Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

We will discuss a new quasi-isometry invariant of metric
spaces which we call thickness. We show that any thick metric space is
not (strongly) relatively hyperbolic with respect to any non-trivial
collection of subsets. The class of thick groups includes many
important examples such as mapping class groups of all surfaces
(except those few that are virtually free), the outer automorphism
group of the free group on at least 3 generators, fundamental groups
of graph manifolds, $SL(n,{\bf Z})$ with $n>2$, and others. We shall also
discuss some ways in which thick groups behave rigidly under
quasi-isometries. This work is joint with Cornelia Drutu and Lee
Mosher.