On metric relative hyperbolicity

TL;DR

This paper characterizes relative hyperbolicity in metric spaces through multiple equivalent conditions, explores geodesic behavior, and examines divergence properties of relatively hyperbolic groups, providing new insights into their geometric structure.

Contribution

It establishes new equivalences for relative hyperbolicity, analyzes geodesic properties, and studies divergence in relatively hyperbolic groups, extending Bowditch's results.

Findings

Multiple characterizations of relative hyperbolicity are equivalent.

Geodesics in relatively hyperbolic spaces exhibit specific behaviors.

Relatively hyperbolic groups have at least exponential divergence.

Abstract

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.