Differentials on Forested and Hairy Graph Complexes with Dishonest Hairs

TL;DR

This paper investigates the cohomology of forested graph complexes with hairs, providing new differentials and spectral sequences to understand the cohomology of groups generalizing automorphism groups of free groups.

Contribution

It introduces new differentials on forested graph complexes and constructs spectral sequences to analyze their cohomology, extending understanding of related automorphism groups.

Findings

Constructed additional differentials for graph complexes.

Developed spectral sequences connecting cohomologies with different hairs.

Computed the limit of a spectral sequence relating classes with varying hairs.

Abstract

We study the cohomology of forested graph complexes with ordered and unordered hairs whose cohomology computes the cohomology of a family of groups $\Gamma_{g,r}$ 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.