Thick metric spaces,relative hyperbolicity,and quasi-isometric rigidity |
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Authors: | Jason Behrstock Cornelia Druţu Lee Mosher |
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Institution: | (1) Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA;(2) Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford, OX1 3LB, UK;(3) Department of Mathematics and Computer Science, Rutgers University at Newark, Newark, NJ 07102, USA |
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Abstract: | 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 Drutu 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. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 20F69 20F65 Secondary 20F28 20F36 30F60 |
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