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: ${\tilde{A}}_1$, ${\tilde{A}}_n(n\ge 2)$, ${\tilde{B}}_n(n\ge 3)$, ${\tilde{C}}_n(n\ge 2)$, ${\tilde{D}}_n(n\ge 4)$, ${\tilde{E}}_6$, ${\tilde{E}}_7$, ${\tilde{E}}_8$, ${\tilde{F}}_4$ or ${\tilde{G}}_2$. Moreover, we show that just infinite Coxeter groups are profinitely rigid among all Coxeter groups.

13 pages