Given a group of isometries of a metric space, one can draw a random sequence of group elements, and look at its action on the space. 

What are the asymptotic properties of such a random walk? 

The answer depends on the geometry of the space. 

Starting from Furstenberg, people considered random walks of this type, and in particular they focused on the case of spaces of negative curvature. 

On the other hand, several groups of interest in geometry and topology act on spaces which are not quite negatively curved (e.g., Teichmuller space) or on spaces which are hyperbolic, but not proper (such as the complex of curves).

 

We shall explore some results on the geometric properties of such random walks. 

For instance, we shall see a multiplicative ergodic theorem for mapping classes (which proves a conjecture of Kaimanovich), as well as convergence and positive drift for random walks on general Gromov hyperbolic spaces. This also yields the identification of the measure-theoretic  boundary with the topological boundary.