Subgroups in Artin Groups and Lattices in Products of Trees

群是编码对象对称性的代数结构。 它可以被抽象地定义为字母串的集合，其中某些方程描述了哪两个字符串对应于相同的对称性。 这些字母称为生成器，方程称为关系，它们一起形成所谓的群表示。 几何群论研究对象的几何形状与其对称性群的性质之间的联系。群的一个例子是整数集，它可以被视为一条线的对称性，其中正数将线上的点向右移动，负数将线向左移动。 群的子群是对称性的较小集合，在合成下封闭。在整数群中，子群的一个例子是移动偶数距离的对称性的集合。了解子群结构对于研究整个群至关重要。 该项目将解决有关两个群族中具有规定属性的子群的问题：Artin 群和树积中的格子。这两个家族中的群体都可以通过简单的演示来描述，但有关它们的许多问题仍未得到解答。 该项目还将通过指导和推广来促进女性参与数学。该项目的第一个目标是检查 Artin 小组在 CAT(0) 立方体复合体上的行动。 该项目将研究哪些 Artin 群的每个群元素都被某个余维 1 子群分隔，以及对于其中哪些 Artin 群，这会导致对 CAT(0) 立方体复合体产生适当的作用。 CAT(0) 立方体复合体理论，特别是特殊立方体复合体理论，一直是理解群的富有成效的工具。 证明 Artin 群对 CAT(0) 立方体复合体的正确作用将回答有关 Artin 群的许多悬而未决的问题；例如，它可以提供应用题的解决方案。 PI 还将在该项目中继续研究 Artin 群的剩余有限性。 在第二个项目中，PI 将研究树及其子群结构乘积中的协紧格子。 特别是，PI 将确定所有这些群体是否不一致。 显示树的乘积中的所有格子都是不相干的，这表明相干性是准等距不变量。 该项目还将确定树乘积中晶格中的任何两个无限阶元素在提高到高幂时是否可交换或生成自由子群。 该项目还包括对本科生和研究生的培训和指导，重点是扩大女性对数学的参与。 PI 还计划与 Jankiewicz Studio 合作开展一个教育项目，Jankiewicz Studio 是一家专门从事设计、艺术、科学和技术交叉领域的教育和文化项目的设计公司。该奖项反映了 NSF 的法定使命，并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。

A group is an algebraic structure encoding symmetries of an object. It can be defined abstractly, as a collection of strings of letters, where certain equations describe which two strings correspond to the same symmetry. Such letters are called generators, and the equations are called relations, and together they form what is called a group presentation. Geometric group theory studies the connection between the geometry of the object, and the properties of the group of its symmetries. An example of a group is the set of integers, which can be viewed as symmetries of a line, where a positive number moves points on the line to the right, and a negative number to the left. A subgroup of a group is a smaller collection of symmetries, closed under composition. In the group of integers, an example of a subgroup is the collection of the symmetries moving by an even distance. Understanding the subgroup structure is essential in studying the whole group. This project will address questions about subgroups with prescribed properties in two families of groups: Artin groups and lattices in products of trees. Groups in both of those families can be described by simple looking presentations, but many questions about them remain unanswered. The project will also promote the participation of women in mathematics via mentoring and outreach.The first goal of this project is to examine the actions of Artin groups on CAT(0) cube complexes. This project will investigate for which Artin groups is every group element is separated by some codimension-1 subgroup, and for which of them this leads to proper actions on CAT(0) cube complexes. The theory of CAT(0) cube complexes, and special cube complexes in particular, has been a fruitful tool in understanding groups. Proving that Artin groups act properly on CAT(0) cube complexes would answer many outstanding questions about Artin groups; for example, it could provide a solution to the word problem. The PI will also continue her work on the residual finiteness of Artin groups in this project. In the second project, the PI will study cocompact lattices in products of trees and their subgroup structures. In particular, the PI will determine if all such groups are incoherent. Showing that all lattices in a product of trees are incoherent would be an indication that coherence is a quasi-isometry invariant. The project will also determine if any two infinite order elements in a lattice in a product of trees either commute or generate a free subgroup, when raised to high powers. The project also includes training and mentoring of undergraduate and graduate students with an emphasis on broadening participation of women in mathematics. The PI is also planning a collaborative educational project with Jankiewicz Studio, a design firm specializing in educational and cultural projects at the intersection of design, art, science and technology.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.