Characterizing hyperbolic spaces and real trees

TL;DR

This paper provides an elementary proof of Gromov's characterization of hyperbolic spaces via triangle conditions, estimates the involved constants, and explores the implications for real trees and asymptotic cones.

Contribution

It offers a simplified proof of Gromov's theorem, estimates the constant k, and connects hyperbolicity with real trees and asymptotic cones.

Findings

Elementary proof of Gromov's hyperbolicity characterization

Estimate for the constant k in the Rips condition

Hyperbolicity linked to asymptotic cones being real trees

Abstract

Let X be a geodesic metric space. Gromov proved that there exists k>0 such that if every sufficiently large triangle T satisfies the Rips condition with constant k times pr(T), where pr(T) is the perimeter T, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for k. We also show that if all the triangles T in X satisfy the Rips condition with constant k times pr(T), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is hyperbolic if and only if every asymptotic cone of X is a real tree.