  The `upper rank' of a group is the supremum of the (Pr\"{u}fer) ranks of its
finite quotients, and for a prime $p$, the `upper $p$-rank' is the supremum of
the sectional $p$-ranks of those quotients. The former is finite if and only if
the latter are finitely bounded as $p$ ranges over all primes (a deep fact).
Here we discuss the question: if the upper $p$-ranks of a finitely generated
group $G$ are all finite, are they necessarily bounded? The case where $G$ is a
soluble group is still an open problem.
