Properties T, Tau and Kazhdan constants

该项目的主要内容是研究Cayley图的展开性质，并理解pro-finite和离散群的各种表示论性质。更具体地说，PI将研究自由群和泛格的自同构群的表示论;并将利用这些结果构造扩展图的新例子。这可能会导致更好地理解计算群论中广泛使用的算法例如乘积替换算法。此外，该项目的发现可能与更遥远的主题有关，例如通过Thurston猜想和Virtual Haken猜想的三维流形。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。非正式地，如果一个图不能通过去掉少量的顶点和相邻的边而分成两大块，则它是一个扩展图。一个标准的计数参数表明，一个随机选择的图是一个很好的扩展器。不幸的是，这个论点不允许显式构造扩展器，这是许多应用所需要的。第一个显式扩展器图是由Margulis使用Kazhdan性质T构造的。本项目的目的是通过估算相关的Kazdan常数来了解这些膨胀剂的性能。除了既有深度又有实际用途的目标外，该项目还有一些基本方面，可以在本科阶段提出，从而达到教育目的。

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.