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Geometry of Free Group Automorphisms: Beyond the Frontier

自由群自同构的几何：超越边界

基本信息

批准号：
1905641
负责人：
Ilya Kapovich
金额：
$ 14.35万
依托单位：
CUNY Hunter College
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-27 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1905641&HistoricalAwards=false
关键词：
Geometry Free Group Automorphisms Beyond

项目摘要

Group is a fundamental concept in many areas of modern Mathematics. The project will continue the study of dynamics and geometry of the automorphism group of a free group, a key object in modern mathematics exhibiting a rich variety of new and unexplored phenomena. The project will capitalize on the recent progress in the area. It also aims to open up new directions of research, such as a systematic study of endomorphisms of free groups. The PI will continue his work on preparing the next generation of STEM graduate and undergraduate students, through running an innovative blended seminar "Geometry, Groups and Dynamics", supervising vertically integrated undergraduate research projects in the Illinois Geometry Lab, directing PhD thesis research in mathematics, and so on. The PI will also continue his active participation in Wikipedia editing on mathematical topics, aiming to reach broad segments of general public and of other sciences and disciplines, and to educate them about the current state of mathematical research. The project will continue developing a free group counterpart of the "fibered face" theory for 3-manifolds and the study of mapping tori of automorphisms and endomorphisms of free groups. A key tool used in this study is the folded mapping torus associated to a train track map, together with a semi-flow, which is then studied as a dynamical system and replaces a fibered 3-manifold in the free group setting. Specific questions to be investigated include refining the understanding of the Bieri-Neumann-Strebel (BNS) invariant for the mapping tori groups, constructing a canonical version of the McMullen polynomial for them, investigating stable trees and Cannon-Thurston maps, and developing a systematic train track theory for free group endomorphisms.

群是现代数学许多领域的一个基本概念。该项目将继续研究自由群的自同构群的动力学和几何，这是现代数学中的一个关键对象，展示了丰富多样的新现象和未被探索的现象。该项目将利用该地区最近取得的进展。它还旨在开辟新的研究方向，例如对自由群的自同态的系统研究。PI将继续他的工作，准备下一代的STEM研究生和本科生，通过运行一个创新的混合研讨会“几何，群和动力学”，监督伊利诺伊州几何实验室垂直整合的本科生研究项目，指导数学博士论文研究，等等。PI还将继续积极参与维基百科关于数学主题的编辑工作，目的是让广大公众和其他科学和学科的人了解数学研究的现状。该项目将继续发展3流形的“纤维面”理论的自由群对应物，并研究自由群的自同构和自同态的映射环面。本研究中使用的一个关键工具是与火车轨道图相关联的折叠映射环面，以及半流，然后将其作为一个动力系统进行研究，并取代自由群设置中的纤维3流形。要研究的具体问题包括改进对映射环面群的Bieri-Neumann-Strebel （BNS）不变量的理解，为它们构造一个规范版本的McMullen多项式，研究稳定树和Cannon-Thurston映射，以及为自由群自同态发展系统的火车轨道理论。