# The Euler characteristic of Out($F_n$)

### Michael Borinsky

National Institute for Subatomic Physics, Amsterdam, Netherlands### Karen Vogtmann

University of Warwick, Coventry, UK

## Abstract

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.

## Cite this article

Michael Borinsky, Karen Vogtmann, The Euler characteristic of Out($F_n$). Comment. Math. Helv. 95 (2020), no. 4, pp. 703–748

DOI 10.4171/CMH/501