Abstract: A triple of finite von Neumann algebras is said to have the relative weak asymptotic homomorphism property if there exists a net of unitaries such that
for all . Recently, J. Fang, M. Gao and R. Smith proved that the triple has the relative weak asymptotic homomorphism property if and only if contains the set of all such that for finitely many elements . Furthermore, if is a pair of groups, they get a purely algebraic characterization of the weak asymptotic homomorphism property for the pair of von Neumann algebras , but their proof requires a result which is very general and whose proof is rather long. We extend the result to the case of a triple of groups , we present a direct and elementary proof of the above-mentioned characterization, and we introduce three more equivalent conditions on the triple , one of them stating that the subspace of -compact vectors of the quasi-regular representation of on is contained in .