Math 758 - My Favorite Groups

Blurb

This class will be an introduction to geometric group theory by means of examples. It will be accessible to first year students. Here is the list of groups that I want to discuss:

• Coxeter and Artin groups, in particular braid groups
• mapping class groups of closed oriented surfaces
• (outer) automorphism groups of free groups
• arithmetic and S-arithmetic groups, in particular SL(n,Z)

These groups are not just appealing to me, they can be considered important in a more objective way: These groups serve as benchmarks for methods and techniques; what applies to these groups will be tried on other groups, too. Moreover, some of these groups are ubiquitous and show up (unexpectedly) in a variety of settings, e.g., Coxeter groups arise as Weil groups from semisimple Lie groups or linear algebraic groups.

A unifying theme in this class will be the idea that you can study a group by means of a nice action of the group (e.g., an action with small stabilizers) on a nice space (e.g., a highly connected CW-complex). For instance, the mapping class group of a surface acts on Teichmüller space and on the complex of curves; braid groups act on configuration spaces; and, most prominently, SL(2,Z) acts on the hyperbolic plane. We will emphasize this interplay of topology and group theory.

Prerequisites: Algebra [631] and some advanced Topology [e.g.: 661 or 651 (can be taken along with this class)]. Basically, you should know (or learn) groups and actions, fundamental groups of spaces and covering spaces; the concepts of homology and homotopy should ring a bell.

Remark: I lectured on "Important Groups" in the spring of 2002. That class is not a prerequisite for this course. In fact, this class has only very little overlap with the past lecture, since the groups I want to present now are of a different flavor.