Note on residual finiteness of Artin groups

Abstract

Let $A$ be an Artin group. A partition $\mathcal{P}$ of the set of standard generators of $A$ is called admissible if, for all $X,Y\in \mathcal{P}$, $X\ne Y$, there is at most one pair $(s,t)\in X\times Y$ which has a relation. An admissible partition $\mathcal{P}$ determines a quotient Coxeter graph $\Gamma /\mathcal{P}$. We prove that, if $\Gamma /\mathcal{P}$ is either a forest or an even triangle free Coxeter graph and $A_X$ is residually finite for all $X\in \mathcal{P}$, then $A$ is residually finite.