The Euler characteristic of Out($F_n$)

Michael Borinsky; Karen Vogtmann

doi:10.4171/cmh/501

JournalscmhVol. 95, No. 4pp. 703–748

The Euler characteristic of Out( $F_{n}$ )

Michael Borinsky
National Institute for Subatomic Physics, Amsterdam, Netherlands
Karen Vogtmann
University of Warwick, Coventry, UK

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Abstract

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

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