This is probably well known to algebraic geometers... Brill-Noether theory gives us Brill-Noether divisors in $\overline{M}_g$. Given [C] ∈ Mg and a line bundle $L$ on $C$, the failure of the Petri map $$ \mu_{0,L} : H^0(C,L) \otimes H^0(C,\mathcal{K}_C \otimes K) \to H^0(C,\mathcal{K}_C) $$ to be injective gives us Gieseker-Petri loci... In GW-theory, we have the double ramification cycle (see: Hain, and Janda-Pandrepande-Pixton-Zvonkine) in $A^*(\overline{M}_{g,n})$ which is defined by pushing forward the VFC of the moduli space of stable maps to rubber. This has a nice description via Hurwitz theory.

My question is:

Are there any other interesting cycles in $A^*(\overline{M}_{g,n})$ coming from Hurwitz theory?