Group Actions on Hyperbolic Spaces

一个对象的对称性的集合形成一个代数对象，称为群。 一个简单的例子是一个正方形的反射和旋转，就像90度旋转一样，它保持不变。 研究小组可以引出许多有趣的问题。例如，人们可以问，有多少不同的群体在一个正方形上活动？ 这些群体有什么共同点？ 几何群论旨在通过将群作用于其上的空间的几何性质转化为群的代数性质来回答这些问题。该项目将使用这些技术来理解某些类别的群体，所有这些群体都作用于具有特定几何结构的空间，称为双曲面。 该项目还寻求支持和鼓励学生参与数学，通过支持研究生，推广到当地社区，并支持一系列研讨会。更详细地说，该项目符合理解双曲或负弯曲空间上的群体的广泛目标。 这个目标是接近在三个不同的方式：首先，通过了解所有的行动，一个给定的组双曲度量空间;其次，通过深入研究两个特定的行动大映射类组，一类的群体，其中最近有一个爆炸的兴趣;最后，通过试图证明一个强稳定性的结果，一类可以完全由它们在双曲度量空间上的作用来描述的群。 该项目的部分内容涉及数学其他领域的工具，包括描述性集合论和（通常是非交换的）环理论。该奖项反映了NSF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。

The collection of symmetries of an object form an algebraic object called a group. A simple example of a group is that of reflections and rotations of a square, as in the case of a 90 degree rotation, which leaves it unchanged. Studying groups can lead to many interesting questions. One could ask, for instance, how many different groups act on a square? What do such groups have in common? Geometric group theory aims to answer such questions by translating the geometric properties of spaces on which a group acts into algebraic properties of the group. The project will use these techniques to work towards understanding certain classes of groups, all of which act on spaces that have a particular geometric structure, called hyperbolicity. This project also seeks to support and encourage student involvement in mathematics, through support for graduate students, outreach to the local community, and support for a seminar series.In more detail, this projects fits into the broad goal of understanding groups that act on hyperbolic, or negatively curved, spaces. This goal is approached in three distinct ways: first, through understanding all actions of a given group on hyperbolic metric spaces; next, through an in-depth study of two particular actions of big mapping class groups, a class of groups in which there has recently been an explosion of interest; and finally, by seeking to prove a strong stability result for quotients of hierarchically hyperbolic groups, a class of groups which can be completely described by their actions on hyperbolic metric spaces. Parts of this project involve tools from other areas of mathematics, including descriptive set theory and (often non-commutative) ring theory.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.