Math 739 — Spring 2002 Topics in Algebra: Important Groups

We will discuss the guiding examples in the theory of infinite discrete groups:

• arithmetic and S-arithmetic groups in characteristic 0, in particular Sl_n(Z)
• S-arithmetic groups in positive characteristic
• mapping class groups of surfaces
• (outer) automorphism groups of free groups
• Thompson's groups
• groups of tree automorphisms, in particular Grigorchuk's group.

These groups are important in the sense that if someone presents a new technique it is considered interesting if it adds to our knowledge about one of these groups: these are groups about which mathematicians want to know everything.

A more or less 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 stabilisers) on a nice space (e.g., a highly connected CW-complex). We will emphasise this interplay of topology and group theory.

Prerequisites: Algebra (631) and some advanced Topology (e.g.: 651). Basically you should know groups and actions, fundamental groups of spaces and covering spaces; the concepts of homology and homotopy should ring a bell.