The Euler characteristic of $\operatorname{Out}(F_n)$

Michael Borinsky; Karen Vogtmann

doi:10.4171/CMH/501

The Euler characteristic of $\operatorname{Out}(F_n)$

Group Theory 2022-02-21 v2 Mathematical Physics math.MP Quantum Algebra

Authors: Michael Borinsky , Karen Vogtmann

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Abstract

We prove that the rational Euler characteristic of $\operatorname{Out}(F_n)$ is always negative and its asymptotic growth rate is $\Gamma(n- \frac32)/\sqrt{2\pi} \log^2 n$ . This settles a 1987 conjecture of J. Smillie and the second author. We establish connections with the Lambert $W$ -function and the zeta function.

Keywords

out(f_n) euler characteristic rational euler characteristic lambert w-function zeta function asymptotic growth

Cite

@article{arxiv.1907.03543,
  title  = {The Euler characteristic of $\operatorname{Out}(F_n)$},
  author = {Michael Borinsky and Karen Vogtmann},
  journal= {arXiv preprint arXiv:1907.03543},
  year   = {2022}
}

Comments

final version, to appear in Commentarii Math. Helv

The Euler characteristic of $\operatorname{Out}(F_n)$

Abstract

Keywords

Cite

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