On metric relative hyperbolicity

By Alessandro Sisto. 2012

68 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d1eae3df48dd18073f15aad849f7b4988351d483

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Growth tight actions (2014, background): “sing our main theorems,Theorem 6.3andTheorem 6.4, we generalize their results to all groups acting on relatively hyperbolic metric spaces. 8.1. Relatively Hyperbolic Metric Spaces. Denition 8.1 (cf. [27,57]). Let Xbe a geodesic metric space and let Pbe a collection of uniformly coarsely connected su”
Hierarchically hyperbolic spaces II: Combination theorems and the distance formula (2015, background): “ in the context of metrically relatively hyperbolic spaces (e.g., what Drutu{Sapir call asymptotically tree-graded [DS05], or spaces that satisfy the equivalent condition on projections formulated in [Sis12]). For instance, arguing exactly as in the proof of Theorem 9.1 shows that if the space X is ”
Infinitely presented graphical small cancellation groups are acylindrically hyperbolic (2014, background): “2 are not non-trivially relatively hyperbolic, which can also be deduced from the fact that the divergence function of any non-trivially relatively hyperbolic group is at least exponential [Sis12].”