Artin groups, CAT(0) 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.

The proposed research lies at the interface of non-commutative geometry with geometric group theory, a research area inspired by the work of Connes and Gromov, and the impact of the proposed research may be felt in both areas. The recent growth in international interest of this exciting area together with the complementary research interests of the PI and the host in these two areas will maximise the dissemination of the results. Drawing together tools from graph theory, geometry and functional analysis the area is extremely fertile in mathematical terms, but also has the potential to find new and exciting applications in science and engineering. For example, together with Brodzki and Wright, the PI for this proposal currently holds EPSRC grant EP/G059101/1, studying wide area blackouts in large scale energy grids using the tools from this relatively new discipline to study power flows in networks in order to identify safe islanding strategies to protect the national and trans-national power grids from catastrophic failure. He has also been invited to join a proposal to design innovative new battery packs using similar technology. In order to maximise the impact of this research it will be published in preprint form using the ArXiV and Soton ePrints servers, and will be submitted to leading mathematics journals. The PI and host will also promote the results by speaking at leading departments and at important national and international meetings. The work promises to positively impact the EPSRC funded research project EP/F031947/1: Analysis and geometry of metric spaces with applications in geometric group theory and topology. The existing project is largely concerned with a more analytic approach to related questions and the geometric tools we propose to develop during the visit with Guentner should provide further examples for analysis.