Artin Groups, CAT (O) geometry and property A

Yu's property A is a wide ranging generalisation of the notion of amenability. It was used by Yu, by Higson and Kasparov, and by others, to establish the strong Novikov conjecture for many important classes of groups. For example word hyperbolic groups and finitely generated linear groups are all known to satisfy property A.Property A is a geometric property which may be demonstrated for a given group by constructing certain weighting functions on the points of a space on which the group acts properly (and usually co-compactly) so that the functions are almost invariant under the action. This was done by the PI with his collaborators for groups admitting a proper action on a CAT(0) cube complex. The weighting functions in this case are explicit, and have attractive growth properties. It is the priniciple aim of this project to exploit that fact to generalise the known result to cover certain non-proper actions, and in particular to apply the technique to establish property A to the natural class of Artin groups. These arise in the study of hyperplane arrangements in algebraic geometry and have been extensively studied in geometric group theory. The request is to fund a visit by the Principal Investigator to Prof. Erik Guentner at the University of Hawaii for 10 days in February/March 2010. The purpose of the visit is to extend and complete an existing collaboration on analytic properties of Artin groups and 2-dimensional CAT(0) spaces, specifically a study of Yu's property A, exploring the interaction of geometry and cohomology in this context.The methods developed are likely to extend to other classes of groups of interest to geometers and to those working on the Baum Connes conjecture.