Characterizing hyperbolic spaces and real trees

Let X be a geodesic metric space. Gromov proved that there exists ε ₀ > 0 such that if every sufficiently large triangle Δ satisfies the Rips condition with constant ε ₀ · pr(Δ), where pr(Δ) is the perimeter of Δ, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for ε ₀. We also show that if all the triangles D í X{\Delta \subseteq X} satisfy the Rips condition with constant ε ₀ · pr(Δ), 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.