Hyperbolic Manifolds: Geometry, Topology, Group Theory and Arithmetic

出于对拓扑学、几何学、算术和群论的考虑，该建议旨在研究真实的、复和四元数双曲流形。这将涉及离散群的研究，以及它们与其他研究领域的联系，例如数论和扩展图理论。我们还将探讨这些几何对象之间的各种互连。这在过去是富有成效的，因为以前的工作已经表明了对高维双曲流形的理解如何影响三维流形的拓扑。该提案旨在进一步理解现代几何和拓扑中的基本对象，即允许某种类型的双曲度量的流形。 提案中的许多问题都是跨学科的，并且可以使用广泛的数学技术进行研究;例如，从数论，几何学，拓扑学和群论。由于其性质，提案中问题的进展和解决方案将产生更广泛的影响。所提出的问题在数论（类数）、宇宙学（及其与四维双曲流形拓扑的关系）和计算机科学（扩展图）等不同领域都有应用。所有这些都有可能成为研究生教育的沃土。

Motivated by considerations in topology, geometry, arithmetic andgroup theory the proposal intends to study real, complex andquaternionic hyperbolic manifolds. This will involve the study ofdiscrete groups, and their connections with other research active areas; forexample number theory, and the theory of expandinggraphs. We will also explore various interconnections between thesegeometric objects. This has been fruitful in the past, since previous work has shown how an understanding of higher dimensionalhyperbolic manifolds can impact the topology of 3-dimensional manifolds.The proposal seeks to further understand basic objects in moderngeometry and topology, namely manifolds that admit a hyperbolic metricof some type. Many of the problems in the proposal arecross-disciplinary, and are amenable to study using a broad spectrumof mathematical technqiues; for example from number theory, geometry,topology and group theory.By their nature, progress and solutions to problems in the proposalwill have a broader impact. The problems suggested have applicationsin fields as diverse as number theory (class numbers) and cosmology(and its relation to the topology of 4-dimensional hyperbolicmanifolds) and computer science (expanding graphs). All of these havethe potential to be fertile grounds for the education of graduatestudents.