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.

群是一个代数对象，是几何对象的对称性集合。例如，对于一个正方形，在平面上连续旋转 90 度的集合是对称性的集合，并产生大小为 4 的群。描述群的一种有用方法是通过所谓的表示法，其中包含生成器和关系完全描述该组的生成器之间。例如，所有整数的集合是一个由单个元素（即整数 1）生成的无限群。如果一个群具有有限表示（具有有限数量的生成器和有限数量的关系），则更容易理解并更容易解决该群体的问题。此外，如果一个群具有更强的性质，即每个有限生成的子群都是有限呈现的，则该群被称为相干群。许多最容易理解的群是相干的：自由群、表面群和三流形群都是相干的。该项目旨在了解群体何时具有一致性，并找到见证不一致性的子群体的几何指标。该项目通过研究看似行为良好的群体的几何野生子群（很难找到）来推进该领域的发展。该项目包括教育工作，PI 正在帮助组织会议和学校，以加强对下一代的培训。 PI 计划调查几个有趣的群体类别的一致性，包括预期一致的群体和预期不一致的群体。该项目解决了有关子群的不相干性和几何形状的基本和困难问题。其中一个项目是实现众所周知的猜想的关键一步，即具有谢尔宾斯基地毯边界的双曲群实际上是克莱因群。另一个是一个较新的猜想，即对两棵树的乘积进行几何作用的群是不连贯的。解决这些猜想将有助于理解群何时是三流形的基本群，以及重要群类的复杂子群结构。该奖项反映了 NSF 的法定使命，并通过评估被认为值得支持利用基金会的智力优势和更广泛的影响审查标准。