Actions of Relatively Hyperbolic Groups on Cube Complexes

在数学中，“群”是一个代数对象，它编码了一个对象的对称性。因此，群论提供了一种代数语言和一套研究数学问题的技术。近年来，三维几何中的一些最重要的问题（如虚拟哈肯猜想）已经由Agol解决，使用几何群论的结果，特别是Wise的工作，以及Kahn， Markovic, Haglund, Hsu, Bergeron， Manning和主要研究者的工作。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，人们通过沿着表面切割来理解三维空间（二维空间）。这就是“哈肯”3流形和虚拟哈肯猜想的背景。事实证明，排列低维切割子空间的方法是通过由立方体构成的空间。该项目的主要重点是了解这些立方体复合体的对称群，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支持。在项目的前两部分，我们研究了紧作用于CAT(0)立方配合物上的相对双曲群。在适当的情况下，曼宁和首席研究员打算证明阿戈尔定理的一个版本，即所有这些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CAT(0)立方络合物上相对双曲群的“相对适当”作用理论，这一背景比适当和紧化作用的设置更广泛适用。我们打算在这种情况下证明Agol定理和Wise的拟凸层次定理的版本，并证明适当和紧化理论是相对适当理论的一个子集。在本工作的最后一部分，我们将探讨该理论在具有平面边界的双曲和相对双曲群的问题中的应用。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。

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.