Abstract:  We will review some basic results about random walks on 
hyperbolic groups.  Examples include free groups, surface groups, and the 
fundamental groups of hyperbolic manifolds, and results include the fact 
that random walks converge to the boundary, and that generic elements are 
loxodromic. We will discuss to what extent these results can be 
generalized to groups which are not hyperbolic, but act by isometries on a 
hyperbolic space.  Examples include the mapping class groups of closed 
orientable surfaces, Out(F_n) and even some uncountable groups, such as 
the Cremona group, which is the group of birational automorphisms of the 
complex projective plane. This is joint work with Giulio Tiozzo.