Differentials on Forested and Hairy Graph Complexes with Dishonest Hairs

We study the cohomology of forested graph complexes with ordered and unordered hairs whose cohomology computes the cohomology of a family of groups ${\mathrm{\Gamma }}_{g,r}$ that generalize the (outer) automorphism group of free groups. We give examples and a recipe for constructing additional differentials on these complexes. These differentials can be used to construct spectral sequences that start with the cohomology of the standard complexes. We focus on one such sequence that relates cohomology classes of graphs with different numbers of hairs and compute its limit.