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

TL;DR

This paper proves that the rational Euler characteristic of the outer automorphism group of free groups is always negative, providing its asymptotic growth rate and confirming a longstanding conjecture from 1987.

Contribution

It establishes the negativity and asymptotic behavior of the Euler characteristic of Out(F_n), settling a conjecture and linking to special functions like Lambert W and zeta.

Findings

Euler characteristic of Out(F_n) is always negative.

Asymptotic growth rate is a6(n-3/2)/b7 \,log^2 n.

Connections made with Lambert W and zeta functions.

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.