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We construct bordifications of the moduli spaces of tropical curves and oftropical abelian varieties, and show that the tropical Torelli map extends totheir bordifications. We prove that the classical bi-invariant differentialforms studied by Cartan extend to these bordifications by studying theirbehaviour at infinity, and consequently deduce infinitely many new non-zerounstable cohomology classes in the cohomology of the general and special lineargroups $\mathrm{GL}_g(\mathbb{Z})$ and $\mathrm{SL}_g(\mathbb{Z})$. Inaddition, we completely determine the cohomology of the link of the modulispace of tropical abelian varieties within a certain range, and show that itcontains the stable cohomology of the general linear group.  In the process, we define new transcendental invariants associated to theminimal vectors of quadratic forms, and show that part of the cohomology of thegeneral linear group $\mathrm{GL}_g(\mathbb{Z})$ admits the structure of amotive.

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