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

这个项目主要涉及Golod-Shafarevich组，这是一类接受有少量关系的演示的组。Golod-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--的工具而引入的，并继续在数学的各个领域发挥重要作用，包括几何群论、代数数论和三维流形拓扑学。首席调查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)以及它们表示理论的某些方面(主要是性质(T)和(Tau))。我们将特别关注起源于数论的Golod-Shafarevich群，例如具有受限分支的数域的Prop扩张的Galois群，以及有限域上的Kac-Moody Prop群。首席调查员还将继续他的工作，用指定的大性质来构造具有性质(T)的新群。群通过描述各种对象的对称性，如几何图形或数字系统，在数学中起着基本的作用。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。这个项目将开发新的工具，帮助更好地理解基于生成器和关系表示的群，这反过来可以产生关于群描述其对称性的对象的新信息。该项目的发现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。

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.