Geometry of spaces and groups with negative eurvature

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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