The geometry of non-positively curved groups

群的概念在数学中自然产生，在物理、化学和材料科学等领域都有广泛的应用。一个例子是物体对称的集合。这些对称性也可以“相乘”，即一个接一个地进行，以获得新的对称性。有时，不同的对称组合可能会产生相同的整体效果，从而产生“关系”。抽象地说，可以用有限的“生成器”列表和生成器之间的关系来指定一个群，这些生成器统称为表示。数学中的一个重要问题是两个表示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体，并研究它们的大规模特征。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。这个项目的主要重点是研究在非正或负弯曲空间上适当地不连续和紧的群，特别是直角Coxeter群（racg）、自由群的几何混合和双曲群。新出现的技术，如层次双曲线和收缩边界将用于进一步分类的计划，以达到准等距和可通约性。自由群的几何混合将被研究到初等等价，这是一个来自逻辑的群等价的概念。这将提供一个启发性的例子，说明Sela的深层定理。此外，该项目将研究racg中的稳定换向子长度，为与拟同态理论和有界上同调理论有联系并在各种领域有应用的发展计划做出贡献。最后一部分的建议试图理解集合可能的扭曲函数的双曲群的子群，并推测这一集合是巨大的。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。

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.