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# Commentarii Mathematici Helvetici

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**Volume 95, Issue 4, 2020, pp. 703–748**

**DOI: 10.4171/CMH/501**

Published online: 2020-12-07

The Euler characteristic of Out($F_n$)

Michael Borinsky^{[1]}and Karen Vogtmann

^{[2]}(1) National Institute for Subatomic Physics, Amsterdam, Netherlands

(2) University of Warwick, Coventry, UK

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

*Keywords: *Automorphisms of free groups, Euler characteristic, asymptotic expansions

Borinsky Michael, Vogtmann Karen: The Euler characteristic of Out($F_n$). *Comment. Math. Helv.* 95 (2020), 703-748. doi: 10.4171/CMH/501