The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Commentarii Mathematici Helvetici


Full-Text PDF (410 KB) | Metadata | Table of Contents | CMH summary
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