Most of this proposal lies under the umbrella of a vast program proposed by Gromov: classify finitely generated groups by their quasi-isometric geometry. One way to study the geometry of a group is via a space called the asymptotic cone; in several important cases, interesting algebraic structure of groups has been found encoded by the topology of their asymptotic cones. Through a study of asymptotic cones, the PI will work towards understanding the geometry of several particular classes of groups, including mapping class groups, relatively hyperbolic groups, and three-manifold groups. In particular, the goal of the project is to develop new geometric invariants with an eye towards proving one of the main geometric conjectures concerning mapping class groups, namely quasi-isometric rigidity. This conjecture asserts that no other groups geometrically look like mapping class groups.
Groups are an algebraic way of encoding the symmetries of a space. It turns out that an important class of groups (the finitely generated ones) are themselves associated with a certain geometric object. This object is built by taking each symmetry to be a point and declaring the distance between a pair of points to be the number of simple symmetries one has to apply to get from one to the other. In the study of finitely generated groups, insight into their geometry can often be gained by looking at the group from ``infinitely far away'' -- this notion can be made mathematically precise and yields what is called an asymptotic cone of the group. The PI proposes certain ways to study groups from this asymptotic viewpoint.
