Rigidity of discrete groups and index theorems

本课题的目的是从负曲线空间的理想边界的几何和离散群的上同调的角度研究离散群的刚性。我们的主要结果概括如下：设Γ是作用在n-球面上的克莱因群。如果Γ是凸余紧的，则不连续域的商按定义是紧的。然而，通常情况下，情况并非如此。Izeki(首席研究员)证明了如果Γ的极限集的Hausdorff维小于n/2，且不连续区域的商是紧的，则Γ是凸余紧的。因此，这样的Γ是准共形稳定的。我们还给出了具有正数量曲率的共形平坦流形的几个应用。当极限集的Hausdorff维数小于(n-2)/2时，我们利用高A-代数的指标定理得到了证明，并利用从单纯形复形到负曲度量空间的调和映射得到了刚性问题的另一种方法.我们得到了p-进李群中格的一个不动点定理，它应该被看作是Marguis超刚性的推广。

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.