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

Michael Borinsky, Karen Vogtmann

We prove that the rational Euler characteristic of $\mathrm{Out}(F_n)$ is always negative and its asymptotic growth rate is $\Gamma (n-\frac{3}{2})/\sqrt{2\pi }{\log }^{2}n$. This settles a…