Subgroups in Artin Groups and Lattices in Products of Trees

群是对对象的对称性进行编码的代数结构。它可以抽象地定义为字母串的集合，其中某些方程描述了哪两个串对应于相同的对称性。这样的字母被称为生成器，方程被称为关系，它们一起形成了所谓的群表示。几何群论研究对象的几何和它的对称性群的性质之间的联系。组的一个例子是整数集，它可以被视为直线的对称，其中正数将直线上的点向右移动，负数向左移动。群的子群是一个较小的对称集合，在合成下闭合。在整数群中，子群的一个例子是以偶数距离移动的对称的集合。了解子群的结构对于研究整个群是至关重要的。这个项目将讨论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。这两个家庭中的群体都可以用简单的陈述来描述，但关于他们的许多问题仍然没有答案。该项目还将通过辅导和外展促进妇女参与数学。该项目的第一个目标是检查Artin小组对CAT(0)立方体复合体的行动。这个项目将调查每个群元素都被某个余维-1子群分开的Artin基团，以及对于它们中的哪个基团，这会导致对CAT(0)立方体络合物的适当操作。CAT(0)立方体复合体理论，特别是特殊立方体复合体的理论，在理解群方面是一个卓有成效的工具。证明Artin群在CAT(0)立方体复形上的作用是正确的，这将回答许多关于Artin群的悬而未决的问题；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还将继续她关于Artin群的剩余有限性的工作。在第二个项目中，PI将研究树的乘积中的余紧格及其子群结构。具体地说，PI将确定是否所有这些组都是不连贯的。证明树的乘积中的所有格都是非相干的，这将表明一致性是准等距不变量。该项目还将确定树的乘积中的格中的任何两个无限阶元，当提升到高次方时，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合作一个教育项目，Jankiewicz Studio是一家专门从事设计、艺术、科学和技术相交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优势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.