本项目拟利用粗几何与算子代数作为工具来研究离散群的几何结构及其在“粗Baum-Connes猜想”、“粗 Novikov 猜测”中的应用。具体的研究内容包括：利用离散度量空间的几何特征，给出它具有几何性质(T)的几何刻画；证明两个具有几何性质(T)的空间，它们的zig-zag乘积也具有几何性质(T);对具有有界几何可以纤维化粗嵌入到一致凸Banach空间的几何空间，证明其上的粗Novikov猜想成立；给出盒子空间可以纤维化粗嵌入到一致凸Banach空间的充分必要条件；利用盒子空间去构造可以粗嵌入到l^p空间但不能粗嵌入到Hilbert空间的反例；若群G等距、真性的作用到离散度量空间X上，并且商空间以及群自身可以粗嵌入到Hilbert空间中，证明X上的等变Novikov猜想成立。通过本项目的研究，希望对几何群理论、非紧空间上的算子代数与指标理论的若干前沿问题作出重要的贡献。

In this proposal, we are going to use tools in coarse geometry and operator algebras to study the geometric structure of discrete groups and its applications to the coarse Baum-Connes conjecture and coarse Novikov conjecture. We will study the geometric property (T) and their characterizations in term of geometry and will prove that the zig-zag product has geometric property (T) if both spaces have geometric property (T). We will show the coarse Novikov conjecture holds for spaces which admits a fibred coarse embedding into uniformly Banach spaces. Moreover, we will study the structure of Box spaces and find a necessary and sufficient condition on fibred coarsely embeddable into uniformly Banach spaces. We will construct an example by using the structure of box spaces, which does not coarsely embed into Hilbert space, but does coarsely embed into l^p space. We will show the equivariant Novikov conjecture holds for X if metric space X admit a proper and isometric action by a group G and the quotient space X/G and G are coarsely embeddable into Hilbert spaces. We expect to achieve important contributions to geometric group theory and the main problems in higher index theory on non-compact spaces.