Excess-zero concentration for the 3-edge-connected forested graph complex

Prove that the homology \(H(FG^{3e}(n,0),d_c)\) of the 3-edge-connected, zero-hair forested graph complex with respect to the contracting differential is concentrated in excess zero for every loop order \(n\).

Background

The paper introduces the subcomplex FG3e(n,0)FG^{3e}(n,0) spanned by 3-edge-connected graphs and explains that it is substantially smaller than the full forested graph complex. For the full complex, concentration of the contracting-differential homology in excess zero permits computations to be restricted to the excess-zero part. The authors conjecture that the same reduction is valid for the 3-edge-connected subcomplex in every loop order; they report numerical verification only for n≤7n\leq 7.

References

We conjecture that the analogous result for FG{3e}(n,0) is true as well:

— The homology of $\mathrm{Out}(F_8)$ and $\mathrm{Aut}(F_7)$  (2609.30872 - Grego et al., 25 Sep 2026) in Appendix, Section “On the 3-edge connected subcomplex,” Conjecture 3 exc 0