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

该奖项是根据2009年《美国复苏和再投资法》(公法111-5)资助的。该项目将调查由发电机和关系定义的几类群的结构，包括Golod-Shafarevich群和Kac-Moody群。首席调查员将研究这些群的各种渐近不变量(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)。项目中使用的方法涉及不同领域的技术，包括组合群论、Prop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，在三维流形拓扑和图论领域。群通过描述各种对象的对称性，如几何图形或数字系统，在数学中扮演着基本的角色。组通常可以由生成器和关系来表示，它们提供了一种简单的方法来定义组，但通常不能提供对其结构的深入了解。该项目将开发新的工具，基于生成器和关系的表示来帮助更好地理解一个群，这反过来可以产生关于该群描述其对称性的对象的新信息。

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.