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

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。该项目旨在通过表面、图形和群体之间的相互作用，更好地理解它们的几何形状。表面的几何形状类似于一个地区的地形图。在这种情况下，许多实际问题，如货物的最佳运输，都需要了解该地区的道路是如何连接的。从数学上讲，一条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。该项目包括通过曲线（道路）来理解曲面（区域），通过记录一条曲线如何连接到另一条曲线（构建图形或简单复合体）。由此产生的一些问题包括：仅研究区域的外围（简单综合体的边界）是否足够，是否可以从市中心到达边界（通过迭代群元素下的一个点）？在这种方法中，一个组合对象，如简单复形，将曲线彼此联系起来，一个代数工具，一组在表面上移动它们。这种方法可以解决许多与表面相关的问题。在这个项目中，研究者将包括本科生和研究生，并将与其他部门的教师合作。此外，这项工作的一些视觉方面将被纳入研究者的外展项目，该项目将涉及来自数学科学中代表性不足群体的中学生。最简单的群是自由群和面群（曲面的基本群）。几十年来，这些类型的子群体被用来理解更大群体的结构。本项目包括两个研究方向。首先利用简单配合物边界上的动力学方法，研究了自由群的外自同构群的子群结构。为此，将使用克莱因群和映射类群的研究技术；例如，提出了某些Cannon-Thurston地图的存在和构造。第二个方案涉及自由群的外自同构群的表面子群的显式构造，其中包括某些类型的自同构，称为iwips，这些自同构具有动力学重要性。研究者研究的新颖之处在于引入了某3—流形的拓扑结构来理解自由群及其外部自同构群。这种方法的目的是在动力学和拓扑工具之间进行转换，以解决几何拓扑和几何群论中一些长期存在的问题，如Gromov的“双曲化”猜想。该项目还包括对学生进行培训，并为来自代表性不足社区的中学生开设为期6周的暑期数学课程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.