Tropical curves, graph complexes, and top weight cohomology of $\mathcal{M}_g$

We study the topology of a space $\Delta _{g}$ parametrizing stable tropical curves of genus $g$ with volume $1$, showing that its reduced rational homology is canonically identified with both the top weight cohomology of $\mathcal {M}_g$ and also with the genus $g$ part of the homology of Kontsevich's graph complex. Using a theorem of Willwacher relating this graph complex to the Grothendieck-Teichmüller Lie algebra, we deduce that $H^{4g-6}(\mathcal {M}_g;\mathbb{Q})$ is nonzero for $g=3$, $g=5$, and $g \geq 7$, and in fact its dimension grows at least exponentially in $g$. This disproves a recent conjecture of Church, Farb, and Putman as well as an older, more general conjecture of Kontsevich. We also give an independent proof of another theorem of Willwacher, that homology of the graph complex vanishes in negative degrees.