Note on residual finiteness of Artin groups

TL;DR

This paper investigates conditions under which Artin groups are residually finite, showing that certain partitions and properties of associated Coxeter graphs imply residual finiteness.

Contribution

It establishes new residual finiteness criteria for Artin groups based on admissible partitions and properties of quotient Coxeter graphs.

Findings

Residual finiteness holds if quotient Coxeter graph is a forest or even triangle-free.

Residually finite subgroups imply residual finiteness of the entire Artin group.

Provides a framework linking graph properties to algebraic residual finiteness.

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 $X,Y \in \mathcal{P}$, $X \neq Y$, 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.