J an 2 00 6 THICK METRIC SPACES , RELATIVE HYPERBOLICITY , AND QUASI - ISOMETRIC RIGIDITY

In this paper we introduce a quasi-isometric invariant class of metric spaces which we call metrically thick. We show that any metrically thick 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. We show that the class of thick groups includes mapping class groups of all surfaces (except those few that are virtually free), the outer automor-phism group of the free group on at least 3 generators, non-uniform lattices in higher rank, certain Artin groups, and others. The mapping class groups thus provide the first examples of groups that are asymptotically with cut-points, but not relatively hyperbolic—this allows us to resolve several questions about the structure of relatively hyperbolic groups. As a further application of these techniques we show that Artin groups are relatively hyperbolic if and only if they are freely decomposable. We also show that Teichmüller spaces for surfaces of sufficiently large complexity are thick, this contrasts with Brock–Farb's hyperbolicity result in low complexity.