  We describe a bigraded cocommutative Hopf algebra structure on the weight
zero compactly supported rational cohomology of the moduli space of principally
polarized abelian varieties. By relating the primitives for the coproduct to
graph cohomology, we deduce that $\dim H^{2g+k}_c(\mathcal{A}_g)$ grows at
least exponentially with $g$ for $k = 0$ and for all but finitely many positive
integers $k$. Our proof relies on a new result of independent interest; we use
a filtered variant of the Waldhausen construction to show that Quillen's
spectral sequence abutting to the cohomology of $BK(\mathbb{Z})$ is a spectral
sequence of Hopf algebras. From the same construction, we also deduce that
$\dim H^{\binom{n}{2} - n - k}(\mathrm{SL}_n(\mathbb{Z}))$ grows at least
exponentially with $n$, for $k = -1$ and for all but finitely many non-negative
integers $k$.
