Pseudo-Sylvester domains and skew laurent polynomials over firs

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