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:
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
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"