  We prove that, under the continuum hypothesis $\frak c=\aleph_1$, any
ultraproduct II$_1$ factor $M= \prod_{\omega} M_n$ of separable finite factors
$M_n$ contains more than $\frak c$ many mutually disjoint singular MASAs, in
other words the {\it singular abelian rank of} $M$, $\text{\rm r}(M)$, is
larger than $ \frak c$. Moreover, if the strong continuum hypothesis $2^{\frak
c}=\aleph_2$ is assumed, then ${\text{\rm r}}(M) = 2^{\frak c}$. More
generally, these results hold true for any II$_1$ factor $M$ with unitary group
of cardinality $\frak c$ that satisfies the bicommutant condition $(A_0'\cap
M)'\cap M=M$, for all $A_0\subset M$ separable abelian.
