



BlurbThis 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:
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 CWcomplex). 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. 

