Automorphism groups of Coxeter groups do not have Kazhdan's property (T)

Abstract

We show that for a large class $\mathcal{W}$ of Coxeter groups the following holds: Given a group $W_\Gamma$ in $\mathcal{W}$, the automorphism group ${\rm Aut}(W_\Gamma)$ virtually surjects onto $W_\Gamma$. In particular, the group ${\rm Aut}(G_\Gamma)$ is virtually indicable and therefore does not satisfy Kazhdan's property (T). Moreover, if $W_\Gamma$ is not virtually abelian, then the group ${\rm Aut}(W_\Gamma)$ is large