Artin groups, CAT(0) geometry and property A

余的性质A是对顺从性概念的广泛概括。它是由余，希格森和卡斯帕罗夫，和其他人，以建立强诺维科夫猜想的许多重要类别的群体。性质A是一个几何性质，可以通过在一个空间的点上构造某些加权函数来证明，在这个空间上，这个群适当地作用（并且通常是协紧的），使得这些函数在这个作用下几乎是不变的。这是由PI与他的合作者为承认CAT（0）立方体复合体上的正确动作的组完成的。在这种情况下的权重函数是明确的，并具有吸引人的增长特性。本项目的主要目的是利用这一事实来推广已知的结果，以涵盖某些非适当的行动，特别是应用技术来建立性质A的自然类阿丁群。这些出现在代数几何中的超平面安排的研究中，并在几何群论中得到了广泛的研究。申请是为了资助首席研究员于2010年2月/3月访问夏威夷大学的Erik Guentner教授，为期10天。这次访问的目的是扩展和完成现有的合作分析性质的阿廷集团和2维CAT（0）空间，特别是研究俞的性质A，探索相互作用的几何和上同调在这种情况下。开发的方法可能会扩展到其他类的群体感兴趣的几何学家和那些工作的鲍姆康纳斯猜想。

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.