Subgroups in Artin Groups and Lattices in Products of Trees

一个群是一个代数结构编码对象的对称性。 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。 这样的字母称为生成元，方程称为关系，它们一起构成了所谓的群表示。 几何群论研究对象的几何形状与其对称群的性质之间的联系。群的一个例子是整数的集合，它可以被看作是一条线的对称性，其中正数将线上的点向右移动，负数将线上的点向左移动。 一个群的子群是一个较小的对称集合，在复合下是封闭的。在整数群中，子群的一个例子是移动偶数距离的对称的集合。了解子群结构是研究整个群的关键。 这个项目将解决两个群族中具有规定性质的子群的问题：Artin群和树的乘积中的格。这两个家族中的群体都可以通过简单的介绍来描述，但关于他们的许多问题仍然没有答案。 该项目还将通过指导和外联促进妇女参与数学，其第一个目标是审查阿廷小组在CAT（0）立方体复合体方面的行动。 这个项目将研究哪些Artin群是每个群元素都被某个余维1子群分开的，以及它们中的哪一个导致CAT（0）立方体复形的正确操作。 CAT（0）立方复形理论，特别是特殊立方复形理论，已经成为理解群的有效工具。 证明阿廷群在CAT（0）立方体复形上的正确作用将回答许多关于阿廷群的悬而未决的问题;例如，它可以提供一个解决单词问题的方法。 PI也将继续她的工作，在这个项目中的剩余有限的Artin群。 在第二个项目中，PI将研究树的乘积及其子群结构中的余紧格。 特别是，PI将确定是否所有这些组都是不相干的。 如果一个树的乘积中的所有格都是不相干的，这就表明相干性是一个准等距不变量。 该项目还将确定，如果任何两个无限阶元素在一个产品的树格要么交换或生成一个自由子群，当提高到高的权力。 该项目还包括对本科生和研究生进行培训和辅导，重点是扩大妇女对数学的参与。 PI还计划与Jankiewicz Studio合作开展一个教育项目。Jankiewicz Studio是一家设计公司，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。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.