$k$-differentials on curves and rigid cycles in moduli space

For $g\ge 2$, $j=1,\dots ,g$ and $n\ge g+j$ we exhibit infinitely many new rigid and extremal effective codimension $j$ cycles in ${\bar{\mathcal{M}}}_{g,n}$ from the strata of quadratic differentials and projections of these strata under forgetful morphisms and show the same holds for $k$-differentials with $k\ge 3$ if the strata are irreducible. We compute the class of the divisors in the case of quadratic differentials which contain the first known examples of effective divisors on ${\bar{\mathcal{M}}}_{g,n}$ with negative ${\psi }_i$ coefficients.