LEAPS-MPS: Cubulation and Property (T) in Random Groups

群是描述几何对象的对称性的数学结构。它们广泛应用于科学领域：研究晶体结构、分子对称性、粒子物理标准模型、公钥加密系统等。这个项目旨在理解群本身；数学家可以从一个抽象的群开始，然后研究它可以描述其对称性的空间(S)，而不是像化学家或物理学家那样从一个物体开始计算它的对称群。一个自然的问题是：“典型”群体满足什么性质？本项目将通过引入和研究一种新的随机群体模型，重点了解群体的这些“典型”性质。该项目还将支持本科生项目和PI正在进行的领导和组织努力，以促进本科生女性和数学领域其他代表性不足群体的包容性和联系，例如学生群体数学/统计中的性别少数群体(GEMM)、导师/被辅导者计划、读书俱乐部和卡尔顿学院的其他社区建设活动。格罗莫夫随机群体是几何群论中丰富的例子来源。本项目的第一个分支引入并探索了群的自由积的随机商的一个新模型的性质，该模型与Gromov模型有组合关系。在与爱因斯坦、克里希纳、Ng和Steenbock的合作下，少年派将在这种情况下研究(相对)幼崽。PI还将探索这些组的属性(T)。在另一个方向上，PI将与本科生一起工作，以加深我们对Gromov随机组中的孵化的理解；特别是他们将增加CAT(0)立方体复合体上的协紧作用的已知界限。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Groups are mathematical structures that describe the symmetry of a geometric object. They are used throughout the sciences: in the study of crystalline structures, molecular symmetry, the Standard Model of particle physics, public-key encryption systems, and more. This project aims to understand groups in their own right; instead of starting with an object and calculating its symmetry group as a chemist or physicist might, a mathematician can start with an abstract group and then study the space(s) whose symmetries it could describe. A natural question to ask is: What properties does a `typical’ group satisfy? This project will focus on understanding these `typical’ properties of groups by introducing and investigating a new model of random groups. The project will also support undergraduate projects and the PI’s ongoing leadership and organizational efforts to promote inclusivity and connections for undergraduate women and other under-represented groups within mathematics, such as the student group Gender Minorities in Math/Stats (GeMMs), mentor/mentee programs, book clubs and other community building activities at Carlton College.Gromov random groups have been a rich source of examples in geometric group theory. The first branch of this project introduces and explores properties of a new model of random quotients of free products of groups, which is combinatorially related to Gromov’s model. In collaboration with Einstein, Krishna, Ng, and Steenbock the PI will investigate (relative) cubulation in this setting. The PI will also explore Property (T) for these groups. In another direction, the PI will work with undergraduates to further our understanding of cubulation in Gromov random groups; in particular they will increase the known bound for cocompact actions on CAT(0) cube complexes.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.