Group Actions on Trees and Boundaries of Trees

几何群论连接着数学的两个基础领域，即群论和几何。一个群可以被认为是一个物体的对称性的集合，比如水分子或魔方。一个群可以表示许多几何或拓扑空间的对称性。如果所选择的空间足够好，并且可以足够好地理解，空间的特征可以揭示群体固有的属性。人们也可以采取相反的方法。通常，人们可以通过研究描述其对称性的群来检测拓扑或几何空间的性质。这个项目旨在探索群体和他们活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继续指导，对初中生和高中生进行公共宣传，以及通过会议组织传播知识。该项目的重点主要是行动在无限树上的团体和无限树的边界。这包括大类群；例如，它包含所有剩余有限群，但也包含许多已知的无限单群的例子。该项目的第一个目标是扩展PI过去的工作，以更好地理解无限简单群的宇宙。在过去的十年里，涌入了大量新的、令人惊讶的定理，阐明了对这门课中各种不同群体的理解。PI将研究扩展的Thompson族中的群，使用它们在有根树上的部分作用，它们在Stein-Farley复形上的全部作用，以及它们在Cantor空间的同胚群中的嵌入。该项目的第二个目标是更好地了解分支和自动机组。在过去的40年里，这类群体一直是异国情调但又容易驯服的群体的丰富来源。PI将使用自动机理论来研究增长和扭转问题，并应用长期发展的分支群理论来工作，以建立分支群的极大子群的一般理论。该项目的最后一个焦点是来自拓扑构造的群性质。PI将开发一个类似于分支覆盖的拓扑理论的几何群论，并探索同调稳定性的一般理论和拓扑有限性质之间的联系，首先通过一些大映射类群的某些自然子群，利用与曲线复数相关的高度连通复合体上的动作。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。

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.