  Building on recent work of Jaikin-Zapirain, we provide a homological
criterion for a ring to be a pseudo-Sylvester domain, that is, to admit a
division ring of fractions over which all stably full matrices become
invertible. We use the criterion to study skew Laurent polynomial rings over
free ideal rings (firs). As an application of our methods, we prove that
crossed products of division rings with free-by-{infinite cyclic} and surface
groups are pseudo-Sylvester domains unconditionally and Sylvester domains if
and only if they admit stably free cancellation. This relies on the recent
proof of the Farrell--Jones conjecture for normally poly-free groups and
extends previous results of Linnell--L\"uck and Jaikin-Zapirain on universal
localizations and universal fields of fractions of such crossed products.
