Boundaries and Nonpositive Curvature

一个物体的对称集合有一个数学结构，称为群。群论在物理世界和理想的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。如果一个几何空间不以任何重要的方式“圆”，那么它是非正弯曲的。也就是说，它要么是平坦的（如欧几里得平面），要么是鞍形的，要么是两者的某种组合。给定一个无限的几何物体，我们可以在无限处附加一个边界，让我们从外到内观察它。本项目主要研究非正弯曲空间中的边界理论。研究结果有望扩大这一理论可触及的群体类别，并有助于界定其局限性。该奖项包括对参与该项目的研究生的支持。该项目主要研究双曲空间和CAT(0)立方体复合体及其相关边界的乘积的保结构自同构。研究者和合作者将：(1)介绍和研究双曲空间积上的不可约非圆柱形作用，并检验非圆柱形双曲群已知性质的持久性；(2)建立了CAT(0)立方体复合体的三个关键自然边界（Roller边界、simplicial边界和Tits边界）之间的自同构-等变关系；(3)检验CAT(0)立方复上随机游走的收敛性，具体建立中心极限定理。该奖项反映了美国国家科学基金会的法定使命，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。

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.