非交换几何是以泛函分析、算子代数的工具来研究其他领域的问题。随着非交换几何的发展，其理论日益丰富，应用日益广泛，产生了很多新的观点和问题。其中Guentner-Tessra-Yu提出的有限分解复杂度性质是一个新的研究热点。这一性质可以用来证明拓扑学中的稳定Borel猜想。本项目希望研究与这一性质的相关的两类问题。一类是哪些群具有或者不具有这一性质，重点研究Thompson群和Grigorchuk群是否具有这一性质。这两个群在几何群论的研究中有重要意义，与Von Neumann猜想等问题相关。一类是这一性质在拓扑学中的应用。我们知道流形分类问题一直是流形拓扑学研究的中心问题。其中非球面流形是重点研究对象。我们对非球面流形了解的很少。M.Davis用Coxeter群的办法构造了一大类非球面流形。这些流形的分类问题至今没有得到解决。用有限分解复杂度这一新的性质可以来研究这类流形的分类问题。

Noncommutative geometry uses the theory of functional analysis and operator theory to study problems in other mathmatical areas. New ideas and problems arise along with the develpopement of noncommutative geometry. One of the research spot is the property called the finite decomposition complexity defined by Guentner-Tessra-Yu.This property can be used to study stable Borel conjecture. This project intend to study two kinds of problems related to this property. One is to study what kinds of groups have or do not have this property, especially the Thompson group and Grigorchuk group. These two groups are of great importantance in the research of geometric group theory。They are related to many problems such as Von Neumann conjecture. The other is to study the application of the finite decomposition property. We know that the classification of topological manifolds is a long standing problem in the research of topology. Among those manifolds, aspherical manifolds may be the most important classes. We know little about this kind of manifold yet. M.Davis uses the Coxeter group to build many such manifolds. The classification of those manifolds is still under consideration. Now we can apply the new propertys coming from the noncommutative geometry to study those manifold.