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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个基本机制，其示例包括加法、乘法、空间旋转等等。任何群都可以被描述为某个对象的对称性，而群运算来自于组合对称性。在这一点和许多其他方面，群与几何和拓扑空间有着深刻的联系，并且群本身就是空间。几何群论就是从这一观点发展起来的。本项目的研究集中在映射类群的子群上，映射类群是由拓扑曲面的对称性产生的无限群。它们的子群包括所有直角阿廷群，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，在连接中总是重要的（一只鸣鸟不是一只鸟），只有在乘法中有时才重要（在普通数之间不重要，在线性代数的矩阵之间是重要的）。直角Artin组包括前两个极端并在前两个极端之间插值。映射类群和直角阿廷群在几何群论中都很重要，并且足够丰富，以至于它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，几何群论的工作对社会有潜在的实际利益。几何群理论家的工作，为群建立路线图，对密码学产生了影响，密码学是基于追溯一个人的步骤的困难。此外，直角Artin群与任何算法任务相关，其中某些步骤之间的顺序很重要，而其他步骤之间的顺序无关，一个有据可查的例子是机器人运动规划。这个项目是一个几何导向的调查三个相互关联的家庭的子群映射类组：直角Artin组，正常的子群，和“凸余紧”或“稳定”的子群。该项目的目标是推进知识都具体到映射类组和相关的几何群论整体。在正规子群中，该项目旨在了解从自由的无限秩正规子群（其自同构群尽可能大）到自同构群由映射类群本身组成的正规子群（即尽可能小）的频谱，直角Artin群作为正规子群出现在这两个极端之间。具有与映射类组相等的自同构组的对象可以被认为是映射类组的几何模型。这项工作旨在进一步阐明这种几何模型可能是什么。该项目还旨在推进映射类群的凸余紧子群的研究，以及它们对其他类型群的推广，包括直角Artin群，以及更一般的作用于CAT（0）空间的群。所采用的方法依赖于各种有趣的空间上的群作用，包括CAT（0）立方体复形、曲面的曲线复形、投影复形和作用于双曲空间的群内的“旋转族”机制。后两个是公理化的结构，因此有关映射类子群作用于这些复合体的结果很容易转化为更一般的settings.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。