We will discuss the guiding examples in the theory of infinite discrete
- arithmetic and S-arithmetic groups in characteristic 0, in
- 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.