Complexes and exactness of certain Artin groups

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–Čech 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.

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Keywords

Property $A$, exactness, Artin group, $\mathrm{CAT}(0)$ cube complex

Mathematical Subject Classification 2000

Primary: 20F36, 20F65, 43A99

Secondary: 51F15

References

Publication

Received: 23 August 2010

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

Volume 11, issue 3 (2011)

Erik Guentner and Graham A Niblo

Abstract