Subgroups in Artin Groups and Lattices in Products of Trees

群是一种编码对象对称性的代数结构。它可以抽象地定义为字母串的集合，其中某些方程描述哪两个字符串对应于相同的对称性。这样的字母被称为生成器，这些方程被称为关系，它们一起构成了所谓的群表示。几何群论研究的是物体的几何形状与其对称群的性质之间的联系。群的一个例子是一组整数，它可以被看作是一条直线的对称性，其中正数使直线上的点向右移动，负数使直线上的点向左移动。群的子群是一个较小的对称集合，在组合下封闭。在整数群中，子群的一个例子是每偶数距离移动的对称的集合。了解子群体的结构对研究整个群体至关重要。这个项目将解决关于两个群族中具有规定性质的子群的问题：树的乘积中的Artin群和格。这两个家族的群体都可以通过简单的演示来描述，但关于它们的许多问题仍未得到解答。该项目还将通过指导和外联促进妇女参与数学。该项目的第一个目标是研究CAT(0)立方配合物上的Artin基团的作用。该项目将研究哪些Artin基团是每个基团元素被一些codimension-1子群分开，以及其中哪些导致对CAT(0)立方体配合物的适当作用。CAT(0)立方配合物的理论，特别是特殊的立方配合物，已经成为理解群的有效工具。证明Artin基团在CAT(0)立方配合物上的适当作用将回答许多关于Artin基团的悬而未决的问题；例如，它可以为单词问题提供一个解决方案。在这个项目中，PI还将继续她对Artin组的剩余有限性的研究。在第二个项目中，PI将研究树积中的紧致格及其子群结构。特别是，PI将确定所有这些组是否都是不连贯的。证明树的乘积中的所有格都是不相干的，将表明相干是一个准等距不变量。该项目还将确定在树的乘积中的晶格中的任意两个无限阶元素，当提高到高幂时，是否可以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，重点是扩大妇女对数学的参与。PI还计划与Jankiewicz工作室合作一个教育项目，Jankiewicz工作室是一家设计公司，专门从事设计、艺术、科学和技术交叉的教育和文化项目。该奖项反映了美国国家科学基金会的法定使命，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。

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.