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Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. (English) Zbl 1220.20037
Summary: We study the geometry of non-relatively 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 non-relatively 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 non-hyperbolic relative to any non-trivial 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. Non-uniform lattices in higher rank semisimple Lie groups are thick and hence non-relatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of non-relatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Druţu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller 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.

MSC:
20F67 Hyperbolic groups and nonpositively curved groups
57M07 Topological methods in group theory
57M50 General geometric structures on low-dimensional manifolds
20F65 Geometric group theory
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