Subgroup Separability of Artin Groups

TL;DR

This paper characterizes when Artin groups are subgroup separable based on their underlying graph, showing they can be constructed from low-rank Artin groups using free and direct products.

Contribution

It provides a complete graph-based criterion for subgroup separability of Artin groups, extending the known results for Right-Angled Artin groups.

Findings

Subgroup separability depends on the underlying graph structure.

Artin groups are subgroup separable iff constructed from rank ≤ 2 groups via specific products.

Generalizes the Metaftsis-Raptis criterion for Right-Angled Artin groups.

Abstract

We find a condition on the underlying graph of an Artin group that fully determines if it is subgroup separable. As a consequence, an Artin group is subgroup separable if and only if it can be obtained from Artin groups of ranks at most 2 via a finite sequence of free products and direct products with the infinite cyclic group. This result generalizes the Metaftsis-Raptis criterion for Right-Angled Artin groups.