Hyperbolic geometry in group theory

群论中的双曲几何

• 批准号: 1308961
• 负责人: Denis Osin
• 金额: $ 20.65万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308961&HistoricalAwards=false
• 关键词: Hyperbolic; geometry; group; theory

The proposed project consists of two parts. The central theme of the first part is a conjecture, which can roughly be stated as follows: the class of groups admitting a non-elementary and "reasonably proper" action on a hyperbolic space is the same for any reasonable interpretation of the phrase "reasonably proper"; moreover, it coincides with the class of groups with non-degenerate hyperbolically embedded subgroups introduced by Dahmani, Guirardel, and the PI. The second part of the proposal is devoted to a more detailed study of groups with hyperbolically embedded subgroups. In particular, the PI plans to study questions related to rigidity, small cancellation theory, and applications of the general theory to groups acting on trees, one-relator groups, graph products, and fundamental groups of 3-manifolds.The general goal of the project is to better understand the scope and limitations of geometric methods based on negative curvature in group theory. As a first step towards this goal, the PI proposes to analyze various negative curvature phenomena of algebraic, geometric, dynamical, and analytic nature occurring in different branches of group theory. This analysis is expected to lead to new general results with possible applications in group theory, 3-dimensional geometry, the theory of operator algebras, and other branches of mathematics.

拟建项目由两部分组成。第一部分的中心主题是一个猜想，它大致可以表述为：在双曲空间上承认非初等作用和“合理适当”作用的群类对于“合理适当”一词的任何合理解释都是相同的；此外，它与Dahmani， Guirardel和PI引入的具有非简并双曲嵌入子群的群类相一致。该建议的第二部分致力于对具有双曲线嵌入子群的群体进行更详细的研究。特别是，PI计划研究刚性、小抵消理论、一般理论在树群、单相关群、图积、3流形基本群上的应用等相关问题。该项目的总体目标是更好地理解群论中基于负曲率的几何方法的范围和局限性。作为实现这一目标的第一步，PI建议分析在群论的不同分支中出现的各种代数、几何、动态和解析性质的负曲率现象。这一分析有望在群论、三维几何、算子代数理论和其他数学分支中产生新的一般结果。