In a Gromov-hyperbolic geodesic metric space, every quasigeodesic segment stays within a uniformly bounded Hausdorff distance of a geodesic segment with the same endpoints. The bound depends only on the hyperbolicity and quasigeodesic constants.
The bounded-distance conclusion is a property of a Gromov-hyperbolic metric space, under the geodesic and quasigeodesic hypotheses stated here.
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