Geometry of spaces and groups with negative eurvature

主要研究了负曲率奇异空间的简单体积、等距群、有限性，建立了下曲率界下3流形的坍缩理论。(1)证明了广义伪多流形的基本同调类的质量与总体积重合，并得到了简单体积、曲率界与体积之间的关系。在奇异度为余维1的空间中，我们也得到了Gromov-Thurston不等式的推广。(2)我们观察了曲率上有界的Alexandrov空间的等距群，并证明了在变紧集上留下的子群是李群。基本利用这一结果，我们发现了等距群的一些类似于树木的性质。作为一个应用，我们证明了非正曲率单连通空间的理想边界相对于Titz度规紧致时的等距群是李群。(3)与Takashi Shioya共同研究了三流形在下曲率界下的坍缩现象。更确切地说，通过找出一些关于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)证明了曲率<或等于> -1且直径和体积有界的空间同伦型的有限性。现在我们试图将这个结果推广到在层维各层体积的加法假设下的同胚的有限性

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