Group Geometry and Mapping Class Groups

组几何和映射类组

基本信息

批准号：
1812021
负责人：
Johanna Mangahas
金额：
$ 17.36万
依托单位：
SUNY at Buffalo
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812021&HistoricalAwards=false
关键词：
Group Geometry Mapping Class Groups

项目摘要

The mathematical notion of a group captures a fundamental mechanism whose examples include addition, multiplication, rotations in space, and more. Any group can be described as the symmetries of some object, where the group operation comes from combining symmetries. In this and many other ways, groups are deeply related to geometric and topological spaces, and are spaces in their own right. Geometric group theory grows out of this insight. The research in this project centers on subgroups of mapping class groups, which are infinite groups arising from symmetries of topological surfaces. Their subgroups include all right-angled Artin groups, which are themselves fundamental objects. For example, notice that order never matters in addition, always matters in concatenation (a songbird is not a birdsong), and only sometimes matters in multiplication (no between ordinary numbers, yes between the matrices of linear algebra). Right-angled Artin groups include and interpolate between the first two extremes. Both mapping class groups and right-angled Artin groups are important in geometric group theory, and rich enough that their study has applications to larger families of groups, as well as to low-dimensional manifolds. As fundamental mathematics, work in geometric group theory has potential for practical benefits to society. The work of geometric group theorists, building road maps for groups, has had ramifications to cryptography, which is based on the difficulty of retracing one's steps. In addition, right-angled Artin groups are relevant to any algorithmic task in which order matters between some steps and not between others, a well-documented example being robot motion planning. This project is a geometrically-oriented investigation of three interrelated families of subgroups of mapping class groups: right-angled Artin groups, normal subgroups, and 'convex cocompact' or 'stable' subgroups. The goal of the project is to advance knowledge both specific to mapping class groups and relevant to geometric group theory overall. Among normal subgroups, the project aims to understand the spectrum from free, infinite-rank normal subgroups (whose group of automorphisms is large as possible), to normal subgroups with automorphism group consisting of the mapping class group itself (that is, as small as possible), with right-angled Artin groups appearing as normal subgroups between these two extremes. Objects with automorphism group equal to the mapping class group can be considered geometric models for the mapping class group. This work aims to further elucidate what such geometric models may be. The project also aims to advance the study of convex cocompact subgroups of the mapping class group, and their generalizations to other kinds of groups, including right-angled Artin groups, and more generally, groups acting on CAT(0) spaces. The approaches to be employed rely on group actions on various interesting spaces, including CAT(0) cube complexes, curve complexes of surfaces, projection complexes, and "rotating family" machinery within groups acting on hyperbolic spaces. The latter two are axiomatic constructions, so that results about mapping class subgroups acting on these complexes readily translate to more general settings.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.

群的数学概念捕捉了一种基本机制，其示例包括加法、乘法、空间旋转等。任何群都可以描述为某个对象的对称性，其中群运算来自组合对称性。在这方面以及许多其他方面，群与几何和拓扑空间密切相关，并且本身就是空间。几何群论正是源于这种见解。该项目的研究集中在映射类群的子群上，这些子群是由拓扑表面的对称性产生的无限群。它们的子群包括所有直角 Artin 群，它们本身就是基本对象。例如，请注意，顺序在加法中从来不重要，在串联中总是很重要（鸣禽不是鸟鸣），只有有时在乘法中才重要（普通数字之间没有，线性代数矩阵之间有）。直角 Artin 群包含并在前两个极端之间进行插值。映射类群和直角 Artin 群在几何群论中都很重要，并且足够丰富，以至于他们的研究可以应用于更大的群族以及低维流形。作为基础数学，几何群论的工作具有给社会带来实际利益的潜力。几何群理论家为群构建路线图的工作对密码学产生了影响，密码学的基础是回溯的难度。此外，直角 Artin 群与任何算法任务相关，其中顺序在某些步骤之间而不是其他步骤之间重要，一个有据可查的例子是机器人运动规划。该项目是对映射类群的三个相互关联的子群族的几何导向研究：直角 Artin 群、正规子群和“凸协紧”或“稳定”子群。该项目的目标是推进特定于映射类群以及与总体几何群论相关的知识。在正规子群中，该项目旨在理解从自由、无限秩正规子群（其自同构群尽可能大）到具有由映射类群本身组成的自同构群的正规子群（即，小至可能），直角 Artin 群作为这两个极端之间的正常子群出现。自同构群等于映射类群的对象可以被认为是映射类群的几何模型。这项工作旨在进一步阐明此类几何模型可能是什么。该项目还旨在推进映射类群的凸协紧子群的研究，以及它们对其他类型群的推广，包括直角 Artin 群，以及更普遍的作用于 CAT(0) 空间的群。所采用的方法依赖于各种有趣空间上的群作用，包括 CAT(0) 立方体复合体、曲面的曲线复合体、投影复合体以及作用于双曲空间的群内的“旋转族”机械。后两者是公理化结构，因此映射作用于这些复合体的类子组的结果很容易转化为更一般的设置。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。