The Euler characteristic of the moduli space of graphs

The moduli space of rank $n$ graphs, the outer automorphism group of the free group of rank $n$ and Kontsevich’s Lie graph complex have the same rational cohomology. We show that the associated Euler characteristic grows like $-e^{-1\mathrm{/}4}(n\mathrm{/}e{)}^n\mathrm{/}(n\log n{)}^2$ as $n$ goes to infinity, and thereby prove that the total dimension of this cohomology grows rapidly with $n$.