Groups acting on hyperbolic spaces

奖：DMS 1612473，首席研究员：Denis Osin几何群论通过将代数对象(群)可视化为几何对象(度量空间)的变换集来研究它们。在20世纪80年代，格罗莫夫引入了双曲空间的概念，并勾勒出了研究这类空间的等距群的广泛方案，从而使这一领域发生了革命性的变化。在这个方向上的密集工作导致了双曲群和相对双曲群的丰富的理论。首席研究员最近提出了一个进一步的推广，即非线性双曲群；在过去的几年里，它在PI和其他人的论文中受到了相当大的关注。该项目的主要目标是继续这项工作，并在双曲空间上作用的群的研究方面取得进一步的进展。第一部分的主要目的是更好地理解几何、分析和拓扑动力学中负曲率的各种表现形式之间的关系。第二部分是关于群论Dehn运算的研究。群论Dehn运算是早期PI论文中介绍的瑟斯顿双曲Dehn填充理论的代数推广。PI指出了进一步的研究方向，并为群von Neumann代数的研究提供了潜在的应用。第三部分是几何方法在置换群研究中的应用。特别是，PI提出了一种解决关于可分解群的几个长期未解决的问题的方法。在最后一部分中，PI定义了双曲空间上给定群的作用的偏序集，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.