The Euler characteristic of $\mathrm{Out}(F_n)$

Abstract

We prove that the rational Euler characteristic of $\mathrm{Out}(F_n)$ is always negative and its asymptotic growth rate is $\Gamma (n-\frac{3}{2})/\sqrt{2\pi }{\log }^{2}n$. This settles a 1987 conjecture of J. Smillie and the second author. We establish connections with the Lambert $W$-function and the zeta function.

final version, to appear in Commentarii Math. Helv

References in corpus (3)

• Dimension of the Torelli group for Out(F_n)
• On the bordification of outer space
• Graphs in perturbation theory: Algebraic structure and asymptotics

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• The Euler characteristic of the moduli space of graphs
• On the top-dimensional ${\ell }^2$-Betti numbers
• Algebraic Interplay between Renormalization and Monodromy
• The ${\mathbb{S}}_n$-equivariant Euler characteristic of the moduli space of graphs