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

群是数学中最基本的抽象符号系统，起源于数学、物理、化学和计算机科学的不同分支。例如，群出现在几何对象的形状、晶体和准晶体的形状、多项式的根的结构、密码学、算法设计等研究中。有限群，即具有有限多个元素的群的研究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。20世纪80年代，格罗莫夫提出了群论的几何方法。一种想法是将这个神秘的群实现为具有有趣的曲率性质的一些几何对象的对称性的集合，从而允许我们从几何的角度研究群。这已经演变成一个非常活跃的领域，称为几何群论。这一建议旨在研究几何群论的前沿问题，并寻求在一些长期存在的拓扑学问题上的应用。该建议还旨在为俄亥俄州立大学在该领域工作的研究生和博士后提供培训资源，重点是支持俄亥俄州立大学在该领域工作的未被充分代表的早期职业阶段数学家。该建议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的思想和技术，研究了一些无限离散群的刚性和曲率性质。该项目有两个更具体的研究目标。第一个是在Artin群的一个主要猜想上取得进展，使用一种新的策略，该策略源于度量图论中的思想。二是了解离散群刚性的基本形式，即准等距刚性和度量等价刚性。该项目强调了这些形式的刚性与奇异度量空间和群的曲率性质之间的密切联系。这个奖项反映了NSF的法定使命，通过使用基金会的智力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.