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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