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

TL;DR

This paper uncovers a Hopf algebra structure in the cohomology of moduli spaces of abelian varieties and special linear groups, revealing exponential growth patterns in their cohomology dimensions.

Contribution

It introduces a new Hopf algebra framework for understanding the cohomology of these spaces and relates primitives to graph cohomology, establishing exponential growth results.

Findings

Cohomology of moduli space of abelian varieties has a Hopf algebra structure.

Dimensions of certain cohomology groups grow exponentially with genus and dimension.

A filtered Waldhausen construction shows Quillen's spectral sequence is a Hopf algebra spectral sequence.

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 $\dim H^{2g+k}_c(\mathcal{A}_g)$ 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 $k$.