We study the geometry of nonrelatively hyperbolic groups. Generalizing a
result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic
space in a relatively hyperbolic space is contained in a bounded neighborhood
of a single peripheral subgroup. This implies that a group being relatively
hyperbolic with nonrelatively hyperbolic peripheral subgroups is a
quasi-isometry invariant. As an application, Artin groups are relatively
hyperbolic if and only if freely decomposable.
We also introduce a new quasi-isometry invariant of metric spaces called
metrically thick, which is sufficient for a metric space to be nonhyperbolic
relative to any nontrivial collection of subsets. Thick finitely generated
groups include: mapping class groups of most surfaces; outer automorphism
groups of most free groups; certain Artin groups; and others. Nonuniform
lattices in higher rank semisimple Lie groups are thick and hence nonrelatively
hyperbolic, in contrast with rank one which provided the motivating examples of
relatively hyperbolic groups. Mapping class groups are the first examples of
nonrelatively hyperbolic groups having cut points in any asymptotic cone,
resolving several questions of Drutu and Sapir about the structure of
relatively hyperbolic groups. Outside of group theory, Teichmuller spaces for
surfaces of sufficiently large complexity are thick with respect to the
Weil-Peterson metric, in contrast with Brock--Farb's hyperbolicity result in
low complexity.Comment: To appear in Mathematische Annale
