Bordifications of the moduli spaces of tropical curves and abelian varieties, and unstable cohomology of $\mathrm{GL}_g(\mathbb{Z})$ and $\mathrm{SL}_g(\mathbb{Z})$

TL;DR

This paper constructs bordifications of tropical moduli spaces, extends classical forms to these, and uses this to reveal new unstable cohomology classes of $ ext{GL}_g(Z)$ and $ ext{SL}_g(Z)$, providing geometric insights into their cohomology.

Contribution

It introduces new bordifications of tropical moduli spaces, extends classical differential forms, and links these to unstable cohomology of linear groups with novel geometric methods.

Findings

Extended classical forms to tropical bordifications.

Discovered new unstable cohomology classes in $ ext{GL}_g(Z)$ and $ ext{SL}_g(Z)$.

Connected tropical geometry with the stable cohomology of linear groups.

Abstract

We construct bordifications of the moduli spaces of tropical curves and of tropical abelian varieties, and show that the tropical Torelli map extends to their bordifications. We prove that the classical bi-invariant differential forms studied by Cartan and others extend to these bordifications by studying their behaviour at infinity, and consequently deduce infinitely many new non-zero unstable cohomology classes in the cohomology of the general and special linear groups $\mathrm{GL}_g(\mathbb{Z})$ and $\mathrm{SL}_g(\mathbb{Z})$. In particular, we obtain a new and geometric proof of Borel's theorem on the stable cohomology of these groups. In addition, we completely determine the cohomology of the link of the moduli space of tropical abelian varieties within a certain range, and show that it contains the stable cohomology of the general linear group. In the process, we define new transcendental invariants associated to the minimal vectors of quadratic forms, and also show that a certain part of the cohomology of the general linear group $\mathrm{GL}_g(\mathbb{Z})$ admits the structure of a motive. In an appendix, we give an algebraic construction of the Borel-Serre compactification by embedding it in the real points of an iterated blow-up of a projective space along linear subspaces, which may have independent applications.