probabilistic methods in geometric group theory

in this course we will build methods from probability theory and apply them to study geometric questions regarding finitely generated groups. we will ultimately aim to provide a proof of Gromov’s celebrated theorem: a finitely generated group is virtually nilpotent if and only if it has polynomial growth. we will also bring up open questions for further research.

Topics:

1. conditional expectation and martingales
2. random walk on groups
3. Cayley graphs
4. entropy
5. harmonic functions
6. unitary actions
7. nilpotent and solvable groups
8. Milnor-Wolf theorem
9. Gromov’s theorem ** time permitting:
10. bounded harmonic functions
11. Choquet-Deny theorem
12. positive harmonic functions