Note on the residual finiteness of Artin groups

Abstract

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