本项目的第一个研究目标是给出弱顺从性的不动点刻划和测度群论刻划，从而证明Cowling猜想(CCAP隐含Haagerup性质)或给出反例；第二个研究目标是利用Hilbert C* -模理论研究离散群的粗嵌入性与Roe代数具有Hilbert C*-模Haagerup性质的等价性；第三个研究目标是本项目的主要研究内容，我们计划从测度群论的角度给出离散群的正合性和粗嵌入性的刻画，从而期望利用测度群论彻底解决离散群的正合性和粗嵌入性的相互关系；第四个研究目标是利用算子空间和测度群论构造Kirchberg猜想反例和非正合群的例子。其意义在于希望将测度群论和算子空间的技巧应用于解决群算子代数和离散群的问题，对沟通算子空间、群算子代数、几何群论、测度群论和离散群这几个数学分支作出有价值的贡献。

This Project is consisted of four parts: (1) we want to give the fixed point characterization and measure group theoretic characterization for weakly amenable groups. Thus we try to prove Cowling Conjecture (CCAP implies Haagerup property) or give some counter-examples; (2) we will investigate the equivalence of coarse embedding for discrete groups and Hilbert C*-module Haagerup property for the corresponding Roe algebras; (3) we mainly wish to obtain the characterizations of exactness and coarse embedding in the view of Measure Group Theory. Thus we try to understand the relationship between exactness and coarse embedding for discrete groups in term of Measure Group Theory; (4) By use of Operator Spaces and Measure Group Theory, we try to construct some counter-examples for Kirchberg conjecture and some non-exact groups. The main aim of this projcet is to solve some problems in Operator Algebras and Discrete Groups in term of Measure Group Theory and Operator Spaces.