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

arXiv comment43 pages, v2: typos and a reference fixed; v3: corrected sign issue in section 5; v4: minor corrections, version to appear in Advances in Mathematics

References in corpus (7)

• On a theorem of Kontsevich
• M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra
• Tropical curves, graph complexes, and top weight cohomology of M_g
• Assembling homology classes in automorphism groups of free groups
• The Euler characteristic of $\mathrm{Out}(F_n)$
• Feynman amplitudes on moduli spaces of graphs
• Integral Euler characteristic of Out F_{11}

Cited by in corpus (3)

• The stable cohomology of self-equivalences of connected sums of products of spheres
• Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group
• The ${\mathbb{S}}_n$-equivariant Euler characteristic of the moduli space of graphs