On the singular abelian rank of ultraproduct II₁ factors

We prove that, under the continuum hypothesis c=₁, any ultraproduct II₁ factor M= _ Mₙ of separable finite factors Mₙ contains more than c many mutually disjoint singular MASAs, in other words the singular abelian rank of M, r (M), is larger than c. Moreover, if the strong continuum hypothesis 2^ c=₂ is assumed, then r (M) = 2^ c. More generally, these results hold true for any II₁ factor M with unitary group of cardinality c that satisfies the bicommutant condition (A₀' M) ' M=M, for all A₀ M separable abelian.