Groups acting on hyperbolic spaces

奖项：DMS 1612473，首席研究员：Denis osingi几何群论通过将代数对象（群）可视化为几何对象（度量空间）的变换集合来研究代数对象（群）。在20世纪80年代，Gromov引入了双曲空间的概念，并概述了研究双曲空间等距群的广泛计划，从而彻底改变了这一领域。在这个方向上的密集工作产生了丰富的双曲和相对双曲群理论。进一步的推广，一类非圆柱形双曲群，最近由首席研究员提出；在过去几年中，它在PI和其他人的论文中受到了相当大的关注。所提议的项目的主要目标是继续这项工作，并在研究双曲空间上的群方面取得进一步的进展。更具体地说，拟议的项目由四个部分组成。第一部分的主要目的是更好地理解负曲率在几何、分析和拓扑动力学中的各种表现形式之间的关系。第二部分致力于研究群论Dehn手术，这是对早先在PI论文中介绍的Thurston双曲Dehn填充理论的代数推广。PI为研究群冯·诺伊曼代数提供了一些潜在的应用方向。第三部分是几何方法在置换群研究中的应用。特别地，PI提出了一种解决几个长期存在的关于可分解群的开放问题的方法。在最后一部分，PI定义了给定群在双曲空间上的作用的偏序集，并提出了关于它的几个自然问题。这里特别感兴趣的一个方向是研究类似于双曲流形的标记长度谱刚性的各种刚性现象。

Award: DMS 1612473, Principal Investigator: Denis OsinGeometric group theory studies algebraic objects (groups) by visualizing them as sets of transformations of geometric objects (metric spaces). In the 1980s, Gromov revolutionized the field by introducing the notion of a hyperbolic space and outlining a broad program of study of isometry groups of such spaces. Intensive work in this direction has resulted in the rich theory of hyperbolic and relatively hyperbolic groups. A further generalization, the class of acylindrically hyperbolic groups, was recently suggested by the Principal investigator; it received considerable attention in the papers of the PI and others over the past few years. The main goal of the proposed project is to continue this work and to make further advances in the study of groups acting on hyperbolic spaces.More specifically, the proposed project consists of 4 parts. The main objective of the first part is to better understand the relation between various manifestations of negative curvature in geometry, analysis, and topological dynamics. The second part is devoted to the study of group theoretic Dehn surgery, an algebraic generalization of Thurston's theory of hyperbolic Dehn filling introduced in earlier papers of the PI. The PI suggests some further directions with potential applications to the study of group von Neumann algebras. The third part is focused on applications of geometric methods to the study of permutation groups. In particular, the PI proposes a way of solving several long standing open problems about factorizable groups. In the last part, the PI defines the poset of actions of a given group on hyperbolic spaces and proposes several natural questions about it. One direction of particular interest here is the study of various rigidity phenomena analogous to the marked length spectrum rigidity of hyperbolic manifolds.