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

戈洛德-沙法列维奇集团和卡兹丹的财产 (T)

• 批准号: 1201452
• 负责人: Mikhail Ershov
• 金额: $ 16.13万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201452&HistoricalAwards=false
• 关键词: Golod; Shafarevich; groups; Kazhdan; property

This project is primarily concerned with Golod-Shafarevich groups, a class of groups which admit a presentation with a small set of relations. Golod-Shafarevich groups have been originally introduced as a tool for solving two outstanding problems -- the class field tower problem and the general Burnside problem -- and continue to play an important role in various areas of mathematics, including geometric group theory, algebraic number theory and three-manifold topology. The Principal Investigator will continue studying the subgroup structure of these groups, their asymptotic invariants (e.g., rank gradient and subgroup growth) and certain aspects of their representation theory (primarily properties (T) and (tau)). Special attention will be devoted to Golod-Shafarevich groups of number-theoretic origin, e.g. Galois groups of pro-p extensions of number fields with restricted ramification, and Kac-Moody pro-p groups over finite fields. The Principal Investigator will also continue his work on constructing new groups with property (T) with prescribed largeness properties.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. This 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. The findings of the project will likely have applications beyond group theory, e.g., in the areas of three-manifold topology, number theory and graph theory.

该项目主要关注gold - shafarevich群，这是一类允许具有少量关系集的表示的群。gold - shafarevich群最初是作为解决两个突出问题（类场塔问题和一般Burnside问题）的工具引入的，并继续在数学的各个领域发挥重要作用，包括几何群论，代数数论和三流形拓扑。首席研究员将继续研究这些群的子群结构，它们的渐近不变量（例如，秩梯度和子群增长）和它们的表示理论的某些方面(主要是性质（T)和（tau））。特别注意将致力于数论起源的gold - shafarevich群，例如具有限制分支的数域的pro-p扩展的Galois群，以及有限域上的Kac-Moody pro-p群。首席研究员还将继续他的工作，以构建具有规定的大属性的具有属性(T)的新群。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。这个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，反过来，可以产生关于组所描述的对称对象的新信息。该项目的研究结果可能会有超越群论的应用，例如，在三流形拓扑，数论和图论领域。