The geometry, rigidity and combinatorics of spaces and groups with non-positive curvature feature

群是数学中的基本抽象符号系统，它起源于数学、物理、化学和计算机科学的不同分支。例如，群出现在几何物体的形状、晶体和准晶体、多项式根的结构、密码学、算法设计等的研究中。有限群的研究，即具有多个元素的群，已经达到相当成熟的阶段，积累到有限单群的完整分类。然而，大多数无限群是相当神秘和难以理解的。在20世纪80年代，Gromov提出了一种几何方法来研究群论。 一个想法是将神秘的群实现为一些具有有趣曲率性质的几何对象的对称性的集合，使我们能够从几何的角度研究群。这已经发展成为一个非常活跃的领域，称为几何群论。这个提议旨在研究几何群论的前沿问题，并寻求应用于拓扑学中一些长期存在的问题。该提案还旨在为培养俄亥俄州州立大学在该领域工作的研究生和博士后提供资源，重点是支持在该领域工作的代表性不足的早期职业生涯阶段的数学家。该提案从几何群论的观点出发，结合遍历理论，度量几何、度量图论和组合群论。该项目有两个更具体的研究目标。首先是取得进展的一个重大猜想阿廷组，使用一种新的战略动机思想度量图论。二是了解离散群刚性的基本形式，即拟等距刚性和测度等价刚性。该项目强调了这些形式的刚性和奇异度量空间和群的曲率性质之间的密切联系。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。

Groups are fundamental abstract symbolic systems in mathematics arising from different branches of mathematics, physics, chemistry and computer science. For example, groups appear in the study of shapes of geometric objects, crystals and quasi-crystals, structure of roots of polynomials, cryptography, algorithm design etc. The study of finite groups, i.e. groups with finitely many elements, has reached a fairly mature stage, accumulating to a complete classification of finite simple groups. However, most infinite groups are fairly mysterious and hard to understand. In the 1980s, Gromov proposed a geometric approaches to group theory. One idea was to realize the mysterious group as a collection of symmetries of some geometric objects with interesting curvature properties, allowing us to study groups from the viewpoint of geometry. This has evolved into a very active field called geometric group theory. This proposal aims to study problems in the frontier of geometric group theory, and seeks applications to some long-standing problems in topology. The proposal also aims to provide resources for training graduate students and postdocs working in the area at the Ohio State University, with an emphasis on supporting under-represented early career stage mathematicians at OSU working in this filed.This proposal is concerned with rigidity and curvature properties of some infinite discrete groups from the viewpoint of geometric group theory, combined with ideas and techniques from ergodic theory, metric geometry, metric graph theory and combinatorial group theory. The project has two more specific research goals. The first is to make progress on a major conjecture on Artin groups, using a new strategy motivated from ideas in metric graph theory. The second is to understand fundamental forms of rigidity for discrete groups, namely quasi-isometric rigidity and measure equivalence rigidity. The project emphasizes the close connections between these forms of rigidity and the curvature properties of singular metric spaces and groups. Several classes of groups of fundamental importance are studied, including Artin groups, CAT(0) groups, graph products etc.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.