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.
