Actions of Groups on Hyperbolic Spaces

虽然这个项目是关于几何群论的，但许多方法和问题的灵感来自几何拓扑学，特别是三维流形的几何和拓扑学。该项目的主要部分是关于相对双曲群的。在与几位数学家的合作中，PI将应用不同的方法(包括群论的“Dehn填充”)来理解相对双曲和双曲群。这些探索的目的是为了阐明双曲群中的一些基本问题，特别是哪些双曲群包含曲面子群的问题。这个问题与Waldhausen的虚拟Haken猜想密切相关。该项目的第二部分旨在发展一个固定群在负弯曲(但不一定是适当的)空间上作用的“多样性”的一般理论。几何群论是利用几何和拓扑技术研究无限群(抽象代数中的对象)。一个核心思想是，理解抽象群的最好方法是将其具体地视为某个几何对象的一组对称，例如晶格或有根的树。虽然这个想法允许将技术从数学的不同领域(包括理论计算机科学、分析、几何、拓扑和动力学)转移到群论，但它也导致了对这些其他领域的新见解。

Although this project is about geometric group theory, the inspiration for many of the methods and questions comes from geometric topology, particularly the geometry and topology of three-manifolds. The main part of the project is on relatively hyperbolic groups. In collaborations with several mathematicians, the PI will apply diverse methods (including group theoretic "Dehn filling") to the understanding of relatively hyperbolic and hyperbolic groups. The goal of these explorations is to shed light on some fundamental problems in hyperbolic groups, especially the question of which hyperbolic groups contain surface subgroups. This question is closely related to the virtual Haken conjecture of Waldhausen. A second part of the project is aimed at developing a general theory of the "variety" of actions of a fixed group on negatively curved (but not necessarily proper) spaces.Geometric group theory is the study of infinite groups (objects from abstract algebra) using the techniques of geometry and topology. A central idea is that the best way to understand an abstract group is to see it concretely as a group of symmetries of some geometric object, such as a crystal lattice or a rooted tree. While this idea allows the transfer of techniques from diverse areas of mathematics (including theoretical computer science, analysis, geometry, topology, and dynamics) to group theory, it also has led to new insight into these other fields.