On quotients of Coxeter groups

Abstract: A group $G$ is said to be just infinite if $G$ itself is infinite but all proper quotients of $G$ are finite. We show that a Coxeter group $W_\Gamma$ is just infinite if and only if $\Gamma$ is isomorphic to one of the following graphs: $\widetilde{A}_1$, $\widetilde{A}_n (n\geq 2)$, $\widetilde{B}_n (n\geq 3)$, $\widetilde{C}_n (n\geq 2)$, $\widetilde{D}_n(n\geq 4)$, $\widetilde{E}_6$, $\widetilde{E}_7$, $\widetilde{E}_8$, $\widetilde{F}_4$ or $\widetilde{G}_2$. Moreover, we show that just infinite Coxeter groups are profinitely rigid among all Coxeter groups.