Giulio Tiozzo: Introduction to Random Walks on Groups

This is a 21-lecture course, with each lecture being either one or two hours, given by Giulio Tiozzo. It gives an introduction to random walks on groups. This class will focus on properties of group actions from a probabilistic point of view, investigating the relations between the dynamics, measure theory and geometry of groups.

We will start with a brief introduction to ergodic theory, discussing measurable transformations and the basic ergodic theorems. Then we will approach random walks on matrix groups and lattices in Lie groups, following the work of Furstenberg. Topics of discussion will be: positivity of drift and Lyapunov exponents. Stationary measures. Geodesic tracking. Entropy of random walks. The Poisson-Furstenberg boundary. Applications to rigidity. We will then turn to a similar study of group actions which do not arise from homogeneous spaces, but which display some features of negatively curved spaces: for instance, hyperbolic groups (in the sense of Gromov) and groups acting on hyperbolic spaces. This will lead us to applications to geometric topology: in particular, to the study of mapping class groups and Out(F_N).

Prerequisites: An introduction to measure theory and/or probability, basic topology and basic group theory. No previous knowledge of geometric group theory or Teichmüller theory is needed.

1. Lecture 1
2. Lecture 2
3. Lecture 3
4. Lecture 4
5. Lecture 5
6. Lecture 6
7. Lecture 7
8. Lecture 8
9. Lecture 9
10. Lecture 10
11. Lecture 11
12. Lecture 12
13. Lecture 13
14. Lecture 14
15. Lecture 15
16. Lecture 16
17. Lecture 17
18. Lecture 18
19. Lecture 19
20. Lecture 20
21. Lecture 21

These videos were produced by the Fields Institute, as a Fields Academy course.