= Hyperbolicity is invariant under quasi-isometry
For <geodesic metric spaces>, a <quasi-isometry> preserves the property of being a <Gromov-hyperbolic metric space>. Images of <metric geodesic> sides are uniform <quasigeodesics>. The <Morse lemma for quasi-geodesics> keeps them uniformly close to <metric geodesic> sides in the hyperbolic target. Thinness there and the coarse lower distance bound pull a uniform triangle-thinness constant back to the source. Coarse surjectivity supplies a quasi-inverse for the reverse implication. Consequently a <metric geodesic> space quasi-isometric to a tree is hyperbolic, with no local finiteness assumption on the tree.
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