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arXiv · 1907.03543

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

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.

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Michael Borinsky, Karen Vogtmann. 2019-07-08. The Euler characteristic of $\operatorname{Out}(F_n)$. https://doi.org/10.4171/cmh/501

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