Group Geometry and Mapping Class Groups

群的数学概念捕捉到了一个基本的机制，其例子包括加法、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，其中群运算来自于组合对称性。在这方面和许多其他方面，群与几何和拓扑空间密切相关，它们本身就是空间。几何群论就是在这一认识的基础上发展起来的。本课题的研究主要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Artin群，这些群本身就是基本对象。例如，请注意，顺序从来都不重要，而且在连接时总是重要的(鸣禽不是鸟鸣)，只有有时在乘法中才重要(普通数字之间不重要，线性代数的矩阵之间是)。直角Artin群包括前两个极端之间的和内插。映射类群和直角Artin群在几何群论中都是重要的，并且足够丰富，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，几何群论的工作有可能给社会带来实际利益。几何群论家为群构建路线图的工作对密码学产生了影响，密码学的基础是难以追溯自己的脚步。此外，直角Artin组与某些步骤之间的顺序重要而其他步骤之间不重要的任何算法任务相关，机器人运动规划就是一个很好的例子。这个项目是对映射类群的三个相互关联的子群族的几何导向的研究：直角Artin群、正规子群和‘凸余紧’或‘稳定’子群。该项目的目标是促进关于绘制类群的具体知识和与几何群论的整体相关的知识。在正规子群中，该项目旨在了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(即，尽可能小)的正规子群的谱，其中直角Artin群表现为介于这两个极端之间的正规子群。自同构组等于映射类组的对象可以被视为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，并将其推广到其他类型的群，包括直角Artin群，以及更一般地，作用在CAT(0)空间上的群。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、投影复合体和作用在双曲空间上的群内的“旋转族”机器。后两种是公理构造，因此关于映射作用于这些复合体的类子群的结果很容易转化为更一般的背景。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.