In this talk, we consider the action of the group of affine transformations on certain nilmanifolds (for example, a torus or a Heisenberg nilmanifold). Given a probability measure on the transformation group, a random walk is defined. I will present some recent progress aiming at answering the following questions. Does the random walk converge in law to the Haar measure? If yes, how fast ? If not, what is the obstruction ?
This talk is based on joint works with Tsviqa Lakrec and Elon Lindenstrauss.