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

Pseudo-Sylvester domains and skew Laurent polynomials over firs