Properties T, Tau and Kazhdan constants

该项目的主要内容是研究Cayley图的扩张性质，了解准有限群和离散群的各种表示理论性质。更具体地说，PI将研究自由群和泛格的自同构群的表示理论；这可能有助于更好地理解计算群论中广泛使用的算法，如乘积替换算法。此外，这个项目的发现可能通过瑟斯顿猜想和虚拟哈肯猜想与更遥远的主题如三维流形有关。扩展器是高度连通的有限图，在组合学和理论计算机科学中有许多应用。在形式上，如果一个图不能通过去除少量顶点和邻接边而被分成两大块，那么它就是一个扩展器。标准的计数论证表明，随机选择的图是一个好的扩展器。不幸的是，这个论点不允许显式地构造扩展器Marguis使用Kazhdan性质T构造了第一个显式扩展器图。这个项目的目的是通过估计相关的Kazdan常数来了解这些扩展器有多好。除了既有深度又有实际用途的目标，该项目有基本的方面，可以在本科生水平上提出，从而服务于教育目的。

The main aspect of the project is to study the expansion properties ofCayley graphs and to understand various representationtheoretic properties of pro-finite and discrete groups.More specifically, the PI will study the representation theoryof the automorphism groups of free groups and the universal lattices;and will use the results to construct new examples of expander graphs.This may lead to better understanding widely used algorithms incomputational group theory like the product replacement algorithm.Furthermore, findings from this project may have connectionsto more distant topics like three-dimensional manifoldsvia the Thurston Conjecture and the Virtual Haken Conjecture.Expanders are highly connected finite graphs which havemany applications in combinatorics and theoretical computer science.Informally, a graph is an expander if it cannot beseparated into two large pieces by removing asmall number of vertices and the adjoining edges.A standard counting argument shows thata randomly chosen graph is a good expander.Unfortunately this argument does not allow for theexplicit construction of expanders, required for many applications.The first explicit expander graphs were constructed byMargulis using Kazhdan property T. The aim of this project is tounderstand how good these expanders are by estimating the relatedKazdan constants. As well as objectives which would be both deepand of practical use, the project has elementary aspectswhich can be presented at an undergraduate leveland thus serve educational purposes.