Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity.
by
Jason Behrstock
University of Utah
Coauthors: Cornelia Drutu and Lee Mosher

In this talk we will introduce a new quasi-isometry invariant of metric spaces which we call thick. We show that any thick metric space is not (strongly) relatively hyperbolic with respect to any non-trivial collection of subsets. Further, we show that the property of being (strongly) relatively hyperbolic with thick peripheral subgroups is a quasi-isometry invariant. 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, SL(n,Z) with n>2, and others. We shall also discuss some ways in which thick groups behave rigidly under quasi-isometries.

Paper reference: arXiv:math.GT/0512592

Date received: January 19, 2006