The Euler characteristic of the moduli space of graphs

TL;DR

This paper investigates the Euler characteristic of the moduli space of graphs, revealing its rapid growth with the rank n, which implies a significant increase in the total cohomology dimension.

Contribution

It establishes the asymptotic behavior of the Euler characteristic for the moduli space of graphs, connecting it to the cohomology of related algebraic structures.

Findings

Euler characteristic grows like -e^{-1/4}(n/e)^n/(n log n)^2 as n increases

Total cohomology dimension increases rapidly with n

Links between moduli space, automorphism groups, and graph complexes are clarified

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 $-e^{-1/4}\,(n/e)^n/(n\log n)^2$ as $n$ goes to infinity, and thereby prove that the total dimension of this cohomology grows rapidly with $n$.