Math 758 — Topics in Topology

Spring 2003

Instructor: Kai-Uwe Bux

Time: MWF 1:25-2:15

Room: MT 205

Topic: My Favorite Groups

Web Page: pi.math.cornell.edu/~bux/teaching/2003_S-758/index.html

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.