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

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 classes in the cohomology of the general and special linear groups $\mathrm{GL}_{g}(\mathbb{Z})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>GL</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:math> and $\mathrm{SL}_{g}(\mathbb{Z})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>SL</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:math> . In particular, we obtain a new geometric proof of Borel’s theorem on the stable cohomology of these groups. 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 addition, we define new transcendental invariants associated to the minimal vectors of quadratic forms, and show that a certain part of the cohomology of the general linear group $\mathrm{GL}_{g}(\mathbb{Z})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>GL</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:math> 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.