# Topics in Algebra: Random Walks on Groups (2023S)

The topic is random walks on infinite, finitely generated groups.

We will cover some classical results such as growth of the group vs transience of walks and Kesten's amenability condition. The main focus will be on connections between the geometry of the group and the behavior of random walks. The model result is the theorem of Kaimanovich that the Poisson boundary of a random walk on a hyperbolic group coincides with the Gromov boundary, so the 'random walk boundary' and the 'geometric boundary' agree. We will cover the recent construction of Qing-Rafi-Tiozzo of a sublinear Morse boundary that plays the role of the Gromov boundary in the non-hyperbolic case, with a proof that for CAT(0) groups the Poisson boundary agrees with the sublinear Morse boundary.

Prior experience with hyperbolic and CAT(0) groups is not required---these concepts will be developed in the lectures.

The Wednesday meetings will be lectures. The Thursday meetings will be discussion/exercise.
## Syllabus

Syllabus
## Exercises:

Exercises will be a mix of random walks problems and introductory
material in geometric group theory, including hyperbolic and CAT(0)
metric spaces.
## Literature:

- Wolfgang Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000.
- Yulan Qing, Kasra Rafi, Sublinearly Morse boundary I: CAT(0) spaces, Advances in Mathematics, Volume 404, Part B, 2022, https://doi.org/10.1016/j.aim.2022.108442.

Last updated June 30, 2023.

http://www.mat.univie.ac.at/~cashen/Classes/RandomWalks.html