatorname{cd}_{k}(G)=h(G)+n+1$$

if $G$ is uncountable of cardinality $\aleph_n$ with $0<n<\omega$. This conjecture aims to characterize cohomological dimension for elementary amenable groups over arbitrary coefficient rings; the source presents technical reductions and proves substantial special cases, including the nilpotent-by-polycyclic case for countable groups over $\mathbb Q$.