Acylindrical hyperbolicity and the centers of Artin groups that are not free of infinity

TL;DR

This paper advances understanding of acylindrical hyperbolicity in infinite-type Artin groups by introducing reduced clique-cube complexes, leading to new results on their centers and triviality in many cases.

Contribution

It introduces reduced clique-cube complexes and proves acylindrical hyperbolicity for a broader class of irreducible Artin groups, clarifying their algebraic properties.

Findings

Irreducible Artin groups not being cliques are acylindrically hyperbolic.

Centers of these Artin groups are finite, often trivial.

The study extends previous results to groups containing infinite type Artin groups of type FC.

Abstract

Charney and Morris-Wright showed acylindrical hyperbolicity of Artin groups of infinite type associated with graphs that are not joins, by studying clique-cube complexes and the actions on them. The authors developed their study and clarified when acylindrical hyperbolicity holds for Artin groups of infinite type associated with graphs that are not cones. In this paper, we introduce reduced clique-cube complexes. By using them, we show acylindrical hyperbolicity of irreducible Artin groups associated with graphs that are not cliques, that is, irreducible Artin groups that are not free of infinity. Such Artin groups contain infinite type Artin groups of type FC. As an application, we see that the centers of such Artin groups are finite, and that actually they are trivial in many cases.