Right-angled Artin subgroups of Artin groups

TL;DR

This paper explores the structure of right-angled Artin subgroups within Artin groups, extending known results and verifying a conjecture for specific classes, with implications for hyperbolic surface subgroups.

Contribution

It introduces a conjecture about larger right-angled Artin subgroups generated by powers of central elements and verifies it for several classes of Artin groups.

Findings

Verified the conjecture for locally reducible Artin groups.

Confirmed the conjecture for spherical Artin groups excluding types E6, E7, E8.

Established the existence of hyperbolic surface subgroups in certain Artin groups.

Abstract

The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible spherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious right-angled Artin subgroup. This alleged right-angled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of locally reducible Artin groups, which includes all $2$-dimensional Artin groups, and for spherical Artin groups of any type other than $E_6$, $E_7$, $E_8$. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.