Actions of Relatively Hyperbolic Groups on Cube Complexes

在数学中，“群”是对对象的对称性进行编码的代数对象。因此，群论提供了一种代数语言和一套在整个数学中研究问题的技术。近年来，Agol利用几何群论的结果，特别是Wise，以及Kahn，Markovic，Haglund，Hsu，Bergeron，Manning和主要研究人员的工作，解决了三维几何中的一些最重要的问题(如虚哈肯猜想)。一个关键的组织原则是通过沿着一个较低维度的子空间切割空间来理解空间。例如，一个人通过沿着表面切割来理解三维空间(二维空间)。这是“Haken”3-流形和虚拟Haken猜想的背景。事实证明，安排一个较低维度的切割子空间的方法是通过由立方体组成的空间。这个项目的主要焦点是了解这些立方体复形的对称群，建立新的工具来研究它们，并将这些新工具应用于几何和群论中的许多问题。该奖项为通过研究培养研究生提供了支持。在项目的前两部分，我们研究了作用在CAT(0)立方体复形上的相对双曲群。在行动是适当的情况下，曼宁和首席研究员打算证明阿戈尔定理的一个版本，即所有这样的群体实际上都是特殊的。与爱因斯坦一起，首席研究员将发展相对双曲群在CAT(0)立方体复数上的“相对适当”作用的理论，这一背景比适当和余紧作用的设定更适用。在这一背景下，我们将证明Agol定理和Wise拟凸层次定理的不同版本，并证明真紧理论是相对真理论的一个子集。在这项工作的最后部分，我们将继续将这一理论应用于双曲以及具有平面边界的相对双曲群的问题。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。

In mathematics, a "group" is an algebraic object which encodes the symmetries of an object. As such, group theory provides an algebraic language and a set of techniques for studying problems throughout mathematics. In recent years, some of the most important problems in 3-dimensional geometry (such as the Virtual Haken Conjecture) have been solved by Agol, using results from geometric group theory, and particularly the work of Wise, as well as that of Kahn, Markovic, Haglund, Hsu, Bergeron, Manning and the principal investigator. A key organizational principle is to understand spaces by cutting them along subspaces of one lower dimension. For example, one understands 3-dimensional spaces by cutting along surfaces (two dimensional spaces). This is the context of "Haken" 3-manifolds and of the Virtual Haken Conjecture. It turns out that the way to arrange the cutting subspaces of one lower dimension is via spaces made out of cubes. The major focus of this project is to understand the symmetry groups of these cube complexes, to build new tools to study them and to apply these new tools to many problems in geometry and group theory. The award provides support of training graduate students through research.In the first two parts of the project, we study relatively hyperbolic groups acting cocompactly on CAT(0) cube complexes. In the case the action is proper, Manning and the principal investigator intend to prove a version of Agol's Theorem, that all such groups are virtually special. Along with Einstein, the principal investigator will develop the theory of "relatively proper" actions of relatively hyperbolic groups on CAT(0) cube complexes, a context much more broadly applicable than the setting of proper and cocompact actions. We intend to prove versions of Agol's Theorem, and of Wise's Quasi-convex Hierarchy Theorem in this setting, and also to show that the proper and cocompact theory is a subset of the relatively proper theory. In the final part of this work, we will pursue applications of this theory to questions about hyperbolic and relatively hyperbolic groups with planar boundary.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.