Geometry of Subgroups

子群的几何

基本信息

批准号：
2005353
负责人：
Genevieve Walsh
金额：
$ 26.48万
依托单位：
Tufts University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005353&HistoricalAwards=false
关键词：
Geometry Subgroups

项目摘要

A group is an algebraic object that is the collection symmetries of a geometrical object. For instance, for a square the set of successive rotations by 90 degrees in the plane is a collection of symmetries and yields a group of size 4. One useful way of describing a group is via what is known as a presentation, with generators and relations between the generators completely describing the group. For example, the collection of all integers is an infinite group that is generated by a single element, namely the integer 1. If a group has a finite presentation (with a finite number of generators and a finite number of relations) then it is easier to understand, and easier to solve problems about that group. In addition, a group is called coherent if it enjoys the stronger property that every finitely generated subgroup is finitely presented. Many of the best understood groups are coherent: free groups, surface groups, and 3-manifold groups are all coherent. This project aims to understand when groups are coherent and to find geometric indicators of subgroups that are witnesses to incoherence. This project advances the field by investigating geometrically wild subgroups of seemingly well-behaved groups, which can be difficult to find. The project includes work in education, and the PI is helping to organize conferences and schools that enhance the training of the next generation. The PI plans to investigate the coherence of several interesting classes of groups, including groups that are expected to be coherent and groups which are expected to be incoherent. This project addresses fundamental and difficult problems about incoherence and the geometry of subgroups. One of the projects is a key step to a well-known conjecture that hyperbolic groups with Sierpinski carpet boundary are virtually Kleinian. Another is a newer conjecture, that groups which act geometrically on the product of two trees are incoherent. Solving these conjectures will shed substantial light on understanding when a group is the fundamental group of a 3-manifold, and also on the intricate subgroup structure of important classes of groups.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.

组是一个代数对象，是几何对象的集合对称。例如，对于平方，一组连续的旋转在平面上乘坐90度是对称的集合，并且产生了4个尺寸4的集合。描述一组的有用方法是通过所谓的演示文稿，生成器和生成器之间的关系完全描述了该组。例如，所有整数的集合是由单个元素生成的无限基团，即整数1。如果组具有有限的呈现（具有有限数量的发电机和有限的关系），那么更容易理解，并且更容易理解该组的问题。此外，如果一个组享有一个有限生成的亚组有限呈现的更强属性，则称为连贯。许多最好的理解组都是连贯的：自由组，表面组和3个manifold组都是连贯的。该项目旨在了解何时连贯，并找到证明不连贯的亚组的几何指标。该项目通过调查表现出良好的群体的几何野生亚组来推进该领域，这可能很难找到。该项目包括教育工作，PI正在帮助组织会议和学校，以增强下一代培训。 PI计划调查几个有趣的群体的连贯性，包括预计将是连贯的群体和预计将是不连贯的群体。该项目解决了有关不一致和亚组几何形状的基本问题和困难问题。其中一个项目是一个众所周知的猜想的关键步骤，即具有Sierpinski地毯边界的双曲线群实际上是克莱尼人。另一个是一个较新的猜想，即在两棵树的产物上几何作用的组是不连贯的。解决这些猜想将大大了解一个小组何时是3个策略的基本小组，还介绍了重要类别的复杂亚组结构。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来通过评估来支持的。