Finite covers of hyperbolic 3-manifolds

出于对拓扑学、几何学、算术和群论的考虑，该提案打算研究真实的、复和四元数双曲流形。这将涉及离散群的研究，它们与数论的联系，以及扩展图。我们还将探讨这些主题之间的各种相互联系，以期更好地理解双曲三维流形的拓扑结构。三维流形在局部上就像我们生活的空间一样，理解这些物体一直是过去30年来研究的中心主题之一。 这些对象的重要性远远超出了它们的内在兴趣，因为它们的研究与数学物理，数学生物学和计算机科学有关。这个提议的目的之一是通过“扩展图”族来探索其中的一些联系。这些图在计算机科学中很有名，因为它们在构建高效网络中很重要。值得注意的是，这与提案的主要研究“三维流形的有限覆盖”有关。提案中的许多问题本质上是跨学科的，因此进展将产生更广泛的影响。此外，事实证明，这些都是培养研究生的肥沃土壤。

Motivated by considerations in topology, geometry, arithmetic and group theory the proposal intends to study real, complex and quaternionic hyperbolic manifolds. This will involve the study of discrete groups, their connections with number theory, and expanding graphs. We will also explore various interconnections between these topics with a view to better understanding the topology of hyperbolic 3-manifolds.Three dimensional manifolds are locally like the space we live in and understanding these objects have been one of the central themes of research in the last 30 years. The importance of these objects extends far beyond their intrinsic interest, since their study connects to mathematical physics, mathematical biology and computer science. One of the aims of this proposal is to explore some of these connections via families of "expanding graphs". These graphs are well-known in computer science because of their importance in building efficient networks. Remarkably, this is connected to the main study of the proposal, "finite covers of 3-manifolds".By their nature, many of the problems in the proposal are cross-disciplinary, and hence progress will have a broader impact. In addition, these have proved to be fertile grounds for the training of graduate students.