Artin groups of infinite type: trivial centers and acylindical hyperbolicity

TL;DR

This paper investigates infinite type Artin groups, demonstrating that under certain conditions they have trivial centers and are acylindrically hyperbolic, expanding understanding beyond well-studied finite and right-angled cases.

Contribution

It proves that infinite type Artin groups have trivial centers and are acylindrically hyperbolic under specific graph conditions, using actions on CAT(0) cube complexes.

Findings

Infinite type Artin groups have trivial centers if the graph is not a star.

They are acylindrically hyperbolic if the graph is not a join.

The study extends properties known for finite and right-angled Artin groups.

Abstract

While finite type Artin groups and right-angled Artin groups are well-understood, little is known about more general Artin groups. In this paper we use the action of an infinite type Artin group $A_\Gamma$ on a CAT(0) cube complex to prove that $A_\Gamma$ has trivial center providing the graph $\Gamma$ is not the star of a single vertex, and is acylindrically hyperbolic providing $\Gamma$ is not a join.