The motivic structures ${\mathsf{L}\mathsf{S}}_{12}$ and ${\mathsf{S}}_{16}$ in the cohomology of moduli spaces of curves

We study the appearances of ${\mathsf{L}\mathsf{S}}_{12}$ and ${\mathsf{S}}_{16}$ in the weight-graded compactly supported cohomology of moduli spaces of curves. As applications, we prove new nonvanishing results for the middle cohomology groups of ${\mathcal{M}}_9$ and ${\mathcal{M}}_{11}$ and give evidence to support the conjecture that the dimension fo $H_c^{2g+k}({\mathcal{M}}_g)$ grows at least exponentially with $g$ for almost all $k$.