Non-amenability and visual Gromov hyperbolic spaces

TL;DR

This paper demonstrates that certain hyperbolic cones over bounded metric spaces are non-amenable and applies this result to visual Gromov hyperbolic spaces, advancing understanding of their geometric properties.

Contribution

It establishes non-amenability of hyperbolic cones over specific metric spaces and extends this to visual Gromov hyperbolic spaces, providing new insights into their structure.

Findings

Hyperbolic cones over bounded spaces are non-amenable.

Application of non-amenability to visual Gromov hyperbolic spaces.

Conditions on the metric space components for non-amenability.

Abstract

We prove that a uniformly coarsely proper hyperbolic cone over a bounded metric space consisting of a finite union of uniformly coarsely connected components each containing at least two points is non-amenable and apply this to visual Gromov hyperbolic spaces.