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案（公法111-5）资助的。这个项目将研究由生成器和关系定义的几类群的结构，包括gold - shafarevich群和Kac-Moody群。首席研究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，包括组合群理论、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，在三流形拓扑和图论领域。群在数学中扮演着重要的角色，它描述了各种物体的对称性，比如几何图形或数字系统。组通常可以通过生成器和关系来表示，这些生成器和关系提供了一种定义组的简单方法，但通常无法深入了解其结构。该项目将开发新的工具，可以帮助更好地理解基于生成器和关系的组的表示，反过来，可以产生关于组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.