Generic conformal dimension estimates for random groups.

Twenty years ago, Gromov introduced his density model for random groups,
and showed when the density parameter is less than one half a random
group is, with overwhelming probability, (Gromov) hyperbolic. Just as
the classical hyperbolic plane has a circle as its boundary at infinity,
hyperbolic groups have a boundary at infinity which carries a canonical
conformal structure.

In this talk, I will survey some of what is known about random groups
and their boundaries. I will outline how Pansu's conformal dimension, a
variation on Hausdorff dimension, can be used to give a more refined
geometric picture of random groups at small densities.