Discrete groups and geometry of ideal boundary

本课题从负弯曲空间理想边界的几何角度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a ^^<^>-格的指数定理给出了证明。将指数定理应用于不连续定义域的商。我们注意到这里所说的理想边界就是不连续域的商。更高的A^^<^>-格携带了基本群的信息，在我们的例子中它是与Γ同构的，这就是更高的A^^<^>-格所知道的全部信息。它是由Γ的上同调决定的。

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.