Boundaries and Nonpositive Curvature

边界和非正曲率

• 批准号: 2005640
• 负责人: Talia Fernos
• 金额: $ 28.31万
• 依托单位: University of North Carolina Greensboro
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005640&HistoricalAwards=false
• 关键词: Boundaries; Nonpositive; Curvature

The collection of symmetries of an object has a mathematical structure known as a group. Group theory has enjoyed a wide range of applications both in the physical world and the ideal mathematical universe. A particularly fruitful direction in the study of groups has been to consider symmetries of geometric spaces of nonpositive curvature and their associated boundaries. A geometric space is nonpositively curved if it is not "rounded" in any significant way. That is, it is either flat (such as the Euclidean plane), saddle-shaped, or some combination of the two. Given an infinite geometric object, one can attach a boundary at infinity in a way that allows us to view it from the outside in. This project centers on investigation of boundary theory in the context of nonpositively curved spaces. The results are expected to both expand the class of groups that are accessible through this theory and help define its limitations. The award includes support for graduate students involved in the project.The project focuses on structure-preserving automorphisms of products of hyperbolic spaces and CAT(0) cube complexes together with their associated boundaries. The investigator and collaborators will: (1) introduce and study irreducibly-acylindrical actions on products of hyperbolic spaces and examine persistence of known properties of acylindrically hyperbolic groups; (2) establish automorphism-equivariant relationships among three of the key natural boundaries associated to CAT(0) cube complexes: the Roller, simplicial, and Tits boundaries; and (3) examine the quality of convergence of a random walk on a CAT(0) cube complex, specifically to establish a Central Limit Theorem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

一个物体的对称性的集合有一个数学结构，称为群。群论在物理世界和理想数学世界中都有广泛的应用。在群的研究中，一个特别富有成果的方向是考虑非正曲率几何空间及其相关边界的对称性。一个几何空间是非正曲的，如果它不是以任何显著的方式“圆化”的。也就是说，它要么是平面（如欧几里得平面），要么是马鞍形，或者是两者的某种组合。给定一个无限大的几何物体，我们可以在无穷远处附加一个边界，使我们能够从外到内观察它。这个项目的中心是在非正弯曲空间的背景下研究边界理论。预计结果将扩大通过该理论可访问的群体类别，并帮助定义其局限性。 该奖项包括对参与该项目的研究生的支持。该项目专注于双曲空间和CAT（0）立方体复形及其相关边界的乘积的结构保持自同构。研究者和合作者将：（1）介绍和研究双曲空间乘积上的不可约-非圆柱线性作用，并研究非圆柱双曲群已知性质的持久性;（2）建立与CAT（0）立方体复形相关的三个关键自然边界之间的自同构-等变关系：Roller边界，单纯边界和Tits边界;以及（3）检验CAT（0）立方体复合体上随机游走的收敛质量，特别是建立中心极限定理。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。