Characterizing hyperbolic spaces and real trees

Details Characterizing hyperbolic spaces and real trees Characterizing hyperbolic spaces and real trees

2008-10-08

arxiv.org/abs/0810.1526v1

Mathematics

Metric Geometry

13 pages, 3 figures. Comments are welcome

Scientific paper

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.

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