Abstract
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 \({\Delta \subseteq X}\) 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.
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References
Bonk M.: Quasi-geodesic segments and Gromov hyperbolic spaces. Geom. Dedicata 62, 281–298 (1996)
Drutu C.: Quasi-isometry invariants and asymptotic cones. Int. J. Alg. Comp. 12, 99–135 (2002)
Gromov M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)
Gromov M.: Hyperbolic groups, Essays in group theory (Springer, New York). Math. Sci. Res. Inst. Publ. 8, 75–263 (1987)
Gromov, M.: Asymptotic invariants of infinite groups. Geometric group theory, vol. 2 (Cambridge Univ. Press, Cambridge). London Math. Soc. Lecture Note Ser. 8, 1–295 (1993).
van den Dries L., Wilkie J.: Gromov’s theorem on groups of polynomial growth and elementary logic. J. Algebra 89, 349–374 (1984)
Wenger S.: Gromov hyperbolic spaces and the sharp isoperimetric constant. Invent. Math. 171, 227–255 (2008)
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Frigerio, R., Sisto, A. Characterizing hyperbolic spaces and real trees. Geom Dedicata 142, 139–149 (2009). https://doi.org/10.1007/s10711-009-9363-4
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DOI: https://doi.org/10.1007/s10711-009-9363-4