On the Euler characteristic of the commutative graph complex and the top weight cohomology of ${\mathcal{M}}_g$

We prove an asymptotic formula for the Euler characteristic of Kontsevich’s commutative graph complex. This formula implies that the total amount of commutative graph homology grows super-exponentially with the rank and, via a theorem of Chan, Galatius, and Payne, that the dimension of the top weight cohomology of the moduli space of curves, ${\mathcal{M}}_g$, grows super-exponentially with the genus $g$.