Discrete groups and geometry of ideal boundary

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标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^&lt；^&gt；-亏格的指数定理得到了证明。我们把指数定理应用到不连续域的商上。我们在这里注意到，我们所说的理想边界就是不连续区域的商。更高的A^^&lt；^&gt；-亏格携带了基本群的信息，在我们的例子中，它与Γ同构，这就是更高的A^^&lt；^&gt；-亏格所知道的全部。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.