Geometry of spaces and groups with negative eurvature

主要研究了负曲率奇异空间的单纯体积、等距群、有限性，建立了三维流形在曲率下界下的塌缩理论。(1)证明了广义拟流形的基本同调类的质量与总体积一致，并得到了单纯体积、曲率界与体积之间的关系.我们还得到了Gromov-Thurston不等式在余维为1的奇性空间中的推广。(2)我们观察了曲率上有界的Alexandrov空间的等距群，并证明了在变体中离开紧集的子群是李群。利用这个结果，我们基本上找到了等距群与树相似的一些性质。作为应用，我们证明了具有非正曲率的单连通空间的等距群是李群，如果理想边界关于Titz度量是紧的。(3)在与Takashi Shioya的联合工作中，我们研究了下曲率界下的三流形的坍缩现象。更确切地说，通过寻找距离函数临界点的方法，我们成功地用极限空间描述了坍缩流形。(4)我们证明了曲率为<less than or equal>-1且维数和体积有界的空间的同伦型的有限性。现在我们试图把这个结果推广到在维数分层的每一层的体积的加法假设下同胚的有限性。

We have mainly studied the simplicial volumes, the isometry groups, the finiteness property for singular spaces with negative curvature, and established the theory of collapsing for 3-manifolds under a lower curvature bound. (1) We have shown that for a generalized psuedomanifold the mass of its fundamental homology class coincides with the total volume, and obtained a relation between the simplicial volume, the curvature bound and the volume. We have also obtained the generalization of Gromov-Thurston inequality for spaces with singularities of codimension one. (2) We have observed the isometry group of an Alexandrov space with curvature bounded above, and proved that the subgroup leaving in variant a compact set is a Lie group. By using this result basically, we have found out some properties of the isometry group similar to those of trees. As an application, we have shown that the isometry group of a simply connected space with nonpositive curvature is a Lie group if the ideal boundary is compact with respect to the Titz metric. (3) In the joint work with Takashi Shioya, we study the collapsing phenomena of three-manifolds under a lower curvature bound. More precisely, by finding out some methods about critical points of distance functions, we have succeeded to describe collapsed manifolds in terms of the limit space. (4) We have shown that the finiteness of homotopy types of spaces with curvature <less than or equal> -1 and with bounds on dimameter and volume. Now we are trying to extend this result to the finiteness of homeomorphisms under an addition hypothesis on the volume of each strata of the dimension stratific