The distribution of sums of real-valued random variables is determined by the classical theorems of probability (law of large numbers, central limit theorem). Starting in the 1960’s Furstenberg, Oseledets and others have generalized such results tor noncommutative groups, e.g. groups of matrices.

In this course, we will consider random walks on groups of isometries of delta-hyperbolic spaces, and establish their asymptotic properties: for instance, sample paths almost surely converge to the boundary and have positive drift. In recent years, this has had many applications to low-dimensional topology, as e.g. the mapping class group and Out(F_n) act on certain (non-locally compact) hyperbolic spaces. We will discuss some such applications and their relations to Teichmuller theory.

Lecture plan:

1) Introduction to delta-hyperbolic spaces and random walks

2) The horofunction boundary

3) Convergence to the boundary and positive drift

4) Genericity of hyperbolic elements

The course is mostly based on my joint work with J. Maher.