The Euler characteristic of the moduli space of graphs

Abstract

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−1/4(n/e)n/(nlog⁡n)2 as n goes to infinity, and thereby prove that the total dimension of this cohomology grows rapidly with n.