Properties T, Tau and pro-finite groups

T、Tau 和亲有限群的性质

• 批准号: 0900932
• 负责人: Martin Kassabov
• 金额: $ 19.6万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900932&HistoricalAwards=false
• 关键词: Properties; Tau; pro; finite; groups

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.

PI 计划继续研究群表示论中的各种主题，特别强调与 Kazhdan 性质 T、性质 Tau 和亲有限群相关的问题。该项目的主要目的是研究凯莱图的展开性质，并理解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，例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。作为该项目的副产品，这些领域的问题可能会得到解答。扩展器是高度连接的稀疏图，广泛应用于计算机科学中，涉及从并行计算到密码学的各个领域。继 Pinsker 观察到随机稀疏图是扩展器之后，Margulis 给出了扩展器的第一个显式构造。八十年代，Margulis、Lubotzky、Phillips 和 Sarnak 改进了这种结构，他们利用深度数论结果构建了拉马努金图（最优展开器）。在接下来的几年中，膨胀机的应用范围和深度不断增加——在过去的十年中，出现了一些全新的、意想不到的发展路线，并且该领域经历了爆炸性增长。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相互作用。