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.

本课题从负弯曲空间的理想边界和离散群的上同调的几何角度研究了离散群的刚性。我们的主要结果总结如下。设Γ为作用于n球的Kleinian群。如果Γ是凸紧的，则不连续域的商根据定义是紧的。然而，相反的情况通常并不成立。Izeki（首席研究者）证明了如果Γ的极限集的Hausdorff维数小于n/2且不连续域的商是紧致的，则Γ是凸紧致的。因此，这样的Γ是准共形稳定的。我们还给出了正标量曲率共形平面流形的拓扑和几何的几个应用。在极限集的Hausdorff维数小于(n - 2) /2的情况下，利用高a属的指标定理给出了证明。我们还开发了另一种解决刚性问题的方法，它使用从简单复形到负弯曲度量空间的调和映射。我们得到了p进李群中格的不动点定理，它可以看作是马古利斯超刚性的推广。

项目成果

藤原 耕二: "On the outer automorphism group of a hyperbolic group"Israel J. of Math.. 131. 277-284 (2002)

藤原浩二：“论双曲群的外自同构群”Israel J. of Math.. 131. 277-284 (2002)

井関 裕靖: "高次元のクライン群の極限集合のハウスドルフ次元-収束指数と凸ココンパクト性-"数理解析研究所講究録. 1223. 61-68 (2001)

Hiroyasu Iseki：“高维克莱因群极限集的豪斯多夫维数 - 收敛指数和凸协紧性 -”数学分析研究所 Kokyuroku。1223. 61-68 (2001)

Y.Nakagawa and T.Mabuchi: "An obstruction to semistability of manifolds"Proc. Japan Acad.. 77, Ser.A. 47-49 (2001)

Y.Nakakawa 和 T.Mabuchi：“对流形半稳定性的阻碍”Proc。

納谷 信: "Quaternionic analogue of CR geometry(with H.kondo)"Semin.Theor.Spectr.Geom.. 19. 41-52 (2001)

Shin Naya：“CR 几何的四元模拟（与 H.kondo）”Semin.Theor.Spectr.Geom.. 19. 41-52 (2001)

Shin Nayatani: "Quaternionic analogue of CR geowetry"Seminalre de theorie spectrale et geometrie GRENOBLE. 19. 41-52 (2001)

Shin Nayatani：“CR 几何学的四元模拟”格勒诺布尔光谱与几何理论研讨会。