Filling invariants for groups

几何群论的一个中心主题是对有限呈现的群进行分类直至拟等距。这就引起了对准等距不变群的性质的研究。本项目的主要重点是研究各种拟等距不变量，这些不变量来自于几何空间中的“填充”来建模群。这些函数包括Dehn函数，它捕捉到用圆盘或其他表面填充环的难度，以及发散函数，它测量从一点出现的测地线彼此远离的速率。该项目的一个方面是研究这些不变量及其在各种群类中的高维类似物，如非正弯曲群、Coxeter群和映射类群。另一个是了解给定组中不同填充功能之间的关系。群是数学研究的基本对象。群的一个很好的例子是一个物体对称性的集合。组成对称的结果（即一个接一个地执行它们）仍然是一个对称。许多组可以通过有限表示来指定：有限的生成器列表，以及指示组合元素的两个顺序何时给出相同答案的关系。当然，理解两个演示文稿何时定义了相同或相似的组是很重要的，但通过检查演示文稿并不总是很明显。几何群论的主题之一是使用几何工具来确定两个演示所给出的群是否具有相同的大规模特征。这个项目的目标是对一类称为填充函数的工具进行详细的研究。这些研究使我们对有限表示群的结构有了更好的了解。

A central theme in geometric group theory is to classify finitely presented groups up to quasi-isomtery. This gives rise to the study of properties of groups which are invariant up to quasi-isometry. The primary focus of this project is to study various quasi-isometry invariants which come from ``fillings'' in geometric spaces modeling the group. These include Dehn functions, which capture the difficulty of filling loops with disks or other surfaces, and divergence functions, which measure the rate at which geodesics emerging from a point move away from each other. One aspect of the project is to study these invariants and their higher dimensional analogs in a variety of classes of groups, such as non-positively curved groups, Coxeter groups and mapping class groups. Another is to understand the relationships between different filling functions in a given group. Groups are fundamental objects of study in mathematics. A good example of a group is the collection of symmetries of an object. The result of composing symmetries (i.e. performing them one after another) is again a symmetry. Many groups can be specified by giving a finite presentation: a finite list of generators, and relations which indicate when two orders of composing elements give the same answer. Naturally, it is important to understand when two presentations define the same, or similar groups, but this is not always evident by examining the presentations. One of the themes in geometric group theory is to use geometric tools to address the question of determining whether the groups given by two presentations have the same large-scale features. The goal of this project is to conduct a detailed study of a class of such tools called filling functions. These investigations give us better insight into the structure of finitely presented groups.