Artin groups, CAT(0) geometry and property A

俞敏洪的房产A是对宜人概念的广泛概括。它被Yu、Higson和Kasparov以及其他人用来为许多重要的群类建立强Novikov猜想。例如，字双曲群和有限生成的线性群都满足性质A。性质A是一种几何性质，它可以通过在群作用适当(通常是余紧的)的空间的点上构造某些加权函数来证明，从而使函数在作用下几乎不变。这是由PI和他的合作者为承认对CAT(0)立方体复合体的适当操作的小组完成的。在这种情况下，权函数是显式的，并且具有吸引人的增长性质。这个项目的主要目的是利用这一事实来推广已知的结果，以涵盖某些不适当的行为，特别是将建立性质A的技巧应用于Artin群的自然类。它们产生于对代数几何中超平面排列的研究，并在几何群论中得到了广泛的研究。这项请求是为了资助首席调查员在2010年2月/3月对夏威夷大学Erik Guentner教授进行为期10天的访问。这次访问的目的是扩展和完成在Artin群和二维CAT(0)空间的分析性质方面的现有合作，特别是对Yu性质A的研究，探索本文中几何和上同调之间的相互作用。所开发的方法很可能扩展到几何和Baum Connes猜想工作的其他感兴趣的群类。

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.