Geometric Group Theory

What is Geometric Group Theory?
The basic idea is to study a group (an algebraic object) using geometric techniques. Sometimes this means endowing the group with its own geometry (via the word metric, for example). Often it means allowing the group to act geometrically on some metric space. Surprisingly, many algebraic facts can be deduced about a group by knowing the sorts of spaces on which it acts in various nice ways. Results in geometric group theory are important not only in topology, geometry, and group theory, but also in logic, decision theory, etc. Many of the techniques of geometric group theory have been around since the early 1900s, primarily in the work of topologists and combinatorial group theorists. The subject truly came into its own in the 1980s in the work of Gromov, Cannon, etc. Geometric group theory has been one of the most active fields in all of mathematics over the past three decades, and many powerful tools and techniques have been developed. On the other hand, there are still many fundamental questions that are totally open.
Course Description:   This course will be a gentle introduction to the subject. One nice characteristic of Geometric Group Theory is that some of the most interesting examples and deepest questions are actually quite accessible. In particular, the textbook we will use has been written expressly for undergraduates (meaning no assuming you know any graduate or even senior level undergraduate mathematics), and yet still has some truly significant mathematics. Prerequisites:   The most important prerequisite is a familiarity and facility with mathematical proof. At least one proofs course beyond linear algebra is strongly recommended. I will begin the course with a review of groups. If you have never seen groups before, you will find this first part rather fast, but not impossibly so. On the other hand, even if you have had Algebra I & II, there will be very little in this course that you will have seen before, as the focus of a geometric group theorist tends to be quite different form that of an algebraist. In particular, graduate students are invited to take the course as a directed study, MA 594. Your Burden:   Your grade in the course will be based on homework to be turned in and graded. Problems of particular interest and/or difficulty may be discussed in class with student participation. The textbook for the course is Groups, Graphs and Trees, by John Meier. Here is a link to the official course syllabus (pdf) and to the homework assignments.

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USA Math Dept

USA

Last Modified 24th Aug 09