LEAPS MPS: Surface subgroups of outer automorphism group of the free group and dynamics on the boundary

LEAPS MPS：自由群外自同构群的表面子群和边界动力学

• 批准号: 2137611
• 负责人: Funda Gultepe
• 金额: $ 16.56万
• 依托单位: University of Toledo
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-01-01 至 2024-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2137611&HistoricalAwards=false
• 关键词: LEAPS; MPS; Surface; subgroups; outer

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project aims to achieve a better understanding of the geometry of surfaces, graphs, and groups via their interactions with each other. The geometry of a surface is similar to the topographical map of a region. In this setting, many practical problems, such as the optimal transport of goods, require an understanding of how the roads in the region are connected. Mathematically, a road corresponds to a curve on the surface and the list of all of the roads and their intersections corresponds to a graph. The project involves understanding surfaces (regions) through their curves (roads) by recording how one curve is connected to another (building a graph or a simplicial complex). Some of the questions that arise include: is it enough to study only the outskirts of the region (boundaries of simplicial complexes) and can one get to the boundary from downtown (by iterating a point under a group element)? In this approach, a combinatorial object such as a simplicial complex relates curves with each other, and an algebraic tool, a group moves them around the surface. This approach can resolve many of the problems related to surfaces. In this project, the investigator will include undergraduate and graduate students and will collaborate with faculty from other departments. Moreover, some visual aspects of the work will be integrated into the investigator’s outreach project which will involve middle school students from under-represented groups in the mathematical sciences.The simplest groups are the free groups and the surface groups (the fundamental groups of surfaces). For many decades, these types of subgroups have been used to understand the structure of larger groups. This project includes two research directions. The first focuses on understanding the subgroup structure of the outer automorphism group of the free group via dynamics on the boundaries of some simplicial complexes. To this end, techniques from the study of Kleinian groups and mapping class groups will be used; for example, the existence and construction of certain Cannon–Thurston maps are proposed. The second project involves the explicit construction of surface subgroups of the outer automorphism group of the free group which include certain type of automorphisms, called iwips, which are of dynamical importance. The novelty of investigator’s research is the introduction of the topology of a certain 3--manifold to understand the free group and its group of outer automorphisms. This approach aims at translating between dynamical and topological tools to resolve some of the long standing problems in geometric topology and geometric group theory such as Gromov’s “ hyperbolization” conjecture. The project also includes training of students and the establishment of a 6 week summer program in mathematics for middle school students drawn from underrepresented communities.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.

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。这个项目的目的是通过曲面、图形和组的相互作用来更好地理解它们的几何图形。曲面的几何形状类似于区域的地形图。在这种情况下，许多实际问题，如货物的最佳运输，需要了解该区域的道路是如何连接的。在数学上，一条道路对应于曲面上的一条曲线，所有道路及其交叉点的列表对应于一个图形。该项目涉及通过记录一条曲线如何连接到另一条曲线来理解曲面(区域)(通过曲线(道路))(构建图形或单纯复合体)。出现的一些问题包括：只研究区域的郊区(单纯复合体的边界)是否足够？人们能否从市中心到达边界(通过迭代群元素下的一个点)？在这种方法中，组合对象，如单纯复形，将曲线彼此联系起来，而代数工具，一组将它们在曲面上移动。这种方法可以解决许多与曲面相关的问题。在这个项目中，调查员将包括本科生和研究生，并将与其他系的教职员工合作。此外，这项工作的一些视觉方面将被整合到调查者的外展项目中，该项目将涉及来自数学科学中代表性不足的群体的中学生。最简单的群体是自由群体和表面群体(表面的基本群体)。几十年来，这些类型的子组一直被用来理解较大组的结构。本项目包括两个研究方向。第一种是通过一些单纯复形边界上的动力学来理解自由群的外自同构群的子群结构。为此，将使用研究Kleian群和映射类群的技巧；例如，提出了某些Cannon-瑟斯顿映射的存在和构造。第二个项目涉及显式构造自由群的外自同构群的表面子群，其中包括具有动力学重要性的某些类型的自同构群，称为iwip。研究人员的新奇之处在于引入了一类3-流形的拓扑来理解外自同构群及其自由群。这种方法的目的是在动力学工具和拓扑工具之间进行转换，以解决几何拓扑学和几何群论中的一些长期存在的问题，如格罗莫夫的“双曲化”猜想。该项目还包括对学生的培训，以及为来自代表不足的社区的中学生建立一个为期6周的数学暑期项目。这一奖项反映了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。