Hopf algebras in the cohomology of ${\mathcal{A}}_g$, ${\mathrm{GL}}_n(\mathbb{Z})$, and ${\mathrm{SL}}_n(\mathbb{Z})$

Abstract

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 $\dim H_c^{2g+k}({\mathcal{A}}_g)$ 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 $\dim H^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)-n-k}({\mathrm{SL}}_n(\mathbb{Z}))$ grows at least exponentially with $n$, for $k=-1$ and for all but finitely many non-negative integers $k$.

v2: 87 pages. Minor revision, improved exposition