The jump-periodicity-cohomological-dimension equivalence conjecture

Let $R$ be a commutative ring with unit, and let $G$ be a discrete group without $R$-torsion. The group $G$ has jump cohomology of height $k$ over $R$ if there is an integer $k\geq 0$ such that every subgroup $H\leq G$ with finite $R$-cohomological dimension satisfies $cd_R(H)\leq k$. It has periodic cohomology over $R$ starting in dimension $k+1$ when cup-product periodicity holds in all dimensions at least $k+1$. The jump-periodicity-cohomological-dimension conjecture. The following are equivalent: (1) $G$ has jump cohomology of height $k$ over $R$; (2) $G$ has periodic cohomology over $R$ starting in dimension $k+1$; and (3) $cd_R(G)\leq k$. The conjecture would generalize Talelli's conjecture. It is proved in the paper for ${\sl H}{\mathcal F}$-groups and, for solvable groups, when $R$ is a domain of characteristic zero; the general statement remains open.