Rigidity of discrete groups and index theorems

离散群的刚性和指数定理

• 批准号: 13640057
• 负责人: IZEKI Hiroyasu
• 金额: $ 2.18万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640057/
• 关键词: discrete groups; rigidity; index theorem; harmonic map; conformally flat; Kleinian group; ユニタリ表現

The purpose of this project was to investigate the rigidity of discrete groups from the viewpoint of geometry of the ideal boundary of negatively curved spaces and the cohomology of discrete 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 also developed another approach to rigidity problems, which uses harmonic maps from a simplicial complex to a negatively curved metric space. We obtained a fixed-point theorem for a lattice in a p-adic Lie group, which should be regarded as a generalization of Margulis superrigidity.

该项目的目的是从负弯曲空间理想边界的几何角度和离散群的上同调性来研究离散群的刚性。我们的主要结果总结如下。 令 Γ 为作用于 n 的克莱因群。 -球面。如果 Г 是凸协紧的，则不连续域的商根据定义是紧的，但是，反之则不然。Izeki（首席研究员）表明，如果 Hausdorff极限集的维数小于 n/2 并且不连续域的商是紧的，则 Г 是凸协紧的，因此，这样的 Г 是拟共形稳定的。具有正标量曲率的共形平坦流形 如果极限集的 Hausdorff 维数小于 (n - 2) /2，我们使用以下方法找到了证明。我们还开发了另一种解决刚性问题的方法，该方法使用从单纯复形到负弯曲度量空间的调和映射，我们获得了 p 进李群中格子的不动点定理，这应该被视为 Margulis 超刚性的概括。