Geometric Group Theory

点击翻译按钮获取中文摘要

This project will focus on the geometry of two important classes ofgroups: mapping class groups of surfaces; and outer automorphism groupsof free groups. Mapping class groups will be studied using the tools oflarge scale geometry, with the aim of learning about quasi-isometricproperties of mapping class groups and proving quasi-isometric rigidity.Along the way we expect to develop new tools for understandingquasi-isometric properties of relatively hyperbolic groups. Outerautomorphism groups are not as well understood as mapping class groups,and will be studied from a more foundational point of view. The focuswill be to develop analogies with the theory of Teichmuller spaces whichis a very well developed tool for studying mapping class groups, with theparticular aim of developing analogies with the ending laminationconjecture.Geometric group theory is the study of symmetry groups associated tohighly symmetric geometries of infinite extent, such as a crystal latticein 3-dimensional space, whose symmetry group is the set of all motions ofspace which move each point of the lattice to another point of thelattice. The theory of groups was developed in the 19th century forstudying symmetry, and it quickly developed into a deep and richmathematical tool. Groups and geometries go hand-in-hand: every group hasa corresponding geometry, and one can use tools of geometry to learn aboutthe group. This project will focus on two classes of groups which haveassumed importance in the field of topology. Given a surface, such as thesurface of a doughnut with one or more holes, the "mapping class group" ofthat surface describes all the ways in which that surface can be mappedback onto itself. Similarly, the "outer automorphism group of a freegroup" is a group that describes all of the ways in which a finite1-dimensional network or graph can be mapped back onto itself. In bothcases, we will study geometries associated to these groups, with the aimof learning more about the groups themselves. In the case of mappingclass groups our aim is to prove one of the central geometric conjectures,known as quasi-isometric rigidity. In the case of outer automorphismgroups of free groups, where the geometry is less well understood, we willattempt to build new tools and make a deeper study of existing tools, inorder to reach a better understanding of the geometry.