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.
|Autori:||FRIGERIO R; ALESSANDRO SISTO|
|Titolo:||Characterizing hyperbolic spaces and real trees|
|Anno del prodotto:||2009|
|Digital Object Identifier (DOI):||10.1007/s10711-009-9363-4|
|Appare nelle tipologie:||1.1 Articolo in rivista|