Geometric Group Theory

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

This project will focus on the geometry of two important classes of groups: mapping class groups of surfaces; and outer automorphism groups of free groups. Mapping class groups will be studied using the tools of large scale geometry, with the aim of learning about quasi-isometric properties of mapping class groups and proving quasi-isometric rigidity. Along the way we expect to develop new tools for understanding quasi-isometric properties of relatively hyperbolic groups. Outer automorphism groups are not as well understood as mapping class groups, and will be studied from a more foundational point of view. The focus will be to develop analogies with the theory of Teichmuller spaces which is a very well developed tool for studying mapping class groups, with the particular aim of developing analogies with the ending lamination conjecture. Geometric group theory is the study of symmetry groups associated to highly symmetric geometries of infinite extent, such as a crystal lattice in 3-dimensional space, whose symmetry group is the set of all motions of space which move each point of the lattice to another point of the lattice. The theory of groups was developed in the 19th century for studying symmetry, and it quickly developed into a deep and rich mathematical tool. Groups and geometries go hand-in-hand: every group has a corresponding geometry, and one can use tools of geometry to learn about the group. This project will focus on two classes of groups which have assumed importance in the field of topology. Given a surface, such as the surface of a doughnut with one or more holes, the "mapping class group" of that surface describes all the ways in which that surface can be mapped back onto itself. Similarly, the "outer automorphism group of a free group" is a group that describes all of the ways in which a finite 1-dimensional network or graph can be mapped back onto itself. In both cases, we will study geometries associated to these groups, with the aim of learning more about the groups themselves. In the case of mapping class groups our aim is to prove one of the central geometric conjectures, known as quasi-isometric rigidity. In the case of outer automorphism groups of free groups, where the geometry is less well understood, we will attempt to build new tools and make a deeper study of existing tools, in order to reach a better understanding of the geometry.