A Serre spectral sequence for the moduli space of tropical curves

We construct, for all \(g\geq 2\) and \(n\geq 0\), a spectral sequence of rational \(S_n\)-representations which computes the \(S_n\)-equivariant reduced rational cohomology of the tropical moduli spaces of curves \(\Delta_{g,n}\) in terms of compactly supported cohomology groups of configuration spaces of \(n\) points on graphs of genus \(g\). Using the canonical \(S_n\)-equivariant isomorphisms \(\widetilde{H}^{i-1}(\Delta_{g,n};\mathbb{Q}) \cong W_0 H^i_c(\mathcal{M}_{g,n};\mathbb{Q})\), we calculate the weight \(0\), compactly supported rational cohomology of the moduli spaces \(\mathcal{M}_{g,n}\) in the range \(g=3\) and \(n\leq 9\), with partial computations available for \(n\leq 13\).