Complexes and Exactness of certain Artin Groups

TL;DR

This paper investigates the property of exactness in Artin groups, especially those acting on finite-dimensional CAT(0) cube complexes, establishing exactness for type FC Artin groups and discussing the broader context of group exactness.

Contribution

The paper proves that Artin groups of type FC are exact and explores the relationship between group actions on CAT(0) cube complexes and exactness properties.

Findings

Artin groups of type FC are exact.

Exactness relates to group actions on CAT(0) cube complexes.

Open question remains whether all Artin groups are exact.

Abstract

In his work on the Novikov conjecture, Yu introduced Property $A$ as a readily verified criterion implying coarse embeddability. Studied subsequently as a property in its own right, Property $A$ for a discrete group is known to be equivalent to exactness of the reduced group $C^*$-algebra and to the amenability of the action of the group on its Stone-Cech compactification. In this paper we study exactness for groups acting on a finite dimensional $\CAT(0)$ cube complex. We apply our methods to show that Artin groups of type FC are exact. While many discrete groups are known to be exact the question of whether every Artin group is exact remains open.