Group Actions on Trees and Boundaries of Trees

几何群论连接了数学的两个基础领域，即群论和几何。群可以被认为是一个物体的对称集合，比如一个水分子或一个魔方。一个群可以表示许多几何或拓扑空间的对称性。如果选择的空间足够好，并且可以很好地理解，那么空间的特征可以揭示群体固有的属性。我们也可以采取相反的方法。人们通常可以通过研究描述其对称性的群来探测拓扑或几何空间的性质。这个项目旨在探索群体之间的联系以及他们所处的空间。该项目的更广泛影响包括通过各种计划和网络继续提供指导，为初高中学生提供公共服务，以及通过组织会议传播知识。这个项目的重点主要是作用于无限树和无限树的边界上的群。这包括大类别的群体；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，以更好地理解无限单群的宇宙。在过去的十年里，我们看到了大量新的令人惊讶的定理，它们照亮了我们对这门课中各种群体的理解。PI将研究扩展汤普森家族中的群，使用它们在根树上的部分作用，它们在Stein-Farley复合体上的完全作用，以及它们在Cantor空间的同纯群中的嵌入。该项目的第二个目标是更好地理解分支和自动机组。在过去的四十年里，这类群体已经成为丰富的外来但易于处理的群体的来源。PI将使用自动机理论来研究生长和扭转问题，并应用长期发展的分支群理论来研究分支群的极大子群的一般理论。项目的最后一个重点是来自拓扑结构的组属性。PI将开发一个分支覆盖拓扑理论的几何群论类比，并探索一般同调稳定性理论与拓扑有限性之间的联系，首先通过利用与曲线复合体相关的高度连接复合体上的作用，通过一些大映射类群的某些自然子群。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。

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.