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On metric relative hyperbolicity

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arxiv 1210.8081 v1 pith:TBZMN3YB submitted 2012-10-30 math.GR math.GT

classification math.GRmath.GT
keywords hyperbolicrelativelyspacegeodesicsspaceshyperbolicitymetricpaths
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We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space. We apply this to characterize hyperbolically embedded subgroups in terms of nice actions on (relatively) hyperbolic spaces. We also study the divergence of (properly) relatively hyperbolic groups, in particular showing that it is at least exponential. Our main tool is the generalization of a result proved by Bowditch for hyperbolic spaces: if a family of paths in a space satisfies a list of properties specific to geodesics in a relatively hyperbolic space then the space is relatively hyperbolic and the paths are close to geodesics.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Distinguishing Gromov-Thurston manifolds using algebraic Dehn fillings

    math.GT 2026-06 unverdicted novelty 7.0 of 10

    High-degree Gromov-Thurston manifolds from the same or non-isometric GT-pairs are homotopy inequivalent when branching degrees are I-unrelated or have large prime factors.

  2. On combination theorems and Bowditch boundaries of relatively hyperbolic TDLC groups

    math.GR 2025-08 conditional novelty 7.0 of 10

    Relatively hyperbolic TDLC groups are closed under amalgams and HNN extensions over compact subgroups, and their Bowditch boundaries are determined by the boundaries of the vertex groups.

  3. The local-to-global property for Morse quasi-geodesics

    math.GR 2019-08 conditional novelty 7.0 of 10

    Morse quasi-geodesics behave locally-to-globally in mapping class groups, CAT(0) groups, 3-manifold groups, and relatively hyperbolic groups, giving new combination theorems for stable subgroups.

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