Kazhdan's property (T), Golod-Shafarevich groups and Kac-Moody groups

Kazhdan 性质 (T)、Golod-Shafarevich 群和 Kac-Moody 群

• 批准号: 0901703
• 负责人: Mikhail Ershov
• 金额: $ 11.43万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901703&HistoricalAwards=false
• 关键词: Kazhdan; property; Golod; Shafarevich; groups

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project will investigate the structure of several classes of groups defined by generators and relations, including Golod-Shafarevich groups and Kac-Moody groups. The Principal Investigator will study various asymptotic invariants for these groups (e.g., subgroup growth, rank gradient) and certain aspects of their representation theory with the emphasis on Kazhdan's property (T). The methods used in the project involve techniques from different areas, including combinatorial group theory, pro-p groups, arithmetic groups and Lie algebras. The findings of this project will likely have applications beyond group theory, e.g., in the areas of three-manifold topology and graph theory.Groups play a fundamental role in mathematics by describing symmetries of various objects like geometric figures or number systems. A group can often be presented by generators and relations which provide a simple way to define the group but usually offer little insight into its structure. The project will develop new tools that can help better understand a group based on its presentation by generators and relations which, in turn, can yield new information about the object whose symmetries the group describes.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目将调查由生成器和关系定义的几类群体的结构，包括Golod-Shafarevich群体和Kac-Moody群体。主要研究者将研究这些组的各种渐近不变量（例如，子群增长，秩梯度）及其表示理论的某些方面，重点是Kazhdan的性质（T）。该项目中使用的方法涉及不同领域的技术，包括组合群论，pro-p群，算术群和李代数。这个项目的发现可能会有超越群论的应用，例如，在三流形拓扑学和图论领域，群通过描述各种对象的对称性（如几何图形或数字系统）在数学中起着基础性的作用。一个群通常可以用生成元和关系来表示，它们提供了一种简单的定义群的方法，但通常对它的结构没有什么洞察力。该项目将开发新的工具，可以帮助更好地了解一个组的基础上，其表示的发电机和关系，这反过来又可以产生新的信息，对象的对称性组描述。