Group Actions on Trees and Boundaries of Trees

树木和树木边界的集体行动

• 批准号: 2343739
• 负责人: Rachel Skipper
• 金额: $ 13.3万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2343739&HistoricalAwards=false
• 关键词: Group; Actions; Trees; Boundaries

Geometric group theory connects two foundational fields of mathematics, namely group theory and geometry. A group can be thought of as the set of symmetries of an object such as a water molecule or a Rubik's Cube. A single group can represent the symmetries of many geometric or topological spaces. If the chosen space is nice enough and can be sufficiently well understood, the characteristics of the spaces can reveal properties inherent to the group. One can take the opposite approach as well. Often one can detect properties of a topological or geometric space by studying groups which describe their symmetries. This project aims to explore these connections between groups and the spaces on which they act. Broader impacts of the project include continued mentoring through various programs and networks, public outreach for middle and high school students, and dissemination of knowledge through conference organization.The focus of this project is primarily on groups acting on infinite trees and on boundaries of infinite trees. This includes large classes of groups; for instance it contains all residually finite groups but also many of the known examples of infinite simple groups. The first goal of the project is to extend the PI's past work to better understand the universe of infinite simple groups. The last 10 years have seen an influx of new and surprising theorems illuminating the understanding of the variety of groups in this class. The PI will study groups in the extended Thompson family using their partial actions on a rooted tree, their full action on the Stein-Farley complex, and their embeddings into the homeomorphism group of the Cantor space. The second goal of the project is to better understand branch and automata groups. Over the last forty years, this class of groups has served as a rich source of exotic yet tractable groups. The PI will use automata theory to investigate questions of growth and torsion and apply the long developed theory of branch groups to work towards a general theory of maximal subgroups of branch groups. The final focus of the project is on group properties coming from topological constructions. The PI will develop a geometric group theory analog of the topological theory of branch coverings as well as explore connections between the general theory of homological stability and topological finiteness properties, first through certain natural subgroups of some big mapping class groups by exploiting actions on highly connected complexes related to the curve complex.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.

几何群体理论连接了两个数学基础领域，即组理论和几何学。 可以将一组视为水分子或rubik的立方体等物体的对称性集。 一组可以代表许多几何或拓扑空间的对称性。 如果所选的空间足够好并且可以充分理解，则空间的特征可以揭示该组固有的属性。 一个人也可以采用相反的方法。 通常，人们可以通过研究描述其对称性的组来检测拓扑或几何空间的性质。 该项目旨在探索群体和他们行动的空间之间的这些联系。 该项目的更广泛影响包括通过各种计划和网络，中学和高中生的公共宣传以及通过会议组织传播知识。该项目的重点主要是在无限树和无限树的边界上进行的团体。 这包括大量的小组；例如，它包含所有残留有限的组，但也包含许多无限简单组的已知示例。 该项目的第一个目标是扩展PI过去的工作，以更好地了解无限简单群体的宇宙。 在过去的十年中，新的和令人惊讶的定理涌入，阐明了对该班级各种群体的理解。 PI将使用对植根树的部分行动，对Stein-Farley综合体的全部作用以及将其嵌入到Cantor Space的同质形态群中的全部作用，研究汤普森家族的小组。 该项目的第二个目标是更好地了解分支机构和自动机组。 在过去的四十年中，这类群体一直是异国情调但可拖延的群体的丰富来源。 PI将使用自动机理论来研究生长和扭转问题，并运用长期发展的分支组理论来致力于分支组最大亚组的一般理论。该项目的最终重点是来自拓扑结构的组属性。 PI将开发出分支覆盖拓扑理论的几何群体理论类似物，并探索同源稳定性和拓扑有限属性的一般理论之间的联系，首先是通过某些大型映射类群体的某些自然亚组，通过利用与曲线相关的高度连接的综合体来利用与曲线相关的高度连接的综合体，反映了NSF的构建范围，这是NSF的构建范围，这是NSF的范围。影响审查标准。