#### Volume 11, issue 3 (2011)

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Complexes and exactness of certain Artin groups

### Erik Guentner and Graham A Niblo

Algebraic & Geometric Topology 11 (2011) 1471–1495
##### 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 ${C}^{\ast }$–algebra and to the amenability of the action of the group on its Stone–Čech compactification. In this paper we study exactness for groups acting on a finite dimensional $CAT\left(0\right)$ 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.

##### Keywords
Property $A$, exactness, Artin group, $\mathrm{CAT}(0)$ cube complex
##### Mathematical Subject Classification 2000
Primary: 20F36, 20F65, 43A99
Secondary: 51F15
##### Publication
Revised: 4 January 2011
Accepted: 24 January 2011
Published: 23 May 2011
##### Authors
 Erik Guentner Department of Mathematics University of Hawai’i at Manoa 2565 McCarthy Mall Honolulu HI 96822 USA Graham A Niblo School of Mathematics University of Southampton Highfield Southampton SO17 1BJ UK