Discrete Groups, Expanding Graphs and Pro-P Methods

该项目从一个统一的角度看待几个看似不相关的问题。讨论了有限群的Cayley图的展开问题、黎曼流形的特征值问题和pro-p群的特征值问题。出乎意料的是，一个领域的方法和结果给其他领域带来了新的启示。这项工作很有希望是跨学科的，如果成功的话，应该会对几个数学领域产生重大影响，无论是纯粹的还是应用的。在纯方面，它有望为三维流形上的瑟斯顿猜想提供一种全新的方法。在应用方面，它将带来对扩展图的更好理解，这些图是各种通信网络的基本构建块。

The project looks at several seemingly unrelated problems from a unified perspective. It deals with questions of expansion of Cayley graphs of finite groups, eigenvalues of Riemannian manifolds and pro-p groups. Somewhat unexpectedly, methods and results from one field shed a new light on the other areas. The work is promising to be very interdisciplinary and should have, if successful, a serious impact on several mathematical fields, pure and applied. On the pure side, it is expected to have a completely new approach toward the Thurston conjecture on three-dimensional manifolds. On the applied side, it will bring a better understanding of expanding graphs, graphs which serve as basic building blocks for various communication networks.