  We prove that under some conditions on the monodromy, families of abelian
covers of the projective line do not give rise to (higher dimensional) Shimura
subvarieties in $A_g$. This is achieved by a reduction to $p$ argument. We also
use another method based on monodromy computations to show that two dimensional
subvarieties in the above locus are not special. In particular it is shown that
such families have usually large monodromy groups. Together with our earlier
results, the above mentioned results contribute to classifying the special
families in the moduli space of abelian varieties and partially completes the
work of several authors including the author's previous work.
