On modules of finite upper rank

For a group $G$ and a prime $p$, the upper $p$-rank of $G$ is the supremum of the sectional $p$-ranks of all finite quotients of $G$. It is unknown whether, for a finitely generated group $G$, these numbers can be finite but unbounded as $p$ ranges over all primes. The conjecture that this cannot happen if $G$ is soluble is reduced to an analogous `relative' conjecture about the upper $p$-ranks of a `quasi-finitely-generated' module $M$for a soluble minimax group $\Gamma$. The main result establishes a special case of this relative conjecture, namely when the module $M$ is finitely generated and the minimax group $\Gamma$ is abelian-by-polycyclic. The proof depends on generalising results of Roseblade on group rings of polycyclic groups to group rings of soluble minimax groups. (If true in general, the above-stated conjecture would imply the truth of Lubotzky's `Gap Conjecture' for subgroup growth, in the case of soluble groups; the Gap Conjecture is known to be false for non-soluble groups.)