Coxeter quotients of the automorphism group of a Coxeter group

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 $\mathrm{Aut}(W_{\Gamma })$ virtually surjects onto some infinite Coxeter group. In particular, the group $\mathrm{Aut}(W_{\Gamma })$ is virtually indicable and therefore does not have Kazhdan's property (T).

Completely rewritten because the proof of Theorem A in the previous version is incomplete