The Euler characteristic of the moduli space of graphs (Q6073602)

The moduli space \(\mathcal{MG}_{n}\) of finite metric graphs with fundamental group \(F_{n}\) was introduced by \textit{M. Culler} and the second author in [Invent. Math. 84, 91--119 (1986; Zbl 0589.20022)] as a tool for studying the group \(\mathrm{Out}(F_{n})\) of outer automorphisms of a free group. In the paper under review, the authors prove a formula for the Euler characteristic of \(\mathcal{MG}_{n}\) and then determine its asymptotic growth rate. In particular, they 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 \(n\).