JournalscmhVol. 95, No. 4pp. 703–748

The Euler characteristic of Out(FnF_n)

  • Michael Borinsky

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

    University of Warwick, Coventry, UK
The Euler characteristic of Out($F_n$) cover

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Abstract

We prove that the rational Euler characteristic of Out(FnF_n) is always negative and its asymptotic growth rate is Γ(n32)/2πlog2n\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 WW-function and the zeta function.

Cite this article

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

DOI 10.4171/CMH/501