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Discrete groups and geometry of ideal boundary

理想边界的离散群和几何

基本信息

批准号：
11640056
负责人：
IZEKI Hiroyasu
金额：
$ 2.05万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640056/
关键词：
discrete groups rigidity ideal boundary Kleinian groups conformally flat index theorem scalar curvature 負曲率空間

项目摘要

The purpose of this project was to investigate the stability/rigidity of discrete groups from the viewpoint of geometry of the ideal boundary of negatively curved spaces and the cohomology of deiscrete groups. Our main result is summarized as follows.Let Γ be a Kleinian group acting on n-sphere. If Γ is convex cocompact, the quotient of the domain of discontinuity is compact by definition. However, the converse is not true in general. Izeki (head investigator) showed that if the Hausdorff dimension of the limit set of Γ is less than n/2 and the quotient of the domain of discontinuity is compact, then Γ is convex cocompact. As a consequence, such a Γ is quasiconformally stable. We also gave several applications to topology and geometry of conformally flat manifolds with positive scalar curvature.In case the Hausdorff dimension of the limit set is less than (n-2)/2, we found a proof using the index theorem for higher A^^<^>-genus. We applied the index theorem to the quotient of the domain of discontinuity. We note here what we mean by the ideal bounary is just the quotient of the domain of discontinuity. The higher A^^<^>-genus carries the information of the fundamental group, which turns out to be isomorphic to Γ in our case, and that is all that the higher A^^<^>-genus knows. And it is determined by the cohomology of Γ.

该项目的目的是从负弯曲空间的理想边界的几何学和离散群的上同调的角度研究离散群的稳定性/刚性。我们的主要结果概括如下：设Γ是作用在n-球面上的Kleinian群。若Γ是凸余紧的，则不连续区域的商根据定义是紧的。然而，一般情况下，匡威并不正确。Izeki（首席研究员）证明，如果Γ的极限集的Hausdorff维数小于n/2，并且不连续域的商是紧的，则Γ是凸余紧的。因此，这样的Γ是拟共形稳定的。我们还给出了具有正数量曲率的共形平坦流形在拓扑学和几何学中的应用，在极限集的Hausdorff维数小于（n-2）/2的情况下，利用高阶A^^<^>-亏格的指标定理给出了一个证明.我们将指数定理应用于不连续区域的商。这里我们注意到理想边界的意思就是不连续区域的商。高级A^^<^>-亏格携带基本群的信息，在我们的例子中，基本群同构于Γ，这就是高级A^^<^>-亏格所知道的全部。它是由Γ的上同调决定的。