  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 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
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 show that part of the cohomology of the
general linear group $\mathrm{GL}_g(\mathbb{Z})$ admits the structure of a
motive.
