The Euler characteristic of the moduli space of graphs. (English) Zbl 1523.20068
The moduli space \(\mathcal{MG}_{n}\) of finite metric graphs with fundamental group \(F_{n}\) was introduced by 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 \(n\).
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 \(n\).
Reviewer: Egle Bettio (Venezia)
MSC:
| 20F65 | Geometric group theory |
| 20E36 | Automorphisms of infinite groups |
| 05A16 | Asymptotic enumeration |
| 20F28 | Automorphism groups of groups |
Keywords:
geometric group theory; moduli space of graphs; graph complex; Euler characteristic; \(\mathrm{Out}(F_{n})\); tropical geometryCitations:
Zbl 0589.20022Software:
FORMOnline Encyclopedia of Integer Sequences:
Number of unlabeled connected multigraphs with circuit rank n and degree >= 3 at each node, loops allowed.References:
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