3 Dimensional Geometry, Heegaard Splittings and Rank of the Fundamental Group

在佩雷尔曼的工作之后，大多数三维流形都被认为是双曲度量，即常负曲率的度量。 不幸的是，除了少数例外，没有进一步的属性，这个指标是已知的。在这个项目中，PI。将研究三维流形的拓扑与双曲几何的性质之间的关系。第一个目标是获得显式估计的几何数据的拓扑和组合信息，如基本组的Heegaard属的流形的秩。第二个目标是利用这些几何信息在适当的假设下重建双曲度量本身。三维流形是一个基本的数学对象。例如，我们生活的空间是一个三维流形。三维流形研究中的一个最近的趋势是在有限的组合模型中尽可能多地编码它们的精细的、无限复杂的几何结构。在这个过程中会丢失一定数量的信息。该项目的目标是量化实际丢失的信息量。因此，获得具体的先验估计至关重要。例如，他们打开了大门，预测方面的有限模型的现象发生在3-流形。令人惊讶的是，似乎很可能在许多情况下，足够精确的先验估计将允许恢复所有的几何信息。具体的估计将使准确的计算机模拟成为可能。

After the work of Perelman, most 3-manifolds are known to admit a hyperbolic metric, i.e. a metric of constant negative curvature. Unfortunately, with few exceptions, no further properties of this metric are known. In this project the P.I. will study how the topology of the 3-manifold and the properties of the hyperbolic geometry are related to each other. A first goal is to obtain explicite estimates of geometric data in terms of topological and combinatorial information such as the rank of the fundamental group of the Heegaard genus of the manifold. The second goal is to use these geometric information to reconstruct, under suitable assumptions, the hyperbolic metric itself.A 3-manifold is a mathematical object of fundamental interest. For example, the space we live in is a 3-manifold. A recent trend in the study of 3-manifolds is to encode as much of their fine, infinitely complicated, geometric structure in finite combinatorial models. A certain amount of information is lost in the process. The goal of the project is to quantify how much information actually gets lost. Obtaining concrete a priori estimates is then crucial. For example, they open the door to predictions in terms of finite models of phenomena occurig in 3-manifolds. Surprisingly, it seems very likely that in many situations sufficiently precise a priori estimates will allow to recover all the geometric information. Concrete estimates will make possible to have accurate computer simulations.