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Complexes and exactness of certain Artin groups

Erik Guentner and Graham A Niblo

Algebraic & Geometric Topology 11 (2011) 1471–1495

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.

Property $A$, exactness, Artin group, $\mathrm{CAT}(0)$ cube complex
Mathematical Subject Classification 2000
Primary: 20F36, 20F65, 43A99
Secondary: 51F15
Received: 23 August 2010
Revised: 4 January 2011
Accepted: 24 January 2011
Published: 23 May 2011
Erik Guentner
Department of Mathematics
University of Hawai’i at Manoa
2565 McCarthy Mall
Honolulu HI 96822
Graham A Niblo
School of Mathematics
University of Southampton
Southampton SO17 1BJ