Spring 2020

MAT 280: Geometric Group Theory.  

MWF, 10:00-10:50 AM.  Officially, OLSON 151, but, in reality, online, most likely, via zoom.  First lecture: Monday, March 30.

Office hours: Monday, 11AM-noon. Online (probably via zoom).

CRN for this class is: 70765


Broadly speaking, geometric group theory deals with interaction of algebraic and geometric properties of groups. The prerequisites for this class are basic group theory, point set topology and algebraic topology (fundamental groups and covering spaces). 

Topics to be covered: 

1. Generating sets, group presentations. Cayley graphs and quasi-isometries.

2. Coarse geometry: Ends of spaces and groups, growth of spaces and groups, isoperimetric inequalities and Dehn function.

3. Stallings' theorem on ends of groups.

4. Hyperbolic groups: Definitions, basic properties, constructions. 

5. Stability of quasigeodesics: Morse Lemma.

6. Elements of small cancellation theory.

7. Gromov boundary of hyperbolic groups.

8. Extensions of quasi-isometries and quasiconformal maps.

9. Mostow Rigidity Theorem.

10. Tukia's theorem on uniformly quasiconformal groups.

11. Group actions on trees and amalgams of groups.

12. Ultralimit and asymptotic cones.

13. Applications of real trees to compactifications of group actions on 

hyperbolic spaces.

14. Applications to the automorphism groups of hyperbolic groups, in particular, surface groups and free groups. 

There will be no homework and no tests. Passing/non-passing will be determined by the (online) attendance in the class. 

There is no textbook. Feel free to download the following (I will add more sources): 

1. M. Kapovich, "Lectures on Quasi-isometric Rigidity"

2. J.Vaisala "Lectures on Gromov-Hyperbolic Spaces"

3. Notes on Rips Machine

4. B. Bowditch "Course in Geometric Group Theory"

5. P. Papasoglu "Lectures on Hyperbolic Groups"