Math 757 — Fall 2001 Topics in Topology

Prerequisite: Mathematics 651 (Introduction to Algebraic Topology) or its equivalent.

This will be an introductory course in Geometric Group Theory. This is a field in which groups are studied by geometric and/or topological methods. For example,

• one can study a group given by a presentation P in terms of maps of the 2-disk D^2 into a certain 2-complex K_P and the combinatorics of handle-decompositions of the 2-disk induced by these maps. The subfields of small cancellation theory and equations over groups are centered in this activity. 
   
• one can study the spaces (cell complexes or metric spaces) which a group acts on, and the nature of these actions. This is perhaps the most prototypical activity of the field of geometric group theory. An example is the study of group actions on trees (simplicial trees or R-trees), or more generally on negatively curved spaces, or even more generally, on non-positively curved spaces.

  An important fact is that some of the properties of groups that arise are purely geometric: while the question "is that group abelian" is still important, the question "is the boundary of that group connected" can be central to a discussion*.

A general introduction to geometric group theory will be given for a few weeks and then one of the subjects named above will be studied intensively. At this writing (4/18/01) the subject most favored is that of equations over groups. The input of those thinking of taking the course will, of course, weight heavily in the final content.

*A poetic attempt to capture the spirit of the subject is posted on the door to Malott 551.