The geometry of non-positively curved groups

群的概念自然产生于数学，在物理、化学和材料科学中有着广泛的应用，仅举几例。一个例子是对象的对称性的集合。这些对称也可以相乘，即一个接一个地执行，以获得新的对称。有时，不同的对称性组合可能会产生相同的整体效果，从而产生“关系”。抽象地说，一个组可以用有限的“生成器”列表和生成器之间的关系来指定，统称为表示。数学中的一个重要问题是，两个演示文稿是否定义了相同或相似的组。几何群论寻求使用几何工具来区分群，并研究它们的大尺度特征。这个项目将开发新的工具来探索这种大规模的特征，并根据这些特征对群体进行分类。这个项目的主要焦点是研究在非正或负曲线空间上适当地不连续和余紧作用的群，特别是直角Coxeter群(RACG)、自由群的几何汞和双曲群。新出现的技术，如等级双曲性和收缩边界，将被用来进一步将RACG分类到准等距和可公度。自由群的几何混合将被研究到基本等价，这是一个来自逻辑的群的等价概念。这将提供一类有启发性的例子来说明SELA的深层定理。此外，该项目还将研究RACG中的稳定换位子长度，这将有助于一个与拟同态和有界上同调理论有关的不断发展的计划，并在各种领域得到应用。提案的最后部分试图了解双曲群的子群的可能扭曲函数的集合，并猜想这个集合是巨大的。这一奖项反映了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。

The concept of a group arises naturally in mathematics, and has wide applications in physics, chemistry and materials science to name a few. One example is the collection of symmetries of an object. These symmetries can also be "multiplied", that is performed one after another, to obtain new symmetries. Sometimes different combinations of symmetries might have the same overall effect, leading to "relations". Abstractly, a group may be specified with a finite list of "generators" and relations among the generators, collectively called a presentation. An important question in mathematics is whether two presentations define the same or similar groups. Geometric group theory seeks to use geometric tools to differentiate groups, and to study their large-scale features. This project will develop new tools to explore such large-scale features and to classify groups according to these features. The primary focus of this project is to study groups which act properly discontinuously and cocompactly on non-positively or negatively curved spaces, particularly right-angled Coxeter groups (RACGs), geometric amalgams of free groups, and hyperbolic groups. Newly emerging techniques such as hierarchical hyperbolicity and contracting boundaries will be used to further the program of classifying RACGs up to quasi-isometry and commensurability. Geometric amalgams of free groups will be studied up to elementary equivalence, a notion of equivalence of groups coming from logic. This will provide an illuminating class of examples illustrating deep theorems of Sela. Furthermore, the project will study stable commutator length in RACGs, contributing to a growing program which has connections to the theory of quasimorphisms and bounded cohomology, and has applications in a variety of areas. The last part of the proposal seeks to understand the set possible distortion functions of subgroups of hyperbolic groups, and conjectures that this set is vast.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.