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

群是描述几何对象对称性的数学结构。 它们被用于整个科学：在晶体结构，分子对称性，粒子物理学的标准模型，公钥加密系统的研究，等等。这个项目的目的是了解群体本身;而不是从一个对象开始，计算其对称群作为化学家或物理学家可能，数学家可以从一个抽象的群体，然后研究空间（S）的对称性，它可以描述。一个自然的问题是：一个“典型”群体满足什么性质？本项目将通过介绍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，以促进包容性和联系本科妇女和其他代表性不足的群体在数学，如学生团体性别少数群体在数学/统计（Gestival），导师/学员计划，在卡尔顿学院的读书俱乐部和其他社区建设活动。格罗莫夫随机群已经成为几何群论中例子的丰富来源。本项目的第一个分支介绍并研究了群的自由积的随机乘积的一个新模型的性质，它与Gromov模型有组合关系。PI将与爱因斯坦、克里希纳、Ng和Steenbock合作，研究这种情况下的（相对）cu。PI还将为这些组探索财产（T）。在另一个方向上，PI将与本科生合作，以进一步我们对Gromov随机群中的cuplex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。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.