Properties T, Tau and pro-finite groups

PI计划继续研究群表示理论中的各种主题，特别强调与哈萨克性质T，性质Tau和亲有限群有关的问题。项目的主要目的是研究Cayley图的展开性质，并了解亲有限群和离散群的各种表示理论性质。该项目还涉及研究几何群论的核心对象，如自由群的自同构群。PI将使用代数、组合、几何和概率工具将提出的问题简化为组合学和随机游走理论中的问题，并将结果应用于图论中的重要开放问题。作为项目的副产品，这些领域的问题可能会得到解答。展开器是高度连通的稀疏图，广泛应用于计算机科学，从并行计算到密码学等领域。马古利斯在Pinsker观察到随机稀疏图是展开器之后，给出了展开器的第一个明确构造。这种构造在八十年代由Margulis、Lubotzky、Phillips和Sarnak改进，他们利用深度数论结果构造了拉马努金图（最优展开器）。在随后的几年里，膨胀器的应用范围和深度都有所增加——在过去的十年里，出现了一些全新的、意想不到的发展方向，该领域经历了爆炸式的增长。首席研究员计划开展以扩展器为中心的项目，涉及算术、群论和组合学之间的互利互动。

The PI plans to continue working on various topics in the representation theory of groups, with special emphasis on problems related to the Kazhdan property T, property Tau and pro-finite groups. The main aims of the project are to study the expansion properties of Cayley graphs and to understand various representation theoretic properties of pro-finite and discrete groups. The project also involves studying objects central to the geometric group theory, like automorphism groups of free groups. The PI will use algebraic, combinatorial, geometric and probabilistic tools to reduce the proposed problems to questions in combinatorics and theory of random walks and apply the results to important open problems in graph theory. As a byproduct of the project, questions in these areas are likely be answered.Expanders are highly-connected sparse graphs widely used in computer science, in areas ranging from parallel computation to cryptography. Margulis gave the first explicit construction of expanders, following Pinsker's observation that random sparse graphs are expanders. This construction was improved in the eighties by Margulis, Lubotzky, Phillips, and Sarnak who constructed Ramanujan graphs (optimal expanders) using deep number-theoretic results. In the ensuing years the scope and depth of applications of expanders has increased -- over the past decade several completely new and unexpected lines of development have emerged and the field has undergone explosive growth. The principal investigator plans to pursue projects centered on expanders and involving mutually beneficial interactions between arithmetic, group theory and combinatorics.