Carolyn Abbott (Columbia), Random walks on groups acting on hyperbolic spaces. Imagine you are standing at the point 0 on a number line, and you take a step forward or a step backwards, each with probability 1/2. If you take a large number of steps, is it likely that you will end up back where you started? What if you are standing at a vertex of an 4-valent tree, and you take a step in each of the 4 possible directions with probability 1/4? This process is special case of what is called a random walk on a space. If the space you choose is the Cayley graph of a group (as these examples are), then a random walk allows you to choose a “random” or “generic” element of the group by taking a large number of steps and considering the label of the vertex where you end up. One can ask what properties a generic element of the group is likely to have: for example, is it likely that the element you land on has infinite order? In this talk, I will discuss the algebraic and geometric properties of generic elements of groups which act “nicely” on hyperbolic metric spaces, with a focus on how such elements interact with certain subgroups of the group. These results will apply to generic elements of hyperbolic groups, relatively hyperbolic groups, mapping class groups, many fundamental groups of 3–manifolds, the outer automorphism group of a free group of rank at least two, and CAT(0) groups with a rank one element, among many others. This is joint work with Michael Hull.