Geometry and Topology of Arithmetic 3-manifolds.

提案：DMS-0072515PI：Anneke Bart和Kevin Scanell摘要提出者感兴趣的是由Bianci群产生的双曲三维流形的几何和拓扑性质，更广泛地说，是对算术定义的三维流形的性质感兴趣。作者打算统一这一领域的两个主要研究主题，即具有非平凡系数的尖顶上同调问题和浸入全测地曲面的计数和描述。作者打算探索尖顶上同调中的非零化结果与全测地曲面上的广义弯曲变形之间的可能联系。比安奇群是特别有趣的数学对象，因为它们位于纯数学的几个不同领域的十字路口：数论、无限群论、几何学和拓扑学。一百多年来，人们一直从这些不同的角度对它们进行研究。某些三维空间以一种自然的方式从比安奇群中产生；这些空间配备了一种“双曲几何”。简而言之，这意味着与欧几里德几何不同，成对的直线通常彼此发散。具有双曲几何的三维空间是一般三维空间研究的中心，因为在某种意义上，它们代表了“一般”的例子。双曲空间是最难理解的空间之一，因此，提出者的目的是阐明上述重要例子的几何。

Proposal: DMS-0072515PI: Anneke Bart and Kevin ScannellAbstract The proposers are interested in geometric and topological properties of hyperbolic 3-manifolds arising from the Bianchi groups, and more generally in the properties of arithmetically-defined 3-manifolds. The proposers intend to unify two of the main themes of research in this area, namely, the cuspidal cohomology problem with non-trivial coefficients, and the enumeration and description of immersed totally geodesic surfaces. The proposers intend to explore possible connections between non-vanishing results in cuspidal cohomology and generalized bending deformations supported on totally geodesic surfaces. The Bianchi groups are particularly interesting mathematical objects as they lie at the crossroads of several distinct area of pure mathematics: number theory, infinite group theory, geometry, and topology. They have been studies from these diverse points of view for more than one hundred years. Certain three-dimensional spaces arise in a natural way from the Bianchi groups; these spaces come equipped with a "hyperbolic geometry". This, in a nutshell, means that pairs of straight lines generically diverge from one another unlike Euclidean geometry. Three-dimensional spaces with a hyperbolic geometry are central to the study of general three-dimensional spaces as they represent, in a certain sense, the "generic" examples. Hyperbolic spaces are among the most difficult to understand, and therefore the proposers aim to elucidate the geometry of these important examples mentioned above.